Research Article A Bayesian Method for Water Resources ...downloads.hindawi.com/journals/mpe/2015/120873.pdf · Research Article A Bayesian Method for Water Resources Vulnerability

Research ArticleA Bayesian Method for Water ResourcesVulnerability Assessment A Case Study ofthe Zhangjiakou Region North China

Xuan Wang12 Fangbing Ma13 Chunhui Li12 and Jie Zhu12

1State Key Laboratory of Water Environment Simulation School of Environment Beijing Normal University Beijing 100875 China2Key Laboratory of Water and Sediment Sciences of Ministry of Education School of Environment Beijing Normal UniversityBeijing 100875 China3Shidu Town Peoplersquos Government Beijing 102411 China

Correspondence should be addressed to Xuan Wang wangxbnueducn

Received 18 June 2014 Accepted 23 August 2014

Academic Editor Jian Li

Copyright copy 2015 Xuan Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Water resources vulnerability (WRV) assessment is an important basis for maintaining water resources security in a basin In thispaper considering the complexity of the water resources system and the uncertainty of the assessment information a methodbased on the Bayesian theory was developed for performing WRV assessments while using the constructed indicator systemThis system includes four subsystems the hydrological subsystem the socioeconomic subsystem the ecoenvironmental subsystemand the hydraulic engineering subsystem The WRV degree for each subsystem and the integrated water resources system wereassessed Finally the assessment results and the characteristics of the Bayesian method were compared with those of the greyrelational analysis method and the parametric-system method The results showed the following (1) The WRV of the integratedwater resources system of the entire Zhangjiakou region was very high Zhangjiakou City and Xuanhua County have tendenciesto belong to Extreme WRV with probabilities of 268 and 25 respectively while the other seven administrative counties havetendencies to belong to High WRV with probabilities ranging from 246 to 27 (2) Compared with the parametric-systemmethod and the grey relational analysis method the Bayesian method is simple and can effectively address the uncertainty issueswith the reliable WRV assessment results

1 Introduction

Under the influence of climate change and human activ-ities many countries in the world are faced with waterproblems such as water shortages water pollution andfrequent occurrences of extreme hydrological events whichhave put pressure on water availability [1 2] Thus it isof importance to study the recovery capability of a waterresources system after being disturbed Besides reliabilityresilience and sustainability vulnerability is an importantindicator to evaluate the performance of a water resourcessystem [3ndash5] Water resources vulnerability (WRV) can bedefined as the ease with which a water resources systemcan be threatened and damaged by both natural disastersand human activities Once this damage occurs it is difficult

for the system to recover to its former status WRV isgenerally reflected by such factors as the quantity and qualityof the surface water and groundwater and the carryingcapacity of the water resources [6] The purpose of WRVassessment is to understand the water resources status in acertain region to find the main influencing factors of theWRV and then to propose appropriate countermeasures toreduce the WRV In recent years following the continuousdevelopment of economic and social progress the growth ofthe population as well as the development of the urbanizationand industrialization the demand for water resources isincreasing and the supply is tending toward shortages At thesame time excessive production activities lead to a declinein waterrsquos ecological value Having addressed water securityissues that are associated with WRV and provided references

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 120873 16 pageshttpdxdoiorg1011552015120873

2 Mathematical Problems in Engineering

for the protection rational development and utilization ofregional water resources WRV assessment has been drawingmore attention

Since the late 1960s many scholars and water managersaround the world have studied the WRV but most of themconcentrated on groundwater vulnerability assessment [7ndash10] With respect to surface water vulnerability most of theliterature focused on the supply and demand balance ofwater resources or selected several indicators and methodsto evaluate the carrying capacity or sustainability of thewater resources [11] Given the uneven distribution of waterresources in space and time it is necessary to pay closeattention to the WRV that is caused by climate changeand human activities It is also important to enhance thejoint assessment of surface water and groundwater waterquantity and water quality to ensure a sustainable utilizationof water resources of a given region In recent years manyapproaches have been developed to assess WRV such asoverlay and indicator methods [12ndash14] process-based sim-ulation methods [15 16] statistical methods [9 17] andfuzzy comprehensive judgment methods [18] For obtainingmore reliable assessment results these methods required alarge amount of natural environmental and socioeconomicdata with high accuracy In [19] Li quantitatively analyzedthe regional WRV by using the fuzzy optimization (FO)method the grey relational analysis (GRA) method and theparametric-system (PS) method by constructing an indicatorsystem with a comprehensive consideration of surface waterand groundwater water quantity and water quality Thesemethods however cannot address the uncertainty that isinherent in all of the vulnerability assessments [20] With theincrease in the pressure exerted by socioeconomic develop-ment on a natural environment the complexity and uncertaincharacteristics of WRV that arise from both climate changeand human activities has become even more obvious Toimprove the water resources management and protectionthe regional WRV research that considers fuzziness anduncertainty of the assessment results is highly desired

With increasingly in-depth research on uncertain eventsthe Bayesian method has received extensive attention [21]It uses objective probability estimations and incompleteinformation to estimate the probability of some unknownstates This method can also describe uncertain informationflexibly and conduct uncertainty reasoning [22] Comparedwith the existing assessment methods the Bayesian methodrsquossuperiority lies in giving the probability or tendency to acertain grade to which the system belongs not just simplygiving the grade In addition it has the advantages of a simpleprinciple and calculation process and can obtain reliableresults for small samples Thus so far it has been widely usedfor water quality assessment [23ndash25] fault diagnosis and riskassessment [26 27] but it has rarely been used for WRVassessmentThe application of the Bayesian method in actualWRV assessment is expected for the effective management ofregional water resources

The objectives of this research were to (a) develop aBayesian method for WRV assessment (b) apply the pro-posed method to assess the WRV condition of the nine

counties of the Zhangjiakou region and describe the prob-ability distribution characteristics at different counties and(c) compare the results with the existing methods includingthe GRA method and the PS method The proposed methodwill improve the existing method in addressing uncertaininformation and the assessment results can help to identifyspecific water resources problems of the Guanting Reservoirin such a way that appropriate measures can be taken foreffective environmental management

2 Study Area

21 Overview of the Study Area The Zhangjiakou regionwhich is located in the upper reaches of the Guanting Reser-voir Basin is selected as a case to assess the regional WRV(Figure 1) It is confrontedwith the dual task of developing theeconomy and protecting the natural ecological environmentThe concerned city and counties are Zhangjiakou CityHuairsquoan County Wanquan County Xuanhua County YuxianCounty Yangyuan County Zhuolu County Huailai Countyand Chongli County with 178 thousand km2 accountingfor 405 of the area of Guanting Reservoir Basin TheZhangjiakou region has a continental monsoon climate withan annual mean temperature ranging from 7∘C to 10∘CThe annual mean precipitation is rather low (approximately400mm) and 80 of the rainfall occurs between Juneand September The topography in the Zhangjiakou regionis very complex mountains hills and rivers account forapproximately 30of the area respectivelyThepopulation ofthe region is 459 million and the water resources per capitaare 399m3 which is less than the internationally acceptedstandard of extreme water scarcity of 500m3 [28]

The region has a backward economic developmentExcessive consumption and pollution of water resourceshave exerted an enormous pressure on local water resourcessystem and have directly influenced the security of the watersupply of Beijing [29] Therefore an effective WRV assess-ment is necessary and desired for regional water resourcesprotection and management The hydrological data camefrom Zhangjiakou Water Conservancy Bureau the waterquality data came from Environmental Protection Agency ofZhangjiakou and the ecoenvironmental and socioeconomicdata were from the statistical yearbooks of the Zhangjiakouregion [19]

22 Key Factors InfluencingWRV RegionalWRV is reflectedby the carrying capacity recovery capacity and sustainabilityof the water resources [6 30 31] The key factors thatinfluence the WRV of the Zhangjiakou region include thefollowing aspects

(1)Hydrology Influenced by the continentalmonsoon climateof the arid and semiarid area the evaporation and runoff ofthis region have the characteristic of uneven distribution ata spatial and temporal scale The amount of water availableis not enough The influencing factors that are associatedwith these hydrological aspects mainly include the naturalconditions of the regional water resources such as the

Mathematical Problems in Engineering 3

5

7

9

4

6

82

3 1

1

1

1

Guanting Reservoir Basin

Other province

Beijing

Figure 1 Location of the Zhangjiakou regionNorthChina (1 ZhangjiakouCity 2 HuairsquoanCounty 3WanquanCounty 4 XuanhuaCounty5 Yuxian County 6 Yangyuan County 7 Zhuolu County 8 Huailai County 9 Chongli County)

stability of the water resources system the water abundancethe ability to renew the underground water and the regionalclimate conditions

(2) Socioeconomy With economic development and popu-lation increase the water resources per capita (399m3) ofthe Zhangjiakou are much lower than the national average(2100m3) thus the region is suffering from serious watershortage issues At the same time the economy of HebeiProvince is backward as well as its science and technol-ogy Extensive pattern of economic growth leads to a lowutilization rate of the water resources and the increasingdemand for water resources to meet the requirements of bothhuman life and industrial production which has imposedgreat pressure on the water resourcesThe influencing factorsthat are associated with the socioeconomic aspects mainlyinclude the population pressure on the water resources thewater management level the water supply and water demandstatus the water exploitation and utilization conditions andthe water reuse rate

(3) Ecoenvironment The area lies in a fragile belt in termsof the ecoenvironment The agricultural application of pes-ticides is approximately 2000 tons a year and fertilizer isapproximately 60000 tons a year which leads to seriousagricultural nonpoint source pollution of the region Sincethe late 1980s rapid development of industrial and miningenterprises together with low treatment of effluent sewages

has brought severe industrial point source pollution to theregion At the same time because of the low coverage of thevegetation the water losses and soil erosion here are veryserious and the ecological environment is vulnerable Theinfluencing factors that are associated with the ecoenviron-ment mainly include the impact of land exploitation andutilization on water resources potential pollution intensityof agricultural activities on water quality of regional surfacewater and groundwater water loss and soil erosion

(4) Hydraulic Engineering and TechnicalManagement A largenumber of water conservancy systems have been constructedin this region since the late 1950s including water stor-age structures abstraction systems and other water supplyprojects Hydraulic engineering can regulate and allocatewater resources and prevent flood disasters tomeet the needsof the water resources of the people and the productionThusmaking use of hydraulic engineering can enhance the anti-interference ability of the water resources system In additionconsidering that agriculture is the largest supporting industryof this region it is necessary to increase the water-savingirrigation area and improve the water use efficiency throughtechnological advances and consciousness enhancement toreduce its influence on WRV The influencing factors thatare associated with the hydraulic engineering and technicalmanagement mainly include the agricultural water-savingcapability as well as the regulation and storage capacity of thereservoirs


Table 1 Assessment standard of the indicators for different WRV types (Li 2007) [19]

Subsystem Indicator Descriptionreflection NoWRV

LowWRV

MediumWRV

HighWRV

ExtremeWRV

A

a1 Regional water resources income 1500 1200 800 400 100a2 System stability 10 20 30 40 50a3 Water abundance 40 30 20 10 5a4 Ground water recharge capacity 80 50 20 10 5a5 Regional climate 05 1 3 5 7

B

b1 Population pressure on water resources 2500 2000 1500 1000 500

b2 Water utilization efficiency 100 350 600 850 1100

b3 Water utilization level 100 300 500 700 900

b4 Water manage level 5 15 25 35 45

b5 Water supply and demand balance 1 09 08 07 06

b6 Status of water resources development 10 30 50 70 90

b7 Repetitive utilization degree of industrial water 80 65 50 35 20

C

c1

Impacts of land development on waterresources 30 350 65 80 90

c2

Agricultural pollution intensity on waterquality 10 25 280 1000 1500

c3 Surface water quality 0 01 02 05 1c4 Groundwater quality 0 5 35 60 85c5 Area changes of soil and water loss 0 10 20 30 40c6 Degree of sewage disposal 90 75 60 45 30

Dd1 Water saving capacity of agriculture 80 60 40 30 20

d2 Regulating capacity of hydraulic engineering 50 40 30 20 10

d3 Storage and release capacity of reservoirs 80 65 50 40 30

Note A hydrological subsystem B socioeconomic subsystem C ecoenvironmental subsystem D hydraulic engineering subsystem a1 mean annualprecipitation (mm) a2 annual relative variability of precipitation () a3 mean water resources amount of unit area (104m3km2) a4 groundwater rechargeof unit area (104m3km2) a5 drought index b1 water resources per capita (m3capita) b2 water consumption per unit GDP (m3104 yuan) b3 waterconsumption per capita (m3capita) b4 difference between planned and actual water utilization amount () b5 water supply and demand ratio b6 waterresources development and utilization ratio () b7 repetitive utilization ratio of industrial water () c1 land development ratio () c2 fertilizer use intensity(kgha) c3 pollution index of surface water c4 ratio of polluted groundwater exceeding the standard () c5 ratio of soil and water loss () c6 disposal ratioof sewage () d1 water-saving irrigation ratio () d2 ratio of design utilizable storage of reservoir to surface water quantity () d3 ratio of effective storageto total storage of reservoirs ()

The four aspects interact with each other for examplethe increase in the population and water demand and thedecrease in the water resources per capita canmake the watershortage even more serious At the same time the increase inthe pollutant emissions leads to a significant deterioration inthe water environment Over disafforestation farming andhusbandry lead to a low coverage of the vegetation thusaggravating the water losses and soil erosion The disharmo-nious development of the natural resources and environmenthas reduced the carrying capacity the recovery capacity andthe sustainability of the water resources system and increasedthe WRV Therefore it is necessary to consider the abovefour aspects comprehensively to establish an indicator systemthat considers the supply and demand balance of the waterresources and that combines thewater quantitywith thewaterquality to fully reflect the influence of the climate change andhuman activities

3 Methodology

31 Conceptual Model Diagram for WRV Assessment Basedon Bayesian Theory In [19] Li divided WRV into four sub-systems the hydrological subsystem the socioeconomic sub-system the ecoenvironmental subsystem and the hydraulicengineering subsystem and then constructed an indicatorsystem for the WRV assessment The vulnerable degree ofevery indicator was divided into five levels which were NoWRV Low WRV Medium WRV High WRV and ExtremeWRV (see Table 1)

Based on Bayesian theory we calculated the posteriorprobability of every indicator and then we calculated theposterior probability of the multi-indicator comprehensiveWRV assessment of each subsystem For the multilayeredWRVassessment the subsystem in one layer can be seen as anindicator of the next layer then we recalculated the posterior


Hydrological subsystem

Socioeconomic subsystem subsystem

Hydraulic engineering subsystem

Integrated water resources system

Ecoenvironmental

a1 a5 b1 b7 c1 c6 d1 d3middot middot middot middot middot middot middot middot middot middot middot middot

Figure 2 The relationship among the indicators subsystems and integrated system

Weight

Procedures Methods

Posterior probability Bayesian formula

Likelihood Normal distribution sampling method

AHP method

Bayesian formulaAHP method

WRV assessment of each subsystem of the Zhangjiakou region

WRV assessment of the integrated water resources system ofZhangjiakou region

Figure 3 Procedures for WRV assessment by using the Bayesian method

probability of the multisubsystem WRV assessment of theintegrated water resources system with the Bayesian formula(Figure 2)

32 Procedures for the WRV Assessment Model Based onthe Bayesian Theory We used the Bayesian formula [32] tocalculate the probability of a system belonging to a certainWRV type to identify the vulnerable degree of the waterresources system The WRV assessment procedures withthe Bayesian formula are shown in Figure 3 First we usedthe normal distribution sampling method to calculate thelikelihood and used the Bayesian formula to calculate theposterior probability of every indicator Second we adoptedthe analytic hierarchy process (AHP) method to obtain theindicator weights and calculated the posterior probability

of the multi-indicator comprehensive WRV assessment ofeach subsystem Next we used the AHP method again tocalculate theweight of every subsystem andwe calculated theposterior probability of themultisubsystemWRV assessmentof the integrated water resources system with the Bayesianformula The calculation steps that were taken with theBayesian method were as follows

(1) Calculate the prior probability of every indicator thatbelongs to a certain standard type Due to insuffi-cient information about the indicators this researchassumed that the probability of each type was equalSpecifically we had the following

119875 (1199101198951) = 119875 (119910

1198952) = 119875 (119910

1198953) = 119875 (119910

1198954) = 119875 (119910

1198955) =

1

5 (1)


where 119875(119910119895119894)119894=125

is the prior probability of the 119895thindicator that belongs to the 119894th standard type

(2) Calculate the likelihood of every indicator Here weadopted the normal distribution sampling methodto estimate the likelihoods The calculation processesof the normal distribution sampling method were asfollows

(i) Calculate the variation coefficient119862V119895 for the 119895thindicator with (2)

119886119895=

(sum5

119894=1119910119895119894)

5

120590119895=radic

sum5

119894=1(119910119895119894minus 119886119895)2

4

119862V119895 =120590119895

119886119895

(2)

where 119894 is the number of standard types (119894 =

1 2 5) 119895 is the number of indicators of asubsystem (119895 = 1 2 119899) 119910

119895119894is the standard

value of the 119895th indicator of the 119894th standardtype 119886

119895is the mean value of the normal distri-

bution and 120590119895is the standard deviation

(ii) Use 119862V119895 to obtain the variation coefficient of thesampling values by assuming the following

119862V119895119894 = 119862V119895 (3)

where 119862V119895119894 is the variation coefficient of thesampling values of the 119894th type

(iii) Calculate the standard deviation with (4) withtaking 119910

119895119894as the mean value of the 119894th type

120590119895119894= 119862V119895119894 sdot 119910119895119894 (4)

where 120590119895119894is the standard deviation of the 119894th

type(iv) Standardize the sampling value with (5)

119905119895119896

=

119909119895119896

minus 119910119895119894

120590119895119894

(5)

where 119909119895119896is the value of the 119895th indicator of the

119896th subsystem and 119905119895119896is the standardized value

of 119909119895119896

(v) Calculate the likelihood 119875(119909119895119896

| 119910119895119894) of every

indicator with (6)

119875 (119909119895119896

| 119910119895119894) = 2 (1 minus Φ (

10038161003816100381610038161003816119905119895119896

10038161003816100381610038161003816)) (6)

where 119896 is the number of subsystems (119896 =

1 2 4) and Φ(|119905119895119896|) = int

119905119895119896

minusinfin(1radic2120587)119890

minus1199062119889119906

(3) Calculate the posterior probability with (7)

119875 (119910119895119894| 119909119895119896) =

119875 (119910119895119894) 119875 (119909

119895119896| 119910119895119894)

sum5

119894=1119875 (119910119895119894) 119875 (119909

119895119896| 119910119895119894)

(7)

where 119875(119910119895119894| 119909119895119896) is the likelihood of every indicator

that belongs to 119894th type(4) Calculate the comprehensive probability of the multi-

indicator WRV assessment of each subsystem with(8)

119875119894=

119898

sum

119895=1

120596119895119875 (119910119895119894| 119909119895119896) (8)

where 119875119894is the weighted sum of the posterior

probability of the multi-indicator and represents theprobability that WRV of the 119896th subsystem belongsto the 119894th standard type 120596

119895is the weight of the 119895th

indicators and 0 le 120596119895le 1 sum119898

119895=1120596119895= 1 Here we

used the AHP method to determine the weight 120596119895of

every indicator of each subsystem(5) Calculate the posterior probability of the multisub-

system WRV assessment of the integrated waterresources system using (8) again Here 119875

119894is the

weighted sum of the posterior probability of foursubsystems and represents the probability that WRVof the integrated water resources system belongs tothe 119894th standard type 120596


subsystem and 0 le 120596119895

le 1 sum119898119895=1

120596119895

= 1 119898 = 4119896 = 1

33 Other WRV Assessment Methods for Comparison Inorder to verify the effectiveness of the proposed methodand compare the results among different methods we chosethe other two WRV assessment methods for simultaneouscalculation as follows

(1) The GRA Method The GRA method is an impactevaluation model that measures the degree of similarity ordifference between two sequences based on the level ofrelation [33] The calculation steps are as follows

(i) Compose a reference sequence (ie (9)) and anassessment sequence (ie (10))

119909119894(119896) = 119909

119894(1) 119909

119894(2) 119909

119894(119899) (119894 = 1 2 119905) (9)

119910119895(119896) = 119910

119895(1) 119910

119895(2) 119910

119895(119899) (119895 = 1 2 119898)

(10)

where 119899 is the number of parameters119898 is the numberof regions to be assessed 119905 is the number of rangesof parameters 119909

119894(119896) is the observed value of the

119896th parameter of the 119894th range and 119910119895(119896) is the

representative value of the 119896th parameter of the 119895thregion


(ii) Normalize the reference sequence with (11) and theassessment sequence with (12) and (13)

1199091015840

119894(119896) =

119909119894(119896) minus 119909

1(119896)

119909119905(119896) minus 119909

1(119896)

1199091(119896) minus 119909

119894(119896)

1199091(119896) minus 119909

119905(119896)

(11)

For an indicator for which the greater value means ahigher vulnerability normalization can be performed with(12)

1199101015840

119895(119896) =

0 119910119895(119896) le 119909

1(119896)

119910119895(119896) minus 119909

1(119896)

119909119905(119896) minus 119909

1(119896)

1199091(119896) lt 119910

119895(119896) lt 119909

119905(119896)

1 119910119895(119896) ge 119909

119905(119896)

(12)

For an indicator for which the smaller value means ahigher vulnerability normalization can be performed with(13)

1199101015840

119895(119896) =

0 119910119895(119896) ge 119909

1(119896)

1199091(119896) minus 119910

119895(119896)

1199091(119896) minus 119909

119905(119896)

1199091(119896) gt 119910

119895(119896) gt 119909

119905(119896)

1 119910119895(119896) le 119909

119905(119896)

(13)

(iii) Identify Δ119895119894(119896) Δmax and Δmin with (14)

Δ119895119894(119896) =

100381610038161003816100381610038161199101015840

119895(119896) minus 119909

1015840

119894(119896)

10038161003816100381610038161003816

Δmax = max max 100381610038161003816100381610038161199101015840

119895(119896) minus 119909

1015840

119894(119896)

10038161003816100381610038161003816

Δmin = min min 100381610038161003816100381610038161199101015840

119895(119896) minus 119909

1015840

119894(119896)

10038161003816100381610038161003816

(14)

where Δ119895119894(119896) is the difference between 119910

1015840

119895(119896) and

1199091015840

119894(119896) and Δmax and Δmin represent the maximum

and minimum values among all the Δ119895119894(119896) respec-

tively(iv) Calculate the grey relational coefficient 120577

119895119894(119896)

120577119895119894(119896) =

Δmin + 05ΔmaxΔ119895119894(119896) + 05Δmax

(15)

(v) Calculate the grey relational grade 119903119895119894

119903119895119894=

119899

sum

119896=1

120582119894120577119895119894(119896) (16)

where 120582119894is the weight of the 119894th parameter

(vi) Determine the maximum value 119903max of the greyrelational grades The range of 119903max represents thevulnerability type which a region belongs to For themultilayer indicator system 119903

119895119894for one layer should

be regarded as 120577119895119894

for its next layer and then berecalculated with (16)

(2) The PS Method The PS method can be used to calculatethe vulnerability indicator to identify the degree of theWRVThe steps that are taken with this method are as follows

(i) Establish a reference system119862119894= (1198621 1198622 119862

119899) that

can be represented by the median of every indicatorand an assessment system 119875

119894= (1198751 1198752 119875

119899) that

uses the observed indicator values in the study region(ii) Normalize the values of the assessment system by (17)

and (18)

For the indicator that the greater the value the morevulnerable the system normalization can be performed asfollows

119883119894=

119875119894

119862119894

(17)

For the indicator that the smaller the value the morevulnerable the system normalization is performed as follows

119883119894=

119862119894

119875119894

(18)

(iii) Calculate the WRV degree by (19)

119881119894= 120582119894times 119883119894 (19)

where 119881119894is the WRV value of a region and 120582

119894is the

weight of indicator

For the multilayer indicator system119881119894obtained from one

layer appears as 119883119894of its next layer then we recalculate 119881

119894

by using (19) Finally we can obtain the WRV degree of theintegrated water resource system The greater the value of 119881

119894

is the more vulnerable the region is

4 Results and Discussion

41 WRV Assessment Results of Subsystems and the IntegratedWater Resources System Based on the Bayesian Method Tocompare the calculation results of the Bayesian method withthose of the PS method and the GRA method [19] wechose the corresponding data in 2003 (see Table 2) With(7) the probability that each indicator of the subsystemsbelonged to a certain WRV level were obtained Then theWRV probabilities of the subsystems were obtained with (8)For the indicators of the hydrological subsystem (Figure 4)the lower the indicator value was the larger the tendencyof a county to belong to a higher WRV was Some WRVlevels to which indicators of the subsystems belonged werethe same for most indicators of all nine counties (ie HighWRV for the mean annual precipitation Extreme WRVfor the groundwater recharge of unit area and MediumWRV for the drought indicator) with the exception of twoindicators (ie the annual relative variability of precipitationand the mean water resources amount of unit area) seeFigures 4(a

1) 4(a

2) 4(a

3) 4(a

4) and 4(a

5) However the

probabilities that each indicator belonged to some WRV


Table 2 Actual values of indicators for study area

Subsystem Indicator 1 2 3 4 5 6 7 8 9

A

a1

38720 40640 40640 40910 40570 37360 45650 38580 44680a2

2083 2775 2271 2265 1412 2190 2277 1883 1830a3

452 684 560 656 650 406 790 827 345a4

258 476 309 480 243 294 295 568 165a5

300 242 235 279 205 267 221 219 182

B

b1

6854 47445 29477 45845 43882 27144 66783 43890 65250b2

19826 115804 131714 73619 41231 31141 57093 46404 52304b3

19444 33677 42260 48515 19368 13814 29607 37202 25080b4

149 196 925 781 381 1230 185 1719 644b5

087 098 094 093 096 109 137 096 099b6

17441 6761 10664 10058 4833 4546 4819 8601 3369b7

7000 6500 6800 7000 6200 6000 4500 4300 4100

C

c1

7624 5094 4759 5918 7683 5836 7169 5100 5637c2

18086 19140 12931 9061 28019 8288 36239 50588 4866c3

346 113 032 256 040 042 023 147 062c4

4982 5003 7139 6365 6154 8999 4286 0 6664c5

6574 6332 6037 7078 6357 6382 6451 6218 5038c6

7000 5000 6500 6800 5000 5500 3500 3500 3000

Dd1

6174 6077 3252 5832 2383 3298 4716 5828 3629d2

710 5248 1379 5681 3941 1977 107 15978 106d3

8964 8262 7607 8147 562 7858 8800 6020 7500Note The symbolic descriptions for the 1st column of AsimD and the 2nd column of a1simd3 are the same as those in Table 1 administrative divisions 1ndash9 areshown in Figure 1

level were different for the nine counties for examplea probability of 4008ndash4651 belonging to High WRVfor the mean annual precipitation while 2656ndash3649belonging to Extreme WRV for the groundwater rechargeof unit area For the indicator of the mean water resourcesamount of unit area six counties belonged to Extreme WRVwith the exception of Huairsquoan Zhuolu and Huailai Countieswhich belonged to High WRV (see Figure 4(a

3)) The WRV

assessment results of the hydrological subsystem are shownin Figure 4(A) We can see that all of the nine counties hada high probability of belonging to High WRV (ranging from2707 to 3348) followed byMediumWRV(223ndash254)and Extreme WRV (144ndash245) This finding meant thatthe WRV of the hydrological subsystem of the Zhangjiakouregion was high The hydrological subsystem reflected thenatural situations of regionalwater resourcesThe abundancestability and hydrographic and meteorological conditionshad a comprehensive impact on the WRV degree of thehydrological subsystem Being located in arid and semiaridareas water supply available of the Zhangjiakou regionwas insufficient although the stability of water resourcessystem was not too bad Specifically water abundance andgroundwater recharge capacity were very weak resulting ina high WRV probability of the hydrological subsystem Theresults were consistent with those of the GRA method andthe actual situations [19]

For indicators of the socioeconomic subsystem(Figure 5) water resources per capita of eight counties werelow (approximately 399m3) which had a high probabil-ity of belonging to Extreme WRV (2667ndash5402) except

for Zhuolu County which tended to belong to High WRV(3080) (see Figure 5(b

1)) This finding meant that the

regional population increase had put great pressure on thewater resources system For the other indicators exceptfor the water resources per capita there were differentWRV levels for each county In general the repetitive waterutilization ratio of industry in this region tended to be LowWRV or Medium WRV (89 altogether) which meant thatthe industrial water resources consumption was not seriousIt is partly because the economic development mode of theregion was mainly based on agriculture Although the watersupply and demand ratio for counties had a tendency tobelong to Low WRV (1195ndash4120) or No WRV (4306ndash8789) from Figure 5(b

5) the water demand had been met

by overextracting groundwater for a long time by the analysiscombined with Figure 4(a

4) In this way the water resources

use is not sustainable Through the probability calculationwith (7) while Yuxian and Chongli counties tended to belongto Low WRV Yangyuan and Chongli counties belonged toNo WRV while the other counties belonged to ExtremeWRV (see Figure 5(B)) It can be concluded that the rapidincrease of population and low water utilization efficiencyresulted in a high WRV of socioeconomic subsystem Asa basic water resources security concern the groundwateroverdraft needs more attention for the purpose of sustainablewater use in the future although it temporarily alleviated theshortage of water resources

For the indicators of the ecoenvironmental subsystem(Figure 6) themost possible twoWRV levels for the indicatorof the ratio of soil and water loss were the same for all nine


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty

Administrative division(a1) Mean annual precipitation

1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty

Administrative division

1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty

1 2 3 4 5 6 7 8 9Administrative division

(a3) Mean water resourcesamount of unit area

05

00

01

02

03

04

Prob

abili

ty


(a5) Drought index


000

005

010

015

020

025

030

035

Prob

abili

ty

(A) Hydrological subsystem

(a2) Annual relative variabilityof precipitation

(a4) Groundwater rechargeof unit area

No WRVLow WRVMedium WRV

High WRVExtreme WRV


High WRVExtreme WRV

Figure 4 Each indicatorrsquos WRV and integratedWRV of the hydrological subsystem in the Zhangjiakou region (administrative divisions 1ndash9are shown in Figure 1)


1 2 3 4 5 6 7 8 900

01

02

03

04

05

06

Prob

abili

ty

Administrative division1 2 3 4 5 6 7 8 9

00

01

02

03

04

05

06

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



00

01

02

03

04

05

Prob

abili

ty


00

01

02

03

04

05

06

07

08

09

Prob

abili

ty


00

01

02

03

04

05

06

07

08

09

Prob

abili

ty

(b1) Water resources per capita

(b3) Water consumption per capita

(b5) Water supply and demand ratio

(b2) Water consumption per unit GDP

(b4) Difference between planned and

(b6) Water resources development

actual water utilization amount

and utilization ratio


High WRVExtreme WRV


High WRVExtreme WRV

Figure 5 Continued



000

005

010

015

020

025

030

035

Prob

abili

ty


(B) Socioeconomic subsystem

00

01

02

03

04

05

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV

(b7) Repetitive utilization ratio of industrial water

Figure 5 Each indicatorrsquos WRV and integrated WRV of the socioeconomic subsystem in the Zhangjiakou region (administrative divisions1ndash9 are shown in Figure 1)

counties (ie Extreme WRV ranked first with a high prob-ability of 6254ndash7918 and High WRV ranked second)The WRV levels were different from each other for otherindicators including the land development ratio the fertilizeruse intensity the pollution index of surface water the ratioof polluted groundwater exceeding the standard and the dis-posal ratio of sewage due to their respective ecoenvironmentconditions Because most of the large counties were locatedin remote mountainous areas their fertilizer use intensitywas not excessively large Therefore as for the indicator ofthe fertilizer use intensity except for the fact that HuailaiCounty tended to belong to High WRV others belonged toMedium WRV (Figure 6(c

2)) In addition the pressure of

economic development had greatly aggravated the pollutionof the surface water leading to a high probability for theindicator of the pollution of the surface water belonging toMedium WRV (3952) High WRV (3868ndash5108) andExtremeWRV (7768ndash9999) (Figure 6(c

3)) For example

all of the threemost prosperous counties in this region that isZhangjiakou City Xuanhua County andHuailai County hadExtreme WRV Among the nine counties Zhangjiakou Cityhad the largest population and the largest economic outputExcept for the fact that the indicator of the disposal ratioof sewage for Zhangjiakou City had a Low WRV level itsWRV levels for the other indicators were usually higher bymore than 1-2 levels compared with other counties leadingto an Extreme WRV for the comprehensive assessmentof the ecoenvironmental subsystem Overall the ecologicalenvironment of the Zhangjiakou region was very vulnerableexcept for Zhuolu County belonging to the Medium WRVall of the other counties belonged to the Extreme WRV orHigh WRV (Figure 6(C)) With the rapid development ofindustry and agriculture waste water discharge and extensiveuse of fertilizers and pesticides in agricultural production had

increased leading to the deterioration of water environmentin Zhangjiakou region In addition polluted groundwaterwould cause the deterioration in ecoenvironment All thesefactors had resulted inWRV of ecoenvironmental subsystemTo decrease the WRV an extensive mode of economicdevelopment should be changed

For the indicators of the hydraulic engineering subsystem(Figure 7) the water-saving irrigation ratio was usually highwith 67 of the counties belonging to MediumWRV or LowWRV and 22 belonging to High WRV (Wanquan Countyand Yangyuan County) (see Figure 7(d

1)) Because the sit-

uations of water resources and hydraulic engineering weredifferent for the nine counties the WRV for the indicatorsof the regulating capacity of the surface water for hydraulicengineering and storage was different (see Figure 7(d

2)) For

the indicator of the storage and release capacity of reservoirsexcept forHuailai County belonging to LowWRVandYuxianCounty belonging to Medium WRV the other countiesbelonged to No WRV (see Figure 7(d

3)) Overall hydraulic

engineering played an important role in the decline of theWRV more than 50 of the counties including ZhangjiakouCity Huairsquoan County Xuanhua County Zhuolu County andHuailai County had Low orNoWRV in this region due to thereallocation and adjustment of the flood by means of waterconservancy facilities (see Figure 7(D))

With (8) the WRV of the integrated water resourcessystemof theZhangjiakou regionwas obtained (see Figure 8)We can see the following (1) WRV of the ecoenvironmentalsubsystem of Zhangjiakou City and XuanHua County wasmore vulnerable than other counties which led to a hightendency to belong to Extreme WRV (268 and 250resp) other counties belonged to High WRV ranging from246 to 270 (2) there were no areas belonging toNoWRVLowWRV andMediumWRV indicating that theWRVof the


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


(c1) Land development ratio

1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


(c2) Fertilizer use intensity

1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


(c6) Disposal ratio of sewage


(c4) Ratio of polluted groundwaterexceeding the standard

06

05

04

03

02

01

00

Prob

abili

ty


(c5) Ratio of soil and water loss

08

07

06

05

04

03

02

01

00

Prob

abili

ty


(c3) Pollution index of surface water

10

09

08

07

06

05

04

03

02

01

00

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV

Figure 6 Continued


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



High WRVExtreme WRV

(C) Ecoenvironmental subsystem

Figure 6 Each indicatorrsquosWRV and integratedWRVof the ecoenvironmental subsystem in the Zhangjiakou region (administrative divisions1ndash9 are shown in Figure 1)

Table 3 Results comparison between the Bayesian method the GRAmethod and the PS method for the integrated water resources system

Administrative division Bayesian method GRA method PS methodZhangjiakou City Extreme WRV Extreme WRV Extreme WRVHuairsquoan County High WRV High WRV Extreme WRVWanquan County High WRV High WRV Extreme WRVXuanhua County Extreme WRV High WRV Extreme WRVYuxian County High WRV High WRV Extreme WRVYangyuan County High WRV High WRV Extreme WRVZhuolu County High WRV High WRV High WRVHuailai County High WRV High WRV Extreme WRVChongli County High WRV High WRV Extreme WRV

integrated water resources system of the Zhangjiakou regionwas very seriousWRVwas an objective attribute of the waterresources system whose vulnerability came from its internalcharacteristics and external driving forces The assessmentresults above were consistent with the actual situation of thewater resources in the Zhangjiakou region which showedthat the Bayesian method can be well used for regional WRVassessment

42 Results Comparison with the PS Method and the GRAMethod A comparison of the assessment results among theBayesian method the GRA method and the PS methodwas conducted (Table 3) It can be seen that the WRV of 6regions with the PS method was higher than those with theBayesian method and 7 regions were higher than those withthe GRA method The PS method had the advantages of asimple principle and simple linear calculation procedure forboth the assessment and reference systems but it can easilygenerate distorted results due to magnifying the influence of

some indicators and the resulting WRV levels were higherthan those use the other two methods

The results of the Bayesian method and the GRAmethodwere almost the same with the exception of the assessmentfor Xuanhua County The two methods divided the sampledata into several grades to determine the vulnerable degreeof the water resources in a large range which can fully reflectthe uncertainties in the water resources system Howeverthe GRA method had a high requirement of assessmentand standard data The assessment results for WRV werethe possibility or tendency of the adverse impact of naturaldisasters and human activities on the water resources systemwhich can reflect the uncertain and fuzzy characteristicsof the WRV reasonably At the same time it was verysimple and quick which made it more convenient andreliable for theWRV assessment In the practical calculationmultiple methods should be used comprehensively for WRVassessment to provide a more reliable decision-making basisfor water resources management


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


Administrative division(d1) Water-saving irrigation ratio (d2) Ratio of design utilizable storage of

(d3) Ratio of effective storage to total storage of reservoirs (D) Hydraulic engineering subsystem


High WRVExtreme WRV


High WRVExtreme WRV

00

01

02

03

04

05

06

07

08

09

10

Prob

abili

ty

reservoir to surface water quantity

00

01

02

03

04

05

06

07

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty


Figure 7 Each indicatorrsquos WRV and integrated WRV of the hydraulic engineering subsystem in the Zhangjiakou region (administrativedivisions 1ndash9 are shown in Figure 1)

5 Conclusions

A Bayesian method that used the sampling normal distribu-tion method and the AHP method was developed to assessthe WRV of the Zhangjiakou region The results showed that(1) the vulnerability of the region was very high ZhangjiakouCity and Xuanhua County belonged to Extreme WRV withthe probabilities of 268 and 250 respectively while othercounties belonged to HighWRV (246ndash270)This findingindicated that the socioeconomic development had put alarge amount of pressure on the water resources Underthe approximately same natural conditions for these nineadjacent areas the economy developed more rapidly and theWRV will become more serious (2) Additionally the resultsof WRV assessment of the Zhangjiakou region by using the

Bayesian method were consistent with the actual situationThis finding indicated that the Bayesian method was feasiblefor the WRV assessment with a simple calculation process

The Zhangjiakou region undertook the important mis-sion of providing a water supply for Beijing City but thewater resources condition was not good Water scarcity andwater pollution had seriously affected the sustainable devel-opment of the regional agricultural production and economicdevelopment To reduce the WRV and enhance the abilityof self-recovery of the water environment and to improvesustainable utilization of water resources of the Zhangjiakouregion effective countermeasures must be taken from thefollowing aspects (1) developing a water-saving industry andtechnology for the sewage purification to improve the reuselevel of the waste water and to construct a water-saving


000

005

010

015

020

025

030

035

Prob

abili

ty



High WRVExtreme WRV

(a)

5

7

9

4

6

82

3 1

1

11

Extreme WRVHigh WRV

(b)

Figure 8 WRV of the integrated water resources system in the Zhangjiakou region (a) WRV probabilities (b) WRV spatial distribution(administrative divisions 1ndash9 are shown in Figure 1)

society which can reduce the water environment deterio-ration and shortage of water resources in the Zhangjiakouregion (2) increasing the vegetation coverage which wouldenhance the capacity for soil and water preservation andstrengthening controls on nonpoint pollution and populationgrowth to promote the sustainable development of a socialeconomy and ecoenvironment (3) raising public environ-mental awareness for the effective protection and sustainableutilization of water resources

Due to insufficient information we assumed that theprior probability of an indicator that belonged to the differentvulnerability grades was equal However their prior proba-bilities were different in practice The quantity of indicatordata that was available largely limits the accuracy of theassessment Thus acquiring more indicator data could helpto improve the precision of theWRV assessment In additionthere is a threshold of the damage to the water resourcessystem the WRV assessment can provide only a level range(such as No WRV and Extreme WRV) and it is a future taskto determine the damage threshold through a mechanismstudy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research was funded by the National Science Founda-tion for Innovative Research Group (no 51121003) and theNational Natural Science Foundation of China (50509001)The authors would like to extend special thanks to the editorand the anonymous reviewers for their valuable comments ingreatly improving the quality of this paper

References

[1] Z J Li QW Chen Q Xu andK Blanckaert ldquoGeneralized like-lihooduncertainty estimationmethod in uncertainty analysis ofnumerical eutrophication models take bloom as an examplerdquoMathematical Problems in Engineering vol 2013 Article ID701923 9 pages 2013

[2] X Wang F B Ma and J Y Li ldquoWater resources vulnerabilityassessment based on the parametric-system method a casestudy of the Zhangjiakou region of guanting reservoir BasinNorth Chinardquo Procedia Environmental Sciences vol 13 pp1204ndash1212 2012

[3] K Jinno Z X Xu A Kawamura andK Tajiri ldquoRisk assessmentof water supply system during droughtrdquo International Journal ofWater Resources Development vol 11 no 2 pp 185ndash204 1995

[4] Z Xu K Jinno A Kawamura S Takesaki and K Ito ldquoPerfor-mance risk analysis for Fukuoka water supply systemrdquo WaterResources Management vol 12 no 1 pp 13ndash30 1998


[5] Z X Xu K Takeuchi H Ishidaira and X W Zhang ldquoSus-tainability analysis for Yellow River water resources using thesystem dynamics approachrdquoWater Resources Management vol16 no 3 pp 239ndash261 2002

[6] L L Liu ldquoConcept and quantitative assessment of vulnerabilityof water resourcerdquo Bulletin of Soil and Water Conservation vol22 no 2 pp 41ndash44 2002 (Chinese)

[7] J D Arthur H A R Wood A E Baker J R Cichon and G LRaines ldquoDevelopment and implementation of a Bayesian-basedaquifer vulnerability assessment in Floridardquo Natural ResourcesResearch vol 16 no 2 pp 93ndash107 2007

[8] B Kattaa W Al-Fares and A R Al Charideh ldquoGroundwatervulnerability assessment for the BanyasCatchment of the Syriancoastal area using GIS and the RISKE methodrdquo Journal ofEnvironmental Management vol 91 no 5 pp 1103ndash1110 2010

[9] A Sorichetta M Masetti C Ballabio S Sterlacchini andG P Beretta ldquoReliability of groundwater vulnerability mapsobtained through statistical methodsrdquo Journal of EnvironmentalManagement vol 92 no 4 pp 1215ndash1224 2011

[10] A Sorichetta M Masetti C Ballabio and S SterlacchinildquoAquifer nitrate vulnerability assessment using positive andnegative weights of evidence methods Milan Italyrdquo Computersand Geosciences vol 48 pp 199ndash210 2012

[11] DMirauda andMOstoich ldquoSurface water vulnerability assess-ment applying the integrity model as a decision support systemfor quality improvementrdquo Environmental Impact AssessmentReview vol 31 no 3 pp 161ndash171 2011

[12] C Yu Y Yao G Hayes B Zhang and C Zheng ldquoQuantitativeassessment of groundwater vulnerability using index systemand transport simulation Huangshuihe catchment ChinardquoScience of the Total Environment vol 408 no 24 pp 6108ndash61162010

[13] J J Martınez-Bastida M Arauzo and M Valladolid ldquoIntrinsicand specific vulnerability of groundwater in central Spain therisk of nitrate pollutionrdquoHydrogeology Journal vol 18 no 3 pp681ndash698 2010

[14] J Zhou G Li F Liu YWang andXGuo ldquoDRAVmodel and itsapplication in assessing groundwater vulnerability in arid areaa case study of pore phreatic water in Tarim Basin XinjiangNorthwest Chinardquo Environmental Earth Sciences vol 60 no 5pp 1055ndash1063 2010

[15] P Martınez-Santos M R Llamas and P E Martınez-AlfaroldquoVulnerability assessment of groundwater resources a model-ling-based approach to the Mancha Occidental aquifer SpainrdquoEnvironmental Modelling and Software vol 23 no 9 pp 1145ndash1162 2008

[16] C Butscher and P Huggenberger ldquoEnhanced vulnerabilityassessment in karst areas by combiningmapping withmodelingapproachesrdquo Science of the Total Environment vol 407 no 3 pp1153ndash1163 2009

[17] A K Antonakos and N J Lambrakis ldquoDevelopment andtesting of three hybrid methods for the assessment of aquifervulnerability to nitrates based on the drasticmodel an examplefrom NE Korinthia Greecerdquo Journal of Hydrology vol 333 pp288ndash304 2007

[18] H-N Zhu X-Z Yuan J Liang G-M Zeng and H-W JiangldquoAn integrated model for assessing the risk of water environ-mental pollution based on fuzzinessrdquo China EnvironmentalScience vol 31 no 3 pp 516ndash521 2011 (Chinese)

[19] J Y Li Study on vulnerability assessment of water resources ofguanting reservoir basin [MS thesis] Beijing Normal Univer-sity Beijing China 2007 (chinese)

[20] F B Ma X Wang and C H Li ldquoResearch progress in waterresources vulnerability assessmentrdquo Journal of Water Resourcesand Water Engineering vol 23 no 1 pp 30ndash37 2012 (Chinese)

[21] F Zong H G Xu and H Y Zhang ldquoPrediction for trafficaccident severity comparing the Bayesian network and regres-sion modelsrdquo Mathematical Problems in Engineering vol 2013Article ID 475194 9 pages 2013

[22] T Chan H Ross S Hoverman and B Powell ldquoParticipatorydevelopment of a Bayesian networkmodel for catchment-basedwater resourcemanagementrdquoWater Resources Research vol 46no 7 Article IDW07544 2010

[23] S S Qian A Schulman J Koplos A Kotros and P KellarldquoA hierarchical modeling approach for estimating nationaldistributions of c hemicals in public drinking water systemsrdquoEnvironmental Science and Technology vol 38 no 4 pp 1176ndash1182 2004

[24] R Wu S S Qian F Hao H Cheng D Zhu and J ZhangldquoModeling contaminant concentration distributions in Chinarsquoscentralized sourcewatersrdquoEnvironmental ScienceampTechnologyvol 45 no 14 pp 6041ndash6048 2011

[25] F BMa C H Li XWang Z F Yang C C Sun and P Y LiangldquoABayesianmethod for comprehensivewater quality evaluationof the Danjiangkou Reservoir water source area for the middleroute of the South-to-NorthWater Diversion Project in ChinardquoFrontiers of Earth Science vol 8 no 2 pp 242ndash250 2014

[26] M Kalantarnia F Khan and K Hawboldt ldquoDynamic riskassessment using failure assessment and Bayesian theoryrdquoJournal of Loss Prevention in the Process Industries vol 22 no5 pp 600ndash606 2009

[27] D L Kelly and C L Smith ldquoBayesian inference in probabilisticrisk assessment-the current state of the artrdquoReliability Engineer-ing and System Safety vol 94 no 2 pp 628ndash643 2009

[28] ldquoZhangjiakou bureau of hydrology and water resources sur-veyrdquo Second Water Resources Assessment Report of Zhangji-akou City Hebei Institute of Architectural Engineering HebeiProvince 2004 (chinese)

[29] X Wang Q Cui and S Y Li ldquoAn optimal water allo-cation model based on water resources security assessmentand its application in Zhangjiakou Region Northern ChinardquoResources Conservation and Recycling vol 69 pp 57ndash65 2012

[30] M A Hamouda M M Nour El-Din and F I MoursyldquoVulnerability assessment of water resources systems in theEastern Nile Basinrdquo Water Resources Management vol 23 no13 pp 2697ndash2725 2009

[31] P E Waggoner Climate Change and US Water Resources JohnWiley amp Sons New York NY USA 1990

[32] F V JensenAn Introduction to BayesianNetworks vol 210 UCLPress London UK 1996

[33] J W K Chan and T K L Tong ldquoMulti-criteria material selec-tions and end-of-life product strategy grey relational analysisapproachrdquo Materials and Design vol 28 no 5 pp 1539ndash15462007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


for the protection rational development and utilization ofregional water resources WRV assessment has been drawingmore attention

Since the late 1960s many scholars and water managersaround the world have studied the WRV but most of themconcentrated on groundwater vulnerability assessment [7ndash10] With respect to surface water vulnerability most of theliterature focused on the supply and demand balance ofwater resources or selected several indicators and methodsto evaluate the carrying capacity or sustainability of thewater resources [11] Given the uneven distribution of waterresources in space and time it is necessary to pay closeattention to the WRV that is caused by climate changeand human activities It is also important to enhance thejoint assessment of surface water and groundwater waterquantity and water quality to ensure a sustainable utilizationof water resources of a given region In recent years manyapproaches have been developed to assess WRV such asoverlay and indicator methods [12ndash14] process-based sim-ulation methods [15 16] statistical methods [9 17] andfuzzy comprehensive judgment methods [18] For obtainingmore reliable assessment results these methods required alarge amount of natural environmental and socioeconomicdata with high accuracy In [19] Li quantitatively analyzedthe regional WRV by using the fuzzy optimization (FO)method the grey relational analysis (GRA) method and theparametric-system (PS) method by constructing an indicatorsystem with a comprehensive consideration of surface waterand groundwater water quantity and water quality Thesemethods however cannot address the uncertainty that isinherent in all of the vulnerability assessments [20] With theincrease in the pressure exerted by socioeconomic develop-ment on a natural environment the complexity and uncertaincharacteristics of WRV that arise from both climate changeand human activities has become even more obvious Toimprove the water resources management and protectionthe regional WRV research that considers fuzziness anduncertainty of the assessment results is highly desired

With increasingly in-depth research on uncertain eventsthe Bayesian method has received extensive attention [21]It uses objective probability estimations and incompleteinformation to estimate the probability of some unknownstates This method can also describe uncertain informationflexibly and conduct uncertainty reasoning [22] Comparedwith the existing assessment methods the Bayesian methodrsquossuperiority lies in giving the probability or tendency to acertain grade to which the system belongs not just simplygiving the grade In addition it has the advantages of a simpleprinciple and calculation process and can obtain reliableresults for small samples Thus so far it has been widely usedfor water quality assessment [23ndash25] fault diagnosis and riskassessment [26 27] but it has rarely been used for WRVassessmentThe application of the Bayesian method in actualWRV assessment is expected for the effective management ofregional water resources

The objectives of this research were to (a) develop aBayesian method for WRV assessment (b) apply the pro-posed method to assess the WRV condition of the nine

counties of the Zhangjiakou region and describe the prob-ability distribution characteristics at different counties and(c) compare the results with the existing methods includingthe GRA method and the PS method The proposed methodwill improve the existing method in addressing uncertaininformation and the assessment results can help to identifyspecific water resources problems of the Guanting Reservoirin such a way that appropriate measures can be taken foreffective environmental management

2 Study Area

21 Overview of the Study Area The Zhangjiakou regionwhich is located in the upper reaches of the Guanting Reser-voir Basin is selected as a case to assess the regional WRV(Figure 1) It is confrontedwith the dual task of developing theeconomy and protecting the natural ecological environmentThe concerned city and counties are Zhangjiakou CityHuairsquoan County Wanquan County Xuanhua County YuxianCounty Yangyuan County Zhuolu County Huailai Countyand Chongli County with 178 thousand km2 accountingfor 405 of the area of Guanting Reservoir Basin TheZhangjiakou region has a continental monsoon climate withan annual mean temperature ranging from 7∘C to 10∘CThe annual mean precipitation is rather low (approximately400mm) and 80 of the rainfall occurs between Juneand September The topography in the Zhangjiakou regionis very complex mountains hills and rivers account forapproximately 30of the area respectivelyThepopulation ofthe region is 459 million and the water resources per capitaare 399m3 which is less than the internationally acceptedstandard of extreme water scarcity of 500m3 [28]

The region has a backward economic developmentExcessive consumption and pollution of water resourceshave exerted an enormous pressure on local water resourcessystem and have directly influenced the security of the watersupply of Beijing [29] Therefore an effective WRV assess-ment is necessary and desired for regional water resourcesprotection and management The hydrological data camefrom Zhangjiakou Water Conservancy Bureau the waterquality data came from Environmental Protection Agency ofZhangjiakou and the ecoenvironmental and socioeconomicdata were from the statistical yearbooks of the Zhangjiakouregion [19]

22 Key Factors InfluencingWRV RegionalWRV is reflectedby the carrying capacity recovery capacity and sustainabilityof the water resources [6 30 31] The key factors thatinfluence the WRV of the Zhangjiakou region include thefollowing aspects

(1)Hydrology Influenced by the continentalmonsoon climateof the arid and semiarid area the evaporation and runoff ofthis region have the characteristic of uneven distribution ata spatial and temporal scale The amount of water availableis not enough The influencing factors that are associatedwith these hydrological aspects mainly include the naturalconditions of the regional water resources such as the


5

7

9

4

6

82

3 1

1

1

1


Other province

Beijing










LowWRV

MediumWRV

HighWRV

ExtremeWRV

A


B








C

c1


c2








3 Methodology








Ecoenvironmental



Weight

Procedures Methods



AHP method









119875 (1199101198951) = 119875 (119910

1198952) = 119875 (119910

1198953) = 119875 (119910

1198954) = 119875 (119910

1198955) =

1

5 (1)


where 119875(119910119895119894)119894=125




119886119895=

(sum5

119894=1119910119895119894)

5

120590119895=radic

sum5

119894=1(119910119895119894minus 119886119895)2

4

119862V119895 =120590119895

119886119895

(2)








119862V119895119894 = 119862V119895 (3)




120590119895119894= 119862V119895119894 sdot 119910119895119894 (4)



119905119895119896

=

119909119895119896

minus 119910119895119894

120590119895119894

(5)



of 119909119895119896


| 119910119895119894) of every

indicator with (6)

119875 (119909119895119896

| 119910119895119894) = 2 (1 minus Φ (

10038161003816100381610038161003816119905119895119896

10038161003816100381610038161003816)) (6)


1 2 4) and Φ(|119905119895119896|) = int

119905119895119896


minus1199062119889119906


119875 (119910119895119894| 119909119895119896) =

119875 (119910119895119894) 119875 (119909

119895119896| 119910119895119894)

sum5

119894=1119875 (119910119895119894) 119875 (119909

119895119896| 119910119895119894)

(7)




119875119894=

119898

sum

119895=1

120596119895119875 (119910119895119894| 119909119895119896) (8)





119895=1120596119895= 1 Here we




119894is the




le 1 sum119898119895=1

120596119895

= 1 119898 = 4119896 = 1




119909119894(119896) = 119909

119894(1) 119909

119894(2) 119909

119894(119899) (119894 = 1 2 119905) (9)

119910119895(119896) = 119910

119895(1) 119910

119895(2) 119910

119895(119899) (119895 = 1 2 119898)

(10)







1199091015840

119894(119896) =

119909119894(119896) minus 119909

1(119896)

119909119905(119896) minus 119909

1(119896)

1199091(119896) minus 119909

119894(119896)

1199091(119896) minus 119909

119905(119896)

(11)


1199101015840

119895(119896) =

0 119910119895(119896) le 119909

1(119896)

119910119895(119896) minus 119909

1(119896)

119909119905(119896) minus 119909

1(119896)

1199091(119896) lt 119910

119895(119896) lt 119909

119905(119896)

1 119910119895(119896) ge 119909

119905(119896)

(12)


1199101015840

119895(119896) =

0 119910119895(119896) ge 119909

1(119896)

1199091(119896) minus 119910

119895(119896)

1199091(119896) minus 119909

119905(119896)

1199091(119896) gt 119910

119895(119896) gt 119909

119905(119896)

1 119910119895(119896) le 119909

119905(119896)

(13)


Δ119895119894(119896) =

100381610038161003816100381610038161199101015840

119895(119896) minus 119909

1015840

119894(119896)

10038161003816100381610038161003816

Δmax = max max 100381610038161003816100381610038161199101015840

119895(119896) minus 119909

1015840

119894(119896)

10038161003816100381610038161003816

Δmin = min min 100381610038161003816100381610038161199101015840

119895(119896) minus 119909

1015840

119894(119896)

10038161003816100381610038161003816

(14)


1015840

119895(119896) and

1199091015840




119895119894(119896)

120577119895119894(119896) =

Δmin + 05ΔmaxΔ119895119894(119896) + 05Δmax

(15)


119903119895119894=

119899

sum

119896=1

120582119894120577119895119894(119896) (16)




be regarded as 120577119895119894




119899) that


119894= (1198751 1198752 119875

119899) that


and (18)


119883119894=

119875119894

119862119894

(17)


119883119894=

119862119894

119875119894

(18)


119881119894= 120582119894times 119883119894 (19)


119894is the

weight of indicator



119894


119894




1) 4(a

2) 4(a

3) 4(a

4) and 4(a

5) However the





A

a1

38720 40640 40640 40910 40570 37360 45650 38580 44680a2

2083 2775 2271 2265 1412 2190 2277 1883 1830a3

452 684 560 656 650 406 790 827 345a4

258 476 309 480 243 294 295 568 165a5

300 242 235 279 205 267 221 219 182

B

b1

6854 47445 29477 45845 43882 27144 66783 43890 65250b2

19826 115804 131714 73619 41231 31141 57093 46404 52304b3

19444 33677 42260 48515 19368 13814 29607 37202 25080b4

149 196 925 781 381 1230 185 1719 644b5

087 098 094 093 096 109 137 096 099b6

17441 6761 10664 10058 4833 4546 4819 8601 3369b7

7000 6500 6800 7000 6200 6000 4500 4300 4100

C

c1

7624 5094 4759 5918 7683 5836 7169 5100 5637c2

18086 19140 12931 9061 28019 8288 36239 50588 4866c3

346 113 032 256 040 042 023 147 062c4

4982 5003 7139 6365 6154 8999 4286 0 6664c5

6574 6332 6037 7078 6357 6382 6451 6218 5038c6

7000 5000 6500 6800 5000 5500 3500 3500 3000

Dd1

6174 6077 3252 5832 2383 3298 4716 5828 3629d2

710 5248 1379 5681 3941 1977 107 15978 106d3



3)) The WRV












1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



05

00

01

02

03

04

Prob

abili

ty


(a5) Drought index


000

005

010

015

020

025

030

035

Prob

abili

ty





High WRVExtreme WRV


High WRVExtreme WRV



1 2 3 4 5 6 7 8 900

01

02

03

04

05

06

Prob

abili

ty


00

01

02

03

04

05

06

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



00

01

02

03

04

05

Prob

abili

ty


00

01

02

03

04

05

06

07

08

09

Prob

abili

ty


00

01

02

03

04

05

06

07

08

09

Prob

abili

ty










High WRVExtreme WRV


High WRVExtreme WRV

Figure 5 Continued



000

005

010

015

020

025

030

035

Prob

abili

ty



00

01

02

03

04

05

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV






3)) For example






2)) For






1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





06

05

04

03

02

01

00

Prob

abili

ty



08

07

06

05

04

03

02

01

00

Prob

abili

ty



10

09

08

07

06

05

04

03

02

01

00

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV

Figure 6 Continued


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



High WRVExtreme WRV










1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





High WRVExtreme WRV


High WRVExtreme WRV

00

01

02

03

04

05

06

07

08

09

10

Prob

abili

ty


00

01

02

03

04

05

06

07

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



5 Conclusions





000

005

010

015

020

025

030

035

Prob

abili

ty



High WRVExtreme WRV

(a)

5

7

9

4

6

82

3 1

1

11

Extreme WRVHigh WRV

(b)






Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5

7

9

4

6

82

3 1

1

1

1


Other province

Beijing










LowWRV

MediumWRV

HighWRV

ExtremeWRV

A


B








C

c1


c2








3 Methodology








Ecoenvironmental



Weight

Procedures Methods



AHP method









119875 (1199101198951) = 119875 (119910

1198952) = 119875 (119910

1198953) = 119875 (119910

1198954) = 119875 (119910

1198955) =

1

5 (1)


where 119875(119910119895119894)119894=125




119886119895=

(sum5

119894=1119910119895119894)

5

120590119895=radic

sum5

119894=1(119910119895119894minus 119886119895)2

4

119862V119895 =120590119895

119886119895

(2)








119862V119895119894 = 119862V119895 (3)




120590119895119894= 119862V119895119894 sdot 119910119895119894 (4)



119905119895119896

=

119909119895119896

minus 119910119895119894

120590119895119894

(5)



of 119909119895119896


| 119910119895119894) of every

indicator with (6)

119875 (119909119895119896

| 119910119895119894) = 2 (1 minus Φ (

10038161003816100381610038161003816119905119895119896

10038161003816100381610038161003816)) (6)


1 2 4) and Φ(|119905119895119896|) = int

119905119895119896


minus1199062119889119906


119875 (119910119895119894| 119909119895119896) =

119875 (119910119895119894) 119875 (119909

119895119896| 119910119895119894)

sum5

119894=1119875 (119910119895119894) 119875 (119909

119895119896| 119910119895119894)

(7)




119875119894=

119898

sum

119895=1

120596119895119875 (119910119895119894| 119909119895119896) (8)





119895=1120596119895= 1 Here we




119894is the




le 1 sum119898119895=1

120596119895

= 1 119898 = 4119896 = 1




119909119894(119896) = 119909

119894(1) 119909

119894(2) 119909

119894(119899) (119894 = 1 2 119905) (9)

119910119895(119896) = 119910

119895(1) 119910

119895(2) 119910

119895(119899) (119895 = 1 2 119898)

(10)







1199091015840

119894(119896) =

119909119894(119896) minus 119909

1(119896)

119909119905(119896) minus 119909

1(119896)

1199091(119896) minus 119909

119894(119896)

1199091(119896) minus 119909

119905(119896)

(11)


1199101015840

119895(119896) =

0 119910119895(119896) le 119909

1(119896)

119910119895(119896) minus 119909

1(119896)

119909119905(119896) minus 119909

1(119896)

1199091(119896) lt 119910

119895(119896) lt 119909

119905(119896)

1 119910119895(119896) ge 119909

119905(119896)

(12)


1199101015840

119895(119896) =

0 119910119895(119896) ge 119909

1(119896)

1199091(119896) minus 119910

119895(119896)

1199091(119896) minus 119909

119905(119896)

1199091(119896) gt 119910

119895(119896) gt 119909

119905(119896)

1 119910119895(119896) le 119909

119905(119896)

(13)


Δ119895119894(119896) =

100381610038161003816100381610038161199101015840

119895(119896) minus 119909

1015840

119894(119896)

10038161003816100381610038161003816

Δmax = max max 100381610038161003816100381610038161199101015840

119895(119896) minus 119909

1015840

119894(119896)

10038161003816100381610038161003816

Δmin = min min 100381610038161003816100381610038161199101015840

119895(119896) minus 119909

1015840

119894(119896)

10038161003816100381610038161003816

(14)


1015840

119895(119896) and

1199091015840




119895119894(119896)

120577119895119894(119896) =

Δmin + 05ΔmaxΔ119895119894(119896) + 05Δmax

(15)


119903119895119894=

119899

sum

119896=1

120582119894120577119895119894(119896) (16)




be regarded as 120577119895119894




119899) that


119894= (1198751 1198752 119875

119899) that


and (18)


119883119894=

119875119894

119862119894

(17)


119883119894=

119862119894

119875119894

(18)


119881119894= 120582119894times 119883119894 (19)


119894is the

weight of indicator



119894


119894




1) 4(a

2) 4(a

3) 4(a

4) and 4(a

5) However the





A

a1

38720 40640 40640 40910 40570 37360 45650 38580 44680a2

2083 2775 2271 2265 1412 2190 2277 1883 1830a3

452 684 560 656 650 406 790 827 345a4

258 476 309 480 243 294 295 568 165a5

300 242 235 279 205 267 221 219 182

B

b1

6854 47445 29477 45845 43882 27144 66783 43890 65250b2

19826 115804 131714 73619 41231 31141 57093 46404 52304b3

19444 33677 42260 48515 19368 13814 29607 37202 25080b4

149 196 925 781 381 1230 185 1719 644b5

087 098 094 093 096 109 137 096 099b6

17441 6761 10664 10058 4833 4546 4819 8601 3369b7

7000 6500 6800 7000 6200 6000 4500 4300 4100

C

c1

7624 5094 4759 5918 7683 5836 7169 5100 5637c2

18086 19140 12931 9061 28019 8288 36239 50588 4866c3

346 113 032 256 040 042 023 147 062c4

4982 5003 7139 6365 6154 8999 4286 0 6664c5

6574 6332 6037 7078 6357 6382 6451 6218 5038c6

7000 5000 6500 6800 5000 5500 3500 3500 3000

Dd1

6174 6077 3252 5832 2383 3298 4716 5828 3629d2

710 5248 1379 5681 3941 1977 107 15978 106d3



3)) The WRV












1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



05

00

01

02

03

04

Prob

abili

ty


(a5) Drought index


000

005

010

015

020

025

030

035

Prob

abili

ty





High WRVExtreme WRV


High WRVExtreme WRV



1 2 3 4 5 6 7 8 900

01

02

03

04

05

06

Prob

abili

ty


00

01

02

03

04

05

06

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



00

01

02

03

04

05

Prob

abili

ty


00

01

02

03

04

05

06

07

08

09

Prob

abili

ty


00

01

02

03

04

05

06

07

08

09

Prob

abili

ty










High WRVExtreme WRV


High WRVExtreme WRV

Figure 5 Continued



000

005

010

015

020

025

030

035

Prob

abili

ty



00

01

02

03

04

05

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV






3)) For example






2)) For






1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





06

05

04

03

02

01

00

Prob

abili

ty



08

07

06

05

04

03

02

01

00

Prob

abili

ty



10

09

08

07

06

05

04

03

02

01

00

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV

Figure 6 Continued


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



High WRVExtreme WRV










1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





High WRVExtreme WRV


High WRVExtreme WRV

00

01

02

03

04

05

06

07

08

09

10

Prob

abili

ty


00

01

02

03

04

05

06

07

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



5 Conclusions





000

005

010

015

020

025

030

035

Prob

abili

ty



High WRVExtreme WRV

(a)

5

7

9

4

6

82

3 1

1

11

Extreme WRVHigh WRV

(b)






Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












LowWRV

MediumWRV

HighWRV

ExtremeWRV

A


B








C

c1


c2








3 Methodology








Ecoenvironmental



Weight

Procedures Methods



AHP method









119875 (1199101198951) = 119875 (119910

1198952) = 119875 (119910

1198953) = 119875 (119910

1198954) = 119875 (119910

1198955) =

1

5 (1)


where 119875(119910119895119894)119894=125




119886119895=

(sum5

119894=1119910119895119894)

5

120590119895=radic

sum5

119894=1(119910119895119894minus 119886119895)2

4

119862V119895 =120590119895

119886119895

(2)








119862V119895119894 = 119862V119895 (3)




120590119895119894= 119862V119895119894 sdot 119910119895119894 (4)



119905119895119896

=

119909119895119896

minus 119910119895119894

120590119895119894

(5)



of 119909119895119896


| 119910119895119894) of every

indicator with (6)

119875 (119909119895119896

| 119910119895119894) = 2 (1 minus Φ (

10038161003816100381610038161003816119905119895119896

10038161003816100381610038161003816)) (6)


1 2 4) and Φ(|119905119895119896|) = int

119905119895119896


minus1199062119889119906


119875 (119910119895119894| 119909119895119896) =

119875 (119910119895119894) 119875 (119909

119895119896| 119910119895119894)

sum5

119894=1119875 (119910119895119894) 119875 (119909

119895119896| 119910119895119894)

(7)




119875119894=

119898

sum

119895=1

120596119895119875 (119910119895119894| 119909119895119896) (8)





119895=1120596119895= 1 Here we




119894is the




le 1 sum119898119895=1

120596119895

= 1 119898 = 4119896 = 1




119909119894(119896) = 119909

119894(1) 119909

119894(2) 119909

119894(119899) (119894 = 1 2 119905) (9)

119910119895(119896) = 119910

119895(1) 119910

119895(2) 119910

119895(119899) (119895 = 1 2 119898)

(10)







1199091015840

119894(119896) =

119909119894(119896) minus 119909

1(119896)

119909119905(119896) minus 119909

1(119896)

1199091(119896) minus 119909

119894(119896)

1199091(119896) minus 119909

119905(119896)

(11)


1199101015840

119895(119896) =

0 119910119895(119896) le 119909

1(119896)

119910119895(119896) minus 119909

1(119896)

119909119905(119896) minus 119909

1(119896)

1199091(119896) lt 119910

119895(119896) lt 119909

119905(119896)

1 119910119895(119896) ge 119909

119905(119896)

(12)


1199101015840

119895(119896) =

0 119910119895(119896) ge 119909

1(119896)

1199091(119896) minus 119910

119895(119896)

1199091(119896) minus 119909

119905(119896)

1199091(119896) gt 119910

119895(119896) gt 119909

119905(119896)

1 119910119895(119896) le 119909

119905(119896)

(13)


Δ119895119894(119896) =

100381610038161003816100381610038161199101015840

119895(119896) minus 119909

1015840

119894(119896)

10038161003816100381610038161003816

Δmax = max max 100381610038161003816100381610038161199101015840

119895(119896) minus 119909

1015840

119894(119896)

10038161003816100381610038161003816

Δmin = min min 100381610038161003816100381610038161199101015840

119895(119896) minus 119909

1015840

119894(119896)

10038161003816100381610038161003816

(14)


1015840

119895(119896) and

1199091015840




119895119894(119896)

120577119895119894(119896) =

Δmin + 05ΔmaxΔ119895119894(119896) + 05Δmax

(15)


119903119895119894=

119899

sum

119896=1

120582119894120577119895119894(119896) (16)




be regarded as 120577119895119894




119899) that


119894= (1198751 1198752 119875

119899) that


and (18)


119883119894=

119875119894

119862119894

(17)


119883119894=

119862119894

119875119894

(18)


119881119894= 120582119894times 119883119894 (19)


119894is the

weight of indicator



119894


119894




1) 4(a

2) 4(a

3) 4(a

4) and 4(a

5) However the





A

a1

38720 40640 40640 40910 40570 37360 45650 38580 44680a2

2083 2775 2271 2265 1412 2190 2277 1883 1830a3

452 684 560 656 650 406 790 827 345a4

258 476 309 480 243 294 295 568 165a5

300 242 235 279 205 267 221 219 182

B

b1

6854 47445 29477 45845 43882 27144 66783 43890 65250b2

19826 115804 131714 73619 41231 31141 57093 46404 52304b3

19444 33677 42260 48515 19368 13814 29607 37202 25080b4

149 196 925 781 381 1230 185 1719 644b5

087 098 094 093 096 109 137 096 099b6

17441 6761 10664 10058 4833 4546 4819 8601 3369b7

7000 6500 6800 7000 6200 6000 4500 4300 4100

C

c1

7624 5094 4759 5918 7683 5836 7169 5100 5637c2

18086 19140 12931 9061 28019 8288 36239 50588 4866c3

346 113 032 256 040 042 023 147 062c4

4982 5003 7139 6365 6154 8999 4286 0 6664c5

6574 6332 6037 7078 6357 6382 6451 6218 5038c6

7000 5000 6500 6800 5000 5500 3500 3500 3000

Dd1

6174 6077 3252 5832 2383 3298 4716 5828 3629d2

710 5248 1379 5681 3941 1977 107 15978 106d3



3)) The WRV












1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



05

00

01

02

03

04

Prob

abili

ty


(a5) Drought index


000

005

010

015

020

025

030

035

Prob

abili

ty





High WRVExtreme WRV


High WRVExtreme WRV



1 2 3 4 5 6 7 8 900

01

02

03

04

05

06

Prob

abili

ty


00

01

02

03

04

05

06

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



00

01

02

03

04

05

Prob

abili

ty


00

01

02

03

04

05

06

07

08

09

Prob

abili

ty


00

01

02

03

04

05

06

07

08

09

Prob

abili

ty










High WRVExtreme WRV


High WRVExtreme WRV

Figure 5 Continued



000

005

010

015

020

025

030

035

Prob

abili

ty



00

01

02

03

04

05

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV






3)) For example






2)) For






1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





06

05

04

03

02

01

00

Prob

abili

ty



08

07

06

05

04

03

02

01

00

Prob

abili

ty



10

09

08

07

06

05

04

03

02

01

00

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV

Figure 6 Continued


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



High WRVExtreme WRV










1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





High WRVExtreme WRV


High WRVExtreme WRV

00

01

02

03

04

05

06

07

08

09

10

Prob

abili

ty


00

01

02

03

04

05

06

07

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



5 Conclusions





000

005

010

015

020

025

030

035

Prob

abili

ty



High WRVExtreme WRV

(a)

5

7

9

4

6

82

3 1

1

11

Extreme WRVHigh WRV

(b)






Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Ecoenvironmental



Weight

Procedures Methods



AHP method









119875 (1199101198951) = 119875 (119910

1198952) = 119875 (119910

1198953) = 119875 (119910

1198954) = 119875 (119910

1198955) =

1

5 (1)


where 119875(119910119895119894)119894=125




119886119895=

(sum5

119894=1119910119895119894)

5

120590119895=radic

sum5

119894=1(119910119895119894minus 119886119895)2

4

119862V119895 =120590119895

119886119895

(2)








119862V119895119894 = 119862V119895 (3)




120590119895119894= 119862V119895119894 sdot 119910119895119894 (4)



119905119895119896

=

119909119895119896

minus 119910119895119894

120590119895119894

(5)



of 119909119895119896


| 119910119895119894) of every

indicator with (6)

119875 (119909119895119896

| 119910119895119894) = 2 (1 minus Φ (

10038161003816100381610038161003816119905119895119896

10038161003816100381610038161003816)) (6)


1 2 4) and Φ(|119905119895119896|) = int

119905119895119896


minus1199062119889119906


119875 (119910119895119894| 119909119895119896) =

119875 (119910119895119894) 119875 (119909

119895119896| 119910119895119894)

sum5

119894=1119875 (119910119895119894) 119875 (119909

119895119896| 119910119895119894)

(7)




119875119894=

119898

sum

119895=1

120596119895119875 (119910119895119894| 119909119895119896) (8)





119895=1120596119895= 1 Here we




119894is the




le 1 sum119898119895=1

120596119895

= 1 119898 = 4119896 = 1




119909119894(119896) = 119909

119894(1) 119909

119894(2) 119909

119894(119899) (119894 = 1 2 119905) (9)

119910119895(119896) = 119910

119895(1) 119910

119895(2) 119910

119895(119899) (119895 = 1 2 119898)

(10)







1199091015840

119894(119896) =

119909119894(119896) minus 119909

1(119896)

119909119905(119896) minus 119909

1(119896)

1199091(119896) minus 119909

119894(119896)

1199091(119896) minus 119909

119905(119896)

(11)


1199101015840

119895(119896) =

0 119910119895(119896) le 119909

1(119896)

119910119895(119896) minus 119909

1(119896)

119909119905(119896) minus 119909

1(119896)

1199091(119896) lt 119910

119895(119896) lt 119909

119905(119896)

1 119910119895(119896) ge 119909

119905(119896)

(12)


1199101015840

119895(119896) =

0 119910119895(119896) ge 119909

1(119896)

1199091(119896) minus 119910

119895(119896)

1199091(119896) minus 119909

119905(119896)

1199091(119896) gt 119910

119895(119896) gt 119909

119905(119896)

1 119910119895(119896) le 119909

119905(119896)

(13)


Δ119895119894(119896) =

100381610038161003816100381610038161199101015840

119895(119896) minus 119909

1015840

119894(119896)

10038161003816100381610038161003816

Δmax = max max 100381610038161003816100381610038161199101015840

119895(119896) minus 119909

1015840

119894(119896)

10038161003816100381610038161003816

Δmin = min min 100381610038161003816100381610038161199101015840

119895(119896) minus 119909

1015840

119894(119896)

10038161003816100381610038161003816

(14)


1015840

119895(119896) and

1199091015840




119895119894(119896)

120577119895119894(119896) =

Δmin + 05ΔmaxΔ119895119894(119896) + 05Δmax

(15)


119903119895119894=

119899

sum

119896=1

120582119894120577119895119894(119896) (16)




be regarded as 120577119895119894




119899) that


119894= (1198751 1198752 119875

119899) that


and (18)


119883119894=

119875119894

119862119894

(17)


119883119894=

119862119894

119875119894

(18)


119881119894= 120582119894times 119883119894 (19)


119894is the

weight of indicator



119894


119894




1) 4(a

2) 4(a

3) 4(a

4) and 4(a

5) However the





A

a1

38720 40640 40640 40910 40570 37360 45650 38580 44680a2

2083 2775 2271 2265 1412 2190 2277 1883 1830a3

452 684 560 656 650 406 790 827 345a4

258 476 309 480 243 294 295 568 165a5

300 242 235 279 205 267 221 219 182

B

b1

6854 47445 29477 45845 43882 27144 66783 43890 65250b2

19826 115804 131714 73619 41231 31141 57093 46404 52304b3

19444 33677 42260 48515 19368 13814 29607 37202 25080b4

149 196 925 781 381 1230 185 1719 644b5

087 098 094 093 096 109 137 096 099b6

17441 6761 10664 10058 4833 4546 4819 8601 3369b7

7000 6500 6800 7000 6200 6000 4500 4300 4100

C

c1

7624 5094 4759 5918 7683 5836 7169 5100 5637c2

18086 19140 12931 9061 28019 8288 36239 50588 4866c3

346 113 032 256 040 042 023 147 062c4

4982 5003 7139 6365 6154 8999 4286 0 6664c5

6574 6332 6037 7078 6357 6382 6451 6218 5038c6

7000 5000 6500 6800 5000 5500 3500 3500 3000

Dd1

6174 6077 3252 5832 2383 3298 4716 5828 3629d2

710 5248 1379 5681 3941 1977 107 15978 106d3



3)) The WRV












1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



05

00

01

02

03

04

Prob

abili

ty


(a5) Drought index


000

005

010

015

020

025

030

035

Prob

abili

ty





High WRVExtreme WRV


High WRVExtreme WRV



1 2 3 4 5 6 7 8 900

01

02

03

04

05

06

Prob

abili

ty


00

01

02

03

04

05

06

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



00

01

02

03

04

05

Prob

abili

ty


00

01

02

03

04

05

06

07

08

09

Prob

abili

ty


00

01

02

03

04

05

06

07

08

09

Prob

abili

ty










High WRVExtreme WRV


High WRVExtreme WRV

Figure 5 Continued



000

005

010

015

020

025

030

035

Prob

abili

ty



00

01

02

03

04

05

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV






3)) For example






2)) For






1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





06

05

04

03

02

01

00

Prob

abili

ty



08

07

06

05

04

03

02

01

00

Prob

abili

ty



10

09

08

07

06

05

04

03

02

01

00

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV

Figure 6 Continued


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



High WRVExtreme WRV










1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





High WRVExtreme WRV


High WRVExtreme WRV

00

01

02

03

04

05

06

07

08

09

10

Prob

abili

ty


00

01

02

03

04

05

06

07

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



5 Conclusions





000

005

010

015

020

025

030

035

Prob

abili

ty



High WRVExtreme WRV

(a)

5

7

9

4

6

82

3 1

1

11

Extreme WRVHigh WRV

(b)






Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 119875(119910119895119894)119894=125




119886119895=

(sum5

119894=1119910119895119894)

5

120590119895=radic

sum5

119894=1(119910119895119894minus 119886119895)2

4

119862V119895 =120590119895

119886119895

(2)








119862V119895119894 = 119862V119895 (3)




120590119895119894= 119862V119895119894 sdot 119910119895119894 (4)



119905119895119896

=

119909119895119896

minus 119910119895119894

120590119895119894

(5)



of 119909119895119896


| 119910119895119894) of every

indicator with (6)

119875 (119909119895119896

| 119910119895119894) = 2 (1 minus Φ (

10038161003816100381610038161003816119905119895119896

10038161003816100381610038161003816)) (6)


1 2 4) and Φ(|119905119895119896|) = int

119905119895119896


minus1199062119889119906


119875 (119910119895119894| 119909119895119896) =

119875 (119910119895119894) 119875 (119909

119895119896| 119910119895119894)

sum5

119894=1119875 (119910119895119894) 119875 (119909

119895119896| 119910119895119894)

(7)




119875119894=

119898

sum

119895=1

120596119895119875 (119910119895119894| 119909119895119896) (8)





119895=1120596119895= 1 Here we




119894is the




le 1 sum119898119895=1

120596119895

= 1 119898 = 4119896 = 1




119909119894(119896) = 119909

119894(1) 119909

119894(2) 119909

119894(119899) (119894 = 1 2 119905) (9)

119910119895(119896) = 119910

119895(1) 119910

119895(2) 119910

119895(119899) (119895 = 1 2 119898)

(10)







1199091015840

119894(119896) =

119909119894(119896) minus 119909

1(119896)

119909119905(119896) minus 119909

1(119896)

1199091(119896) minus 119909

119894(119896)

1199091(119896) minus 119909

119905(119896)

(11)


1199101015840

119895(119896) =

0 119910119895(119896) le 119909

1(119896)

119910119895(119896) minus 119909

1(119896)

119909119905(119896) minus 119909

1(119896)

1199091(119896) lt 119910

119895(119896) lt 119909

119905(119896)

1 119910119895(119896) ge 119909

119905(119896)

(12)


1199101015840

119895(119896) =

0 119910119895(119896) ge 119909

1(119896)

1199091(119896) minus 119910

119895(119896)

1199091(119896) minus 119909

119905(119896)

1199091(119896) gt 119910

119895(119896) gt 119909

119905(119896)

1 119910119895(119896) le 119909

119905(119896)

(13)


Δ119895119894(119896) =

100381610038161003816100381610038161199101015840

119895(119896) minus 119909

1015840

119894(119896)

10038161003816100381610038161003816

Δmax = max max 100381610038161003816100381610038161199101015840

119895(119896) minus 119909

1015840

119894(119896)

10038161003816100381610038161003816

Δmin = min min 100381610038161003816100381610038161199101015840

119895(119896) minus 119909

1015840

119894(119896)

10038161003816100381610038161003816

(14)


1015840

119895(119896) and

1199091015840




119895119894(119896)

120577119895119894(119896) =

Δmin + 05ΔmaxΔ119895119894(119896) + 05Δmax

(15)


119903119895119894=

119899

sum

119896=1

120582119894120577119895119894(119896) (16)




be regarded as 120577119895119894




119899) that


119894= (1198751 1198752 119875

119899) that


and (18)


119883119894=

119875119894

119862119894

(17)


119883119894=

119862119894

119875119894

(18)


119881119894= 120582119894times 119883119894 (19)


119894is the

weight of indicator



119894


119894




1) 4(a

2) 4(a

3) 4(a

4) and 4(a

5) However the





A

a1

38720 40640 40640 40910 40570 37360 45650 38580 44680a2

2083 2775 2271 2265 1412 2190 2277 1883 1830a3

452 684 560 656 650 406 790 827 345a4

258 476 309 480 243 294 295 568 165a5

300 242 235 279 205 267 221 219 182

B

b1

6854 47445 29477 45845 43882 27144 66783 43890 65250b2

19826 115804 131714 73619 41231 31141 57093 46404 52304b3

19444 33677 42260 48515 19368 13814 29607 37202 25080b4

149 196 925 781 381 1230 185 1719 644b5

087 098 094 093 096 109 137 096 099b6

17441 6761 10664 10058 4833 4546 4819 8601 3369b7

7000 6500 6800 7000 6200 6000 4500 4300 4100

C

c1

7624 5094 4759 5918 7683 5836 7169 5100 5637c2

18086 19140 12931 9061 28019 8288 36239 50588 4866c3

346 113 032 256 040 042 023 147 062c4

4982 5003 7139 6365 6154 8999 4286 0 6664c5

6574 6332 6037 7078 6357 6382 6451 6218 5038c6

7000 5000 6500 6800 5000 5500 3500 3500 3000

Dd1

6174 6077 3252 5832 2383 3298 4716 5828 3629d2

710 5248 1379 5681 3941 1977 107 15978 106d3



3)) The WRV












1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



05

00

01

02

03

04

Prob

abili

ty


(a5) Drought index


000

005

010

015

020

025

030

035

Prob

abili

ty





High WRVExtreme WRV


High WRVExtreme WRV



1 2 3 4 5 6 7 8 900

01

02

03

04

05

06

Prob

abili

ty


00

01

02

03

04

05

06

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



00

01

02

03

04

05

Prob

abili

ty


00

01

02

03

04

05

06

07

08

09

Prob

abili

ty


00

01

02

03

04

05

06

07

08

09

Prob

abili

ty










High WRVExtreme WRV


High WRVExtreme WRV

Figure 5 Continued



000

005

010

015

020

025

030

035

Prob

abili

ty



00

01

02

03

04

05

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV






3)) For example






2)) For






1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





06

05

04

03

02

01

00

Prob

abili

ty



08

07

06

05

04

03

02

01

00

Prob

abili

ty



10

09

08

07

06

05

04

03

02

01

00

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV

Figure 6 Continued


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



High WRVExtreme WRV










1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





High WRVExtreme WRV


High WRVExtreme WRV

00

01

02

03

04

05

06

07

08

09

10

Prob

abili

ty


00

01

02

03

04

05

06

07

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



5 Conclusions





000

005

010

015

020

025

030

035

Prob

abili

ty



High WRVExtreme WRV

(a)

5

7

9

4

6

82

3 1

1

11

Extreme WRVHigh WRV

(b)






Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1199091015840

119894(119896) =

119909119894(119896) minus 119909

1(119896)

119909119905(119896) minus 119909

1(119896)

1199091(119896) minus 119909

119894(119896)

1199091(119896) minus 119909

119905(119896)

(11)


1199101015840

119895(119896) =

0 119910119895(119896) le 119909

1(119896)

119910119895(119896) minus 119909

1(119896)

119909119905(119896) minus 119909

1(119896)

1199091(119896) lt 119910

119895(119896) lt 119909

119905(119896)

1 119910119895(119896) ge 119909

119905(119896)

(12)


1199101015840

119895(119896) =

0 119910119895(119896) ge 119909

1(119896)

1199091(119896) minus 119910

119895(119896)

1199091(119896) minus 119909

119905(119896)

1199091(119896) gt 119910

119895(119896) gt 119909

119905(119896)

1 119910119895(119896) le 119909

119905(119896)

(13)


Δ119895119894(119896) =

100381610038161003816100381610038161199101015840

119895(119896) minus 119909

1015840

119894(119896)

10038161003816100381610038161003816

Δmax = max max 100381610038161003816100381610038161199101015840

119895(119896) minus 119909

1015840

119894(119896)

10038161003816100381610038161003816

Δmin = min min 100381610038161003816100381610038161199101015840

119895(119896) minus 119909

1015840

119894(119896)

10038161003816100381610038161003816

(14)


1015840

119895(119896) and

1199091015840




119895119894(119896)

120577119895119894(119896) =

Δmin + 05ΔmaxΔ119895119894(119896) + 05Δmax

(15)


119903119895119894=

119899

sum

119896=1

120582119894120577119895119894(119896) (16)




be regarded as 120577119895119894




119899) that


119894= (1198751 1198752 119875

119899) that


and (18)


119883119894=

119875119894

119862119894

(17)


119883119894=

119862119894

119875119894

(18)


119881119894= 120582119894times 119883119894 (19)


119894is the

weight of indicator



119894


119894




1) 4(a

2) 4(a

3) 4(a

4) and 4(a

5) However the





A

a1

38720 40640 40640 40910 40570 37360 45650 38580 44680a2

2083 2775 2271 2265 1412 2190 2277 1883 1830a3

452 684 560 656 650 406 790 827 345a4

258 476 309 480 243 294 295 568 165a5

300 242 235 279 205 267 221 219 182

B

b1

6854 47445 29477 45845 43882 27144 66783 43890 65250b2

19826 115804 131714 73619 41231 31141 57093 46404 52304b3

19444 33677 42260 48515 19368 13814 29607 37202 25080b4

149 196 925 781 381 1230 185 1719 644b5

087 098 094 093 096 109 137 096 099b6

17441 6761 10664 10058 4833 4546 4819 8601 3369b7

7000 6500 6800 7000 6200 6000 4500 4300 4100

C

c1

7624 5094 4759 5918 7683 5836 7169 5100 5637c2

18086 19140 12931 9061 28019 8288 36239 50588 4866c3

346 113 032 256 040 042 023 147 062c4

4982 5003 7139 6365 6154 8999 4286 0 6664c5

6574 6332 6037 7078 6357 6382 6451 6218 5038c6

7000 5000 6500 6800 5000 5500 3500 3500 3000

Dd1

6174 6077 3252 5832 2383 3298 4716 5828 3629d2

710 5248 1379 5681 3941 1977 107 15978 106d3



3)) The WRV












1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



05

00

01

02

03

04

Prob

abili

ty


(a5) Drought index


000

005

010

015

020

025

030

035

Prob

abili

ty





High WRVExtreme WRV


High WRVExtreme WRV



1 2 3 4 5 6 7 8 900

01

02

03

04

05

06

Prob

abili

ty


00

01

02

03

04

05

06

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



00

01

02

03

04

05

Prob

abili

ty


00

01

02

03

04

05

06

07

08

09

Prob

abili

ty


00

01

02

03

04

05

06

07

08

09

Prob

abili

ty










High WRVExtreme WRV


High WRVExtreme WRV

Figure 5 Continued



000

005

010

015

020

025

030

035

Prob

abili

ty



00

01

02

03

04

05

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV






3)) For example






2)) For






1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





06

05

04

03

02

01

00

Prob

abili

ty



08

07

06

05

04

03

02

01

00

Prob

abili

ty



10

09

08

07

06

05

04

03

02

01

00

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV

Figure 6 Continued


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



High WRVExtreme WRV










1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





High WRVExtreme WRV


High WRVExtreme WRV

00

01

02

03

04

05

06

07

08

09

10

Prob

abili

ty


00

01

02

03

04

05

06

07

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



5 Conclusions





000

005

010

015

020

025

030

035

Prob

abili

ty



High WRVExtreme WRV

(a)

5

7

9

4

6

82

3 1

1

11

Extreme WRVHigh WRV

(b)






Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












A

a1

38720 40640 40640 40910 40570 37360 45650 38580 44680a2

2083 2775 2271 2265 1412 2190 2277 1883 1830a3

452 684 560 656 650 406 790 827 345a4

258 476 309 480 243 294 295 568 165a5

300 242 235 279 205 267 221 219 182

B

b1

6854 47445 29477 45845 43882 27144 66783 43890 65250b2

19826 115804 131714 73619 41231 31141 57093 46404 52304b3

19444 33677 42260 48515 19368 13814 29607 37202 25080b4

149 196 925 781 381 1230 185 1719 644b5

087 098 094 093 096 109 137 096 099b6

17441 6761 10664 10058 4833 4546 4819 8601 3369b7

7000 6500 6800 7000 6200 6000 4500 4300 4100

C

c1

7624 5094 4759 5918 7683 5836 7169 5100 5637c2

18086 19140 12931 9061 28019 8288 36239 50588 4866c3

346 113 032 256 040 042 023 147 062c4

4982 5003 7139 6365 6154 8999 4286 0 6664c5

6574 6332 6037 7078 6357 6382 6451 6218 5038c6

7000 5000 6500 6800 5000 5500 3500 3500 3000

Dd1

6174 6077 3252 5832 2383 3298 4716 5828 3629d2

710 5248 1379 5681 3941 1977 107 15978 106d3



3)) The WRV












1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



05

00

01

02

03

04

Prob

abili

ty


(a5) Drought index


000

005

010

015

020

025

030

035

Prob

abili

ty





High WRVExtreme WRV


High WRVExtreme WRV



1 2 3 4 5 6 7 8 900

01

02

03

04

05

06

Prob

abili

ty


00

01

02

03

04

05

06

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



00

01

02

03

04

05

Prob

abili

ty


00

01

02

03

04

05

06

07

08

09

Prob

abili

ty


00

01

02

03

04

05

06

07

08

09

Prob

abili

ty










High WRVExtreme WRV


High WRVExtreme WRV

Figure 5 Continued



000

005

010

015

020

025

030

035

Prob

abili

ty



00

01

02

03

04

05

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV






3)) For example






2)) For






1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





06

05

04

03

02

01

00

Prob

abili

ty



08

07

06

05

04

03

02

01

00

Prob

abili

ty



10

09

08

07

06

05

04

03

02

01

00

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV

Figure 6 Continued


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



High WRVExtreme WRV










1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





High WRVExtreme WRV


High WRVExtreme WRV

00

01

02

03

04

05

06

07

08

09

10

Prob

abili

ty


00

01

02

03

04

05

06

07

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



5 Conclusions





000

005

010

015

020

025

030

035

Prob

abili

ty



High WRVExtreme WRV

(a)

5

7

9

4

6

82

3 1

1

11

Extreme WRVHigh WRV

(b)






Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



05

00

01

02

03

04

Prob

abili

ty


(a5) Drought index


000

005

010

015

020

025

030

035

Prob

abili

ty





High WRVExtreme WRV


High WRVExtreme WRV



1 2 3 4 5 6 7 8 900

01

02

03

04

05

06

Prob

abili

ty


00

01

02

03

04

05

06

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



00

01

02

03

04

05

Prob

abili

ty


00

01

02

03

04

05

06

07

08

09

Prob

abili

ty


00

01

02

03

04

05

06

07

08

09

Prob

abili

ty










High WRVExtreme WRV


High WRVExtreme WRV

Figure 5 Continued



000

005

010

015

020

025

030

035

Prob

abili

ty



00

01

02

03

04

05

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV






3)) For example






2)) For






1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





06

05

04

03

02

01

00

Prob

abili

ty



08

07

06

05

04

03

02

01

00

Prob

abili

ty



10

09

08

07

06

05

04

03

02

01

00

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV

Figure 6 Continued


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



High WRVExtreme WRV










1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





High WRVExtreme WRV


High WRVExtreme WRV

00

01

02

03

04

05

06

07

08

09

10

Prob

abili

ty


00

01

02

03

04

05

06

07

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



5 Conclusions





000

005

010

015

020

025

030

035

Prob

abili

ty



High WRVExtreme WRV

(a)

5

7

9

4

6

82

3 1

1

11

Extreme WRVHigh WRV

(b)






Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 5 6 7 8 900

01

02

03

04

05

06

Prob

abili

ty


00

01

02

03

04

05

06

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



00

01

02

03

04

05

Prob

abili

ty


00

01

02

03

04

05

06

07

08

09

Prob

abili

ty


00

01

02

03

04

05

06

07

08

09

Prob

abili

ty










High WRVExtreme WRV


High WRVExtreme WRV

Figure 5 Continued



000

005

010

015

020

025

030

035

Prob

abili

ty



00

01

02

03

04

05

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV






3)) For example






2)) For






1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





06

05

04

03

02

01

00

Prob

abili

ty



08

07

06

05

04

03

02

01

00

Prob

abili

ty



10

09

08

07

06

05

04

03

02

01

00

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV

Figure 6 Continued


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



High WRVExtreme WRV










1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





High WRVExtreme WRV


High WRVExtreme WRV

00

01

02

03

04

05

06

07

08

09

10

Prob

abili

ty


00

01

02

03

04

05

06

07

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



5 Conclusions





000

005

010

015

020

025

030

035

Prob

abili

ty



High WRVExtreme WRV

(a)

5

7

9

4

6

82

3 1

1

11

Extreme WRVHigh WRV

(b)






Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











000

005

010

015

020

025

030

035

Prob

abili

ty



00

01

02

03

04

05

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV






3)) For example






2)) For






1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





06

05

04

03

02

01

00

Prob

abili

ty



08

07

06

05

04

03

02

01

00

Prob

abili

ty



10

09

08

07

06

05

04

03

02

01

00

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV

Figure 6 Continued


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



High WRVExtreme WRV










1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





High WRVExtreme WRV


High WRVExtreme WRV

00

01

02

03

04

05

06

07

08

09

10

Prob

abili

ty


00

01

02

03

04

05

06

07

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



5 Conclusions





000

005

010

015

020

025

030

035

Prob

abili

ty



High WRVExtreme WRV

(a)

5

7

9

4

6

82

3 1

1

11

Extreme WRVHigh WRV

(b)






Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





06

05

04

03

02

01

00

Prob

abili

ty



08

07

06

05

04

03

02

01

00

Prob

abili

ty



10

09

08

07

06

05

04

03

02

01

00

Prob

abili

ty


High WRVExtreme WRV


High WRVExtreme WRV

Figure 6 Continued


1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



High WRVExtreme WRV










1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





High WRVExtreme WRV


High WRVExtreme WRV

00

01

02

03

04

05

06

07

08

09

10

Prob

abili

ty


00

01

02

03

04

05

06

07

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



5 Conclusions





000

005

010

015

020

025

030

035

Prob

abili

ty



High WRVExtreme WRV

(a)

5

7

9

4

6

82

3 1

1

11

Extreme WRVHigh WRV

(b)






Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty



High WRVExtreme WRV










1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





High WRVExtreme WRV


High WRVExtreme WRV

00

01

02

03

04

05

06

07

08

09

10

Prob

abili

ty


00

01

02

03

04

05

06

07

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



5 Conclusions





000

005

010

015

020

025

030

035

Prob

abili

ty



High WRVExtreme WRV

(a)

5

7

9

4

6

82

3 1

1

11

Extreme WRVHigh WRV

(b)






Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 5 6 7 8 900

01

02

03

04

05

Prob

abili

ty





High WRVExtreme WRV


High WRVExtreme WRV

00

01

02

03

04

05

06

07

08

09

10

Prob

abili

ty


00

01

02

03

04

05

06

07

Prob

abili

ty


00

01

02

03

04

05

Prob

abili

ty



5 Conclusions





000

005

010

015

020

025

030

035

Prob

abili

ty



High WRVExtreme WRV

(a)

5

7

9

4

6

82

3 1

1

11

Extreme WRVHigh WRV

(b)






Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










000

005

010

015

020

025

030

035

Prob

abili

ty



High WRVExtreme WRV

(a)

5

7

9

4

6

82

3 1

1

11

Extreme WRVHigh WRV

(b)






Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article A Bayesian Method for Water Resources ...downloads.hindawi.com/journals/mpe/2015/120873.pdf · Research Article A Bayesian Method for Water Resources Vulnerability