Evaluation of various distributions for flood frequency analysis · Evaluation of various distributions for flood frequency analysis TEFARUK HAKTANIR Department of Civil Engineering,

Bydrological Sciences -Journal- des Sciences Hydrologiques,3S,l,2/1993 15

Evaluation of various distributions for flood frequency analysis

TEFARUK HAKTANIR Department of Civil Engineering, Cukurova University, 01330 Adana, Turkey

HANS B. HORLACHER Institute fur Wasserbau, Universitât Stuttgart, Ffaffenwaldring 61, 7000, Stuttgart 80, Germany

Abstract A statistical model comprising nine different probability distributions used especially for flood frequency analysis was applied to annual flood peak series with at least 30 observations for 11 unregulated streams in the Rhine Basin in Germany and two streams in Scotland. The parameters of most of those distributions were estimated by the methods of maximumlikelihoodandprobability-weightedmoments. The distributions were first compared by classical goodness-of-fit tests on the observed series. Next, the goodness of predictions of the extreme right-tail events by all the models were evaluated through detailed analyses of long synthetically generated series. The general extreme value and 3-para-meter lognormal distributions were found to predict the rare floods of return periods of 100 years or more better than the other distributions used. The general extreme value type 2 and log-Pearson type 3 (when skewness is positive) would usually yield slightly conservative peaks. The Wakeby distribution also gave peaks mostly on the conservative side. The log-logistic distribution with the method of maximum likelihood was found to overestimate greatly high return period floods.

Evaluation de diverses distributions pour l'analyse des fréquences des crues Résumé Un modèle statistique comportant neuf lois de distribution de probabilité, spécialement utilisé pour l'analyse des fréquences des crues est appliqué aux séries de débits maximaux annuels pour 31 stations sur 11 rivières non-régularisées du bassin du Rhin, Allemagne, et deux rivières en Ecosse. La plupart des paramètres des distributions sont estimés par la méthode (a) du maximum vraisemblance et (b) de probabilité des moments pondérés. Les distributions sont d'abord comparées par le test classique d'ajustement sur les séries observées. Puis, la qualité de la prédétermination par ces modèles des événements vers l'extrémité droite de la distribution est évaluée par l'analyse détaillée sur des longues séries générées par synthèse. On a constaté que la loi de distribution des valeurs extrêmes et la loi log-normale à trois paramètres donnaient pour les crues de période de retour de 100 ans une prédétermination meilleure que les autres lois de distribution. La loi générale des valeurs extrêmes de type 2 et la loi log-Pearson III (lorsque l'asymétrie est positive) conduiraient habituellement à des estimations légèrement surestimées des débits de pointe. La distribution de Wakeby a donnée également des pointes pour la plupart du côté de la sécurité. La distribution log-logistique avec la méthode du maximum de vraisemblance surestimait largement les débits de crues de longues périodes de retour.

Open for discussion until I August 1993

16 Tefaruk Haktanir & Hans B. Horlacher

INTRODUCTION

Because many probability distributions (PDs) and parameter estimation methods have been proposed, especially in the last twenty years, the question of best fit has always been of concern. Many studies of this issue have been reported (Ahmad et al., 1988; Arora & Singh, 1989; Cunnane, 1989; Hosking et al., 1985; Jain & Singh, 1987; Phien, 1987; USGS, 1982; Rao & Arora, 1987; Wallis & Wood, 1985).

This paper presents another study of comparisons of PDs and parameter estimation methods. A single computer package has been compiled incorporating the PDs of the 2-parameter lognormal (LN2), 3-parameter lognormal (LN3), Gumbel, general extreme value (GEV), Pearson 3 (P3), log-Pearson 3 (LP3), log-logistic (LL) and Wakeby distributions and the transformations for normalization of the two-step-power transformation (TSPT) and the LN3 with zero skewness (LN3-CSX = 0). The parameter estimation methods of maximum likelihood (ML) and probability-weighted moments (PWM) were applied to most of those PDs. Methods of mixed moments (MM1) and entropy were also employed for the LP3 distribution. The location parameter of the LN3 PD was computed so that the reduced variate, x (where x = ln(Q - c); Q = annual peak, c = location parameter), had a skewness coefficient equalling zero exactly (CSX = 0). Considering the same PD with a different parameter estimation method as a "different model", a total of 19 different models resulted.

Three classical goodness-of-fit (GOF) tests, viz. chi-squared, Kolmogorov-Smirnov (K-S) and Cramer von Mises (CvM), were also included in the package. The GOF evaluations of all 19 models were made first by straightforward assessments of the tests on observed annual flood peak series of 11 natural streams in Germany and two in Scotland with record lengths greater than 30 years. Next, synthetic series of 100 000 element lengths were generated for six of the 13 streams, with two different base distributions (GEV and Wakeby). Further, right-tail events (QT) of 100, 1000 and 10 000 year return periods (7) from the base model were compared with those of the 19 models computed from one thousand synthetic series with 30 and 100 elements and from two hundred similar series with 500 elements. Box plots of the relative errors and of the square errors were investigated.

SUMMARY OF PROBABILITY DISTRIBUTIONS USED

Many references are available containing detailed information about the PDs used in flood frequency analysis (FFA) and their parameter estimation methods, e.g. Abramowitz & Stegun, 1972; Ahmad et al, 1988; Arora & Singh, 1989; Benjamin & Cornel, 1972; Cohen & Whitten, 1980; Cunnane, 1987, 1989; Greenwood et al., 1979; Gumbel, 1958; Gupta et al., 1987; Hosking et al., 1985; Jing et al., 1989; Kappenman, 1985; Landwehr et al, 1979a,b; NERC,

Evaluation of distributions for flood frequency analysis 17

1975; USGS, 1982; and many others. Some of those reference are reviewed by Haktanir (1991).

Computation of the parameters of the GEV and especially of the log-logistic PDs by the method of ML is the most difficult and time consuming of all. Gumbel parameters by the ML and entropy methods (e.g. Phien, 1987) require the solution of two simultaneous nonlinear equations which always yield convergent paths with the straightforward Newton algorithm without too much computational load. In this study, the values given by the PWM method were taken as the initial estimates for the parameters in the ML method, which in turn were used as the initial estimates for the entropy method.

Computation of the parameters of the P3 and LP3 PDs by the ML method was not so cumbersome as that of the LL PD, and was performed by a sound numerical algorithm summarized by Haktanir (1991). Solution by the MM1 method for the LP3 PD was executed by an algorithm similar to the one suggested by Arora & Singh (1989). Parameters of the LP3 PD were also computed by the entropy method, which required an iterative algorithm similar to the ML method (Arora & Singh, 1989). The entropy method always yields two sets of parameters. Here, the set whose skewness coefficient was of the same sign as that of the ML method was taken and the other set was discarded.

Details of the parameter estimation methods and the analytical expressions of the probability function (cpf) of each PD used in this study will not be repeated here, being available in the relevant literature listed above. For example, computation of the five parameters of the Wakeby PD was performed here in exactly the same way as fully explained in Landwehr et al., (1979b). Other researchers such as Houghton (1978), Kuczera (1982), Kumar & Chander (1987), Rao & Arora (1987) and Cunnane (1989) praised this particular PD on convincing grounds.

Because the histograms of the recorded annual flood peak series of the 13 streams taken here were all skewed to the right, and because three of these histograms exhibited a slight second peak irregularity at the end of the right tail, the possibility of a practical transformation which would result in a variate with a symmetrically shaped histogram was also examined. First, the two-step-power transformation (TSPT) was employed. Again, although not very popular, the TSPT is based on the famous Box-Cox transformation, and was presented in the 1986 Louisiana Symposium (Gupta et al., 1987). In TSPT, the original Q series is transformed to an x variate such that both the skewness and the excess coefficients of the x series (CSX and CEX) are exactly zero. The allegation is that when both those coefficients are equal to zero the normal distribution should be a good fit to the x series. The necessary equations to compute the QT <-> Trelationships by the TSPT are given by Gupta et al. (1987). The two parameters necessary for the transformation are computed as the solution of the two equations CSX = 0 and CEX = 0, which must be performed by an iterative numerical algorithm.

Along with the ML method, two other methods are also employed for the LN3 PD here: one is the method by Kappenman (1985) and the other is a


method which determines the location parameter such that the skewness coefficient of the reduced variate, x, (x = ln(Q - c)) is exactly equal to zero. This last method was not used before, and it was hoped herein that the LN3 PD so determined should be a better fit, because it would mean that the CS of the x variate which obeys the standard 2-parameter normal distribution is zero as is the CS of the model. Therefore, the LN3-CSX = 0 model is considered here as another transformation for normalization.

ACQUISITION OF ANNUAL FLOOD PEAKS DATA

Eleven streams in the Rhine Basin in Southern Germany with record lengths greater than 30 years were picked (Landesanstalt, 1900-1987). They are all free from regulation by any sizeable structure. Some are in the Schwarzwald (Black Forest) region with an appreciable average annual precipitation (1000-2000 mm year"1). For the purpose of comparison, two other streams in Scotland, whose annual flood peak series are taken from Ahmad et al. (1988), were also included. Details of all 13 streams are given in Table 1. The original values of the observed records for five of the streams are given in Table 2.

Table 1 Some characteristics of the stream gauging stations considered Number

1

2

3

4

5

6

7

8

9

10

11

12

13

Identification

Giessenbriicke on Argen

Wehr on Wehra

St Roman on Kinzig

Oberlauchringen on Wutach

Schwaibach on Kinzig

Lohrbriicke on Schussen

Zell on Wiese

Oberuhldingen on S.Aach

Biihl on Biihlot

Gutach on Elz

Ebnet on Dreisam

Spey at Kinrara

Kelvin at Killermont

Drainage Average area Qp

(km2)

646.1

60.1

289.6

617.0

957.0

792.0

209.0

276.0

33.3

303.0

257.3

(m3 s-')

241.5

32.1

86.0

94.0

261.3

87.7

91.0

29.4

14.8

94.5

59.0

145.3

81.6

Coefficient of vanation

0.41

0.45

0.31

0.47

0.45

0.34

0.55

0.39

0.89

0.44

0.43

0.46

0.20

Coefficient of skewness

0.86

0.84

0.24

0.86

0.67

0.39

1.78

0.70

3.48

0.91

0.79

2.01

0.80

Record length

(years)

64

58

31

61

60

46

57

61

42

45

42

31

35

HISTOGRAMS OF ORIGINAL AND TRANSFORMED SERBES

First, to have a visual opinion of the distributions of the recorded series, the histograms of all 13 stations were drawn. The histograms for Giessenbriicke, Wehr and St Roman had a slight second peak at the end of the right tail. The histograms of all the other 10 series, however, revealed the usual skewed-to-


Table 2 Observed annual flood peaks series for five of the streams used (in nr s'1) 120023400, Giessenbrûcke, Argen, Rhein Basin

551.0 318.0 275.0 237.7 191.0 165.0 116.0

520.0 316.0 274.1 236.0 188.0 158.0 116.0

491.0 315.0 273.8 235.0 183.0 153.8 88.7

377.0 312.2 267.0 234.0 181.6 146.4 55.0

365.0 309.1 265.3 228.3 180.3 133.0

120003740, Wehr, Wehra, Rhein Basin

75.8 42.7 36.3 29.7 23.3 16.2

68.3 42.0 35.0 29.3 23.0 16.2

66.1 41.4 34.8 28.9 21.7 15.3

54.0 41.0 34.1 27.6 21.7 12.7

51.1 40.5 33.6 25.4 21.4 11.9

352.0 309.0 255.0 227.0 180.3 124.0

48.5 40.5 32.8 25.1 21.1 11.5

120003890, St. Roman, Kinzig, Rhein Basin

142.0 93.1 74.7 33.1

136.8 92.2 72.1

133.6 88.9 71.7

114.1 88.0 66.7

110.0 86.0 65.0

108.1 84.0 60.2

20003570, Oberlauchringen, Wutach, Rhein Basin

228.0 136.0 103.5 89.3 72.6 51.9 26.4

190.0 128.0 101.0 88.0 71.0 49.3

185.0 128.0 100.0 87.0 67.3 48.4

183.0 114.0 99.1 86.4 65.1 43.0

173.3 107.6 99.0 85.4 64.6 42.6

173.3 107.2 95.7 84.0 63.0 40.0

Kelvin at Killermont, Scotland (Data in Ahmad et

128.2 90.1 77.0 65.1

114.9 87.2 76.4 64.5

107.1 86.5 76.4 60.4

105.6 80.6 73.6 57.3

98.3 80.6 72.3 53.9

98.3 80.6 69.6

351.0 298.0 253.0 222.0 176.0 122.0

48.3 39.6 32.6 24.4 20.2 10.9

105.0 82.6 57.6

158.4 107.0 95.2 80.0 58.8 39.1

348.0 296.0 251.0 219.0 175.1 119.0

47.4 38.6 32.3 24.4 19.0 10.7

103.0 82.3 51.3

154.0 106.0 95.0 78.8 54.7 38.9

al., 1988)

94.1 79.9 68.5

91.8 78.7 68.3

336.0 293.0 247.5 214.0 174.0 119.0

47.4 37.8 31.5 24.0 18.5

103.0 81.8 51.0

139.0 105.3 94.0 77.8 53.0 32.1

91.0 77.5 68.3

320.1 277.2 244.9 205.0 172.0 119.0

45.3 36.8 29.7 23.4 17.6

99.8 81.8 45.8

137.0 104.0 91.2 75.5 51.9 28.0

91.0 77.0 65.8

the-right type as is common with most streams of the world. In Figs 1, 2, 3 and 4, the histograms of the recorded and TSP and LN3-CSX = 0 transformed series are shown for Giessenbrûcke, Wehr, Oberlauchringhen and Kelvin.

Histograms of the TSP and LN3-CS,. = 0 transformed series of some of the stations did not really form ideal symmetrical shapes (Figs 1 and 2). Therefore, for some series, the TPST created a new series whose skewness coefficient was zero but it still did not yeild a symmetrically shaped histogram. In other words, the transformation was producing a set of numbers whose CS was made numerically equal to zero without really rendering them symmetrically

ha - i = ;

Fig. 1 Giessenbrûcke: histograms of (a) recorded series; (b) TSPT; and (c) LN3-CSX transformed series.


£X S] Fig. 2 Wehr: histograms of (a) recorded series; (b) TSPT; and (c) LN3-CSX transformed series.

distributed, a numerical achievement only. Therefore, confidence in TSPT was not good at this stage. Further the LN3-CS; = 0 was no better than the TSPT. The TSP and LN3-C5-,. = 0 transformed histograms for some of the other ten series, however, did reveal symmetrically unimodal shapes (as in Fig. 4). Analyses of goodness of fit of both those transformations by conventional GOF tests and by the right-tail QT comparisons on synthetic series were still carried out, as will be summarized later.

In order to assess the significance of this slight perturbation in the right tail of the observed histograms, the 100, 1000 and 10 000 year return period peaks were computed from a double peaked distribution, which was assumed to be the superposition of two Gumbel PDs as defined below, and as suggested by Heinhold & Gaede (1972, pp. 146-150):

Probability of non-exceedence = (1-1/7) = clFl(Q7) + c2F2(QJ) (1)

where c1 and c2 are weights which add up to 1. For example, in the case of Giessenbriicke, c1 = 61/64 and c2 = 3/64 since Fl{Q^ is a Gumbel cpf determined from the 61 points on the left, and F2(QT) is another Gumbel cpf fitted to those three extreme observations in the last slice (Fig. 1). Later, the 100, 1000 and 10 000 year QT s were also computed from the single Gumbel PD taking into account all the 64 points. The parameters of all the Gumbel PDs were computed by the method of PWM. In Table 3, the 100, 1000 and 10 000 year QT s from mixed and single PDs are given. The differences in QT s are

Fig. 3 Oherlauchringen: histograms of (a) recorded series; (b) TSPT; and (c) LN3-CSX transformed series.


Fig. 4 Kelvin: histograms of (a) recorded series; (b) TSPT; and (c) LN3-CSX transformed series.

about 10% or less. Because the second Gumbel PD was estimated from three elements only, that much difference should not be considered too high. Besides, the histograms of many other streams in the same region having exhibited positively skewed shapes further supported the assumption that this perturbation is negligible, and is there only because of sampling variation.

Table 3(a) Parameters of the Gumbel PDs fitted (i) to the entire single series; (ii) to the first part; and (Hi) to the second part of the observed record for three streams in the Rhine basin Series

Original single

First part

Second part

Stations and parameters

Giessenbriicke

Scale Location

0.012716 196.10

0.015255 189.92

0.010599 466.21

Wehr

Scale

0.086585

0.10358

0.075696

Location

25.417

24.44

62.411

St. Roman

Scale Location

0.045039 73.164

0.055108 69.989

0.045294 118.88

Table 3(b) 100, 1000 and 10 000 year peaks computed by the single Gumbel PD vs those computed by mixed PD from two Gumbel PDs for the three streams Stations

Giessenbriicke

Wehr

St. Roman

Return

100

Single

557.8

78.5

175.3

periods (7s)

Mixed

616.9

85.5

179.2

and Tyear

1000

Single

739.3

105.2

226.5

peaks

Mixed

833.3

115.5

228.7

10 000

Single

920.4

131.8

277.7

Mixed

1048.2

145.4

278.4

FFA COMPUTATIONS AND CONVENTIONAL GOF TESTS ON OBSERVED SERIES

A computer program executed all the computations leading to the results of the detailed FFA package summarized above. The classical GOF tests, chi squared and K-S, were performed in the same way as outlined in Haktanir (1991). Additionally the Cramer von Mises (CvM) GOF test was also included as explained by Conover (1971, pp. 306-308) and by Dyck (1976, pp. 127-129). The output consists of: (a) parameter values of the PDs taken by the methods considered;


(b) 19 columns of QT < - > Tfor all the resulting 19 models; and (c) a summary of the GOF tests (a copy of the program can be sent to anybody interested).

SYNTHETIC DATA AND QT COMPARISONS

Determination of high return period floods such as those of 100, 500 and 1000 years is clearly important. However, none of the three classical GOF tests can really be sound indicators of the behaviour of a particular PD at the extreme right-tail portion of the PDs since all these GOF tests quantify the degree of match of either the observed histogram to the expected histogram of the model used or the observed cpf estimated by a plotting position formula to the cpf of the model, both within the range of the observed data. It is self-evident that they indicate goodness of fit within the range of the recorded data and hence they cannot be true indicators of the goodness of prediction of the extreme right-tail quantiles such as QTs of 100 year return periods or greater.

As is commonly known, the meteorological factors and various basin characteristics such as forest cover and géomorphologie structure of the main and tributary channels will most probably not remain constant over long periods and so long annual peak sequences are not really statistically stationary. However, for important hydraulic structures whose lifetime risk must be small, a design peak with a very small probability of exceedence may have to be chosen. As such, the concept of a high return period should be viewed as a relation between a rare quantile versus a very small exceedence probability. Moreover, comparison of a peak with a very high return period such as 10 000 years with the magnitude of the probable maximum flood may also be of interest. Therefore the performance of the various PDs in the region of return periods 100 years or greater is of the utmost importance. Here it is relevant to quote Kuczera (1982): "... The hydrologist's interest resides in the extreme tail of the flood distribution, where contending distributions that fit observed data satisfactorily may differ embarrassingly and where estimates of extreme floods are unstable. "

In order to assess the deviations of the right-tail QT s of 100 years or greater among the PDs examined, and to have an opinion of the goodness of any model to predict rare events soundly, analyses based on long synthetic data series were planned. For the moment, accepting the views of Houghton (1978) that: "Most other distributions can be mimicked by the Wakeby but not always vice versa," and by Kuczera (1982) that: "Because of its versatility, the five-parameter Wakeby distribution can credibly be considered a parent flood distribution" , the Wakeby PD was selected as the base PD in the generation of the long synthetic series. The Wakeby PD had indeed a convincing lead over the other 18 PDs in classical GOF tests for the recorded annual flood peak series of the 13 streams taken. To bring versatility, GEV-PWM was also accepted as another base PD for synthetic data. As a result of the GEV PDs having been advocated among many others for classical FFA analysis in the UK by NERC


(1975), and of Burn's (1990) identification of the GEV-PWM PD as an efficient basis for combining extreme flow data, all the analyses on synthetic data were repeated twice, once with synthetic data using the Wakeby PD as the parent, and the other using the GEV-PWM PD as the parent.

The uniform random number generator, RANI, developed by Press et al., (1986) was chosen. Sub-program RANI is based on three linear congruential generators, and is claimed to be one of the best: "... its period for all practical purposes is infinite, and it ought to have no sensible sequential correlations" (Pressera/., 1986). Because of the total number of computations required, only four of the 11 streams in Germany together with the two in Scotland were included in analyses with synthetic data. For any one of the six stations, the values of the parameters of either the Wakeby or the GEV-PWM PDs computed from the recorded sample series were assumed to be those of the parent (base) PD for the synthetic series also (Table 4). The same beginning seed value of -23 (any arbitrary negative integer is required) was assigned for each of the six stations, and thus the same set of uniform random numbers was obtained for each stream. First, RANI gave a random number uniformly distributed in (0,1). Assuming this was the probability of nonexceedence, the value of that particular element of the synthetic series was computed from the known relationship of quantile versus probability for the PD used as the parent distribution, whose parameters were initially given. For each station, a total of 100 000 elements were thus computed. Later, out of each of the long sequences of data, 1000 non-overlapping series either 30 elements or 100 elements long, and finally 200 series 500 elements long, were taken. Now that the parameter values of the parent PD which generated the long synthetic data were known, the 100, 1000 and 10 000 year QTs were initially and easily computed and stored, forming three reference QTs of the parent model (QTp s). A PD which would compute QTs of the same return period from each of many finite-length synthetic segments closest to those parent QTp s would be the "best" PD in the sense of predicting the extreme right-tail event. Therefore, the relative error commonly used for assessing the degree of prediction as defined below was computed:

REjm = (QT -QTp)/QTp (2)

Table 4 Parameter values of the parent distributions for the four streams in Germany and two streams in Scotland for the synthetic data analyses Distribution

GEV

Wakeby

Parameters

Shape Scale Location

m * a b c d

Stations

Buhl on Biihlot

-0.334389 5.3375 9.1455 3.1673 6.2617 2.2148 10.548 0.41057

Gutach on Elz

0.065040 35.220 76.295 26.069 59.970 2.6575 118.31 0.17432

Swaibach on Kinzig 0.044924 98.712 208.49 66.063 125.07 3.70 21301.0 0.004521

i Zell on Wiese -0.129603 32.579 67.425 19.902 36.021 5.6367 237.24 0.14577

Spey at Kinrara -0.311627 33.056 111.69 44.186 42.083 49.90 273.33 0.17966

Kelvin at Killermont

0.027312 14.236 73.769 46.067 21.911 9.3886 3457.6 0.0045323

* symbols as used in Landwehr et al. (1979b).


where REjm is the relative error of the QTjm computed by thejth PD from the rath synthetic series of finite length (three lengths: 30, 100 and 500 were taken), and QTp is the originally computed QT of the parent PD (for Js 100, 1000 and 10 000 years only). So, at the end of a particular run, 1000 REs resulted for any one of the 19 models and for any one of the three extreme Is. Next, the arithmetic average of those 1000 REs, commonly known as "bias" (e.g. Phien, 1987; Wallis & Wood, 1985), together with the boundary values of the middle 50% (middle 500) and the maximum and minimum values of all those ordered 1000 REs were also computed. Thus a summarized distribution of the 1000 REs for each of the 19 PDs suitable for box plots as in Bum (1990) and Moss & Tasker (1991) was obtained. Each particular run, performing all the calculations, first for parameters of the 19 PDs, next for the 3 x 19 REs of each synthetic series, repeating these calculations for all the 1000 non-overlapping series, and finally computing the critical values of the 1000 REs after ordering them, took about 4 h on a 33 Mhz 386+387 PC. The two different parent PDs (Wakeby and GEV), six stations (Table 4) and three different sample series lengths (30, 100, 500) needed 36 such runs in total.

The criteria of goodness were both the smallness in magnitude of the bias (arithmetic average (« median) of 1000 REs) and, equally importantly, the narrowness of the Box plots and of the max-min ranges of all the REs. Eighteen Box plot figures with Wakeby as the parent PD, and 18 other Box plots with GEV as the parent PD were carefully inspected after all had been plotted with the help of a special program. In Figs 5, 6 and 7, the Box plots of the REs of the Spey at Kinrara (station no. 12 in Table 1), for sample series

fflïïffl 1 2 3 -t 5 6 7 9 10 11 12 13 It 15 16 1? 18 19 1 2 3 4 5 6 7 8 8 10 11 12 13 M 15 16 17 18 18

PDs corresponding to the numbers are: 1. LN2-ML 2. Gumbel-ML 3. Gumbel-PWM 4. Gumbel-entropy 5. TSPT 6. log-logistic-ML

(LL-ML) 7. LL-PWM 8. P3-ML 9. P3-PWM

10. LP3-ML 11. LP3-PWM 12. LP3-MM1 13. LP3-entropy 14. LN3-ML 15. LN3-Kappenman 16. LN3-CSX = 0 17. GEV-ML 18.GEV-PWM 19. Wakeby-PWM

Fig. 5 Box plots of the relative errors (REs) of the Spey at Kinrara for sample series length 30, with Wakeby as the parent PD.




10. LP3-ML 11. LP3-PWM 12. LP3-MM1 13. LP3-entropy 14. LN3-ML 15. LN3-Kappenman 16. LN3-CSX = 0 17.GEV-ML 18.GEV-PWM 19. Wakeby-PWM

Fig. 6 Box plots of the relative errors (REs) of the Spey atKinrara for sample series length 100, with Wakeby as the parent PD.

lengths of n = 30, 100 and 500, respectively, with Wakeby as the parent PD are given. In Figs 8 and 9, the same plots with the GEV as the parent PD are presented (excluding n = 500). In Figs 10, 11 and 12 the Box plots for the

10 11 12 13 11 IS 16 17 18 19

PDs corresponding to the numbers are:

10 11 12 13 11 15 16 17 18 19

1 . LN2-ML 2 . Gumbel-ML 3. Gumbel-PWM 4 . Gumbel-entropy 5. TSPT 6. log-logist ic-ML


10. LP3-ML 1 1 . LP3-PWM 12. LP3-MM1 13. LP3-entropy 14. LN3-ML 15. LN3-Kappenman 16. LN3-CSX = 0 17. GEV-ML 18. GEV-PWM 19. Wakeby-PWM

Fig. 7 Box plots of the relative errors (REs) of the Spey at Kinrara for sample series length 500, with Wakeby as the parent PD.


9 10 11 12 13 14 15 16 17 18 19 1 2 3 4 S 6 7 9 10 11 12 13 14 15 16 17 18 19

PDs corresponding to the numbers are: 1.LN2-ML 2. Gumbel-ML 3. Gumbel-PWM 4. Gumbel-entropy 5. TSPT 6. log-logistic-ML


10. LP3-ML 11 . LP3-PWM 12. LP3-MM1 13. LP3-entropy 14. LN3-ML 15. LN3-Kappenman 16. LN3-CSX = 0 17.GEV-ML 18. GEV-PWM 19. Wakeby-PWM

Fig. 8 Box plots of the relative errors (REs) of the Spey at Kinrarafor sample series length 30, with GEV as the parent PD.

10 11 12 13 14 15 16 17 18 19


9 10 11 12 13 14 15 16 17 18 19

1 . LN2-ML 2. Gumbel-ML 3. Gumbel-PWM 4 . Gumbel-entropy 5. TSPT 6. log-logist ic-ML

(LL-ML) 7 . LL-PWM 8. P3-ML 9. P3-PWM

10. LP3-ML 1 1 . LP3-PWM 12. LP3-MM1 13. LP3-entropy 14. LN3-ML 15 . LN3-Kappenman 16 . LN3-CSX = 0 17 . GEV-ML 18. GEV-PWM 19. Wakeby-PWM

Fig. 9 Box plots of the relative errors (REs) of the Spey at Kinrarafor sample series length 100, with GEV as the parent PD.

Zell on Wiese station with Wakeby as the parent PD are also given. Because of space limitation, only those few typical Figures are included here. (All the computer programs developed together with the data used and all the rest of the


1 2 3 1 5 6 7 8 9 10 1 1 1 2 13 1 1 1 3 16 17 18 19 1 2 3 1 5 6 7 8 9 10 11 12 13 11 I S I S 17 18 19

PDs corresponding to the numbers are: LN2-ML Gumbel-ML Gumbel-PWM Gumbel-entropy TSPT log-logistic-ML

(IL-MU LL-PWM P3-ML P3-PWM

10. 11. 12. 13. 14. 15. 16. 17. 18.

LP3-ML LP3-PWM LP3-MM1 LP3-entropy LN3-ML LN3-Kappenman LN3-CSX = 0 GEV-ML GEV-PWM 19. Wakeby-PWM

Fig. 10 Box plots of the relative errors (REs) of the Zell on Wiese for sample series length 30, with Wakeby as the parent PD.

3 10 11 12 13 1-t IE 16 17 18 19

6 7 8 9 10 11 12 13 11 IS 16 17 1

1 . LN2-ML 2. Gumbel-ML 3. Gumbel-PWM 4 . Gumbel-entropy 5. TSPT 6. log-logist ic-ML

(LL-ML) 7 . LL-PWM 8. P3-ML 9. P3-PWM

10 . LP3-ML 1 1 . LP3-PWM 12 . LP3-MM1 13 . LP3-entropy 14 . LN3-ML 15 . LN3-Kappenman 16 . LN3-CS„ = 0 17. GEV-ML 18 . GEV-PWM 19. Wakeby-PWM

Fig. 11 Box plots of the relative errors (REs) of the Zell on Wiese station for sample series length 100, with Wakeby as the parent PD.

Box plot figures can be sent to anybody interested). Together with the REs, square errors (SEs) as defined below were also

computed.


t i i ffiiiiiiiiiiiii L

i ï ï Mtm 1 2 3 4 5 6 7 8 9 10 H 12 13 I t 15 I S 17 18 13 1 2 3 1 5 6 7 3 10 11 12 13 11 15 la 17 18 19


1 2 3 4 S . 5 7 8 9 10 11

1 . LN2-ML 2. Gumbel-ML 3. Gumbel-PWM 4 . Gumbel-entropy 5. TSPT 6. log-logistic-ML

(LL-ML) 7. LL-PWM 8 . P3-ML 9. P3-PWM

10 . LP3-ML 1 1 . LP3-PWM 12 . LP3-MM1 13 . LP3-entropy 14 . LN3-ML 15 . LN3-Kappenman 16 . LN3-CS„ = 0 17 . GEV-ML 18 . GEV-PWM 19. Wakeby-PWM

Fig. 12 Box plots of the relative errors (REs) of the Zell on Wiese station for sample series length 500, with Wakeby as the parent PD.

SEjm = [{QTjm-QTp)IQTpf (3)

The square root of the arithmetic average of the 1000 SEs, the root mean square error (RMSE), is another comparison statistic used (Phien, 1987; Wallis & Wood, 1985). The distribution of the REs and bias are given more consideration in attempting to assess the predictive ability of any PD for the right-tail quantities in this study. However, the RMSE values and the Box plots of the SEs were also always in parallel to the evaluations of the Box plots of the REs. In Fig. 13, the Box plots of the SEs for the Zell on Wiese station with n = 30 and Wakeby as the parent PD are given.

RESULTS AND DISCUSSION

Inspecting the Box plots formed as a result of the analyses outlined above and considering the conventional GOF testing evaluations, the following conclusions can be made. (a) Both with the recorded series of lengths between 31 and 64 elements and with the synthetic series of lengths of 30, 100 and 500 elements, the Wakeby-PWM model scored much better than the other PDs by the conventional GOF tests. The second best of the GOF tests was the TSPT, followed by LL-PWM, GEV-PWM, LL-ML, GEV-ML, LP3-entropy, LP3-ML, P3-PWM and the others. (b) Although the narrowness of the Box plots for the Gumbel PDs was


ï i i i S u u o ù o B o a ê a ù a y f l Q

1 2 3 4 5 6 7

raHiil iiiM

1 2 3 4 5 é 7 S 10 11 12 13 14 IS 16 17 18 19

um Immimli

9 10 11 12 13 14 15 IS 17 18 19 1 2 3 4 E & 7 10 11 12 13 14 15 1« 17 18 19



10. LP3-ML 11. LP3-PWM 12. LP3-MM1 13. LP3-entropy 14. LN3-ML 15. LN3-Kappenman 16. LN3-CSX = 0 17.GEV-ML 18.GEV-PWM 19. Wakeby-PWM

Fig. 13 Box plots of the square errors (SEs)for the Zell on Wiese station for sample series length 30, with Wakeby as the parent PD.

considerably smaller than that of all the other PDs, this particular model underestimated the right-tail events a greater number of times. The right-tail QTs of all the three Gumbel models (Gumbel-ML, Gumbel-PWM, and Gumbel-entropy) were almost always close to each other. They underestimated most of the time when the parent PD was Wakeby. They always underestimated when the parent PD was GEV type 2 (as expected), and they overestimated when the parent PD was GEV type 3 (again as expected). (c) Similarly, the LN2 PD underestimated the right-tail events a greater number of times than it predicted them correctly. (d) The TSPT always had a rather wide distribution of REs, with biases appreciably different from zero. This particular model is also discouragingly difficult to compute. One further disadvantage is that computation of real values for high return period floods is sometimes problematic, due probably to the influence of round-off or cumulative errors for those series with high skewness. Considering all these facts, although it scores fairly well in the classical GOF tests, the TSPT is not recommended. (e) The GEV-PWM model was found to slightly overestimate a greater number of times than underestimating the parent QT s of 100, 1000 and 10 000 year return periods. The bias of the GEV-PWM PD was consistently close to zero whereas those of some other models revealed sharp fluctuations in either the positive or negative direction. Therefore, the GEV-PWM model seems to be a recommendable PD for a conservative design. (f) All the P3 and LP3 models underestimated the high return period parent QTS (QTP

S) a greater number of times than they overestimated, except for the


LP3-PWM. The LP3-PWM model gave QT values of the extreme right tail usually higher than the other LP3s. In general, the greater the adopted skew-ness coefficient of the ln(0 series, the higher the values of the right tail QTs with the LP3 model. An LP3 PD with a highly negative valued C&, such as —0.7 or -0.9 or so, would yield a frequency curve increasing at low rates in the right tail zone. Therefore a conservative-minded design would always avoid an LP3 PD with a large negative CS (possible under-design). (g) Solution for the parameters of the log-logistic (LL) PD by the method of ML was the most difficult and time-consuming of all. In spite of this numerical burden, the LL-ML model largely overestimated the right tail QTs of the parent PD. One final disadvantage was that for some series, especially those with large skewness, it did not even give a convergent solution. Therefore, it is the strong conviction of this study that the LL-ML model is not recommendable at all. Computation of the parameters of the LL model by the PWM method was much easier. However, the right tail QTs from the LL-PWM method still overestimated the parent right tail QTs. The degree of overestimation of the LL-PWM PD, however, was much less than that of the LL-ML PD. The LL-PWM model seems to be acceptable for Ts up to 100 years. (h) The behaviour of the GEV-ML PD was fairly similar to that of the GE V-PWM PD. The likelihood of the GEV-ML model yielding a convergent solution was much higher than that of the LL-ML model. (i) The results of all three conventional GOF tests were considerably consistent among themselves, but the results of those GOF tests were not parallel with the QT comparisons analyses. Therefore the classical GOF tests could not be true guides in deciding on a particular model. Instead of concentrating on the idea of searching for a strict goodness of fit, an analyst should be more aware of the behaviour of the many different PDs in the right tail part of return periods of 100 years or greater. For example, considering the GEV model, if it results in type 3, a conservative-minded analyst should favour the Gumbel PD instead (GEV type 1). Similarly, the LN2 model could be favoured over an LP3 model when the skewness coefficient of the LP3 PD is negative. (j) The LN3 models slightly underestimated the right tail QTs of the parent distribution more than they overestimated. They performed better than the LP3 ones, however. All the LN3 models (ML, Kappenman and CSX = 0) were rather close to each other. Especially, the LN3-CSX = 0 PD gave QTs quite close to those of the LN3-ML. Therefore, the CSX = 0 method does not yield an appreciable improvement for this particular PD. It could be used instead of the ML method when the latter fails to yield a convergent solution. The LN3-Kappenman did not produce a convincing improvement over the other two LN3 models. In any case, computation of the location parameter by this particular procedure is not as easy as is claimed by Kappenman (1985). Comparison of the Box plots of the REs clearly indicated, however, that the LN3 PD is definitely better than the LN2 PD. (k) The Wakeby distribution did not reveal a superiority over the other 18


models in the Box plots. It was one of the better ones, but definitely not the best. Considering the non-negligible numerical burden in obtaining its parameters, even with the method of PWM, usage of GEV-PWM instead may be more feasible for a single-site analysis. Cunnane (1989) suggests that the Wakeby PD is more appropriate to be used as a regional frequency model rather than a single-site PD. (1) For any PD, the numerical effort of the ML method is much greater than that of the PWM method. Further, the PWM models seem to be either as good as, or even a little better than, the ML ones. Moreover, the ML method has a real probability of diverging and thus yielding no solution at all, especially for short and highly skewed series. Therefore, the PWM method should be preferred over the ML method, whatever the type of probability distribution. (m) Because of the ample availability of computers nowadays, a single-site flood frequency analysis should be done with the inclusion of many standard PDs, and a final decision should be made combining experience with engineering judgement. As more and more observed flood peak data become accumulated over the coming years, flood frequency analyses may be performed with regionalized parameters of proven models such as the Wakeby (Cunnane, 1989).

Acknowledgements The suggestions by two reviewers for the improvement of this paper, and the research fellowship provided to the first author by the Av-Humboldt Foundation during this study, are gratefully acknowledged.

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Received 13 December 1991; accepted 25 July 1992

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Evaluation of various distributions for flood frequency analysis · Evaluation of various distributions for flood frequency analysis TEFARUK HAKTANIR Department of Civil Engineering,