Research Article Calibration of Conceptual Rainfall-Runoff ...downloads.hindawi.com/journals/amete/2015/545376.pdf · global optimization method was used to calibrate the Xinanjiang

Research ArticleCalibration of Conceptual Rainfall-Runoff Models UsingGlobal Optimization

Chao Zhang12 Ru-bin Wang12 and Qing-xiang Meng123

1Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering Hohai University Nanjing 210098 China2Research Institute of Geotechnical Engineering Hohai University Nanjing 210098 China3Department of Civil amp Environmental Engineering University of Waterloo Waterloo ON Canada N2L 3G1

Correspondence should be addressed to Chao Zhang zchohaifoxmailcom

Received 19 November 2014 Revised 9 February 2015 Accepted 19 February 2015

Academic Editor Hann-Ming H Juang

Copyright copy 2015 Chao Zhang 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

Parameter optimization for the conceptual rainfall-runoff (CRR) model has always been the difficult problem in hydrology sincewatershed hydrological model is high-dimensional and nonlinear with multimodal and nonconvex response surface and itsparameters are obviously related and complementary In the research presented here the shuffled complex evolution (SCE-UA)global optimization method was used to calibrate the Xinanjiang (XAJ) model We defined the ideal data and applied the methodto observed data Our results show that in the case of ideal data the data length did not affect the parameter optimization for thehydrological model If the objective function was selected appropriately the proposed method found the true parameter values Inthe case of observed data we applied the technique to different lengths of data (1 2 and 3 years) and compared the results withideal data We found that errors in the data and model structure lead to significant uncertainties in the parameter optimization

1 Introduction

Over the past few decades a wide range of conceptualrainfall-runoff (CRR) models have been developed CRRmodels are oftenpreferable to other types ofwatershedhydro-logical models (eg physically basedmodels)This is becausethey have a reasonable accuracy and are simpler to computein many practical cases because we may only require runoffprocess estimates from rainfall at the watershed outlet or agiven location

CRR models that simulate watershed hydrological pro-cesses using methods from mathematical physics can beexpressed in terms of their model structure and param-eters In other words in addition to the rationality ofthe model structure the parameters directly determine theaccuracy of the model and its forecasts In theory most CRRmodel parameters could be determined directly or indirectlythrough measurements or a physical method because theyhave physical meaning The process of model parameter

conditioning to historical system response data is calledcalibration

There are generally two types of optimization proceduresmanual selection and automatic optimization The manualselection process relies on certain subjectivity so the resultscan vary from person to person It requires some experienceand understanding of the model structure Additionallymanual selection can be laborious and time consumingespecially for inexperienced hydrology workers Automaticoptimization methods have become increasingly popularbecause of rapid developments in computing technology Forexample genetic algorithms the shuffled complex evolutionmethod (SCE-UA) the multiple start simplex adaptive ran-dom search particle swarm optimization (PSO) multiobjec-tive shuffled complex evolutionmetropolis (MOSCEM) algo-rithm anddynamically dimensioned search (DDS) have beensuccessfully applied to model calibration [1ndash11] However ifa model is calibrated by either of these two procedures wecannot be certain that we will obtain a unique set of optimalparameters for a CRR model

Hindawi Publishing CorporationAdvances in MeteorologyVolume 2015 Article ID 545376 12 pageshttpdxdoiorg1011552015545376

2 Advances in Meteorology

Despite nearly 20 years of research little progress hasbeen made on the parameter estimation problem for CRRmodels since objective function response surface typicallycontains hundreds if not thousands of local optima nestedat several scales [3] In the research presented here we usedthe SCE-UA method to calibrate the XAJ model The XAJmodel is extensively used throughout the world to analyzethe stability of parameters from automatic optimizationmethods It considers the hydrological data parameter searchinterval objective functions and other aspects

2 XAJ Model

21 The Model Structure and Its Main Characteristics TheXAJ [12 13] model was developed in 1973 by the East ChinaCollege of Hydraulic Engineering (now Hohai University)Its underlying aim was to forecast flows to the XinanjiangreservoirThemodel has been successfully andwidely appliedin humid and semihumid regions It is based on ldquorunoffformation at the natural storagerdquo which is the distinguishingfeature of the XAJ model when compared to other modelsThe basin is divided into a set of subbasins using a methodsuch as Thiessen polygon modification considering theuneven distribution of rainfall and the underlying surfaceThen the discharge curve on the outlet section of eachsubbasin is simulated and flood rooting is determinedFinally the total discharge is obtained using a simple sumTheXAJmodel is composed of fourmodules the evaporationmodule the runoff productionmodule the runoff separationmodule and the runoff concentration module

Figure 1 shows the flow chart of the XAJ model Rainfall(119875) and water-surface evaporation (EM) are the input dataand the discharge curve for the outlet section (119876) and theevaporation of the watershed (119864) are the output results Thestate variables are in boxes and the model parameters areoutside the boxes (Figure 1) For this research 119882 is thetension water storage WU is the upper layer tension waterstorage WL is the lower layer tension water storage FR is therunoff contributing area factor 119878 is the free water storageRS is the surface runoff RI is the interflow runoff RG isthe ground water runoff QS is the surface flow QI is theinterflow and QG is the ground water flow

The XAJ model has several characteristics that can besummarized as follows

(1) The rainfall-runoff process is divided into two stagesrunoff generation and concentration in the water-shed It is thought that in the runoff yield stagerunoff is produced only after the deficit of the vadosezone is satisfied A homogeneous vadose zone issubject to ground water flow and excess surface flowinfiltration Interflow will be produced in a vadosezone with a relatively impermeable layer in additionto ground water flow and saturated overland flowIn the runoff concentration stage the river networkand subbasin concentration can be considered astwo types of watershed concentration The subbasincombined with a river network can fully embody thewatershed concentration Because the XAJ model is

very compatible in the treatment of the watershedconcentration the subbasin and river network con-centration are commonly represented by the Shermanunit hydrograph and Muskingum methods for suc-cessive routing by subreaches [14] as well as othermethods (such as Clark method etc)

(2) A three-layer evaporation model is used Here theldquolayerrdquo takes soil moisture constants such as thefield capacity and wilting point as thresholds Inaddition to the soil moisture constants the soilevaporation ability is an important factor in the three-layer evaporation model and has a great effect onthe accuracy It is generally difficult to directly obtainan accurate value using instrumental observationsTherefore in the XAJ model the measured water-surface evaporation is revised by a correction factorand improved by the water balance of the watershedIn this way we can avoid using empirical formulas tocalculate the soil evaporationThis method of dealingwith evaporation is the only one used in practicalapplications

(3) The XAJ model considers runoff separation whichmeans that the calculated flood process is more inline with the actual situation Because the runoffproduction components have different flow velocitiesthe runoff concentration results are more accurate ifwe calculate the runoff separation using the appro-priate velocities The XAJ model uses a ldquodownwardrdquostructure for the runoff separation whereas otherCRR models generally use an ldquoupwardrdquo structure

(4) The XAJ model uses a statistically significant water-shed storage capacity curve and a watershed freewater capacity curve Note that these curves are onlyapplicable to the analysis of runoff area variabilitiescaused by an uneven distribution of the underlyingsurface under the condition of a uniform rainfall spa-tial distribution Additionally for a closed watershedwe should use the upper limits of both curves Foran unclosed watershed the upper limit is infiniteThe infiltration capacity area distribution curve isused by Stanford model to account for the influenceof the uneven distribution of the underlying surfaceon the infiltration of the excess surface runoff Thisis because the watershed storage and watershed freewater capacity curves are set in the model Howeverthe XAJ model is more advanced than the other CRRmodels

22 Model Parameters The XAJ model has 17 parametersthat must be determined by the user (XAJ Model Parameterssection) when computing the flood using the Muskingummethod for successive routing by subreachesWe temporarilyset the feasible parameter space by fixing the upper and lowerparameter bounds (see Table 1) The Muskingum methodparameters were predetermined based on the observedhydrograph and were not included in the optimization

Advances in Meteorology 3

E

C

ST Q

W

EU WU

RS

RI

RG

QS

QI

QG

WL

WUM

IMP

WLMEL

ED

P EM

Wminus (WU + WL)

1 minus FR

R(1 minus IMP)

FR

SM CS

CI

CG

EX

KI

KG

R middot IMP

NKE

XE

WMK B (1 minus IMP)

Figure 1 Flow chart for the XAJ model

Table 1 Lower and upper bounds on the parameters

Parameter 119870 119861 119862 WM WUM WLM SM EX KI CS CI CG IMPLower bound 01 01 01 50 1 50 2 01 001 001 05 05 0001Upper bound 15 04 09 200 50 100 100 2 07 09 0999 0999 005

We divided the model parameters for each subbasin intofour categories according to the characteristics of the modelTheir physical significances are explained in the following

(1) Evaporation parameters are 119870 WUM WLM and119862 119870 is the reduction coefficient of the evaporationwhich is equal to the ratio of the potential evap-otranspiration to the pan evaporation WUM andWLM are areal mean water capacity tensions for theupper and lower layers of the watershed and are inthe ranges of 5ndash20mm and 60ndash90mm respectively[12 13] Their values depend on the condition of thesoil and vegetation rich soil and vegetation result inlarger values119862 depends on the proportion of the areathat is covered by vegetation with deep roots a largervalue means less evaporation It takes values in 009ndash012 in semihumid and semiarid regions and in 015ndash020 in humid regions [12 13]

(2) Runoff production parameters are IMP 119861 and WMIMP is the ratio of impervious areas (includingsaturated areas) to the total area of the basin 119861 is theinhomogeneity distribution of the water deficit in thevadose zone and is proportional to the inhomogeneityof the water deficit in the watershed WM representsthe severity of a drought and is the sum of WUMWLM and WDM (in the deepest layer) It takesvalues in the range 120ndash180mm Note that 119861 is relatedtoWM ifWM increases119861 decreases (and vice versa)

(3) Runoff separation parameters are SM EX KG andKI SM is the areal mean of the free water capacityof the surface soil layer EX is the spatial distributionof the watershedrsquos free water storage capacity KGand KI are the outflow coefficients for the free waterstorage to groundwater and interflow relationshipsrespectively The sum of KG and KI is the free waterflow velocity Because it lasts 3 days to fall we takeKG + KI = 07 so that the independence of theparameter problem can be solved

(4) Runoff concentration parameters are CG CI andCS These are regression constants for the surfacerunoff interflow runoff and groundwater runoffrespectively

The above classification of the parameters into fourgroups corresponds to different optimization characteristicsThe first group influences the total runoff production Theiraim is to balance the rainfall evaporation and runoff Thesecond and third groups determine the runoff productionand separation The fourth group is the most sensitive tochanges over time and determines the discharge process

Parameters within the same group tend to be mutuallydependent whereas those in different groups are relativelyindependent Based on the above analysis parameters of thelower numbered groups should be optimized before thoseof the higher numbered groups as was suggested by Zhao[12 13] The parameters determined for groups X-X can bedirectly applied to groups X-X This is the method used inthis paper


Table 2 Yandu River watershed

Catchmentarea (km2)

Number ofrainfallstations

Averageannual

precipitation

Averageannual

evaporation

Averageannualrunoffdepth

601 5 1657 741 1281

3 Materials and Methods

31 Study Area Yandu River watershed is located in theThree Gorges Region of Yangtze River and has a catchmentarea of 601 km2 It has a large watershed slope with anaverage gradient of 287 and an elevation drop of 2800mThewatershed is described in Table 2

There is plentiful precipitation in Yandu River watershedwith an average annual precipitation of 1337mm 68 ofwhich occurs in summer and autumnThemaximum annualrainfall is 24482mm and theminimum is 8084mm Runoffis mainly due to rain The flood season is from April toOctober and the annual maximum flood peak flow typicallyoccurs between May and July With high mountains steepslopes deep valleys and narrow rivers the floods are char-acterized by rapid confluence and sharp peak flows

Yandu River watershed belongs to mountain terrainAffected by human activities since 1990s obvious change inthe climate and underlying surface conditions is happen-ing and confluence process changes accordingly [15] Themeasured data does not present the natural condition ofwatershed that is the degree of response of runoff to rainfallis reducedHistorical records of 1981 to 1987 used in this papercould basically meet the demand of the research

32 Definition of Ideal Data Based on the physical signifi-cance of the parameters of the hydrological models and thewatershed characteristics we randomly generated a set ofparameters within the parameter search interval We tookthese as the true values of the model parameters Withthis true parameter set and the hydrologic input data (con-tinuous daily rainfall and evaporation data) we generateda sequence of streamflows which we considered the idealdata (or ldquoobservedrdquo streamflows) for the calibration timeperiod Obviously by using this ideal data in the parameteroptimization we avoid the influences of errors in the inputand output data and the model structure

33 SCE-UA Method The selection of an automatic param-eter optimization algorithm in calibration of CCR modelshas been studied extensively Virtually a lot of parameteroptimization studies have used ldquolocal-searchrdquo proceduressuch as the downhill simplex method the pattern searchmethod and the rotating directions method The ldquooptimalrdquoparameters provided by them vary with the choice of startingpoint It is now known that the objective function responsesurface contains hundreds of thousands of local optima Inthis study we used a relatively common algorithm called theSCE-UA method (an abbreviation for the shuffled complexevolution method) developed at The University of Arizona

by Duan et al [3 16 17] The SCE-UA strategy combinesthe strengths of the simplex procedure [18] with controlledrandom search [19] competitive evolution [20] and complexshuffling The algorithm with global search has proved to beboth effective and relatively efficient and provided superiorperformance compared with other optimization algorithmssuch as the downhill simplex method the genetic algorithmandmultistart versions of the downhill simplex for parameteridentification [3 4 16 17]

The SCE-UA method regards a global search as anevolutionary process Detailed descriptions and explanationsof themethod were given by Duan et al [3 16] and so will notbe repeated here Briefly themethod beginswith a populationof points sampled randomly from the feasible parameterspaceThe population is partitioned into several ldquocomplexesrdquowith each complex containing 2119899 + 1 points where 119899 is thedimension of the problem Each complex ldquoevolvesrdquo accordingto a statistical reproduction process that uses the shape ofthe simplex to direct the search in an appropriate directionAt periodic stages in the evolution the entire population isshuffled and points are reassigned to complexes to ensureinformation sharing This procedure enhances survivabilityby sharing search space information that was independentlygained by each complex The processes of competitive evo-lution and complex shuffling are inherent to the SCE-UAalgorithm and help to ensure that the information containedin the sample is efficiently and thoroughly exploited Theseproperties endow the SCE-UA method with good globalconvergence properties over a broad range of problemsIn other words given a prespecified number of functionevolutions (ie a fixed level of efficiency) the SCE-UAmethod should have a high probability of succeeding in itsobjective of finding the global optimum

The SCE-UA method contains some certain and randomfactors that are controlled by the algorithmrsquos parametersThese parameters must be carefully chosen so that themethod performs optimally They include the number ofpoints in each complex (119898) the number of points in eachsubcomplex (119902) the number of offspring that can be gener-ated by each subcomplex (120572) the number of evolution stepsfor each complex (120573) and the number of complexes (119901) Intheory 119898 can take any value greater than 1 However if 119898is too small or there are too few points in a single complexthe search process is the same as the general simplex methodThis reduces the possibility of a global search Conversely if119898 is too big the method takes too long to compute and is noteffective The SCE-UA algorithm used by Duan et al (19921994) used the values 119898 = 2119899 + 1 119902 = 119899 + 1 120572 = 1 and 120573 =2119899+1 Hence the only variable to be specified is the number ofcomplexes (119901) which depends on the optimization problemIn theory 119901 can take any integer greater than or equal to1 However if 119901 is too small for high-dimension problemthe global optima may not be found and if 119901 is too bigthe method takes too long to compute and is not effectiveGenerally speaking amore complicated problem needsmorecomplexes to obtain the global optimum in other words thevalue of 119901 should be increased The optimization results of 119901for the ideal and measured data are shown in Tables 3(a) and3(b) By fixing somemodel parameters (119862 = 015WM = 120


WUM = 20WLM = 70 and IMP = 001) we could calculatethe other optimum parameters for the ideal data as shown inTable 4

As can be seen from Tables 3(a) and 3(b) the algorithmcould not find the true parameter values when 119901 = 1 and 119901 =2 for both the ideal and measured data When 119901 = 4 or 119901 =10 besides some insensitive parameters we determined thetrue values of most parameters for the measured data Whenusing the ideal data we required 119901 gt 4

Table 4 shows that when we fixed some parameters (119862WM WUM WLM and IMP) we could determine the trueparameter values if 119901 ge 2 This further illustrates thatthe SCE-UA parameter depends on the dimensions of theoptimization problem In this paper we used 119901 = 4 for theideal data and 119901 = 10 for the measured data

34 Objective Functions The selection of an appropriateobjective function has been discussed extensively The selec-tion is full of subjectivity It would affect the calibration ofmodel if the function lacks considering random factors ofdata The most commonly used functions have been simpleleast squares (SLS) heteroscedastic error maximum likeli-hood estimation (HMLE) determination coefficient (DY)and multiobjective function For CRR model multiobjectivefunction usually is converted into some specific operationalfunction like that which considers the total runoff errorand evaluates the water balance or the relative error of thecalculated discharge and so on XAJ model generally couldbe divided into daily rainfall-runoff simulation and floodsimulation Due to the different task background there isalso a difference of the selection of objective function Theformer focuses on the water balance and the latter in additionto the consideration of water balance also focuses on theflood peak simulation Different functions evaluate differentcharacteristics of the hydrological process The objectivefunctions directly influence the results of the optimizationThe most commonly used functions are as follows Thefirst considers the total runoff error and evaluates the waterbalance It is

Obj1= min

100381610038161003816100381610038161003816100381610038161003816

(sum119899

119894=1119876obs (119894) minus sum

119899

119894=1119876cal (119894))

sum119899

119894=1119876obs (119894)

100381610038161003816100381610038161003816100381610038161003816

(1)

where 119876obs(119894) is the measured discharge 119876cal(119894) is the calcu-lated discharge and 119899 is the length of the measured data Thesecond considers the relative error of the calculated dischargeand is defined as

Obj2= min

sum119899

119894=1

1003816100381610038161003816119876obs (119894) minus 119876cal (119894)1003816100381610038161003816

sum119899

119894=1119876obs (119894)

(2)

The third is a determination coefficient proposed by [21]which reflects the accuracy of the simulation results It isdefined as

DY = max

1 minussum119899

119894=1[119876obs(119894) minus 119876cal(119894)]

2

sum119899

119894=1[119876obs(119894) minus 119876obs]

2

(3)

A larger DY indicates a better fit between the measuredand calculated discharges

35 Stopping Criterion for the Iterative Process Parameteroptimization methods are generally iterative In theory thereis a point in the parameter space that achieves the optimumvalue of the objective function However in practice we donot know when we have reached this point and we maynot find it at all We thus require a termination conditionso that method finds an optimal parameter value within theoptimum running time The most commonly used stoppingcriteria have been objective function convergence criterionparameter convergence and maximum iteration number

(1) The convergence of the objective functionthat is

100381610038161003816100381610038161003816100381610038161003816

(119891119894minus119896minus 119891)

119891119894

100381610038161003816100381610038161003816100381610038161003816

le TOL (4)

where 119891119894and 119891

119894minus119896are objective function values for

the 119894th and the (119894 minus 119896)th iterations and TOL isthe tolerance The values of TOL and 119896 are bothdetermined according to practical conditions If thefunction value after 119896 iterations does not improve theprecision we stop the search The optimal position ismost likely on a relatively flat response surface Thefunctionrsquos convergence is useful when consideringinaccurate solutions

(2) The convergence of the parametersthat is

100381610038161003816100381610038161003816100381610038161003816

(120582119894minus119896(119895) minus 120582

119894(119895))

(120582max (119895) minus 120582min (119895))

100381610038161003816100381610038161003816100381610038161003816

le TOL120582 (5)

where 120582119894(119895) and 120582

119894minus119896(119895) are the 119895th parameter values

at the 119894th and 119894 minus 119896th iterations 120582max(119895) and 120582min(119895)are the maximum and minimum values of 119895 for allparameter combinations and TOL

120582is a tolerance

value If the parameter does not obviously change over119896 iterations we stop the search

(3) The maximum iteration numberto prevent a dead cycle we set a maximum numberof iterations in advance If we exceed this we stop thesearch

The parameter convergence criterion is more suitablefor parameter optimization because it terminates when theparameters are not obviously changing [22] The maximumiteration criterion should also be used to avoid wastingcomputational time If the algorithm does not terminatewithin a reasonable number of iterations we must check thecomputational procedure

We controlled the iterations using the following values

(i) for the objective functions 119896 = 10 and TOL = 10minus6(ii) for the parameter convergence 119896 = 10 and TOL

120582=

10minus4

(iii) for the maximum iteration number 119896max = 106


Table 3 (a) Parameter optimization results using ideal data (b) Parameter optimization results using measured data

(a)

Parameter 119870 119861 119862 WM WUM WLM SM EX KI CS CI CG IMPTrue value 1027 0204 0285 162077 35262 71240 11888 1436 0420 0497 0608 0610 0039119901 = 1 1026 0207 0102 162919 32687 79639 11995 1443 0681 0497 0603 0767 0037119901 = 2 1027 0204 0280 162167 35036 72046 11889 1437 0222 0497 0609 0609 0039119901 = 4 1027 0204 0280 162077 35263 71240 11888 1436 0420 0497 0609 0610 0039119901 = 10 1027 0204 0285 162077 35262 71240 11888 1436 0420 0497 0608 0610 0039

(b)


Table 4 Optimization results for the remaining parameters after fixing some parameters using the ideal data

Parameter 119870 119861 SM EX KI CS CI CGTrue value 1025 0192 6036 1825 0511 0215 0607 0894119901 = 1 1025 0192 604 1825 051 0215 0606 0894119901 = 2 1025 0192 6036 1825 0511 0215 0607 0894119901 = 4 1025 0192 6036 1825 0511 0215 0607 0894119901 = 10 1025 0192 6036 1825 0511 0215 0607 0894

If any criterion is satisfied the optimization terminatesThe experience with the SCE-UA algorithm indicated thatafter about a set number of shuffling iterations the parameterestimates would stabilize in a region where search wouldsubsequently terminate due to parameter convergence ormaximum iteration criterion to avoid wasting computationaltime

4 Results and Discussion

41 Parameter Optimization Using Ideal Data

411 The Effect of the Objective Function We considered theeffect of the objective function on parameter optimization fordaily rainfall-runoff and flood simulations

For the daily rainfall-runoff simulation we used Obj1

Obj2 DY and a simple combination of Obj

1and Obj

2

as the objective functions We used default values for thealgorithm parameters to calibrate the model and ran thealgorithm 10 times for each objective function Table 5 showsthe parameter optimization results for the ideal data from1981ndash1985 for the Yandu River watershed as calculated by theSCE-UA method

From Table 5 we can see that the objective functions canall be applied to the parameter optimization using ideal dataexcept for Obj

1 which could not find the true values

Table 5 shows that the parameter optimization results areconsistent with the true values in addition to 119862 We ran theoptimization on the ideal data to remove the effect of errors

in the data and model structure There is a big differencebetween the calibrated and true values for 119862 mainly becauseit is a coefficient for deep evapotranspiration and depends onthe coverage area of deep-rooted plants Deep evaporationis rare in humid areas so 119862 is not sensitive enough for theoptimization to obtain its true value

Less iterations were required when using DY HoweverObj2produced objective function values that were closer

to the true values The results obtained using Obj2were

(0000010 0000010 0000010 0000010 0000009 00000090000009 0000010 0000010 0000010) and using DY were(0000073 0000071 0000048 0000033 0000067 00001050000081 0000040 0000054 0000079) Considering theoptimization speed DY was quicker to achieve the optimalvalue

It is difficult to find the true value using only Obj1

because it reflects the total error between the measured andcalculated total flow and focuses on the overall water balanceBecause of the mutual compensation between parametersmore than one set of parameters can satisfy the total waterbalance That is the optimization results are not uniqueTo obtain a good combination of parameters we shouldsimultaneously use other functions

These results show that correlations between parametersare important to parameter optimizationThe choice of objec-tive function has an impact on the parameter optimizationand the results vary with different functions To exclude theinfluence of errors in the data and model structure we usedthe ideal data to investigate the daily rainfall-runoff model


Table 5 Characteristics of the optimization results for the daily rainfall-runoff simulation

Parameter Truevalue

Obj1 Obj2 DYMaximum Average Variance Maximum Average Variance Maximum Average Variance

119870 0533 0538 0533 92119864 minus 06 0533 0533 0 0533 0533 0119861 0306 0323 0306 53119864 minus 05 0306 0306 0 0306 0306 0119862 0149 0253 0195 95119864 minus 04 03 0177 54119864 minus 03 0286 0203 30119864 minus 03

WM 153246 155029 153828 83119864 minus 01 153246 153246 0 153246 153246 0WUM 18286 20798 19499 39119864 minus 01 18286 18286 0 18286 18286 0WLM 89436 86524 80176 26119864 + 01 89436 89436 0 89436 89436 0SM 14942 24203 15175 23119864 + 01 14942 14942 0 14942 14942 0EX 072 1908 1479 74119864 minus 02 072 072 0 072 072 0KI 0665 0693 0508 14119864 minus 02 0665 0665 0 0665 0665 0CS 0058 0408 0222 71119864 minus 03 0058 0058 0 0058 0058 0CI 054 0656 0559 18119864 minus 03 054 054 0 054 054 0CG 0548 0854 0735 58119864 minus 03 0548 0548 0 0548 0548 0IMP 003 0042 0029 72119864 minus 05 003 003 0 003 003 0Averageiterations 84 522 139

Note that the average iterations are the average number of evolutions of complexes

The SCE-UA method found the true parameter values whenusing an appropriate objective function

For flood simulation we used Obj2 DY and a simple

combination of Obj1andObj

2as objective functionsWe first

fixed the parameters that did not depend on the computingtime (119870 = 0533 WUM = 18286 WLM = 89436119862 = 0149 WM = 153246 119861 = 0306 IMP = 003 andEX = 072) We then optimized the remaining parameters forflood simulation The results calculated using these objectivefunctions are listed in Table 6

Table 6 shows that the global optimum was achievedwith these objective functions because the targets of theseobjective functions are consistent with the flood simulation

412 The Effect of the Data Length We randomly generateda set of parameters within the parameter search space for theXAJ model and used these as the true values (see Table 7)Then we calculated the ideal data using the true values andthe daily areal rainfall for 1981ndash1985 to study the effect ofthe data length on parameter optimizationThe results of theparameters optimized usingObj

2for different lengths of ideal

data are shown in Table 7The results are generally consistent with the true values

(except for 119862) regardless of the length of the data (1 3 or 5years) If we exclude the impact of measured data and modelstructure errors automatic optimization methods can obtainthe global optimumwhen using reasonable algorithmparam-eters That is for ideal data global optimization methodsobtained the global optimum

Table 7 shows that the results calculated for 1 3 and 5years of data are almost equal So the length of the data didnot affect the optimization method

As previously mentioned it is difficult to obtain thetrue value of 119862 However as mentioned above avoiding the

influences of errors in the input and output data and themodel structure by using ideal data the result for KI is stilldifferent from its true value which cannot be attributed todata or model structure errors The large differences in thecalibrated and true values for KI are mainly because thewater source has been divided by the structural constraintthat KI and KG are independent This demonstrates that thecorrelations between parameters are important to parameteroptimization

To more clearly reveal the parameter optimization pro-cess Figure 2 shows the optimization results using ideal datafrom 1981 (1 year) 1981ndash1983 (3 years) and 1981ndash1985 (5 years)For illustrative purposes we normalized all the parametervalues using (119909 minus 119909min)(119909max minus 119909min) = 119909new so that theyare between 0 and 1

Figure 2 demonstrates that for ideal data the parametersultimately converged to the global optimums regardless ofthe data length However some parameters had long timefluctuations and were not stable (eg WUM and WLM)Although KI was constant for a long period there were signsof volatility in later iterations 119862 had an irregular shock in theoptimization process and did not converge This illustratesthat insensitive parameters and the correlations betweenparameters have important influences on the results of theoptimization

42 Parameter Optimization Using Measured Data

421 The Effect of Objective Function As previously men-tioned if the objective function is selected appropriately thetrue parameter values can be obtained by a direct searchWe used Obj

1 Obj2 and their combination as objective

functions and used observed data from 1981ndash1985 We ran


Table 6 Parameter optimization results for the flood simulation

Parameter SM KI CS CI CG KE XETrue value 34958 0135 0284 0631 0875 1137 0313Obj2 34958 0135 0284 0631 0875 1137 0313DY 34958 0135 0284 0631 0875 1137 0313Obj1 + Obj2 34958 0135 0284 0631 0875 1137 0313

Table 7 Parameter optimization results

Parameter True value Bound Optimization results of parametersLower Upper 1 year 3 years 5 years

119870 0681 02 15 0681 0681 0681119861 049 01 09 049 049 049119862 0254 01 09 0632 0625 0627WM 152334 50 200 152334 152334 152334WUM 67516 10 100 67516 67516 67516WLM 53000 10 100 53000 53000 53000SM 74913 1 100 74913 74913 74913EX 1356 01 2 1356 1356 1356KI 0683 001 07 0513 0187 0604CS 0801 001 09 0801 0801 0801CI 0891 05 0999 0891 0891 0891CG 0988 05 0999 0988 0988 0988IMP 0033 0001 005 0033 0033 0033

the algorithm ten times for each objective function Theoptimization results are listed in Table 8

Table 8 shows that the results were different for theconsidered objective functions and that the values of someparameters (C and WM) were different when using thesame objective functionUsingmeasured data the parameteroptimization results were diverse because of the influences oferrors in the data and the model structure

422TheEffect ofData Length Aspreviouslymentioned thedata length has no effect on the optimizationwhen using idealdata We selected the combination function as the objectivefunction and applied it to measured data from 1981 to 1987We ran the algorithm ten times and compared the results Weconsidered that if the algorithm converged to the same groupparameters all ten times we had found the optimum valueThe results are listed in Table 9

The parameter optimization results using differentlengths of measured data were different In some casesall instances of the same data length were different (sevensituations when using one year of measured data six timeswhen using two consecutive years and five times when usingthree consecutive years)

The cumulative distributions of some parameters usingdifferent lengths of measured data are shown in Figure 3The119910-coordinates indicate the cumulative distribution functionsand the 119909-coordinates indicate the parameter The verticaldotted line represents the optimal parameter calculated usingthe measured data from 1981 to 1987 ldquoOne yearrdquo indicates

the optimal parameter calculated using one year of measureddata and so on

Figure 3 shows that the cumulative distributions of theparameters changed irregularly as the length of the measureddata increased That is the parameter optimization did notconverge or become stable as the length of the measured dataincreased

5 Conclusions

Several observations are noted from calibrating XAJ modelwith the SCE-UA optimization method

Models represent the core of hydrological forecastingand parameter identification is key to applying a model Inthis paper we investigated parameter optimization for theXAJ model using the SCE-UA algorithm On the basis ofprevious research we used ideal data to study the relation-ships between the data length search range of the parametersobjective functions and parameter stability The results indi-cated that the calculated global optimal parameters did notdepend on the data length but did depend on the objectivefunctions For the daily rainfall-runoff model the optimumvalues could not be determined using only the water balanceobjective function However the global optimal values couldbe found when using the appropriate objective functionThe global optimization algorithms did not have the sameaccuracy For flood simulation we fixed the parameters thatdo not depend on time and found that the choice of objectivefunction did not affect the results


0

02

04

06

08

1

0 400 800 1200

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3

(a) 1981 year

0

02

04

06

08

1

0 500 1000 1500

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3

(b) 1981sim1983 years

0

02

04

06

08

1

0 500 1000 1500

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3

(c) 1981sim1985 years

Figure 2 Convergence of the optimization using ideal data

When using observed data the optimization results weredifferent when using different objective functions and wereeven different when using the same objective function Thevalues were significantly different because of the influenceof errors in the data and model structure Although thedata length had no effect when using ideal data when usingobserved data the parameter optimization results were differ-ent That is the parameter optimization did not converge toa certain value as the length of the measured data increasedIn summary errors in the data and model structure lead touncertainties in the parameter optimization

An approach that has the flexibility of emphasizingdifferent portions of the model residuals according to onersquospreference is multiobjective optimizationThe advantage of a

multiobjective approach is to explicitly focus attention on themodel performance We will focus on this issue in the future

XAJ Model Parameters

119870 Ratio of potential evapotranspiration topan evaporation

WM Areal mean tension water capacityWUM Upper layer areal mean tension water

capacityWLM Lower layer areal mean tension water

capacity119862 Coefficient of deep evapotranspiration


00

02

04

06

08

10

0 03 06 09 12 15

CDF

K

(a) CDF versus119870

00

02

04

06

08

10

0 01 02 03 04

CDF

B

(b) CDF versus 119861

00

02

04

06

08

10

0 01 02 03

CDF

C

(c) CDF versus 119862

00

02

04

06

08

10

80 100 120 140 160 180

CDF

WM

(d) CDF versus WM

00

02

04

06

08

10

5 20 35 50

CDF

SM

1 year2 years

3 years7 years

(e) CDF versus SM

00

02

04

06

08

10

05 06 07 08 09 1

CDF

CI

1 year2 years

3 years7 years

(f) CDF versus CI

Figure 3 Cumulative distribution functions of the parameters


Table 8 Characteristics of the optimization results

Parameter 119870 119861 119862 WM WUM CS CIObj1

Minimum 0589 034 0117 119567 11729 082 0836Maximum 0604 0395 0264 122744 13454 082 0851Average 0595 0377 0191 121112 12394 082 0841Square deviation 250119864 minus 05 210119864 minus 04 160119864 minus 03 130119864 + 00 220119864 minus 01 0 250119864 minus 05

Obj2Minimum 0556 0175 0103 100 10 0138 0753Maximum 0556 0175 0265 100 10 0166 0753Average 0556 0175 0171 100 10 0155 0753Square deviation 0 0 230119864 minus 03 0 0 100119864 minus 04 0

Obj1 + Obj2Minimum 0537 0192 0134 100574 10574 001 0598Maximum 0537 0192 0279 100574 10574 001 0598Average 0537 0192 0207 100574 10574 001 0598Square deviation 0 0 180119864 minus 03 0 0 0 0

Table 9 Parameter optimization results using measured data

Parameter 1981ndash1987 One year Two consecutive years Three consecutive years1981 sdot sdot sdot 1987 1981-1982 sdot sdot sdot 1986-1987 1981ndash1983 sdot sdot sdot 1985ndash1987

119870 0571 0449

0418 0444

0529 0454

0583119861 0114 0108 0325 0226 01 01 0148119862 0247 0194 0274 0243 0284 0222 0102WM 101539 100001 100 112963 10084 111657 10118WUM 11539 10 10 22962 10839 21656 1118WLM 60 60 60 60 60 60 60SM 38901 23895 43129 45968 41708 3755 43673EX 1494 0251 1311 2 1458 1823 1464KI 0447 0482 0507 0504 0503 0475 0543CS 001 001 0081 001 0056 001 0053CI 0598 0666 0728 0613 0717 062 0718CG 0957 0987 0999 0976 0998 0964 0998IMP 005 005 005 0032 0045 005 005

IMP Ratio of impervious area to total area of basin119861 Exponent of the tension water capacity curveSM Areal mean of the free water capacity of the

surface soil layerEX Exponent of free water capacity curveKI Outflow coefficients of free water storage to

interflow relationshipsKG Outflow coefficients of free water storage to

groundwater relationshipsCS Recession constant for routing through the

channel system within each subbasinCI Recession constant of the lower interflow

storageCG Recession constant of groundwater119873 Number of subreachesKE Travel time of flood wave through subreachXE Flow ration of subreaches

Conflict of Interests

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

Acknowledgments

The study is financially supported by the National NaturalScience Foundation of China (no 51409082) the ChinaPostdoctoral Science Foundation (2013M531264) and theFundamental Research Funds for the Central Universities(no 2014B04914)

References

[1] Q J Wang ldquoThe genetic algorithms and its application tocalibrating conceptual rainfall-runoff modelsrdquoWater ResourcesResearch vol 27 no 9 pp 2467ndash2471 1991


[2] M Franchini ldquoUse of a genetic algorithm combined with alocal search method for the automatic calibration of conceptualrainfall-runoff modelsrdquo Hydrological Sciences Journal vol 41no 1 pp 21ndash37 1996

[3] S Sorooshian and V Gupta ldquoEffective and efficient globaloptimization for conceptual rainfall- runoff modelsrdquo WaterResources Research vol 28 no 4 pp 1015ndash1031 1992

[4] S Sorooshian Q Duan and V K Gupta ldquoCalibration ofrainfall-runoffmodels application of global optimization to theSacramento soil moisture accounting modelrdquo Water ResourcesResearch vol 29 no 4 pp 1185ndash1194 1993

[5] P O Yapo H V Gupta and S Sorooshian ldquoAutomatic cal-ibration of conceptual rainfall-runoff models sensitivity tocalibration datardquo Journal of Hydrology vol 181 no 1ndash4 pp 23ndash48 1996

[6] T Y Gan E M Dlamini and G F Biftu ldquoEffects of modelcomplexity and structure data quality and objective functionson hydrologic modelingrdquo Journal of Hydrology vol 192 no 1ndash4pp 81ndash103 1997

[7] V A Cooper V-T-V Nguyen and J A Nicell ldquoCalibrationof conceptual rainfall-runoff models using global optimisationmethods with hydrologic process-based parameter constraintsrdquoJournal of Hydrology vol 334 no 3-4 pp 455ndash466 2007

[8] M K Gill Y H Kaheil A Khalil M McKee and L BastidasldquoMultiobjective particle swarm optimization for parameterestimation in hydrologyrdquoWater Resources Research vol 42 no7 Article IDW07417 2006

[9] C Zhang and Y-Y Sun ldquoAutomatic calibration of conceptualrainfall-runoff modelsrdquo Applied Mechanics and Materials vol405ndash408 pp 2185ndash2189 2013

[10] J A Vrugt H V Gupta L A Bastidas W Bouten and SSorooshian ldquoEffective and efficient algorithm formultiobjectiveoptimization of hydrologic modelsrdquo Water Resources Researchvol 39 no 8 pp SWC51ndashSWC519 2003

[11] B A Tolson and C A Shoemaker ldquoDynamically dimensionedsearch algorithm for computationally efficient watershedmodelcalibrationrdquoWater Resources Research vol 43 no 1 2007

[12] R J Zhao Y L Zhuang L R Fang et al ldquoThe Xinrsquoanjiangmodel in hydrological forecastingrdquo in Proceeding of the OxfordSymposium vol 129 pp 351ndash356 IAHS Publication April 1980

[13] R J Zhao ldquoThe Xinrsquoanjiang model applied in Chinardquo Journal ofHydrology vol 135 no 1ndash4 pp 371ndash381 1992

[14] V T Chow Handbook of Applied Hydrology McGaw-Hill NewYork NY USA 1964

[15] N J Potter and F H S Chiew ldquoAn investigation into changes inclimate characteristics causing the recent very low runoff in thesouthern Murray-Darling Basin using rainfall-runoff modelsrdquoWater Resources Research vol 47 no 10 2011

[16] Q Y Duan V K Gupta and S Sorooshian ldquoShuffled complexevolution approach for effective and efficient global minimiza-tionrdquo Journal of Optimization Theory and Applications vol 76no 3 pp 501ndash521 1993

[17] Q Duan S Sorooshian and V K Gupta ldquoOptimal use of theSCE-UA global optimization method for calibrating watershedmodelsrdquo Journal of Hydrology vol 158 no 3-4 pp 265ndash2841994

[18] J A Nelder and R Mead ldquoA simplex method for functionminimizationrdquoThe Computer Journal vol 7 no 4 pp 308ndash3131965

[19] W L Price ldquoGlobal optimization algorithms for a CAD work-stationrdquo Journal of Optimization Theory and Applications vol55 no 1 pp 133ndash146 1987

[20] J H Holland Adaptation in Natural and Artificial SystemsUniversity of Michigan Press Ann Arbor Mich USA 1975

[21] J E Nash and J V Sutcliffe ldquoRiver flow forecasting throughconceptual models part Imdasha discussion of principlesrdquo Journalof Hydrology vol 10 no 3 pp 282ndash290 1970

[22] V P Singh Computer Models of Watershed Hydrology WaterResources Publication Boulder Colo USA 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of


EarthquakesJournal of


Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining


Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of


Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering


GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of


OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in


MineralogyInternational Journal of


MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Geological ResearchJournal of


Geology Advances in


Despite nearly 20 years of research little progress hasbeen made on the parameter estimation problem for CRRmodels since objective function response surface typicallycontains hundreds if not thousands of local optima nestedat several scales [3] In the research presented here we usedthe SCE-UA method to calibrate the XAJ model The XAJmodel is extensively used throughout the world to analyzethe stability of parameters from automatic optimizationmethods It considers the hydrological data parameter searchinterval objective functions and other aspects

2 XAJ Model

21 The Model Structure and Its Main Characteristics TheXAJ [12 13] model was developed in 1973 by the East ChinaCollege of Hydraulic Engineering (now Hohai University)Its underlying aim was to forecast flows to the XinanjiangreservoirThemodel has been successfully andwidely appliedin humid and semihumid regions It is based on ldquorunoffformation at the natural storagerdquo which is the distinguishingfeature of the XAJ model when compared to other modelsThe basin is divided into a set of subbasins using a methodsuch as Thiessen polygon modification considering theuneven distribution of rainfall and the underlying surfaceThen the discharge curve on the outlet section of eachsubbasin is simulated and flood rooting is determinedFinally the total discharge is obtained using a simple sumTheXAJmodel is composed of fourmodules the evaporationmodule the runoff productionmodule the runoff separationmodule and the runoff concentration module

Figure 1 shows the flow chart of the XAJ model Rainfall(119875) and water-surface evaporation (EM) are the input dataand the discharge curve for the outlet section (119876) and theevaporation of the watershed (119864) are the output results Thestate variables are in boxes and the model parameters areoutside the boxes (Figure 1) For this research 119882 is thetension water storage WU is the upper layer tension waterstorage WL is the lower layer tension water storage FR is therunoff contributing area factor 119878 is the free water storageRS is the surface runoff RI is the interflow runoff RG isthe ground water runoff QS is the surface flow QI is theinterflow and QG is the ground water flow

The XAJ model has several characteristics that can besummarized as follows

(1) The rainfall-runoff process is divided into two stagesrunoff generation and concentration in the water-shed It is thought that in the runoff yield stagerunoff is produced only after the deficit of the vadosezone is satisfied A homogeneous vadose zone issubject to ground water flow and excess surface flowinfiltration Interflow will be produced in a vadosezone with a relatively impermeable layer in additionto ground water flow and saturated overland flowIn the runoff concentration stage the river networkand subbasin concentration can be considered astwo types of watershed concentration The subbasincombined with a river network can fully embody thewatershed concentration Because the XAJ model is

very compatible in the treatment of the watershedconcentration the subbasin and river network con-centration are commonly represented by the Shermanunit hydrograph and Muskingum methods for suc-cessive routing by subreaches [14] as well as othermethods (such as Clark method etc)

(2) A three-layer evaporation model is used Here theldquolayerrdquo takes soil moisture constants such as thefield capacity and wilting point as thresholds Inaddition to the soil moisture constants the soilevaporation ability is an important factor in the three-layer evaporation model and has a great effect onthe accuracy It is generally difficult to directly obtainan accurate value using instrumental observationsTherefore in the XAJ model the measured water-surface evaporation is revised by a correction factorand improved by the water balance of the watershedIn this way we can avoid using empirical formulas tocalculate the soil evaporationThis method of dealingwith evaporation is the only one used in practicalapplications

(3) The XAJ model considers runoff separation whichmeans that the calculated flood process is more inline with the actual situation Because the runoffproduction components have different flow velocitiesthe runoff concentration results are more accurate ifwe calculate the runoff separation using the appro-priate velocities The XAJ model uses a ldquodownwardrdquostructure for the runoff separation whereas otherCRR models generally use an ldquoupwardrdquo structure

(4) The XAJ model uses a statistically significant water-shed storage capacity curve and a watershed freewater capacity curve Note that these curves are onlyapplicable to the analysis of runoff area variabilitiescaused by an uneven distribution of the underlyingsurface under the condition of a uniform rainfall spa-tial distribution Additionally for a closed watershedwe should use the upper limits of both curves Foran unclosed watershed the upper limit is infiniteThe infiltration capacity area distribution curve isused by Stanford model to account for the influenceof the uneven distribution of the underlying surfaceon the infiltration of the excess surface runoff Thisis because the watershed storage and watershed freewater capacity curves are set in the model Howeverthe XAJ model is more advanced than the other CRRmodels

22 Model Parameters The XAJ model has 17 parametersthat must be determined by the user (XAJ Model Parameterssection) when computing the flood using the Muskingummethod for successive routing by subreachesWe temporarilyset the feasible parameter space by fixing the upper and lowerparameter bounds (see Table 1) The Muskingum methodparameters were predetermined based on the observedhydrograph and were not included in the optimization


E

C

ST Q

W

EU WU

RS

RI

RG

QS

QI

QG

WL

WUM

IMP

WLMEL

ED

P EM

Wminus (WU + WL)

1 minus FR

R(1 minus IMP)

FR

SM CS

CI

CG

EX

KI

KG

R middot IMP

NKE

XE

WMK B (1 minus IMP)













Catchmentarea (km2)


Averageannual

precipitation

Averageannual

evaporation


601 5 1657 741 1281















Obj1= min

100381610038161003816100381610038161003816100381610038161003816

(sum119899

119894=1119876obs (119894) minus sum

119899

119894=1119876cal (119894))

sum119899

119894=1119876obs (119894)

100381610038161003816100381610038161003816100381610038161003816

(1)


Obj2= min

sum119899

119894=1

1003816100381610038161003816119876obs (119894) minus 119876cal (119894)1003816100381610038161003816

sum119899

119894=1119876obs (119894)

(2)


DY = max

1 minussum119899

119894=1[119876obs(119894) minus 119876cal(119894)]

2

sum119899

119894=1[119876obs(119894) minus 119876obs]

2

(3)




100381610038161003816100381610038161003816100381610038161003816

(119891119894minus119896minus 119891)

119891119894

100381610038161003816100381610038161003816100381610038161003816

le TOL (4)

where 119891119894and 119891




100381610038161003816100381610038161003816100381610038161003816

(120582119894minus119896(119895) minus 120582

119894(119895))

(120582max (119895) minus 120582min (119895))

100381610038161003816100381610038161003816100381610038161003816

le TOL120582 (5)

where 120582119894(119895) and 120582









120582=

10minus4




(a)


(b)










1and Obj

2














Parameter Truevalue





































5 Conclusions




0

02

04

06

08

1

0 400 800 1200

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3

(a) 1981 year

0

02

04

06

08

1

0 500 1000 1500

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3


0

02

04

06

08

1

0 500 1000 1500

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3












00

02

04

06

08

10

0 03 06 09 12 15

CDF

K


00

02

04

06

08

10

0 01 02 03 04

CDF

B


00

02

04

06

08

10

0 01 02 03

CDF

C


00

02

04

06

08

10

80 100 120 140 160 180

CDF

WM

(d) CDF versus WM

00

02

04

06

08

10

5 20 35 50

CDF

SM

1 year2 years

3 years7 years

(e) CDF versus SM

00

02

04

06

08

10

05 06 07 08 09 1

CDF

CI

1 year2 years

3 years7 years

(f) CDF versus CI










119870 0571 0449

0418 0444

0529 0454










Acknowledgments


References

































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


E

C

ST Q

W

EU WU

RS

RI

RG

QS

QI

QG

WL

WUM

IMP

WLMEL

ED

P EM

Wminus (WU + WL)

1 minus FR

R(1 minus IMP)

FR

SM CS

CI

CG

EX

KI

KG

R middot IMP

NKE

XE

WMK B (1 minus IMP)













Catchmentarea (km2)


Averageannual

precipitation

Averageannual

evaporation


601 5 1657 741 1281















Obj1= min

100381610038161003816100381610038161003816100381610038161003816

(sum119899

119894=1119876obs (119894) minus sum

119899

119894=1119876cal (119894))

sum119899

119894=1119876obs (119894)

100381610038161003816100381610038161003816100381610038161003816

(1)


Obj2= min

sum119899

119894=1

1003816100381610038161003816119876obs (119894) minus 119876cal (119894)1003816100381610038161003816

sum119899

119894=1119876obs (119894)

(2)


DY = max

1 minussum119899

119894=1[119876obs(119894) minus 119876cal(119894)]

2

sum119899

119894=1[119876obs(119894) minus 119876obs]

2

(3)




100381610038161003816100381610038161003816100381610038161003816

(119891119894minus119896minus 119891)

119891119894

100381610038161003816100381610038161003816100381610038161003816

le TOL (4)

where 119891119894and 119891




100381610038161003816100381610038161003816100381610038161003816

(120582119894minus119896(119895) minus 120582

119894(119895))

(120582max (119895) minus 120582min (119895))

100381610038161003816100381610038161003816100381610038161003816

le TOL120582 (5)

where 120582119894(119895) and 120582









120582=

10minus4




(a)


(b)










1and Obj

2














Parameter Truevalue





































5 Conclusions




0

02

04

06

08

1

0 400 800 1200

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3

(a) 1981 year

0

02

04

06

08

1

0 500 1000 1500

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3


0

02

04

06

08

1

0 500 1000 1500

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3












00

02

04

06

08

10

0 03 06 09 12 15

CDF

K


00

02

04

06

08

10

0 01 02 03 04

CDF

B


00

02

04

06

08

10

0 01 02 03

CDF

C


00

02

04

06

08

10

80 100 120 140 160 180

CDF

WM

(d) CDF versus WM

00

02

04

06

08

10

5 20 35 50

CDF

SM

1 year2 years

3 years7 years

(e) CDF versus SM

00

02

04

06

08

10

05 06 07 08 09 1

CDF

CI

1 year2 years

3 years7 years

(f) CDF versus CI










119870 0571 0449

0418 0444

0529 0454










Acknowledgments


References

































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in



Catchmentarea (km2)


Averageannual

precipitation

Averageannual

evaporation


601 5 1657 741 1281















Obj1= min

100381610038161003816100381610038161003816100381610038161003816

(sum119899

119894=1119876obs (119894) minus sum

119899

119894=1119876cal (119894))

sum119899

119894=1119876obs (119894)

100381610038161003816100381610038161003816100381610038161003816

(1)


Obj2= min

sum119899

119894=1

1003816100381610038161003816119876obs (119894) minus 119876cal (119894)1003816100381610038161003816

sum119899

119894=1119876obs (119894)

(2)


DY = max

1 minussum119899

119894=1[119876obs(119894) minus 119876cal(119894)]

2

sum119899

119894=1[119876obs(119894) minus 119876obs]

2

(3)




100381610038161003816100381610038161003816100381610038161003816

(119891119894minus119896minus 119891)

119891119894

100381610038161003816100381610038161003816100381610038161003816

le TOL (4)

where 119891119894and 119891




100381610038161003816100381610038161003816100381610038161003816

(120582119894minus119896(119895) minus 120582

119894(119895))

(120582max (119895) minus 120582min (119895))

100381610038161003816100381610038161003816100381610038161003816

le TOL120582 (5)

where 120582119894(119895) and 120582









120582=

10minus4




(a)


(b)










1and Obj

2














Parameter Truevalue





































5 Conclusions




0

02

04

06

08

1

0 400 800 1200

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3

(a) 1981 year

0

02

04

06

08

1

0 500 1000 1500

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3


0

02

04

06

08

1

0 500 1000 1500

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3












00

02

04

06

08

10

0 03 06 09 12 15

CDF

K


00

02

04

06

08

10

0 01 02 03 04

CDF

B


00

02

04

06

08

10

0 01 02 03

CDF

C


00

02

04

06

08

10

80 100 120 140 160 180

CDF

WM

(d) CDF versus WM

00

02

04

06

08

10

5 20 35 50

CDF

SM

1 year2 years

3 years7 years

(e) CDF versus SM

00

02

04

06

08

10

05 06 07 08 09 1

CDF

CI

1 year2 years

3 years7 years

(f) CDF versus CI










119870 0571 0449

0418 0444

0529 0454










Acknowledgments


References

































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in






Obj1= min

100381610038161003816100381610038161003816100381610038161003816

(sum119899

119894=1119876obs (119894) minus sum

119899

119894=1119876cal (119894))

sum119899

119894=1119876obs (119894)

100381610038161003816100381610038161003816100381610038161003816

(1)


Obj2= min

sum119899

119894=1

1003816100381610038161003816119876obs (119894) minus 119876cal (119894)1003816100381610038161003816

sum119899

119894=1119876obs (119894)

(2)


DY = max

1 minussum119899

119894=1[119876obs(119894) minus 119876cal(119894)]

2

sum119899

119894=1[119876obs(119894) minus 119876obs]

2

(3)




100381610038161003816100381610038161003816100381610038161003816

(119891119894minus119896minus 119891)

119891119894

100381610038161003816100381610038161003816100381610038161003816

le TOL (4)

where 119891119894and 119891




100381610038161003816100381610038161003816100381610038161003816

(120582119894minus119896(119895) minus 120582

119894(119895))

(120582max (119895) minus 120582min (119895))

100381610038161003816100381610038161003816100381610038161003816

le TOL120582 (5)

where 120582119894(119895) and 120582









120582=

10minus4




(a)


(b)










1and Obj

2














Parameter Truevalue





































5 Conclusions




0

02

04

06

08

1

0 400 800 1200

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3

(a) 1981 year

0

02

04

06

08

1

0 500 1000 1500

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3


0

02

04

06

08

1

0 500 1000 1500

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3












00

02

04

06

08

10

0 03 06 09 12 15

CDF

K


00

02

04

06

08

10

0 01 02 03 04

CDF

B


00

02

04

06

08

10

0 01 02 03

CDF

C


00

02

04

06

08

10

80 100 120 140 160 180

CDF

WM

(d) CDF versus WM

00

02

04

06

08

10

5 20 35 50

CDF

SM

1 year2 years

3 years7 years

(e) CDF versus SM

00

02

04

06

08

10

05 06 07 08 09 1

CDF

CI

1 year2 years

3 years7 years

(f) CDF versus CI










119870 0571 0449

0418 0444

0529 0454










Acknowledgments


References

































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in



(a)


(b)










1and Obj

2














Parameter Truevalue





































5 Conclusions




0

02

04

06

08

1

0 400 800 1200

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3

(a) 1981 year

0

02

04

06

08

1

0 500 1000 1500

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3


0

02

04

06

08

1

0 500 1000 1500

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3












00

02

04

06

08

10

0 03 06 09 12 15

CDF

K


00

02

04

06

08

10

0 01 02 03 04

CDF

B


00

02

04

06

08

10

0 01 02 03

CDF

C


00

02

04

06

08

10

80 100 120 140 160 180

CDF

WM

(d) CDF versus WM

00

02

04

06

08

10

5 20 35 50

CDF

SM

1 year2 years

3 years7 years

(e) CDF versus SM

00

02

04

06

08

10

05 06 07 08 09 1

CDF

CI

1 year2 years

3 years7 years

(f) CDF versus CI










119870 0571 0449

0418 0444

0529 0454










Acknowledgments


References

































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in



Parameter Truevalue





































5 Conclusions




0

02

04

06

08

1

0 400 800 1200

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3

(a) 1981 year

0

02

04

06

08

1

0 500 1000 1500

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3


0

02

04

06

08

1

0 500 1000 1500

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3












00

02

04

06

08

10

0 03 06 09 12 15

CDF

K


00

02

04

06

08

10

0 01 02 03 04

CDF

B


00

02

04

06

08

10

0 01 02 03

CDF

C


00

02

04

06

08

10

80 100 120 140 160 180

CDF

WM

(d) CDF versus WM

00

02

04

06

08

10

5 20 35 50

CDF

SM

1 year2 years

3 years7 years

(e) CDF versus SM

00

02

04

06

08

10

05 06 07 08 09 1

CDF

CI

1 year2 years

3 years7 years

(f) CDF versus CI










119870 0571 0449

0418 0444

0529 0454










Acknowledgments


References

































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in














5 Conclusions




0

02

04

06

08

1

0 400 800 1200

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3

(a) 1981 year

0

02

04

06

08

1

0 500 1000 1500

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3


0

02

04

06

08

1

0 500 1000 1500

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3












00

02

04

06

08

10

0 03 06 09 12 15

CDF

K


00

02

04

06

08

10

0 01 02 03 04

CDF

B


00

02

04

06

08

10

0 01 02 03

CDF

C


00

02

04

06

08

10

80 100 120 140 160 180

CDF

WM

(d) CDF versus WM

00

02

04

06

08

10

5 20 35 50

CDF

SM

1 year2 years

3 years7 years

(e) CDF versus SM

00

02

04

06

08

10

05 06 07 08 09 1

CDF

CI

1 year2 years

3 years7 years

(f) CDF versus CI










119870 0571 0449

0418 0444

0529 0454










Acknowledgments


References

































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


0

02

04

06

08

1

0 400 800 1200

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3

(a) 1981 year

0

02

04

06

08

1

0 500 1000 1500

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3


0

02

04

06

08

1

0 500 1000 1500

Para

met

er st

anda

rdiz

ed v

alue

Iteration number

K

C

WMWUM

WLMSMEXKI

Obj3












00

02

04

06

08

10

0 03 06 09 12 15

CDF

K


00

02

04

06

08

10

0 01 02 03 04

CDF

B


00

02

04

06

08

10

0 01 02 03

CDF

C


00

02

04

06

08

10

80 100 120 140 160 180

CDF

WM

(d) CDF versus WM

00

02

04

06

08

10

5 20 35 50

CDF

SM

1 year2 years

3 years7 years

(e) CDF versus SM

00

02

04

06

08

10

05 06 07 08 09 1

CDF

CI

1 year2 years

3 years7 years

(f) CDF versus CI










119870 0571 0449

0418 0444

0529 0454










Acknowledgments


References

































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


00

02

04

06

08

10

0 03 06 09 12 15

CDF

K


00

02

04

06

08

10

0 01 02 03 04

CDF

B


00

02

04

06

08

10

0 01 02 03

CDF

C


00

02

04

06

08

10

80 100 120 140 160 180

CDF

WM

(d) CDF versus WM

00

02

04

06

08

10

5 20 35 50

CDF

SM

1 year2 years

3 years7 years

(e) CDF versus SM

00

02

04

06

08

10

05 06 07 08 09 1

CDF

CI

1 year2 years

3 years7 years

(f) CDF versus CI










119870 0571 0449

0418 0444

0529 0454










Acknowledgments


References

































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in









119870 0571 0449

0418 0444

0529 0454










Acknowledgments


References

































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in
































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in










Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in

Documents

Research Article Calibration of Conceptual Rainfall-Runoff ...downloads.hindawi.com/journals/amete/2015/545376.pdf · global optimization method was used to calibrate the Xinanjiang