IDENTIFICATION OF MODEL STRUCTURAL STABILITY THROUGH .
Annual Journal of Hydraulic Engineering, JSCE, Vol.51, 2007, FebruaryAnnual Journal of Hydraulic Engineering, JSCE, Vol.51, 2007, FebruaryIDENTIFICATION OF MODEL STRUCTURALSTABILITY THROUGH COMPARISON OFHYDROLOGIC MODELSGiha LEE 1, Yasuto TACHIKAWA2 and Kaoru TAKARA31Student Member, Graduate student, Dept. of Urban and Environmental Eng., Kyoto University(Kyoto 606-8501, Japan) E-mail:leegiha@flood.dpri.kyoto-u.ac.jp2Member of JSCE, Dr. Eng., Associate Professor, DPRI, Kyoto University (Gokasho, Uji 611-0011, Japan)3Fellow of JSCE, Dr. Eng., Professor, DPRI, Kyoto University (Gokasho, Uji 611-0011, Japan)No matter how sophisticated and accurate hydrologic models may be, prediction uncertainty isunavoidable problem in rainfall-runoff modeling and it stems from various components. Therefore, it isessential to identify and reduce sources of uncertainty for more precise agreement between modelprediction and observations in the real system. In the context of uncertainty issues, the problem of modelstructural uncertainty or stability is an issue of increasing interest in recent research. This paper examinesa nature of model structural inadequacy using a single-objective global optimization method inhydrological modeling and proposes a framework to assess model structural stability through acomparison of two hydrologic models.Key Words: prediction uncertainty, model structural stability, single-objective global optimization1. INTRODUCTIONThe principal and intense task of modelers (e.g.,hydrologists and engineers) is to identify ahydrologic model through the estimation of anoptimal parameter set within a specific modelstructure, which is suitable for given catchmentcharacteristics, historical data and intendedmodeling purpose. The model identification processtherefore consists of two steps generally1). The firststep is the selection of a proper hydrologic modelstructure and the second one is the identification ofan appropriate parameter set. However, this work isnot easy due to an influence of uncertaintiesinvolved in modeling procedure that are alsounavoidably propagated into the prediction of modeloutput variables (e.g., stream flow). This modeloutput uncertainty originates from various sources:input uncertainty; measurement errors; parameteruncertainty; and model structure uncertainty.A great deal of research has been proposed torecognize the propagation of the uncertainties whichappear in the different components of therainfall-runoff modeling into model prediction2),3),4).Beven and Binely2) suggest a method of estimatingparameter uncertainty and its propagation,Generalized Likelihood Uncertainty Estimation(GLUE) approach. On the other hand, Kavetski etal.4) review the influence of forcing inputuncertainty on model output in hydrologicalmodeling.Especially, the problem of model structuraluncertainty with advanced automatic calibrationmethods is an issue of increasing interest in recentresearches3),5),6). Structure error is unavoidableproblem in hydrological modeling since hydrologicmodels are conversion and simplification of reality,thus no matter how sophisticated and accurate theymay be those models only represent aspects ofconceptualization or empiricism of modelers. Inconsequence, output time series of hydrologicmodels are as reliable as hypothesis, structure ofmodels, quantity and quality of available forcingdata, parameter estimates7).Gupta et al.3) pointed out that one parameter setmay be insufficient to represent the behavior of thecatchment due to the inadequacy of modelstructures. In other words, a subjective selection ofobjective functions for calibration of conceptualhydrologic models results in an overemphasis ondifferent response modes such as low and high flowperiods. This fact implies different parametercombinations can be existent according to variousobjective functions due to the presence of structural- 49 -
uncertainty.Hydrologists have concentrated their effort ondevelopment of more rigorous and efficient schemeto assess the suitability of model structure forrepresenting the natural system and identifyingmodel structural inadequacy3),6),8). However, theirresearch is limited to improve a classical calibrationstrategy, single-objective optimization algorithm,coupled with their own conceptual model (e.g.,SAC-SMA model). Hence, a method to identify themodel structural stability, not only limited tolumped models but also applicable to various typeof distributed models is required.As reported in previous studies3),5),6),8), the resultof variation of optimal parameter combinationcalibrated by a single-objective optimization methodcan be employed as one of the well-foundedindicators to account for model structural stability.Accordingly, we think that a more reliable modelstructure leads to the identical optimal parameter setwithout regard of any objective functions selectedsubjectively. Moreover, such model structuremaintains high degree of accuracy for simulatedhydrographs when applying parameter set forvarious type and magnitude of floods in the samestudy site. It means that a structurally-stable modelhas high parameter transferability from event toevent. As a result, model structural stability can beestimated as a degree of capability which enables toreduce the influence of objective functions andflood events on model parameter sets.From this point of view, this study is conductedto investigate answers to the following questions: 1)What kinds of models are stable in terms of modelstructure for description of rainfall-runoffprocesses? 2) How can modelers identify modelstructural stability and suitability? In section 2, aframework to assess the model structural stability isproposed and different kinds of two models areevaluated through the framework. In section 3, thecomparison results with respect to the assessment ofmodel structural stability are discussed andconclusions are presented in section 4.2. METHODOLOGY FOR MODELSTURCTURE ANALYSISOur purpose of this study is to establish aframework for how to assess the model structuralstability. This work is summarized by two mainprocedures. The first step is an identification ofmodel stability according to selection of objectivefunctions. The second procedure is an assessment ofmodel structural stability through the analysis ofparameter transferability. Figure 1 illustrates theschematic process of the framework for assessmentof model structural stability.This assessment procedure is based on thefollowing ideas:1) If hydrologic observed data (e.g., rainfall, streamflow) used for model calibration are noterroneous, the calibrated parameter set canreflect the structure of hydrologic model.2) An ideally-structured model can be regarded as astable model which has the identical bestparameter set regardless of objective functionsand various flood events.Therefore, analysis of the variation of singleoptimal parameter sets calibrated by the single-objective global optimization method for variousflood events can be used as an indicator of thedegree of structural stability.The Shuffled Complex Evolution (SCE-UA)algorithm with three different objective functions(SLS, HMLE, MIA) is used to calibrate aconceptual lumped model (Storage FunctionMethod, SFM) and a physically based distributedmodel (Cell Distributed Runoff Model Version 3,CDRMV3, http://fmd.dpri.kyto-u.ac.jp/ flood/product/cellModel.html)9). Five historical flood events atKamishiiba catchment (210 ) located in theKyushu area are used to compare model structuralstability for verification of our framework. Appliedhydrologic models, optimization method andobjective functions are introduced in followingsub-sections.(1) Hydrologic Modelsa) Conceptual lumped model, Storage FunctionMethod (SFM)This model is a lumped model with the reflectionof nonlinear characteristics of hydrologic responsebehavior. SFM is used for the rainfall-runoffsimulation in a small watershed usually less thanfive hundred square kilometers. The form of SFM isexpressed as:dS re (t Tl ) q,dt f r ,re r ,ififS kq p r R r RSA(1)(2)SAwhere S water storage; re effective rainfallintensity; r rainfall intensity; q runoff; t time;k storage coefficient; p coefficient ofnonlinearity; f primary runoff ratio; Tl lag time;and RSA cumulative observed rainfall from thebeginning of the studied storm. Four parameters (k,p, f and RSA ) are optimized in this model.b) Physically based distributed model, CDRMV3CDRMV3 is a physically based distributedhydrologic model developed by Kojima et al.9)including discharge-stage relationship with saturated-unsaturated flow10). The model solves the onedimensional kinematic wave equation with thedischarge-stage equation using the Lax-Wendrofffinite difference scheme according to the flow- 50 -
Selection ofmodel structureSensitivityanalysisDecision of parametersto be optimized1θ OFEvent 1Event 1 θ OFnModel1OF1 ,, OF nθEvent 1 θ OFnModel2OF1 ,, OF nEvent 1OF1 Objective FunctionSelection (OFs)Initial boundary ofparametersAutomatic Calibrationθ OFEventN 1EventN θ OFnModel1OF1 ,, OF nθEventN θ OFnModel2OF1 ,, OF nEventNOF1θ Event 1θ Event 2θ Model1,2 (Event1)Model1,2 (Event2)EventNModel1,2 (EventN)1Index θEventEvent 1 ,2Index θEvent,Event 1IndexEventNθ Event 1,Analysis of optimalparameter set basedon various OFs1, Index θEventEventN2, Index θEventEventN, IndexEventNθ EventNAnalysis of parametertransferability fromevent to eventQualitative andQuantitativeAssessment of modelstructural stability1EventNEvent 1EventNUnstable Model Structure : θθ OFEvent θ OF θ OFOF11nnFig.1 Schematic illustration of a framework to assess model structural stability; θ i j optimal parameter set; Indexkj measurement value for assessment of parameter transferability; i Objective Function; j storm event; k optimalparameter set of each event.Ideal Model Structure :(a) h q r (t ) t x(b)Radar Rainfallor Gage RainfallGrid lationsurface )(d)(Option)Discharge atNodes and EdgesFig.2 Schematic representation of CDRMV3 (a) Modularstructure of CDRMV3 (b) Distributed grid rainfall data (c)Slope and channel components extracted from DEM (d)Model structure for the hillslope soil layer and discharge-stage relationship.direction map (see Figure 2). All geomorphologicinformation are extracted from 250m based DEMdata. Channel routing is also carried out by thekinematic routing scheme as well as calculation ofslope elements reflecting contributing areas.The model assumes that a permeable soil layercovers the hillslope as illustrated in Figure 2(d). Thesoil layer consists of a capillary layer in whichunsaturated flow occurs and a non-capillary layer inwhich saturated flow occurs. According to thismechanism, if the depth of water is higher than thesoil depth, then overland flow occurs. Thedischarge-stage relationship is expressed by equation(3) corresponding to water levels (see Figure 2(d))defined as: vc d c (h / d c ) β ,0 h dc q vc d c va (h d c ),dc h ds v d v (h d ) α (h d ) m , d hcss c c a(3)(4)Flow rate of each slope segment are calculated byabove governing equations combined with thecontinuity equation (4), where v c k c i ; v a k a i ;k c k a / β ; α i / n ; i is slope gradient, k c issaturated hydraulic conductivity of the capillary soillayer, k a is hydraulic conductivity of thenon-capillary soil layer, n is roughness coefficient,d c is the depth of the capillary soil layer and d s issoil depth. Detailed explanations of model structureappear in Tachikawa et al.10). There are 5 parameters(n, k c , β , d c and d s ), which are assumed to havehomogeneous values spatially to be optimized inCDRMV3.(2) Shuffled Complex Evolution (SCE-UA)AlogrithmThe Shuffled Complex Evolution Algorithm(SCE)11),12) is used to identify the best fittedparameter set, which is a single-objective globaloptimization method designed to handle high-parameter dimensionality encountered in calibrationof a nonlinear hydrologic simulation models. Thisevolutionary approach method has been performedby a number of researchers on a variety of modelswith outstanding positive results7) and has proved tobe an efficient, powerful method for the automaticoptimization13),14). SCE algorithm is basicallysynthesized by following three concepts: (1)combination of a simplex procedure with theconcepts of controlled random search approaches;(2) competitive evolution; and (3) complex shuffling.The integration of these steps above mentionedmakes the SCE method effective and flexible12).Initial state variables of both models are determinedby initial observed discharge assuming steady-statecondition.- 51 -
(3) Applied Objective FunctionsThe aim of computer-based automatic calibrationis to find the values of the model parameters thatminimize or maximize the numerical value of theobjective functions15). In general, the mostcommonly utilized objective functions inhydrological modeling are variations of the SimpleLeast Squares (SLS) function defined as:NSLS (qobst qt (θ )) 2(5)t 1where qtobs is observed stream flow value at time t;qt (θ ) is model simulated stream flow value at time tusing parameter set θ ; N is the number of flowvalues available. SLS has a feature that largedischarge is emphasized due to squared errors whilelow flows are neglected16), thus the parameter setfitting around peak discharge value is likely toobtain.Krause et al.17) proposed the Modified Index ofAgreement (MIA)18) to reduce the influence of thesquared term during high flows as putting an weighton flow values. This objective function is calculatedas:n t 1MIA 1 n qtobs qt (θ ) qtobs qt (θ ) q t 1meantq qobst2(6) qmeantmeant 2where qtmean is mean value of observed time series.Sorooshian and Dracup19) proposed a differentobjective function to consider entire behavior ofhydrograph,theHeteroscedasticMaximumLikelihood Estimator (HMLE), which enables toestimate the most likely weights through the use ofthe maximum estimation theory. This new measurecan eliminate some of the subjectivity involved inthe selection of transformation and/or a weightingscheme by handling heteroscedastic error, so that ityields a more balanced performance over the entirehydrograph3),5). It is calculated as:HMLE minθ ,λ1NNN wε wtt 1tt(7)t 1where ε t q tobs qt (θ ) is the model residual at time t;wt is the weight assigned to time t, computed aswt f t 2 ( λ 1) ; f t q ttrue is the expected true flow attime t; λ is the transformation parameter whichstabilizes the variance.In this study, above mentioned three objectivefunctions are used for the calibration trials and theanalysis of model structural stability.illustrated in Figure 3. From Figure 3, we notice that:1) In SFM cases, the simulated hydrographs basedon the parameters calibrated by SLS are close tothe observed ones over all events while otherparameters optimized by HMLE, MIA lead toless magnitude than the measured stream flow inlarge flood, Event 4 and 5 (peak flow 500 /s).The results show that the optimized parameter setis dependent on objective functions. On the otherhand, the computed hydrographs in small flood,Event 3 (peak flow 500 /s) have a good“goodness-of-fit” measurement value regardlessof objective functions.2) In CDRMV3 cases, there is no influence ofobjective functions on model performance. Inother words, all predicted hydrographs shown inFigure 3(b) are close to observed discharge forany objective functions. However, constantparameter set is not observed for small floodevents (Event 2, 3) while approximate values ofparameter sets are obtained for large flood events(Event 1, 4, 5), i.e., θ OF1 θ OFn .This result implies that the lumped model used inthis study is structurally unstable in terms ofdependency of objective function, so that we need tochange the model parameter set according to themodeling purpose. On the other hand, the problem ofsubjectivity related with selection of objectivefunctions for automatic calibration can be ignoredfor the distributed hydrological modeling used here.(2) The Assessment of Parameter Transferabilityfrom Event to EventFrom event to event, transferability of theidentified parameter set also can be guideline toassess model stability. If optimal parameter setsobtained from various flood events are occupied in asimilar location on feasible parameter space, i.e.,θ Event1 θ EventN , undoubtedly, each optimalparameter set could be applicable for different floodevents and model performances would be good. Wecan regard that such model structure has highparameter transferability. Each model performancewith transferred parameter sets is evaluated by PeakDischarge Ratio (PDR) and Nash-Sutcliffe (NS)statistics of the residuals as guideline indexes formeasurement of parameter transferability, definedas:(8)PDR Peaksim / PeakobsNNS 1 3. RESULTS AND DISCUSSIONS(1) The Influence of Objective Functions onModel PerformanceThe plots of comparisons between the simulatedand the observed hydrographs according to the threeobjective functions (SLS, MIA and HMLE) are (qobst qt (θ )) 2t 1(9)N (qobst qmeant)2t 1where Peaksim is the simulated peak discharge,Peakobs is the observe peak discharge. PDR measurestendency of the simulated peak discharge to be largeror smaller than the observed peak discharge. NSmeasures a relative magnitude of the residual- 52 -
(a) SFMEvent3Event4Event5(b) CDRMV3Event4Event3Event5Fig.3 Comparison between the simulated and the observed hydrographs according to three objective functions;(a) SFM cases, (b) CDRMV3 cases.variance to the variance of the observed stream Instead, the different identified parameter sets areapplicable to reproduce other flood events. It meansflows; the optimal value of both measures is 1.0.The identified parameter set for each event is applied that the different parameter combinations can lead toto different events and their model performances are acceptable model performances with proper valuesplotted in the Figure 4. This figure illustrates the of NS or PDR. This effect is often calledquantified results of parameter transferability. Each “equifinality”2). Equifinality makes it difficult tooptimal parameter set of SFM is not applicable for identify a suitable model parameter set fordifferent flood events (i.e., low parameter distributed hydrological modeling. Therefore,transferability; NS, PDR are scattered far from 1.0) analysis of equifinality is an urgent problem to bewhile those of CDRMV3 are applicable for solved for more reliable rainfall-runoff simulation.simulations of different flood events (i.e., highparameter transferability; NS, PDR are plotted near 4. CONCLUSIONoptimal value 1.0). Interestingly, in CDRMV3, theresults evaluated by optimal parameters of Event 2In this paper, we have demonstrated a frameworkover entire cases tend to be inapplicable and for assessment of model structural stability through ainaccurate; NS values are usually less than single-objective global optimization (SCE-UA)approximately 0.75 and PDR values are method with three objective functions for variousunderestimated or overestimated irregularly. flood events and compared two hydrologic modelsSimulated results for Event 2 with the optimal (SFM, CDRMV3) as an example. The results underparameter sets of other events are also inapplicable. our framework lead to following conclusions:This result indicates that the observed data for Event 1) The simulated results of CDRMV3 are not2 is unreliable, so that it is impossible to extractaffected by objective functions while theuseful information from this kind of uninformativepredicted hydrographs of SFM depend on thedata. This finding shows that the proposedobjective functions and magnitude of flood.framework for assessment of model stability is also 2) The structural stability of CDRMV3 is superior topossible to detect a low quality observed data.
structural uncertainty or stability is an issue of increasing interest in recent research. This paper examines a nature of model structural inadequacy using a single-objective global optimization method in hydrological modeling and proposes a framework to assess model structural stability through a comparison of two hydrologic models.
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