Research ArticleRegional Frequency Analysis of Extremes Precipitation UsingL-Moments and Partial L-Moments

Said Arab Khan12 Ijaz Hussain1 Tajammal Hussain3 Muhammad Faisal45

Yousaf ShadMuhammad1 and Alaa Mohamd Shoukry67

1Department of Statistics Quaid-i-Azam University Islamabad Pakistan2Government Degree College Lahore Swabi Khyber Pakhtunkhwa Pakistan3Department of Statistics COMSATS Institute of Information Technology Lahore Pakistan4Faculty of Health Studies University of Bradford Bradford UK5Bradford Institute for Health Research Bradford Teaching Hospitals NHS Foundation Trust Bradford UK6Arriyadh Community College King Saud University Riyadh Saudi Arabia7Workers University Cairo Egypt

Correspondence should be addressed to Ijaz Hussain ijazqauedupk

Received 21 December 2016 Revised 26 January 2017 Accepted 7 February 2017 Published 7 March 2017

Academic Editor Mouleong Tan

Copyright copy 2017 Said Arab Khan et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Extremes precipitation may cause a series of social environmental and ecological problems Estimation of frequency of extremeprecipitations and its magnitude is vital for making decisions about hydraulic structures such as dams spillways and dikes In thisstudy we focus on regional frequency analysis of extreme precipitation based on monthly precipitation records (1999ndash2012) at 17stations of Northern areas and Khyber Pakhtunkhwa Pakistan We develop regional frequency methods based on L-moment andpartial L-moments (L- and PL-moments) The L- and PL-moments are derived for generalized extreme value (GEV) generalizedlogistic (GLO) generalized normal (GNO) and generalized Pareto (GPA) distributions The 119885-statistics and L- and PL-momentsratio diagrams of GNO GEV and GPA distributions were identified to represent the statistical properties of extreme precipitationin Northern areas and Khyber Pakhtunkhwa Pakistan We also perform a Monte Carlo simulation study to examine the samplingproperties of L- and PL-moments The results show that PL-moments perform better than L-moments for estimating large returnperiod events

1 Introduction

Hydraulic and hydrologic designs are key steps in planningof any water project Any problem pitched at designingstage will result in the failure of design irrespective of thefact how correctly the other steps are taken Hydrologistsdeal with water-related issues problems of quantity qualityand availability in the society that known as hydrologicevents Stochastic methods are often used to understandsources of uncertainties in physical processes that give riseto observed hydrologic events as precipitation and streamflow estimates depend on the past or future events Severalstatistical methods offered to minimize and summarize theuncertainties of observed data and frequency analysis is oneof them It determines that how often a particular event will

occur by estimating the quantile 119876119879 for return period of 119879where 119876 is the magnitude of the event that occurs at a giventime and location

Dalrymple [1] proposed regional frequency analysis(RFA) method for pooling various data samples It is index-flood procedure in hydrology Hosking et al [2] studiedthe properties of probability-weighted moments (PWMs)method based on RFA method This method is first used byGreis and Wood [3] and Wallis [4] Cunnane [5] reviewedtwelve methods of RFA and related regional PWMs algo-rithm

Initially PWMs are considered as an alternative parame-ter estimation method however it was difficult to interpretdirectly as measures of the shape and scale parameters ofdistribution RFA can forecast the flood flow using empirical

HindawiAdvances in MeteorologyVolume 2017 Article ID 6954902 20 pageshttpsdoiorg10115520176954902

2 Advances in Meteorology

formula and unit-hydrograph procedure Subramanya [6]and it can also estimate the quantiles of extreme precipitation

Hosking and Wallis [7] showed that RFA method basedon L-moments is used to detect homogeneous regions toselect suitable regional frequency distribution and to predictextreme precipitation quantiles at region of interest

Whilst L-moment methodology is effective in estimatingparameters it may not valid for predicting high returnperiod events Wang [8] suggested that relatively small floodsmight create disturbance in the analysis To overcome suchsituation a censored sample can be used as by Cunnane [9]

Wang [8] proposed partial probability-weightedmoments (PPWMs) for fitting the probability distributionfunction to the censored sample Partial L-moments(PL-moments) are variants of the L-moments and similar toPPWMs PL-momentsmethod has used in fitting generalizedextreme value (GEV) distribution for censored flood samples(see Wang [8 10 11] and Bhattarai [12]) Bhattarai [12] foundthat censoring flood samples are nearly thirty percent ofbasic L-moments

Shabri et al [13] used Trim L-moments (TL-moments)for the RFA and compared its performance with L-momentsSaf [14] determined hydrologically homogeneous region andregional flood frequency estimates by using index-floodtechnique along with L-moments for the West Mediter-ranean River Turkey L-moments method is also used toassign a suitable regional distribution for the individualsubregions and to assess their homogeneity see Abolverdiand Khalili [15] Hussain and Pasha [16] suggested theregional flood frequency analysis based on L-momentsTheyused discordancy measure for data screening and used thefour-parameter Kappa distribution with 500 simulationsfor the heterogeneity analysis Zakaria et al [17] used thePL-moments technique and found another link for thehomogeneity analysis Shahzadi et al [18] showed that thegeneralized normal (GNO) distribution is suitable for theregional quantile estimation at maximum return period andthe GEV distribution for the overall regions at low returnperiod based on relative RMSE and relative absolute biasMost commonly used statistical distributions for high climatemodeling are as follows the logistic distribution with threeparameters lognormal distribution with three parametersLog Pearson type III GEV and generalized Pareto (GPA)(Coles [19] Katz et al [20] Abida and Ellouze [21] Feng et al[22] Yang et al [23] Villarini et al [24] Zakaria et al [17] Sheet al [25])Moreover GEV andGPAdistributions are suitableif data contains extremes values Over the years the GNOGEV generalized logistic (GLO) and GPA distributionshave been widely employed in the extreme value estimationof annual flood peaks In this study we aim to developRFA method based on L- and PL-moments approach Ourproposed method can be used at all levels of regional analysissuch as identification of homogeneous regions identificationand also testing of the suitable probability regional frequencydistribution based on 119885-statistic and the L- and PL-momentratio diagram and estimation of the flood quantile at siteof interest We use 17 sites of Northern areas and KhyberPakhtunkhwa as a case study to perform the analysis

We explain the methodology of the L-moments and PL-moments in Section 22 Section 23 shows the applicationof the RFA where we choose the appropriate distributionfor the regional analysis Section 33 provides the estimationof quantile for both the small and large return period Theresults of our simulations studywill be presented in Section 4

2 Material and Methods

21 Study Area and Data Sources The data was collectedfor this study from Karachi Data Processing Center throughPakistan Meteorological (PMD) Islamabad Monthly precip-itation data has been recorded from 1999 to 2012 There are17 meteorological stations of Northern areas and KhyberPakhtunkhwa Pakistan These stations are full precipitationregions that affect areas in Pakistan where water is essentialfor hydropower and flood plains Figure 1 shows the locationof the study area and geographic distribution of precipitationstations There is no missing value in this data set

22 Statistical Methods

221 The L-Moments Conventional moments method maybe used for estimating the parameters of probability distri-bution However this approach has some serious drawbacksRatio ofmoment estimators is biased and often assumption ofbeing normally distributed is violated Wallis et al [26] Fur-thermore it is sensitive to outliers Pearson [27]Therefore itis unreliable for skewed distributions

Hosking [28] proposed L-moments approach to over-come the above problems The L-moments may preciselydescribe the statistical properties of hydrological informa-tion and it can write as a linear function of the PWMs ThePWMs of order 119903 were properly described by Greenwood etal [29] as

119872119901119903119904 = 119864 [119909119901 119865 (119909)119903 1 minus 119865 (119909)119904] (1)

where 119903 119904 and 119901 are the real numbers and 119865(119909) is thecumulative distribution function of 119909 A functional case is

120573119903 = 11987211199030 (2)

Therefore

120573119903 = int10119909 (119865) 119865119903119889119865 (3)

where 119865 = 119865(119909) is the cumulative distribution function of arandom variable 119909 and 119903 is a nonnegative integer of the realnumber that is 119903 = 0 1 2 3 Therefore the first four L-moments which are the linear combinations of the PWMsare

1205821 = 12057301205822 = 21205731 minus 12057301205823 = 1205732 minus 61205731 + 12057301205824 = 201205733 minus 301205732 + 121205731 minus 1205730

(4)

Advances in Meteorology 3

37∘0㰀0㰀㰀N

36∘0㰀0㰀㰀N

35∘0㰀0㰀㰀N

34∘0㰀0㰀㰀N

33∘0㰀0㰀㰀N

32∘0㰀0㰀㰀N

37∘0㰀0㰀㰀N

36∘0㰀0㰀㰀N

35∘0㰀0㰀㰀N

34∘0㰀0㰀㰀N

33∘0㰀0㰀㰀N

32∘0㰀0㰀㰀N

71∘0㰀0㰀㰀E 72∘0㰀0㰀㰀E 73∘0㰀0㰀㰀E 74∘0㰀0㰀㰀E 75∘0㰀0㰀㰀E 76∘0㰀0㰀㰀E

71∘0㰀0㰀㰀E 72∘0㰀0㰀㰀E 73∘0㰀0㰀㰀E 74∘0㰀0㰀㰀E 75∘0㰀0㰀㰀E 76∘0㰀0㰀㰀E

W

N

E

S

0 30 60 120 180 240(km)

StationsWater bodiesStudy area

Figure 1 Locations of Northern areas and Khyber Pakhtunkhwa and the meteorological stations (119909 axis indicates Longitude [E] 119910 axisindicates Latitude [N])

The L-moments have no units of measurement which arecalled the L-moments ratios The L-moments ratios pro-posed by Hosking [28] are computed as

120591 = 12058221205821

1205913 = 12058231205822

1205914 = 12058241205822 (5)

4 Advances in Meteorology

where 120591 represents the L-coefficient of variation (L-Cv)1205913 represents the L-coefficient of skewness (L-Cs) and 1205914represents the L-coefficient of kurtosis (L-Ck)

The arranged sample is given as 119909(1) le 119909(2) le 119909(3) sdot sdot sdot le119909(119899) Wang [8] stated that the statistic

119887119903 = 1119899119899sum119894=1

(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909(119894) (6)

is an unbiased estimator of 120573119903So

1198971 = 11988701198972 = 21198871 minus 11988701198973 = 61198872 minus 61198871 + 11988701198974 = 201198873 minus 301198872 + 121198871 minus 1198870

(7)

where 1198971 1198972 1198973 and 1198974 are the first four L-moments of thesample And similarly

119905 = 11989721198971 1199053 = 11989731198972 1199054 = 11989741198972

(8)

are the sample L-moments ratios

222 The Partial L-Moments Wang [8 10 11] introduced aconcept of partial probability-weighted moments (PPWMs)that will estimate the higher quantiles of flood flows Data canbe censored to the right tail or left tail

Initially PPWMswere to take out the smaller values fromthe process of distribution fitting because such values haveslight influence on the frequency analysis and are nuisance tothe fitting process

The left tail PPWMs are defined by Wang [8 11] as

120573lowast119903 = int11198650

119909 (119865) 119865119903119889119865 (9)

where 1198650 = 119865(1199090) which is the lower limit of the censoredobservations and 1199090 is the censoring threshold value

The PPWMs elongated form described by Wang [10] areto be given a censored sample as

120573119903 = 11 minus 119865119903+10 int11198650

119909 (119865) 119865119903119889119865 (10)

If the value of 1198650 is starting from the zero then the result ofPPWMswill be the same as the usual PWMs As 119909(1) le 119909(2) le119909(3) sdot sdot sdot le 119909(119899) is the arranged sample Wang [10] describes theunbiased estimator of 120573119903 as

119903 = 1(1 minus 119865119903+10 ) 119899119899sum119894=1

(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909lowast(119894) (11)

where

119909lowast(119894) = 0 for 119909(119894) le 119909(0)119909lowast(119894) = 119909(119894) for 119909(119894) gt 119909(0) (12)

The censoring level 1198650 is the prior selection of the numberof censored sample data The procedure that determines thenumber of sample data points are to be censored

1198650 = 1198990119899 (13)

where 1198990 and 119899 are the lengths of sample which are to becensored and uncensored respectively Similarly 1199090 is thehighest value of the censored sample The first four PL-moments for the PPWMs are

1 = 12057302 = 2 1205731 minus 12057303 = 1205732 minus 6 1205731 + 12057304 = 20 1205733 minus 30 1205732 + 12 1205731 minus 1205730

(14)

Similarly the PL-moments ratios can be written as

120591 = 21

1205913 = 32

1205914 = 42

(15)

where 120591 1205913 and 1205914 denote the partial L-coefficient of variation(PL-Cv) partial L-coefficient of skewness (PL-Cs) and par-tial L-coefficient of kurtosis (PL-Ck) respectively Thereforethe first four sample PL-moments can be computed as

1198971 = 01198972 = 21 minus 01198973 = 62 minus 61 + 01198974 = 203 minus 302 + 121 minus 0

(16)

And the first four sample PL-moments ratios can be com-puted as

119905 = 11989721198971 1199053 = 11989731198972 1199054 = 11989741198972

(17)

Advances in Meteorology 5

where 119905 1199053 and 1199054 represent the sample partial L-momentsratios of the PL-Cv PL-Cs and PL-Ck respectively Thederivation is L-moment and PL-moments are given inAppendix In the present study different level of censoringthreshold is selected

23 Regional Frequency Analysis Hosking and Wallis [7 30]identified the following four steps to explain the procedure ofthe RFA

(1) Data screening(2) Designing of the homogeneous region(3) Selection of an appropriate probability distribution(4) Parameters estimation of the appropriate probability

distribution

231 Data Screening We screened data anomalies beforeapplying any statistical analysis

232 Discordance Test Hosking and Wallis [7] suggested adiscordancy measure (119863119894) test that recognizes the locationswhere sample L-moments are marked contrarily from themost other locations Locations with the large flaws in thedata will be flagged as discordant

The discordancy test for a region containing119873 locationsfor site 119894 is proposed by Hosking and Wallis [7] as follows

119863119894 = 13119873 (119906119894 minus 119906)119879 119878minus1 (119906119894 minus 119906) 119894 = 1 2 119873 (18)

where 119906119894 is the vector containing the three sample L-momentsratios for the site 119894 expressed as

119906119894 = [119905(119894)2 119905(119894)3 119905(119894)4 ]119879 (19)

119906 is the average vector of 119906119894 for the overall region that is

119906 = 1119873119873sum119894=1

119906119894 (20)

and 119878 is the covariance matrix for the sample that can beexpressed as

119878 = 119873sum119894=1

(119906119894 minus 119906) (119906119894 minus 119906)119879 (21)

Broadly speaking a location or a site is considered to bediscordant from the whole region or group if the value of 119863119894is larger than the critical value

233 Heterogeneity Test The homogeneity measure (119867119895)identifies homogenous regions Hosking and Wallis [7] Itis also useful to tag the locations if they are plausible tohandle as a homogeneous region It estimates the amountof heterogeneity in the overall region The heterogeneity test(119867119895) is computed as

119867119895 = 119881119895 minus 120583V119895120590V119895 119895 = 1 2 3 (22)

where 120583V119895 and 120590V119895 are representing the mean and standarddeviation of the simulated 119881119895 values Also

1198811 = 119873sum119894=1

119899119894 (119905(119894)2 minus 1199051198772 )sum119873119894=1 119899119894

12

1198812 =sum119873119894=1 [119899119894 (119905(119894)2 minus 1199051198772 )2 + (119905(119894)3 minus 1199051198773 )212]

sum119873119894=1 119899119894

1198813 =sum119873119894=1 [119899119894 (119905(119894)3 minus 1199051198773 )2 + (119905(119894)4 minus 1199051198774 )212]

sum119873119894=1 119899119894

(23)

Here 1199051198772 1199051198773 and 1199051198774 are region average L-moments or PL-moments ratios We assessed the heterogeneity of a region assuggested by Hosking and Wallis [7]

Region is acceptably homogeneous if119867 lt 1Region is possibly heterogeneous if 1 le 119867 lt 2Region is definitely heterogeneous if119867 ge 2

24 Selection of the Appropriate Probability DistributionHosking andWallis [7] proposed two approaches to select thedistribution that fitted best the data the L-moment ratios dia-gram and the 119885-test The L-moment ratios diagram is usingthe unbiased estimators Hosking [28] Stedinger et al [31]Vogel and Fennessey [32] and Hosking [33]The L-momentsratio diagram is a plot of the computed values L-Cs andthe observed values L-Ck of the distribution function Thecurves indicate the hypothetical connections between L-Csand L-Ck of the candidate distribution The L-moment ratiodiagrams have been proposed for discriminating between thecandidate probability distributions in describing the regionalinformation (Hosking [28] Stedinger et al [31] Hoskingand Wallis [7]) The L-moments ratio diagrams have beenused as a component of probability distribution process forregional information (Schaefer [34] Pearson [35] Vogel andFennessey [32] Vogel et al [36] Chow and Watt [37] OnOzand Bayazit [38] Vogel and Wilson [39] Peel et al [40])

Hosking and Wallis [7] suggested a measure to see howwell the L-Cs and L-Ck of the fitted probability distributionmatch the regional average L-Cs and L-Ck of the observedinformation

The measure goodness of fit for every single selectedprobability distribution is computed as follows

119885Dis = (120591Dis4 minus 1199051198774 )1205904 (24)

where 120591Dis4 represents the value of the L-Ck of the fitteddistribution 1199051198774 represents the weighted regional averageL-Ck and 1205904 represents standard deviation of the 1199051198774 whichis obtained from the simulation of the Kappa probabilitydistribution

If the computed value of 119885Dis is equal to zero the proba-bility distributionwill be themost suitable fit If the computed

6 Advances in Meteorology

Table 1 Statistics of annual extreme monthly precipitation for study region based on L-moments and PL-moments

Sites L-moments PL-momentsMean 119905 1199053 1199054 Mean 119905 1199053 1199054

Astore 37624 0518 0366 0193 41752 0467 0377 0183Balakot 120450 0478 0322 0192 133259 0428 0335 0198Bunji 14297 0609 0456 0259 16339 0553 0450 0257Chilas 15091 0585 0392 0194 17603 0516 0375 0200Cherat 49038 0580 0375 0162 55289 0526 0355 0161DI khan 28885 0689 0516 0262 32351 0651 0487 0250Dir 105694 0432 0216 0098 116741 0378 0222 0097Drosh 45284 0525 0317 0136 50316 0474 0307 0137Garhi Dupatta 111352 0454 0227 0111 123374 0398 0229 0120Gilgit 12677 0591 0412 0228 14290 0539 0399 0239Gupis 23446 0678 0529 0331 30299 0583 0506 0350Kakul 99431 0453 0247 0102 110068 0399 0254 0093Kotli 95070 0542 0332 0136 106903 0487 0314 0138Muzaffarabad 118439 0459 0284 0182 131122 0405 0301 0194Peshawar 46216 0572 0398 0216 51761 0520 0392 0222Saidu Sharif 85667 0455 0265 0181 94978 0399 0283 0200Skardu 20438 0616 04564 0238 22885 0570 0444 0231

value of 119885-statistic is less than 164 at 90 confidence level(ie |119885Dis| le 164) it will indicate that the distributionqualifies the goodness of fit criteria If there are more thanone distribution that qualify the criteria the most suitabledistribution has the minimum |119885Dis| value3 Estimation

The sitesrsquo information and statistic by using L-moments forthe present study are presented in Table 1 In Table 1 meanrepresents the first sample LPL-moments and 119905 1199053 and 1199054are the sample LPL-moments ratios of the L-CvPL-CvL-CsPL-Cs and L-CkPL-Ck respectively The lower levelcensoring threshold is selected from 10 to 23 Table 2expresses the feasible threshold values according to thepercentile technique along with Average Annual OccurrenceNumber (AAON) Jiang et al [41] and Yuguo [42] suggestedthat the optimal threshold can be obtained if the values ofAAON lie between 1 and 2 Table 2 shows that 90th percentileobservations are suitable for the optimum threshold selectionof most of areas in the present study We have 168 valuesin each station according to the above table Astore stationhas 32 threshold values due to which 16 values are beingcensored in 168 According to censored level 102 censoredlevel was selected Similarly 105 was selected for Balakotand Muzaffarabad According to this process maximumthreshold level 223 was selected for Gupis By using theabove process for each station 17 different censoring levelswere selected So we decide that the range from 10 to 23of censoring level should be kept for selecting thresholdvalues

31 Regional Frequency Analysis The following four steps areconsidered as prerequisite for frequency analysis Hoskingand Wallis [7 30]

(1) Data screening(2) Designing of the homogeneous region(3) Selection of an appropriate probability distribution(4) Parameters estimation of the appropriate probability

distribution

311 Data Screening In this study we use secondary dataafter carefully examining all locations for abnormalities andmissing observations Therefore we use 14 years of data forRFA that were obtained from seventeen locations

312 Discordance Test Table 3 shows 119863119894 result of (18) for17 locations of this study region It can be observed fromthe results of LPL-moments in Table 3 that the value of 119863-statistic varies from 007 to 244 If 119863119894 is greater than 3 thelocation is considered to be discordant from the rest of theregional data Hosking and Wallis [7] In this study regionno location is diagnosed as discordant (119863119894 ge 3) Thereforewe use all data for the development of the RFA based on L-moments and PL-moments

313 Regional Heterogeneity Measure The next step is theformation of the homogeneous region which is conven-tionally tougher and needs the higher number of subjectivejudgments The homogeneity conditions are defined as thelocations that have the same frequency distributions

Advances in Meteorology 7

Table 2 Precipitation threshold selection in GPA distribution for 17 stations

Station 80th 90th 95th 975th 99thAstore

Threshold 98 32 02 0 0AAON 243 121 064 043 043

BalakotThreshold 325 148 51 3 0AAON 243 129 057 029 014

MuzaffarabadThreshold 304 118 5 1 0AAON 25 121 057 029 021

Garhi DupattaThreshold 288 10 38 0 0AAON 243 121 057 029 029

DirThreshold 31 20 5 005 0AAON 243 121 057 029 021

KakulThreshold 289 134 21 05 0AAON 164 129 057 029 021

KotliThreshold 13 6 005 0 0AAON 243 135 057 036 036

Saidu SharifThreshold 256 83 2 0 0AAON 243 121 057 036 036

cheratThreshold 5 005 0 0 0AAON 243 143 093 093 093

PeshawarThreshold 6 005 0 0 0AAON 243 129 057 057 057

DroshThreshold 76 15 02 005 0AAON 243 121 057 05 021

D I KhanThreshold 03 0 0 0 0AAON 257 129 129 129 129

GupisThreshold 0 0 0 0 0AAON 271 271 271 271 271

SkarduThreshold 2 03 0 0 0AAON 243 071 071 071 071

ChilasThreshold 1 05 0 0 0AAON 171 107 107 107 107

BunjiThreshold 13 0 0 0 0AAON 15 15 15 15 15

GilgitThreshold 12 005 0 0 0AAON 243 135 071 071 071

8 Advances in Meteorology

04

03

02

01

00

04

03

02

01

00

00 01 02 03 04 05 06 00 01 02 03 04 05 06

GLOGEVGPA

GNOAverageData

GLOGEVGPA

GNOAverageData

L-ku

rtos

is

L㰀-k

urto

sis

L-skewness L-skewness

Figure 2 L-diagram and PL-diagram of the GLO GEV GPA and GNO distributions

Table 3 Discordance test result based on L-moments and PL-moments

Name of site L-moments (119863119894) PL-moments (119863119894)Astore 205 244Balakot 142 085Bunji 055 060Chilas 029 007Cherat 079 064DI khan 129 190Dir 082 086Drosh 051 043Garhi Dupatta 143 133Gilgit 026 059Gupis 237 231Kakul 082 122Kotli 077 059Muzaffarabad 099 085Peshawar 007 008Saidu Sharif 191 157Skardu 067 065

For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments

Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different

values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary

32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO

We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area

Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments

Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions

However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions

Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution

Advances in Meteorology 9

Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments

Heterogeneity measures L-moment PL-momentHeterogeneity measure1198671

Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060

Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800

Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170

Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments

Method Distribution Parameters120585 120572 119870L-moments

GEV 04660 05644 minus02751GNO 06671 07565 minus07594GPA minus00551 09944 minus00575

PL-momentsGEV 05220 05116 minus02686GNO 07046 06837 minus07487GPA 00478 09073 minus00472

is larger than the critical value of 164 (at 90 confidencelevel)

It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile

Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution

33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7

based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)

Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is

119876 (119865) = 1198971198941119902 (119865) (25)

The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3

Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions

Therefore it is necessary to assess the performance ofregional quantile estimates

4 Accuracy of the Estimated Quantiles and theRegional Growth Curve

A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten

10 Advances in Meteorology

Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865

01000 05000 08000 09000 09500 09800 09900 09975 099875 09990

L-momentsGEV 00454 06836 15139 22247 30591 44161 56871 90748 113165 121338GNO 00473 06671 15585 23072 31448 44098 54997 80676 95666 100823GPA 00500 06481 16217 23931 31958 43073 51878 70578 80502 83782

PL-momentsGEV 01397 07190 14670 21034 28469 40497 51703 81363 10086 107950GNO 01412 07046 15062 21752 29203 40410 50032 72608 85739 90249GPA 01436 06871 15649 22547 29674 39460 47149 63304 71786 74577

Qua

ntile

Regional growth curve (L-moments)

GPAGNOGEV

GPAGNOGEV

Qua

ntile

Return period

Regional growth gurve (PL-moments)

5

4

3

2

1

0

5

4

3

2

1

0

minus2 minus1 0 1 2 3 4 5

2 5 10 20 50 100

Return period2 5 10 20 50 100

Reduced variate minuslog (minus log (F))

minus2 minus1 0 1 2 3 4 5

Reduced variate minuslog (minus log (F))

Figure 3 L-moments and PL-moments regional growth curves

as 119876(119898)119894 (119865) for the nonexceedance probability 119865 The relativeerror for this estimator is

119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) (26)

The bias and the RMSE of the above quantity over all 119872repetition are

Bias = 119861119894 (119865) = 1119872119872sum119898=1

119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)

= 1119872 [[119872sum119898=1

(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12

(27)

Also for the estimated quantile the regional average bias andthe relative RMSE are

119861119877 (119865) = 1119873119873sum119894=1

119861119894 (119865)

119877119877 (119865) = 1119873119873sum119894=1

119877119894 (119865) (28)

We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval

119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)

Advances in Meteorology 11

Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le

119902 (119865)119871005 (119865) (30)

The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves

We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]

We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively

Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds

Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion

This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail

GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments

Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents

Appendix

The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows

PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by

119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (minus ln119865)119896 119896 = 0 (A2)

Wang [10] developed the partial probability-weightedmoments (PWMs) of the GEV as

(119903 + 1) 1205731015840119903 = 120577 + 120572119867 (120574 1198650 119896) (A3)

where

119867(120574 1198650 119896) = 1119896 [1 minus119875 1 + 119896 minus (119903 + 1) ln1198650(1 minus 119865119903+10 ) (119903 + 1)119896 ] (A4)

12 Advances in Meteorology

Table8Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f30

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus0540

0minus19

7674

minus195328

minus193602

minus191579

minus188659

minus186376

minus1840

97minus18

1165

minus179056

119877(119865)

00900

00695

00963

01952

03334

06038

09057

13261

21337

300

48119877119877 (119865)

19873

01019

00635

00876

01090

01369

01595

01839

02191

02478

119871119861119864lowast

minus20937

08829

09296

08842

08411

07862

07460

07065

06557

06177

119880119861119864lowast

18714

11949

11208

11580

11884

12244

12516

12818

13285

13683

L-mom

ents

GNO

119861119877 (119865)

08124

minus196279

minus193846

minus192837

minus191816

minus190506

minus189573

minus188699

minus187636

minus186899

119877(119865)

00986

00696

00917

02051

03520

06027

08421

11312

16008

20312

119877119877 (119865)

20766

01045

00587

00889

01120

01368

01533

01686

01878

02016

119871119861119864lowast

minus21331

08683

09271

08780

08379

07958

07691

07453

07172

06978

119880119861119864lowast

16569

11974

11077

11569

11954

12355

12615

12851

13149

13369

L-mom

ents

GPA

119861119877 (119865)

38654

minus194351

minus193564

minus193226

minus192616

minus191706

minus191060

minus190506

minus189966

minus189727

119877(119865)

01242

006

6900998

02136

03552

05968

08313

11188

15941

20368

119877119877 (119865)

24785

01033

00614

00892

01111

01386

01602

01833

02162

02431

119871119861119864lowast

minus25192

08591

09215

08817

08444

07979

07646

07327

06919

06623

119880119861119864lowast

14272

11911

11115

11588

11976

12459

12841

13257

13859

14367

PL-m

oments

GEV

119861119877 (119865)

minus02662

minus197059

minus194770

minus193340

minus191600

minus188996

minus186909

minus184797

minus182052

minus1800

66119877(119865)

05379

006

8700947

02023

03467

06171

09087

13058

20551

2854

119877119877 (119865)

38117

00948

006

4900960

01210

01508

01736

01977

02324

02609

119871119861119864lowast

minus77814

08892

09252

08725

08259

07697

07296

06911

064

1706050

119880119861119864lowast

45516

11827

11207

11716

12117

12554

12881

13211

13693

14104

PL-m

oments

GNO

119861119877 (119865)

35142

minus195922

minus193485

minus192704

minus191894

minus190806

minus1900

03minus18

9235

minus188284

minus187617

119877(119865)

06203

006

8900939

02141

03644

06143

08483

11272

15747

19809

119877119877 (119865)

43522

00971

00626

00980

01238

01504

01675

01831

02024

02162

119871119861119864lowast

minus10618

08772

09204

08655

08233

07800

07530

07290

07012

06822

119880119861119864lowast

46353

11849

11122

11734

12189

12653

12949

13218

13546

13780

PL-m

oments

GPA

119861119877 (119865)

1212

3minus19

4300

minus193238

minus193113

minus192716

minus192046

minus191555

minus19114

1minus19

0768

minus1906

49119877(119865)

08513

006

6601041

02250

03684

06021

08215

10853

15144

19094

119877119877 (119865)

58811

00962

006

6900996

01233

01509

01720

01941

02258

02518

119871119861119864lowast

minus17507

08698

09130

08680

08299

07846

07524

07220

06831

06552

119880119861119864lowast

47307

11791

11189

11774

12215

12739

1314

13567

14195

14722

Advances in Meteorology 13

Table9Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f60

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus13271

minus194230

minus193551

minus193241

minus192649

minus191612

minus190736

minus189839

minus188690

minus187888

119877(119865)

00846

00624

00851

01873

03250

05841

08652

12505

19839

27721

119877119877 (119865)

19155

00833

00612

00855

01024

01213

01360

01523

01770

01983

119871119861119864lowast

minus19593

08756

09254

08851

08491

08074

07796

07501

07118

06834

119880119861119864lowast

16242

11675

11015

11472

11879

12309

12678

13038

13659

14141

L-mom

ents

GNO

119861119877 (119865)

08417

minus193335

minus192795

minus192984

minus192999

minus192879

minus192742

minus192593

minus192399

minus192264

119877(119865)

00968

00634

00858

02000

03439

05850

08123

10845

15235

19239

119877119877 (119865)

21055

00881

00582

00861

01050

01235

01354

01463

01599

01699

119871119861119864lowast

minus22887

08647

09244

08794

08466

08141

07957

07798

07597

07443

119880119861119864lowast

15863

11714

1096

11511

11934

1240

512

666

12939

13253

13505

L-mom

ents

GPA

119861119877 (119865)

46371

minus192512

minus192631

minus193160

minus193417

minus193606

minus193745

minus193933

minus194306

minus194708

119877(119865)

01280

00624

00950

02094

03468

05740

07903

10536

14878

18928

119877119877 (119865)

264

4200915

00614

00882

01064

01270

01429

01599

01849

02059

119871119861119864lowast

minus38088

08628

09186

08815

08510

08167

07924

07703

07422

07178

119880119861119864lowast

14699

11721

11020

11534

11925

12454

12791

13236

13869

1440

5

PL-m

oments

GEV

119861119877 (119865)

minus16968

minus193984

minus193601

minus193623

minus193319

minus192576

minus191861

minus191081

minus190026

minus189255

119877(119865)

05431

00596

00888

01978

03366

05844

08417

11838

18179

24868

119877119877 (119865)

38656

00783

00634

0095

01166

01387

01541

01700

01928

02120

119871119861119864lowast

minus92921

08907

09201

08750

08403

07984

07714

07434

07043

06783

119880119861119864lowast

53553

11521

11099

1166

412

098

12572

12869

13216

13697

14105

PL-m

oments

GNO

119861119877 (119865)

23785

minus193306

minus192979

minus193417

minus193628

minus193678

minus193617

minus193509

minus193331

minus193185

119877(119865)

06278

006

0100911

02100

03523

05804

0788

10303

14113

17516

119877119877 (119865)

43743

00812

00623

00965

01186

01391

01513

01618

01742

01830

119871119861119864lowast

minus12083

08856

09188

08709

08389

08061

07870

07706

07517

07380

119880119861119864lowast

52707

11561

11059

11704

12152

12613

12898

13138

13361

1356

PL-m

oments

GPA

119861119877 (119865)

1118

7minus19

2590

minus192910

minus193681

minus1940

84minus19

4322

minus1944

04minus19

4463

minus194560

minus1946

73119877(119865)

08596

00597

01009

02229

03581

05615

07391

09418

12558

15341

119877119877 (119865)

58310

00840

006

6000991

0119

501391

01522

01650

01828

01971

119871119861119864lowast

minus18977

08800

09127

08699

08413

08098

07912

07682

07447

07255

119880119861119864lowast

51281

11551

11117

11769

12173

12615

12887

13227

1360

913

899

14 Advances in Meteorology

Table10A

ccuracymeasure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f90

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus00117

minus194226

minus193109

minus192926

minus192542

minus191809

minus191151

minus1904

48minus18

9497

minus188790

119877(119865)

00787

00556

00823

01850

03151

05459

07840

10989

16793

22883

119877119877 (119865)

17329

00814

00544

00832

01031

01238

01381

01525

01727

01892

119871119861119864lowast

minus19303

08935

09287

08885

08587

08246

08006

07772

07463

07229

119880119861119864lowast

15717

11499

10973

11461

11804

12156

12393

12635

12966

13242

L-mom

ents

GNO

119861119877 (119865)

15851

minus193606

minus192517

minus192695

minus192747

minus192678

minus192569

minus192437

minus192245

minus192098

119877(119865)

00917

00567

00846

01967

03305

05446

07392

09660

13223

1640

0119877119877 (119865)

19328

00850

00543

00853

01052

01237

01347

01442

01554

01632

119871119861119864lowast

minus23268

08848

09261

08844

08568

08301

08139

08000

07833

07722

119880119861119864lowast

15168

11542

10942

11489

11851

12197

1240

112

575

12779

12925

L-mom

ents

GPA

119861119877 (119865)

49081

minus192995

minus192417

minus192865

minus193078

minus193175

minus193195

minus193209

minus193256

minus193330

119877(119865)

01234

00571

00935

02081

03365

05322

07054

09053

12180

14975

119877119877 (119865)

246

8800882

00576

00870

01054

01239

01364

01489

01660

01798

119871119861119864lowast

minus45353

08783

09210

08838

08594

08326

08141

07961

07731

07557

119880119861119864lowast

14833

11574

10996

11532

11870

12223

1246

712

711

1304

413

312

PL-m

oments

GEV

119861119877 (119865)

minus00877

minus194145

minus193060

minus193062

minus192847

minus192278

minus191698

minus191038

minus190100

minus189374

119877(119865)

05458

00565

00855

01963

03338

05712

08090

11156

16650

22276

119877119877 (119865)

39079

00784

00585

00934

0117

201406

01558

01703

01897

02050

119871119861119864lowast

minus10190

08983

09236

08765

08431

08067

07818

07579

07271

07041

119880119861119864lowast

58730

1144

811036

11652

12081

12504

12771

13026

13364

13623

PL-m

oments

GNO

119861119877 (119865)

38414

minus193665

minus192536

minus192859

minus193044

minus193102

minus193057

minus192966

minus192804

minus192665

119877(119865)

06308

00578

00896

02088

03488

05690

07659

09926

13434

16524

119877119877 (119865)

44651

00819

00596

00961

0119

301405

01526

01628

01744

01822

119871119861119864lowast

minus13011

08910

09192

08716

08411

08122

07949

07801

07634

07519

119880119861119864lowast

56756

11491

11030

11693

12128

12539

12775

12974

13197

13347

PL-m

oments

GPA

119861119877 (119865)

126398

minus193142

minus192484

minus193077

minus193404

minus193589

minus193637

minus193655

minus193680

minus193720

119877(119865)

08641

00588

00999

02220

03558

05535

07227

09125

1200

914

521

119877119877 (119865)

60116

00855

006

4000985

0119

901400

01529

01651

01812

01937

119871119861119864lowast

minus19889

08842

09130

08700

08435

08156

07970

07789

07566

07399

119880119861119864lowast

53743

11528

11103

11755

12158

12552

12807

13055

13385

13636

Advances in Meteorology 15

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

0 2 4 6

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

Regional growth curve for GPA

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F))

0

5

10

15

20

Qua

ntile

2 4 60Reduced variate minuslog (minus log (F))

0

5

10

15

Qua

ntile

(based on L-moments) Regional growth curve for GPA

(based on PL-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30

16 Advances in Meteorology

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60

Advances in Meteorology 17

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90

18 Advances in Meteorology

The first four PL-moments of the GEV are defined as

12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)

(A5)

In previous equation 119875(sdot sdot) is an Incomplete Gammafunction

119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500

120579119896expminus120579119889Θ (A6)

Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution

The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by

119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)

The partial PWMs of the GLO are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)

(A9)

where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function

1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650

0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)

The first four PL-moments of the GLO are defined as

12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520

minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530

minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896

201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

(A11)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution

The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by

119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)

The partial PWMs of the GPA are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int

1

1198650

(1 minus 119865)119896 119865119903119889119865 (A14)

The first four PL-moments of the GPA are defined as

12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)

12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)

(A15)

where

119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution

Advances in Meteorology 19

Ethical Approval

The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975

Conflicts of Interest

The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest

Acknowledgments

The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027

References

[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960

[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985

[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981

[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982

[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988

[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007

[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997

[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990

[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987

[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996

[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990

[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004

[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011

[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009

[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010

[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009

[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012

[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013

[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001

[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002

[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008

[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007

[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010

[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011

[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013

[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974

[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991

[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990

[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979

[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993

[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993

20 Advances in Meteorology

[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993

[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995

[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990

[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993

[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993

[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994

[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995

[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996

[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001

[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009

[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mining

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GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

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MineralogyInternational Journal of

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MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Advances in Meteorology 3

37∘0㰀0㰀㰀N

36∘0㰀0㰀㰀N

35∘0㰀0㰀㰀N

34∘0㰀0㰀㰀N

33∘0㰀0㰀㰀N

32∘0㰀0㰀㰀N

37∘0㰀0㰀㰀N

36∘0㰀0㰀㰀N

35∘0㰀0㰀㰀N

34∘0㰀0㰀㰀N

33∘0㰀0㰀㰀N

32∘0㰀0㰀㰀N

71∘0㰀0㰀㰀E 72∘0㰀0㰀㰀E 73∘0㰀0㰀㰀E 74∘0㰀0㰀㰀E 75∘0㰀0㰀㰀E 76∘0㰀0㰀㰀E

71∘0㰀0㰀㰀E 72∘0㰀0㰀㰀E 73∘0㰀0㰀㰀E 74∘0㰀0㰀㰀E 75∘0㰀0㰀㰀E 76∘0㰀0㰀㰀E

W

N

E

S

0 30 60 120 180 240(km)

StationsWater bodiesStudy area

Figure 1 Locations of Northern areas and Khyber Pakhtunkhwa and the meteorological stations (119909 axis indicates Longitude [E] 119910 axisindicates Latitude [N])

The L-moments have no units of measurement which arecalled the L-moments ratios The L-moments ratios pro-posed by Hosking [28] are computed as

120591 = 12058221205821

1205913 = 12058231205822

1205914 = 12058241205822 (5)

4 Advances in Meteorology

where 120591 represents the L-coefficient of variation (L-Cv)1205913 represents the L-coefficient of skewness (L-Cs) and 1205914represents the L-coefficient of kurtosis (L-Ck)

The arranged sample is given as 119909(1) le 119909(2) le 119909(3) sdot sdot sdot le119909(119899) Wang [8] stated that the statistic

119887119903 = 1119899119899sum119894=1

(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909(119894) (6)

is an unbiased estimator of 120573119903So

1198971 = 11988701198972 = 21198871 minus 11988701198973 = 61198872 minus 61198871 + 11988701198974 = 201198873 minus 301198872 + 121198871 minus 1198870

(7)

where 1198971 1198972 1198973 and 1198974 are the first four L-moments of thesample And similarly

119905 = 11989721198971 1199053 = 11989731198972 1199054 = 11989741198972

(8)

are the sample L-moments ratios

222 The Partial L-Moments Wang [8 10 11] introduced aconcept of partial probability-weighted moments (PPWMs)that will estimate the higher quantiles of flood flows Data canbe censored to the right tail or left tail

Initially PPWMswere to take out the smaller values fromthe process of distribution fitting because such values haveslight influence on the frequency analysis and are nuisance tothe fitting process

The left tail PPWMs are defined by Wang [8 11] as

120573lowast119903 = int11198650

119909 (119865) 119865119903119889119865 (9)

where 1198650 = 119865(1199090) which is the lower limit of the censoredobservations and 1199090 is the censoring threshold value

The PPWMs elongated form described by Wang [10] areto be given a censored sample as

120573119903 = 11 minus 119865119903+10 int11198650

119909 (119865) 119865119903119889119865 (10)

If the value of 1198650 is starting from the zero then the result ofPPWMswill be the same as the usual PWMs As 119909(1) le 119909(2) le119909(3) sdot sdot sdot le 119909(119899) is the arranged sample Wang [10] describes theunbiased estimator of 120573119903 as

119903 = 1(1 minus 119865119903+10 ) 119899119899sum119894=1

(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909lowast(119894) (11)

where

119909lowast(119894) = 0 for 119909(119894) le 119909(0)119909lowast(119894) = 119909(119894) for 119909(119894) gt 119909(0) (12)

The censoring level 1198650 is the prior selection of the numberof censored sample data The procedure that determines thenumber of sample data points are to be censored

1198650 = 1198990119899 (13)

where 1198990 and 119899 are the lengths of sample which are to becensored and uncensored respectively Similarly 1199090 is thehighest value of the censored sample The first four PL-moments for the PPWMs are

1 = 12057302 = 2 1205731 minus 12057303 = 1205732 minus 6 1205731 + 12057304 = 20 1205733 minus 30 1205732 + 12 1205731 minus 1205730

(14)

Similarly the PL-moments ratios can be written as

120591 = 21

1205913 = 32

1205914 = 42

(15)

where 120591 1205913 and 1205914 denote the partial L-coefficient of variation(PL-Cv) partial L-coefficient of skewness (PL-Cs) and par-tial L-coefficient of kurtosis (PL-Ck) respectively Thereforethe first four sample PL-moments can be computed as

1198971 = 01198972 = 21 minus 01198973 = 62 minus 61 + 01198974 = 203 minus 302 + 121 minus 0

(16)

And the first four sample PL-moments ratios can be com-puted as

119905 = 11989721198971 1199053 = 11989731198972 1199054 = 11989741198972

(17)

Advances in Meteorology 5

where 119905 1199053 and 1199054 represent the sample partial L-momentsratios of the PL-Cv PL-Cs and PL-Ck respectively Thederivation is L-moment and PL-moments are given inAppendix In the present study different level of censoringthreshold is selected

23 Regional Frequency Analysis Hosking and Wallis [7 30]identified the following four steps to explain the procedure ofthe RFA

(1) Data screening(2) Designing of the homogeneous region(3) Selection of an appropriate probability distribution(4) Parameters estimation of the appropriate probability

distribution

231 Data Screening We screened data anomalies beforeapplying any statistical analysis

232 Discordance Test Hosking and Wallis [7] suggested adiscordancy measure (119863119894) test that recognizes the locationswhere sample L-moments are marked contrarily from themost other locations Locations with the large flaws in thedata will be flagged as discordant

The discordancy test for a region containing119873 locationsfor site 119894 is proposed by Hosking and Wallis [7] as follows

119863119894 = 13119873 (119906119894 minus 119906)119879 119878minus1 (119906119894 minus 119906) 119894 = 1 2 119873 (18)

where 119906119894 is the vector containing the three sample L-momentsratios for the site 119894 expressed as

119906119894 = [119905(119894)2 119905(119894)3 119905(119894)4 ]119879 (19)

119906 is the average vector of 119906119894 for the overall region that is

119906 = 1119873119873sum119894=1

119906119894 (20)

and 119878 is the covariance matrix for the sample that can beexpressed as

119878 = 119873sum119894=1

(119906119894 minus 119906) (119906119894 minus 119906)119879 (21)

Broadly speaking a location or a site is considered to bediscordant from the whole region or group if the value of 119863119894is larger than the critical value

233 Heterogeneity Test The homogeneity measure (119867119895)identifies homogenous regions Hosking and Wallis [7] Itis also useful to tag the locations if they are plausible tohandle as a homogeneous region It estimates the amountof heterogeneity in the overall region The heterogeneity test(119867119895) is computed as

119867119895 = 119881119895 minus 120583V119895120590V119895 119895 = 1 2 3 (22)

where 120583V119895 and 120590V119895 are representing the mean and standarddeviation of the simulated 119881119895 values Also

1198811 = 119873sum119894=1

119899119894 (119905(119894)2 minus 1199051198772 )sum119873119894=1 119899119894

12

1198812 =sum119873119894=1 [119899119894 (119905(119894)2 minus 1199051198772 )2 + (119905(119894)3 minus 1199051198773 )212]

sum119873119894=1 119899119894

1198813 =sum119873119894=1 [119899119894 (119905(119894)3 minus 1199051198773 )2 + (119905(119894)4 minus 1199051198774 )212]

sum119873119894=1 119899119894

(23)

Here 1199051198772 1199051198773 and 1199051198774 are region average L-moments or PL-moments ratios We assessed the heterogeneity of a region assuggested by Hosking and Wallis [7]

Region is acceptably homogeneous if119867 lt 1Region is possibly heterogeneous if 1 le 119867 lt 2Region is definitely heterogeneous if119867 ge 2

24 Selection of the Appropriate Probability DistributionHosking andWallis [7] proposed two approaches to select thedistribution that fitted best the data the L-moment ratios dia-gram and the 119885-test The L-moment ratios diagram is usingthe unbiased estimators Hosking [28] Stedinger et al [31]Vogel and Fennessey [32] and Hosking [33]The L-momentsratio diagram is a plot of the computed values L-Cs andthe observed values L-Ck of the distribution function Thecurves indicate the hypothetical connections between L-Csand L-Ck of the candidate distribution The L-moment ratiodiagrams have been proposed for discriminating between thecandidate probability distributions in describing the regionalinformation (Hosking [28] Stedinger et al [31] Hoskingand Wallis [7]) The L-moments ratio diagrams have beenused as a component of probability distribution process forregional information (Schaefer [34] Pearson [35] Vogel andFennessey [32] Vogel et al [36] Chow and Watt [37] OnOzand Bayazit [38] Vogel and Wilson [39] Peel et al [40])

Hosking and Wallis [7] suggested a measure to see howwell the L-Cs and L-Ck of the fitted probability distributionmatch the regional average L-Cs and L-Ck of the observedinformation

The measure goodness of fit for every single selectedprobability distribution is computed as follows

119885Dis = (120591Dis4 minus 1199051198774 )1205904 (24)

where 120591Dis4 represents the value of the L-Ck of the fitteddistribution 1199051198774 represents the weighted regional averageL-Ck and 1205904 represents standard deviation of the 1199051198774 whichis obtained from the simulation of the Kappa probabilitydistribution

If the computed value of 119885Dis is equal to zero the proba-bility distributionwill be themost suitable fit If the computed

6 Advances in Meteorology

Table 1 Statistics of annual extreme monthly precipitation for study region based on L-moments and PL-moments

Sites L-moments PL-momentsMean 119905 1199053 1199054 Mean 119905 1199053 1199054

Astore 37624 0518 0366 0193 41752 0467 0377 0183Balakot 120450 0478 0322 0192 133259 0428 0335 0198Bunji 14297 0609 0456 0259 16339 0553 0450 0257Chilas 15091 0585 0392 0194 17603 0516 0375 0200Cherat 49038 0580 0375 0162 55289 0526 0355 0161DI khan 28885 0689 0516 0262 32351 0651 0487 0250Dir 105694 0432 0216 0098 116741 0378 0222 0097Drosh 45284 0525 0317 0136 50316 0474 0307 0137Garhi Dupatta 111352 0454 0227 0111 123374 0398 0229 0120Gilgit 12677 0591 0412 0228 14290 0539 0399 0239Gupis 23446 0678 0529 0331 30299 0583 0506 0350Kakul 99431 0453 0247 0102 110068 0399 0254 0093Kotli 95070 0542 0332 0136 106903 0487 0314 0138Muzaffarabad 118439 0459 0284 0182 131122 0405 0301 0194Peshawar 46216 0572 0398 0216 51761 0520 0392 0222Saidu Sharif 85667 0455 0265 0181 94978 0399 0283 0200Skardu 20438 0616 04564 0238 22885 0570 0444 0231

value of 119885-statistic is less than 164 at 90 confidence level(ie |119885Dis| le 164) it will indicate that the distributionqualifies the goodness of fit criteria If there are more thanone distribution that qualify the criteria the most suitabledistribution has the minimum |119885Dis| value3 Estimation

The sitesrsquo information and statistic by using L-moments forthe present study are presented in Table 1 In Table 1 meanrepresents the first sample LPL-moments and 119905 1199053 and 1199054are the sample LPL-moments ratios of the L-CvPL-CvL-CsPL-Cs and L-CkPL-Ck respectively The lower levelcensoring threshold is selected from 10 to 23 Table 2expresses the feasible threshold values according to thepercentile technique along with Average Annual OccurrenceNumber (AAON) Jiang et al [41] and Yuguo [42] suggestedthat the optimal threshold can be obtained if the values ofAAON lie between 1 and 2 Table 2 shows that 90th percentileobservations are suitable for the optimum threshold selectionof most of areas in the present study We have 168 valuesin each station according to the above table Astore stationhas 32 threshold values due to which 16 values are beingcensored in 168 According to censored level 102 censoredlevel was selected Similarly 105 was selected for Balakotand Muzaffarabad According to this process maximumthreshold level 223 was selected for Gupis By using theabove process for each station 17 different censoring levelswere selected So we decide that the range from 10 to 23of censoring level should be kept for selecting thresholdvalues

31 Regional Frequency Analysis The following four steps areconsidered as prerequisite for frequency analysis Hoskingand Wallis [7 30]

(1) Data screening(2) Designing of the homogeneous region(3) Selection of an appropriate probability distribution(4) Parameters estimation of the appropriate probability

distribution

311 Data Screening In this study we use secondary dataafter carefully examining all locations for abnormalities andmissing observations Therefore we use 14 years of data forRFA that were obtained from seventeen locations

312 Discordance Test Table 3 shows 119863119894 result of (18) for17 locations of this study region It can be observed fromthe results of LPL-moments in Table 3 that the value of 119863-statistic varies from 007 to 244 If 119863119894 is greater than 3 thelocation is considered to be discordant from the rest of theregional data Hosking and Wallis [7] In this study regionno location is diagnosed as discordant (119863119894 ge 3) Thereforewe use all data for the development of the RFA based on L-moments and PL-moments

313 Regional Heterogeneity Measure The next step is theformation of the homogeneous region which is conven-tionally tougher and needs the higher number of subjectivejudgments The homogeneity conditions are defined as thelocations that have the same frequency distributions

Advances in Meteorology 7

Table 2 Precipitation threshold selection in GPA distribution for 17 stations

Station 80th 90th 95th 975th 99thAstore

Threshold 98 32 02 0 0AAON 243 121 064 043 043

BalakotThreshold 325 148 51 3 0AAON 243 129 057 029 014

MuzaffarabadThreshold 304 118 5 1 0AAON 25 121 057 029 021

Garhi DupattaThreshold 288 10 38 0 0AAON 243 121 057 029 029

DirThreshold 31 20 5 005 0AAON 243 121 057 029 021

KakulThreshold 289 134 21 05 0AAON 164 129 057 029 021

KotliThreshold 13 6 005 0 0AAON 243 135 057 036 036

Saidu SharifThreshold 256 83 2 0 0AAON 243 121 057 036 036

cheratThreshold 5 005 0 0 0AAON 243 143 093 093 093

PeshawarThreshold 6 005 0 0 0AAON 243 129 057 057 057

DroshThreshold 76 15 02 005 0AAON 243 121 057 05 021

D I KhanThreshold 03 0 0 0 0AAON 257 129 129 129 129

GupisThreshold 0 0 0 0 0AAON 271 271 271 271 271

SkarduThreshold 2 03 0 0 0AAON 243 071 071 071 071

ChilasThreshold 1 05 0 0 0AAON 171 107 107 107 107

BunjiThreshold 13 0 0 0 0AAON 15 15 15 15 15

GilgitThreshold 12 005 0 0 0AAON 243 135 071 071 071

8 Advances in Meteorology

04

03

02

01

00

04

03

02

01

00

00 01 02 03 04 05 06 00 01 02 03 04 05 06

GLOGEVGPA

GNOAverageData

GLOGEVGPA

GNOAverageData

L-ku

rtos

is

L㰀-k

urto

sis

L-skewness L-skewness

Figure 2 L-diagram and PL-diagram of the GLO GEV GPA and GNO distributions

Table 3 Discordance test result based on L-moments and PL-moments

Name of site L-moments (119863119894) PL-moments (119863119894)Astore 205 244Balakot 142 085Bunji 055 060Chilas 029 007Cherat 079 064DI khan 129 190Dir 082 086Drosh 051 043Garhi Dupatta 143 133Gilgit 026 059Gupis 237 231Kakul 082 122Kotli 077 059Muzaffarabad 099 085Peshawar 007 008Saidu Sharif 191 157Skardu 067 065

For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments

Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different

values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary

32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO

We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area

Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments

Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions

However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions

Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution

Advances in Meteorology 9

Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments

Heterogeneity measures L-moment PL-momentHeterogeneity measure1198671

Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060

Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800

Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170

Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments

Method Distribution Parameters120585 120572 119870L-moments

GEV 04660 05644 minus02751GNO 06671 07565 minus07594GPA minus00551 09944 minus00575

PL-momentsGEV 05220 05116 minus02686GNO 07046 06837 minus07487GPA 00478 09073 minus00472

is larger than the critical value of 164 (at 90 confidencelevel)

It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile

Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution

33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7

based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)

Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is

119876 (119865) = 1198971198941119902 (119865) (25)

The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3

Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions

Therefore it is necessary to assess the performance ofregional quantile estimates

4 Accuracy of the Estimated Quantiles and theRegional Growth Curve

A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten

10 Advances in Meteorology

Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865

01000 05000 08000 09000 09500 09800 09900 09975 099875 09990

L-momentsGEV 00454 06836 15139 22247 30591 44161 56871 90748 113165 121338GNO 00473 06671 15585 23072 31448 44098 54997 80676 95666 100823GPA 00500 06481 16217 23931 31958 43073 51878 70578 80502 83782

PL-momentsGEV 01397 07190 14670 21034 28469 40497 51703 81363 10086 107950GNO 01412 07046 15062 21752 29203 40410 50032 72608 85739 90249GPA 01436 06871 15649 22547 29674 39460 47149 63304 71786 74577

Qua

ntile

Regional growth curve (L-moments)

GPAGNOGEV

GPAGNOGEV

Qua

ntile

Return period

Regional growth gurve (PL-moments)

5

4

3

2

1

0

5

4

3

2

1

0

minus2 minus1 0 1 2 3 4 5

2 5 10 20 50 100

Return period2 5 10 20 50 100

Reduced variate minuslog (minus log (F))

minus2 minus1 0 1 2 3 4 5

Reduced variate minuslog (minus log (F))

Figure 3 L-moments and PL-moments regional growth curves

as 119876(119898)119894 (119865) for the nonexceedance probability 119865 The relativeerror for this estimator is

119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) (26)

The bias and the RMSE of the above quantity over all 119872repetition are

Bias = 119861119894 (119865) = 1119872119872sum119898=1

119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)

= 1119872 [[119872sum119898=1

(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12

(27)

Also for the estimated quantile the regional average bias andthe relative RMSE are

119861119877 (119865) = 1119873119873sum119894=1

119861119894 (119865)

119877119877 (119865) = 1119873119873sum119894=1

119877119894 (119865) (28)

We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval

119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)

Advances in Meteorology 11

Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le

119902 (119865)119871005 (119865) (30)

The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves

We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]

We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively

Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds

Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion

This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail

GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments

Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents

Appendix

The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows

PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by

119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (minus ln119865)119896 119896 = 0 (A2)

Wang [10] developed the partial probability-weightedmoments (PWMs) of the GEV as

(119903 + 1) 1205731015840119903 = 120577 + 120572119867 (120574 1198650 119896) (A3)

where

119867(120574 1198650 119896) = 1119896 [1 minus119875 1 + 119896 minus (119903 + 1) ln1198650(1 minus 119865119903+10 ) (119903 + 1)119896 ] (A4)

12 Advances in Meteorology

Table8Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f30

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus0540

0minus19

7674

minus195328

minus193602

minus191579

minus188659

minus186376

minus1840

97minus18

1165

minus179056

119877(119865)

00900

00695

00963

01952

03334

06038

09057

13261

21337

300

48119877119877 (119865)

19873

01019

00635

00876

01090

01369

01595

01839

02191

02478

119871119861119864lowast

minus20937

08829

09296

08842

08411

07862

07460

07065

06557

06177

119880119861119864lowast

18714

11949

11208

11580

11884

12244

12516

12818

13285

13683

L-mom

ents

GNO

119861119877 (119865)

08124

minus196279

minus193846

minus192837

minus191816

minus190506

minus189573

minus188699

minus187636

minus186899

119877(119865)

00986

00696

00917

02051

03520

06027

08421

11312

16008

20312

119877119877 (119865)

20766

01045

00587

00889

01120

01368

01533

01686

01878

02016

119871119861119864lowast

minus21331

08683

09271

08780

08379

07958

07691

07453

07172

06978

119880119861119864lowast

16569

11974

11077

11569

11954

12355

12615

12851

13149

13369

L-mom

ents

GPA

119861119877 (119865)

38654

minus194351

minus193564

minus193226

minus192616

minus191706

minus191060

minus190506

minus189966

minus189727

119877(119865)

01242

006

6900998

02136

03552

05968

08313

11188

15941

20368

119877119877 (119865)

24785

01033

00614

00892

01111

01386

01602

01833

02162

02431

119871119861119864lowast

minus25192

08591

09215

08817

08444

07979

07646

07327

06919

06623

119880119861119864lowast

14272

11911

11115

11588

11976

12459

12841

13257

13859

14367

PL-m

oments

GEV

119861119877 (119865)

minus02662

minus197059

minus194770

minus193340

minus191600

minus188996

minus186909

minus184797

minus182052

minus1800

66119877(119865)

05379

006

8700947

02023

03467

06171

09087

13058

20551

2854

119877119877 (119865)

38117

00948

006

4900960

01210

01508

01736

01977

02324

02609

119871119861119864lowast

minus77814

08892

09252

08725

08259

07697

07296

06911

064

1706050

119880119861119864lowast

45516

11827

11207

11716

12117

12554

12881

13211

13693

14104

PL-m

oments

GNO

119861119877 (119865)

35142

minus195922

minus193485

minus192704

minus191894

minus190806

minus1900

03minus18

9235

minus188284

minus187617

119877(119865)

06203

006

8900939

02141

03644

06143

08483

11272

15747

19809

119877119877 (119865)

43522

00971

00626

00980

01238

01504

01675

01831

02024

02162

119871119861119864lowast

minus10618

08772

09204

08655

08233

07800

07530

07290

07012

06822

119880119861119864lowast

46353

11849

11122

11734

12189

12653

12949

13218

13546

13780

PL-m

oments

GPA

119861119877 (119865)

1212

3minus19

4300

minus193238

minus193113

minus192716

minus192046

minus191555

minus19114

1minus19

0768

minus1906

49119877(119865)

08513

006

6601041

02250

03684

06021

08215

10853

15144

19094

119877119877 (119865)

58811

00962

006

6900996

01233

01509

01720

01941

02258

02518

119871119861119864lowast

minus17507

08698

09130

08680

08299

07846

07524

07220

06831

06552

119880119861119864lowast

47307

11791

11189

11774

12215

12739

1314

13567

14195

14722

Advances in Meteorology 13

Table9Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f60

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus13271

minus194230

minus193551

minus193241

minus192649

minus191612

minus190736

minus189839

minus188690

minus187888

119877(119865)

00846

00624

00851

01873

03250

05841

08652

12505

19839

27721

119877119877 (119865)

19155

00833

00612

00855

01024

01213

01360

01523

01770

01983

119871119861119864lowast

minus19593

08756

09254

08851

08491

08074

07796

07501

07118

06834

119880119861119864lowast

16242

11675

11015

11472

11879

12309

12678

13038

13659

14141

L-mom

ents

GNO

119861119877 (119865)

08417

minus193335

minus192795

minus192984

minus192999

minus192879

minus192742

minus192593

minus192399

minus192264

119877(119865)

00968

00634

00858

02000

03439

05850

08123

10845

15235

19239

119877119877 (119865)

21055

00881

00582

00861

01050

01235

01354

01463

01599

01699

119871119861119864lowast

minus22887

08647

09244

08794

08466

08141

07957

07798

07597

07443

119880119861119864lowast

15863

11714

1096

11511

11934

1240

512

666

12939

13253

13505

L-mom

ents

GPA

119861119877 (119865)

46371

minus192512

minus192631

minus193160

minus193417

minus193606

minus193745

minus193933

minus194306

minus194708

119877(119865)

01280

00624

00950

02094

03468

05740

07903

10536

14878

18928

119877119877 (119865)

264

4200915

00614

00882

01064

01270

01429

01599

01849

02059

119871119861119864lowast

minus38088

08628

09186

08815

08510

08167

07924

07703

07422

07178

119880119861119864lowast

14699

11721

11020

11534

11925

12454

12791

13236

13869

1440

5

PL-m

oments

GEV

119861119877 (119865)

minus16968

minus193984

minus193601

minus193623

minus193319

minus192576

minus191861

minus191081

minus190026

minus189255

119877(119865)

05431

00596

00888

01978

03366

05844

08417

11838

18179

24868

119877119877 (119865)

38656

00783

00634

0095

01166

01387

01541

01700

01928

02120

119871119861119864lowast

minus92921

08907

09201

08750

08403

07984

07714

07434

07043

06783

119880119861119864lowast

53553

11521

11099

1166

412

098

12572

12869

13216

13697

14105

PL-m

oments

GNO

119861119877 (119865)

23785

minus193306

minus192979

minus193417

minus193628

minus193678

minus193617

minus193509

minus193331

minus193185

119877(119865)

06278

006

0100911

02100

03523

05804

0788

10303

14113

17516

119877119877 (119865)

43743

00812

00623

00965

01186

01391

01513

01618

01742

01830

119871119861119864lowast

minus12083

08856

09188

08709

08389

08061

07870

07706

07517

07380

119880119861119864lowast

52707

11561

11059

11704

12152

12613

12898

13138

13361

1356

PL-m

oments

GPA

119861119877 (119865)

1118

7minus19

2590

minus192910

minus193681

minus1940

84minus19

4322

minus1944

04minus19

4463

minus194560

minus1946

73119877(119865)

08596

00597

01009

02229

03581

05615

07391

09418

12558

15341

119877119877 (119865)

58310

00840

006

6000991

0119

501391

01522

01650

01828

01971

119871119861119864lowast

minus18977

08800

09127

08699

08413

08098

07912

07682

07447

07255

119880119861119864lowast

51281

11551

11117

11769

12173

12615

12887

13227

1360

913

899

14 Advances in Meteorology

Table10A

ccuracymeasure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f90

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus00117

minus194226

minus193109

minus192926

minus192542

minus191809

minus191151

minus1904

48minus18

9497

minus188790

119877(119865)

00787

00556

00823

01850

03151

05459

07840

10989

16793

22883

119877119877 (119865)

17329

00814

00544

00832

01031

01238

01381

01525

01727

01892

119871119861119864lowast

minus19303

08935

09287

08885

08587

08246

08006

07772

07463

07229

119880119861119864lowast

15717

11499

10973

11461

11804

12156

12393

12635

12966

13242

L-mom

ents

GNO

119861119877 (119865)

15851

minus193606

minus192517

minus192695

minus192747

minus192678

minus192569

minus192437

minus192245

minus192098

119877(119865)

00917

00567

00846

01967

03305

05446

07392

09660

13223

1640

0119877119877 (119865)

19328

00850

00543

00853

01052

01237

01347

01442

01554

01632

119871119861119864lowast

minus23268

08848

09261

08844

08568

08301

08139

08000

07833

07722

119880119861119864lowast

15168

11542

10942

11489

11851

12197

1240

112

575

12779

12925

L-mom

ents

GPA

119861119877 (119865)

49081

minus192995

minus192417

minus192865

minus193078

minus193175

minus193195

minus193209

minus193256

minus193330

119877(119865)

01234

00571

00935

02081

03365

05322

07054

09053

12180

14975

119877119877 (119865)

246

8800882

00576

00870

01054

01239

01364

01489

01660

01798

119871119861119864lowast

minus45353

08783

09210

08838

08594

08326

08141

07961

07731

07557

119880119861119864lowast

14833

11574

10996

11532

11870

12223

1246

712

711

1304

413

312

PL-m

oments

GEV

119861119877 (119865)

minus00877

minus194145

minus193060

minus193062

minus192847

minus192278

minus191698

minus191038

minus190100

minus189374

119877(119865)

05458

00565

00855

01963

03338

05712

08090

11156

16650

22276

119877119877 (119865)

39079

00784

00585

00934

0117

201406

01558

01703

01897

02050

119871119861119864lowast

minus10190

08983

09236

08765

08431

08067

07818

07579

07271

07041

119880119861119864lowast

58730

1144

811036

11652

12081

12504

12771

13026

13364

13623

PL-m

oments

GNO

119861119877 (119865)

38414

minus193665

minus192536

minus192859

minus193044

minus193102

minus193057

minus192966

minus192804

minus192665

119877(119865)

06308

00578

00896

02088

03488

05690

07659

09926

13434

16524

119877119877 (119865)

44651

00819

00596

00961

0119

301405

01526

01628

01744

01822

119871119861119864lowast

minus13011

08910

09192

08716

08411

08122

07949

07801

07634

07519

119880119861119864lowast

56756

11491

11030

11693

12128

12539

12775

12974

13197

13347

PL-m

oments

GPA

119861119877 (119865)

126398

minus193142

minus192484

minus193077

minus193404

minus193589

minus193637

minus193655

minus193680

minus193720

119877(119865)

08641

00588

00999

02220

03558

05535

07227

09125

1200

914

521

119877119877 (119865)

60116

00855

006

4000985

0119

901400

01529

01651

01812

01937

119871119861119864lowast

minus19889

08842

09130

08700

08435

08156

07970

07789

07566

07399

119880119861119864lowast

53743

11528

11103

11755

12158

12552

12807

13055

13385

13636

Advances in Meteorology 15

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

0 2 4 6

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

Regional growth curve for GPA

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F))

0

5

10

15

20

Qua

ntile

2 4 60Reduced variate minuslog (minus log (F))

0

5

10

15

Qua

ntile

(based on L-moments) Regional growth curve for GPA

(based on PL-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30

16 Advances in Meteorology

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60

Advances in Meteorology 17

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90

18 Advances in Meteorology

The first four PL-moments of the GEV are defined as

12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)

(A5)

In previous equation 119875(sdot sdot) is an Incomplete Gammafunction

119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500

120579119896expminus120579119889Θ (A6)

Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution

The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by

119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)

The partial PWMs of the GLO are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)

(A9)

where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function

1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650

0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)

The first four PL-moments of the GLO are defined as

12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520

minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530

minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896

201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

(A11)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution

The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by

119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)

The partial PWMs of the GPA are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int

1

1198650

(1 minus 119865)119896 119865119903119889119865 (A14)

The first four PL-moments of the GPA are defined as

12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)

12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)

(A15)

where

119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution

Advances in Meteorology 19

Ethical Approval

The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975

Conflicts of Interest

The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest

Acknowledgments

The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027

References

[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960

[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985

[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981

[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982

[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988

[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007

[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997

[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990

[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987

[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996

[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990

[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004

[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011

[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009

[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010

[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009

[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012

[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013

[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001

[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002

[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008

[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007

[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010

[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011

[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013

[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974

[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991

[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990

[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979

[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993

[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993

20 Advances in Meteorology

[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993

[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995

[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990

[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993

[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993

[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994

[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995

[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996

[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001

[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009

[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012

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4 Advances in Meteorology

where 120591 represents the L-coefficient of variation (L-Cv)1205913 represents the L-coefficient of skewness (L-Cs) and 1205914represents the L-coefficient of kurtosis (L-Ck)

The arranged sample is given as 119909(1) le 119909(2) le 119909(3) sdot sdot sdot le119909(119899) Wang [8] stated that the statistic

119887119903 = 1119899119899sum119894=1

(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909(119894) (6)

is an unbiased estimator of 120573119903So

1198971 = 11988701198972 = 21198871 minus 11988701198973 = 61198872 minus 61198871 + 11988701198974 = 201198873 minus 301198872 + 121198871 minus 1198870

(7)

where 1198971 1198972 1198973 and 1198974 are the first four L-moments of thesample And similarly

119905 = 11989721198971 1199053 = 11989731198972 1199054 = 11989741198972

(8)

are the sample L-moments ratios

222 The Partial L-Moments Wang [8 10 11] introduced aconcept of partial probability-weighted moments (PPWMs)that will estimate the higher quantiles of flood flows Data canbe censored to the right tail or left tail

Initially PPWMswere to take out the smaller values fromthe process of distribution fitting because such values haveslight influence on the frequency analysis and are nuisance tothe fitting process

The left tail PPWMs are defined by Wang [8 11] as

120573lowast119903 = int11198650

119909 (119865) 119865119903119889119865 (9)

where 1198650 = 119865(1199090) which is the lower limit of the censoredobservations and 1199090 is the censoring threshold value

The PPWMs elongated form described by Wang [10] areto be given a censored sample as

120573119903 = 11 minus 119865119903+10 int11198650

119909 (119865) 119865119903119889119865 (10)

If the value of 1198650 is starting from the zero then the result ofPPWMswill be the same as the usual PWMs As 119909(1) le 119909(2) le119909(3) sdot sdot sdot le 119909(119899) is the arranged sample Wang [10] describes theunbiased estimator of 120573119903 as

119903 = 1(1 minus 119865119903+10 ) 119899119899sum119894=1

(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909lowast(119894) (11)

where

119909lowast(119894) = 0 for 119909(119894) le 119909(0)119909lowast(119894) = 119909(119894) for 119909(119894) gt 119909(0) (12)

The censoring level 1198650 is the prior selection of the numberof censored sample data The procedure that determines thenumber of sample data points are to be censored

1198650 = 1198990119899 (13)

where 1198990 and 119899 are the lengths of sample which are to becensored and uncensored respectively Similarly 1199090 is thehighest value of the censored sample The first four PL-moments for the PPWMs are

1 = 12057302 = 2 1205731 minus 12057303 = 1205732 minus 6 1205731 + 12057304 = 20 1205733 minus 30 1205732 + 12 1205731 minus 1205730

(14)

Similarly the PL-moments ratios can be written as

120591 = 21

1205913 = 32

1205914 = 42

(15)

where 120591 1205913 and 1205914 denote the partial L-coefficient of variation(PL-Cv) partial L-coefficient of skewness (PL-Cs) and par-tial L-coefficient of kurtosis (PL-Ck) respectively Thereforethe first four sample PL-moments can be computed as

1198971 = 01198972 = 21 minus 01198973 = 62 minus 61 + 01198974 = 203 minus 302 + 121 minus 0

(16)

And the first four sample PL-moments ratios can be com-puted as

119905 = 11989721198971 1199053 = 11989731198972 1199054 = 11989741198972

(17)

Advances in Meteorology 5

where 119905 1199053 and 1199054 represent the sample partial L-momentsratios of the PL-Cv PL-Cs and PL-Ck respectively Thederivation is L-moment and PL-moments are given inAppendix In the present study different level of censoringthreshold is selected

23 Regional Frequency Analysis Hosking and Wallis [7 30]identified the following four steps to explain the procedure ofthe RFA

(1) Data screening(2) Designing of the homogeneous region(3) Selection of an appropriate probability distribution(4) Parameters estimation of the appropriate probability

distribution

231 Data Screening We screened data anomalies beforeapplying any statistical analysis

232 Discordance Test Hosking and Wallis [7] suggested adiscordancy measure (119863119894) test that recognizes the locationswhere sample L-moments are marked contrarily from themost other locations Locations with the large flaws in thedata will be flagged as discordant

The discordancy test for a region containing119873 locationsfor site 119894 is proposed by Hosking and Wallis [7] as follows

119863119894 = 13119873 (119906119894 minus 119906)119879 119878minus1 (119906119894 minus 119906) 119894 = 1 2 119873 (18)

where 119906119894 is the vector containing the three sample L-momentsratios for the site 119894 expressed as

119906119894 = [119905(119894)2 119905(119894)3 119905(119894)4 ]119879 (19)

119906 is the average vector of 119906119894 for the overall region that is

119906 = 1119873119873sum119894=1

119906119894 (20)

and 119878 is the covariance matrix for the sample that can beexpressed as

119878 = 119873sum119894=1

(119906119894 minus 119906) (119906119894 minus 119906)119879 (21)

Broadly speaking a location or a site is considered to bediscordant from the whole region or group if the value of 119863119894is larger than the critical value

233 Heterogeneity Test The homogeneity measure (119867119895)identifies homogenous regions Hosking and Wallis [7] Itis also useful to tag the locations if they are plausible tohandle as a homogeneous region It estimates the amountof heterogeneity in the overall region The heterogeneity test(119867119895) is computed as

119867119895 = 119881119895 minus 120583V119895120590V119895 119895 = 1 2 3 (22)

where 120583V119895 and 120590V119895 are representing the mean and standarddeviation of the simulated 119881119895 values Also

1198811 = 119873sum119894=1

119899119894 (119905(119894)2 minus 1199051198772 )sum119873119894=1 119899119894

12

1198812 =sum119873119894=1 [119899119894 (119905(119894)2 minus 1199051198772 )2 + (119905(119894)3 minus 1199051198773 )212]

sum119873119894=1 119899119894

1198813 =sum119873119894=1 [119899119894 (119905(119894)3 minus 1199051198773 )2 + (119905(119894)4 minus 1199051198774 )212]

sum119873119894=1 119899119894

(23)

Here 1199051198772 1199051198773 and 1199051198774 are region average L-moments or PL-moments ratios We assessed the heterogeneity of a region assuggested by Hosking and Wallis [7]

Region is acceptably homogeneous if119867 lt 1Region is possibly heterogeneous if 1 le 119867 lt 2Region is definitely heterogeneous if119867 ge 2

24 Selection of the Appropriate Probability DistributionHosking andWallis [7] proposed two approaches to select thedistribution that fitted best the data the L-moment ratios dia-gram and the 119885-test The L-moment ratios diagram is usingthe unbiased estimators Hosking [28] Stedinger et al [31]Vogel and Fennessey [32] and Hosking [33]The L-momentsratio diagram is a plot of the computed values L-Cs andthe observed values L-Ck of the distribution function Thecurves indicate the hypothetical connections between L-Csand L-Ck of the candidate distribution The L-moment ratiodiagrams have been proposed for discriminating between thecandidate probability distributions in describing the regionalinformation (Hosking [28] Stedinger et al [31] Hoskingand Wallis [7]) The L-moments ratio diagrams have beenused as a component of probability distribution process forregional information (Schaefer [34] Pearson [35] Vogel andFennessey [32] Vogel et al [36] Chow and Watt [37] OnOzand Bayazit [38] Vogel and Wilson [39] Peel et al [40])

Hosking and Wallis [7] suggested a measure to see howwell the L-Cs and L-Ck of the fitted probability distributionmatch the regional average L-Cs and L-Ck of the observedinformation

The measure goodness of fit for every single selectedprobability distribution is computed as follows

119885Dis = (120591Dis4 minus 1199051198774 )1205904 (24)

where 120591Dis4 represents the value of the L-Ck of the fitteddistribution 1199051198774 represents the weighted regional averageL-Ck and 1205904 represents standard deviation of the 1199051198774 whichis obtained from the simulation of the Kappa probabilitydistribution

If the computed value of 119885Dis is equal to zero the proba-bility distributionwill be themost suitable fit If the computed

6 Advances in Meteorology

Table 1 Statistics of annual extreme monthly precipitation for study region based on L-moments and PL-moments

Sites L-moments PL-momentsMean 119905 1199053 1199054 Mean 119905 1199053 1199054

Astore 37624 0518 0366 0193 41752 0467 0377 0183Balakot 120450 0478 0322 0192 133259 0428 0335 0198Bunji 14297 0609 0456 0259 16339 0553 0450 0257Chilas 15091 0585 0392 0194 17603 0516 0375 0200Cherat 49038 0580 0375 0162 55289 0526 0355 0161DI khan 28885 0689 0516 0262 32351 0651 0487 0250Dir 105694 0432 0216 0098 116741 0378 0222 0097Drosh 45284 0525 0317 0136 50316 0474 0307 0137Garhi Dupatta 111352 0454 0227 0111 123374 0398 0229 0120Gilgit 12677 0591 0412 0228 14290 0539 0399 0239Gupis 23446 0678 0529 0331 30299 0583 0506 0350Kakul 99431 0453 0247 0102 110068 0399 0254 0093Kotli 95070 0542 0332 0136 106903 0487 0314 0138Muzaffarabad 118439 0459 0284 0182 131122 0405 0301 0194Peshawar 46216 0572 0398 0216 51761 0520 0392 0222Saidu Sharif 85667 0455 0265 0181 94978 0399 0283 0200Skardu 20438 0616 04564 0238 22885 0570 0444 0231

value of 119885-statistic is less than 164 at 90 confidence level(ie |119885Dis| le 164) it will indicate that the distributionqualifies the goodness of fit criteria If there are more thanone distribution that qualify the criteria the most suitabledistribution has the minimum |119885Dis| value3 Estimation

The sitesrsquo information and statistic by using L-moments forthe present study are presented in Table 1 In Table 1 meanrepresents the first sample LPL-moments and 119905 1199053 and 1199054are the sample LPL-moments ratios of the L-CvPL-CvL-CsPL-Cs and L-CkPL-Ck respectively The lower levelcensoring threshold is selected from 10 to 23 Table 2expresses the feasible threshold values according to thepercentile technique along with Average Annual OccurrenceNumber (AAON) Jiang et al [41] and Yuguo [42] suggestedthat the optimal threshold can be obtained if the values ofAAON lie between 1 and 2 Table 2 shows that 90th percentileobservations are suitable for the optimum threshold selectionof most of areas in the present study We have 168 valuesin each station according to the above table Astore stationhas 32 threshold values due to which 16 values are beingcensored in 168 According to censored level 102 censoredlevel was selected Similarly 105 was selected for Balakotand Muzaffarabad According to this process maximumthreshold level 223 was selected for Gupis By using theabove process for each station 17 different censoring levelswere selected So we decide that the range from 10 to 23of censoring level should be kept for selecting thresholdvalues

31 Regional Frequency Analysis The following four steps areconsidered as prerequisite for frequency analysis Hoskingand Wallis [7 30]

(1) Data screening(2) Designing of the homogeneous region(3) Selection of an appropriate probability distribution(4) Parameters estimation of the appropriate probability

distribution

311 Data Screening In this study we use secondary dataafter carefully examining all locations for abnormalities andmissing observations Therefore we use 14 years of data forRFA that were obtained from seventeen locations

312 Discordance Test Table 3 shows 119863119894 result of (18) for17 locations of this study region It can be observed fromthe results of LPL-moments in Table 3 that the value of 119863-statistic varies from 007 to 244 If 119863119894 is greater than 3 thelocation is considered to be discordant from the rest of theregional data Hosking and Wallis [7] In this study regionno location is diagnosed as discordant (119863119894 ge 3) Thereforewe use all data for the development of the RFA based on L-moments and PL-moments

313 Regional Heterogeneity Measure The next step is theformation of the homogeneous region which is conven-tionally tougher and needs the higher number of subjectivejudgments The homogeneity conditions are defined as thelocations that have the same frequency distributions

Advances in Meteorology 7

Table 2 Precipitation threshold selection in GPA distribution for 17 stations

Station 80th 90th 95th 975th 99thAstore

Threshold 98 32 02 0 0AAON 243 121 064 043 043

BalakotThreshold 325 148 51 3 0AAON 243 129 057 029 014

MuzaffarabadThreshold 304 118 5 1 0AAON 25 121 057 029 021

Garhi DupattaThreshold 288 10 38 0 0AAON 243 121 057 029 029

DirThreshold 31 20 5 005 0AAON 243 121 057 029 021

KakulThreshold 289 134 21 05 0AAON 164 129 057 029 021

KotliThreshold 13 6 005 0 0AAON 243 135 057 036 036

Saidu SharifThreshold 256 83 2 0 0AAON 243 121 057 036 036

cheratThreshold 5 005 0 0 0AAON 243 143 093 093 093

PeshawarThreshold 6 005 0 0 0AAON 243 129 057 057 057

DroshThreshold 76 15 02 005 0AAON 243 121 057 05 021

D I KhanThreshold 03 0 0 0 0AAON 257 129 129 129 129

GupisThreshold 0 0 0 0 0AAON 271 271 271 271 271

SkarduThreshold 2 03 0 0 0AAON 243 071 071 071 071

ChilasThreshold 1 05 0 0 0AAON 171 107 107 107 107

BunjiThreshold 13 0 0 0 0AAON 15 15 15 15 15

GilgitThreshold 12 005 0 0 0AAON 243 135 071 071 071

8 Advances in Meteorology

04

03

02

01

00

04

03

02

01

00

00 01 02 03 04 05 06 00 01 02 03 04 05 06

GLOGEVGPA

GNOAverageData

GLOGEVGPA

GNOAverageData

L-ku

rtos

is

L㰀-k

urto

sis

L-skewness L-skewness

Figure 2 L-diagram and PL-diagram of the GLO GEV GPA and GNO distributions

Table 3 Discordance test result based on L-moments and PL-moments

Name of site L-moments (119863119894) PL-moments (119863119894)Astore 205 244Balakot 142 085Bunji 055 060Chilas 029 007Cherat 079 064DI khan 129 190Dir 082 086Drosh 051 043Garhi Dupatta 143 133Gilgit 026 059Gupis 237 231Kakul 082 122Kotli 077 059Muzaffarabad 099 085Peshawar 007 008Saidu Sharif 191 157Skardu 067 065

For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments

Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different

values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary

32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO

We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area

Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments

Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions

However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions

Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution

Advances in Meteorology 9

Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments

Heterogeneity measures L-moment PL-momentHeterogeneity measure1198671

Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060

Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800

Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170

Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments

Method Distribution Parameters120585 120572 119870L-moments

GEV 04660 05644 minus02751GNO 06671 07565 minus07594GPA minus00551 09944 minus00575

PL-momentsGEV 05220 05116 minus02686GNO 07046 06837 minus07487GPA 00478 09073 minus00472

is larger than the critical value of 164 (at 90 confidencelevel)

It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile

Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution

33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7

based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)

Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is

119876 (119865) = 1198971198941119902 (119865) (25)

The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3

Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions

Therefore it is necessary to assess the performance ofregional quantile estimates

4 Accuracy of the Estimated Quantiles and theRegional Growth Curve

A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten

10 Advances in Meteorology

Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865

01000 05000 08000 09000 09500 09800 09900 09975 099875 09990

L-momentsGEV 00454 06836 15139 22247 30591 44161 56871 90748 113165 121338GNO 00473 06671 15585 23072 31448 44098 54997 80676 95666 100823GPA 00500 06481 16217 23931 31958 43073 51878 70578 80502 83782

PL-momentsGEV 01397 07190 14670 21034 28469 40497 51703 81363 10086 107950GNO 01412 07046 15062 21752 29203 40410 50032 72608 85739 90249GPA 01436 06871 15649 22547 29674 39460 47149 63304 71786 74577

Qua

ntile

Regional growth curve (L-moments)

GPAGNOGEV

GPAGNOGEV

Qua

ntile

Return period

Regional growth gurve (PL-moments)

5

4

3

2

1

0

5

4

3

2

1

0

minus2 minus1 0 1 2 3 4 5

2 5 10 20 50 100

Return period2 5 10 20 50 100

Reduced variate minuslog (minus log (F))

minus2 minus1 0 1 2 3 4 5

Reduced variate minuslog (minus log (F))

Figure 3 L-moments and PL-moments regional growth curves

as 119876(119898)119894 (119865) for the nonexceedance probability 119865 The relativeerror for this estimator is

119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) (26)

The bias and the RMSE of the above quantity over all 119872repetition are

Bias = 119861119894 (119865) = 1119872119872sum119898=1

119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)

= 1119872 [[119872sum119898=1

(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12

(27)

Also for the estimated quantile the regional average bias andthe relative RMSE are

119861119877 (119865) = 1119873119873sum119894=1

119861119894 (119865)

119877119877 (119865) = 1119873119873sum119894=1

119877119894 (119865) (28)

We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval

119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)

Advances in Meteorology 11

Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le

119902 (119865)119871005 (119865) (30)

The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves

We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]

We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively

Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds

Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion

This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail

GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments

Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents

Appendix

The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows

PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by

119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (minus ln119865)119896 119896 = 0 (A2)

Wang [10] developed the partial probability-weightedmoments (PWMs) of the GEV as

(119903 + 1) 1205731015840119903 = 120577 + 120572119867 (120574 1198650 119896) (A3)

where

119867(120574 1198650 119896) = 1119896 [1 minus119875 1 + 119896 minus (119903 + 1) ln1198650(1 minus 119865119903+10 ) (119903 + 1)119896 ] (A4)

12 Advances in Meteorology

Table8Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f30

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus0540

0minus19

7674

minus195328

minus193602

minus191579

minus188659

minus186376

minus1840

97minus18

1165

minus179056

119877(119865)

00900

00695

00963

01952

03334

06038

09057

13261

21337

300

48119877119877 (119865)

19873

01019

00635

00876

01090

01369

01595

01839

02191

02478

119871119861119864lowast

minus20937

08829

09296

08842

08411

07862

07460

07065

06557

06177

119880119861119864lowast

18714

11949

11208

11580

11884

12244

12516

12818

13285

13683

L-mom

ents

GNO

119861119877 (119865)

08124

minus196279

minus193846

minus192837

minus191816

minus190506

minus189573

minus188699

minus187636

minus186899

119877(119865)

00986

00696

00917

02051

03520

06027

08421

11312

16008

20312

119877119877 (119865)

20766

01045

00587

00889

01120

01368

01533

01686

01878

02016

119871119861119864lowast

minus21331

08683

09271

08780

08379

07958

07691

07453

07172

06978

119880119861119864lowast

16569

11974

11077

11569

11954

12355

12615

12851

13149

13369

L-mom

ents

GPA

119861119877 (119865)

38654

minus194351

minus193564

minus193226

minus192616

minus191706

minus191060

minus190506

minus189966

minus189727

119877(119865)

01242

006

6900998

02136

03552

05968

08313

11188

15941

20368

119877119877 (119865)

24785

01033

00614

00892

01111

01386

01602

01833

02162

02431

119871119861119864lowast

minus25192

08591

09215

08817

08444

07979

07646

07327

06919

06623

119880119861119864lowast

14272

11911

11115

11588

11976

12459

12841

13257

13859

14367

PL-m

oments

GEV

119861119877 (119865)

minus02662

minus197059

minus194770

minus193340

minus191600

minus188996

minus186909

minus184797

minus182052

minus1800

66119877(119865)

05379

006

8700947

02023

03467

06171

09087

13058

20551

2854

119877119877 (119865)

38117

00948

006

4900960

01210

01508

01736

01977

02324

02609

119871119861119864lowast

minus77814

08892

09252

08725

08259

07697

07296

06911

064

1706050

119880119861119864lowast

45516

11827

11207

11716

12117

12554

12881

13211

13693

14104

PL-m

oments

GNO

119861119877 (119865)

35142

minus195922

minus193485

minus192704

minus191894

minus190806

minus1900

03minus18

9235

minus188284

minus187617

119877(119865)

06203

006

8900939

02141

03644

06143

08483

11272

15747

19809

119877119877 (119865)

43522

00971

00626

00980

01238

01504

01675

01831

02024

02162

119871119861119864lowast

minus10618

08772

09204

08655

08233

07800

07530

07290

07012

06822

119880119861119864lowast

46353

11849

11122

11734

12189

12653

12949

13218

13546

13780

PL-m

oments

GPA

119861119877 (119865)

1212

3minus19

4300

minus193238

minus193113

minus192716

minus192046

minus191555

minus19114

1minus19

0768

minus1906

49119877(119865)

08513

006

6601041

02250

03684

06021

08215

10853

15144

19094

119877119877 (119865)

58811

00962

006

6900996

01233

01509

01720

01941

02258

02518

119871119861119864lowast

minus17507

08698

09130

08680

08299

07846

07524

07220

06831

06552

119880119861119864lowast

47307

11791

11189

11774

12215

12739

1314

13567

14195

14722

Advances in Meteorology 13

Table9Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f60

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus13271

minus194230

minus193551

minus193241

minus192649

minus191612

minus190736

minus189839

minus188690

minus187888

119877(119865)

00846

00624

00851

01873

03250

05841

08652

12505

19839

27721

119877119877 (119865)

19155

00833

00612

00855

01024

01213

01360

01523

01770

01983

119871119861119864lowast

minus19593

08756

09254

08851

08491

08074

07796

07501

07118

06834

119880119861119864lowast

16242

11675

11015

11472

11879

12309

12678

13038

13659

14141

L-mom

ents

GNO

119861119877 (119865)

08417

minus193335

minus192795

minus192984

minus192999

minus192879

minus192742

minus192593

minus192399

minus192264

119877(119865)

00968

00634

00858

02000

03439

05850

08123

10845

15235

19239

119877119877 (119865)

21055

00881

00582

00861

01050

01235

01354

01463

01599

01699

119871119861119864lowast

minus22887

08647

09244

08794

08466

08141

07957

07798

07597

07443

119880119861119864lowast

15863

11714

1096

11511

11934

1240

512

666

12939

13253

13505

L-mom

ents

GPA

119861119877 (119865)

46371

minus192512

minus192631

minus193160

minus193417

minus193606

minus193745

minus193933

minus194306

minus194708

119877(119865)

01280

00624

00950

02094

03468

05740

07903

10536

14878

18928

119877119877 (119865)

264

4200915

00614

00882

01064

01270

01429

01599

01849

02059

119871119861119864lowast

minus38088

08628

09186

08815

08510

08167

07924

07703

07422

07178

119880119861119864lowast

14699

11721

11020

11534

11925

12454

12791

13236

13869

1440

5

PL-m

oments

GEV

119861119877 (119865)

minus16968

minus193984

minus193601

minus193623

minus193319

minus192576

minus191861

minus191081

minus190026

minus189255

119877(119865)

05431

00596

00888

01978

03366

05844

08417

11838

18179

24868

119877119877 (119865)

38656

00783

00634

0095

01166

01387

01541

01700

01928

02120

119871119861119864lowast

minus92921

08907

09201

08750

08403

07984

07714

07434

07043

06783

119880119861119864lowast

53553

11521

11099

1166

412

098

12572

12869

13216

13697

14105

PL-m

oments

GNO

119861119877 (119865)

23785

minus193306

minus192979

minus193417

minus193628

minus193678

minus193617

minus193509

minus193331

minus193185

119877(119865)

06278

006

0100911

02100

03523

05804

0788

10303

14113

17516

119877119877 (119865)

43743

00812

00623

00965

01186

01391

01513

01618

01742

01830

119871119861119864lowast

minus12083

08856

09188

08709

08389

08061

07870

07706

07517

07380

119880119861119864lowast

52707

11561

11059

11704

12152

12613

12898

13138

13361

1356

PL-m

oments

GPA

119861119877 (119865)

1118

7minus19

2590

minus192910

minus193681

minus1940

84minus19

4322

minus1944

04minus19

4463

minus194560

minus1946

73119877(119865)

08596

00597

01009

02229

03581

05615

07391

09418

12558

15341

119877119877 (119865)

58310

00840

006

6000991

0119

501391

01522

01650

01828

01971

119871119861119864lowast

minus18977

08800

09127

08699

08413

08098

07912

07682

07447

07255

119880119861119864lowast

51281

11551

11117

11769

12173

12615

12887

13227

1360

913

899

14 Advances in Meteorology

Table10A

ccuracymeasure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f90

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus00117

minus194226

minus193109

minus192926

minus192542

minus191809

minus191151

minus1904

48minus18

9497

minus188790

119877(119865)

00787

00556

00823

01850

03151

05459

07840

10989

16793

22883

119877119877 (119865)

17329

00814

00544

00832

01031

01238

01381

01525

01727

01892

119871119861119864lowast

minus19303

08935

09287

08885

08587

08246

08006

07772

07463

07229

119880119861119864lowast

15717

11499

10973

11461

11804

12156

12393

12635

12966

13242

L-mom

ents

GNO

119861119877 (119865)

15851

minus193606

minus192517

minus192695

minus192747

minus192678

minus192569

minus192437

minus192245

minus192098

119877(119865)

00917

00567

00846

01967

03305

05446

07392

09660

13223

1640

0119877119877 (119865)

19328

00850

00543

00853

01052

01237

01347

01442

01554

01632

119871119861119864lowast

minus23268

08848

09261

08844

08568

08301

08139

08000

07833

07722

119880119861119864lowast

15168

11542

10942

11489

11851

12197

1240

112

575

12779

12925

L-mom

ents

GPA

119861119877 (119865)

49081

minus192995

minus192417

minus192865

minus193078

minus193175

minus193195

minus193209

minus193256

minus193330

119877(119865)

01234

00571

00935

02081

03365

05322

07054

09053

12180

14975

119877119877 (119865)

246

8800882

00576

00870

01054

01239

01364

01489

01660

01798

119871119861119864lowast

minus45353

08783

09210

08838

08594

08326

08141

07961

07731

07557

119880119861119864lowast

14833

11574

10996

11532

11870

12223

1246

712

711

1304

413

312

PL-m

oments

GEV

119861119877 (119865)

minus00877

minus194145

minus193060

minus193062

minus192847

minus192278

minus191698

minus191038

minus190100

minus189374

119877(119865)

05458

00565

00855

01963

03338

05712

08090

11156

16650

22276

119877119877 (119865)

39079

00784

00585

00934

0117

201406

01558

01703

01897

02050

119871119861119864lowast

minus10190

08983

09236

08765

08431

08067

07818

07579

07271

07041

119880119861119864lowast

58730

1144

811036

11652

12081

12504

12771

13026

13364

13623

PL-m

oments

GNO

119861119877 (119865)

38414

minus193665

minus192536

minus192859

minus193044

minus193102

minus193057

minus192966

minus192804

minus192665

119877(119865)

06308

00578

00896

02088

03488

05690

07659

09926

13434

16524

119877119877 (119865)

44651

00819

00596

00961

0119

301405

01526

01628

01744

01822

119871119861119864lowast

minus13011

08910

09192

08716

08411

08122

07949

07801

07634

07519

119880119861119864lowast

56756

11491

11030

11693

12128

12539

12775

12974

13197

13347

PL-m

oments

GPA

119861119877 (119865)

126398

minus193142

minus192484

minus193077

minus193404

minus193589

minus193637

minus193655

minus193680

minus193720

119877(119865)

08641

00588

00999

02220

03558

05535

07227

09125

1200

914

521

119877119877 (119865)

60116

00855

006

4000985

0119

901400

01529

01651

01812

01937

119871119861119864lowast

minus19889

08842

09130

08700

08435

08156

07970

07789

07566

07399

119880119861119864lowast

53743

11528

11103

11755

12158

12552

12807

13055

13385

13636

Advances in Meteorology 15

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

0 2 4 6

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

Regional growth curve for GPA

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F))

0

5

10

15

20

Qua

ntile

2 4 60Reduced variate minuslog (minus log (F))

0

5

10

15

Qua

ntile

(based on L-moments) Regional growth curve for GPA

(based on PL-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30

16 Advances in Meteorology

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60

Advances in Meteorology 17

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90

18 Advances in Meteorology

The first four PL-moments of the GEV are defined as

12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)

(A5)

In previous equation 119875(sdot sdot) is an Incomplete Gammafunction

119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500

120579119896expminus120579119889Θ (A6)

Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution

The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by

119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)

The partial PWMs of the GLO are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)

(A9)

where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function

1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650

0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)

The first four PL-moments of the GLO are defined as

12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520

minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530

minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896

201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

(A11)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution

The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by

119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)

The partial PWMs of the GPA are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int

1

1198650

(1 minus 119865)119896 119865119903119889119865 (A14)

The first four PL-moments of the GPA are defined as

12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)

12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)

(A15)

where

119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution

Advances in Meteorology 19

Ethical Approval

The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975

Conflicts of Interest

The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest

Acknowledgments

The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027

References

[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960

[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985

[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981

[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982

[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988

[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007

[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997

[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990

[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987

[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996

[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990

[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004

[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011

[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009

[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010

[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009

[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012

[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013

[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001

[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002

[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008

[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007

[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010

[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011

[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013

[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974

[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991

[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990

[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979

[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993

[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993

20 Advances in Meteorology

[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993

[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995

[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990

[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993

[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993

[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994

[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995

[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996

[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001

[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009

[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Geological ResearchJournal of

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Geology Advances in

Advances in Meteorology 5

where 119905 1199053 and 1199054 represent the sample partial L-momentsratios of the PL-Cv PL-Cs and PL-Ck respectively Thederivation is L-moment and PL-moments are given inAppendix In the present study different level of censoringthreshold is selected

23 Regional Frequency Analysis Hosking and Wallis [7 30]identified the following four steps to explain the procedure ofthe RFA

(1) Data screening(2) Designing of the homogeneous region(3) Selection of an appropriate probability distribution(4) Parameters estimation of the appropriate probability

distribution

231 Data Screening We screened data anomalies beforeapplying any statistical analysis

232 Discordance Test Hosking and Wallis [7] suggested adiscordancy measure (119863119894) test that recognizes the locationswhere sample L-moments are marked contrarily from themost other locations Locations with the large flaws in thedata will be flagged as discordant

The discordancy test for a region containing119873 locationsfor site 119894 is proposed by Hosking and Wallis [7] as follows

119863119894 = 13119873 (119906119894 minus 119906)119879 119878minus1 (119906119894 minus 119906) 119894 = 1 2 119873 (18)

where 119906119894 is the vector containing the three sample L-momentsratios for the site 119894 expressed as

119906119894 = [119905(119894)2 119905(119894)3 119905(119894)4 ]119879 (19)

119906 is the average vector of 119906119894 for the overall region that is

119906 = 1119873119873sum119894=1

119906119894 (20)

and 119878 is the covariance matrix for the sample that can beexpressed as

119878 = 119873sum119894=1

(119906119894 minus 119906) (119906119894 minus 119906)119879 (21)

Broadly speaking a location or a site is considered to bediscordant from the whole region or group if the value of 119863119894is larger than the critical value

233 Heterogeneity Test The homogeneity measure (119867119895)identifies homogenous regions Hosking and Wallis [7] Itis also useful to tag the locations if they are plausible tohandle as a homogeneous region It estimates the amountof heterogeneity in the overall region The heterogeneity test(119867119895) is computed as

119867119895 = 119881119895 minus 120583V119895120590V119895 119895 = 1 2 3 (22)

where 120583V119895 and 120590V119895 are representing the mean and standarddeviation of the simulated 119881119895 values Also

1198811 = 119873sum119894=1

119899119894 (119905(119894)2 minus 1199051198772 )sum119873119894=1 119899119894

12

1198812 =sum119873119894=1 [119899119894 (119905(119894)2 minus 1199051198772 )2 + (119905(119894)3 minus 1199051198773 )212]

sum119873119894=1 119899119894

1198813 =sum119873119894=1 [119899119894 (119905(119894)3 minus 1199051198773 )2 + (119905(119894)4 minus 1199051198774 )212]

sum119873119894=1 119899119894

(23)

Here 1199051198772 1199051198773 and 1199051198774 are region average L-moments or PL-moments ratios We assessed the heterogeneity of a region assuggested by Hosking and Wallis [7]

Region is acceptably homogeneous if119867 lt 1Region is possibly heterogeneous if 1 le 119867 lt 2Region is definitely heterogeneous if119867 ge 2

24 Selection of the Appropriate Probability DistributionHosking andWallis [7] proposed two approaches to select thedistribution that fitted best the data the L-moment ratios dia-gram and the 119885-test The L-moment ratios diagram is usingthe unbiased estimators Hosking [28] Stedinger et al [31]Vogel and Fennessey [32] and Hosking [33]The L-momentsratio diagram is a plot of the computed values L-Cs andthe observed values L-Ck of the distribution function Thecurves indicate the hypothetical connections between L-Csand L-Ck of the candidate distribution The L-moment ratiodiagrams have been proposed for discriminating between thecandidate probability distributions in describing the regionalinformation (Hosking [28] Stedinger et al [31] Hoskingand Wallis [7]) The L-moments ratio diagrams have beenused as a component of probability distribution process forregional information (Schaefer [34] Pearson [35] Vogel andFennessey [32] Vogel et al [36] Chow and Watt [37] OnOzand Bayazit [38] Vogel and Wilson [39] Peel et al [40])

Hosking and Wallis [7] suggested a measure to see howwell the L-Cs and L-Ck of the fitted probability distributionmatch the regional average L-Cs and L-Ck of the observedinformation

The measure goodness of fit for every single selectedprobability distribution is computed as follows

119885Dis = (120591Dis4 minus 1199051198774 )1205904 (24)

where 120591Dis4 represents the value of the L-Ck of the fitteddistribution 1199051198774 represents the weighted regional averageL-Ck and 1205904 represents standard deviation of the 1199051198774 whichis obtained from the simulation of the Kappa probabilitydistribution

If the computed value of 119885Dis is equal to zero the proba-bility distributionwill be themost suitable fit If the computed

6 Advances in Meteorology

Table 1 Statistics of annual extreme monthly precipitation for study region based on L-moments and PL-moments

Sites L-moments PL-momentsMean 119905 1199053 1199054 Mean 119905 1199053 1199054

Astore 37624 0518 0366 0193 41752 0467 0377 0183Balakot 120450 0478 0322 0192 133259 0428 0335 0198Bunji 14297 0609 0456 0259 16339 0553 0450 0257Chilas 15091 0585 0392 0194 17603 0516 0375 0200Cherat 49038 0580 0375 0162 55289 0526 0355 0161DI khan 28885 0689 0516 0262 32351 0651 0487 0250Dir 105694 0432 0216 0098 116741 0378 0222 0097Drosh 45284 0525 0317 0136 50316 0474 0307 0137Garhi Dupatta 111352 0454 0227 0111 123374 0398 0229 0120Gilgit 12677 0591 0412 0228 14290 0539 0399 0239Gupis 23446 0678 0529 0331 30299 0583 0506 0350Kakul 99431 0453 0247 0102 110068 0399 0254 0093Kotli 95070 0542 0332 0136 106903 0487 0314 0138Muzaffarabad 118439 0459 0284 0182 131122 0405 0301 0194Peshawar 46216 0572 0398 0216 51761 0520 0392 0222Saidu Sharif 85667 0455 0265 0181 94978 0399 0283 0200Skardu 20438 0616 04564 0238 22885 0570 0444 0231

value of 119885-statistic is less than 164 at 90 confidence level(ie |119885Dis| le 164) it will indicate that the distributionqualifies the goodness of fit criteria If there are more thanone distribution that qualify the criteria the most suitabledistribution has the minimum |119885Dis| value3 Estimation

The sitesrsquo information and statistic by using L-moments forthe present study are presented in Table 1 In Table 1 meanrepresents the first sample LPL-moments and 119905 1199053 and 1199054are the sample LPL-moments ratios of the L-CvPL-CvL-CsPL-Cs and L-CkPL-Ck respectively The lower levelcensoring threshold is selected from 10 to 23 Table 2expresses the feasible threshold values according to thepercentile technique along with Average Annual OccurrenceNumber (AAON) Jiang et al [41] and Yuguo [42] suggestedthat the optimal threshold can be obtained if the values ofAAON lie between 1 and 2 Table 2 shows that 90th percentileobservations are suitable for the optimum threshold selectionof most of areas in the present study We have 168 valuesin each station according to the above table Astore stationhas 32 threshold values due to which 16 values are beingcensored in 168 According to censored level 102 censoredlevel was selected Similarly 105 was selected for Balakotand Muzaffarabad According to this process maximumthreshold level 223 was selected for Gupis By using theabove process for each station 17 different censoring levelswere selected So we decide that the range from 10 to 23of censoring level should be kept for selecting thresholdvalues

31 Regional Frequency Analysis The following four steps areconsidered as prerequisite for frequency analysis Hoskingand Wallis [7 30]

(1) Data screening(2) Designing of the homogeneous region(3) Selection of an appropriate probability distribution(4) Parameters estimation of the appropriate probability

distribution

311 Data Screening In this study we use secondary dataafter carefully examining all locations for abnormalities andmissing observations Therefore we use 14 years of data forRFA that were obtained from seventeen locations

312 Discordance Test Table 3 shows 119863119894 result of (18) for17 locations of this study region It can be observed fromthe results of LPL-moments in Table 3 that the value of 119863-statistic varies from 007 to 244 If 119863119894 is greater than 3 thelocation is considered to be discordant from the rest of theregional data Hosking and Wallis [7] In this study regionno location is diagnosed as discordant (119863119894 ge 3) Thereforewe use all data for the development of the RFA based on L-moments and PL-moments

313 Regional Heterogeneity Measure The next step is theformation of the homogeneous region which is conven-tionally tougher and needs the higher number of subjectivejudgments The homogeneity conditions are defined as thelocations that have the same frequency distributions

Advances in Meteorology 7

Table 2 Precipitation threshold selection in GPA distribution for 17 stations

Station 80th 90th 95th 975th 99thAstore

Threshold 98 32 02 0 0AAON 243 121 064 043 043

BalakotThreshold 325 148 51 3 0AAON 243 129 057 029 014

MuzaffarabadThreshold 304 118 5 1 0AAON 25 121 057 029 021

Garhi DupattaThreshold 288 10 38 0 0AAON 243 121 057 029 029

DirThreshold 31 20 5 005 0AAON 243 121 057 029 021

KakulThreshold 289 134 21 05 0AAON 164 129 057 029 021

KotliThreshold 13 6 005 0 0AAON 243 135 057 036 036

Saidu SharifThreshold 256 83 2 0 0AAON 243 121 057 036 036

cheratThreshold 5 005 0 0 0AAON 243 143 093 093 093

PeshawarThreshold 6 005 0 0 0AAON 243 129 057 057 057

DroshThreshold 76 15 02 005 0AAON 243 121 057 05 021

D I KhanThreshold 03 0 0 0 0AAON 257 129 129 129 129

GupisThreshold 0 0 0 0 0AAON 271 271 271 271 271

SkarduThreshold 2 03 0 0 0AAON 243 071 071 071 071

ChilasThreshold 1 05 0 0 0AAON 171 107 107 107 107

BunjiThreshold 13 0 0 0 0AAON 15 15 15 15 15

GilgitThreshold 12 005 0 0 0AAON 243 135 071 071 071

8 Advances in Meteorology

04

03

02

01

00

04

03

02

01

00

00 01 02 03 04 05 06 00 01 02 03 04 05 06

GLOGEVGPA

GNOAverageData

GLOGEVGPA

GNOAverageData

L-ku

rtos

is

L㰀-k

urto

sis

L-skewness L-skewness

Figure 2 L-diagram and PL-diagram of the GLO GEV GPA and GNO distributions

Table 3 Discordance test result based on L-moments and PL-moments

Name of site L-moments (119863119894) PL-moments (119863119894)Astore 205 244Balakot 142 085Bunji 055 060Chilas 029 007Cherat 079 064DI khan 129 190Dir 082 086Drosh 051 043Garhi Dupatta 143 133Gilgit 026 059Gupis 237 231Kakul 082 122Kotli 077 059Muzaffarabad 099 085Peshawar 007 008Saidu Sharif 191 157Skardu 067 065

For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments

Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different

values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary

32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO

We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area

Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments

Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions

However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions

Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution

Advances in Meteorology 9

Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments

Heterogeneity measures L-moment PL-momentHeterogeneity measure1198671

Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060

Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800

Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170

Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments

Method Distribution Parameters120585 120572 119870L-moments

GEV 04660 05644 minus02751GNO 06671 07565 minus07594GPA minus00551 09944 minus00575

PL-momentsGEV 05220 05116 minus02686GNO 07046 06837 minus07487GPA 00478 09073 minus00472

is larger than the critical value of 164 (at 90 confidencelevel)

It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile

Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution

33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7

based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)

Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is

119876 (119865) = 1198971198941119902 (119865) (25)

The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3

Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions

Therefore it is necessary to assess the performance ofregional quantile estimates

4 Accuracy of the Estimated Quantiles and theRegional Growth Curve

A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten

10 Advances in Meteorology

Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865

01000 05000 08000 09000 09500 09800 09900 09975 099875 09990

L-momentsGEV 00454 06836 15139 22247 30591 44161 56871 90748 113165 121338GNO 00473 06671 15585 23072 31448 44098 54997 80676 95666 100823GPA 00500 06481 16217 23931 31958 43073 51878 70578 80502 83782

PL-momentsGEV 01397 07190 14670 21034 28469 40497 51703 81363 10086 107950GNO 01412 07046 15062 21752 29203 40410 50032 72608 85739 90249GPA 01436 06871 15649 22547 29674 39460 47149 63304 71786 74577

Qua

ntile

Regional growth curve (L-moments)

GPAGNOGEV

GPAGNOGEV

Qua

ntile

Return period

Regional growth gurve (PL-moments)

5

4

3

2

1

0

5

4

3

2

1

0

minus2 minus1 0 1 2 3 4 5

2 5 10 20 50 100

Return period2 5 10 20 50 100

Reduced variate minuslog (minus log (F))

minus2 minus1 0 1 2 3 4 5

Reduced variate minuslog (minus log (F))

Figure 3 L-moments and PL-moments regional growth curves

as 119876(119898)119894 (119865) for the nonexceedance probability 119865 The relativeerror for this estimator is

119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) (26)

The bias and the RMSE of the above quantity over all 119872repetition are

Bias = 119861119894 (119865) = 1119872119872sum119898=1

119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)

= 1119872 [[119872sum119898=1

(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12

(27)

Also for the estimated quantile the regional average bias andthe relative RMSE are

119861119877 (119865) = 1119873119873sum119894=1

119861119894 (119865)

119877119877 (119865) = 1119873119873sum119894=1

119877119894 (119865) (28)

We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval

119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)

Advances in Meteorology 11

Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le

119902 (119865)119871005 (119865) (30)

The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves

We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]

We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively

Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds

Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion

This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail

GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments

Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents

Appendix

The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows

PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by

119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (minus ln119865)119896 119896 = 0 (A2)

Wang [10] developed the partial probability-weightedmoments (PWMs) of the GEV as

(119903 + 1) 1205731015840119903 = 120577 + 120572119867 (120574 1198650 119896) (A3)

where

119867(120574 1198650 119896) = 1119896 [1 minus119875 1 + 119896 minus (119903 + 1) ln1198650(1 minus 119865119903+10 ) (119903 + 1)119896 ] (A4)

12 Advances in Meteorology

Table8Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f30

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus0540

0minus19

7674

minus195328

minus193602

minus191579

minus188659

minus186376

minus1840

97minus18

1165

minus179056

119877(119865)

00900

00695

00963

01952

03334

06038

09057

13261

21337

300

48119877119877 (119865)

19873

01019

00635

00876

01090

01369

01595

01839

02191

02478

119871119861119864lowast

minus20937

08829

09296

08842

08411

07862

07460

07065

06557

06177

119880119861119864lowast

18714

11949

11208

11580

11884

12244

12516

12818

13285

13683

L-mom

ents

GNO

119861119877 (119865)

08124

minus196279

minus193846

minus192837

minus191816

minus190506

minus189573

minus188699

minus187636

minus186899

119877(119865)

00986

00696

00917

02051

03520

06027

08421

11312

16008

20312

119877119877 (119865)

20766

01045

00587

00889

01120

01368

01533

01686

01878

02016

119871119861119864lowast

minus21331

08683

09271

08780

08379

07958

07691

07453

07172

06978

119880119861119864lowast

16569

11974

11077

11569

11954

12355

12615

12851

13149

13369

L-mom

ents

GPA

119861119877 (119865)

38654

minus194351

minus193564

minus193226

minus192616

minus191706

minus191060

minus190506

minus189966

minus189727

119877(119865)

01242

006

6900998

02136

03552

05968

08313

11188

15941

20368

119877119877 (119865)

24785

01033

00614

00892

01111

01386

01602

01833

02162

02431

119871119861119864lowast

minus25192

08591

09215

08817

08444

07979

07646

07327

06919

06623

119880119861119864lowast

14272

11911

11115

11588

11976

12459

12841

13257

13859

14367

PL-m

oments

GEV

119861119877 (119865)

minus02662

minus197059

minus194770

minus193340

minus191600

minus188996

minus186909

minus184797

minus182052

minus1800

66119877(119865)

05379

006

8700947

02023

03467

06171

09087

13058

20551

2854

119877119877 (119865)

38117

00948

006

4900960

01210

01508

01736

01977

02324

02609

119871119861119864lowast

minus77814

08892

09252

08725

08259

07697

07296

06911

064

1706050

119880119861119864lowast

45516

11827

11207

11716

12117

12554

12881

13211

13693

14104

PL-m

oments

GNO

119861119877 (119865)

35142

minus195922

minus193485

minus192704

minus191894

minus190806

minus1900

03minus18

9235

minus188284

minus187617

119877(119865)

06203

006

8900939

02141

03644

06143

08483

11272

15747

19809

119877119877 (119865)

43522

00971

00626

00980

01238

01504

01675

01831

02024

02162

119871119861119864lowast

minus10618

08772

09204

08655

08233

07800

07530

07290

07012

06822

119880119861119864lowast

46353

11849

11122

11734

12189

12653

12949

13218

13546

13780

PL-m

oments

GPA

119861119877 (119865)

1212

3minus19

4300

minus193238

minus193113

minus192716

minus192046

minus191555

minus19114

1minus19

0768

minus1906

49119877(119865)

08513

006

6601041

02250

03684

06021

08215

10853

15144

19094

119877119877 (119865)

58811

00962

006

6900996

01233

01509

01720

01941

02258

02518

119871119861119864lowast

minus17507

08698

09130

08680

08299

07846

07524

07220

06831

06552

119880119861119864lowast

47307

11791

11189

11774

12215

12739

1314

13567

14195

14722

Advances in Meteorology 13

Table9Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f60

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus13271

minus194230

minus193551

minus193241

minus192649

minus191612

minus190736

minus189839

minus188690

minus187888

119877(119865)

00846

00624

00851

01873

03250

05841

08652

12505

19839

27721

119877119877 (119865)

19155

00833

00612

00855

01024

01213

01360

01523

01770

01983

119871119861119864lowast

minus19593

08756

09254

08851

08491

08074

07796

07501

07118

06834

119880119861119864lowast

16242

11675

11015

11472

11879

12309

12678

13038

13659

14141

L-mom

ents

GNO

119861119877 (119865)

08417

minus193335

minus192795

minus192984

minus192999

minus192879

minus192742

minus192593

minus192399

minus192264

119877(119865)

00968

00634

00858

02000

03439

05850

08123

10845

15235

19239

119877119877 (119865)

21055

00881

00582

00861

01050

01235

01354

01463

01599

01699

119871119861119864lowast

minus22887

08647

09244

08794

08466

08141

07957

07798

07597

07443

119880119861119864lowast

15863

11714

1096

11511

11934

1240

512

666

12939

13253

13505

L-mom

ents

GPA

119861119877 (119865)

46371

minus192512

minus192631

minus193160

minus193417

minus193606

minus193745

minus193933

minus194306

minus194708

119877(119865)

01280

00624

00950

02094

03468

05740

07903

10536

14878

18928

119877119877 (119865)

264

4200915

00614

00882

01064

01270

01429

01599

01849

02059

119871119861119864lowast

minus38088

08628

09186

08815

08510

08167

07924

07703

07422

07178

119880119861119864lowast

14699

11721

11020

11534

11925

12454

12791

13236

13869

1440

5

PL-m

oments

GEV

119861119877 (119865)

minus16968

minus193984

minus193601

minus193623

minus193319

minus192576

minus191861

minus191081

minus190026

minus189255

119877(119865)

05431

00596

00888

01978

03366

05844

08417

11838

18179

24868

119877119877 (119865)

38656

00783

00634

0095

01166

01387

01541

01700

01928

02120

119871119861119864lowast

minus92921

08907

09201

08750

08403

07984

07714

07434

07043

06783

119880119861119864lowast

53553

11521

11099

1166

412

098

12572

12869

13216

13697

14105

PL-m

oments

GNO

119861119877 (119865)

23785

minus193306

minus192979

minus193417

minus193628

minus193678

minus193617

minus193509

minus193331

minus193185

119877(119865)

06278

006

0100911

02100

03523

05804

0788

10303

14113

17516

119877119877 (119865)

43743

00812

00623

00965

01186

01391

01513

01618

01742

01830

119871119861119864lowast

minus12083

08856

09188

08709

08389

08061

07870

07706

07517

07380

119880119861119864lowast

52707

11561

11059

11704

12152

12613

12898

13138

13361

1356

PL-m

oments

GPA

119861119877 (119865)

1118

7minus19

2590

minus192910

minus193681

minus1940

84minus19

4322

minus1944

04minus19

4463

minus194560

minus1946

73119877(119865)

08596

00597

01009

02229

03581

05615

07391

09418

12558

15341

119877119877 (119865)

58310

00840

006

6000991

0119

501391

01522

01650

01828

01971

119871119861119864lowast

minus18977

08800

09127

08699

08413

08098

07912

07682

07447

07255

119880119861119864lowast

51281

11551

11117

11769

12173

12615

12887

13227

1360

913

899

14 Advances in Meteorology

Table10A

ccuracymeasure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f90

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus00117

minus194226

minus193109

minus192926

minus192542

minus191809

minus191151

minus1904

48minus18

9497

minus188790

119877(119865)

00787

00556

00823

01850

03151

05459

07840

10989

16793

22883

119877119877 (119865)

17329

00814

00544

00832

01031

01238

01381

01525

01727

01892

119871119861119864lowast

minus19303

08935

09287

08885

08587

08246

08006

07772

07463

07229

119880119861119864lowast

15717

11499

10973

11461

11804

12156

12393

12635

12966

13242

L-mom

ents

GNO

119861119877 (119865)

15851

minus193606

minus192517

minus192695

minus192747

minus192678

minus192569

minus192437

minus192245

minus192098

119877(119865)

00917

00567

00846

01967

03305

05446

07392

09660

13223

1640

0119877119877 (119865)

19328

00850

00543

00853

01052

01237

01347

01442

01554

01632

119871119861119864lowast

minus23268

08848

09261

08844

08568

08301

08139

08000

07833

07722

119880119861119864lowast

15168

11542

10942

11489

11851

12197

1240

112

575

12779

12925

L-mom

ents

GPA

119861119877 (119865)

49081

minus192995

minus192417

minus192865

minus193078

minus193175

minus193195

minus193209

minus193256

minus193330

119877(119865)

01234

00571

00935

02081

03365

05322

07054

09053

12180

14975

119877119877 (119865)

246

8800882

00576

00870

01054

01239

01364

01489

01660

01798

119871119861119864lowast

minus45353

08783

09210

08838

08594

08326

08141

07961

07731

07557

119880119861119864lowast

14833

11574

10996

11532

11870

12223

1246

712

711

1304

413

312

PL-m

oments

GEV

119861119877 (119865)

minus00877

minus194145

minus193060

minus193062

minus192847

minus192278

minus191698

minus191038

minus190100

minus189374

119877(119865)

05458

00565

00855

01963

03338

05712

08090

11156

16650

22276

119877119877 (119865)

39079

00784

00585

00934

0117

201406

01558

01703

01897

02050

119871119861119864lowast

minus10190

08983

09236

08765

08431

08067

07818

07579

07271

07041

119880119861119864lowast

58730

1144

811036

11652

12081

12504

12771

13026

13364

13623

PL-m

oments

GNO

119861119877 (119865)

38414

minus193665

minus192536

minus192859

minus193044

minus193102

minus193057

minus192966

minus192804

minus192665

119877(119865)

06308

00578

00896

02088

03488

05690

07659

09926

13434

16524

119877119877 (119865)

44651

00819

00596

00961

0119

301405

01526

01628

01744

01822

119871119861119864lowast

minus13011

08910

09192

08716

08411

08122

07949

07801

07634

07519

119880119861119864lowast

56756

11491

11030

11693

12128

12539

12775

12974

13197

13347

PL-m

oments

GPA

119861119877 (119865)

126398

minus193142

minus192484

minus193077

minus193404

minus193589

minus193637

minus193655

minus193680

minus193720

119877(119865)

08641

00588

00999

02220

03558

05535

07227

09125

1200

914

521

119877119877 (119865)

60116

00855

006

4000985

0119

901400

01529

01651

01812

01937

119871119861119864lowast

minus19889

08842

09130

08700

08435

08156

07970

07789

07566

07399

119880119861119864lowast

53743

11528

11103

11755

12158

12552

12807

13055

13385

13636

Advances in Meteorology 15

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

0 2 4 6

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

Regional growth curve for GPA

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F))

0

5

10

15

20

Qua

ntile

2 4 60Reduced variate minuslog (minus log (F))

0

5

10

15

Qua

ntile

(based on L-moments) Regional growth curve for GPA

(based on PL-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30

16 Advances in Meteorology

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60

Advances in Meteorology 17

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90

18 Advances in Meteorology

The first four PL-moments of the GEV are defined as

12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)

(A5)

In previous equation 119875(sdot sdot) is an Incomplete Gammafunction

119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500

120579119896expminus120579119889Θ (A6)

Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution

The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by

119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)

The partial PWMs of the GLO are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)

(A9)

where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function

1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650

0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)

The first four PL-moments of the GLO are defined as

12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520

minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530

minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896

201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

(A11)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution

The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by

119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)

The partial PWMs of the GPA are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int

1

1198650

(1 minus 119865)119896 119865119903119889119865 (A14)

The first four PL-moments of the GPA are defined as

12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)

12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)

(A15)

where

119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution

Advances in Meteorology 19

Ethical Approval

The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975

Conflicts of Interest

The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest

Acknowledgments

The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027

References

[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960

[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985

[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981

[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982

[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988

[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007

[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997

[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990

[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987

[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996

[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990

[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004

[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011

[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009

[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010

[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009

[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012

[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013

[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001

[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002

[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008

[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007

[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010

[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011

[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013

[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974

[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991

[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990

[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979

[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993

[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993

20 Advances in Meteorology

[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993

[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995

[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990

[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993

[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993

[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994

[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995

[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996

[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001

[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009

[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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EarthquakesJournal of

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Applied ampEnvironmentalSoil Science

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Mining

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Journal of

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OceanographyInternational Journal of

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GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Atmospheric SciencesInternational Journal of

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OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

6 Advances in Meteorology

Table 1 Statistics of annual extreme monthly precipitation for study region based on L-moments and PL-moments

Sites L-moments PL-momentsMean 119905 1199053 1199054 Mean 119905 1199053 1199054

Astore 37624 0518 0366 0193 41752 0467 0377 0183Balakot 120450 0478 0322 0192 133259 0428 0335 0198Bunji 14297 0609 0456 0259 16339 0553 0450 0257Chilas 15091 0585 0392 0194 17603 0516 0375 0200Cherat 49038 0580 0375 0162 55289 0526 0355 0161DI khan 28885 0689 0516 0262 32351 0651 0487 0250Dir 105694 0432 0216 0098 116741 0378 0222 0097Drosh 45284 0525 0317 0136 50316 0474 0307 0137Garhi Dupatta 111352 0454 0227 0111 123374 0398 0229 0120Gilgit 12677 0591 0412 0228 14290 0539 0399 0239Gupis 23446 0678 0529 0331 30299 0583 0506 0350Kakul 99431 0453 0247 0102 110068 0399 0254 0093Kotli 95070 0542 0332 0136 106903 0487 0314 0138Muzaffarabad 118439 0459 0284 0182 131122 0405 0301 0194Peshawar 46216 0572 0398 0216 51761 0520 0392 0222Saidu Sharif 85667 0455 0265 0181 94978 0399 0283 0200Skardu 20438 0616 04564 0238 22885 0570 0444 0231

value of 119885-statistic is less than 164 at 90 confidence level(ie |119885Dis| le 164) it will indicate that the distributionqualifies the goodness of fit criteria If there are more thanone distribution that qualify the criteria the most suitabledistribution has the minimum |119885Dis| value3 Estimation

The sitesrsquo information and statistic by using L-moments forthe present study are presented in Table 1 In Table 1 meanrepresents the first sample LPL-moments and 119905 1199053 and 1199054are the sample LPL-moments ratios of the L-CvPL-CvL-CsPL-Cs and L-CkPL-Ck respectively The lower levelcensoring threshold is selected from 10 to 23 Table 2expresses the feasible threshold values according to thepercentile technique along with Average Annual OccurrenceNumber (AAON) Jiang et al [41] and Yuguo [42] suggestedthat the optimal threshold can be obtained if the values ofAAON lie between 1 and 2 Table 2 shows that 90th percentileobservations are suitable for the optimum threshold selectionof most of areas in the present study We have 168 valuesin each station according to the above table Astore stationhas 32 threshold values due to which 16 values are beingcensored in 168 According to censored level 102 censoredlevel was selected Similarly 105 was selected for Balakotand Muzaffarabad According to this process maximumthreshold level 223 was selected for Gupis By using theabove process for each station 17 different censoring levelswere selected So we decide that the range from 10 to 23of censoring level should be kept for selecting thresholdvalues

31 Regional Frequency Analysis The following four steps areconsidered as prerequisite for frequency analysis Hoskingand Wallis [7 30]

(1) Data screening(2) Designing of the homogeneous region(3) Selection of an appropriate probability distribution(4) Parameters estimation of the appropriate probability

distribution

311 Data Screening In this study we use secondary dataafter carefully examining all locations for abnormalities andmissing observations Therefore we use 14 years of data forRFA that were obtained from seventeen locations

312 Discordance Test Table 3 shows 119863119894 result of (18) for17 locations of this study region It can be observed fromthe results of LPL-moments in Table 3 that the value of 119863-statistic varies from 007 to 244 If 119863119894 is greater than 3 thelocation is considered to be discordant from the rest of theregional data Hosking and Wallis [7] In this study regionno location is diagnosed as discordant (119863119894 ge 3) Thereforewe use all data for the development of the RFA based on L-moments and PL-moments

313 Regional Heterogeneity Measure The next step is theformation of the homogeneous region which is conven-tionally tougher and needs the higher number of subjectivejudgments The homogeneity conditions are defined as thelocations that have the same frequency distributions

Advances in Meteorology 7

Table 2 Precipitation threshold selection in GPA distribution for 17 stations

Station 80th 90th 95th 975th 99thAstore

Threshold 98 32 02 0 0AAON 243 121 064 043 043

BalakotThreshold 325 148 51 3 0AAON 243 129 057 029 014

MuzaffarabadThreshold 304 118 5 1 0AAON 25 121 057 029 021

Garhi DupattaThreshold 288 10 38 0 0AAON 243 121 057 029 029

DirThreshold 31 20 5 005 0AAON 243 121 057 029 021

KakulThreshold 289 134 21 05 0AAON 164 129 057 029 021

KotliThreshold 13 6 005 0 0AAON 243 135 057 036 036

Saidu SharifThreshold 256 83 2 0 0AAON 243 121 057 036 036

cheratThreshold 5 005 0 0 0AAON 243 143 093 093 093

PeshawarThreshold 6 005 0 0 0AAON 243 129 057 057 057

DroshThreshold 76 15 02 005 0AAON 243 121 057 05 021

D I KhanThreshold 03 0 0 0 0AAON 257 129 129 129 129

GupisThreshold 0 0 0 0 0AAON 271 271 271 271 271

SkarduThreshold 2 03 0 0 0AAON 243 071 071 071 071

ChilasThreshold 1 05 0 0 0AAON 171 107 107 107 107

BunjiThreshold 13 0 0 0 0AAON 15 15 15 15 15

GilgitThreshold 12 005 0 0 0AAON 243 135 071 071 071

8 Advances in Meteorology

04

03

02

01

00

04

03

02

01

00

00 01 02 03 04 05 06 00 01 02 03 04 05 06

GLOGEVGPA

GNOAverageData

GLOGEVGPA

GNOAverageData

L-ku

rtos

is

L㰀-k

urto

sis

L-skewness L-skewness

Figure 2 L-diagram and PL-diagram of the GLO GEV GPA and GNO distributions

Table 3 Discordance test result based on L-moments and PL-moments

Name of site L-moments (119863119894) PL-moments (119863119894)Astore 205 244Balakot 142 085Bunji 055 060Chilas 029 007Cherat 079 064DI khan 129 190Dir 082 086Drosh 051 043Garhi Dupatta 143 133Gilgit 026 059Gupis 237 231Kakul 082 122Kotli 077 059Muzaffarabad 099 085Peshawar 007 008Saidu Sharif 191 157Skardu 067 065

For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments

Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different

values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary

32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO

We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area

Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments

Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions

However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions

Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution

Advances in Meteorology 9

Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments

Heterogeneity measures L-moment PL-momentHeterogeneity measure1198671

Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060

Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800

Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170

Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments

Method Distribution Parameters120585 120572 119870L-moments

GEV 04660 05644 minus02751GNO 06671 07565 minus07594GPA minus00551 09944 minus00575

PL-momentsGEV 05220 05116 minus02686GNO 07046 06837 minus07487GPA 00478 09073 minus00472

is larger than the critical value of 164 (at 90 confidencelevel)

It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile

Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution

33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7

based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)

Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is

119876 (119865) = 1198971198941119902 (119865) (25)

The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3

Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions

Therefore it is necessary to assess the performance ofregional quantile estimates

4 Accuracy of the Estimated Quantiles and theRegional Growth Curve

A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten

10 Advances in Meteorology

Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865

01000 05000 08000 09000 09500 09800 09900 09975 099875 09990

L-momentsGEV 00454 06836 15139 22247 30591 44161 56871 90748 113165 121338GNO 00473 06671 15585 23072 31448 44098 54997 80676 95666 100823GPA 00500 06481 16217 23931 31958 43073 51878 70578 80502 83782

PL-momentsGEV 01397 07190 14670 21034 28469 40497 51703 81363 10086 107950GNO 01412 07046 15062 21752 29203 40410 50032 72608 85739 90249GPA 01436 06871 15649 22547 29674 39460 47149 63304 71786 74577

Qua

ntile

Regional growth curve (L-moments)

GPAGNOGEV

GPAGNOGEV

Qua

ntile

Return period

Regional growth gurve (PL-moments)

5

4

3

2

1

0

5

4

3

2

1

0

minus2 minus1 0 1 2 3 4 5

2 5 10 20 50 100

Return period2 5 10 20 50 100

Reduced variate minuslog (minus log (F))

minus2 minus1 0 1 2 3 4 5

Reduced variate minuslog (minus log (F))

Figure 3 L-moments and PL-moments regional growth curves

as 119876(119898)119894 (119865) for the nonexceedance probability 119865 The relativeerror for this estimator is

119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) (26)

The bias and the RMSE of the above quantity over all 119872repetition are

Bias = 119861119894 (119865) = 1119872119872sum119898=1

119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)

= 1119872 [[119872sum119898=1

(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12

(27)

Also for the estimated quantile the regional average bias andthe relative RMSE are

119861119877 (119865) = 1119873119873sum119894=1

119861119894 (119865)

119877119877 (119865) = 1119873119873sum119894=1

119877119894 (119865) (28)

We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval

119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)

Advances in Meteorology 11

Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le

119902 (119865)119871005 (119865) (30)

The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves

We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]

We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively

Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds

Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion

This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail

GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments

Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents

Appendix

The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows

PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by

119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (minus ln119865)119896 119896 = 0 (A2)

Wang [10] developed the partial probability-weightedmoments (PWMs) of the GEV as

(119903 + 1) 1205731015840119903 = 120577 + 120572119867 (120574 1198650 119896) (A3)

where

119867(120574 1198650 119896) = 1119896 [1 minus119875 1 + 119896 minus (119903 + 1) ln1198650(1 minus 119865119903+10 ) (119903 + 1)119896 ] (A4)

12 Advances in Meteorology

Table8Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f30

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus0540

0minus19

7674

minus195328

minus193602

minus191579

minus188659

minus186376

minus1840

97minus18

1165

minus179056

119877(119865)

00900

00695

00963

01952

03334

06038

09057

13261

21337

300

48119877119877 (119865)

19873

01019

00635

00876

01090

01369

01595

01839

02191

02478

119871119861119864lowast

minus20937

08829

09296

08842

08411

07862

07460

07065

06557

06177

119880119861119864lowast

18714

11949

11208

11580

11884

12244

12516

12818

13285

13683

L-mom

ents

GNO

119861119877 (119865)

08124

minus196279

minus193846

minus192837

minus191816

minus190506

minus189573

minus188699

minus187636

minus186899

119877(119865)

00986

00696

00917

02051

03520

06027

08421

11312

16008

20312

119877119877 (119865)

20766

01045

00587

00889

01120

01368

01533

01686

01878

02016

119871119861119864lowast

minus21331

08683

09271

08780

08379

07958

07691

07453

07172

06978

119880119861119864lowast

16569

11974

11077

11569

11954

12355

12615

12851

13149

13369

L-mom

ents

GPA

119861119877 (119865)

38654

minus194351

minus193564

minus193226

minus192616

minus191706

minus191060

minus190506

minus189966

minus189727

119877(119865)

01242

006

6900998

02136

03552

05968

08313

11188

15941

20368

119877119877 (119865)

24785

01033

00614

00892

01111

01386

01602

01833

02162

02431

119871119861119864lowast

minus25192

08591

09215

08817

08444

07979

07646

07327

06919

06623

119880119861119864lowast

14272

11911

11115

11588

11976

12459

12841

13257

13859

14367

PL-m

oments

GEV

119861119877 (119865)

minus02662

minus197059

minus194770

minus193340

minus191600

minus188996

minus186909

minus184797

minus182052

minus1800

66119877(119865)

05379

006

8700947

02023

03467

06171

09087

13058

20551

2854

119877119877 (119865)

38117

00948

006

4900960

01210

01508

01736

01977

02324

02609

119871119861119864lowast

minus77814

08892

09252

08725

08259

07697

07296

06911

064

1706050

119880119861119864lowast

45516

11827

11207

11716

12117

12554

12881

13211

13693

14104

PL-m

oments

GNO

119861119877 (119865)

35142

minus195922

minus193485

minus192704

minus191894

minus190806

minus1900

03minus18

9235

minus188284

minus187617

119877(119865)

06203

006

8900939

02141

03644

06143

08483

11272

15747

19809

119877119877 (119865)

43522

00971

00626

00980

01238

01504

01675

01831

02024

02162

119871119861119864lowast

minus10618

08772

09204

08655

08233

07800

07530

07290

07012

06822

119880119861119864lowast

46353

11849

11122

11734

12189

12653

12949

13218

13546

13780

PL-m

oments

GPA

119861119877 (119865)

1212

3minus19

4300

minus193238

minus193113

minus192716

minus192046

minus191555

minus19114

1minus19

0768

minus1906

49119877(119865)

08513

006

6601041

02250

03684

06021

08215

10853

15144

19094

119877119877 (119865)

58811

00962

006

6900996

01233

01509

01720

01941

02258

02518

119871119861119864lowast

minus17507

08698

09130

08680

08299

07846

07524

07220

06831

06552

119880119861119864lowast

47307

11791

11189

11774

12215

12739

1314

13567

14195

14722

Advances in Meteorology 13

Table9Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f60

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus13271

minus194230

minus193551

minus193241

minus192649

minus191612

minus190736

minus189839

minus188690

minus187888

119877(119865)

00846

00624

00851

01873

03250

05841

08652

12505

19839

27721

119877119877 (119865)

19155

00833

00612

00855

01024

01213

01360

01523

01770

01983

119871119861119864lowast

minus19593

08756

09254

08851

08491

08074

07796

07501

07118

06834

119880119861119864lowast

16242

11675

11015

11472

11879

12309

12678

13038

13659

14141

L-mom

ents

GNO

119861119877 (119865)

08417

minus193335

minus192795

minus192984

minus192999

minus192879

minus192742

minus192593

minus192399

minus192264

119877(119865)

00968

00634

00858

02000

03439

05850

08123

10845

15235

19239

119877119877 (119865)

21055

00881

00582

00861

01050

01235

01354

01463

01599

01699

119871119861119864lowast

minus22887

08647

09244

08794

08466

08141

07957

07798

07597

07443

119880119861119864lowast

15863

11714

1096

11511

11934

1240

512

666

12939

13253

13505

L-mom

ents

GPA

119861119877 (119865)

46371

minus192512

minus192631

minus193160

minus193417

minus193606

minus193745

minus193933

minus194306

minus194708

119877(119865)

01280

00624

00950

02094

03468

05740

07903

10536

14878

18928

119877119877 (119865)

264

4200915

00614

00882

01064

01270

01429

01599

01849

02059

119871119861119864lowast

minus38088

08628

09186

08815

08510

08167

07924

07703

07422

07178

119880119861119864lowast

14699

11721

11020

11534

11925

12454

12791

13236

13869

1440

5

PL-m

oments

GEV

119861119877 (119865)

minus16968

minus193984

minus193601

minus193623

minus193319

minus192576

minus191861

minus191081

minus190026

minus189255

119877(119865)

05431

00596

00888

01978

03366

05844

08417

11838

18179

24868

119877119877 (119865)

38656

00783

00634

0095

01166

01387

01541

01700

01928

02120

119871119861119864lowast

minus92921

08907

09201

08750

08403

07984

07714

07434

07043

06783

119880119861119864lowast

53553

11521

11099

1166

412

098

12572

12869

13216

13697

14105

PL-m

oments

GNO

119861119877 (119865)

23785

minus193306

minus192979

minus193417

minus193628

minus193678

minus193617

minus193509

minus193331

minus193185

119877(119865)

06278

006

0100911

02100

03523

05804

0788

10303

14113

17516

119877119877 (119865)

43743

00812

00623

00965

01186

01391

01513

01618

01742

01830

119871119861119864lowast

minus12083

08856

09188

08709

08389

08061

07870

07706

07517

07380

119880119861119864lowast

52707

11561

11059

11704

12152

12613

12898

13138

13361

1356

PL-m

oments

GPA

119861119877 (119865)

1118

7minus19

2590

minus192910

minus193681

minus1940

84minus19

4322

minus1944

04minus19

4463

minus194560

minus1946

73119877(119865)

08596

00597

01009

02229

03581

05615

07391

09418

12558

15341

119877119877 (119865)

58310

00840

006

6000991

0119

501391

01522

01650

01828

01971

119871119861119864lowast

minus18977

08800

09127

08699

08413

08098

07912

07682

07447

07255

119880119861119864lowast

51281

11551

11117

11769

12173

12615

12887

13227

1360

913

899

14 Advances in Meteorology

Table10A

ccuracymeasure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f90

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus00117

minus194226

minus193109

minus192926

minus192542

minus191809

minus191151

minus1904

48minus18

9497

minus188790

119877(119865)

00787

00556

00823

01850

03151

05459

07840

10989

16793

22883

119877119877 (119865)

17329

00814

00544

00832

01031

01238

01381

01525

01727

01892

119871119861119864lowast

minus19303

08935

09287

08885

08587

08246

08006

07772

07463

07229

119880119861119864lowast

15717

11499

10973

11461

11804

12156

12393

12635

12966

13242

L-mom

ents

GNO

119861119877 (119865)

15851

minus193606

minus192517

minus192695

minus192747

minus192678

minus192569

minus192437

minus192245

minus192098

119877(119865)

00917

00567

00846

01967

03305

05446

07392

09660

13223

1640

0119877119877 (119865)

19328

00850

00543

00853

01052

01237

01347

01442

01554

01632

119871119861119864lowast

minus23268

08848

09261

08844

08568

08301

08139

08000

07833

07722

119880119861119864lowast

15168

11542

10942

11489

11851

12197

1240

112

575

12779

12925

L-mom

ents

GPA

119861119877 (119865)

49081

minus192995

minus192417

minus192865

minus193078

minus193175

minus193195

minus193209

minus193256

minus193330

119877(119865)

01234

00571

00935

02081

03365

05322

07054

09053

12180

14975

119877119877 (119865)

246

8800882

00576

00870

01054

01239

01364

01489

01660

01798

119871119861119864lowast

minus45353

08783

09210

08838

08594

08326

08141

07961

07731

07557

119880119861119864lowast

14833

11574

10996

11532

11870

12223

1246

712

711

1304

413

312

PL-m

oments

GEV

119861119877 (119865)

minus00877

minus194145

minus193060

minus193062

minus192847

minus192278

minus191698

minus191038

minus190100

minus189374

119877(119865)

05458

00565

00855

01963

03338

05712

08090

11156

16650

22276

119877119877 (119865)

39079

00784

00585

00934

0117

201406

01558

01703

01897

02050

119871119861119864lowast

minus10190

08983

09236

08765

08431

08067

07818

07579

07271

07041

119880119861119864lowast

58730

1144

811036

11652

12081

12504

12771

13026

13364

13623

PL-m

oments

GNO

119861119877 (119865)

38414

minus193665

minus192536

minus192859

minus193044

minus193102

minus193057

minus192966

minus192804

minus192665

119877(119865)

06308

00578

00896

02088

03488

05690

07659

09926

13434

16524

119877119877 (119865)

44651

00819

00596

00961

0119

301405

01526

01628

01744

01822

119871119861119864lowast

minus13011

08910

09192

08716

08411

08122

07949

07801

07634

07519

119880119861119864lowast

56756

11491

11030

11693

12128

12539

12775

12974

13197

13347

PL-m

oments

GPA

119861119877 (119865)

126398

minus193142

minus192484

minus193077

minus193404

minus193589

minus193637

minus193655

minus193680

minus193720

119877(119865)

08641

00588

00999

02220

03558

05535

07227

09125

1200

914

521

119877119877 (119865)

60116

00855

006

4000985

0119

901400

01529

01651

01812

01937

119871119861119864lowast

minus19889

08842

09130

08700

08435

08156

07970

07789

07566

07399

119880119861119864lowast

53743

11528

11103

11755

12158

12552

12807

13055

13385

13636

Advances in Meteorology 15

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

0 2 4 6

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

Regional growth curve for GPA

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F))

0

5

10

15

20

Qua

ntile

2 4 60Reduced variate minuslog (minus log (F))

0

5

10

15

Qua

ntile

(based on L-moments) Regional growth curve for GPA

(based on PL-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30

16 Advances in Meteorology

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60

Advances in Meteorology 17

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90

18 Advances in Meteorology

The first four PL-moments of the GEV are defined as

12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)

(A5)

In previous equation 119875(sdot sdot) is an Incomplete Gammafunction

119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500

120579119896expminus120579119889Θ (A6)

Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution

The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by

119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)

The partial PWMs of the GLO are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)

(A9)

where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function

1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650

0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)

The first four PL-moments of the GLO are defined as

12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520

minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530

minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896

201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

(A11)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution

The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by

119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)

The partial PWMs of the GPA are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int

1

1198650

(1 minus 119865)119896 119865119903119889119865 (A14)

The first four PL-moments of the GPA are defined as

12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)

12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)

(A15)

where

119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution

Advances in Meteorology 19

Ethical Approval

The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975

Conflicts of Interest

The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest

Acknowledgments

The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027

References

[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960

[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985

[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981

[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982

[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988

[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007

[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997

[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990

[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987

[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996

[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990

[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004

[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011

[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009

[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010

[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009

[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012

[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013

[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001

[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002

[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008

[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007

[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010

[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011

[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013

[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974

[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991

[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990

[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979

[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993

[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993

20 Advances in Meteorology

[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993

[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995

[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990

[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993

[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993

[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994

[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995

[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996

[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001

[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009

[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

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Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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International Journal of

Geophysics

OceanographyInternational Journal of

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Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

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MineralogyInternational Journal of

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MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Advances in Meteorology 7

Table 2 Precipitation threshold selection in GPA distribution for 17 stations

Station 80th 90th 95th 975th 99thAstore

Threshold 98 32 02 0 0AAON 243 121 064 043 043

BalakotThreshold 325 148 51 3 0AAON 243 129 057 029 014

MuzaffarabadThreshold 304 118 5 1 0AAON 25 121 057 029 021

Garhi DupattaThreshold 288 10 38 0 0AAON 243 121 057 029 029

DirThreshold 31 20 5 005 0AAON 243 121 057 029 021

KakulThreshold 289 134 21 05 0AAON 164 129 057 029 021

KotliThreshold 13 6 005 0 0AAON 243 135 057 036 036

Saidu SharifThreshold 256 83 2 0 0AAON 243 121 057 036 036

cheratThreshold 5 005 0 0 0AAON 243 143 093 093 093

PeshawarThreshold 6 005 0 0 0AAON 243 129 057 057 057

DroshThreshold 76 15 02 005 0AAON 243 121 057 05 021

D I KhanThreshold 03 0 0 0 0AAON 257 129 129 129 129

GupisThreshold 0 0 0 0 0AAON 271 271 271 271 271

SkarduThreshold 2 03 0 0 0AAON 243 071 071 071 071

ChilasThreshold 1 05 0 0 0AAON 171 107 107 107 107

BunjiThreshold 13 0 0 0 0AAON 15 15 15 15 15

GilgitThreshold 12 005 0 0 0AAON 243 135 071 071 071

8 Advances in Meteorology

04

03

02

01

00

04

03

02

01

00

00 01 02 03 04 05 06 00 01 02 03 04 05 06

GLOGEVGPA

GNOAverageData

GLOGEVGPA

GNOAverageData

L-ku

rtos

is

L㰀-k

urto

sis

L-skewness L-skewness

Figure 2 L-diagram and PL-diagram of the GLO GEV GPA and GNO distributions

Table 3 Discordance test result based on L-moments and PL-moments

Name of site L-moments (119863119894) PL-moments (119863119894)Astore 205 244Balakot 142 085Bunji 055 060Chilas 029 007Cherat 079 064DI khan 129 190Dir 082 086Drosh 051 043Garhi Dupatta 143 133Gilgit 026 059Gupis 237 231Kakul 082 122Kotli 077 059Muzaffarabad 099 085Peshawar 007 008Saidu Sharif 191 157Skardu 067 065

For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments

Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different

values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary

32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO

We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area

Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments

Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions

However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions

Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution

Advances in Meteorology 9

Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments

Heterogeneity measures L-moment PL-momentHeterogeneity measure1198671

Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060

Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800

Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170

Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments

Method Distribution Parameters120585 120572 119870L-moments

GEV 04660 05644 minus02751GNO 06671 07565 minus07594GPA minus00551 09944 minus00575

PL-momentsGEV 05220 05116 minus02686GNO 07046 06837 minus07487GPA 00478 09073 minus00472

is larger than the critical value of 164 (at 90 confidencelevel)

It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile

Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution

33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7

based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)

Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is

119876 (119865) = 1198971198941119902 (119865) (25)

The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3

Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions

Therefore it is necessary to assess the performance ofregional quantile estimates

4 Accuracy of the Estimated Quantiles and theRegional Growth Curve

A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten

10 Advances in Meteorology

Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865

01000 05000 08000 09000 09500 09800 09900 09975 099875 09990

L-momentsGEV 00454 06836 15139 22247 30591 44161 56871 90748 113165 121338GNO 00473 06671 15585 23072 31448 44098 54997 80676 95666 100823GPA 00500 06481 16217 23931 31958 43073 51878 70578 80502 83782

PL-momentsGEV 01397 07190 14670 21034 28469 40497 51703 81363 10086 107950GNO 01412 07046 15062 21752 29203 40410 50032 72608 85739 90249GPA 01436 06871 15649 22547 29674 39460 47149 63304 71786 74577

Qua

ntile

Regional growth curve (L-moments)

GPAGNOGEV

GPAGNOGEV

Qua

ntile

Return period

Regional growth gurve (PL-moments)

5

4

3

2

1

0

5

4

3

2

1

0

minus2 minus1 0 1 2 3 4 5

2 5 10 20 50 100

Return period2 5 10 20 50 100

Reduced variate minuslog (minus log (F))

minus2 minus1 0 1 2 3 4 5

Reduced variate minuslog (minus log (F))

Figure 3 L-moments and PL-moments regional growth curves

as 119876(119898)119894 (119865) for the nonexceedance probability 119865 The relativeerror for this estimator is

119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) (26)

The bias and the RMSE of the above quantity over all 119872repetition are

Bias = 119861119894 (119865) = 1119872119872sum119898=1

119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)

= 1119872 [[119872sum119898=1

(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12

(27)

Also for the estimated quantile the regional average bias andthe relative RMSE are

119861119877 (119865) = 1119873119873sum119894=1

119861119894 (119865)

119877119877 (119865) = 1119873119873sum119894=1

119877119894 (119865) (28)

We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval

119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)

Advances in Meteorology 11

Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le

119902 (119865)119871005 (119865) (30)

The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves

We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]

We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively

Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds

Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion

This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail

GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments

Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents

Appendix

The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows

PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by

119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (minus ln119865)119896 119896 = 0 (A2)

Wang [10] developed the partial probability-weightedmoments (PWMs) of the GEV as

(119903 + 1) 1205731015840119903 = 120577 + 120572119867 (120574 1198650 119896) (A3)

where

119867(120574 1198650 119896) = 1119896 [1 minus119875 1 + 119896 minus (119903 + 1) ln1198650(1 minus 119865119903+10 ) (119903 + 1)119896 ] (A4)

12 Advances in Meteorology

Table8Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f30

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus0540

0minus19

7674

minus195328

minus193602

minus191579

minus188659

minus186376

minus1840

97minus18

1165

minus179056

119877(119865)

00900

00695

00963

01952

03334

06038

09057

13261

21337

300

48119877119877 (119865)

19873

01019

00635

00876

01090

01369

01595

01839

02191

02478

119871119861119864lowast

minus20937

08829

09296

08842

08411

07862

07460

07065

06557

06177

119880119861119864lowast

18714

11949

11208

11580

11884

12244

12516

12818

13285

13683

L-mom

ents

GNO

119861119877 (119865)

08124

minus196279

minus193846

minus192837

minus191816

minus190506

minus189573

minus188699

minus187636

minus186899

119877(119865)

00986

00696

00917

02051

03520

06027

08421

11312

16008

20312

119877119877 (119865)

20766

01045

00587

00889

01120

01368

01533

01686

01878

02016

119871119861119864lowast

minus21331

08683

09271

08780

08379

07958

07691

07453

07172

06978

119880119861119864lowast

16569

11974

11077

11569

11954

12355

12615

12851

13149

13369

L-mom

ents

GPA

119861119877 (119865)

38654

minus194351

minus193564

minus193226

minus192616

minus191706

minus191060

minus190506

minus189966

minus189727

119877(119865)

01242

006

6900998

02136

03552

05968

08313

11188

15941

20368

119877119877 (119865)

24785

01033

00614

00892

01111

01386

01602

01833

02162

02431

119871119861119864lowast

minus25192

08591

09215

08817

08444

07979

07646

07327

06919

06623

119880119861119864lowast

14272

11911

11115

11588

11976

12459

12841

13257

13859

14367

PL-m

oments

GEV

119861119877 (119865)

minus02662

minus197059

minus194770

minus193340

minus191600

minus188996

minus186909

minus184797

minus182052

minus1800

66119877(119865)

05379

006

8700947

02023

03467

06171

09087

13058

20551

2854

119877119877 (119865)

38117

00948

006

4900960

01210

01508

01736

01977

02324

02609

119871119861119864lowast

minus77814

08892

09252

08725

08259

07697

07296

06911

064

1706050

119880119861119864lowast

45516

11827

11207

11716

12117

12554

12881

13211

13693

14104

PL-m

oments

GNO

119861119877 (119865)

35142

minus195922

minus193485

minus192704

minus191894

minus190806

minus1900

03minus18

9235

minus188284

minus187617

119877(119865)

06203

006

8900939

02141

03644

06143

08483

11272

15747

19809

119877119877 (119865)

43522

00971

00626

00980

01238

01504

01675

01831

02024

02162

119871119861119864lowast

minus10618

08772

09204

08655

08233

07800

07530

07290

07012

06822

119880119861119864lowast

46353

11849

11122

11734

12189

12653

12949

13218

13546

13780

PL-m

oments

GPA

119861119877 (119865)

1212

3minus19

4300

minus193238

minus193113

minus192716

minus192046

minus191555

minus19114

1minus19

0768

minus1906

49119877(119865)

08513

006

6601041

02250

03684

06021

08215

10853

15144

19094

119877119877 (119865)

58811

00962

006

6900996

01233

01509

01720

01941

02258

02518

119871119861119864lowast

minus17507

08698

09130

08680

08299

07846

07524

07220

06831

06552

119880119861119864lowast

47307

11791

11189

11774

12215

12739

1314

13567

14195

14722

Advances in Meteorology 13

Table9Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f60

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus13271

minus194230

minus193551

minus193241

minus192649

minus191612

minus190736

minus189839

minus188690

minus187888

119877(119865)

00846

00624

00851

01873

03250

05841

08652

12505

19839

27721

119877119877 (119865)

19155

00833

00612

00855

01024

01213

01360

01523

01770

01983

119871119861119864lowast

minus19593

08756

09254

08851

08491

08074

07796

07501

07118

06834

119880119861119864lowast

16242

11675

11015

11472

11879

12309

12678

13038

13659

14141

L-mom

ents

GNO

119861119877 (119865)

08417

minus193335

minus192795

minus192984

minus192999

minus192879

minus192742

minus192593

minus192399

minus192264

119877(119865)

00968

00634

00858

02000

03439

05850

08123

10845

15235

19239

119877119877 (119865)

21055

00881

00582

00861

01050

01235

01354

01463

01599

01699

119871119861119864lowast

minus22887

08647

09244

08794

08466

08141

07957

07798

07597

07443

119880119861119864lowast

15863

11714

1096

11511

11934

1240

512

666

12939

13253

13505

L-mom

ents

GPA

119861119877 (119865)

46371

minus192512

minus192631

minus193160

minus193417

minus193606

minus193745

minus193933

minus194306

minus194708

119877(119865)

01280

00624

00950

02094

03468

05740

07903

10536

14878

18928

119877119877 (119865)

264

4200915

00614

00882

01064

01270

01429

01599

01849

02059

119871119861119864lowast

minus38088

08628

09186

08815

08510

08167

07924

07703

07422

07178

119880119861119864lowast

14699

11721

11020

11534

11925

12454

12791

13236

13869

1440

5

PL-m

oments

GEV

119861119877 (119865)

minus16968

minus193984

minus193601

minus193623

minus193319

minus192576

minus191861

minus191081

minus190026

minus189255

119877(119865)

05431

00596

00888

01978

03366

05844

08417

11838

18179

24868

119877119877 (119865)

38656

00783

00634

0095

01166

01387

01541

01700

01928

02120

119871119861119864lowast

minus92921

08907

09201

08750

08403

07984

07714

07434

07043

06783

119880119861119864lowast

53553

11521

11099

1166

412

098

12572

12869

13216

13697

14105

PL-m

oments

GNO

119861119877 (119865)

23785

minus193306

minus192979

minus193417

minus193628

minus193678

minus193617

minus193509

minus193331

minus193185

119877(119865)

06278

006

0100911

02100

03523

05804

0788

10303

14113

17516

119877119877 (119865)

43743

00812

00623

00965

01186

01391

01513

01618

01742

01830

119871119861119864lowast

minus12083

08856

09188

08709

08389

08061

07870

07706

07517

07380

119880119861119864lowast

52707

11561

11059

11704

12152

12613

12898

13138

13361

1356

PL-m

oments

GPA

119861119877 (119865)

1118

7minus19

2590

minus192910

minus193681

minus1940

84minus19

4322

minus1944

04minus19

4463

minus194560

minus1946

73119877(119865)

08596

00597

01009

02229

03581

05615

07391

09418

12558

15341

119877119877 (119865)

58310

00840

006

6000991

0119

501391

01522

01650

01828

01971

119871119861119864lowast

minus18977

08800

09127

08699

08413

08098

07912

07682

07447

07255

119880119861119864lowast

51281

11551

11117

11769

12173

12615

12887

13227

1360

913

899

14 Advances in Meteorology

Table10A

ccuracymeasure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f90

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus00117

minus194226

minus193109

minus192926

minus192542

minus191809

minus191151

minus1904

48minus18

9497

minus188790

119877(119865)

00787

00556

00823

01850

03151

05459

07840

10989

16793

22883

119877119877 (119865)

17329

00814

00544

00832

01031

01238

01381

01525

01727

01892

119871119861119864lowast

minus19303

08935

09287

08885

08587

08246

08006

07772

07463

07229

119880119861119864lowast

15717

11499

10973

11461

11804

12156

12393

12635

12966

13242

L-mom

ents

GNO

119861119877 (119865)

15851

minus193606

minus192517

minus192695

minus192747

minus192678

minus192569

minus192437

minus192245

minus192098

119877(119865)

00917

00567

00846

01967

03305

05446

07392

09660

13223

1640

0119877119877 (119865)

19328

00850

00543

00853

01052

01237

01347

01442

01554

01632

119871119861119864lowast

minus23268

08848

09261

08844

08568

08301

08139

08000

07833

07722

119880119861119864lowast

15168

11542

10942

11489

11851

12197

1240

112

575

12779

12925

L-mom

ents

GPA

119861119877 (119865)

49081

minus192995

minus192417

minus192865

minus193078

minus193175

minus193195

minus193209

minus193256

minus193330

119877(119865)

01234

00571

00935

02081

03365

05322

07054

09053

12180

14975

119877119877 (119865)

246

8800882

00576

00870

01054

01239

01364

01489

01660

01798

119871119861119864lowast

minus45353

08783

09210

08838

08594

08326

08141

07961

07731

07557

119880119861119864lowast

14833

11574

10996

11532

11870

12223

1246

712

711

1304

413

312

PL-m

oments

GEV

119861119877 (119865)

minus00877

minus194145

minus193060

minus193062

minus192847

minus192278

minus191698

minus191038

minus190100

minus189374

119877(119865)

05458

00565

00855

01963

03338

05712

08090

11156

16650

22276

119877119877 (119865)

39079

00784

00585

00934

0117

201406

01558

01703

01897

02050

119871119861119864lowast

minus10190

08983

09236

08765

08431

08067

07818

07579

07271

07041

119880119861119864lowast

58730

1144

811036

11652

12081

12504

12771

13026

13364

13623

PL-m

oments

GNO

119861119877 (119865)

38414

minus193665

minus192536

minus192859

minus193044

minus193102

minus193057

minus192966

minus192804

minus192665

119877(119865)

06308

00578

00896

02088

03488

05690

07659

09926

13434

16524

119877119877 (119865)

44651

00819

00596

00961

0119

301405

01526

01628

01744

01822

119871119861119864lowast

minus13011

08910

09192

08716

08411

08122

07949

07801

07634

07519

119880119861119864lowast

56756

11491

11030

11693

12128

12539

12775

12974

13197

13347

PL-m

oments

GPA

119861119877 (119865)

126398

minus193142

minus192484

minus193077

minus193404

minus193589

minus193637

minus193655

minus193680

minus193720

119877(119865)

08641

00588

00999

02220

03558

05535

07227

09125

1200

914

521

119877119877 (119865)

60116

00855

006

4000985

0119

901400

01529

01651

01812

01937

119871119861119864lowast

minus19889

08842

09130

08700

08435

08156

07970

07789

07566

07399

119880119861119864lowast

53743

11528

11103

11755

12158

12552

12807

13055

13385

13636

Advances in Meteorology 15

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

0 2 4 6

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

Regional growth curve for GPA

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F))

0

5

10

15

20

Qua

ntile

2 4 60Reduced variate minuslog (minus log (F))

0

5

10

15

Qua

ntile

(based on L-moments) Regional growth curve for GPA

(based on PL-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30

16 Advances in Meteorology

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60

Advances in Meteorology 17

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90

18 Advances in Meteorology

The first four PL-moments of the GEV are defined as

12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)

(A5)

In previous equation 119875(sdot sdot) is an Incomplete Gammafunction

119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500

120579119896expminus120579119889Θ (A6)

Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution

The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by

119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)

The partial PWMs of the GLO are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)

(A9)

where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function

1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650

0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)

The first four PL-moments of the GLO are defined as

12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520

minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530

minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896

201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

(A11)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution

The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by

119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)

The partial PWMs of the GPA are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int

1

1198650

(1 minus 119865)119896 119865119903119889119865 (A14)

The first four PL-moments of the GPA are defined as

12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)

12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)

(A15)

where

119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution

Advances in Meteorology 19

Ethical Approval

The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975

Conflicts of Interest

The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest

Acknowledgments

The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027

References

[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960

[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985

[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981

[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982

[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988

[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007

[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997

[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990

[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987

[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996

[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990

[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004

[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011

[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009

[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010

[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009

[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012

[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013

[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001

[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002

[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008

[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007

[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010

[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011

[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013

[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974

[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991

[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990

[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979

[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993

[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993

20 Advances in Meteorology

[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993

[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995

[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990

[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993

[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993

[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994

[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995

[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996

[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001

[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009

[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

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MineralogyInternational Journal of

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ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

8 Advances in Meteorology

04

03

02

01

00

04

03

02

01

00

00 01 02 03 04 05 06 00 01 02 03 04 05 06

GLOGEVGPA

GNOAverageData

GLOGEVGPA

GNOAverageData

L-ku

rtos

is

L㰀-k

urto

sis

L-skewness L-skewness

Figure 2 L-diagram and PL-diagram of the GLO GEV GPA and GNO distributions

Table 3 Discordance test result based on L-moments and PL-moments

Name of site L-moments (119863119894) PL-moments (119863119894)Astore 205 244Balakot 142 085Bunji 055 060Chilas 029 007Cherat 079 064DI khan 129 190Dir 082 086Drosh 051 043Garhi Dupatta 143 133Gilgit 026 059Gupis 237 231Kakul 082 122Kotli 077 059Muzaffarabad 099 085Peshawar 007 008Saidu Sharif 191 157Skardu 067 065

For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments

Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different

values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary

32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO

We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area

Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments

Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions

However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions

Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution

Advances in Meteorology 9

Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments

Heterogeneity measures L-moment PL-momentHeterogeneity measure1198671

Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060

Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800

Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170

Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments

Method Distribution Parameters120585 120572 119870L-moments

GEV 04660 05644 minus02751GNO 06671 07565 minus07594GPA minus00551 09944 minus00575

PL-momentsGEV 05220 05116 minus02686GNO 07046 06837 minus07487GPA 00478 09073 minus00472

is larger than the critical value of 164 (at 90 confidencelevel)

It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile

Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution

33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7

based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)

Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is

119876 (119865) = 1198971198941119902 (119865) (25)

The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3

Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions

Therefore it is necessary to assess the performance ofregional quantile estimates

4 Accuracy of the Estimated Quantiles and theRegional Growth Curve

A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten

10 Advances in Meteorology

Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865

01000 05000 08000 09000 09500 09800 09900 09975 099875 09990

L-momentsGEV 00454 06836 15139 22247 30591 44161 56871 90748 113165 121338GNO 00473 06671 15585 23072 31448 44098 54997 80676 95666 100823GPA 00500 06481 16217 23931 31958 43073 51878 70578 80502 83782

PL-momentsGEV 01397 07190 14670 21034 28469 40497 51703 81363 10086 107950GNO 01412 07046 15062 21752 29203 40410 50032 72608 85739 90249GPA 01436 06871 15649 22547 29674 39460 47149 63304 71786 74577

Qua

ntile

Regional growth curve (L-moments)

GPAGNOGEV

GPAGNOGEV

Qua

ntile

Return period

Regional growth gurve (PL-moments)

5

4

3

2

1

0

5

4

3

2

1

0

minus2 minus1 0 1 2 3 4 5

2 5 10 20 50 100

Return period2 5 10 20 50 100

Reduced variate minuslog (minus log (F))

minus2 minus1 0 1 2 3 4 5

Reduced variate minuslog (minus log (F))

Figure 3 L-moments and PL-moments regional growth curves

as 119876(119898)119894 (119865) for the nonexceedance probability 119865 The relativeerror for this estimator is

119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) (26)

The bias and the RMSE of the above quantity over all 119872repetition are

Bias = 119861119894 (119865) = 1119872119872sum119898=1

119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)

= 1119872 [[119872sum119898=1

(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12

(27)

Also for the estimated quantile the regional average bias andthe relative RMSE are

119861119877 (119865) = 1119873119873sum119894=1

119861119894 (119865)

119877119877 (119865) = 1119873119873sum119894=1

119877119894 (119865) (28)

We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval

119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)

Advances in Meteorology 11

Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le

119902 (119865)119871005 (119865) (30)

The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves

We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]

We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively

Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds

Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion

This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail

GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments

Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents

Appendix

The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows

PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by

119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (minus ln119865)119896 119896 = 0 (A2)

Wang [10] developed the partial probability-weightedmoments (PWMs) of the GEV as

(119903 + 1) 1205731015840119903 = 120577 + 120572119867 (120574 1198650 119896) (A3)

where

119867(120574 1198650 119896) = 1119896 [1 minus119875 1 + 119896 minus (119903 + 1) ln1198650(1 minus 119865119903+10 ) (119903 + 1)119896 ] (A4)

12 Advances in Meteorology

Table8Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f30

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus0540

0minus19

7674

minus195328

minus193602

minus191579

minus188659

minus186376

minus1840

97minus18

1165

minus179056

119877(119865)

00900

00695

00963

01952

03334

06038

09057

13261

21337

300

48119877119877 (119865)

19873

01019

00635

00876

01090

01369

01595

01839

02191

02478

119871119861119864lowast

minus20937

08829

09296

08842

08411

07862

07460

07065

06557

06177

119880119861119864lowast

18714

11949

11208

11580

11884

12244

12516

12818

13285

13683

L-mom

ents

GNO

119861119877 (119865)

08124

minus196279

minus193846

minus192837

minus191816

minus190506

minus189573

minus188699

minus187636

minus186899

119877(119865)

00986

00696

00917

02051

03520

06027

08421

11312

16008

20312

119877119877 (119865)

20766

01045

00587

00889

01120

01368

01533

01686

01878

02016

119871119861119864lowast

minus21331

08683

09271

08780

08379

07958

07691

07453

07172

06978

119880119861119864lowast

16569

11974

11077

11569

11954

12355

12615

12851

13149

13369

L-mom

ents

GPA

119861119877 (119865)

38654

minus194351

minus193564

minus193226

minus192616

minus191706

minus191060

minus190506

minus189966

minus189727

119877(119865)

01242

006

6900998

02136

03552

05968

08313

11188

15941

20368

119877119877 (119865)

24785

01033

00614

00892

01111

01386

01602

01833

02162

02431

119871119861119864lowast

minus25192

08591

09215

08817

08444

07979

07646

07327

06919

06623

119880119861119864lowast

14272

11911

11115

11588

11976

12459

12841

13257

13859

14367

PL-m

oments

GEV

119861119877 (119865)

minus02662

minus197059

minus194770

minus193340

minus191600

minus188996

minus186909

minus184797

minus182052

minus1800

66119877(119865)

05379

006

8700947

02023

03467

06171

09087

13058

20551

2854

119877119877 (119865)

38117

00948

006

4900960

01210

01508

01736

01977

02324

02609

119871119861119864lowast

minus77814

08892

09252

08725

08259

07697

07296

06911

064

1706050

119880119861119864lowast

45516

11827

11207

11716

12117

12554

12881

13211

13693

14104

PL-m

oments

GNO

119861119877 (119865)

35142

minus195922

minus193485

minus192704

minus191894

minus190806

minus1900

03minus18

9235

minus188284

minus187617

119877(119865)

06203

006

8900939

02141

03644

06143

08483

11272

15747

19809

119877119877 (119865)

43522

00971

00626

00980

01238

01504

01675

01831

02024

02162

119871119861119864lowast

minus10618

08772

09204

08655

08233

07800

07530

07290

07012

06822

119880119861119864lowast

46353

11849

11122

11734

12189

12653

12949

13218

13546

13780

PL-m

oments

GPA

119861119877 (119865)

1212

3minus19

4300

minus193238

minus193113

minus192716

minus192046

minus191555

minus19114

1minus19

0768

minus1906

49119877(119865)

08513

006

6601041

02250

03684

06021

08215

10853

15144

19094

119877119877 (119865)

58811

00962

006

6900996

01233

01509

01720

01941

02258

02518

119871119861119864lowast

minus17507

08698

09130

08680

08299

07846

07524

07220

06831

06552

119880119861119864lowast

47307

11791

11189

11774

12215

12739

1314

13567

14195

14722

Advances in Meteorology 13

Table9Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f60

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus13271

minus194230

minus193551

minus193241

minus192649

minus191612

minus190736

minus189839

minus188690

minus187888

119877(119865)

00846

00624

00851

01873

03250

05841

08652

12505

19839

27721

119877119877 (119865)

19155

00833

00612

00855

01024

01213

01360

01523

01770

01983

119871119861119864lowast

minus19593

08756

09254

08851

08491

08074

07796

07501

07118

06834

119880119861119864lowast

16242

11675

11015

11472

11879

12309

12678

13038

13659

14141

L-mom

ents

GNO

119861119877 (119865)

08417

minus193335

minus192795

minus192984

minus192999

minus192879

minus192742

minus192593

minus192399

minus192264

119877(119865)

00968

00634

00858

02000

03439

05850

08123

10845

15235

19239

119877119877 (119865)

21055

00881

00582

00861

01050

01235

01354

01463

01599

01699

119871119861119864lowast

minus22887

08647

09244

08794

08466

08141

07957

07798

07597

07443

119880119861119864lowast

15863

11714

1096

11511

11934

1240

512

666

12939

13253

13505

L-mom

ents

GPA

119861119877 (119865)

46371

minus192512

minus192631

minus193160

minus193417

minus193606

minus193745

minus193933

minus194306

minus194708

119877(119865)

01280

00624

00950

02094

03468

05740

07903

10536

14878

18928

119877119877 (119865)

264

4200915

00614

00882

01064

01270

01429

01599

01849

02059

119871119861119864lowast

minus38088

08628

09186

08815

08510

08167

07924

07703

07422

07178

119880119861119864lowast

14699

11721

11020

11534

11925

12454

12791

13236

13869

1440

5

PL-m

oments

GEV

119861119877 (119865)

minus16968

minus193984

minus193601

minus193623

minus193319

minus192576

minus191861

minus191081

minus190026

minus189255

119877(119865)

05431

00596

00888

01978

03366

05844

08417

11838

18179

24868

119877119877 (119865)

38656

00783

00634

0095

01166

01387

01541

01700

01928

02120

119871119861119864lowast

minus92921

08907

09201

08750

08403

07984

07714

07434

07043

06783

119880119861119864lowast

53553

11521

11099

1166

412

098

12572

12869

13216

13697

14105

PL-m

oments

GNO

119861119877 (119865)

23785

minus193306

minus192979

minus193417

minus193628

minus193678

minus193617

minus193509

minus193331

minus193185

119877(119865)

06278

006

0100911

02100

03523

05804

0788

10303

14113

17516

119877119877 (119865)

43743

00812

00623

00965

01186

01391

01513

01618

01742

01830

119871119861119864lowast

minus12083

08856

09188

08709

08389

08061

07870

07706

07517

07380

119880119861119864lowast

52707

11561

11059

11704

12152

12613

12898

13138

13361

1356

PL-m

oments

GPA

119861119877 (119865)

1118

7minus19

2590

minus192910

minus193681

minus1940

84minus19

4322

minus1944

04minus19

4463

minus194560

minus1946

73119877(119865)

08596

00597

01009

02229

03581

05615

07391

09418

12558

15341

119877119877 (119865)

58310

00840

006

6000991

0119

501391

01522

01650

01828

01971

119871119861119864lowast

minus18977

08800

09127

08699

08413

08098

07912

07682

07447

07255

119880119861119864lowast

51281

11551

11117

11769

12173

12615

12887

13227

1360

913

899

14 Advances in Meteorology

Table10A

ccuracymeasure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f90

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus00117

minus194226

minus193109

minus192926

minus192542

minus191809

minus191151

minus1904

48minus18

9497

minus188790

119877(119865)

00787

00556

00823

01850

03151

05459

07840

10989

16793

22883

119877119877 (119865)

17329

00814

00544

00832

01031

01238

01381

01525

01727

01892

119871119861119864lowast

minus19303

08935

09287

08885

08587

08246

08006

07772

07463

07229

119880119861119864lowast

15717

11499

10973

11461

11804

12156

12393

12635

12966

13242

L-mom

ents

GNO

119861119877 (119865)

15851

minus193606

minus192517

minus192695

minus192747

minus192678

minus192569

minus192437

minus192245

minus192098

119877(119865)

00917

00567

00846

01967

03305

05446

07392

09660

13223

1640

0119877119877 (119865)

19328

00850

00543

00853

01052

01237

01347

01442

01554

01632

119871119861119864lowast

minus23268

08848

09261

08844

08568

08301

08139

08000

07833

07722

119880119861119864lowast

15168

11542

10942

11489

11851

12197

1240

112

575

12779

12925

L-mom

ents

GPA

119861119877 (119865)

49081

minus192995

minus192417

minus192865

minus193078

minus193175

minus193195

minus193209

minus193256

minus193330

119877(119865)

01234

00571

00935

02081

03365

05322

07054

09053

12180

14975

119877119877 (119865)

246

8800882

00576

00870

01054

01239

01364

01489

01660

01798

119871119861119864lowast

minus45353

08783

09210

08838

08594

08326

08141

07961

07731

07557

119880119861119864lowast

14833

11574

10996

11532

11870

12223

1246

712

711

1304

413

312

PL-m

oments

GEV

119861119877 (119865)

minus00877

minus194145

minus193060

minus193062

minus192847

minus192278

minus191698

minus191038

minus190100

minus189374

119877(119865)

05458

00565

00855

01963

03338

05712

08090

11156

16650

22276

119877119877 (119865)

39079

00784

00585

00934

0117

201406

01558

01703

01897

02050

119871119861119864lowast

minus10190

08983

09236

08765

08431

08067

07818

07579

07271

07041

119880119861119864lowast

58730

1144

811036

11652

12081

12504

12771

13026

13364

13623

PL-m

oments

GNO

119861119877 (119865)

38414

minus193665

minus192536

minus192859

minus193044

minus193102

minus193057

minus192966

minus192804

minus192665

119877(119865)

06308

00578

00896

02088

03488

05690

07659

09926

13434

16524

119877119877 (119865)

44651

00819

00596

00961

0119

301405

01526

01628

01744

01822

119871119861119864lowast

minus13011

08910

09192

08716

08411

08122

07949

07801

07634

07519

119880119861119864lowast

56756

11491

11030

11693

12128

12539

12775

12974

13197

13347

PL-m

oments

GPA

119861119877 (119865)

126398

minus193142

minus192484

minus193077

minus193404

minus193589

minus193637

minus193655

minus193680

minus193720

119877(119865)

08641

00588

00999

02220

03558

05535

07227

09125

1200

914

521

119877119877 (119865)

60116

00855

006

4000985

0119

901400

01529

01651

01812

01937

119871119861119864lowast

minus19889

08842

09130

08700

08435

08156

07970

07789

07566

07399

119880119861119864lowast

53743

11528

11103

11755

12158

12552

12807

13055

13385

13636

Advances in Meteorology 15

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

0 2 4 6

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

Regional growth curve for GPA

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F))

0

5

10

15

20

Qua

ntile

2 4 60Reduced variate minuslog (minus log (F))

0

5

10

15

Qua

ntile

(based on L-moments) Regional growth curve for GPA

(based on PL-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30

16 Advances in Meteorology

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60

Advances in Meteorology 17

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90

18 Advances in Meteorology

The first four PL-moments of the GEV are defined as

12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)

(A5)

In previous equation 119875(sdot sdot) is an Incomplete Gammafunction

119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500

120579119896expminus120579119889Θ (A6)

Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution

The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by

119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)

The partial PWMs of the GLO are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)

(A9)

where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function

1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650

0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)

The first four PL-moments of the GLO are defined as

12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520

minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530

minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896

201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

(A11)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution

The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by

119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)

The partial PWMs of the GPA are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int

1

1198650

(1 minus 119865)119896 119865119903119889119865 (A14)

The first four PL-moments of the GPA are defined as

12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)

12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)

(A15)

where

119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution

Advances in Meteorology 19

Ethical Approval

The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975

Conflicts of Interest

The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest

Acknowledgments

The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027

References

[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960

[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985

[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981

[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982

[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988

[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007

[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997

[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990

[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987

[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996

[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990

[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004

[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011

[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009

[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010

[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009

[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012

[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013

[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001

[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002

[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008

[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007

[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010

[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011

[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013

[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974

[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991

[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990

[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979

[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993

[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993

20 Advances in Meteorology

[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993

[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995

[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990

[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993

[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993

[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994

[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995

[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996

[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001

[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009

[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

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Applied ampEnvironmentalSoil Science

Volume 2014

Mining

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Journal of

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International Journal of

Geophysics

OceanographyInternational Journal of

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GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

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OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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MineralogyInternational Journal of

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ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Advances in Meteorology 9

Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments

Heterogeneity measures L-moment PL-momentHeterogeneity measure1198671

Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060

Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800

Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170

Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments

Method Distribution Parameters120585 120572 119870L-moments

GEV 04660 05644 minus02751GNO 06671 07565 minus07594GPA minus00551 09944 minus00575

PL-momentsGEV 05220 05116 minus02686GNO 07046 06837 minus07487GPA 00478 09073 minus00472

is larger than the critical value of 164 (at 90 confidencelevel)

It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile

Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution

33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7

based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)

Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is

119876 (119865) = 1198971198941119902 (119865) (25)

The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3

Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions

Therefore it is necessary to assess the performance ofregional quantile estimates

4 Accuracy of the Estimated Quantiles and theRegional Growth Curve

A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten

10 Advances in Meteorology

Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865

01000 05000 08000 09000 09500 09800 09900 09975 099875 09990

L-momentsGEV 00454 06836 15139 22247 30591 44161 56871 90748 113165 121338GNO 00473 06671 15585 23072 31448 44098 54997 80676 95666 100823GPA 00500 06481 16217 23931 31958 43073 51878 70578 80502 83782

PL-momentsGEV 01397 07190 14670 21034 28469 40497 51703 81363 10086 107950GNO 01412 07046 15062 21752 29203 40410 50032 72608 85739 90249GPA 01436 06871 15649 22547 29674 39460 47149 63304 71786 74577

Qua

ntile

Regional growth curve (L-moments)

GPAGNOGEV

GPAGNOGEV

Qua

ntile

Return period

Regional growth gurve (PL-moments)

5

4

3

2

1

0

5

4

3

2

1

0

minus2 minus1 0 1 2 3 4 5

2 5 10 20 50 100

Return period2 5 10 20 50 100

Reduced variate minuslog (minus log (F))

minus2 minus1 0 1 2 3 4 5

Reduced variate minuslog (minus log (F))

Figure 3 L-moments and PL-moments regional growth curves

as 119876(119898)119894 (119865) for the nonexceedance probability 119865 The relativeerror for this estimator is

119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) (26)

The bias and the RMSE of the above quantity over all 119872repetition are

Bias = 119861119894 (119865) = 1119872119872sum119898=1

119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)

= 1119872 [[119872sum119898=1

(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12

(27)

Also for the estimated quantile the regional average bias andthe relative RMSE are

119861119877 (119865) = 1119873119873sum119894=1

119861119894 (119865)

119877119877 (119865) = 1119873119873sum119894=1

119877119894 (119865) (28)

We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval

119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)

Advances in Meteorology 11

Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le

119902 (119865)119871005 (119865) (30)

The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves

We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]

We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively

Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds

Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion

This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail

GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments

Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents

Appendix

The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows

PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by

119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (minus ln119865)119896 119896 = 0 (A2)

Wang [10] developed the partial probability-weightedmoments (PWMs) of the GEV as

(119903 + 1) 1205731015840119903 = 120577 + 120572119867 (120574 1198650 119896) (A3)

where

119867(120574 1198650 119896) = 1119896 [1 minus119875 1 + 119896 minus (119903 + 1) ln1198650(1 minus 119865119903+10 ) (119903 + 1)119896 ] (A4)

12 Advances in Meteorology

Table8Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f30

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus0540

0minus19

7674

minus195328

minus193602

minus191579

minus188659

minus186376

minus1840

97minus18

1165

minus179056

119877(119865)

00900

00695

00963

01952

03334

06038

09057

13261

21337

300

48119877119877 (119865)

19873

01019

00635

00876

01090

01369

01595

01839

02191

02478

119871119861119864lowast

minus20937

08829

09296

08842

08411

07862

07460

07065

06557

06177

119880119861119864lowast

18714

11949

11208

11580

11884

12244

12516

12818

13285

13683

L-mom

ents

GNO

119861119877 (119865)

08124

minus196279

minus193846

minus192837

minus191816

minus190506

minus189573

minus188699

minus187636

minus186899

119877(119865)

00986

00696

00917

02051

03520

06027

08421

11312

16008

20312

119877119877 (119865)

20766

01045

00587

00889

01120

01368

01533

01686

01878

02016

119871119861119864lowast

minus21331

08683

09271

08780

08379

07958

07691

07453

07172

06978

119880119861119864lowast

16569

11974

11077

11569

11954

12355

12615

12851

13149

13369

L-mom

ents

GPA

119861119877 (119865)

38654

minus194351

minus193564

minus193226

minus192616

minus191706

minus191060

minus190506

minus189966

minus189727

119877(119865)

01242

006

6900998

02136

03552

05968

08313

11188

15941

20368

119877119877 (119865)

24785

01033

00614

00892

01111

01386

01602

01833

02162

02431

119871119861119864lowast

minus25192

08591

09215

08817

08444

07979

07646

07327

06919

06623

119880119861119864lowast

14272

11911

11115

11588

11976

12459

12841

13257

13859

14367

PL-m

oments

GEV

119861119877 (119865)

minus02662

minus197059

minus194770

minus193340

minus191600

minus188996

minus186909

minus184797

minus182052

minus1800

66119877(119865)

05379

006

8700947

02023

03467

06171

09087

13058

20551

2854

119877119877 (119865)

38117

00948

006

4900960

01210

01508

01736

01977

02324

02609

119871119861119864lowast

minus77814

08892

09252

08725

08259

07697

07296

06911

064

1706050

119880119861119864lowast

45516

11827

11207

11716

12117

12554

12881

13211

13693

14104

PL-m

oments

GNO

119861119877 (119865)

35142

minus195922

minus193485

minus192704

minus191894

minus190806

minus1900

03minus18

9235

minus188284

minus187617

119877(119865)

06203

006

8900939

02141

03644

06143

08483

11272

15747

19809

119877119877 (119865)

43522

00971

00626

00980

01238

01504

01675

01831

02024

02162

119871119861119864lowast

minus10618

08772

09204

08655

08233

07800

07530

07290

07012

06822

119880119861119864lowast

46353

11849

11122

11734

12189

12653

12949

13218

13546

13780

PL-m

oments

GPA

119861119877 (119865)

1212

3minus19

4300

minus193238

minus193113

minus192716

minus192046

minus191555

minus19114

1minus19

0768

minus1906

49119877(119865)

08513

006

6601041

02250

03684

06021

08215

10853

15144

19094

119877119877 (119865)

58811

00962

006

6900996

01233

01509

01720

01941

02258

02518

119871119861119864lowast

minus17507

08698

09130

08680

08299

07846

07524

07220

06831

06552

119880119861119864lowast

47307

11791

11189

11774

12215

12739

1314

13567

14195

14722

Advances in Meteorology 13

Table9Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f60

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus13271

minus194230

minus193551

minus193241

minus192649

minus191612

minus190736

minus189839

minus188690

minus187888

119877(119865)

00846

00624

00851

01873

03250

05841

08652

12505

19839

27721

119877119877 (119865)

19155

00833

00612

00855

01024

01213

01360

01523

01770

01983

119871119861119864lowast

minus19593

08756

09254

08851

08491

08074

07796

07501

07118

06834

119880119861119864lowast

16242

11675

11015

11472

11879

12309

12678

13038

13659

14141

L-mom

ents

GNO

119861119877 (119865)

08417

minus193335

minus192795

minus192984

minus192999

minus192879

minus192742

minus192593

minus192399

minus192264

119877(119865)

00968

00634

00858

02000

03439

05850

08123

10845

15235

19239

119877119877 (119865)

21055

00881

00582

00861

01050

01235

01354

01463

01599

01699

119871119861119864lowast

minus22887

08647

09244

08794

08466

08141

07957

07798

07597

07443

119880119861119864lowast

15863

11714

1096

11511

11934

1240

512

666

12939

13253

13505

L-mom

ents

GPA

119861119877 (119865)

46371

minus192512

minus192631

minus193160

minus193417

minus193606

minus193745

minus193933

minus194306

minus194708

119877(119865)

01280

00624

00950

02094

03468

05740

07903

10536

14878

18928

119877119877 (119865)

264

4200915

00614

00882

01064

01270

01429

01599

01849

02059

119871119861119864lowast

minus38088

08628

09186

08815

08510

08167

07924

07703

07422

07178

119880119861119864lowast

14699

11721

11020

11534

11925

12454

12791

13236

13869

1440

5

PL-m

oments

GEV

119861119877 (119865)

minus16968

minus193984

minus193601

minus193623

minus193319

minus192576

minus191861

minus191081

minus190026

minus189255

119877(119865)

05431

00596

00888

01978

03366

05844

08417

11838

18179

24868

119877119877 (119865)

38656

00783

00634

0095

01166

01387

01541

01700

01928

02120

119871119861119864lowast

minus92921

08907

09201

08750

08403

07984

07714

07434

07043

06783

119880119861119864lowast

53553

11521

11099

1166

412

098

12572

12869

13216

13697

14105

PL-m

oments

GNO

119861119877 (119865)

23785

minus193306

minus192979

minus193417

minus193628

minus193678

minus193617

minus193509

minus193331

minus193185

119877(119865)

06278

006

0100911

02100

03523

05804

0788

10303

14113

17516

119877119877 (119865)

43743

00812

00623

00965

01186

01391

01513

01618

01742

01830

119871119861119864lowast

minus12083

08856

09188

08709

08389

08061

07870

07706

07517

07380

119880119861119864lowast

52707

11561

11059

11704

12152

12613

12898

13138

13361

1356

PL-m

oments

GPA

119861119877 (119865)

1118

7minus19

2590

minus192910

minus193681

minus1940

84minus19

4322

minus1944

04minus19

4463

minus194560

minus1946

73119877(119865)

08596

00597

01009

02229

03581

05615

07391

09418

12558

15341

119877119877 (119865)

58310

00840

006

6000991

0119

501391

01522

01650

01828

01971

119871119861119864lowast

minus18977

08800

09127

08699

08413

08098

07912

07682

07447

07255

119880119861119864lowast

51281

11551

11117

11769

12173

12615

12887

13227

1360

913

899

14 Advances in Meteorology

Table10A

ccuracymeasure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f90

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus00117

minus194226

minus193109

minus192926

minus192542

minus191809

minus191151

minus1904

48minus18

9497

minus188790

119877(119865)

00787

00556

00823

01850

03151

05459

07840

10989

16793

22883

119877119877 (119865)

17329

00814

00544

00832

01031

01238

01381

01525

01727

01892

119871119861119864lowast

minus19303

08935

09287

08885

08587

08246

08006

07772

07463

07229

119880119861119864lowast

15717

11499

10973

11461

11804

12156

12393

12635

12966

13242

L-mom

ents

GNO

119861119877 (119865)

15851

minus193606

minus192517

minus192695

minus192747

minus192678

minus192569

minus192437

minus192245

minus192098

119877(119865)

00917

00567

00846

01967

03305

05446

07392

09660

13223

1640

0119877119877 (119865)

19328

00850

00543

00853

01052

01237

01347

01442

01554

01632

119871119861119864lowast

minus23268

08848

09261

08844

08568

08301

08139

08000

07833

07722

119880119861119864lowast

15168

11542

10942

11489

11851

12197

1240

112

575

12779

12925

L-mom

ents

GPA

119861119877 (119865)

49081

minus192995

minus192417

minus192865

minus193078

minus193175

minus193195

minus193209

minus193256

minus193330

119877(119865)

01234

00571

00935

02081

03365

05322

07054

09053

12180

14975

119877119877 (119865)

246

8800882

00576

00870

01054

01239

01364

01489

01660

01798

119871119861119864lowast

minus45353

08783

09210

08838

08594

08326

08141

07961

07731

07557

119880119861119864lowast

14833

11574

10996

11532

11870

12223

1246

712

711

1304

413

312

PL-m

oments

GEV

119861119877 (119865)

minus00877

minus194145

minus193060

minus193062

minus192847

minus192278

minus191698

minus191038

minus190100

minus189374

119877(119865)

05458

00565

00855

01963

03338

05712

08090

11156

16650

22276

119877119877 (119865)

39079

00784

00585

00934

0117

201406

01558

01703

01897

02050

119871119861119864lowast

minus10190

08983

09236

08765

08431

08067

07818

07579

07271

07041

119880119861119864lowast

58730

1144

811036

11652

12081

12504

12771

13026

13364

13623

PL-m

oments

GNO

119861119877 (119865)

38414

minus193665

minus192536

minus192859

minus193044

minus193102

minus193057

minus192966

minus192804

minus192665

119877(119865)

06308

00578

00896

02088

03488

05690

07659

09926

13434

16524

119877119877 (119865)

44651

00819

00596

00961

0119

301405

01526

01628

01744

01822

119871119861119864lowast

minus13011

08910

09192

08716

08411

08122

07949

07801

07634

07519

119880119861119864lowast

56756

11491

11030

11693

12128

12539

12775

12974

13197

13347

PL-m

oments

GPA

119861119877 (119865)

126398

minus193142

minus192484

minus193077

minus193404

minus193589

minus193637

minus193655

minus193680

minus193720

119877(119865)

08641

00588

00999

02220

03558

05535

07227

09125

1200

914

521

119877119877 (119865)

60116

00855

006

4000985

0119

901400

01529

01651

01812

01937

119871119861119864lowast

minus19889

08842

09130

08700

08435

08156

07970

07789

07566

07399

119880119861119864lowast

53743

11528

11103

11755

12158

12552

12807

13055

13385

13636

Advances in Meteorology 15

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

0 2 4 6

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

Regional growth curve for GPA

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F))

0

5

10

15

20

Qua

ntile

2 4 60Reduced variate minuslog (minus log (F))

0

5

10

15

Qua

ntile

(based on L-moments) Regional growth curve for GPA

(based on PL-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30

16 Advances in Meteorology

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60

Advances in Meteorology 17

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90

18 Advances in Meteorology

The first four PL-moments of the GEV are defined as

12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)

(A5)

In previous equation 119875(sdot sdot) is an Incomplete Gammafunction

119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500

120579119896expminus120579119889Θ (A6)

Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution

The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by

119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)

The partial PWMs of the GLO are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)

(A9)

where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function

1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650

0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)

The first four PL-moments of the GLO are defined as

12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520

minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530

minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896

201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

(A11)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution

The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by

119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)

The partial PWMs of the GPA are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int

1

1198650

(1 minus 119865)119896 119865119903119889119865 (A14)

The first four PL-moments of the GPA are defined as

12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)

12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)

(A15)

where

119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution

Advances in Meteorology 19

Ethical Approval

The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975

Conflicts of Interest

The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest

Acknowledgments

The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027

References

[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960

[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985

[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981

[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982

[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988

[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007

[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997

[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990

[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987

[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996

[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990

[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004

[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011

[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009

[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010

[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009

[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012

[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013

[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001

[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002

[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008

[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007

[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010

[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011

[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013

[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974

[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991

[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990

[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979

[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993

[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993

20 Advances in Meteorology

[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993

[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995

[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990

[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993

[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993

[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994

[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995

[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996

[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001

[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009

[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Geological ResearchJournal of

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Geology Advances in

10 Advances in Meteorology

Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865

01000 05000 08000 09000 09500 09800 09900 09975 099875 09990

L-momentsGEV 00454 06836 15139 22247 30591 44161 56871 90748 113165 121338GNO 00473 06671 15585 23072 31448 44098 54997 80676 95666 100823GPA 00500 06481 16217 23931 31958 43073 51878 70578 80502 83782

PL-momentsGEV 01397 07190 14670 21034 28469 40497 51703 81363 10086 107950GNO 01412 07046 15062 21752 29203 40410 50032 72608 85739 90249GPA 01436 06871 15649 22547 29674 39460 47149 63304 71786 74577

Qua

ntile

Regional growth curve (L-moments)

GPAGNOGEV

GPAGNOGEV

Qua

ntile

Return period

Regional growth gurve (PL-moments)

5

4

3

2

1

0

5

4

3

2

1

0

minus2 minus1 0 1 2 3 4 5

2 5 10 20 50 100

Return period2 5 10 20 50 100

Reduced variate minuslog (minus log (F))

minus2 minus1 0 1 2 3 4 5

Reduced variate minuslog (minus log (F))

Figure 3 L-moments and PL-moments regional growth curves

as 119876(119898)119894 (119865) for the nonexceedance probability 119865 The relativeerror for this estimator is

119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) (26)

The bias and the RMSE of the above quantity over all 119872repetition are

Bias = 119861119894 (119865) = 1119872119872sum119898=1

119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)

= 1119872 [[119872sum119898=1

(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12

(27)

Also for the estimated quantile the regional average bias andthe relative RMSE are

119861119877 (119865) = 1119873119873sum119894=1

119861119894 (119865)

119877119877 (119865) = 1119873119873sum119894=1

119877119894 (119865) (28)

We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval

119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)

Advances in Meteorology 11

Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le

119902 (119865)119871005 (119865) (30)

The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves

We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]

We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively

Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds

Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion

This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail

GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments

Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents

Appendix

The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows

PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by

119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (minus ln119865)119896 119896 = 0 (A2)

Wang [10] developed the partial probability-weightedmoments (PWMs) of the GEV as

(119903 + 1) 1205731015840119903 = 120577 + 120572119867 (120574 1198650 119896) (A3)

where

119867(120574 1198650 119896) = 1119896 [1 minus119875 1 + 119896 minus (119903 + 1) ln1198650(1 minus 119865119903+10 ) (119903 + 1)119896 ] (A4)

12 Advances in Meteorology

Table8Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f30

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus0540

0minus19

7674

minus195328

minus193602

minus191579

minus188659

minus186376

minus1840

97minus18

1165

minus179056

119877(119865)

00900

00695

00963

01952

03334

06038

09057

13261

21337

300

48119877119877 (119865)

19873

01019

00635

00876

01090

01369

01595

01839

02191

02478

119871119861119864lowast

minus20937

08829

09296

08842

08411

07862

07460

07065

06557

06177

119880119861119864lowast

18714

11949

11208

11580

11884

12244

12516

12818

13285

13683

L-mom

ents

GNO

119861119877 (119865)

08124

minus196279

minus193846

minus192837

minus191816

minus190506

minus189573

minus188699

minus187636

minus186899

119877(119865)

00986

00696

00917

02051

03520

06027

08421

11312

16008

20312

119877119877 (119865)

20766

01045

00587

00889

01120

01368

01533

01686

01878

02016

119871119861119864lowast

minus21331

08683

09271

08780

08379

07958

07691

07453

07172

06978

119880119861119864lowast

16569

11974

11077

11569

11954

12355

12615

12851

13149

13369

L-mom

ents

GPA

119861119877 (119865)

38654

minus194351

minus193564

minus193226

minus192616

minus191706

minus191060

minus190506

minus189966

minus189727

119877(119865)

01242

006

6900998

02136

03552

05968

08313

11188

15941

20368

119877119877 (119865)

24785

01033

00614

00892

01111

01386

01602

01833

02162

02431

119871119861119864lowast

minus25192

08591

09215

08817

08444

07979

07646

07327

06919

06623

119880119861119864lowast

14272

11911

11115

11588

11976

12459

12841

13257

13859

14367

PL-m

oments

GEV

119861119877 (119865)

minus02662

minus197059

minus194770

minus193340

minus191600

minus188996

minus186909

minus184797

minus182052

minus1800

66119877(119865)

05379

006

8700947

02023

03467

06171

09087

13058

20551

2854

119877119877 (119865)

38117

00948

006

4900960

01210

01508

01736

01977

02324

02609

119871119861119864lowast

minus77814

08892

09252

08725

08259

07697

07296

06911

064

1706050

119880119861119864lowast

45516

11827

11207

11716

12117

12554

12881

13211

13693

14104

PL-m

oments

GNO

119861119877 (119865)

35142

minus195922

minus193485

minus192704

minus191894

minus190806

minus1900

03minus18

9235

minus188284

minus187617

119877(119865)

06203

006

8900939

02141

03644

06143

08483

11272

15747

19809

119877119877 (119865)

43522

00971

00626

00980

01238

01504

01675

01831

02024

02162

119871119861119864lowast

minus10618

08772

09204

08655

08233

07800

07530

07290

07012

06822

119880119861119864lowast

46353

11849

11122

11734

12189

12653

12949

13218

13546

13780

PL-m

oments

GPA

119861119877 (119865)

1212

3minus19

4300

minus193238

minus193113

minus192716

minus192046

minus191555

minus19114

1minus19

0768

minus1906

49119877(119865)

08513

006

6601041

02250

03684

06021

08215

10853

15144

19094

119877119877 (119865)

58811

00962

006

6900996

01233

01509

01720

01941

02258

02518

119871119861119864lowast

minus17507

08698

09130

08680

08299

07846

07524

07220

06831

06552

119880119861119864lowast

47307

11791

11189

11774

12215

12739

1314

13567

14195

14722

Advances in Meteorology 13

Table9Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f60

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus13271

minus194230

minus193551

minus193241

minus192649

minus191612

minus190736

minus189839

minus188690

minus187888

119877(119865)

00846

00624

00851

01873

03250

05841

08652

12505

19839

27721

119877119877 (119865)

19155

00833

00612

00855

01024

01213

01360

01523

01770

01983

119871119861119864lowast

minus19593

08756

09254

08851

08491

08074

07796

07501

07118

06834

119880119861119864lowast

16242

11675

11015

11472

11879

12309

12678

13038

13659

14141

L-mom

ents

GNO

119861119877 (119865)

08417

minus193335

minus192795

minus192984

minus192999

minus192879

minus192742

minus192593

minus192399

minus192264

119877(119865)

00968

00634

00858

02000

03439

05850

08123

10845

15235

19239

119877119877 (119865)

21055

00881

00582

00861

01050

01235

01354

01463

01599

01699

119871119861119864lowast

minus22887

08647

09244

08794

08466

08141

07957

07798

07597

07443

119880119861119864lowast

15863

11714

1096

11511

11934

1240

512

666

12939

13253

13505

L-mom

ents

GPA

119861119877 (119865)

46371

minus192512

minus192631

minus193160

minus193417

minus193606

minus193745

minus193933

minus194306

minus194708

119877(119865)

01280

00624

00950

02094

03468

05740

07903

10536

14878

18928

119877119877 (119865)

264

4200915

00614

00882

01064

01270

01429

01599

01849

02059

119871119861119864lowast

minus38088

08628

09186

08815

08510

08167

07924

07703

07422

07178

119880119861119864lowast

14699

11721

11020

11534

11925

12454

12791

13236

13869

1440

5

PL-m

oments

GEV

119861119877 (119865)

minus16968

minus193984

minus193601

minus193623

minus193319

minus192576

minus191861

minus191081

minus190026

minus189255

119877(119865)

05431

00596

00888

01978

03366

05844

08417

11838

18179

24868

119877119877 (119865)

38656

00783

00634

0095

01166

01387

01541

01700

01928

02120

119871119861119864lowast

minus92921

08907

09201

08750

08403

07984

07714

07434

07043

06783

119880119861119864lowast

53553

11521

11099

1166

412

098

12572

12869

13216

13697

14105

PL-m

oments

GNO

119861119877 (119865)

23785

minus193306

minus192979

minus193417

minus193628

minus193678

minus193617

minus193509

minus193331

minus193185

119877(119865)

06278

006

0100911

02100

03523

05804

0788

10303

14113

17516

119877119877 (119865)

43743

00812

00623

00965

01186

01391

01513

01618

01742

01830

119871119861119864lowast

minus12083

08856

09188

08709

08389

08061

07870

07706

07517

07380

119880119861119864lowast

52707

11561

11059

11704

12152

12613

12898

13138

13361

1356

PL-m

oments

GPA

119861119877 (119865)

1118

7minus19

2590

minus192910

minus193681

minus1940

84minus19

4322

minus1944

04minus19

4463

minus194560

minus1946

73119877(119865)

08596

00597

01009

02229

03581

05615

07391

09418

12558

15341

119877119877 (119865)

58310

00840

006

6000991

0119

501391

01522

01650

01828

01971

119871119861119864lowast

minus18977

08800

09127

08699

08413

08098

07912

07682

07447

07255

119880119861119864lowast

51281

11551

11117

11769

12173

12615

12887

13227

1360

913

899

14 Advances in Meteorology

Table10A

ccuracymeasure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f90

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus00117

minus194226

minus193109

minus192926

minus192542

minus191809

minus191151

minus1904

48minus18

9497

minus188790

119877(119865)

00787

00556

00823

01850

03151

05459

07840

10989

16793

22883

119877119877 (119865)

17329

00814

00544

00832

01031

01238

01381

01525

01727

01892

119871119861119864lowast

minus19303

08935

09287

08885

08587

08246

08006

07772

07463

07229

119880119861119864lowast

15717

11499

10973

11461

11804

12156

12393

12635

12966

13242

L-mom

ents

GNO

119861119877 (119865)

15851

minus193606

minus192517

minus192695

minus192747

minus192678

minus192569

minus192437

minus192245

minus192098

119877(119865)

00917

00567

00846

01967

03305

05446

07392

09660

13223

1640

0119877119877 (119865)

19328

00850

00543

00853

01052

01237

01347

01442

01554

01632

119871119861119864lowast

minus23268

08848

09261

08844

08568

08301

08139

08000

07833

07722

119880119861119864lowast

15168

11542

10942

11489

11851

12197

1240

112

575

12779

12925

L-mom

ents

GPA

119861119877 (119865)

49081

minus192995

minus192417

minus192865

minus193078

minus193175

minus193195

minus193209

minus193256

minus193330

119877(119865)

01234

00571

00935

02081

03365

05322

07054

09053

12180

14975

119877119877 (119865)

246

8800882

00576

00870

01054

01239

01364

01489

01660

01798

119871119861119864lowast

minus45353

08783

09210

08838

08594

08326

08141

07961

07731

07557

119880119861119864lowast

14833

11574

10996

11532

11870

12223

1246

712

711

1304

413

312

PL-m

oments

GEV

119861119877 (119865)

minus00877

minus194145

minus193060

minus193062

minus192847

minus192278

minus191698

minus191038

minus190100

minus189374

119877(119865)

05458

00565

00855

01963

03338

05712

08090

11156

16650

22276

119877119877 (119865)

39079

00784

00585

00934

0117

201406

01558

01703

01897

02050

119871119861119864lowast

minus10190

08983

09236

08765

08431

08067

07818

07579

07271

07041

119880119861119864lowast

58730

1144

811036

11652

12081

12504

12771

13026

13364

13623

PL-m

oments

GNO

119861119877 (119865)

38414

minus193665

minus192536

minus192859

minus193044

minus193102

minus193057

minus192966

minus192804

minus192665

119877(119865)

06308

00578

00896

02088

03488

05690

07659

09926

13434

16524

119877119877 (119865)

44651

00819

00596

00961

0119

301405

01526

01628

01744

01822

119871119861119864lowast

minus13011

08910

09192

08716

08411

08122

07949

07801

07634

07519

119880119861119864lowast

56756

11491

11030

11693

12128

12539

12775

12974

13197

13347

PL-m

oments

GPA

119861119877 (119865)

126398

minus193142

minus192484

minus193077

minus193404

minus193589

minus193637

minus193655

minus193680

minus193720

119877(119865)

08641

00588

00999

02220

03558

05535

07227

09125

1200

914

521

119877119877 (119865)

60116

00855

006

4000985

0119

901400

01529

01651

01812

01937

119871119861119864lowast

minus19889

08842

09130

08700

08435

08156

07970

07789

07566

07399

119880119861119864lowast

53743

11528

11103

11755

12158

12552

12807

13055

13385

13636

Advances in Meteorology 15

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

0 2 4 6

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

Regional growth curve for GPA

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F))

0

5

10

15

20

Qua

ntile

2 4 60Reduced variate minuslog (minus log (F))

0

5

10

15

Qua

ntile

(based on L-moments) Regional growth curve for GPA

(based on PL-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30

16 Advances in Meteorology

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60

Advances in Meteorology 17

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90

18 Advances in Meteorology

The first four PL-moments of the GEV are defined as

12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)

(A5)

In previous equation 119875(sdot sdot) is an Incomplete Gammafunction

119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500

120579119896expminus120579119889Θ (A6)

Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution

The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by

119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)

The partial PWMs of the GLO are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)

(A9)

where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function

1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650

0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)

The first four PL-moments of the GLO are defined as

12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520

minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530

minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896

201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

(A11)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution

The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by

119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)

The partial PWMs of the GPA are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int

1

1198650

(1 minus 119865)119896 119865119903119889119865 (A14)

The first four PL-moments of the GPA are defined as

12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)

12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)

(A15)

where

119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution

Advances in Meteorology 19

Ethical Approval

The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975

Conflicts of Interest

The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest

Acknowledgments

The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027

References

[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960

[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985

[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981

[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982

[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988

[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007

[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997

[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990

[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987

[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996

[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990

[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004

[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011

[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009

[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010

[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009

[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012

[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013

[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001

[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002

[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008

[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007

[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010

[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011

[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013

[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974

[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991

[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990

[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979

[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993

[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993

20 Advances in Meteorology

[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993

[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995

[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990

[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993

[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993

[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994

[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995

[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996

[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001

[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009

[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

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Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

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MineralogyInternational Journal of

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MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Advances in Meteorology 11

Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le

119902 (119865)119871005 (119865) (30)

The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves

We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]

We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively

Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds

Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion

This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail

GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments

Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents

Appendix

The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows

PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by

119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (minus ln119865)119896 119896 = 0 (A2)

Wang [10] developed the partial probability-weightedmoments (PWMs) of the GEV as

(119903 + 1) 1205731015840119903 = 120577 + 120572119867 (120574 1198650 119896) (A3)

where

119867(120574 1198650 119896) = 1119896 [1 minus119875 1 + 119896 minus (119903 + 1) ln1198650(1 minus 119865119903+10 ) (119903 + 1)119896 ] (A4)

12 Advances in Meteorology

Table8Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f30

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus0540

0minus19

7674

minus195328

minus193602

minus191579

minus188659

minus186376

minus1840

97minus18

1165

minus179056

119877(119865)

00900

00695

00963

01952

03334

06038

09057

13261

21337

300

48119877119877 (119865)

19873

01019

00635

00876

01090

01369

01595

01839

02191

02478

119871119861119864lowast

minus20937

08829

09296

08842

08411

07862

07460

07065

06557

06177

119880119861119864lowast

18714

11949

11208

11580

11884

12244

12516

12818

13285

13683

L-mom

ents

GNO

119861119877 (119865)

08124

minus196279

minus193846

minus192837

minus191816

minus190506

minus189573

minus188699

minus187636

minus186899

119877(119865)

00986

00696

00917

02051

03520

06027

08421

11312

16008

20312

119877119877 (119865)

20766

01045

00587

00889

01120

01368

01533

01686

01878

02016

119871119861119864lowast

minus21331

08683

09271

08780

08379

07958

07691

07453

07172

06978

119880119861119864lowast

16569

11974

11077

11569

11954

12355

12615

12851

13149

13369

L-mom

ents

GPA

119861119877 (119865)

38654

minus194351

minus193564

minus193226

minus192616

minus191706

minus191060

minus190506

minus189966

minus189727

119877(119865)

01242

006

6900998

02136

03552

05968

08313

11188

15941

20368

119877119877 (119865)

24785

01033

00614

00892

01111

01386

01602

01833

02162

02431

119871119861119864lowast

minus25192

08591

09215

08817

08444

07979

07646

07327

06919

06623

119880119861119864lowast

14272

11911

11115

11588

11976

12459

12841

13257

13859

14367

PL-m

oments

GEV

119861119877 (119865)

minus02662

minus197059

minus194770

minus193340

minus191600

minus188996

minus186909

minus184797

minus182052

minus1800

66119877(119865)

05379

006

8700947

02023

03467

06171

09087

13058

20551

2854

119877119877 (119865)

38117

00948

006

4900960

01210

01508

01736

01977

02324

02609

119871119861119864lowast

minus77814

08892

09252

08725

08259

07697

07296

06911

064

1706050

119880119861119864lowast

45516

11827

11207

11716

12117

12554

12881

13211

13693

14104

PL-m

oments

GNO

119861119877 (119865)

35142

minus195922

minus193485

minus192704

minus191894

minus190806

minus1900

03minus18

9235

minus188284

minus187617

119877(119865)

06203

006

8900939

02141

03644

06143

08483

11272

15747

19809

119877119877 (119865)

43522

00971

00626

00980

01238

01504

01675

01831

02024

02162

119871119861119864lowast

minus10618

08772

09204

08655

08233

07800

07530

07290

07012

06822

119880119861119864lowast

46353

11849

11122

11734

12189

12653

12949

13218

13546

13780

PL-m

oments

GPA

119861119877 (119865)

1212

3minus19

4300

minus193238

minus193113

minus192716

minus192046

minus191555

minus19114

1minus19

0768

minus1906

49119877(119865)

08513

006

6601041

02250

03684

06021

08215

10853

15144

19094

119877119877 (119865)

58811

00962

006

6900996

01233

01509

01720

01941

02258

02518

119871119861119864lowast

minus17507

08698

09130

08680

08299

07846

07524

07220

06831

06552

119880119861119864lowast

47307

11791

11189

11774

12215

12739

1314

13567

14195

14722

Advances in Meteorology 13

Table9Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f60

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus13271

minus194230

minus193551

minus193241

minus192649

minus191612

minus190736

minus189839

minus188690

minus187888

119877(119865)

00846

00624

00851

01873

03250

05841

08652

12505

19839

27721

119877119877 (119865)

19155

00833

00612

00855

01024

01213

01360

01523

01770

01983

119871119861119864lowast

minus19593

08756

09254

08851

08491

08074

07796

07501

07118

06834

119880119861119864lowast

16242

11675

11015

11472

11879

12309

12678

13038

13659

14141

L-mom

ents

GNO

119861119877 (119865)

08417

minus193335

minus192795

minus192984

minus192999

minus192879

minus192742

minus192593

minus192399

minus192264

119877(119865)

00968

00634

00858

02000

03439

05850

08123

10845

15235

19239

119877119877 (119865)

21055

00881

00582

00861

01050

01235

01354

01463

01599

01699

119871119861119864lowast

minus22887

08647

09244

08794

08466

08141

07957

07798

07597

07443

119880119861119864lowast

15863

11714

1096

11511

11934

1240

512

666

12939

13253

13505

L-mom

ents

GPA

119861119877 (119865)

46371

minus192512

minus192631

minus193160

minus193417

minus193606

minus193745

minus193933

minus194306

minus194708

119877(119865)

01280

00624

00950

02094

03468

05740

07903

10536

14878

18928

119877119877 (119865)

264

4200915

00614

00882

01064

01270

01429

01599

01849

02059

119871119861119864lowast

minus38088

08628

09186

08815

08510

08167

07924

07703

07422

07178

119880119861119864lowast

14699

11721

11020

11534

11925

12454

12791

13236

13869

1440

5

PL-m

oments

GEV

119861119877 (119865)

minus16968

minus193984

minus193601

minus193623

minus193319

minus192576

minus191861

minus191081

minus190026

minus189255

119877(119865)

05431

00596

00888

01978

03366

05844

08417

11838

18179

24868

119877119877 (119865)

38656

00783

00634

0095

01166

01387

01541

01700

01928

02120

119871119861119864lowast

minus92921

08907

09201

08750

08403

07984

07714

07434

07043

06783

119880119861119864lowast

53553

11521

11099

1166

412

098

12572

12869

13216

13697

14105

PL-m

oments

GNO

119861119877 (119865)

23785

minus193306

minus192979

minus193417

minus193628

minus193678

minus193617

minus193509

minus193331

minus193185

119877(119865)

06278

006

0100911

02100

03523

05804

0788

10303

14113

17516

119877119877 (119865)

43743

00812

00623

00965

01186

01391

01513

01618

01742

01830

119871119861119864lowast

minus12083

08856

09188

08709

08389

08061

07870

07706

07517

07380

119880119861119864lowast

52707

11561

11059

11704

12152

12613

12898

13138

13361

1356

PL-m

oments

GPA

119861119877 (119865)

1118

7minus19

2590

minus192910

minus193681

minus1940

84minus19

4322

minus1944

04minus19

4463

minus194560

minus1946

73119877(119865)

08596

00597

01009

02229

03581

05615

07391

09418

12558

15341

119877119877 (119865)

58310

00840

006

6000991

0119

501391

01522

01650

01828

01971

119871119861119864lowast

minus18977

08800

09127

08699

08413

08098

07912

07682

07447

07255

119880119861119864lowast

51281

11551

11117

11769

12173

12615

12887

13227

1360

913

899

14 Advances in Meteorology

Table10A

ccuracymeasure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f90

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus00117

minus194226

minus193109

minus192926

minus192542

minus191809

minus191151

minus1904

48minus18

9497

minus188790

119877(119865)

00787

00556

00823

01850

03151

05459

07840

10989

16793

22883

119877119877 (119865)

17329

00814

00544

00832

01031

01238

01381

01525

01727

01892

119871119861119864lowast

minus19303

08935

09287

08885

08587

08246

08006

07772

07463

07229

119880119861119864lowast

15717

11499

10973

11461

11804

12156

12393

12635

12966

13242

L-mom

ents

GNO

119861119877 (119865)

15851

minus193606

minus192517

minus192695

minus192747

minus192678

minus192569

minus192437

minus192245

minus192098

119877(119865)

00917

00567

00846

01967

03305

05446

07392

09660

13223

1640

0119877119877 (119865)

19328

00850

00543

00853

01052

01237

01347

01442

01554

01632

119871119861119864lowast

minus23268

08848

09261

08844

08568

08301

08139

08000

07833

07722

119880119861119864lowast

15168

11542

10942

11489

11851

12197

1240

112

575

12779

12925

L-mom

ents

GPA

119861119877 (119865)

49081

minus192995

minus192417

minus192865

minus193078

minus193175

minus193195

minus193209

minus193256

minus193330

119877(119865)

01234

00571

00935

02081

03365

05322

07054

09053

12180

14975

119877119877 (119865)

246

8800882

00576

00870

01054

01239

01364

01489

01660

01798

119871119861119864lowast

minus45353

08783

09210

08838

08594

08326

08141

07961

07731

07557

119880119861119864lowast

14833

11574

10996

11532

11870

12223

1246

712

711

1304

413

312

PL-m

oments

GEV

119861119877 (119865)

minus00877

minus194145

minus193060

minus193062

minus192847

minus192278

minus191698

minus191038

minus190100

minus189374

119877(119865)

05458

00565

00855

01963

03338

05712

08090

11156

16650

22276

119877119877 (119865)

39079

00784

00585

00934

0117

201406

01558

01703

01897

02050

119871119861119864lowast

minus10190

08983

09236

08765

08431

08067

07818

07579

07271

07041

119880119861119864lowast

58730

1144

811036

11652

12081

12504

12771

13026

13364

13623

PL-m

oments

GNO

119861119877 (119865)

38414

minus193665

minus192536

minus192859

minus193044

minus193102

minus193057

minus192966

minus192804

minus192665

119877(119865)

06308

00578

00896

02088

03488

05690

07659

09926

13434

16524

119877119877 (119865)

44651

00819

00596

00961

0119

301405

01526

01628

01744

01822

119871119861119864lowast

minus13011

08910

09192

08716

08411

08122

07949

07801

07634

07519

119880119861119864lowast

56756

11491

11030

11693

12128

12539

12775

12974

13197

13347

PL-m

oments

GPA

119861119877 (119865)

126398

minus193142

minus192484

minus193077

minus193404

minus193589

minus193637

minus193655

minus193680

minus193720

119877(119865)

08641

00588

00999

02220

03558

05535

07227

09125

1200

914

521

119877119877 (119865)

60116

00855

006

4000985

0119

901400

01529

01651

01812

01937

119871119861119864lowast

minus19889

08842

09130

08700

08435

08156

07970

07789

07566

07399

119880119861119864lowast

53743

11528

11103

11755

12158

12552

12807

13055

13385

13636

Advances in Meteorology 15

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

0 2 4 6

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

Regional growth curve for GPA

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F))

0

5

10

15

20

Qua

ntile

2 4 60Reduced variate minuslog (minus log (F))

0

5

10

15

Qua

ntile

(based on L-moments) Regional growth curve for GPA

(based on PL-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30

16 Advances in Meteorology

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60

Advances in Meteorology 17

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90

18 Advances in Meteorology

The first four PL-moments of the GEV are defined as

12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)

(A5)

In previous equation 119875(sdot sdot) is an Incomplete Gammafunction

119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500

120579119896expminus120579119889Θ (A6)

Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution

The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by

119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)

The partial PWMs of the GLO are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)

(A9)

where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function

1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650

0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)

The first four PL-moments of the GLO are defined as

12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520

minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530

minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896

201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

(A11)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution

The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by

119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)

The partial PWMs of the GPA are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int

1

1198650

(1 minus 119865)119896 119865119903119889119865 (A14)

The first four PL-moments of the GPA are defined as

12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)

12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)

(A15)

where

119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution

Advances in Meteorology 19

Ethical Approval

The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975

Conflicts of Interest

The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest

Acknowledgments

The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027

References

[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960

[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985

[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981

[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982

[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988

[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007

[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997

[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990

[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987

[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996

[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990

[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004

[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011

[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009

[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010

[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009

[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012

[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013

[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001

[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002

[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008

[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007

[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010

[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011

[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013

[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974

[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991

[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990

[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979

[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993

[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993

20 Advances in Meteorology

[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993

[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995

[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990

[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993

[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993

[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994

[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995

[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996

[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001

[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009

[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

12 Advances in Meteorology

Table8Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f30

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus0540

0minus19

7674

minus195328

minus193602

minus191579

minus188659

minus186376

minus1840

97minus18

1165

minus179056

119877(119865)

00900

00695

00963

01952

03334

06038

09057

13261

21337

300

48119877119877 (119865)

19873

01019

00635

00876

01090

01369

01595

01839

02191

02478

119871119861119864lowast

minus20937

08829

09296

08842

08411

07862

07460

07065

06557

06177

119880119861119864lowast

18714

11949

11208

11580

11884

12244

12516

12818

13285

13683

L-mom

ents

GNO

119861119877 (119865)

08124

minus196279

minus193846

minus192837

minus191816

minus190506

minus189573

minus188699

minus187636

minus186899

119877(119865)

00986

00696

00917

02051

03520

06027

08421

11312

16008

20312

119877119877 (119865)

20766

01045

00587

00889

01120

01368

01533

01686

01878

02016

119871119861119864lowast

minus21331

08683

09271

08780

08379

07958

07691

07453

07172

06978

119880119861119864lowast

16569

11974

11077

11569

11954

12355

12615

12851

13149

13369

L-mom

ents

GPA

119861119877 (119865)

38654

minus194351

minus193564

minus193226

minus192616

minus191706

minus191060

minus190506

minus189966

minus189727

119877(119865)

01242

006

6900998

02136

03552

05968

08313

11188

15941

20368

119877119877 (119865)

24785

01033

00614

00892

01111

01386

01602

01833

02162

02431

119871119861119864lowast

minus25192

08591

09215

08817

08444

07979

07646

07327

06919

06623

119880119861119864lowast

14272

11911

11115

11588

11976

12459

12841

13257

13859

14367

PL-m

oments

GEV

119861119877 (119865)

minus02662

minus197059

minus194770

minus193340

minus191600

minus188996

minus186909

minus184797

minus182052

minus1800

66119877(119865)

05379

006

8700947

02023

03467

06171

09087

13058

20551

2854

119877119877 (119865)

38117

00948

006

4900960

01210

01508

01736

01977

02324

02609

119871119861119864lowast

minus77814

08892

09252

08725

08259

07697

07296

06911

064

1706050

119880119861119864lowast

45516

11827

11207

11716

12117

12554

12881

13211

13693

14104

PL-m

oments

GNO

119861119877 (119865)

35142

minus195922

minus193485

minus192704

minus191894

minus190806

minus1900

03minus18

9235

minus188284

minus187617

119877(119865)

06203

006

8900939

02141

03644

06143

08483

11272

15747

19809

119877119877 (119865)

43522

00971

00626

00980

01238

01504

01675

01831

02024

02162

119871119861119864lowast

minus10618

08772

09204

08655

08233

07800

07530

07290

07012

06822

119880119861119864lowast

46353

11849

11122

11734

12189

12653

12949

13218

13546

13780

PL-m

oments

GPA

119861119877 (119865)

1212

3minus19

4300

minus193238

minus193113

minus192716

minus192046

minus191555

minus19114

1minus19

0768

minus1906

49119877(119865)

08513

006

6601041

02250

03684

06021

08215

10853

15144

19094

119877119877 (119865)

58811

00962

006

6900996

01233

01509

01720

01941

02258

02518

119871119861119864lowast

minus17507

08698

09130

08680

08299

07846

07524

07220

06831

06552

119880119861119864lowast

47307

11791

11189

11774

12215

12739

1314

13567

14195

14722

Advances in Meteorology 13

Table9Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f60

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus13271

minus194230

minus193551

minus193241

minus192649

minus191612

minus190736

minus189839

minus188690

minus187888

119877(119865)

00846

00624

00851

01873

03250

05841

08652

12505

19839

27721

119877119877 (119865)

19155

00833

00612

00855

01024

01213

01360

01523

01770

01983

119871119861119864lowast

minus19593

08756

09254

08851

08491

08074

07796

07501

07118

06834

119880119861119864lowast

16242

11675

11015

11472

11879

12309

12678

13038

13659

14141

L-mom

ents

GNO

119861119877 (119865)

08417

minus193335

minus192795

minus192984

minus192999

minus192879

minus192742

minus192593

minus192399

minus192264

119877(119865)

00968

00634

00858

02000

03439

05850

08123

10845

15235

19239

119877119877 (119865)

21055

00881

00582

00861

01050

01235

01354

01463

01599

01699

119871119861119864lowast

minus22887

08647

09244

08794

08466

08141

07957

07798

07597

07443

119880119861119864lowast

15863

11714

1096

11511

11934

1240

512

666

12939

13253

13505

L-mom

ents

GPA

119861119877 (119865)

46371

minus192512

minus192631

minus193160

minus193417

minus193606

minus193745

minus193933

minus194306

minus194708

119877(119865)

01280

00624

00950

02094

03468

05740

07903

10536

14878

18928

119877119877 (119865)

264

4200915

00614

00882

01064

01270

01429

01599

01849

02059

119871119861119864lowast

minus38088

08628

09186

08815

08510

08167

07924

07703

07422

07178

119880119861119864lowast

14699

11721

11020

11534

11925

12454

12791

13236

13869

1440

5

PL-m

oments

GEV

119861119877 (119865)

minus16968

minus193984

minus193601

minus193623

minus193319

minus192576

minus191861

minus191081

minus190026

minus189255

119877(119865)

05431

00596

00888

01978

03366

05844

08417

11838

18179

24868

119877119877 (119865)

38656

00783

00634

0095

01166

01387

01541

01700

01928

02120

119871119861119864lowast

minus92921

08907

09201

08750

08403

07984

07714

07434

07043

06783

119880119861119864lowast

53553

11521

11099

1166

412

098

12572

12869

13216

13697

14105

PL-m

oments

GNO

119861119877 (119865)

23785

minus193306

minus192979

minus193417

minus193628

minus193678

minus193617

minus193509

minus193331

minus193185

119877(119865)

06278

006

0100911

02100

03523

05804

0788

10303

14113

17516

119877119877 (119865)

43743

00812

00623

00965

01186

01391

01513

01618

01742

01830

119871119861119864lowast

minus12083

08856

09188

08709

08389

08061

07870

07706

07517

07380

119880119861119864lowast

52707

11561

11059

11704

12152

12613

12898

13138

13361

1356

PL-m

oments

GPA

119861119877 (119865)

1118

7minus19

2590

minus192910

minus193681

minus1940

84minus19

4322

minus1944

04minus19

4463

minus194560

minus1946

73119877(119865)

08596

00597

01009

02229

03581

05615

07391

09418

12558

15341

119877119877 (119865)

58310

00840

006

6000991

0119

501391

01522

01650

01828

01971

119871119861119864lowast

minus18977

08800

09127

08699

08413

08098

07912

07682

07447

07255

119880119861119864lowast

51281

11551

11117

11769

12173

12615

12887

13227

1360

913

899

14 Advances in Meteorology

Table10A

ccuracymeasure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f90

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus00117

minus194226

minus193109

minus192926

minus192542

minus191809

minus191151

minus1904

48minus18

9497

minus188790

119877(119865)

00787

00556

00823

01850

03151

05459

07840

10989

16793

22883

119877119877 (119865)

17329

00814

00544

00832

01031

01238

01381

01525

01727

01892

119871119861119864lowast

minus19303

08935

09287

08885

08587

08246

08006

07772

07463

07229

119880119861119864lowast

15717

11499

10973

11461

11804

12156

12393

12635

12966

13242

L-mom

ents

GNO

119861119877 (119865)

15851

minus193606

minus192517

minus192695

minus192747

minus192678

minus192569

minus192437

minus192245

minus192098

119877(119865)

00917

00567

00846

01967

03305

05446

07392

09660

13223

1640

0119877119877 (119865)

19328

00850

00543

00853

01052

01237

01347

01442

01554

01632

119871119861119864lowast

minus23268

08848

09261

08844

08568

08301

08139

08000

07833

07722

119880119861119864lowast

15168

11542

10942

11489

11851

12197

1240

112

575

12779

12925

L-mom

ents

GPA

119861119877 (119865)

49081

minus192995

minus192417

minus192865

minus193078

minus193175

minus193195

minus193209

minus193256

minus193330

119877(119865)

01234

00571

00935

02081

03365

05322

07054

09053

12180

14975

119877119877 (119865)

246

8800882

00576

00870

01054

01239

01364

01489

01660

01798

119871119861119864lowast

minus45353

08783

09210

08838

08594

08326

08141

07961

07731

07557

119880119861119864lowast

14833

11574

10996

11532

11870

12223

1246

712

711

1304

413

312

PL-m

oments

GEV

119861119877 (119865)

minus00877

minus194145

minus193060

minus193062

minus192847

minus192278

minus191698

minus191038

minus190100

minus189374

119877(119865)

05458

00565

00855

01963

03338

05712

08090

11156

16650

22276

119877119877 (119865)

39079

00784

00585

00934

0117

201406

01558

01703

01897

02050

119871119861119864lowast

minus10190

08983

09236

08765

08431

08067

07818

07579

07271

07041

119880119861119864lowast

58730

1144

811036

11652

12081

12504

12771

13026

13364

13623

PL-m

oments

GNO

119861119877 (119865)

38414

minus193665

minus192536

minus192859

minus193044

minus193102

minus193057

minus192966

minus192804

minus192665

119877(119865)

06308

00578

00896

02088

03488

05690

07659

09926

13434

16524

119877119877 (119865)

44651

00819

00596

00961

0119

301405

01526

01628

01744

01822

119871119861119864lowast

minus13011

08910

09192

08716

08411

08122

07949

07801

07634

07519

119880119861119864lowast

56756

11491

11030

11693

12128

12539

12775

12974

13197

13347

PL-m

oments

GPA

119861119877 (119865)

126398

minus193142

minus192484

minus193077

minus193404

minus193589

minus193637

minus193655

minus193680

minus193720

119877(119865)

08641

00588

00999

02220

03558

05535

07227

09125

1200

914

521

119877119877 (119865)

60116

00855

006

4000985

0119

901400

01529

01651

01812

01937

119871119861119864lowast

minus19889

08842

09130

08700

08435

08156

07970

07789

07566

07399

119880119861119864lowast

53743

11528

11103

11755

12158

12552

12807

13055

13385

13636

Advances in Meteorology 15

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

0 2 4 6

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

Regional growth curve for GPA

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F))

0

5

10

15

20

Qua

ntile

2 4 60Reduced variate minuslog (minus log (F))

0

5

10

15

Qua

ntile

(based on L-moments) Regional growth curve for GPA

(based on PL-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30

16 Advances in Meteorology

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60

Advances in Meteorology 17

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90

18 Advances in Meteorology

The first four PL-moments of the GEV are defined as

12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)

(A5)

In previous equation 119875(sdot sdot) is an Incomplete Gammafunction

119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500

120579119896expminus120579119889Θ (A6)

Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution

The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by

119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)

The partial PWMs of the GLO are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)

(A9)

where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function

1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650

0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)

The first four PL-moments of the GLO are defined as

12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520

minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530

minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896

201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

(A11)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution

The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by

119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)

The partial PWMs of the GPA are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int

1

1198650

(1 minus 119865)119896 119865119903119889119865 (A14)

The first four PL-moments of the GPA are defined as

12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)

12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)

(A15)

where

119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution

Advances in Meteorology 19

Ethical Approval

The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975

Conflicts of Interest

The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest

Acknowledgments

The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027

References

[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960

[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985

[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981

[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982

[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988

[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007

[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997

[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990

[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987

[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996

[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990

[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004

[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011

[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009

[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010

[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009

[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012

[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013

[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001

[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002

[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008

[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007

[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010

[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011

[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013

[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974

[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991

[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990

[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979

[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993

[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993

20 Advances in Meteorology

[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993

[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995

[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990

[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993

[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993

[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994

[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995

[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996

[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001

[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009

[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Advances in Meteorology 13

Table9Ac

curacy

measure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f60

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus13271

minus194230

minus193551

minus193241

minus192649

minus191612

minus190736

minus189839

minus188690

minus187888

119877(119865)

00846

00624

00851

01873

03250

05841

08652

12505

19839

27721

119877119877 (119865)

19155

00833

00612

00855

01024

01213

01360

01523

01770

01983

119871119861119864lowast

minus19593

08756

09254

08851

08491

08074

07796

07501

07118

06834

119880119861119864lowast

16242

11675

11015

11472

11879

12309

12678

13038

13659

14141

L-mom

ents

GNO

119861119877 (119865)

08417

minus193335

minus192795

minus192984

minus192999

minus192879

minus192742

minus192593

minus192399

minus192264

119877(119865)

00968

00634

00858

02000

03439

05850

08123

10845

15235

19239

119877119877 (119865)

21055

00881

00582

00861

01050

01235

01354

01463

01599

01699

119871119861119864lowast

minus22887

08647

09244

08794

08466

08141

07957

07798

07597

07443

119880119861119864lowast

15863

11714

1096

11511

11934

1240

512

666

12939

13253

13505

L-mom

ents

GPA

119861119877 (119865)

46371

minus192512

minus192631

minus193160

minus193417

minus193606

minus193745

minus193933

minus194306

minus194708

119877(119865)

01280

00624

00950

02094

03468

05740

07903

10536

14878

18928

119877119877 (119865)

264

4200915

00614

00882

01064

01270

01429

01599

01849

02059

119871119861119864lowast

minus38088

08628

09186

08815

08510

08167

07924

07703

07422

07178

119880119861119864lowast

14699

11721

11020

11534

11925

12454

12791

13236

13869

1440

5

PL-m

oments

GEV

119861119877 (119865)

minus16968

minus193984

minus193601

minus193623

minus193319

minus192576

minus191861

minus191081

minus190026

minus189255

119877(119865)

05431

00596

00888

01978

03366

05844

08417

11838

18179

24868

119877119877 (119865)

38656

00783

00634

0095

01166

01387

01541

01700

01928

02120

119871119861119864lowast

minus92921

08907

09201

08750

08403

07984

07714

07434

07043

06783

119880119861119864lowast

53553

11521

11099

1166

412

098

12572

12869

13216

13697

14105

PL-m

oments

GNO

119861119877 (119865)

23785

minus193306

minus192979

minus193417

minus193628

minus193678

minus193617

minus193509

minus193331

minus193185

119877(119865)

06278

006

0100911

02100

03523

05804

0788

10303

14113

17516

119877119877 (119865)

43743

00812

00623

00965

01186

01391

01513

01618

01742

01830

119871119861119864lowast

minus12083

08856

09188

08709

08389

08061

07870

07706

07517

07380

119880119861119864lowast

52707

11561

11059

11704

12152

12613

12898

13138

13361

1356

PL-m

oments

GPA

119861119877 (119865)

1118

7minus19

2590

minus192910

minus193681

minus1940

84minus19

4322

minus1944

04minus19

4463

minus194560

minus1946

73119877(119865)

08596

00597

01009

02229

03581

05615

07391

09418

12558

15341

119877119877 (119865)

58310

00840

006

6000991

0119

501391

01522

01650

01828

01971

119871119861119864lowast

minus18977

08800

09127

08699

08413

08098

07912

07682

07447

07255

119880119861119864lowast

51281

11551

11117

11769

12173

12615

12887

13227

1360

913

899

14 Advances in Meteorology

Table10A

ccuracymeasure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f90

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus00117

minus194226

minus193109

minus192926

minus192542

minus191809

minus191151

minus1904

48minus18

9497

minus188790

119877(119865)

00787

00556

00823

01850

03151

05459

07840

10989

16793

22883

119877119877 (119865)

17329

00814

00544

00832

01031

01238

01381

01525

01727

01892

119871119861119864lowast

minus19303

08935

09287

08885

08587

08246

08006

07772

07463

07229

119880119861119864lowast

15717

11499

10973

11461

11804

12156

12393

12635

12966

13242

L-mom

ents

GNO

119861119877 (119865)

15851

minus193606

minus192517

minus192695

minus192747

minus192678

minus192569

minus192437

minus192245

minus192098

119877(119865)

00917

00567

00846

01967

03305

05446

07392

09660

13223

1640

0119877119877 (119865)

19328

00850

00543

00853

01052

01237

01347

01442

01554

01632

119871119861119864lowast

minus23268

08848

09261

08844

08568

08301

08139

08000

07833

07722

119880119861119864lowast

15168

11542

10942

11489

11851

12197

1240

112

575

12779

12925

L-mom

ents

GPA

119861119877 (119865)

49081

minus192995

minus192417

minus192865

minus193078

minus193175

minus193195

minus193209

minus193256

minus193330

119877(119865)

01234

00571

00935

02081

03365

05322

07054

09053

12180

14975

119877119877 (119865)

246

8800882

00576

00870

01054

01239

01364

01489

01660

01798

119871119861119864lowast

minus45353

08783

09210

08838

08594

08326

08141

07961

07731

07557

119880119861119864lowast

14833

11574

10996

11532

11870

12223

1246

712

711

1304

413

312

PL-m

oments

GEV

119861119877 (119865)

minus00877

minus194145

minus193060

minus193062

minus192847

minus192278

minus191698

minus191038

minus190100

minus189374

119877(119865)

05458

00565

00855

01963

03338

05712

08090

11156

16650

22276

119877119877 (119865)

39079

00784

00585

00934

0117

201406

01558

01703

01897

02050

119871119861119864lowast

minus10190

08983

09236

08765

08431

08067

07818

07579

07271

07041

119880119861119864lowast

58730

1144

811036

11652

12081

12504

12771

13026

13364

13623

PL-m

oments

GNO

119861119877 (119865)

38414

minus193665

minus192536

minus192859

minus193044

minus193102

minus193057

minus192966

minus192804

minus192665

119877(119865)

06308

00578

00896

02088

03488

05690

07659

09926

13434

16524

119877119877 (119865)

44651

00819

00596

00961

0119

301405

01526

01628

01744

01822

119871119861119864lowast

minus13011

08910

09192

08716

08411

08122

07949

07801

07634

07519

119880119861119864lowast

56756

11491

11030

11693

12128

12539

12775

12974

13197

13347

PL-m

oments

GPA

119861119877 (119865)

126398

minus193142

minus192484

minus193077

minus193404

minus193589

minus193637

minus193655

minus193680

minus193720

119877(119865)

08641

00588

00999

02220

03558

05535

07227

09125

1200

914

521

119877119877 (119865)

60116

00855

006

4000985

0119

901400

01529

01651

01812

01937

119871119861119864lowast

minus19889

08842

09130

08700

08435

08156

07970

07789

07566

07399

119880119861119864lowast

53743

11528

11103

11755

12158

12552

12807

13055

13385

13636

Advances in Meteorology 15

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

0 2 4 6

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

Regional growth curve for GPA

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F))

0

5

10

15

20

Qua

ntile

2 4 60Reduced variate minuslog (minus log (F))

0

5

10

15

Qua

ntile

(based on L-moments) Regional growth curve for GPA

(based on PL-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30

16 Advances in Meteorology

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60

Advances in Meteorology 17

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90

18 Advances in Meteorology

The first four PL-moments of the GEV are defined as

12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)

(A5)

In previous equation 119875(sdot sdot) is an Incomplete Gammafunction

119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500

120579119896expminus120579119889Θ (A6)

Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution

The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by

119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)

The partial PWMs of the GLO are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)

(A9)

where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function

1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650

0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)

The first four PL-moments of the GLO are defined as

12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520

minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530

minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896

201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

(A11)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution

The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by

119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)

The partial PWMs of the GPA are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int

1

1198650

(1 minus 119865)119896 119865119903119889119865 (A14)

The first four PL-moments of the GPA are defined as

12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)

12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)

(A15)

where

119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution

Advances in Meteorology 19

Ethical Approval

The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975

Conflicts of Interest

The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest

Acknowledgments

The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027

References

[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960

[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985

[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981

[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982

[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988

[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007

[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997

[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990

[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987

[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996

[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990

[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004

[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011

[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009

[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010

[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009

[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012

[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013

[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001

[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002

[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008

[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007

[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010

[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011

[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013

[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974

[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991

[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990

[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979

[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993

[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993

20 Advances in Meteorology

[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993

[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995

[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990

[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993

[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993

[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994

[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995

[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996

[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001

[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009

[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

14 Advances in Meteorology

Table10A

ccuracymeasure

ofthee

stim

ated

grow

thcurves

ofther

egionforsam

ples

izeo

f90

Metho

dDistrib

ution

11986501

05

08

09

095

098

099

0995

0998

0999

L-mom

ents

GEV

119861119877 (119865)

minus00117

minus194226

minus193109

minus192926

minus192542

minus191809

minus191151

minus1904

48minus18

9497

minus188790

119877(119865)

00787

00556

00823

01850

03151

05459

07840

10989

16793

22883

119877119877 (119865)

17329

00814

00544

00832

01031

01238

01381

01525

01727

01892

119871119861119864lowast

minus19303

08935

09287

08885

08587

08246

08006

07772

07463

07229

119880119861119864lowast

15717

11499

10973

11461

11804

12156

12393

12635

12966

13242

L-mom

ents

GNO

119861119877 (119865)

15851

minus193606

minus192517

minus192695

minus192747

minus192678

minus192569

minus192437

minus192245

minus192098

119877(119865)

00917

00567

00846

01967

03305

05446

07392

09660

13223

1640

0119877119877 (119865)

19328

00850

00543

00853

01052

01237

01347

01442

01554

01632

119871119861119864lowast

minus23268

08848

09261

08844

08568

08301

08139

08000

07833

07722

119880119861119864lowast

15168

11542

10942

11489

11851

12197

1240

112

575

12779

12925

L-mom

ents

GPA

119861119877 (119865)

49081

minus192995

minus192417

minus192865

minus193078

minus193175

minus193195

minus193209

minus193256

minus193330

119877(119865)

01234

00571

00935

02081

03365

05322

07054

09053

12180

14975

119877119877 (119865)

246

8800882

00576

00870

01054

01239

01364

01489

01660

01798

119871119861119864lowast

minus45353

08783

09210

08838

08594

08326

08141

07961

07731

07557

119880119861119864lowast

14833

11574

10996

11532

11870

12223

1246

712

711

1304

413

312

PL-m

oments

GEV

119861119877 (119865)

minus00877

minus194145

minus193060

minus193062

minus192847

minus192278

minus191698

minus191038

minus190100

minus189374

119877(119865)

05458

00565

00855

01963

03338

05712

08090

11156

16650

22276

119877119877 (119865)

39079

00784

00585

00934

0117

201406

01558

01703

01897

02050

119871119861119864lowast

minus10190

08983

09236

08765

08431

08067

07818

07579

07271

07041

119880119861119864lowast

58730

1144

811036

11652

12081

12504

12771

13026

13364

13623

PL-m

oments

GNO

119861119877 (119865)

38414

minus193665

minus192536

minus192859

minus193044

minus193102

minus193057

minus192966

minus192804

minus192665

119877(119865)

06308

00578

00896

02088

03488

05690

07659

09926

13434

16524

119877119877 (119865)

44651

00819

00596

00961

0119

301405

01526

01628

01744

01822

119871119861119864lowast

minus13011

08910

09192

08716

08411

08122

07949

07801

07634

07519

119880119861119864lowast

56756

11491

11030

11693

12128

12539

12775

12974

13197

13347

PL-m

oments

GPA

119861119877 (119865)

126398

minus193142

minus192484

minus193077

minus193404

minus193589

minus193637

minus193655

minus193680

minus193720

119877(119865)

08641

00588

00999

02220

03558

05535

07227

09125

1200

914

521

119877119877 (119865)

60116

00855

006

4000985

0119

901400

01529

01651

01812

01937

119871119861119864lowast

minus19889

08842

09130

08700

08435

08156

07970

07789

07566

07399

119880119861119864lowast

53743

11528

11103

11755

12158

12552

12807

13055

13385

13636

Advances in Meteorology 15

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

0 2 4 6

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

Regional growth curve for GPA

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F))

0

5

10

15

20

Qua

ntile

2 4 60Reduced variate minuslog (minus log (F))

0

5

10

15

Qua

ntile

(based on L-moments) Regional growth curve for GPA

(based on PL-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30

16 Advances in Meteorology

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60

Advances in Meteorology 17

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90

18 Advances in Meteorology

The first four PL-moments of the GEV are defined as

12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)

(A5)

In previous equation 119875(sdot sdot) is an Incomplete Gammafunction

119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500

120579119896expminus120579119889Θ (A6)

Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution

The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by

119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)

The partial PWMs of the GLO are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)

(A9)

where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function

1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650

0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)

The first four PL-moments of the GLO are defined as

12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520

minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530

minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896

201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

(A11)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution

The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by

119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)

The partial PWMs of the GPA are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int

1

1198650

(1 minus 119865)119896 119865119903119889119865 (A14)

The first four PL-moments of the GPA are defined as

12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)

12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)

(A15)

where

119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution

Advances in Meteorology 19

Ethical Approval

The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975

Conflicts of Interest

The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest

Acknowledgments

The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027

References

[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960

[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985

[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981

[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982

[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988

[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007

[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997

[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990

[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987

[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996

[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990

[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004

[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011

[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009

[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010

[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009

[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012

[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013

[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001

[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002

[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008

[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007

[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010

[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011

[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013

[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974

[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991

[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990

[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979

[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993

[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993

20 Advances in Meteorology

[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993

[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995

[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990

[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993

[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993

[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994

[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995

[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996

[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001

[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009

[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

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Applied ampEnvironmentalSoil Science

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Mining

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Journal of

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International Journal of

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GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

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MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Advances in Meteorology 15

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

0 2 4 6

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

Regional growth curve for GPA

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F))

0

5

10

15

20

Qua

ntile

2 4 60Reduced variate minuslog (minus log (F))

0

5

10

15

Qua

ntile

(based on L-moments) Regional growth curve for GPA

(based on PL-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30

16 Advances in Meteorology

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60

Advances in Meteorology 17

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90

18 Advances in Meteorology

The first four PL-moments of the GEV are defined as

12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)

(A5)

In previous equation 119875(sdot sdot) is an Incomplete Gammafunction

119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500

120579119896expminus120579119889Θ (A6)

Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution

The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by

119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)

The partial PWMs of the GLO are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)

(A9)

where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function

1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650

0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)

The first four PL-moments of the GLO are defined as

12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520

minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530

minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896

201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

(A11)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution

The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by

119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)

The partial PWMs of the GPA are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int

1

1198650

(1 minus 119865)119896 119865119903119889119865 (A14)

The first four PL-moments of the GPA are defined as

12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)

12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)

(A15)

where

119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution

Advances in Meteorology 19

Ethical Approval

The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975

Conflicts of Interest

The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest

Acknowledgments

The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027

References

[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960

[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985

[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981

[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982

[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988

[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007

[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997

[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990

[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987

[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996

[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990

[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004

[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011

[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009

[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010

[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009

[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012

[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013

[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001

[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002

[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008

[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007

[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010

[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011

[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013

[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974

[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991

[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990

[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979

[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993

[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993

20 Advances in Meteorology

[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993

[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995

[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990

[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993

[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993

[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994

[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995

[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996

[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001

[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009

[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

16 Advances in Meteorology

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60

Advances in Meteorology 17

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90

18 Advances in Meteorology

The first four PL-moments of the GEV are defined as

12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)

(A5)

In previous equation 119875(sdot sdot) is an Incomplete Gammafunction

119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500

120579119896expminus120579119889Θ (A6)

Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution

The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by

119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)

The partial PWMs of the GLO are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)

(A9)

where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function

1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650

0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)

The first four PL-moments of the GLO are defined as

12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520

minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530

minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896

201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

(A11)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution

The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by

119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)

The partial PWMs of the GPA are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int

1

1198650

(1 minus 119865)119896 119865119903119889119865 (A14)

The first four PL-moments of the GPA are defined as

12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)

12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)

(A15)

where

119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution

Advances in Meteorology 19

Ethical Approval

The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975

Conflicts of Interest

The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest

Acknowledgments

The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027

References

[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960

[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985

[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981

[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982

[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988

[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007

[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997

[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990

[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987

[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996

[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990

[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004

[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011

[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009

[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010

[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009

[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012

[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013

[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001

[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002

[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008

[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007

[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010

[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011

[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013

[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974

[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991

[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990

[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979

[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993

[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993

20 Advances in Meteorology

[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993

[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995

[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990

[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993

[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993

[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994

[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995

[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996

[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001

[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009

[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Advances in Meteorology 17

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

5

10

15

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12

Qua

ntile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

12Q

uant

ile

2 5 10 20 50 100 500

Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500Return period

0 2 4 6

0

2

4

6

8

10

Qua

ntile

2 5 10 20 50 100 500

Return period

Regional growth curve for GPA(based on L-moments)

Regional growth curve for GNO(based on L-moments)

Regional growth curve for GEV(based on L-moments)

Regional growth curve for GPA(based on PL-moments)

Regional growth curve for GNO(based on PL-moments)

Regional growth curve for GEV(based on PL-moments)

GEVGEV lower boundGEV upper bound

GEVGEV lower boundGEV upper bound

GNOGNO lower boundGNO upper bound

GNOGNO lower boundGNO upper bound

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

GPAGPA lower boundGPA upper bound

Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))

Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90

18 Advances in Meteorology

The first four PL-moments of the GEV are defined as

12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)

(A5)

In previous equation 119875(sdot sdot) is an Incomplete Gammafunction

119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500

120579119896expminus120579119889Θ (A6)

Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution

The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by

119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)

The partial PWMs of the GLO are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)

(A9)

where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function

1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650

0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)

The first four PL-moments of the GLO are defined as

12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520

minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530

minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896

201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

(A11)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution

The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by

119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)

The partial PWMs of the GPA are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int

1

1198650

(1 minus 119865)119896 119865119903119889119865 (A14)

The first four PL-moments of the GPA are defined as

12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)

12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)

(A15)

where

119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution

Advances in Meteorology 19

Ethical Approval

The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975

Conflicts of Interest

The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest

Acknowledgments

The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027

References

[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960

[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985

[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981

[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982

[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988

[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007

[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997

[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990

[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987

[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996

[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990

[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004

[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011

[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009

[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010

[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009

[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012

[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013

[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001

[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002

[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008

[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007

[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010

[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011

[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013

[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974

[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991

[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990

[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979

[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993

[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993

20 Advances in Meteorology

[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993

[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995

[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990

[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993

[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993

[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994

[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995

[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996

[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001

[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009

[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

18 Advances in Meteorology

The first four PL-moments of the GEV are defined as

12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)

(A5)

In previous equation 119875(sdot sdot) is an Incomplete Gammafunction

119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500

120579119896expminus120579119889Θ (A6)

Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution

The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by

119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)

The partial PWMs of the GLO are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)

(A9)

where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function

1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650

0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)

The first four PL-moments of the GLO are defined as

12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520

minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530

minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896

201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650

(A11)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution

The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by

119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)

and quantile function

119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)

The partial PWMs of the GPA are developed as follows

(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int

1

1198650

(1 minus 119865)119896 119865119903119889119865 (A14)

The first four PL-moments of the GPA are defined as

12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)

12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)

(A15)

where

119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)

Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution

Advances in Meteorology 19

Ethical Approval

The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975

Conflicts of Interest

The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest

Acknowledgments

The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027

References

[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960

[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985

[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981

[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982

[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988

[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007

[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997

[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990

[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987

[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996

[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990

[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004

[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011

[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009

[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010

[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009

[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012

[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013

[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001

[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002

[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008

[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007

[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010

[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011

[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013

[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974

[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991

[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990

[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979

[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993

[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993

20 Advances in Meteorology

[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993

[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995

[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990

[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993

[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993

[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994

[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995

[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996

[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001

[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009

[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Advances in Meteorology 19

Ethical Approval

The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975

Conflicts of Interest

The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest

Acknowledgments

The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027

References

[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960

[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985

[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981

[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982

[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988

[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007

[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997

[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990

[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987

[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996

[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990

[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004

[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011

[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009

[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010

[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009

[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012

[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013

[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001

[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002

[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008

[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007

[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010

[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011

[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013

[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974

[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991

[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990

[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979

[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993

[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993

20 Advances in Meteorology

[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993

[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995

[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990

[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993

[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993

[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994

[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995

[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996

[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001

[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009

[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

20 Advances in Meteorology

[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993

[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995

[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990

[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993

[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993

[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994

[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995

[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996

[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001

[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009

[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in