Issue Paper/

Are Models Too Simple? Arguments forIncreased Parameterizationby Randall J. Hunt1, John Doherty2, and Matthew J. Tonkin3,4

AbstractThe idea that models should be as simple as possible is often accepted without question. However, too much

simplification and parsimony may degrade a model’s utility. Models are often constructed to make predictions;yet, they are commonly parameterized with a focus on calibration, regardless of whether (1) the calibration datacan constrain simulated predictions or (2) the number and type of calibration parameters are commensuratewith the hydraulic property details on which key predictions may depend. Parameterization estimated through thecalibration process is commonly limited by the necessity that the number of calibration parameters be smallerthan the number of observations. This limitation largely stems from historical restrictions in calibration and com-puting capability; we argue here that better methods and computing capabilities are now available and shouldbecome more widely used. To make this case, two approaches to model calibration are contrasted: (1) a trad-itional approach based on a small number of homogeneous parameter zones defined by the modeler a priori and(2) regularized inversion, which includes many more parameters than the traditional approach. We discuss someadvantages of regularized inversion, focusing on the increased insight that can be gained from calibration data.We present these issues using reasoning that we believe has a common sense appeal to modelers; knowledge ofmathematics is not required to follow our arguments. We present equations in an Appendix, however, to illustratethe fundamental differences between traditional model calibration and a regularized inversion approach.

IntroductionAlbert Einstein observed that our approach to prob-

lem solving should be ‘‘as simple as possible but notsimpler.’’ This philosophy appears to be a firm tenet formodeling environmental systems. How this philosophyshould be applied in practice is not, however, universallyagreed upon. Problems associated with too little complex-ity are well known, as evidenced by the popularity ofcomplex simulation tools beyond simple analytical

solutions. However, Anderson (1983) observed that mod-elers often invoke complexity beyond that which is war-ranted from a conceptual understanding gleaned fromfield data, such that often the ‘‘emperor has no clothes.’’That is, the model is not as good as it might appear, givenits cost and reams of output. Freyberg (1988) noted that ina modeling class he taught, predicted system responsewas better simulated with more parsimonious but lesswell-calibrated models than with models calibrated usinga large number of parameters to obtain a good fit (a phe-nomenon often referred to as ‘‘point calibration’’). Thereappear to be diminishing returns whereby some level ofparameter complexity improves our simulation capa-bilities, but too much leads to instability, nonuniqueness,long run times, and an increased potential for predictiveerror.

What is the optimal level of parameterization? Theemphasis within the environmental modeling communityis often ‘‘the simpler the better.’’ For example, Hill (1998,2006) lists parameter parsimony—essentially, the use ofa small number of parameters—as the number one guide-line for effective model calibration. An American

1Corresponding author: U.S. Geological Survey, 8505Research Way, Middleton, WI 53562; (608) 828-9901; fax (608)821-3817; rjhunt@usgs.gov

2Watermark Numerical Computing, 336 Cliveden Avenue,Corinda, 4075, Brisbane, Australia; johndoherty@ozemail.com.au

3Department of Civil Engineering, University of Queensland,St Lucia, Australia 4068.

4S.S. Papadopulos & Associates Inc., 7944 Wisconsin Avenue,Bethesda, MD 20814; matt@sspa.com

Received May 2006, accepted January 2007.No claim to original US government works.Journal compilationª2007National GroundWaterAssociation.doi: 10.1111/j.1745-6584.2007.00316.x

254 Vol. 45, No. 3—GROUND WATER—May–June 2007 (pages 254–262)

Geophysical Union special session on model complexityconcluded that ‘‘if models are kept in the context of theirobjective, we should feel comfortable resisting the sirenof complexity and construct simpler, less encompassingmodels’’ (Hunt and Zheng 1999). This logic seems unas-sailable and can produce models that provide some levelof insight. However, questions linger about the focus ona priori parsimony, such as the following: Did thisapproach get the most information from the calibrationdata? Does the model adequately reflect the response ofthe physical system, or have bias and/or uncertainty beenintroduced into model predictions as a result of ‘‘hand-cuffing’’ to a simplified parameter structure?

Clearly, it is critical to consider the modeling objec-tive(s) when answering these questions. Predictionsrequiring ‘‘ballpark’’ estimates based on bulk systemproperties may be made using simple methods—evenhand calculations (for example, Haitjema 1995, 2006). Forother predictions such as contaminant transport, how-ever, complex simulation capabilities and/or detailed para-meterization may be required. There has been someprevious work on the utility of model calibration undersuch circumstances, with some researchers arguing that thenotion of the calibrated model has no place in environ-mental simulation since uniqueness is always achieved atthe cost of error-inducing simplification(s). For example,Woodbury and Ulrich (2000), Gomez-Hernandez et al.(2003), Gomez-Hernandez (2006), and others argue thata large number of model runs should be used withina probabilistic Bayesian framework to explore the rangeof predictive possibilities rather than making a discreteprediction. Although these arguments have merit, forbetter or for worse calibrated models commonly form thebasis of environmental decision making and it is withinthis context that our discussion takes place.

There appears to be a middle ground between modelcalibration using relatively few zones and very complexparameterization using stochastic or other methods. Toillustrate this middle ground, we contrast a traditionalapproach to calibration based on zones of piecewise con-stancy with an approach based on regularized inversion.The discussion focuses on model calibration, but alsotouches upon the use of calibrated models to evaluate pre-dictive error and assess the potential relative worth of futuredata collection. The discussion is presented in terms ofwords and concepts. For the reader interested in the under-lying mathematics, an Appendix is included that containsmathematical formulations and references that explorethese concepts in greater detail.

The Traditional ApproachIn order to estimate the values of parameters used in

a model, comparisons are made between real-world data(measurements of ground water elevations, baseflows,etc.) and simulated equivalents. This can be accom-plished using manual trial-and-error or inverse codes that‘‘automate’’ the trial-and-error process (Anderson andWoessner 1992). Although the value of intuition devel-oped by manually manipulating model parameters shouldnot be understated, Poeter and Hill (1997) among others

suggest that automated methods are often superior. Ourdiscussion focuses on automated calibration methods thatseek a best fit by minimizing the weighted squared dif-ferences between measured data and their simulatedequivalents.

For the purpose of this discussion, it is sufficient tounderstand that traditional automated approaches requirethat the calibration problem be ‘‘well posed’’ in order toobtain a unique set of parameters that provide the ‘‘bestfit’’ between model outputs and observed equivalents (e.g.,Draper and Smith 1998). This usually means that the num-ber of estimated parameters is fewer than the number ofobservations plus the number of items of independentinformation about parameters (often referred to as ‘‘priorinformation’’—see, for example, Cooley 1982). Thisrequirement is dictated at least in part by (1) historicalrestrictions in computing capability; (2) the fact thatobservation data often contain redundant information forcalibration purposes; and (3) the traditional use of directmatrix inversion to solve the linear system of equations thatis required for estimation of parameters (see Appendix).The traditional restriction of small numbers of parametersoften results in many important but subjective decisionsduring the model construction process.

In truth, the system to be modeled likely contains farmore spatial variability in hydraulic properties than canpossibly be estimated through the calibration process. Toachieve stable and unique calibration using traditionalmethods, a priori, user-defined parameter ‘‘parsimony’’ isused to subdivide the model domain into a small numberof zones with uniform parameter values inferred fromgeological knowledge about the system. In some cases,such zones simply represent areas where few or no dataexist. If this a priori ‘‘lumping’’ leads to more parametersthan observations and/or excessive parameter insensitivityand/or correlation, the modeler combines zones to reducethe number of estimated parameters and/or fixes some pa-rameters at ‘‘reasonable’’ values. Calibration often ceasesonce an acceptable fit is achieved. In the case where dataare plentiful, it may become apparent that the level ofparsimony is too great (for example, if it is obvious thatbetter fits between measured and simulated values canbe obtained). New zones are then introduced in a some-what ad hoc fashion to reduce the misfit incurred by theassumption of piecewise property uniformity on too broada scale. Unless rigorous methods are employed to simul-taneously estimate parameter values and structure (e.g.,Sun and Yeh 1985; Eppstein and Dougherty 1996; Zhengand Wang 1996; Tsai et al. 2003), the modeler faces ago-nizing choices over which parameters to lump, which tofix, what values to assign to fixed parameters, when the fitis good enough, and when the fit is too good—often usingarbitrary criteria that are hard to convey, document, ordefend. All of these decisions are made while recognizingthat every decision on what to fix or lump may introduce‘‘hardwired error’’ into model predictions.

Problems with this approach extend beyond the diffi-culty of identifying the optimal level of parameter simpli-fication. First, rigid a priori parameter parsimony basedon an existing understanding of property variability canmake it difficult to improve that understanding through

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calibration, since the calibration data may be speaking to‘‘ears’’ that are not listening. Second, a fundamental moti-vation for developing models—to test hypotheses thatwould be impossible to test in the real world—may beundermined if parsimony limits the capability of the modelto analyze variations and alternatives. Third, property vari-ability beyond that which can be uniquely inferred isignored: Moore and Doherty (2005) show that this mayundermine efforts to analyze model predictive uncertainty.

Many modelers believe that a well-calibrated modelwill naturally provide accurate predictions. However,Moore and Doherty (2006) demonstrate that transportpredictions made by a model that calibrates perfectly toground water elevation data can be 100% wrong as aconsequence of the simplifications required to achievea unique calibration. Thus, it is important that model pre-dictions be accompanied by an estimate of their likelyerror. One way to evaluate the potential magnitude ofpredictive error is through implementing a constrained pre-dictive maximization/minimization process through whicha prediction of interest is ‘‘stretched’’ while ensuring thata suitably good fit between model outputs and field meas-urements is maintained (Vecchia and Cooley 1987).Unfortunately, when these calculations are based on thetraditional use of a small number of parameters, they canbe susceptible to errors incurred because simplificationsrequired to obtain a unique calibration create a form of‘‘structural noise’’ that commonly exceeds measurementnoise. This structural noise can be evaluated using methodssuch as those developed by Cooley (2004) and Cooleyand Christensen (2006), and this can, in turn, facilitateproper estimation of potential predictive error. Thesemethods can be difficult to implement in complex model-ing contexts, however, and do not directly address theproblem of minimization of that error or help identifyconditions that could lead to extreme values of a pre-diction—despite the fact this might be of primary interestto decision makers.

The ability of models to evaluate the utility of futuredata collection exemplifies the underlying issue. If struc-tural noise is not accounted for, predictive error calcula-tions used to compare the relative worth of differentpotential data gathering strategies (e.g., Tonkin et al.2007) may be misleading. When using traditional meth-ods, it follows that the fewer data available within a studyarea, the greater the need to acquire extra data. Yet, thefewer the data, the greater the level of parsimony requiredto ensure that the number of observations exceeds thenumber of parameters. Moreover, as detailed previously,the fewer the number of parameters, the greater the con-tribution to predictive error from structural noise. Onecould argue that, as a consequence, when the need formore data is most urgent, the use of precalibration parsi-mony is most questionable.

Potential weaknesses of precalibration parsimony maybe most evident when extended from situations wherevery few data are available to the extreme case wherenone is available at all. In this case, should a modelhave no parameters? This contravenes common practice,where on occasions that predictions must be made withno calibration support, highly parameterized models are

often used to account for parameter variability—oftentogether with Monte Carlo and related techniques forexploration of predictive variability (e.g., U.S. EPA2001). Why is it that when a limited number of databecome available, methods that recognize the complexityinherent to real-world systems are disregarded in favor ofmethods based upon ad hoc simplification and parameterparsimony?

An Alternative to the Traditional Approach:Regularized Inversion

Let us be clear that there is no obvious formulationof model calibration that previous generations of model-ers have overlooked. Model calibration is fundamentallynonunique because there is always more spatial propertyvariability—and hence more parameters—than observa-tion data can constrain. The fundamental tension remains:parameterization of true hydraulic property detail isimpossible; yet, ‘‘calibration’’ implies that a unique para-meter field is obtained one way or another. Therefore, thequestion is not whether a model—which is always asimplification of reality—should simplify parameter vari-ability. Rather, it is how this simplification should beperformed.

Advances in computing power, equation solutiontechniques, and techniques for formulating the inverse(i.e., parameter estimation) problem have allowed moresophisticated methodologies to be used in other fieldssuch as medical imaging and geophysical prospectingthan are typically used in ground water modeling. Thesecapabilities present an opportunity to use parsimony toestimate parameters in a more rigorous way than is tradi-tionally done using ad hoc parsimony described pre-viously. Of these, ‘‘regularized inversion’’ (Engl et al. 2006)is described here. ‘‘Inversion’’ simply refers to the use ofmeasured data (such as heads, fluxes, etc.) to estimatemodel parameters. The term ‘‘regularization’’ describes anyprocess that makes a function more regular or smooth;it can be broadly interpreted as any method that helpsprovide an approximate and meaningful answer to an ill-posed problem. In this sense, traditional parsimony is aninformal regularization strategy that reduces a variableworld to a small number of model parameters. Hence, theneed for regularization is not in question—rather thequestion is how should it be accomplished?

The basic tenet of regularized inversion describedhere is that the level of parameterization used in a modelshould not be unnecessarily restricted by ad hoc pre-calibration parsimony because that may contravene theoriginal motivation for building the model. Thus, thecalibration process should be flexible in order to be maxi-mally responsive to information contained in the calibra-tion data and in the assignment of parameters on thebasis of that information. In doing so, regularized inver-sion provides a systematic and quantitative frameworkfor achieving parameter simplifications, whereby therationale for the simplification is formally constructedand decipherable. Regularized inversion incorporates twoprimary differences from the traditional approach de-scribed previously. These are now outlined.

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The Level of ParameterizationThe first difference is that more parameters can be

assigned to a model because alternative techniques todirect matrix inversion are used to solve the system ofequations formed during model calibration—techniquessuch as singular value decomposition. In practice, thereare still practical limits to how many parameters areappropriate. Thus, rather than specifying a parameterfor every model cell, for example, parameterization de-vices such as ‘‘pilot points’’ (de Marsily et al. 1984;RamaRao et al. 1995; Doherty 2003) are often used.In the pilot-point approach, parameter values are esti-mated at a number of discrete locations distributedthroughout the model domain; cell-by-cell parameteri-zation then takes place through spatial interpolationfrom the pilot points to the model grid or mesh. If neces-sary, pilot points can be grouped to represent geologiccontinuity where it is believed to exist and combinedwith the use of zones so that hydraulic property hetero-geneity is preferentially expressed at zone boundaries.This approach provides a zonation scheme that it isnot ‘‘hardwired’’ as in traditional approaches, and add-ing parameters can help the calibration process extractmore information from the calibration data. The result,however, is many more parameters than is typical intraditional model calibration, which can lead to (1) para-meter insensitivity and correlation, which in turn lead tosolution nonuniqueness and (2) long run times. To over-come these issues, ‘‘regularization’’ is required. Somecommonly used regularized inversion approaches aredescribed below, which can be used by themselves or incombination.

Stabilizing the Calibration Process withMathematical Regularization

In contrast to the more hardwired and static nature ofthe traditional approach, regularized inversion can enableknowledge of a study site to form a flexible mathematicalregularization strategy that can be employed to attainparameter uniqueness and stability of the inversion pro-cess. Understanding of a site can enter into the calibrationprocess through definition of a preferred system condi-tion (for example ‘‘the hydraulic conductivity shouldhave a value around 1 m/d’’ or ‘‘the hydraulic conductiv-ity should be uniform in this area’’). This condition isformally injected into the calibration process using‘‘Tikhonov regularization’’ (Tikhonov and Arsenin 1977;Ory and Pratt 1995). In Tikhonov regularization, pre-ferred system conditions are maintained if possible. How-ever, minimized deviation from these conditions isacceptable if this is required in order to obtain a user-specified level of fit (Doherty 2003). Tikhonov regulari-zation has been used in the calibration of a variety ofground water models (e.g., Skaggs and Kabala 1994; Liuand Ball 1999; Doherty 2003; van den Doel and Ascher2006), achieving results that would not have been possi-ble using traditional precalibration parsimony. The use ofTikhonov regularization can enable the estimation ofa large number of parameters in a geologically reasonablemanner. However, it does not relieve the computational

burden of estimating many parameters, nor is it uncondi-tionally numerically stable.

A second approach to stabilizing the estimation ofa large number of parameters is to formally decomposethe information on parameter sensitivities and corre-lations, and eliminate (effectively, fix) inestimablecombinations of parameters that destabilize the inverseproblem, while estimating the remaining parametercombinations. This approach to regularized inversionuses the truncated singular value decomposition (TSVD)matrix analysis technique to identify combinations ofparameters that cannot be estimated using the availablecalibration data (insensitive parameters, for example,are included in this ‘‘calibration null space’’) and combi-nations of parameters that can be estimated on the basisof the available calibration data (these comprising the‘‘calibration solution space’’). The threshold for trunca-tion is specified by the user. If too many combinationsof parameters are estimated, the problem will still benumerically unstable; if too few parameters are esti-mated, the model fit may be unnecessarily poor andpredictive error may be larger than an optimallyparameterized model. While TSVD can provide stableand unique model calibration, it too does not alleviatethe high computational burden incurred by the use ofmany parameters. Nor are parameter fields sometimesas ‘‘aesthetically pleasing’’ or geologically reasonable asthey are for Tikhonov calibration where reasonablenessis built into the regularization process through use ofa preferred condition.

Tikhonov regularization can be combined withTSVD to simultaneously achieve the benefits of both.However, neither TSVD nor Tikhonov regularization, northeir use in combination, reduces the computational costincurred by a highly parameterized model because sensi-tivities of model outputs must be calculated with respectto all parameters during each iteration of the calibrationprocess regardless of whether they are sensitive or not.On some occasions, the use of adjoint sensitivity tech-niques (Townley and Wilson 1985; Clemo et al. 2003) canexpedite the computation of these parameter sensitivities.However, if adjoint sensitivities are not available, thecomputational burden may be reduced through the useof a novel extension of subspace inversion techniques—hybrid Tikhonov-TSVD (Tonkin and Doherty 2005).

This regularization approach is a hybrid of Tikhonovand TSVD regularization that accelerates the process bysolving the inverse problem in a parameter ‘‘subspace’’computed using TSVD rather than solving the problem intrue parameter space where the sensitivity of each param-eter is required independently. Once estimable parametercombinations (referred to as ‘‘super parameters’’) are de-fined, derivatives of model outputs with respect to theseparameter combinations are calculated through directfinite differencing of these combinations. Collectively,these combinations span the calibration solution space. Indistinction from the traditional use of TSVD where thissubdivision occurs in each iteration of the inverse pro-cess, the initial subdivision is retained throughout theinverse process. As the dimensionality of this space isnormally smaller than that of the calibration null space

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(comprising parameter combinations that remain unesti-mated and thus retain their original values), the numberof model runs required per iteration for computation ofderivatives can be reduced enormously. Unconditionalnumerical stability is also obtained, as insensitive andhighly correlated parameter combinations are automati-cally excluded from the parameter solution space throughthe TSVD process through which super parameters aredefined.

The appendix illustrates the form of the basicparameter estimation equations when Tikhonov (Equa-tion A-6), TSVD (Equation A-8), and hybrid Tikhonov-TSVD (Equation A-9) regularization are employed. Inmany cases, solving the problem in a subspace of theimportant parameter combinations reduces the numberof model runs required (often by more than an order ofmagnitude), removes the effects of insensitive param-eters, and can obtain solutions to the inverse problemthat are sufficiently close to the ‘‘true’’ solution to beuseful.

In addition to extracting the more information froma calibration data set, regularized inversion as a calibra-tion methodology allows quantification of potentialprediction error using both linear (Moore and Doherty2005) and nonlinear (Tonkin et al. in review) methods.The nonlinear approach employs an extension of theprediction minimization/maximization technique ofVecchia and Cooley (1987), in which constraints appliedto parameters during the predictive analysis processensure that model-to-measurement fits remain within ex-pectations and that parameters remain geostatisticallyreasonable as predictions are maximized or minimized.Such analyses can be easily undertaken following modelcalibration using any of the regularized inversion meth-ods discussed previously. Because many parameters areincluded in the predictive analysis process, errors thatresult from failure of the calibration process to ‘‘cap-ture’’ system heterogeneity due to limitations in the cali-bration data are explicitly accounted for.

A final note regarding regularized inversion is thatMoore and Doherty (2005, 2006) demonstrate that re-gardless of the method employed, the calibrated hydrau-lic property field is smoother or simpler than the truehydraulic property field. This is simply the cost of obtain-ing a unique solution to the inverse problem of modelcalibration. Nevertheless, this blurred image of the subsur-face (e.g., McLaughlin and Townley 1996) so obtainedoften appears more geologically reasonable than animage comprising traditional zones of piecewise con-stancy since it lacks abrupt and artificial looking dis-continuities. Furthermore, this ‘‘blurred’’ image of thesubsurface encapsulates as much information as possiblefrom the calibration data and is thus as representativea picture as can be expected given the observationsavailable. This image can be improved upon, therebymaking it sharper (and bringing hydraulic propertydetail into focus) by the acquisition of extra data. Putanother way, regularized inversion (and model calibra-tion in general) does not necessarily produce an accuraterepresentation of real-world complexity. Rather, it re-flects the complexity that is supported by the data—and

commonly far more than can be inferred using tradi-tional parsimony.

Comparison of Traditional and RegularizedInversion Approaches

To the authors’ knowledge, only a small number ofpublications directly compare the two methods describedpreviously. For example, Tonkin and Doherty (2005)compare a traditional zone–based parameterization withthe hybrid TSVD regularized inversion, in the context ofa ground water flow and transport model. A brief sum-mary is provided here; the reader is directed to Tonkinand Doherty (2005) for more detailed discussion.

The problem was evaluated using a traditionalapproach using a single zone for each model layer anda regularized inversion approach that included 1195 base

Figure 1. Active area of transport (white) from Tonkin andDoherty (2005) showing water level observation locations(open circles), MTBE observation locations (stars), InterimRemedial Measure (IRM) well (filled circle), and pilot points(crosses). A calibrated pathline from the regularized inver-sion is also shown (black line).

258 R.J. Hunt et al. GROUND WATER 45, no. 3: 254–262

parameters, including horizontal and vertical hydraulicconductivity, recharge, porosity, boundary conditions,and parameters describing the contaminant source term.Each model was calibrated to ground water levels and

contaminant concentrations collected from about 40multilevel monitoring locations (Figure 1). Slug tests andboring logs indicate that there is geological heterogeneitypresent; however, the highly transmissive sand aquifer didnot lend itself to a priori delineation of laterally and/orvertically contiguous parameter zones. Therefore, the re-sults of the lumped parameter calibration were used asinitial conditions for the regularized inversion.

The hydraulic conductivities estimated using regu-larized inversion were generally consistent with thefield data and other independent sources of information(Figure 2), and produced an improved fit to the measureddata (Table 1)—as might be expected. The most signifi-cant improvement in the objective function was achievedthrough improved fit of simulated and measured con-centrations at monitoring wells. Plots of simulated andmeasured concentrations illustrate that the regularizedinversion model reproduced sharp fronts better than thelumped model. This is most evident in ‘‘bubble plots’’ ofsimulated and observed methyl tert-butyl ether (MTBE)concentrations in monitoring wells (Figure 3). In thesefigures, the source area is located at right and the dis-charge area is located at left. The lumped parametermodel produces a simulated plume that gives the appear-ance of simple spreading about a centerline; areas of lowor high measured concentrations are not reflected in theirsimulated equivalents. The regularized inversion model(1) more accurately reproduces areas of higher and lowerconcentrations throughout the plume and (2) better matchesthe true location of the discharge of the plume to the surfacewater body.

Review of the parameters estimated through regular-ized inversion suggests that the improved fit was largelyobtained by the introduction of spatial variability intohydraulic conductivity and porosity parameters. This isexpected since using a large number of pilot pointstogether with Tikhonov regularization enables the hybridscheme to introduce such variability. Nonetheless, param-eter values estimated in the inversion are generally withinthe range of values estimated from field testing. In partic-ular, regularized inversion appeared to identify an area oflow hydraulic conductivity that causes the plume to dis-charge further offshore than in the lumped calibration(Figure 2). Tonkin and Doherty (2005), and the authors ofthe current discussion paper, do not infer that regularizedinversion identified the ‘‘true’’ parameter values and/or

Figure 2. Regularized inversion calibrated horizontalhydraulic conductivity (HHK), model layer 1. For compari-son, the traditional approach used one value of conductivityfor the layer (Tonkin and Doherty 2005).

Table 1Summary of Traditional and Regularized Inversion Calibration (Tonkin and Doherty 2005)

Observation Group

Weighted Residualsfrom Lumped

Calibration UsingTraditional Methods

Weighed Residualsfrom RegularizedInversion UsingHybrid TSVD

Percent Reductionin Weighted Residual

MTBE1 mass removal at interim remedial measure well 5.273 106 2.003 106 62MTBE concentrations in observation wells 9.313 106 1.683 106 82Water levels in observation wells 6.193 102 3.713 102 40Composite objective function 1.463 107 3.683 106 74

1MTBE ¼ methyl tert-butyl ether.

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their distribution—for reasons discussed by Menke(1989) among others. Regularized inversion, however, didlead to greatly improved fits (and hopefully predictions)and produced parameter distributions that are consistentwith field and other independent data, and that could notreasonably be refuted without the collection of additionalsite-specific information.

Discussion and ConclusionsThere is no universally correct way to parameterize

and calibrate a ground water model or indeed any envir-onmental model. Nevertheless, a case can be made thatregularized inversion provides a more rigorous and lesssubjective mechanism for calibrating ground watermodels than do traditional zone–based approaches. Wedo not suggest that mapped geological, morphological,and other boundaries should not be respected where theyare believed to exist; indeed, regularized inversion

accommodates this. However, one must ask why tradi-tional zone–based parsimony should be employed in con-texts where parsimony can be flexibly and formallyimplemented using regularized inversion. This approachis employed as a matter of course in other industries suchas geophysical data analysis and medical imaging. Imag-ine having to draw a zone where an anomaly may residein your kidney or brain in order to interpret the wealth ofinformation that is available in an image of that organ.Using the same reasoning, one might ask why it is stillthe standard approach in ground water modeling?

While many calibration strategies exist, we believethat regularized inversion is attractive since it is theoreti-cally rigorous and sufficiently model-run efficient to bepractical. It offers the potential to include parameter spa-tial variability in a model on a scale commensurate withmodel predictions, at the same time can encapsulate themodeler’s understanding of a system through the use ofpreferred condition constraints. Regularized inversionhelps maximize the insights gained from the field dataand facilitates more encompassing evaluations of predic-tive uncertainty, while providing a mechanism for simu-lating conditions that may lead to extreme outcomes whentesting the bounds of model predictive confidence—oftenthe principal motivation for developing the model in thefirst place.

Why is regularized inversion not ubiquitous inground water modeling? This may be due in part to theperception that it is difficult to do and computationallyexpensive to implement. However, open-source softwarethat implements these approaches is freely available (forexample, see Doherty 2003; Tonkin and Doherty 2005;Hunt et al. in review) and incorporated in several graphi-cal user interfaces, and most current computational re-sources are more than sufficient to allow its use. Perhapsmodelers have grown comfortable with the traditionalapproach; if so, we hope that this discussion will allowexploration of potentially better alternatives. While thereis, indeed, no single way to calibrate a model, we believethat regularized inversion may help ensure that our mod-els are not only ‘‘as simple as possible’’—as with tradi-tional parameter parsimony—but also fulfill the tenet‘‘but not simpler.’’

AcknowledgmentsWe acknowledge the USGS Trout Lake Water,

Energy, and Biogeochemical Budgets program for fund-ing support, and we thank Allen Wylie, Daniel Feinstein,Mike Fienen, Charlie Andrews, and two anonymous re-viewers for reviewing the article. We also thank MaryAnderson for soliciting and providing the soapbox forthis article.

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Appendix

Mathematics of Traditional Calibration and RegularizedInversion

Model Calibration

Let the vector p represent the true values of a sys-tem’s properties, and let h denote the n observations onwhich calibration is based. Let � represent measurementnoise. If the matrix X denotes the linear sensitivities ofobservations of model outputs corresponding to systemstate h to changes in system properties p, the relationshipbetween h and p can be represented as follows:

h ¼ Xp 1 � ðA-1Þ

Let p represent values inferred for m model parame-ters through model calibration. If traditional model cali-bration is employed to limit the number of parameters inp so that a well-posed inverse problem is formed, p canbe estimated using the Gauss-Marquardt-Levenbergmethod as follows:

p ¼ ðXtQXÞ21XtQh ¼ Gh ðA-2Þ

where G denotes the (linear) relationship between h andp, and Q is the observation weight matrix. If Q is pro-portional to the inverse of the measurement error covari-ance matrix C(�), the covariance matrix of the estimatedparameters is as follows:

CðpÞ ¼ r2r ðXtQXÞ21 ðA-3Þ

where r2r , the reference variance that quantifies the levelof measurement uncertainty, is given by:

r2r ¼ F=ðn 2 mÞ ðA-4Þ

where F is the minimized sum of squared weighted re-siduals. Equations A-1 through A-4 apply where themodel is linear. Nonlinear parameter estimation is ach-ieved through extension of these concepts to an iterativesolution process whereby a parameter change vector �pis determined on the basis of the current vector of re-siduals (i.e., model-to-measurement misfits) r using:

�p ¼ ðXtQXÞ21XtQr ¼ Gr ðA-5Þ

In traditional parameter estimation as represented byEquations A-2 and A-5, matrix XtQX must be invertibleto obtain a unique solution—i.e., to estimate the para-meter values comprising p. This requires the number ofobservations be greater than or equal to the number of pa-rameters—often much greater.

Two particular regularized inversion techniques arediscussed in this article, together with a third that com-bines these into a hybrid regularization strategy. Thefirst of these, Tikhonov regularization, supplements cali-bration data with information pertaining to parametersusing regularization equations, the weights for which are

determined during calibration. This ‘‘penalized leastsquares’’ approach sums measurement and regularizationobjective functions to form a global objective function.The expression for G when Tikhonov regularization isemployed is as follows:

G ¼ ðXtQX1b2TtSTÞ21XtQ ðA-6Þ

where T is the vector of Tikhonov regularization con-straints on parameters, S is the regularization weightmatrix, and b2 is the regularization weight factor (deter-mined though the calibration process as that which ach-ieves a user-specified level of model-to-measurement fit).

The second regularization strategy is based uponTSVD (Anderson et al. 1999). In the special case of thesquare symmetric matrix XtQX, TSVD decomposesXtQX into:

XtQX ¼ VEVt ðA-7Þ

where E is diagonal and lists the m singular values ofXtQX, while the m column vectors of V are the eigen-vectors of XtQX (Lawson and Hanson 1995). XtQX pos-sesses m real-valued eigenvalues. In practice, a smallnumber of these dominate, corresponding to calibrationsolution space eigenvectors (Aster et al. 2005). Stableinversion is achieved using regularization as a filter that es-timates parameter combinations that reside in the calibra-tion solution space, while ignoring parameter combinationsthat reside in the calibration null space. The expression forG when TSVD regularization is employed is as follows:

G ¼ V1E211 Vt

1XtQ ðA-8Þ

where V1 and E1 contain the eigenvectors and eigen-values that are retained following the application of theregularization filter.

The hybrid regularization methodology of Tonkinand Doherty (2005) combines Tikhonov and TSVD regu-larization strategies to estimate a limited number of‘‘super parameters,’’ while enforcing Tikhonov constraintson base parameter values as actually employed by themodel. Super parameters are scalar multipliers for thesolution space eigenvectors (column vectors of V1); theseare estimated using classical least squares in a refor-mulated inverse problem. Jacobson (1985) describes asimilar approach to redefining the inverse problem onthe basis of an eigenvalue-eigenvector decompositionreferred to as ‘‘surrogate parameters.’’ The expression forG for the hybrid Tikhonov-TSVD regularization processis as follows:

G ¼ V1ðZtQZ1b2TtSTÞ21ZtQ ðA-9Þ

where Z is the sensitivity matrix of model outputs withrespect to ‘‘super parameters’’, and T is the matrix ofregularization constraints on base parameters after pro-jection of these constraints onto the calibration solutionsubspace.

262 R.J. Hunt et al. GROUND WATER 45, no. 3: 254–262