A spatial regularization approach to parameter estimation for a distributed watershed model

A spatial regularization approach to parameter estimation

for a distributed watershed model

P. Pokhrel,1 H. V. Gupta,1 and T. Wagener2

Received 25 October 2007; revised 30 September 2008; accepted 15 October 2008; published 13 December 2008.

[1] Environmental models have become increasingly complex with greater attentionbeing given to the spatially distributed representation of processes. Distributed modelshave large numbers of parameters to be specified, which is typically done either byrecourse to a priori methods based on observable physical watershed characteristics, bycalibration to watershed input-state-output data, or by some combination of both. In thecase of calibration, the high dimensionality of the parameter search space poses asignificant identifiability problem. This article discusses how this problem can beaddressed, utilizing additional information about the parameters through a process knownas regularization. Regularization, in its broadest sense, is a mathematical technique thatutilizes additional information or constraints about the parameters to reduce problemsrelated to over-parameterization. This article develops and applies a regularizationapproach to the calibration of a version of the Hydrology Laboratory DistributedHydrologic Model (HL-DHM) developed by the US National Weather Service. A prioriparameter estimates derived using the approach by Koren et al. (2000) were used todevelop regularization relationships to constrain the feasible parameter space and enableexisting global optimization techniques to be applied to solve the calibration problem.In a case study for the Blue River basin, the number of unknowns to be estimated wasreduced from 858 to 33, and this calibration strategy improved the model performancewhile preserving the physical realism of the model parameters. Our results alsosuggest that the commonly used parameter field ‘‘multiplier’’ approach may oftennot be appropriate.

Citation: Pokhrel, P., H. V. Gupta, and T. Wagener (2008), A spatial regularization approach to parameter estimation for a

distributed watershed model, Water Resour. Res., 44, W12419, doi:10.1029/2007WR006615.

1. Introduction

[2] Over the past few decades, considerable progress hasbeen made in the development of highly sophisticatedhydrologic models. At the same time, reliable global opti-mization techniques such as the Shuffled Complex Evolu-tion algorithm (SCE-UA [Duan et al., 1992]) have beendeveloped that are known to be effective and efficient atfinding the global minimum of a parameter estimationproblem even for complex watershed models [e.g., Duanet al., 1992, 1993, 1994; Sorooshian et al., 1993; Luce andCundy, 1994; Gan and Biftu, 1996; Tanakamaru, 1995;Kuczera, 1997; Hogue et al., 2000; Boyle et al., 2000;Wagener and Gupta, 2005; among others]. Powerful com-puters now allow us to handle large volumes of data andcalculations. These developments, the increasing availabil-ity of spatial data (such as precipitation estimates providedby NEXRAD weather radar), and the pressing need foraccurate predictions of river stage at all points along the

river network of a watershed as well as the need to providespatial estimates of soil moisture and other hydrologicalvariables, have led to the development of spatially distrib-uted watershed models [see, e.g., Havno et al., 1995; Ivanovet al., 2004; Carpenter and Georgakakos, 2004; Smith etal., 1995; Koren et al., 2004, among many others].[3] In principle, a spatially discretized model of a system

is expected to provide better predictions than its lumpedcounterpart, because by aggregating the data for the latterwe lose valuable information about spatial heterogeneityand non-linearity that can influence the system response.Further, lumped watershed models can only simulate theaggregate average hydrologic behavior of the system, and inparticular only the streamflow response at the watershedoutlet, while a spatially distributed model can simulate thespatial distribution of processes throughout the modeldomain. However, while spatial discretization of the domaincan provide more useful information, it also significantlyincreases the number of unknown model parameters thatmust be specified. When dealing with complex physicalprocesses over a large spatial domain discretized at a finespatial resolution, the number of unknowns can quicklybecome unmanageable. The high dimensionality of thisparameter estimation problem imposes severe restrictionson the use of available optimization schemes for modelcalibration. Without suitable methods to address this prob-

1Department of Hydrology and Water Resources, University of Arizona,Tucson, Arizona, USA.

2Department of Civil and Environmental Engineering, PennsylvaniaState University, University Park, Pennsylvania, USA.

Copyright 2008 by the American Geophysical Union.0043-1397/08/2007WR006615$09.00

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lem, the practical application of distributed watershedhydrologic models is hampered.[4] The solution to this problem lies, of course, in a

recognition that the model parameters are not, in fact,independent entities that can take on arbitrary values inthe parameter space. Rather, the spatial distribution of theirvalues must be related to the spatial variability of hydro-logically relevant watershed properties over the modeldomain, as is recognized, for example, by the literature onparameter regionalization [Hundecha et al., 2008; Wageneret al., 2004; Schumann et al., 2000; Seibert, 1999; Wagenerand Wheater, 2006; Gotzinger and Bardossy, 2007, amongothers]. This spatial variability typically exhibits variouskinds of structures and patterns, being related to the spatialdistribution of watershed characteristics such as geology,soil type, vegetation, and topography [Grayson and Bloschl,2000]. Properly recognized, this fact can enable the imple-mentation of additional relationships that constrain thedimensionality of the parameter estimation problem.[5] In other words we can exploit the fact that the values

of the unknown parameters within a watershed are notspatially independent of each other and can be related toparameter values at nearby locations by means of properlychosen relationships. The objective of this article is todiscuss an approach to establishing regularization relation-ships for distributed environmental models that results in abetter conditioned specification of the parameter estimationproblem and allows the use of existing global optimizationtechniques for model calibration. The approach is demon-strated by application to calibration of the University ofArizona version of the Hydrology Laboratory-DistributedHydrologic Model (HL-DHM) developed by the NationalWeather Service for flood forecasting throughout the con-tinental US [Smith et al., 2004; Reed et al., 2004].[6] Section 2 of the article briefly discusses the principles

of regularization and how available information about therelationships between model parameters and spatial distri-butions of observable (or inferable) physical watershedcharacteristics can be used to condition and reduce thedimensionality of the optimization problem. Section 3describes the study area, data, and model used. Section 4discusses the regularization relationships used in this work.Sections 5 and 6 describe the formulation and application ofthe calibration methodology and discuss the results. Finallysection 7 presents a discussion of this work along with ourconclusions and some suggestions for further research.

2. Regularization

[7] Over-parameterization of a model can result in an illposed and numerically intractable calibration problem[Doherty, 2003]. This problem of over-parameterizationhas been studied in a variety of fields including petroleumengineering and groundwater modeling [Crook et al., 2003;Doherty, 2003, among others] and addressed by use of amathematical technique known as regularization.[8] In its broadest sense, regularization is a technique that

facilitates the inclusion of additional information, in theform of regularization relationships or constraints, to help inthe stabilization and solution of ill-posed problems [Dohertyand Skahill, 2005; Linden et al., 2005]. There are twogeneral approaches by which this can be achieved. The first

is the so-called Penalty Function approach [Fletcher, 1974].Consider the optimization problem:

minimizewrtq

F qð Þ ð1Þ

in which we seek values for the parameters q that minimizethe performance criterion F(q) which measures the distancebetween the system response data and the model output. Ifthe dimensionality Rq of the parameter space in thisoptimization problem is sufficiently large compared to theinformation content RI of the system response data, thisproblem can become poorly conditioned. However, if we canprovide additional information about the values of para-meters q in the form of constraints G(q) � 0, we can use thisinformation to restrict the set of feasible solutions therebyconstructing a better conditioned optimization problem. Animportant example of this approach is Tikhonov regulariza-tion [see Tikhonov and Arsenin, 1977], in which a penalty isapplied on solutions that deviate from satisfying theregularization constraints. Of course, while such impositionof constraints can result in improved parameter reason-ableness and stability, this can come at the cost of a reducedability of the model outputs to match the observations.[9] In the Penalty Function approach we retain the original

parameter dimension Rq (i.e., the optimization search isconducted in the original parameter space) while the shapeof the function F(q) to be optimized is modified (to becomebetter conditioned) via the inclusion of additional informa-tion through the regularization criterion. An alternativeapproach is to constrain the solution through control of (orreduction of) the dimensionality of the solution space[Demoment, 1989]. For example, the Truncated SingularValue Decomposition (TSVD) approach confines the cali-bration problem to a subspace of the full problem [Lawsonand Hanson, 1995; Weiss and Smith, 1998] by excludingsearch directions that are associated with little or no functionsensitivity (very small Eigenvalues of the Hessian Matrixr2F(q)). By conducting the search along the directions of‘‘super-parameters’’ that exist in a lower dimensional sub-space, the TVSD approach can dramatically improve thespeed of convergence of the optimization algorithm. How-ever, the result can be sensitive to the initial starting pointand the method offers no guarantee for reasonableness of the‘‘optimal’’ parameters [Tonkin and Doherty, 2005].[10] Each of these approaches has its strengths and

weaknesses, and hybrid approaches have emerged. Forexample, Tonkin and Doherty [2005] present a hybridregularization methodology that uses Tikhonov regulariza-tion (imposing constraints in the original parameters space)to improve the conditioning of the optimization problem,while using TSVD to speed up convergence by searchingonly in the smaller subspace of super-parameters. Suchstrategies have been used to address high dimensionalinverse modeling problems in hydrology [see, e.g., Skaggsand Kabala, 1998; van Loon and Troch, 2002; Tonkin andDoherty, 2005; Doherty and Skahill, 2005, etc.].[11] The two principles at work in the regularization

methodologies discussed above are (1) to use additionalinformation to improve conditioning of the optimizationproblem, and (2) to reduce the dimension of the parametersearch space. Here, we implement a regularization method-

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ology that exploits both of these principles in a differentmanner, by constraining the extent of the search spaceusing additional information about the relationshipsbetween the spatial distributions of static system character-istics X and the model parameters q. These relationshipscan be expressed in the general form:

q ¼ H X=fð Þ ð2Þ

such that their shape is controlled by a small set of super-parameters f, where the dimension Rf � Rq. For example,X could refer to spatial maps of soil properties (e.g., soiltype and depth), and the functional relationship H(./.) coulddescribe how the parameters controlling horizontal andvertical movements of water at the model grid scaledepend on these soil properties. By substituting equation (2)into equation (1) we obtain the regularized optimizationproblem:

min imizewrt q

R fð Þ ¼ F H X=fð Þð Þ ð3Þ

where the regularization relationships (equation (2)) con-strain the q space of feasible parameter solutions, while theactual optimization search is conducted in the lowerdimensional space of super-parameters f. In contrast to theTSVD approach, the relationships between the super-parameters f and the original parameters q will not berestricted to simple linear transformations.[12] Of course, while our approach to spatial regulariza-

tion bears functional similarity to parameter ‘‘regionaliza-tion’’ wherein regression relationships are used to specifytransfer functions that relate model parameters to observablephysical characteristics at the watershed scale via calibration[Hundecha and Bardossy, 2004; Gotzinger and Bardossy,2007]. However, while the goal of regionalization is pri-marily to transfer information regarding parameter valuesfrom gauged/calibrated basins to ungauged ones, the objectof regularization is simply to better condition the optimiza-tion problem within a basin by constraining the degrees offreedom of the search space in a physically meaningfulmanner.[13] In this article, we discuss how this regularization

methodology can be the applied to the problem of modelingthe spatial rainfall-runoff processes in a watershed, anddemonstrate this approach for the case of the HL-DHMmodel applied to the Blue River basin in Oklahoma, one ofthe study areas for the Distributed Model IntercomparisonProject (DMIP; http://www.nws.noaa.gov/oh/hrl/dmip/2/index.html). Information regarding the spatial distributionof a variety of observable (or inferable) static watershedcharacteristics is used to develop regularization constraintsthat constrain the dimension of the conceptual watershedmodel calibration problem and facilitate a practical ap-proach to its solution. These relationships are inferred froma priori estimates of spatial parameter fields provided by themethodology developed by Koren et al. [2000].

3. Study Area and Model Used

[14] TheBlueRiver basin in southernOklahoma is a narrowand elongated gently sloping river valley (see Figure 1),approximately 56 miles long and ranging in elevation from

400 m to 158 m. The basin has a drainage area of approxi-mately 476 squaremiles, and is characterized by shallow depthto bedrock with soil depths of less than 2 m, the dominant soiltypes being sand, clay and loam. A U.S. Geological Surveystream discharge station (number 07332500) is located at theoutlet of the basin (node 14 in Figure 1). The average annualflow at the outlet is 317 cfs, the long-term runoff ratiois approximately 0.2, and the maximum observed flow of65000 cfs occurred in the year 1982. The streamflow obser-vations consist of hourly measurements of instantaneous riverdischarge.[15] Spatially distributed Stage III precipitation estimates

are available for the basin from Next Generation WeatherRadar (NEXRAD) coverage of the area. The estimates werederived using a Weather Surveillance Radar-1988 Dopplersystem (WSR-88D) combined with rain gage data, qualitycontrolled by the NWS. The data are available at a temporalresolution of 1 hour and a spatial resolution of approxi-mately 4 � 4 km2, over a rectilinear HRAP (HydrologicRainfall Analysis Project) grid based on a polar stereo-graphic projection. The HRAP grid is a ‘‘subset’’ of theLimited Fine Mesh (LFM) grid used by the Nested GridModel (NGM) available from the NWS (www.weather.gov/oh/hrl/dmip/2/ok_precip.html). The HRAP grid over theBlue River basin consists of 78 units (see Figure 1) andprecipitation estimates are available for each grid cell.[16] Evaporation data for the basin consists of free water

surface evaporation estimates available from the DMIP Website (www.weather.gov/oh/hrl/dmip/2/evap.html). Theseestimates are based on annual free water surface (FWS)evaporation maps and mean monthly station data (V. Korenet al., NOAA-NWS Hydrology Lab, unpublished report, 13August 1998). One limitation is that the FWS evaporationestimates vary seasonally but not annually, which meansthat the same evaporation data set is repeated every year. Inthe HL-DHM model, the FWS estimates are multiplied bypotential evaporation adjustment (PE) values to account forthe effects of vegetation. The product of FWS evaporationestimates and the PE adjustment values gives estimates ofpotential evapotranspiration demand (PET) for use by themodel.[17] The conceptual distributed watershed model used in

this study is the University of Arizona research version ofthe HL-DHM distributed modeling system, programmedin MATLABTM (version 7.0.1, www.mathworks.com) anddesigned to run on a personal computer. The water balancecomponent consists of the Sacramento Soil MoistureAccounting Model (SACSMA [Burnash et al., 1973]),applied to each HRAP grid cell. A simplification has beenimplemented to the grid routing component by removingthe hillslope routing component and using the Muskingummethod instead of the kinematic wave approach forchannel routing. Using a synthetic study the impact ofthis modification was found to be insignificant for theBlue River basin.[18] The model generates precipitation excess in each

grid, which is then accumulated at the nearest downhill rivernode positioned along the main river channel as shown inFigure 1. The river nodes are defined at cells where two ormore lower order stream branches meet and contribute tothe flow; fourteen such nodes appear in the Blue River basinconnectivity map (Figure 1). The drainage area represented

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by each node is based upon the connectivity scheme usedby the NWS [Koren et al., 1992]. The precipitation excessgenerated at each grid is assumed to travel instantaneouslyto the associated river node, where it is added to the flowfrom the upstream node and routed to the downstream nodevia a Muskingum routing scheme.[19] The water balance component of the SACSMA

model consists of 16 parameters and 6 state variables(see Table 1) for each grid cell. Five of these parametersare assumed to be spatially lumped and fixed at valuesspecified by the NWS. With 11 parameters taking onspatially distributed values at 78 grid locations. Includingthe 2 channel routing parameters, the total number ofunknowns to be estimated is 860 (11 � 78 + 2).

4. Development of the RegularizationRelationships

[20] In our regularization approach, we seek to exploitany relationships that exist among the physically observablestructural properties of the system, which must be preserved

to properly characterize the spatial distribution of functionalresponses of the system. In the case of watershed modelingthese functional responses are controlled by the distributionof surface and subsurface hydraulic properties, and we aretherefore primarily concerned with the distributions of soils,vegetation, and topography etc. For this study, the observ-able watershed characteristics investigated included griddedwatershed elevation and slope, depth of soil, average soilmoisture content of the soil horizon, soil type, topographicaspect and vegetation. Properties derivable from thesecharacteristics include the NRCS based curve numbers,curvature of the landscape, specific catchment area andtopographic index. Meanwhile, a priori estimates of the 11spatially distributed SACSMA model parameters werederived from information about antecedent soil moisture,soil sand-silt-clay factions, depth of soil horizon, vegeta-tion type and land use, using the procedure reported byKoren et al. [2000]; hereafter we call these the KAP(Koren A Priori parameter estimates).[21] To infer the form of the regularization equations, we

assume that the KAP are reasonably representative of the

Figure 1. Model domain showing HRAP grid and locations of nodes for the Blue River basin. Nodes 5,8 and 14 are observation nodes.

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actual spatial relationships among the parameters (althoughperhaps not the magnitudes), and therefore contain infor-mation about the dominant patterns of spatial correlationthat should be preserved during calibration. We then use aregression approach to derive empirical equations that relatethe a priori estimates of each parameter to one or moreobservable (or inferable) watershed characteristics. Thisresults in a set of 11 non-linear regularization equationsthat are valid over the spatial domain (one for each modelparameter field) in the sense that they capture the pattern ofrelative magnitudes among the parameters. We proceed byassuming that these patterns of parameter variability arereasonable, but that the model simulation performance canbe improved by allowing the parameter magnitudes withinthe field to vary in ways that preserve these patterns. Thiseffect can be achieved by allowing the 3 tunable coefficientsof each of the 11 regularization equations to be varied insuch a way that the model provides a better simulation ofthe input-state-output behavior of the watershed.[22] In terms of the language introduced in section 2, the

11 regression equations constitute the regularization rela-tionships described in equation (2), and the 33 regressioncoefficients (3 per equation) constitute the super-parametersf to be optimized. By varying f, we allow the spatiallydistributed values of the parameters q to vary, but only in away that preserves the pattern of spatial relationshipsembedded in the regularization equations. Further, becausethe number of super-parameters f is much smaller than the

number of unknown parameters q, the dimension of thecalibration problem is reduced considerably. Because theregularization equations are tied directly to the watershedcharacteristics, which remain fixed, the spatial relationshipsamong the parameters are only allowed to change to theextent that is allowable by the variations in the values of thesuper-parameters.[23] A comprehensive study of relationships between the

KAP and various watershed characteristics [see Pokhrel,2007] revealed that much of the parameter variability couldbe related to variations in two watershed properties—thesoil depth (ZMAX) and the curve number (CN); this is notsurprising since these two variables play a strong role in theKoren formulation. In particular, the parameters LZTWMand LZFSM showed an increasing linear trend with increas-ing soil depth (see Figure 2) and the parameters UZTWM,UZFWM, UZK, REXP, PFREE and ZPERC showed linear,exponential, logarithmic or quadratic trends with changes incurve number (see Figure 3). We also found that some of theparameters displayed strong interparameter correlations thatcould be exploited; for example LZSK co-varies stronglywith UZK, LZFPM co-varies strongly with PFREE andREXP, and parameter LZPK co-varies strongly withLZTWM (see Figure 4). The outcome of this analysis isthe 11 regularization relationships shown below (see Table 1for the coefficients values derived during the regression).

UZFWM ¼ aUZFWM � ebUZFWM�CN þ gUZFWM ð4Þ

Table 1. List of the SACSMA Parameters and States

List of the SACSMA ParametersThat Were Distributed

SACSMAParameterCalibrationRange

Values of Regularization Super Parameters

a b g

Name Description Min Max A Priori Calibrated A Priori Calibrated A Priori Calibrated

UZTWM Upper zone tension water capacity (mm) 5 300 19304 24719 1.53 1.56 0 13.73UZFWM Upper zone free water capacity (mm) 5 150 561 4478 0.047 0.077 0 86.57UZK Fractional daily upper zone free water

withdrawal rate (mm/hr)0.1 0.95 3.18 3.46 0.032 0.029 0 0.122

REXP Percolation equation exponent 1 5 1.38 1.43 1 1.02 3.08 1.63LZTWM Lower zone tension water capacity (mm) 5 500 0.111 0.107 1 0.832 1.02 41.61LZFSM Lower zone supplemental free water

capacity (mm)5 500 0.028 0.022 1 0.99 8.49 2.89

LZFPM Lower zone primary free water capacity(mm)

5 500 0.583 0.103 5.51 5.25 0 172

LZSK Fractional daily supplemental withdrawalrate (mm/hr)

0.01 0.4 0.366 0.452 1 1.47 0.012 0.001

LZPK Fractional daily primary withdrawal rate(mm/hr)

0.001 0.05 8.8e 06 1.23e 05 1.26 1.17 0 0.007

PFREE Fraction of percolated water going zonefree water storage directly to lower

0 0.8 0.500 0.492 1 0.97 1.72 1.87

ZPERC Maximum percolation rate coefficient 5 350 m y g0.047 0.048 5.2 4.39 185.8 263.9

List of SACSMA Parameters That Were LumpedRSERV Fraction of lower zone free water not transferable

to lower zone tension water storage0.3 ADIMP Additional impervious area (decimal fraction) 0.00

SIDE Ratio of deep recharge to channel base flow 0.00 RIVA Riparian vegetation area (decimal fraction) 0.03PCTIM Minimum impervious area (decimal fraction) 0.005

State Variables of SACSMA ModelADIMC Tension water contents of the ADIMP area (mm) LZTWC Lower zone tension water contents (mm)UZTWC Upper zone tension water contents (mm) LZFSC Lower zone free supplemental contents (mm)UZFWC Upper zone free water contents (mm) LZFPC Lower zone free primary contents (mm)


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UZK ¼ aUZK � ebUZK�CN þ gUZK ð5Þ

UZTWM ¼ aUZTWM � CNbUZTWM þ gUZTWM ð6Þ

REXP ¼ aREXP � ln CNð ÞbREXPþgREXP ð7Þ

LZTWM ¼ aLZTWM � ZMAX bLZTWM þ gLZTWM ð8Þ

LZFSM ¼ aLZFSM � ZMAX bLZFSM þ gLZFSM ð9Þ

LZSK ¼ aLZSK � UZKbLZSK þ gLZSK ð10Þ

PFREE ¼ aPFREE � lnðCNÞbPFREE þ gPFREE ð11Þ

LZPK ¼ aLZPK � LZTWMbLZPK þ gLZPK ð12Þ

LZFPM ¼ aLZFPM � eRatio�bLZFPM þ gLZFPM ð13Þ

where Ratio =log LZFSM=PFREEð Þ

max log LZFSM=PFREEðð ÞZPERC ¼ mZPERC � CN2 þ yZPERC � CN þ gZPERC ð14Þ

[24] As a check on the validity of the regression relation-ships, the model input-state-output response generated usingthese Regularized A Priori parameter estimates (RAP) wascompared with the input-state-output response generated byKAP estimates—the model simulations were found to bevirtually indistinguishable even though there are someobservable differences in the parameter distributions (see

Figure 9). This indicates that any loss of informationregarding spatial parameter variability caused by the regu-larization process is not significant.[25] Note that 10 of the regularization equations can be

expressed in a generalized way using the form expressed inequation (15). Each equation has three coefficients; ‘‘a’’which controls the slope of the equation, ‘‘b’’ (the powerterm) which controls the strength of the non-linearity, and‘‘g’’, which controls the intercept. If we use X to denote aspatially varying watershed characteristic (CN or ZMAX) orone of the model parameters (e.g., UZK, LZTWM), thegeneral form of the regularization equations (4) to (13) canbe expressed as:

qi ¼ ai � ½X �bi þ gi ð15Þ

Only one of the regularization equations, the relation ofZPERC with respect to CN, is better expressed in a differentform, as a quadratic equation as shown below:

f ¼ mX 2 þ yX þ g ð16Þ

Also, note that three of these regularization equations (7), (9)and (10) describe, and therefore help to preserve, thedominant interparameter correlations discovered during ouranalysis.[26] It is interesting to note that the general regression

form expressed by equation (15) has a strong functionalsimilarity to the ‘‘multiplier’’ approach commonly used toreduce the dimensionality of the calibration problem forspatially distributed hydrological models [Bandaragoda etal., 2004; Vieux et al., 2004; Du et al., 2006; NOAA, 2007;Yatheendradas et al., 2008]. For example, a commonapproach is to assume that the a priori parameter fields q p

properly represent the spatial distribution pattern but that therelative magnitudes of the parameters in each field need tobe adjusted up or down via a single multiplier (i.e., super-parameter) a applied to the field, so that q = a � q p. Lesscommon, but also possible, is to assume that the a prioriparameter fields q p properly represent the spatial distribu-tion pattern but that the absolute magnitudes of the param-eters in each field need to be adjusted up or down via a

Figure 2. Plots showing relationships between spatially distributed a priori parameter estimates andmaximum soil depth ZMAX; each circle represents one HRAP grid. The solid lines show the fittedregression equations. Note the end effects caused by NWS imposed feasible limits on the parameterestimates.

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single additive constant g applied to the field, so that q = q p

+ g. The general form of regularization equation providedhere is considerably more flexible because it includes thepossibility of either or both of these effects, plus a non-linear transformation via the b super-parameter. Further, thegeneral approach includes consideration for preservingdominant interparameter relationships.

5. Regularized Calibration of the ModelParameters

[27] The regularization strategy discussed above was usedto conduct a regularized calibration of the HL-DHM modelfor the Blue River basin, where we sought to improve the

simulated input-state-output performance while preservingthe spatial patterns of the model parameter distributions.The 11 spatially distributed SACSMA parameters describedvia equations (4)–(14) were allowed to vary while the fivespatially lumped parameters (ADIMP, RIVA, RSERV,PCTIM and SIDE) were kept fixed at the values previouslydetermined by the NWS via manual calibration (Table 1).With 3 super-parameters per regularization equation, andincluding the 2 channel routing parameters of the Muskin-gum equation, the total number of unknowns to be estimatedwas 35 (11 � 3 + 2).[28] Because two different calibration strategies have

previously been shown to provide good results when

Figure 3. Plots showing relationships between spatially distributed a priori parameter estimates andcurve number CN; each circle represents one HRAP grid. The solid lines show the fitted regressionequations.


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pursuing lumped parameter calibration of the SACSMAmodel, we decided to test and compare both approacheshere. The first is a stagewise multicriterion parameterestimation approach called the Multistep Automatic Cali-bration Scheme (MACS [Hogue et al., 2000]), which usesthe single-objective Shuffled Complex Evolution optimiza-tion algorithm (SCE-UA [Duan et al., 1992]) to search forthe optimal model parameters while varying the perfor-mance criteria in a stagewise manner. The second is asimultaneous multicriterion parameter estimation approachwe call the Multi Criteria Optimization Scheme (MCOS:see, e.g., Gupta et al. [1998]) using the Multi-ObjectiveShuffled Complex Evolution Metropolis optimizationalgorithm (MOSCEM [Vrugt et al., 2003]) to conduct asimultaneous search for the Pareto-optimal set of modelparameters.

5.1. MACS Calibration

[29] MACS is a stage wise multicriterion parameterestimation approach that emulates, in a simple manner,the steps performed during manual parameter estimation.First, all of the super-parameters are adjusted to improve

the model performance as measured by the Log MeanSquared Error criterion (MSEL, equation (17)), whichemphasizes matching recessions and low flows:

MSEL ¼ 1

n

Xn

i¼1

log Obs:flowið Þ log Sim:flowið Þð Þ2 ð17Þ

where n is the number of time steps. Next, only the super-parameters that control the hydrograph peaks are furtheradjusted to improve the model performance as measured bythe Mean Squared Error criterion (MSE, equation (18)),which emphasizes matching of the hydrograph peaks, whilekeeping the other parameters fixed at values determined inthe first step:

MSE ¼ 1

n

Xn

i¼1

Obs:flowi Sim:flowið Þ2 ð18Þ

Finally, only the super-parameters that influence thehydrograph recessions and low flows are re-adjusted toimprove the model performance as measured by the Log

Figure 4. Plots showing interparameter relationships between spatially distributed a priori parameterestimates; each circle represents one HRAP grid. The solid lines show the fitted regression equations.Note the end effects caused by NWS imposed feasible limits on the parameter estimates.

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Mean Squared Error criterion. The approach results in asingle ‘‘optimal’’ set of model parameters.

5.2. MCOS Calibration

[30] MCOS is a simultaneous multicriterion parameterestimation approach that enables global search for thePareto-optimal parameter sets that optimize two or moremodel performance criteria at once; see the studies byGupta et al. [1998, 2003] for a discussion of multicriteriaoptimization in the context of hydrological models. Here,we use the MOSCEM stochastic sampling optimizationscheme developed by Vrugt et al. [2003]. The approachprovides a final set of mutually non-dominated (Pareto-optimal) parameter combinations, in terms of the criteriabeing optimized; that is, moving from parameter combina-tion to another improves at least one criterion while at leastone other criterion gets worse, and it is not possible to findfeasible parameter sets for which all the criteria can besimultaneously improved compared to the final set. In thisarticle, to maintain consistency, we used the same twomodel performance criteria as in the MACS approach—MSE and MSEL.[31] To obtain a better understanding of the sensitivity of

model performance to different model parameter fields, wetested three different variations of the MCOS approach. Inthe first, we varied only one of the sets of regularizationsuper-parameters (either a including m, or b including y , org), while fixing the others at their a priori values, to seewhich provides the most significant improvement; in eachcase we also include optimization of the routing parameters.As discussed in section 4, varying a is similar to theconventional parameter-field ‘‘multiplier’’ approach whilevarying g is similar to adding a spatially constant valueto each parameter-field. This individual super-parameteradjustment approach will be referred to as MCOS-Ind.[32] In the second approach, we use a three-stage strategy

to successively allow all of the super-parameters to bevaried in succession. So, one of the three types of super-parameters was first calibrated (a including m, or b includ-ing y , or g) while keeping the other super-parameters fixedat their a priori values; then the second type of super-parameter was calibrated with the first type fixed to itsnewly calibrated value; and finally the third type of super-parameters was calibrated with both of the other types fixedat their newly calibrated values. With this approach, 13quantities were adjusted at each step (11 super-parametersplus two routing parameters). Given that we have threesuper-parameter types, there were six possible stagewisecombinations (a-b-g, a-g-b, b-a-g, b-g-a, g-b-a and g-a-b) to be tested; perhaps not surprisingly, the best modelcalibration was obtained using the order g-b-a (intercept-power-slope). This three-stage super-parameter adjustmentapproach will be referred to as MCOS-stage.[33] In the third approach, called MCOS-all, all three sets

of super-parameters were calibrated simultaneously. Finally,as an independent basis for comparison, we calibrated themodel using the common multiplier approach, wherein the11 KAP fields were adjusted using scalar multipliers (wecall this MCOS–Mult).[34] For all optimization runs, the period used for cali-

bration was one water year of data from 1 October 2001 to31 September 2002. To initialize the model states, the modelwas run for a five-year warm up period prior to the start of

the simulation using the a priori parameter set; the endingvalue of all the SACSMA states and river heights at eachnode were used as initial state estimates for all model runsin this study.

6. Calibration Results

[35] This section presents and discusses the results of thedifferent regularized calibration runs described above. Firstwe examine the improvement in model performance asmeasured by the two criteria MSE and MSEL; the formeris more biased toward model performance on high-flowevents while the latter is more biased to toward modelperformance on low-flow and recession events. The resultsare summarized in Table 2 and Figure 5a. Notice, that bothcalibration strategies provided significant improvements inthe model performance criteria when compared to the two apriori model runs, with 50–70% reductions in both MSEand MSEL. However, there is a clear trade-off in ability ofthe model to simultaneously reproduce both the peak andrecession portions of the hydrograph, suggesting unresolvederrors in the model structure. The white triangle toward thetop right corner of Figure 5a represents the model perfor-mance using the Koren a priori parameter estimates (KAP)and the black triangle represents the Regularized a prioriparameter estimates (RAP); i.e., before calibration. The graybox with a black cross indicates the MACS calibrationsolution. The dark gray squares represent the MCOS-IndPareto-optimal solutions for the case of varying only thea(plus m) super-parameters; only this case is shown as itresulted in the maximum reduction in the criterion values.The white dots represent the MCOS-stage Pareto-optimalsolutions for the order g-b-a (intercept-power-slope); onlythis case is shown as it gave the best results. The black dotsrepresent the MCOS-all Pareto-optimal solutions when allsuper-parameters are adjusted simultaneously. As can beseen, the lowest MSE value was obtained by the MCOS-stage approach while the lowest MSEL value was obtainedby the MCOS-all calibration.[36] Figure 5a indicates that our implementation of the

MCOS-all approach was unable to find the global Pareto-optimal solution to the problem, since superior results onthe MSE criterion were obtained using the MCOS-stageapproach. To obtain a better estimate of the Pareto-optimalsolution set we next compiled all of the MCOS-stage andMCOS-all solutions into a single set and used these as thestarting point for subsequent MCOS-all multicriterion opti-mization of all of the super-parameters simultaneously. ThePareto solution set for this ‘‘MCOS-Final’’ calibration run isshown in Figure 5b (dark grey circles). This final calibrationachieves 10–20% improvement in the mid Pareto region.For an independent comparison, Figure 5b also shows thecalibration results obtained using a classic ‘‘multiplier’’approach wherein multipliers on each of the 11 KAPparameter fields are optimized using the multicriteriaapproach (MCOS-Mult). Whereas both methods givesimilar performance at the minimum MSEL end of thePareto frontier, it is clear that the regularization approach(MCOS-Final) provides significantly better performancein terms of MSE.[37] Figures 6 and 7 show the model’s performance in

reproducing the hydrograph at the watershed outlet. Singlecompromise solutions were selected from the mid-regions


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of the MCOS-Final and MCOS-Mult Pareto frontiers, in themanner discussed by Boyle et al. [2000], such that theselected solution gives the best (minimal) overall andmonthly flow volume bias. The MCOS-Final hydrographshows significant visual improvement compared to thehydrograph simulated by the a priori parameter estimates.While the MCOS-Mult hydrograph shows fairly goodperformance (including excellent simulation of the magni-tude of the flood peak in Figure 7), we see that the MCOS-Final hydrograph shows better representation of the lowflows (see for example flow periods ranging from 2000 hrsto 7000 hrs; Figure 6), and better timing of the flood peaks(see enlarged part of the hydrograph shown in Figure 7).Note the interesting flood peak timing error in the MCOS-Mult hydrograph (Figure 7), which contributes to the higher

MSE and lower NSE values compared to the MCOS Finalsolution. In general, MCOS-Final underestimates the peakmagnitudes of certain large flood events (see between 4000and 5000 hrs). All of the methods exhibit problems match-ing the last 1500 time steps, indicating some other form ofsystematic error either in the model or the input data.Further, all methods simulate the initial 2000 hours poorly,indicating error in initialization of the model states notresolved by use of a 5-year warm-up period. It is alsoencouraging to note that all the cases considered achieve arelatively small percentage volume bias, with the smallestbias given by MCOS-Final (Table 2). Both MCOS–Finaland MCOS-Mult hydrographs show a slight positive overallbias while the KAP hydrograph shows a negative bias. On amonthly basis (Figure 8), the volume bias is also generally

Table 2. Calibration Results

Effectiveness of the Calibration

MinimumMSE

MinimumMSEL

Volume Biasat Selected Point

Nash Sutcliffe (NSE)at Selected Point

Function Valueat Selected Points

A priori (Koren) 688 0.0663 5.3% 0.73 –A priori (Regularization) 705 0.0639 4.3% 0.72 –MCOS-Mult 351 0.0211 2.4 0.83 437, MSE

0.032, MSELMACS (step 3) 450 0.0248 6.0% 0.82 –MCOS-Ind (am) 338 0.0310 8.2% 0.83 433, MSE

0.040, MSELMCOS-stage 225 0.0294 3.1% 0.88 293, MSE

0.041, MSELMCOS-all 299 0.0207 4.7% 0.85 388, MSE

0.028, MSELMCOS-Final 222 0.0190 0.5% 0.88 296, MSE

0.032, MSEL

Figure 5. Calibration results. (a) Multicriteria plot showing tradeoff solutions for different regularizedcalibration strategies; white triangle, a priori parameter estimates; black triangle, a priori regularizedestimates (before calibration); black dots, MCOS-all calibration; white dots, MCOS-stage calibration;dark grey squares, MCOS-Ind (a and m) calibration; black ‘‘x’’ on grey background, MACS calibration.(b) Multicriteria plot showing the final set of non-dominated solutions (MCOS-Final and MCOS-Mult);light grey dots, calibration using MCOS-Mult; dark grey dots, calibration using MCOS-Final; whitesquare, selected compromise solution from MCOS-Final; grey square, selected compromise solutionfrom MCOS-Mult. The black dashed lines (in both figures) represent the lowest objective function valuesachieved during the calibration process (achieved by the MCOS-Final calibration).

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small except during the month of September when none ofthe solutions perform well (as mentioned earlier), indicatingother problems.[38] Finally we compare the distributions of the MCOS-

Final and MCOS-Mult calibrated (a posterior) parameterfields with the pre-calibration KAP and RAP (a prior)estimates (Figure 9). The left column shows the KAP priorand MCOS-Mult posterior parameter distributions and theright column shows the corresponding RAP prior andMCOS-Final parameter distributions. While both theMCOS-Final and MCOS-Mult calibrations preserve aspectsof the spatial relationships among the parameters there aresome important differences. MCOS-Final allows more flex-ibility to modify the shape of the distribution (allowingchanges in the mean, variance, skew and other properties)by allowing changes in the intercept, slope and curvature ofthe regularization equations, while maintaining direct con-nection with the underlying physical properties of thewatershed, and preserving interparameter relationships. Incontrast, MCOS-Mult allows only for scalar changes to theprior KAP parameter distributions, and the interparameterinteractions are not necessarily preserved. A common trendseen in both calibrations is an increase in the mean valuesfor parameters UZTWM, UZK, ZPERC, REXP and LZSKand a decrease in the mean values for parameters PFREE,LZTWM and LZFSM. The major difference between theMCOS-Final and MCOS-Mult calibrations (apart fromobvious differences in the variances), is the values forparameter UZFWM. For MCOS-Final both UZTWM and

UZFWM increase to result in a larger upper soil zonestorage capacity, while for MCOS-Mult the parameterUZTWM increases slightly while UZFWM decreasessubstantially indicating a rather different soil type andsoil storage capacity. Similarly differences are seen in thelower zone, with MCOS-Final moving toward a largerlower primary free water capacity (the store contributingmainly to long-term base flow).

7. Summary and Discussion

[39] The availability of powerful computers and hydro-logically relevant spatial data sets, and the growing interestin finer temporal- and spatial-scale hydrological predictions,are driving the development of increasingly more sophisti-cated models that simulate the spatial variability of water-shed processes. Although such models have the potential toprovide better predictions than their lumped counterparts,spatial discretization of the domain significantly increasesthe number of unknown model parameters that must beestimated. The parameter estimation problem can thereforebecome severely ill conditioned.[40] In this article we present an approach to spatial

parameter regularization for the calibration of environmen-tal models, which improves conditioning of the optimiza-tion problem by use of additional information and improvesefficiency by reducing the dimension of the parametersearch space. The basic strategy is to exploit additionalsub-watershed scale information about the relationships

Figure 6. Comparison of observed and simulated hydrographs. Grey area, observed data. (top) Solidline, a priori parameter estimates (KAP). (middle) Dash-dot line, MCOS-Mult (selected compromisesolution). (bottom) Dashed line, MCOS-Final (selected compromise solution). The flow is in the Log10space [Log10(Flow)].


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Figure 7. Enlarged portion of the flood hydrographs (between 4000 and 5000 hours) showing thelargest flood peak. Grey area, observed data. (top) Solid line, a priori parameter estimates (KAP).(middle) Dash-dot line, MCOS-Mult (selected solution). (bottom) Dashed line, MCOS-Final (selectedsolution).

Figure 8. Monthly percent volume bias. KAP, monthly volume bias using a priori parameter estimates;MCOS-Mult, monthly volume bias using classic multiplier approach; MCOS-Final, monthly volume biasusing the regularization approach.

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Figure 9. Frequency distribution of parameter estimates. (left) MCOS-Mult versus KAP. (right) MCOS-Final versus RAP. MCOS-Mult, parameter distribution after calibration with classic multiplier approach;MCOS-Final, parameter distribution after calibration with the regularization approach; KAP, a prioriparameter estimates; RAP, regularized a priori parameter estimates.


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between model parameters and typically available staticphysical system characteristics. This information can beexpressed in the form of non-linear regularization equationswhose shape is controlled by a small number of super-parameters. Therefore we seek to exploit any relationshipsthat exist among the physically observable structural prop-erties of the system, which must be preserved in order toproperly characterize the spatial distribution of functionalresponses of the system. These regularization relationshipsconstrain the feasible solution space in the original param-eter dimension, while the actual optimization search isconducted in a much lower dimensional space of super-parameters.[41] The reduced problem dimension allows effective and

efficient global optimization techniques to be used formodel calibration. Further, our approach is considerablymore flexible than the commonly applied technique ofusing either multipliers or additive constants applied to theparameter fields; while including the possibility of either orboth of these effects, it also enables non-linear transforma-tions and the preservation of interparameter relationships.[42] An important point to note here is that the imposition

of regularization constraints is done primarily to obtainimproved parameter realism and stability of the optimiza-tion problem, but this does not mean that model perfor-mance at the outlet (or measurement points, if multiple) willnecessarily be improved. In fact, the benefits stated abovemight actually come at the cost of a slightly reduced abilityof the model outputs to match the observations, whilehopefully providing better spatial ‘‘consistency’’ of themodel performance, resulting in a more plausible model.For example, if a uniform parameter field gave betterresults, we might not necessarily trust it over one withspatially distributed parameters—particularly when the lat-ter is based on sound physical reasoning. Unfortunately, thiscase study did not allow testing at internal points. We planto investigate this in future work with other basins.[43] The usefulness of the approach was demonstrated

using a research version of the HL-DHM hydrologic modeldeveloped by the National Weather Service for floodforecasting throughout the continental US. To infer theform of the regularization equations, we assume that the apriori parameter estimates derived via the method by Korenet al. [2000] are representative of the actual spatial relation-ships among the parameters, and therefore contain informa-tion about the dominant patterns of spatial correlation thatshould be preserved during calibration. We use a regressionapproach to derive 11 empirical equations that are valid overthe spatial domain in the sense that they capture the patternsof relative magnitudes among the parameters. By tuning thesuper-parameters of these regularization equations, we pre-serve the patterns of parameter variability (and their inter-parameter interactions) while allowing the actual parametermagnitudes to vary in ways that improve the match betweenthe model input-state-output response and the observeddata. When applied to the Blue River basin, this reducedthe number of unknowns to be estimated from 860 param-eters to 35 super-parameters.[44] These 35 super-parameters were calibrated for the

Blue River basin using a multicriteria approach that seeks toachieve a balance in the fitting of peak and recessioncomponents of the hydrograph. A Pareto-optimal solution

set was obtained, indicating an inability of the model tosimultaneously reproduce both the peak and recessionportions of the hydrograph, and suggesting unresolvedmodel structural problems. However, the Pareto frontierdid achieve a 50–70% improvement in both measures ofoverall model performance, compared with the Koren apriori parameter set, and about 40% improvement in MSEover the classic multiplier approach. The selected finalsolution was shown to provide a better simulation of thestreamflow hydrograph (within the limitations achievableby the HL-DHM model structure), including improvedNash Sutcliffe efficiency and overall flow volume bias,compared to the classic multiplier approach or the priorparameter estimates provided by the Koren approach.[45] Of interest is the particular form of parameter adjust-

ments achieved by the regularization approach in improvingthe model performance. For several of the parameters, thepredominant change was an upward or downward shift ofthe mean of the parameter frequency distribution,corresponding to an increased depth of the upper soil zoneand a reduced depth of the lower soil zone. Overall, thesechanges manifest themselves as more water available forevapotranspiration and for the quick response componentsof the hydrograph, and less water available for the slowerlong-term hydrograph recession response.[46] Comparison of the regularized calibration approach

with the classical multiplier approach brings into questionthe common use of the latter for adjusting parameter fields.Table 1 lists the a priori and calibrated values of the super-parameters for selected MCOS-Final solution; bolded text ina cell indicates a significant change. For all but three of theparameter fields, the g super-parameter (additive constant)has changed significantly thereby changing the mean of theparameter distribution either upward or downward. Mean-while, the a super-parameter (multiplicative factor) changessignificantly for only five of the parameters; e.g., forLZFPM the a value becomes smaller, narrowing the param-eter range (while g increases shifting the distribution to theright). Similarly, the b super-parameter (non-linearity fac-tor) changes for only 3 of the parameter fields. While theflexibility of the proposed regularization formulation allowsthe optimization algorithm to select appropriate combina-tions of these three kinds of adjustments (additive constant,multiplier, and non-linear transformation), more explorationof this issue seems warranted.[47] In general, the overall regularized calibration results

were not strongly sensitive to the choice of multicriteriacalibration strategy, all performing quite well. However, wenoticed that the stagewise strategy allowed the parametersto vary through a greater range of values, while thesimultaneous strategy facilitated greater freedom of param-eter movement. There may therefore be some advantageto performing multiple calibration runs using differentapproaches.[48] The regularization approach presented here is gener-

ally applicable to any situation where spatial parametervariability can be related to observable properties of thewatershed. Clearly, the strength of the proposed approachdepends on (1) how well the a priori parameters canbe estimated from available watershed information, and(2) how well the spatial variability of the a priori parameterestimates can be modeled using regularization equations.

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Recent publications have indicated that use of high-resolution land use data coupled with the recently availableSoil Survey Geographic Database (SSURGO), consideredto be superior to STATSGO, may provide a more accuratebasis for a priori estimation of model parameters [Andersonet al., 2006]. We plan to examine this hypothesis in ongoingresearch. Further, although there seems to be no reason whythe general form of the regularization equations developedhere should not be applicable for other locations, theirtransferability needs to be studied.[49] In addition to more rigorous testing of the regulari-

zation approach on several basins to investigate the gener-ality of the proposed relationships, future work will explorethe use of signature type objective functions selectedspecifically for their enhanced diagnostic power and infor-mation extraction ability [Gupta et al., 2008; Yilmaz et al.,2008], and the use of parallel computing to enable the use oflarger data sets. As always we invite and encourage ongoingdialogue with other scientists interested in these and relatedmodel identification issues.

[50] Acknowledgments. Partial support for this work providedby the National Weather Service Office of Hydrology under grantNA04NWS462001 and by SAHRA under NSF-STC grant EAR-9876800is gratefully acknowledged. The first author was also partially supported bya grant from the World Laboratory International Center for ScientificCulture.

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H. V. Gupta and P. Pokhrel, Department of Hydrology and Water

Resources, University of Arizona, Harshbarger Building Room 314, 1133East North Campus Drive, P.O. Box 210011, Tucson, AZ 85721-0011,USA. ([email protected])

T. Wagener, Department of Civil and Environmental Engineering,Pennsylvania State University, 212 Sackett Building, University Park, PA16802, USA.

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A spatial regularization approach to parameter estimation for a distributed watershed model