Catchment responses to plausible parameters and...

HYDROLOGICAL PROCESSESHydrol. Process. 26, 893–906 (2012)Published online 19 October 2011 in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/hyp.8303

Catchment responses to plausible parameters and input dataunder equifinality in distributed rainfall-runoff modeling

Giha Lee,1*,† Yasuto Tachikawa,2 Takahiro Sayama3 and Kaoru Takara41 Construction and Disaster Research Center, Department of Civil Engineering, Chungnam National University, Daejeon 305-764, Korea

2 Department of Civil and Earth Resources Engineering, Kyoto University, Kyoto 615-8540, Japan3 ICHARM, PWRI, 1–6, Tsukuba, Ibaraki-ken 305-8516, Japan

4 Disaster Prevention Research Institute, Kyoto University, Uji, Kyoto 611-0011, Japan

*CCenDaE-m†PrCh

Co

Abstract:

In a hydrological simulation using a complex model, equifinality is likely to occur in many different ways, due to various sourcesassociated with the model structure, parameter as well as data uncertainties involved in modeling process. This study aims todemonstrate the equifinality problem in streamflow prediction with a distributed rainfall-runoff model and also investigate theeffect of parameter and input uncertainties on an internal catchment response in time and space under the equifinality condition.For this objective, several experiments were carried out step by step as follows. First, we estimated plausible parameter sets,which lead to similarly good objective function values, using the shuffled complex evolution metropolis (SCEM-UA) algorithmand also generated plausible input scenarios with different spatial patterns from radar rainfall field, which showed minimumrunoff errors. Then, we adopted a computational tracer method based on a time-space accounting scheme, combined with adistributed model for analyzing how a catchment behaves under a given equifinality condition by parameter and inputuncertainties. It was found that the global responses of a catchment to the predefined equifinality factors were very similar; allsimulated hydrographs were almost identical in spite of the different model parameter values and different spatial patterns ofrainfall data. On the other hand, the internal catchment responses, tracked by the computational tracer method, were sensitive toplausible scenarios. The results of the internal catchment behavior show that the distributed rainfall-runoff model has multiplealternative flow pathways, providing identical hydrographs in time and space when the rainfall over the catchment transformsinto runoff. Our approach can be used to understand catchment responses to various uncertainty sources by visualizing thespatiotemporal runoff generation process at the watershed scale. Copyright © 2011 John Wiley & Sons, Ltd.

KEY WORDS catchment responses; computational tracer method; distributed rainfall-runoff model; equifinality; plausibleparameter sets; plausible input scenarios

Received 30 September 2009; Accepted 30 December 2010

INTRODUCTION

In modern hydrology, it is impossible to imagine water-related problems whose solutions do not involve theapplication of hydrological models. Rainfall-runoff modelshave been a standard tool in addressing a wide spectrum ofwater resource and environmental problems. There are anumber of rainfall-runoff models available in the worldtoday (Singh and Frevert, 2002a, 2002b; Reed et al., 2004).Such models are in operational use for various objectivessuch as flood warning, integrated watershed managementand so on (Beven, 2005). In particular, distributed rainfall-runoff models have been widely used in various hydro-logical fields because of remarkable advancements in bothcomputer technology and GIS tools, which are capable ofhandling a large amount of heterogeneous data. Such modelsare more physically based (less conceptualized) in represent-ing the rainfall-runoff mechanism and spatial variability of

orrespondence to: Giha Lee, Construction and Disaster Researchter, Department of Civil Engineering, Chungnam National University,ejeon 305-764, Korea.ail: [email protected] Address: International Water Resources Research Institute,ungnam National University, Daejeon 305-764, Korea.

pyright © 2011 John Wiley & Sons, Ltd.

nature, and thusmodelers have believed that distributedmodelsguarantee superior prediction results to lumped models.

However, Jakeman and Hornberger (1993) pointed outthat making hydrological models more complex does notnecessarily lead to more precise and less uncertainpredictions due to the various sources of uncertaintyinvolved in rainfall-runoff modeling. The three uncertaintycomponents in general are divided as follows: modelparameter, model structure and measurement errorsassociated with the system input and output. In particular,distributed hydrological models require more parametersand data than parsimonious models for operating theirmodules. Therefore, the lack of balance between modelcomplexity (e.g. numerous parameters) and measurementavailability is likely to worsen parameter identifiability(Gupta et al., 2003). As a result, some parametercombinations, which have different values from the optimalparameter set, can lead to equally good performancemeasures or indistinguishable hydrographs in rainfall-runoffmodeling.

Beven and Binley (1992) used the special term ‘equifinal-ity’ to define this phenomenon. After that, it has been anoticeable issue in the hydrological modeling communitysuch as the international working group on Uncertainty

894 G. LEE ET AL.

Analysis in Hydrologic Modeling, a part of the Predictions inUngauged Basins initiative (Sivapalan et al., 2003). Much ofthe research (e.g. Freer et al., 1996; Beven and Freer, 2001;Wagener et al., 2004; and others) regarding equifinality hasmainly focused on parameter uncertainty, that is, theyhighlighted that many behavioral parameter sets could existwithin a feasible parameter space, and those, in turn, led toprediction results with the same level of accuracy obtainedfrom the optimal parameter combination.Moreover, equifinality can occur in many different

ways as Savenije (2001) reported that identically goodmodel simulations can arise from various sources. Forexample, Refsgaard and Knudsen (1996) and Uhlenbrooket al. (1999) demonstrated that different and diverseconceptualizations of a catchment runoff system canproduce equally good numerical outcomes, even thoughsome hydrological models are overly simplified or outsideof a hydrologist’s experience/expertise. Such results canbe regarded as the equifinality due to model structuraluncertainty. Furthermore, Tachikawa et al. (2003) foundthat all runoff simulation results were similar althoughthey applied various input rainfall data with differentspatial patterns for mesoscale mountainous catchmentmodeling. This result can be interpreted as the equifinalitydue to input uncertainty. Also, Lee et al. (2009)demonstrated that model performances in terms ofstreamflow estimation were insensitive to many non-optimal parameter sets in a fine resolution-based rainfall-runoff model. Then, they concluded that the fine spatialdiscretization was a factor that caused the equifinality aswell as over-parameterization in distributed rainfall-runoffmodeling. This result is also seen as evidence ofequifinality which stems from scale uncertainty.As documented earlier, the term equifinality indicates

identical hydrological output that has been restricted onlyto streamflow at a specific outlet. However, it may be moreinteresting to know what happens inside/within a catch-ment rather than at the outlet under the equifinalityconditions in catchment hydrology (McDonnell, 2003). Inother words, we need to change the viewpoint of theequifinality in rainfall-runoff modeling from the aggregatedhydrological response (i.e. discharge at outlets) to theinternal catchment responses in time and space.One of the advantages in distributed hydrological

modeling is that a distributed hydrological model offersspatially distributed hydrological information such as soilmoisture, ground water flux and ground water recharge rateand so on, at any point of interest within a certain studydomain. In addition to providing abundant informationregarding internal state variables, tracing the pathway ofspatiotemporal origin of streamflow computationally pro-vides a new perspective to figure out the mysterious andcomplicated streamflow generation processes.Some models have begun to incorporate flow source and

age components into model development (Seibert andMcDonnell, 2002; Uhlenbrook and Leibundgut, 2002; Dunnet al., 2007) and evaluation (Vache and McDonnell, 2006;Fenecia et al., 2008). However, all of these models simplytrack tracers and do not record the history of solute

Copyright © 2011 John Wiley & Sons, Ltd.

progression in space through the watershed (Sayama andMcDonnell, 2009). The computational tracing of waterthrough a hydrological system without using physiochemicalisotopes has recently started to receive attention (Coulthardand Macklin, 2003; Reaney, 2008). Sayama et al. (2007)developed a computational tracer method, which is based on anew concept, named the spatiotemporal record matrix ofstreamflow, in order to track the potential origins ofstreamflow in time and space at the catchment scale. Theycombined this strategy with the kinematic wave distributedmodel that takes into account the unsaturated subsurface,saturated subsurface and surface flows. This method canestimate a flow component age by temporally splitting thehydrograph into the same number of runoff componentswhich correspond to a selected number of rainfall segmentswithout any hydrochemical measure (e.g. natural/artificialisotope tracers). It can trace the potential geographical originsof the streamflow in space at the time step of interest and thencreate an effective visualization. This method allowsmodelers to investigate how different modeling conditionsinfluence runoff generation by illustrating the spatiotemporalinformation of flow components. Sayama et al. (2007)demonstrated that different model structures or parametervalues showed different patterns with respect to bothspatiotemporally separated runoff components and meanresidence time estimates. Moreover, Sayama and McDonnell(2009) applied this method to capture the hydrographicaldynamics of well-studied watersheds Maimai M8 and HJAndrewsWS10 in the USA. They then demonstrated that thecomputational tracer method can capture flow and transportdynamics for the right dominant process reasons. Newinformation from the computational tracer method is useful togive a better understanding of the internal catchmentresponses under various equifinality conditions.

This paper aims to point out the equifinality problem dueto parameter and input uncertainties in streamflowforecasting with a distributed rainfall-runoff model. Then,using the computational tracer method, we investigated theeffect of both plausible parameter sets and plausible rainfallscenarios on the spatiotemporal variation of streamfloworigins in a small mountainous catchment. This studyconducts the following experiments for the above objec-tives. First, a kinematic wave distributed rainfall-runoff(KWMSS, Tachikawa et al., 2004) model is representedunder the Objective-oriented Hydrologic Modeling System(OHyMoS, Takasao et al., 1996; Ichikawa et al., 2000) andis combined with the computational tracer method (Sayamaet al., 2007). Second, three plausible parameter combina-tions are obtained by the shuffled complex evolutionmetropolis (SCEM-UA; Vrugt et al., 2003) algorithm.Those are then used for comparisons between global andinternal catchment responses under the equifinality condi-tion, due to parameter uncertainty by simulating hydro-graphs and tracing the potential streamflow origin. Third,seven rainfall scenarios with different spatial patterns aregenerated from the original radar rainfall data forcomparisons between the global and internal catchmentresponses under the equifinality condition due to inputuncertainty. We select plausible rainfall fields producing

Hydrol. Process. 26, 893–906 (2012)

895CATCHMENT RESPONSES TO PLAUSIBLE PARAMETERS AND INPUT DATA

similarly minimal errors regarding the runoff property atthe outlet. Then, we apply them to the computational tracermethod in order to analyze the influence of spatialvariability of rainfall on streamflow origin in time andspace. Section 2 deals with a distributed rainfall-runoffmodel and a computational tracer method used in thisstudy. Section 3 introduces basic materials such as thestudy catchment and historical data. The results ofcatchment responses to plausible parameters are analyzedand discussed in Section 4, and Section 5 addresses theresults of catchment responses to plausible input scenarios.Finally, we end this paper with concluding remarks anddiscussions in Section 6.

INTRODUCTION OF HYDROLOGICAL MODEL ANDCOMPUTATIONAL TRACER METHOD

Kinematic wave method for subsurface and surface runoff(KWMSS)

The topography of mountainous slopes is modeled as aset of rectangular slopes (Shiiba et al., 1999). In thistopographical representation, the drainage network of thecatchment is represented by sets of slope elements asillustrated in Figure 1(a). Each element is represented by arectangle formed by two adjacent nodes of grid cells,determined by the steepest gradient. The grey poly lines ofFigure 1(a) indicate the set of slope elements generatedfrom the 250m DEM. By using this topographicalrepresentation, the slope flow is routed by the kinematicwave model with surface and subsurface flow (KWMSS)based on OHyMoS, which is designed by object-orientedprogramming techniques (Ichikawa et al., 2000).Each slope is assumed to be covered with a permeable

soil layer. As shown in Figure 1(b), this soil layer consists ofboth capillary and non-capillary layers. In these conceptualsoil layers, slow and quick flows are simulated as unsaturatedDarcy flow and saturated Darcy flow, respectively. Overland

Figure 1. (a) The modeled drainage network by 250m DEM, (b) schematiKWMSS for runoff simulatio


flow occurred if the water depth, h, exceeds the soil watercapacity. The KWMSS represents these runoff processes bythe stage–discharge relationship as shown in Figure 1(c)(Tachikawa et al., 2004), and it is expressed as:

q ¼vcdc h=dcð Þb; 0≤h≤dcvcdc þ va h� dcð Þ; dc≤h≤dsvcdc þ va h� dcð Þ þ a h� dsð Þm; ds≤h

8<: (1)

where vc= kci; va= kai; kc= ka/b; a ¼ ffiffii

p=n ; m=5/3; i is

the slope gradient; kc is the hydraulic conductivity of thecapillary soil layer; ka is the hydraulic conductivity of thenon-capillary soil layer; n is the roughness coefficient; dsis the water depth corresponding to the water content and dcis the water depth corresponding to maximum water contentin the capillary pore. Equation (1) represents the unsaturatedsubsurface flow, saturated subsurface flow and surface flowwith a stage–discharge relationship. The flow rate of eachslope element, q, is calculated by Equation (1) combinedwith the continuity equation, Equation (2).

@h

@tþ @q

@x¼ r tð Þ (2)

Five process parameters (n, ka, ds, dc and b), need to beoptimized in the KWMSS and those were assumed to bespatially homogeneous over the study catchment. Thisrainfall-runoff model is combined with the computationaltracer method in order to estimate the spatial source ofstreamflow at the catchment scale and discriminate betweenpre-event and event water during the historical flood event.

Computational tracer method based on the spatiotemporalmatrix of streamflow

The goal of the computational tracer method (Sayama et al.,2007; Sayama and McDonnell, 2009) is to trace where thestreamflow comes from when it rains over the catchment; thisconcept is well described in Figures 2(a) and (b). This method

c structure of slope element and (c) a stage–discharge relationship of then (Tachikawa et al., 2004)

Hydrol. Process. 26, 893–906 (2012)

Figure 2. Conceptual diagram of the computational tracer method based on (a) temporal record of streamflow, (b) spatial record of streamflow and (c)spatiotemporal matrix of streamflow at the catchment outlet; NS=6, NT=5 (Sayama et al., 2007; Sayama and McDonnell, 2009)

896 G. LEE ET AL.

can provide the contributive level of the potential streamfloworigin in time and space by using a concept based on thespatiotemporal record matrix as illustrated in Figure 2(c).The dimension of spatiotemporal matrix is given with NS

rows, number of spatial units (e.g. number of sub-catchmentor slope element) within a catchment and NT columns,number of temporal classes, where t is time of interest.Figure 2(c) shows the spatiotemporal matrix at time t at thecatchment outlet. For example, the value belonging to spatialzone C and temporal class 2 implies that the proportion ofrunoff in (C, 2) entry to the streamflow observation at time taccounts for 6%. When summarizing the whole valuesvertically along the columns, the temporal contribution ofrainfall to streamflow can be obtained (see Figure 2(a)). As aresult, modelers can divide a hydrograph into runoffcomponents corresponding to the subjectively separatedrainfall fractions; basically, those are classified into oldwater (i.e. pre-event water at time class 0 possesses 15%)and new water (i.e. event water components are 30% at timeclass 1 and 55% at time class 2, respectively) at the specifictime t. Note that the rainfall segments 3 and 4 do not affectrunoff yet at time t, and thus their contributive level of thematrix is represented by zero values. It is possible to trackthe spatially distributed (or geographic) origin for stream-flow generation using the spatial record matrix withinformation stored in each slope element (see Figure 2(b)).The spatial contribution of each sub-catchment on thestreamflow is calculated by horizontal summation along therows. For instance, downstream spatial zones (e.g. D, E andF) comprise 65% of streamflow whereas upstream spatialzones (e.g. A, B and C) comprise 35% at time t. In thismanner, we implemented the analysis of the effect ofparameter and input uncertainties on spatiotemporal vari-ation of streamflow origin under the equifinality condition.

Derivation of the spatiotemporal record matrix

The runoff, qi, and water depth, hi, of each unit slopeelement, i (i2NS;NSis a set of slope elements in the studyarea. Figure 1(a)) at all time steps during the event arecalculated by Equations (1) and (2). The runoff of unitslope elements is then split into three flow components;surface flow, qOi , saturated flow, qPi , and unsaturated flow,qMi . The runoff of each flow component at the unit slopeelement i is computed by considering the mass balance


among those as shown in Figure 1(b), where inflow fromthe adjacent upstream element is expressed by qini whileoutflow into the downstream element is defined as qouti .Therefore, qinOi , qinPi and qinMi indicate inflows at the surface,non-capillary pore zone and capillary pore zone, respect-ively, while qoutOi , qoutPi and qoutMi are outflows at the surface,non-capillary pore zone and capillary pore zone, respect-ively. Vertical flows are divided into two components asfollows: qOPi is the water from surface to non-capillary zoneand qPMi is the water from non-capillary zone to capillaryzone. These vertical flows are defined as positive andnegative according to its flow direction.

For computing the spatiotemporal record matrix, the massbalance of each flow component corresponding to the waterdepth in the unit slope element is calculated as the followingequations:

First, when the water depth is less than or equal to thedepth of capillary pore layer (0≤ hi≤ dc)

qinOi ¼ 0qoutOi ¼ 0qinPi ¼ 0qoutPi ¼ 0

qinMi ¼ Pj2Ui

qoutOj þ qoutPj þ qoutMj

� �qoutMi ¼ qiqOPi ¼ 0qPMi ¼ 0

8>>>>>>>>>>>><>>>>>>>>>>>>:

(3)

Second, when the water depth is larger than the depth ofthe capillary pore layer and less than or equal to the depthof non-capillary pore layer (dc< hi≤ ds)

qinOi ¼ 0qoutOi ¼ 0

qinPi ¼ Pj2Ui

qoutOj þ qoutPj

� �qoutPi ¼ qi � qoutMiqinMi ¼ P

j2UiqoutMj

qoutMi ¼ vcdcqOPi ¼ 0qPMi ¼ qoutMi � qoutMi

8>>>>>>>>>>>><>>>>>>>>>>>>:

(4)

Third, when the water depth exceeds the depth of soillayer (ds< hi)

Hydrol. Process. 26, 893–906 (2012)


qinOi ¼ Pj2Ui

qoutOj

qoutOi ¼ qi � qoutPi � qoutMiqinPi ¼ P

j2Ui

qoutPj

qoutPi ¼ va ds � dcð ÞqinMi ¼ P

j2Ui

qoutMj

qoutMi ¼ vcdcqOPi ¼ qoutPi þ qoutMi � qinPi � qinMiqPMi ¼ qoutMi � qinMi

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

(5)

where qoutOj , qoutPj and qoutMj are flow components at theupstream slope element, j draining into the slope element i(j2Ui; Ui is a set of upstream slope elements connectedwith a unit slope element, i).Next, the spatiotemporal matrices of each flow compo-

nent of the slope element i, ROi tð Þ, RP

i tð Þ and RMi tð Þ are

computed at time step t using the flow rate calculated byEquations (3–5). Their dimensions are identical to thespatiotemporal matrix at the outlet of the catchment byassigning 0 to the entries, which do not contribute to runoffof i. The matrices of three runoff components areformulated as:

d VOi R

Oi

� �dt

¼Xj2Ui

qoutOj ROj � qoutOi RO

i � qOPi ROPi

þriAiRraini

(6)

d VPi R

Pi

� �dt

¼Xj2Ui

qoutPj RPj � qoutPi RP

i þ qOPi ROPi

�qPMi RPMi

(7)

d VMi RM

i

� �dt

¼Xj2Ui

qoutMj RoutMj � qoutMi RM

i � qPMi RPMi (8)

ROPi ¼ RO

i qOPi ≥0� �

RPi qOPi < 0� �

�(9)

RPMi ¼ RP

i qPMi ≥0� �

RMi qPMi < 0� �

�(10)

Figure 3. The Kamishiiba catchment: (a) elevation map, (b) land use ma


where the water volume stored in each layer are VOi ,V

Pi and

VMi ; Ai is the projected area of slope element i. The

spatiotemporal matrices of upstream slope elements, j(j2Ui), which inflow to each layer of i, areRO

j ,RPj andR

Mj .

Rraini is the spatiotemporal matrix of rainfall, ri at i. Note

that if dc< hi≤ ds, the rainfall inflows indirectly to the non-capillary zone so that riAiRrain

i of Equation (6) is removedbut is added to the left-hand side of Equation (7); if hi≤ dc,rainfall inflows directly into the capillary zone such thatriAiRrain

i is added to the left-hand side of Equation (8).All matrices of the three components at unit slope element

within the study site are calculated and updated in orderfrom upstream to downstream, and then both flow rate andspatiotemporal matrix of the element k, which drains into theriver element, are calculated by the information of all slopeelements within the study catchment. Finally, the spatio-temporal matrix of streamflow at time t, RR tð Þ is calculatedby integrating the information obtained from the element, k(k2C ;C is a set of slope elements, directly connected withriver elements), and it is defined as:

RR tð Þ ¼Pk2C

qoutOk tð ÞROk tð Þ þ qoutPk tð ÞRP

k tð Þ þ qoutMk tð ÞRMk tð Þ� �

Pk2C qoutOk tð Þ þ qoutPk tð Þ þ qoutMk tð Þ� �

(11)

As stated previously, when whole values are summed upvertically along the columns, the contribution of rainfall tostreamflowcan be traced temporally. Likewise, the spatial con-tribution of each slope element to streamflow can be calculatedby a horizontal summation along the rows of the matrix.

STUDY CATCHMENT AND HISTORICAL FLOODEVENT

The study site is the Kamishiiba catchment, an upstreamarea of the Kamishiiba Dam, which lies within the Kyushuregion of Japan (see Figure 3). This catchment covers an

p and (c) spatially distributed total depth of radar rainfall of Event97

Hydrol. Process. 26, 893–906 (2012)

898 G. LEE ET AL.

area of 211 km2 and has a hilly topography with elevationsranging from 431m to 1720m. Most of the land is forestedas shown in Figure 3(b). The observed discharge data andradar rainfall data with a 10min temporal resolution (theEjiroyama X-band radar covering a 128 km radius) wasavailable in the study catchment. The rainfall-runoff modelused in this study was calibrated based on Typhoon No. 9(15–19 September 1997; hereafter Event97), and thecalibrated model was applied to the flood event causedby the continuous Typhoon No. 7 and Typhoon No. 8 (1–7September 1999; hereafter Event 99) for a diversity ofanalysis purposes. Figure 3(c) shows the grid formattedtotal rainfall depth during the Event97, with a 1 km� 1 kmspatial resolution.

CATCHMENT RESPONSES UNDER EQUIFINALITYDUE TO PARAMETER UNCERTAINTY

Identifying plausible parameter sets for catchmentresponse analyses under equifinality

Many studies have been conducted to quantify the param-eter uncertainty and assess its propagation into subsequentprediction results. For example, multinormal approximations(Kuczera and Mroczkowski, 1998), simple uniform randomsampling over the feasible parameter space (Uhlenbrooket al., 1999) and Markov Chain Monte Carlo (MCMC)methods (Kuczera and Parent, 1998; Vrugt et al., 2003)were developed to analyze the parameter uncertaintyinvolved in hydrological modeling. In comparison with thetraditional statistical theory based on first-order approxima-tions and multinormal distributions, the MCMC methodshave become increasingly popular as one of the generalpurpose approximation methods for complex inference,search and optimization problems (Feyen et al., 2006).The MCMC sampler, the SCEM-UA method (Vrugt

et al., 2003), is well suited for the practical assessment of

Figure 4. (a) Evaluation of the Scale Reduction score for Event97, (b) ~ (fperforming parameter set for Event99 (i.e. OPT) and three plausible parame

each p


parameter uncertainty in hydrological models. This samplerincorporates effective features of the SCE-UA method(Duan et al., 1992) such as a controlled random search,competitive evolution and complex shuffling with thestrengths of the Metropolis-Hasting algorithm to evolvesamples to a stationary posterior distribution of parameters.There are two major revisions made in the SCEM-UA inorder to prevent the convergence into a small attractionregion in the original SCE-UA algorithm and obtain astationary posterior target distribution of parameters. Thefirst modification is the replacement of the downhillsimplex method with a Metropolis annealing covariance-based offspring approach, thereby avoiding a determin-istic drift toward a single mode. Second, the SCEM-UAdoes not divide the complex into sub-complexes duringthe generation of the offspring and uses a differentreplacement method to terminate the regions containing alower posterior density in the parameter space. Thisalgorithm can provide not only the optimal parameter setbut also its underlying posterior distribution within asingle optimization run.

The five model parameters of the KWMSS werecalibrated by this stochastic optimization technique withEvent97. A fundamental process of the MCMC sampling isto evaluate a convergence of the sampler to the stationaryposterior distribution. Practically, the Scale Reductionscore (

ffiffiffiffiffiffiSR

p) developed by Gelman and Rubin (1992) has

been used to be a criterion for checking convergence. If theffiffiffiffiffiffiSR

pis less than 1.2, the Markov chain is considered to

have converged into the target posterior distribution;otherwise, the evaluation steps are repeated until thesesequences become stable. Figure 4(a) illustrates thecalculated


pagainst the number of MCMC iterations.

All parameters become stable after approximately 4000iterations (i.e.


p<1.2). Note that the posterior

distributions were obtained by 6000 behavioral parametersets after the convergence of the SCEM algorithm (here,

) marginal posterior parameter probability distributions including the bestter sets (i.e. Sample (1) (2), and (3)), marked in upper horizontal axes ofanel

Hydrol. Process. 26, 893–906 (2012)


the first 4000 simulations of each parallel sequence arediscarded because the


pis larger than 1.2). Using the 6000

parameter combinations, we plotted the marginal posteriorparameter distributions as shown in Figures 4(b)–(f).The feasible (or initial) ranges of each parameter are

constrained through the stochastic optimization procedure.The width of each histogram of Figures 4(b)–(f) corre-sponds to the specified ranges of the estimated parameteruncertainty, and the highest densities of each probabilitydistribution indicate the global optimal values of eachparameter of the KWMSS for Event97. The diamondmarks for each probability distribution denote the best-performing parameter values for the validation event,Event99. Although all calibrated parameter values for bothevents were well identified within predefined feasibleparameter spaces, the parameter values of ds and b are verysensitive to the applied event. The optimal parametervalues for the calibration event, Event97, deviate too muchfrom the optimal values for the validation event, Event99.Here, the best-performing parameter set for Event99 isnamed as the OPT, and it is used as a standard parameterset to compare the simulation results based on the plausibleparameter sets. In addition to the standard parameter set,we selected three different parameter sets belonging to theestimated posterior parameter distributions that provideacceptable model performance measures for Event99. Theyare named as Sample(1), Sample(2) and Sample(3),respectively, and referred to as plausible parameter sets.

Table I. The optimal parameter set and sele

Parameter Data used for calibration n [m�1/3 s]

OPT Event99 0.5Sample(1) Event97 0.495Sample(2) 0.487Sample(3) 0.484

Figure 5. Simulation uncertainty associated with the behavioral parameter setplotted), derived by SCEM-UA and simulated hydrographs associated with pl

sets are plotted as colored


All of the values of the parameter sets used in this study aresummarized in Table I.

The probabilistic results of the hydrograph for Event99were obtained from the ensemble simulation of KWMSSwith 6000 behavior parameter sets, including the threeselected plausible sets, sampled from the posteriorparameter distributions. Figure 5 shows how the parameteruncertainty propagates into the estimates of the predictionuncertainty for the hydrograph. In Figure 5, the blackdotted line indicates the observed streamflow data, and thegrey shaded region represents the 90% prediction uncer-tainty associated with the posterior distribution of theparameter estimates. In spite of the considerable parameteruncertainty, the ensemble simulation results match theobserved runoff well, and the simulation uncertaintyboundary is very narrow. Also, the plausible parametersets can generally produce acceptable hydrographs, whiletheir results slightly underestimate the OPT hydrographafter the peak flow at 132 h.

The results support that the distributed rainfall-runoffmodel, KWMSS, is likely to be exposed to the equifinalityproblem that makes it difficult to determine the uniquelyoptimized parameter set. The behavioral parameter setsmay sometimes lose the physical meaning in terms of thenumerical value but would be meaningful in terms of themodel predictions. If so, when the KWMSS acceptsmultiple flow pathways, leading to acceptable hydrographs,how does spatiotemporal information in each slope element

cted plausible parameter sets for Event99

ka [m/s] ds [m] dc [m] b[�]

0.0130 0.865 0.472 18.60.0191 0.599 0.452 5.780.0196 0.726 0.562 7.990.0195 0.551 0.418 4.99

s having 90% confidence interval (i.e. 5400 (6000� 90%) hydrographs areausible parameter sets. The hydrographs reproduced by plausible parametersolid lines in Figure 5

Hydrol. Process. 26, 893–906 (2012)

Table II. Summary of temporally separated components of thesimulated hydrographs under the equifinality due to parameter

uncertainty

Temporal Class CasePre-eventwater (%)

Event water (%)

1 2 3 4 5

OPT 57 8 9 7 9 10Sample(1) 56 9 9 7 9 10Sample(2) 60 8 8 6 8 10Sample(3) 56 9 9 7 9 10

900 G. LEE ET AL.

change due to plausible parameters? We adopted thecomputational tracer method to show the effect of plausiblescenarios associated with parameter values on the spatio-temporal origin of streamflow. Here, we applied the threepredefined plausible parameter sets (i.e. Sample (1)–(3)) ininvestigating internal catchment responses in time andspace and then compared their results with the result of theparameter set OPT.

Temporal variation of flow source based on plausibleparameter sets

For the temporal hydrograph separation, the rainfall ofEvent99 is split into six temporal classes (NT=6; pre-event,0 ~ 25 h, 26 ~ 96 h, 97 ~ 109 h, 110 ~ 120 h and121 ~ 192 h). Then, the hydrographs simulated by theplausible parameter sets are separated into six componentsthat correspond to the rainfall fractions by a temporalrecord of the matrix described in Figure 2. Figure 6 showsthe temporal hydrograph separation results using theparameter set OPT for Event99. The colors in thehydrographs correspond to the colors in the hyetographswhile the yellow portion corresponds to the stored waterbefore the beginning of the simulation, which is defined aspre-event water in this study. Each runoff component of allapplications, corresponding to the six selected rainfallsegments, is summarized in Table II.This table suggests that the percentage of runoff

components, with respect to both pre-event and eventwater, is subject to the applied parameters despite visuallyindistinguishable runoff components. Sklash et al. (1986)and McGuire (2004) demonstrated in their field survey thatthe large amount of streamflow might have originated frompre-event water. The high pre-event water fractions areconsistent with the results of this simulation study. That is,the pre-event water which comprised the streamflow for theEvent99 period varied in the range of 56% to 60% underthe given equifinality condition while the event waterchanged marginally due to applied parameters. In spite ofnearly identical hydrographs (i.e. global responses) withvery similar objective function values, the temporally

Figure 6. Temporal hydrograph separation result: the yellow portion andthe other colors represent pre-event water runoff and event waterrunoff, respectively, and the black solid line shows the observed hydrograph

of Event99


separated runoff component showed different aspects underthe equifinality. Lee et al. (2008) also demonstrated that thecalibrated parameter values, which were obtained byseveral different objective functions such as the hetero-scedastic maximum likelihood estimate and the modifiedindex of agreement, showed similar discharges at the outletbut led to totally different temporal patterns regarding bothpre-event and event water.

Spatial variation of flow source under equifinality based onplausible parameter sets

The study catchment was represented by 3118 slopeelements (i.e. NS=3118), and each was referred to as thegeographical source of the streamflow. Using the compu-tational tracer method with the spatial record of the matrixduring the flood period, we calculated the ratio of the floworiginating from each slope element at every 1 h time step.The ratio is defined as the contributing ratio (CR) of theevent water for each slope element to the discharge at theoutlet at the specific simulation time step. If we add all ofthe CR values over the catchment at each time step, thetotal would then be 100%, which is equal to the ordinate ofthe hydrograph at the selected time step. Figure 7represents the CR value for all slope elements at fourspecific time steps: 1, 28, 72 and 136 (peak time) h.

The simulation results indicate that the contributing areaexpands from the near stream zone to the upper slope areasas the catchment becomes saturated during the flood event.At the beginning of the rainfall-runoff process, the slopeelements adjacent to the river channel, referred to as theriparian zone, constituted primarily of streamflow while thewater stored in the upstream slope elements did not yetreach the river. As time goes on, the contributing areasspread gradually over the catchment, and eventually, allslope elements influence the streamflow generation.

The variation of geographical event water sources wasmarginally sensitive to the plausible parameter sets usedhere. In Figure 7, even though plausible parameter setsproduce the spatially different distribution of CR valuesfrom the OPT, each slope element over the catchment, ingeneral, shows visually indistinguishable CR values of theevent water regardless of different values of appliedparameter sets. The plausible parameter sets, which wereobtained from the posterior parameter distributions,provide a relatively equivalent geographical source ofstreamflow.

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Figure 7. Spatial source distributions of the event water for OPT, Sample(1), Sample(2) and Sample(3) at specific time steps: 1, 28, 72 and 136 h forEvent99; here, CR indicates the percentage of runoff rate of each slope element to hydrograph ordinates at simulation time step of interest


CATCHMENT RESPONSES UNDER EQUIFINALITYDUE TO INPUT UNCERTAINTY

Identifying plausible rainfall patterns for catchmentresponse analyses under equifinality

In general, the data uncertainty arises from various sources:the errors introduced by the field measurement itself, thespatiotemporal scale (i.e. aggregation and disaggregation)of observations for hydrological applications and data


processing like transforming radar reflectivity into rainfallintensity (Wagener et al., 2004). Among those, the poorobservation of rainfall in time and space leads toinaccurate prediction results in distributed rainfall-runoffmodeling. Therefore, for more reliable prediction results,it is essential to capture the amount and spatiotemporalvariability of rainfall as precisely as possible and thenincorporate it into the hydrological modeling process(O’Connell and Todini, 1996).

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902 G. LEE ET AL.

Most past research regarding runoff errors due to thespatial variability of rainfall (e.g. Ogden and Julien, 1993;Obled et al., 1994; Faurès et al., 1995; Lopez, 1996; Shahet al., 1996) demonstrated that general hydrologicalresponses such as runoff volume, peak flow and the timingof peak flow must have been influenced by the spatialpattern of rainfall, while Tachikawa et al. (2003) found thatthe effect of spatial pattern of rainfall on runoff simulationusing a distributed model was not significant whenapplying various rainfall scenarios to the mesoscalemountainous catchment. Here, we focus on how the inputuncertainty based on spatially various rainfall scenarios(derived from observed radar rainfall data) influencesrainfall-runoff simulations.Five grid cells were selected from the observed radar

field for generating various rainfall scenarios, and then it isassumed that these cells are virtual gauge stationsrepresenting the rainfall aspects of downstream (gauge 1),mid-stream (gauge 2) and upstream (gauge 3, 4 and 5) inthe study area. Figure 8 describes the spatial distributionsof the total rainfall depth for Event99. Five syntheticrainfall fields were produced based on the selected virtualstations using the nearest neighborhood method using adifferent number of virtual gauges for Scenario 2 toScenario 6. Scenario 7 was set up by rearranging theScenario 6 randomly to examine the effects of the spatialarrangement of rain gauges on runoff simulations. Scenario8 changed the origin of the original radar data (i.e. Scenario1) based on the X and Y axes in order to investigate moreclearly the influence of the rainfall spatial pattern due to therelocation of rainfall cells on catchment responses. Thedistributed model was calibrated with Scenario 1 rainfall

Scenario 1 Scenario 2


guage1

guage2

guage3

guage4

guage5

x

y

Figure 8. Spatial patterns of accumulated ra


data, and this data was assumed to provide the best inputestimate to the model.

For each rainfall scenario, the relative errors of rainfalland runoff are computed as:

RE ¼ RS1 � RSi

RS1� 100 i ¼ 2;⋯⋯; 8 (12)

whereRS1 is either the accumulated depth of areal rainfall orthe general properties of runoff such as runoff volume,peak flow and timing of peak for Scenario 1; RSi is thevariable of interest in Scenario i. The relative errors of bothrainfall and discharge to the original data are illustrated inFigures 9(a) and (b), respectively.

Figure 9 shows that the most spatially uniform rainfallfield fails to estimate an accurate amount of areal rainfall; therainfall of Scenario 2 is overestimated at 32%. The errors ofScenarios 6 and 7 are much larger even if their spatialpatterns are more heterogeneous than Scenarios 3, 4 and 5.The reason is that the rainfall depth of rain gauge 4 at eachtime step was recorded at a much higher reading than thoseof the other rain gauges such that the total amounts ofrainfall in the scenarios interpolated using gauge 4 werecomputed considerably higher than the other scenarios.

The difference of rainfall volume results in very similarerror patterns with respect to runoff volume and peakdischarge, while the peak time error is less sensitive to therainfall scenarios used here. The peak times of evenerroneous rainfall data (i.e. Scenarios 2, 6 and 7) led torelative errors of less than 12%, while the errors of runoffvolume and peak flow for these scenarios deviatesignificantly from the original data of Scenario 1. However,



x

y

infall for all rainfall scenarios of Event99

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Figure 9. Relative errors of (a) catchment mean rainfall and (b) runoff foreach scenario of Event99; REV, REP and REPT indicate relative error ofrunoff volume, relative error of peak discharge and relative error of peak

time, respectively


it does not imply that Scenario 4, with only three raingauges, optimally presents the actual spatiotemporalvariability of rainfall. We assume that Scenarios 4 and 8have plausible input data in terms of the small amount ofrelative errors for runoff properties when considering thesuperiority of their results to the other scenarios.

Temporal variation of flow source under equifinality basedon plausible input scenarios

The hydrographs simulated by the plausible rainfall datawere mainly divided into pre-event water and event watercomponents. Each hydrograph component that correspondsto the same rainfall durations of the section Temporalvariation of flow source based on plausible parameter setsis summarized in Table III.Here, the pre-event water that the streamflow comprised

of for the Event99 period varied in the range of 55% to57% under the given equifinality condition. The lumpedinput data, Scenario 4, led to the same amount of pre-eventwater with the original data, while Scenario 8, with the

Table III. Summary of temporally separated components of thesimulated hydrographs under the equifinality due to input

uncertainty

Temporal Class CasePre-eventwater (%)

Event water (%)

1 2 3 4 5

OPT 57 8 9 7 9 10Scenario 4 57 6 9 7 10 11Scenario 8 55 9 9 7 9 11


opposite spatial pattern, led to a 2% decrease in pre-eventwater. Moreover, the event water from rainfall segment 1showed a 2% reduction and a 1% increase in Scenarios 4and 8, respectively. Despite the different spatial patterns ofplausible scenarios, both scenarios could provide a similartemporal variation of runoff generation during the eventwhen compared to the original input data. The next sectiondeals with the geographical origin of streamflow under theequifinality condition, due to the input uncertainty, in orderto clarify the contradiction between global and internalcatchment responses.

Spatial variation of flow source under equifinality based onplausible input scenarios

To show the overall variation trend of CR for all slopeelements through the event period, we calculate thetemporal average value of CR for each slope element,called the average contributive level (ACL). As shown inFigure 10, the most contributive area ultimately follows thespatial pattern of rainfall data. The ACL of each slopeelement across the flood period also supports the highervalues of spatial source contribution that were given at nearstream zones. Moreover, due to the plausible input data, thespatial distributions of the ACL are very sensitive tospatially different rainfall intensity. Figures 10 and 11apparently support that the spatial pattern of forcing inputdata is much more dominant for geographical sources ofstreamflow than the model parameter value in producingevent water. Due to the aggregated spatial pattern ofScenario 4, the most contributive area, the southern part ofthe area does not appear in this case, and the oppositespatial pattern of Scenario 8 leads to contrary geographicalsources of streamflow in the OPT result. Consequently,Scenarios 4 and 8 result in poorly matched ACL dottedplots as shown in Figure 11. On the other hand, the ACLcomparison plot of pre-event water as shown in Figure 11demonstrates that plausible rainfall scenarios provide arelatively equivalent geographical source of streamflow,particularly in potent contributing slope elements withhigher ACL values, although Scenario 8 leads tounmatched values in lower contributing slope elements,due to an extremely changed spatial pattern.

From these application results, it can be seen that thedistributed rainfall-runoff model, KWMSS, permits mul-tiple alternative flow pathways in time and space whenrainfall over catchment transforms into runoff. Theplausible parameter sets and rainfall scenarios resulted inacceptable hydrographs, while the catchment did notrespond identically to them in time and space.

The results also reveal that the commonly usedstreamflow data (i.e. the representative aggregated catch-ment response) is not sufficient enough to evaluate thephysically meaningful flow pathway among a number ofpossible ones. In spite of different insights into theequifinality problem, all research has a consensus on theequifinality problem in rainfall-runoff modeling. A tradi-tional parameter estimation, i.e. just searching for theoptimum, has to be abandoned and replaced by set

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Figure 10. Spatial source distributions of event water for OPT, Scenario 4 and Scenario 8; each color indicates the ACL value of individualslope element

Figure 11. ACL comparison plots between OPT and the plausible input scenarios of (a) event water and (b) pre-event water

904 G. LEE ET AL.

theoretical approaches (e.g. Beven and Binley, 1992; Vrugtet al., 2003), which accept multiple parameter combina-tions. Second, a classic model identification framework,based on a single criterion such as streamflow data, isinsufficient to either identify a reliable parameter set or toreduce parameter uncertainty. Therefore, the multiplehydrological variables are necessary to solve the equifinal-ity problem by conditioning the model parameters withthese additional hydrological constraints (e.g. Ambroiseet al., 1995; Mroczkowski et al., 1997; Franks et al., 1998;Lamb et al., 1998).


In this regard, our approach to the equifinality problemcan be used to give a better understanding regardingdynamic catchment behaviors under various hydrologicaluncertainties and can be used as an evaluative criterion forfurther parameter identification.

CONCLUSIONS

This study presented the equifinality phenomenon instreamflow prediction by a distributed rainfall-runoff

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model. Most of the past research associated withequifinality have concentrated on assessing the global (oraggregated) catchment response (e.g. streamflow) due toparameter uncertainty. On the other hand, our approachdealt with the effect of parameter and input uncertainty onnot only global catchment responses, but also internalcatchment responses in time and space. For this objective,we carried out several experiments progressively with twocategories. At first, we estimated the plausible parametersets of the distributed rainfall-runoff model, KWMSS, usingthe SCEM-UA algorithm and also generated plausible inputrainfall scenarios from the original radar rainfall field inorder to satisfy the equifinality condition in terms of theobjective function value and the runoff property error,respectively. Then, we adopted a computational tracermethod combined with the KWMSS for investigating howa catchment behaves under the given equifinality conditionusing parameter and input uncertainties.The computational tracer method clearly showed the

spatial progress of streamflow generation when rainfallover the catchment flows into the catchment outlet duringthe event. At the beginning of the event, the dominantinfluencing streamflow (from a geographic origin) waslocated near the channel network, and then the contributingarea spread much wider as the catchment became saturateddue to the rainfall. In the case of event water, the plausibleparameter sets used in this study do not excessivelyinfluence the CR values of each slope element while thespatial patterns of the ACL explicitly reflect the spatialpatterns of plausible input data. In addition, the ACLcomparison results of event water showed that the plausibleinput scenarios used here led to less equivalent geograph-ical sources than in the cases of pre-event water.To trace the temporal origin of the streamflow, the

historical event was divided into six segments including pre-event water. Old water components account for theminimum, 55% to the maximum, 60% under the equifinalitycondition due to parameter and input uncertainties whileevent water components were not changed significantly.The equivalent hydrographs, despite different spatio-

temporal source variations due to applied parameters andinput data, support the idea that a distributed rainfall-runoffmodel can be exposed to the equifinality problem. That is,it can permit various alternative flow pathways guarantee-ing good hydrograph simulations. This result suggests thata split sample test using streamflow data should be treatedas the minimum requirement for testing the modelperformance in an operational application. One obviousand well-known way to resolve this impasse is to couplehydrological models with streamflow data augmented byother kinds of hydrological information relevant toprediction tasks. Some examples of data augmentationinclude streamflow and stream solute concentrations atdifferent locations within the study watershed and othermeasurable hydrological fluxes or states such as soilmoisture, piezometric levels and evapotranspiration atselected locations.For the given model complexity, increasing either the

data availability or hydrological information leads to less


predictive uncertainty and better predictive performance upto a point, after which the content does not improve theprediction and reduce uncertainty. In other words,complementary hydrological information presents addi-tional opportunities to test or falsify model hypothesesbased on parameters. In this context, the computationaltracer method can be a useful and physically evaluativecriterion for further model identification.

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