A representer-based inverse method for groundwater

flow and transport applications

Johan R. Valstar

Netherlands Institute of Applied Geoscience TNO–National Geological Survey, Utrecht, Netherlands

Dennis B. McLaughlin

Ralph M. Parsons Laboratory, Massachusetts Institute of Technology Cambridge, Massachussetts, USA

Chris B. M. te Stroet1 and Frans C. van Geer

Netherlands Institute of Applied Geoscience TNO–National Geological Survey, Utrecht, Netherlands

Received 2 December 2003; revised 26 February 2004; accepted 30 March 2004; published 28 May 2004.

[1] Groundwater inverse problems are concerned with the estimation of uncertain modelparameters, such as hydraulic conductivity, from field or laboratory measurements. Inpractice, model and measurement errors compromise the ability of inverse procedures toprovide accurate results. It is important to account for such errors in order to determinethe proper weight to give to each source of information. Probabilistic descriptions ofmodel and measurement errors can be incorporated into classical variational inverseprocedures, but the computational demands are excessive if the model errors vary overtime. An alternative approach based on representer expansions is able to efficientlyaccommodate time-dependent errors for large problems. In the representer approach,unknown variables are expanded in finite series which depend on unknown functionscalled representers. Each representer quantifies the influence of a given measurement onthe estimate of a particular variable. This procedure replaces the original inverse problemby an equivalent problem where the number of independent unknowns is proportional tothe number of measurements. The representer approach is especially advantageous ingroundwater problems, where the total number of measurements is often small. Thisapproach is illustrated with a synthetic flow and transport example that includestime-dependent model errors. The representer algorithm is able to provide good estimatesof a spatially variable hydraulic conductivity field and good predictions of soluteconcentration. Its computational demands are reasonable, and it is relatively easy toimplement. The example reveals that it is beneficial to account for model errors even whenthey are difficult to estimate. INDEX TERMS: 1832 Hydrology: Groundwater transport; 1829

Hydrology: Groundwater hydrology; 1869 Hydrology: Stochastic processes; 1831 Hydrology: Groundwater

quality; KEYWORDS: calibration, groundwater hydrology, groundwater transport, inverse modeling,

representers, uncertainty

Citation: Valstar, J. R., D. B. McLaughlin, C. B. M. te Stroet, and F. C. van Geer (2004), A representer-based inverse method for

groundwater flow and transport applications, Water Resour. Res., 40, W05116, doi:10.1029/2003WR002922.

1. Introduction

[2] Combined groundwater flow and transport modelsare frequently used to predict the movement and fate ofsubsurface contaminants. These models are prone to anumber of errors that need to be considered in applications.First, some of the physical and biochemical processesincluded, such as biodegradation and adsorption, arenot well understood and are frequently represented in asimplified way. Second, the models rely on poorly knowninputs such as hydraulic conductivity, porosity, disper-sivities, adsorption coefficients, and recharge values. Third,

approximate upscaling procedures are generally used torepresent the aggregate or large-scale effects of small-scaleprocesses that cannot be explicitly resolved. Examples ofspatial upscaling are macrodispersion theory [Gelhar andAxness, 1983] and the concept of dual porosity [Harveyand Gorelick, 2000], which account in somewhat differentways for enhanced spreading caused by geological hetero-geneity. Although some model errors are time-invariantothers, particularly those associated with groundwaterrecharge, solute mass fluxes, and upscaling, can vary overtime.[3] Time-dependent model errors can be accounted for

probabilistically in various stages of the modeling process[Schweppe, 1973]. This helps to make model predictions,parameter estimates, and related uncertainty analysesmore realistic since model limitations are acknowledged.Here we are particularly interested in using stochastic

1Also at Faculty of Civil Engineering, Delft University of Technology,Delft, Netherlands.

Copyright 2004 by the American Geophysical Union.0043-1397/04/2003WR002922$09.00

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WATER RESOURCES RESEARCH, VOL. 40, W05116, doi:10.1029/2003WR002922, 2004

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descriptions of model errors to improve the solutions toparameter estimation (inverse) problems. Conventionalinverse methods impose excessive computational demandsfor problems with time-dependent model errors since theproblem dimensionality increases without limit as timeincreases. In this paper we describe a computationallyefficient variational inverse procedure that accounts fortime-dependent model errors. Our discussion begins with ageneral probabilistic problem formulation. We then intro-duce an inverse solution procedure based on representerexpansions. This procedure is illustrated with a syntheticexample of coupled groundwater flow and transport. Weconclude with an assessment of the method’s potentialusefulness for field applications.

2. Formulation of the Inverse Problem

2.1. Forward Model

[4] Here we consider the problem of time-dependentsolute transport in a time-invariant (steady state) velocityfield. This problem is typically solved with numericalmodels based on finite element or finite difference discre-tizations of the governing flow and transport equations. Thesteady state discretized groundwater model may be writtenin a general matrix form as:

Ah� q ¼ 0 ð1Þ

where A is a coefficient matrix that depends on thehydraulic conductivity at the nodes of the spatial grid, h isa vector of nodal heads, and q is a vector of correspondingforcing terms that depend on the flow boundary conditions.The time-dependent discretized groundwater transportmodel may be written as:

Bct � Dct�1 � ut ¼ 0; t ¼ 1; 2; . . . ;Nt

c0 ¼ Cinit

ð2Þ

where B and D are coefficient matrices that depend onvelocities and transport properties such as dispersivities andsorption coefficients, ct is a vector of nodal soluteconcentrations at time t, ut is a vector of correspondingforcing terms that depend on the transport boundaryconditions, Cinit is a vector of initial concentrations, andNt is the number of time steps [Pinder and Gray, 1977]. Thevelocities included in B and D are derived from the headsand conductivities by means of Darcy’s law. We supposethat all of the unknown parameters (hydraulic conductiv-ities, dispersivities, etc.) used to derive A, B, D, and Cinit areassembled in a time-invariant vector a. This is the parametervector that we wish to estimate with the inverse procedure.

2.2. Accounting for Model Errors

[5] In order to account for model errors we must distin-guish the model from reality. One way to do this is tosuppose that the true heads and concentrations obey equa-tions similar to equations (1) and (2), but with randomvariables added to account for imperfectly modeled inputs.Here we assume that the model errors act as additive forcingterms, similar in effect to unmodeled recharge or solutemass fluxes. The flow model errors are assumed to be time-invariant, as our flow model is time-invariant, while the

transport model errors are time-dependent. This formulationof model errors can be represented mathematically if wesuppose that the true heads and concentrations at the gridnodes obey the following stochastic state equations:

Ah� q� wh ¼ 0 ð3Þ

Bct � Dct�1 � ut � wtc ¼ 0; t ¼ 1; 2; . . . ;Nt ð4Þ

c0 ¼ Cinit

where wh are the random flow equation model errors and wct

are the random transport equation model errors over timestep t [Schweppe, 1973]. The head model errors are assumedto have a zero mean and a known covariance matrix Ph.[6] When the true concentration approaches zero the

transport model errors are likely to be smaller than whenthe concentration is large. To account for such state-depen-dent model errors we suppose that wc

t has the form:

wtc ¼ Y ct�1

� ��t ð5Þ

where Y(ct�1) is a diagonal matrix with diagonal element iproportional to the ith element of the ct�1 vector and �t arerandom errors with a zero mean and a known time-invariantcovariance matrix P�. The values of �

t at different times areassumed to be uncorrelated.

2.3. Measurements

[7] Inverse procedures derive estimates of uncertain modelparameters from measurements of observable variables suchas head and solute concentration. In the problem consideredhere the measurements are assumed to be interpolatedvalues of the true nodal head and concentration values withrandom errors added. This is expressed mathematically bythe following stochastic measurement equation:

z ¼ M h; cð Þ þ v ð6Þ

where z is the vector of all measurements taken in the regionof interest over a time period spanning Nt time steps and c(with no superscript) is an extended vector that contains thetrue concentrations at all times. The interpolation function(or measurement operator) M(h, c) describes how eachmeasurement is related to the head and extended concentra-tion vectors. In the groundwater problem considered it isreasonable to assume that M(h, c) is linear, as the modelprediction of either a head or a concentration measurementis linearly related to the nodal heads or concentrationsrespectively. The random measurement errors v are assumedto have a zero mean and known covariance matrix Pv. Thevector v and the matrix Pv contain the measurement errorsand their covariances for the measurements at all time steps.

2.4. Objective Function

[8] With the state and measurement equations defined wecan provide a statement of the inverse problem. We supposethat the parameter vector a is random with a known mean �aand covariance matrix Pa. These statistics summarize ourprior knowledge about the parameter values (if the param-eters are completely unknown the values in the priorparameter covariance matrix approach infinity). We seekan ‘‘optimal’’ estimate of the parameter vector a as well as

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the model errors included in the vectors wh and �t. If theerrors and parameters included in the state and measurementequations are independent Gaussian random vectors themaximum a posteriori estimate is an attractive option. Thisestimate minimizes the following generalized least squaresobjective function [Schweppe, 1973]:

J ¼ z�M h; cð Þ½ �TP�1v z�M h; cð Þ½ � þ a� �að ÞTP�1

a a� �að Þ

þ wTh P

�1whwh þ

XNt

t¼1

�tTP�1� �t ð7Þ

[9] The first term in this expression penalizes the misfitbetween the measurement and model prediction vectors.This term is a function of a, wh and � as the modelpredictions h and c depend on these parameters and modelerrors. The second term penalizes deviations of themodel parameter vector from its prior mean. The third andfourth terms penalize the flow and transport model errors.The relative importance of each term is determined by thecovariances, which act like weighting factors.[10] In the synthetic example in this paper, we assume the

covariance matrices Pv , Pa, Pwhand P� and the prior mean

of the parameters �a are known. For real world problemsobtaining these statistical values is not a straightforwardtask. In practice, they are often based on the subjectivechoice of the modeler and are likely to differ from the actualstatistical values.[11] The generalized least squares objective function is

very difficult to minimize when the number of parameters islarge, especially when the minimization algorithm requiresinversion of nondiagonal covariance matrices. Gradient-based search procedures such as conjugate gradient are themost commonly used minimization options. The gradientsrequired by these procedures can be efficiently derived with aLagrange multiplier approach (also commonly referred to asan adjoint or variational approach). Unfortunately, the com-putational effort can still be substantial for large problemssince convergence can be slow. Moreover, for large sizeinverse problem the storage and inversion of the full matricesPa, Pwh

and P�, needed for these gradient methods, is oftentoo demanding. The number of time-invariant parameters canbe reduced by using techniques such as zonation [Cooley,1977; Sun, 1994;Cooley, 1982;Carrera and Neuman, 1986].However, if time-dependent model errors are included in theproblem formulation the total number of unknowns increaseswith time and the classical variational approach quicklybecomes unmanageable. The representer technique describedin the next section provides a computationally acceptablealternative for situations where time-dependent model errorsare important.

3. Solution of the Inverse Problem

[12] The inverse problem outlined above can be solvedwith an extension of the representer algorithm, a least squaresminimization procedure that has been extensively applied inoceanography [Bennett, 1992; Eknes and Evensen, 1995].Hydrologic applications have been presented by Reichle etal. [2001], Reid [1996] and Sun [1998]. The representermethod introduces an expansion that reduces the number ofunknown parameters to the number of measurements. Thisexpansion is exact when the state andmeasurement equationsare linear and often provides a useful approximation when

these equations are nonlinear. The method is most readilyexplained by first considering the necessary conditions for alocal minimum.

3.1. Euler-Lagrange Equations

[13] The minimium of the generalized least squaresobjective function must be derived subject to the constraintthat the state equations are satisfied. This constraint can beincorporated into the minimization procedure if the stateequations are multiplied by unknown Lagrange multipliersand added (or adjoined) to the objective function. Thenecessary condition for a constrained local minimumrequires that the derivatives of this adjoined objective, takenwith respect to the unknown states (h and c), Lagrangemultipliers (lc and lh), parameters (a) and model errors (wh

and �t), must all be zero. When the individual derivatives arecomputed and set equal to zero the result is a set ofnonlinear coupled equations called the Euler-Lagrangeequations. A short derivation is given in Appendix A; acomplete derivation is provided by Valstar [2001].[14] The Euler-Lagrange equations naturally group into

two adjoint (or Lagrange multiplier) equations, one fortransport and one for flow, a parameter equation, and twoforward equations, one for transport and one for flow. Theforward equations incorporate estimates for the modelerrors. The complete set may be written in indicial notationas follows:

Bjiltci¼ Djiltþ1

ciþ �tþ1

m

@Ymi ctð Þ

@ctjltþ1ci

þ @Mp h; cð Þ@ctj

P�1v

� �pn

� zn �Mn h; cð Þ½ �; t ¼ Nt � 1;Nt � 2; . . . ; 1 ð8Þ

BjilNt

ci¼ @Mp h; cð Þ

@cNt

j

P�1v

� �pn

zn �Mn h; cð Þ½ � ð9Þ

Agf lhf ¼@Mp h; cð Þ

@hgP�1v

� �pn

zn �Mn h; cð Þ½ �

�XNt

t¼1

ctj@Bji

@hg� ct�1

j

@Dji

@hgr

� �ltci

ð10Þ

al ¼ �al � Palkhg

@Agf

@ak

lhf þXNt

t¼1

ctj@Bji

@ak

� ct�1j

@Dji

@ak

� ltci

" #

ð11Þ

Afghg ¼ qf þ whf ð12Þ

whf ¼ Pwhfdlhd ð13Þ

Bijctj ¼ Dijc

t�1j þ uti þ Yir ct�1

� ��tr; t ¼ 1; 2; . . . ;Nt ð14Þ

c0j ¼ Cinitj ð15Þ

�tr ¼ P�rmYms ct�1� �

ltcs; t ¼ 1; 2; . . . ;Nt ð16Þ

where d, f, and g range from 1 to the number of head statevariables; i, j, m, r, and s range from 1 to the number ofconcentration state variables; k and l range from 1 to thenumber of uncertain parameters; and n and p range from1 to the number of measurements. Indices repeated within a

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single product term are assumed to be summed overappropriate ranges. Different indices used for thesame variables (such as the indices n and p in M(h, c) inequation (8)) only denote how matrix-matrix or matrix-vector multiplications should be performed.[15] Note that equations (12) and (14) have the same

form as the flow and transport state equations (3) and (4),with the true model errors replaced by estimates thatdepend on the head and concentration adjoint variables,respectively. The adjoint variables determine how themodel error estimates are distributed over space and/ortime while the model error covariances control theirmagnitude. When the error covariances or adjoint varia-bles are zero the corresponding model error estimates arealso zero. To simplify notation in the remainder of thispaper we use the symbols h, c, lh, lc, a, wh and � torepresent the best available approximate solution to theEuler-Lagrange equations, rather than the unknown truevalues.[16] The Euler-Lagrange equations form a two-point

boundary value problem, with the boundary values givenby the initial condition of the transport equation and theterminal condition of the transport adjoint equation. Thesolution may not be unique since it only defines a local(rather than global) minimum of the least squares objective.The likelihood of multiple solutions decreases when theprior covariances of the parameters and model errors aresmall and the quadratic prior terms in the least squaresfunction that contain the parameters and model errors, aregiven more weight. Techniques for solving two-pointboundary value problems include the sweep method[Gelfand and Fomin, 1963; Bryson and Ho, 1975; Bennett,1992] and the representer method [Bennett, 1992]. In thispaper, we use an extension of the representer method ofBennett.

3.2. Representer Solution

[17] The representer method writes each of the unknownsin the Euler-Lagrange equations as an expansion in a set ofunknown basis functions called representers. The number ofterms in each representer expansion is the total number ofmeasurements. For our problem the representer expansionshave the following form:

ltci¼

XNz

p¼1

Ftipbp ð17Þ

lhf ¼XNz

p¼1

Gfpbp ð18Þ

al ¼ �al þXNz

p¼1

�lpbp ð19Þ

hg ¼ hFgþ hcorrg þ

XNz

p¼1

Xgpbp ð20Þ

ctj ¼ ctFjþ ctcorrj þ

XNz

p¼1

Wtjpbp ð21Þ

where bp is an unknown representer coefficient thatdetermines the weight given to the representer associatedwith measurement zp, Ft

ip is the concentration adjointrepresenter, Gfp is the head adjoint representer, �lp is theparameter representer, Xgp is the head representer, Wt

jp isthe concentration representer, hFg

and ctFjare prior head and

concentration estimates, hcorrg and ctcorrj are correction termswhich allow the head and concentration expansions to betaken about the latest estimates, and Nz is the total numberof measurements. The prior estimates hF and ctF are thesolutions obtained by solving equations (1) and (2) witha = �a.[18] The representer expansions can be shown to be

exact solutions to the Euler-Lagrange equations whenthe original optimization problem has constraints thatare linear in all decision variables. In this case therepresenters are simply the cross covariances betweeneach unknown variable and each measurement, the priorhead and concentration estimates are the mean values, andthe correction terms are zero. So, for example, Xgp is thecross covariance between the head hg and the measure-ment zp. In our problem the linear constraint requirementis not met and the representer expansions are onlyapproximations. However, a representer such as Xgp canstill be viewed as an influence function that specifies howthe unknown head hg should be modified when themeasurement zp differs from the nominal model predic-tions. In this sense, the representer approach provides avery flexible parameterization of the inverse problem. Thespatial variation of each unknown depends on the govern-ing equation, its forcing terms, and specified prior infor-mation. This is in contrast to methods that assume eachunknown has a fixed spatial structure, such as a piecewiseconstant function.[19] In a linear constraint problem the representer

expansions can be substituted into the Euler-Lagrangeequations and closed-form expressions obtained for therepresenters and their coefficients. In the nonlinear prob-lem of interest here an iterative solution approach mustbe used since obtaining the closed-form solutions is notpossible. The basic concept is to approximate each of thenonlinear functions A(a), B(h, a), D(h, a), and Y(c) by afirst-order Taylor series expansion taken round the mostrecently computed estimates of h, c, and a. The resultinglinearized (and approximate) Euler-Lagrange equationscan be solved, the unknowns are updated, and thelinearization is repeated until the iteration procedureconverges. After convergence the Euler-Lagrange equa-tions are solved exactly and the parameter and modelerror estimates are the solution of the original optimiza-tion problem.[20] When the representer definitions are substituted into

the linearized Euler-Lagrange equations they decouple theequation set, making it possible to derive a sequential set ofexpressions for (1) the representers, (2) the correction terms,and (3) the representer coefficients (a short derivation isgiven in Appendix B; an extensive derivation is given byValstar [2001]). In any given iteration the values of h, c, a,lh, lc and � appearing in an update equation are the onescomputed in the previous iteration.[21] The algorithm is initialized with the ‘‘initial guess’’

h = hF, ct = ctF, a = �a, lh = 0, and ltc = 0. Then, the

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updated representers are obtained using the followingformula:

BjiFtip ¼ DjiFtþ1

ip þ @Mp h; cð Þ@ctj

þ �tþ1m

@Ymi ctð Þ

@c tð Þ Ftþ1ip ;

t ¼ Nt � 1;Nt � 2; . . . ; 1 ð22Þ

BjiFNt

ip ¼ @Mp h; cð Þ@cNt

j

ð23Þ

Agf Gfp ¼@Mp h; cð Þ

@hg�XNt

t¼1

ctj@Bji

@hg� ct�1

j

@Dji

@hg

� �Ftip ð24Þ

�lp ¼ �Palkhg

@Agf

@ak

Gfp þXNt

t¼1

ctj@Bji

@ak

� ct�1j

@Dji

@ak

r

� Ftip

" #ð25Þ

AfgXgp ¼ � @Afg

@ak

�kphg þ Pwh½ �fdGdp ð26Þ

BijWtjp ¼ DijWt�1

jp þ Yir ct�1� �

P�½ �rmYms ct�1� �

Ftsp

þ @Yir ct�1ð Þ@ct�1

j

Wt�1jp �tr �

@Bij

@ak

�kp þ@Bij

@hgXgp

� �ctj

þ @Dij

@ak

�kp þ@Dij

@hgXgp

� �ct�1j ; t ¼ 1; 2; . . . ;Nt ð27Þ

W0jp ¼ 0 ð28Þ

[22] Then the updated head and concentration correctionsare obtained from the following equations:

Afghcorrg ¼ qf þ@Afg

@ak

ak � �akð Þhg � AfghFgð29Þ

Bijctcorrj

¼ Dijct�1corrj

þ uti

þ @Bij

@ak

ak � �akð Þ þ @Bij

@hghg � hFg

� hcorrg� �� �

ctj

� @Dij

@ak

ak � �akð Þ þ @Dij

@hghg � hFg

� hcorrg� �� �

ct�1j

� BijctFjþ Dijc

t�1Fj

� @Yir ct�1ð Þ@ct�1

j

ct�1j � ct�1

Fj� ct�1

corrj

� �tr;

t ¼ 1; 2 . . . ;Nt ð30Þ

c0corrj ¼ 0 ð31Þ

[23] Then the updated representer coefficient vector isobtained by solving the following equation:

Pvnp þMn Xp;Wp

� �� �bp ¼ zn �Mn hF þ hcorr; cF þ ccorrð Þ½ � ð32Þ

[24] After the representers, corrections, and representercoefficient vector are found they may be substituted intoequations (17)–(21) to give updated values for the adjointvariables and the parameter vector. Alternatively, the adjoint

updates may be obtained by substituting equation (32) intoequations (8)–(10). The parameter updates can then bederived from equation (11). The latter approach uses lessmemory but requires somewhat more computational effort.In either case, the updated head and concentration arecomputed directly from equations (12)–(16). The A, B, D,and Y derivatives required for the next iteration areevaluated at these updated values. The computationaleffort is dominated by the computation of the representers.For each measurement, the effort is approximatelyequal to solving two flow and transport models. For alarge number of measurements, the computational effortbecomes considerable.

4. Computational Issues

4.1. Eigenvalue Decomposition

[25] In the algorithm outlined above the coefficient vectorb is obtained by solving the set of linear equations given inequation (32). When measurements are highly correlatedthis matrix equation is ill-conditioned and the solution isvery sensitive to small changes in the measurement resid-uals on the right hand side. We resolve this problem byreplacing the original matrix equation by a reduced rankapproximation based on the leading eigenvectors of theoriginal coefficient matrix. However, in groundwater in-verse problems concentration measurements near zero oftenhave small variances. A straightforward reduced rank ap-proximation tends to ignore such small variance measure-ments, even though they convey important informationabout the boundaries of the solute plume. The influenceof small variance measurements can be retained if the rowsand columns of the measurement residual covariance matrix(Pv + M(X, W)) are rescaled before the eigenvalue decom-position is performed. The scale factors are selected to givevalues of unity on the main diagonal of this matrix. In asynthetic example described by Valstar [2001] rescalinggives much better predictions near the low-concentrationedges of the solute plume.

4.2. Convergence

[26] We assume that convergence is achieved when theresiduals between the state variables, obtained from equa-tions (12) and (14), and the linear representer expansions,obtained from equations (20) and (21), are smaller than athreshold value for each measurement. For this thresholdvalue we generally use 1% of the measurement errorstandard deviation. There is no guarantee that the iterativerepresenter solution to the Euler-Lagrange equations con-verges. In practice, it is helpful to introduce a relaxationfactor so that each new parameter and model error estimateis a weighted combination of the estimate from the previousiteration and the estimate derived from the representerexpansion. The optimal relaxation factor is adjusted tominimize the mean squared scaled difference between thehead and concentration estimates obtained from the stateequations (12)–(16) and the corresponding relaxed esti-mates obtained from the representer expansions (20) and(21). Each difference is scaled by its standard deviation. Therelaxation adjustment procedure can be viewed as a linesearch along a search direction determined by the repre-senter approximation. We cannot guarantee that the methodalways converges for highly nonlinear estimation problems.

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In our experience so far, we did not encounter any conver-gence problems.

4.3. Accuracy Estimates

[27] After convergence the accuracy of the parameter andstate estimates can be inferred from approximate posterior(conditional) covariances, which are based on linearizationsof the state equations. These are the covariances of theparameters and states, conditioned on the measurements usedin the inverse procedure. The posterior variances are neverlarger than the prior (unconditional) variances and should besignificantly smaller if the measurements provide usefulinformation about the estimated variables. The approximateposterior covariances of the parameters are given by:

Ppostakl

¼ Pakl��kp M W;Xð Þ þ Pvð Þ�1

� pq�lq ð33Þ

The approximate posterior covariances of the states are

Pposthfg

¼ Phfg � Xfp M W;Xð Þ þ Pvð Þ�1�

pqXgq ð34Þ

Ppost;t1;t2cij

¼ Pt1 ;t2cij

� Wt1ip M W;Xð Þ þ Pvð Þ�1�

pqWt2

jq ð35Þ

The posterior parameter variances (the diagonals of thecovariance matrix Pakl

post) can be readily calculated sincethe prior parameter covariance Pakl

is known. However, theposterior variances of the states depend on the priorcovariances Phfg

and Pcij

t1,t2, which can be readily calculatedonly at the measurement locations. A significant amount ofadditional computational effort is required to obtain thecomplete prior covariances of the states. In our examplewe only consider the posterior variances of the parametersand states. The latter required considerable additionalcomputations.

4.4. Comparison to Other Nonlinear Inverse Methods

[28] The primary factors that influence the computationalrequirements of nonlinear least squares inverse methods arethe number of forward model simulations needed to deter-mine the minimization search step and the number of searchsteps required to converge to an acceptable solution. Thenumber of forward simulations per step in the representeralgorithm is proportional to the number of scalar measure-ments used in the inversion. This is comparable to a Gauss-Newton or iterative cokriging algorithm that relies onmeasurement sensitivity derivatives derived from coupledforward and adjoint model solutions. By contrast, a conju-gate gradient search algorithm using adjoint-based deriva-tives requires only one forward and one adjoint simulationper search step. The number of steps required by any of thesesearch algorithms generally increases with the number ofunknown parameters (i.e., the dimension of the searchspace). Gauss-Newton and iterative cokriging algorithmsmust treat model errors at each time step as unknownparameters. Consequently, the total number of parameters(and the number of iterations required for convergence)increases as the length of the simulation interval increases.The representer method does not require the model errors tobe included in the parameter vector. Instead, they are deriveddirectly from the adjoint vector associated with thecorresponding model state. In our experience, the number

of iterations required for the representer method to convergeis no greater when time-dependent model errors are includedthan when it is neglected. This is a significant benefit inpractical groundwater applications, where recharge ratesand/or contaminant source histories are uncertain.

4.5. Applicability of the Inverse Method for VariousMeasurement Sets

[29] The representer method is applicable for both denselyand sparsely filled measurement sets. For densely filledmeasurements sets, some measurements are likely to bestrongly correlated (short distance in space and time). Thiscorrelation shows up during the eigenvalue decompositionof the measurement residual covariance matrix (Pv +M(X, W)). For sparsely filled measurements sets, somemeasurements may have only a weak correlation with allother measurements. The representer coefficient b of thatmeasurement is obtained almost independently of the othermeasurement values. For regions with no measurements atall its parameters are hardly updated, but their uncertaintydoes not reduce either.

5. Example

[30] We can now illustrate the performance of our repre-senter-based inverse method with a synthetic example thatincludes time-dependent model errors. Such an example isuseful for initial tests since the true answer is known anderrors can be evaluated unambiguously. Of course, addi-tional full-scale tests with real field data are needed toadequately assess the practical applicability of any inverseprocedure.[31] Our test problem considers mobile and immobile

solute transport in a vertical two-dimensional section withrecharge occurring on the top boundary (surface), as shownin Figure 1. A spatially distributed mass of solute enters asource area on the surface at time t = 0 and is subsequentlypartitioned into mobile and immobile fractions. Mobilesolute moves with the prevailing steady state flow field(from right to left) and out the left boundary. Mass transferbetween the mobile and immobile fractions at any time andlocation is proportional to the difference between mobileand immobile concentrations [Brusseau and Rao, 1989;Harvey and Gorelick, 2000]. There are no flow modelerrors but mobile and immobile transport model errorsaccount for deviations from the idealized linear masstransfer model.[32] The flow state equation (3) is obtained by discretiz-

ing the following partial differential equation:

@

@xiKij

@h xð Þ@xj

� �¼ 0 x 2 D ð36Þ

with boundary conditions:

h xð Þ ¼ 0 x 2 @Dh

Kij xð Þ @h xð Þ@xi

ni xð Þ ¼ qb xð Þ x 2 @Dq

where: h(x) is the hydraulic head at location x in thecomputational domain D, qb(x) is the specified boundary

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flux discharge rate per unit area, Kij(x) is the hydraulicconductivity (assumed to be isotropic in our example), @Dh

and @Dq are the specified head and specified flow portionsof the domain boundary @D, and ni(x) is the outward unitvector normal to the boundary @Dq.[33] The concentration state equation (4) is obtained by

discretizing the following partial differential equations,which describe transport in the mobile and immobileportions of the porous medium, respectively:

@cm x; tð Þ@t

þ ui xð Þ @cm x; tð Þ@xi

� @

@xiDij

@cm x; tð Þ@xj

� �þ fr xð Þ

qm� cm x; tð Þ � cim x; tð Þ½ � ¼ cm x; tð Þ�m x; tð Þ x 2 D ð37Þ

@cim x; tð Þ@t

� fr xð Þqim;e

cm x; tð Þ � cim x; tð Þ½ � ¼ cim x; tð Þ�im x; tð Þ x 2 D

ð38Þ

with initial conditions:

cm x; tð Þ ¼ cm0xð Þ t ¼ 0 x 2 D

cim x; tð Þ ¼ cim0xð Þ t ¼ 0 x 2 D

and boundary conditions:

cm x; tð Þ ¼ 0 on @D for inflow boundaries

@cm x; tð Þ@xi

ni xð Þ ¼ 0 on @D for outflow boundaries

where cm(x, t) is the mobile solute concentration at locationx and time t, cim(x, t) is the immobile solute concentration,

ui(x, t) is the pore water velocity (obtained from Darcy’slaw). Dij(x) is the dispersion coefficient, which is computedfrom the Scheidegger relationship using constant dispersiv-ities, fr(x) is the exchange rate between the mobile andimmobile fractions. qm is the mobile porosity, �m(x, t) is themobile fraction of the model errors, qim,e is the effectiveimmobile porosity and �im(x, t) is the immobile fraction ofthe model errors.[34] The spatial domain used in our example has 151 by

41 nodes and uses finite rectangular elements of 2.0 by0.5 m each. In the inverse procedure, the parametersestimated are the hydraulic conductivity and exchangerate coefficient (which vary over space) and the effectiveimmobile porosity (which is assumed to be spatiallyuniform). The groundwater flow boundary conditions forthe example are shown in Figure 1a. These conditions createa nonuniform flow moving generally from right to left, witha downward component near the upper right corner of thedomain. Hydraulic heads are measured in m and soluteconcentrations are measured in g L�1. The initial mobileconcentration distribution is shown in Figure 1b. The initialimmobile concentration is zero everywhere. The simulationconsists of 120 time steps of 5 days each.[35] The spatially variable ln conductivity and ln

exchange rate coefficient used in the synthetic experimenthave mean values of ln(1) and ln(0.01) and variances of2.0 and 1.0, respectively, where the conductivity is mea-sured in m day�1 and the exchange rate is measured inday�1. Both parameters have exponential covariance func-tions with correlation lengths of 50 m and 2 m in thehorizontal and vertical directions, respectively. The simu-lated ln conductivity field is shown in Figure 2a. The naturallogarithm of the effective immobile porosity is a unitlessspatially uniform random parameter with mean ln(0.1) andvariance of 1. The synthetic realization of the effective

Figure 1. (a) Model area and flow boundary conditions and (b) initial concentration contours of 0.01,0.1, 1.0, 2.0 and 5.0 g L�1.

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immobile porosity is 0.13. The other model parameters areassumed to be perfectly known. The horizontal and trans-versal dispersivities are 0.5 m and 0.05 m. The effectivemobile porosity is 0.2.[36] The concentration model errors �m(x, t) and �im(x, t)

are independent random zero mean fields uncorrelated overtime. Their spatial covariance functions are exponentialwith a variance of 10�4 and correlation lengths of 50 min the horizontal direction and 2 m in the vertical direction.Note that these errors are multiplied by their respectiveconcentrations in the transport state equations.[37] The reference (or ‘‘true’’) head and mobile concen-

tration distributions after 120 time steps are shown inFigures 3a and 4a. Prior estimates for the hydraulic con-ductivity, exchange rate coefficients and immobile porosityare set equal to the specified mean values. Values of theprior heads and mobile concentrations after 120 time stepsare shown in Figures 3b and 4b, respectively.[38] Estimates of all uncertain parameters are derived

from 4 noisy head measurements and 185 noisy mobileconcentration measurements. The immobile concentrations

are not measured. The locations of the head measurementsare shown in Figure 3c, and the locations of the concen-tration measurements are shown in Figure 4c. The mea-surement errors are zero mean, with a standard deviationof 10�2 m for the head measurements and 10% of the realconcentration value for the concentration measurements.After 10, 20 and 30 time steps, only the 10 mostupstream wells are monitored; after 40, 50, 60, 70 and80 time steps the 15 most upstream are monitored, andafter 90, 100, 110 and 120 time steps all wells aremonitored.[39] The estimated ln conductivity field is shown in

Figure 2b. The associated posterior (conditional) varianceis shown in Figure 2c. The variance plot indicates that the lnconductivity variance is reduced from 2.0 to less than 0.5 insome locations. The normalized differences (difference be-tween real and estimated ln conductivity divided by thesquare root of the posterior variance) are shown inFigure 2d. If the ln conductivity estimates are optimal (i.e.,the posterior variance values are consistent with reality) thesenormalized differences should have a mean near 0.0, a

Figure 2. (a) Generated (‘‘true’’) hydraulic conductivities, (b) posterior conductivities, (c) approximateposterior conductivity variance, and (d) histogram of the normalized conductivity residuals. Figures 2aand 2b have the same legend.

Figure 3. (a) Generated (‘‘true’’) hydraulic heads, (b) prior hydraulic heads, (c) posterior hydraulicheads, and (d) histogram of the normalized hydraulic head residuals. Hydraulic head measurementlocations are denoted by asterisks in Figures 3a–3c. Solid lines in Figures 3b and 3c denote the truehydraulic head contours from Figure 3a.

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variance near 1.0 and a normal probability distribution. Inthis example, the distribution of the normalized residuals ofthe posterior ln conductivity field shows a bias. The lnexchange rate coefficients showed minor changes for theposterior estimate and relatively little variance reductionafter estimation, suggesting that the available measurementsdo not contain much information about this particularparameter. The immobile porosity estimate improved from0.1 to 0.109.[40] The posterior head and mobile concentration distri-

butions are shown in Figures 3c and 4c. They show a clearimprovement when compared to the prior results. The nor-malized head and concentration residuals are shown inFigures 3d and 4d. In order to calculate these residuals theposterior variances of the heads and concentrations werecalculated. For the concentration residuals only the nodeswith a prior concentration variance larger than 10�10 g2 L�2

was taken into account to prevent numerical inaccuracyeffects. The normalized concentration residuals seem nor-mally distributed, but the head residuals show a clear positivebias. We believe that this bias is due to the strong spatialcorrelation between the head residuals. Figure 3c shows thatin both the left and the right side of the domain the headestimates are consistently higher than the true values.[41] Considering that the posterior variance of the ln

conductivity in the plume area is relatively large (0.5–

1.0), it is remarkable that the predicted concentrationestimates are so close to the true values. This suggests thatthe concentration values in this problem are relativelyinsensitive to small-scale fluctuations in conductivity.[42] It is difficult to obtain accurate estimates of model

errors from the relatively small number of measurementsused in this example. However, the recognition that sucherrors are present reduces the estimator’s dependence on themodel and gives the measurements more influence than theywould otherwise have. In Figure 5 the posterior estimate ofthe mobile concentration after 120 time steps is shown incase model errors are neglected in the inverse model. Theresults are inferior in the region (150, 10) when compared tothe posterior estimate when model errors are taken intoaccount; see Figure 4c. So there is a definite benefit toacknowledge the presence of model errors even whenavailable measurements are too sparse to allow such errorsto be identified.

6. Conclusions

[43] The representer-based inverse method introduced inthis paper provides an efficient way to solve inverse prob-lems that include uncertain model errors. The method’sefficiency is achieved by converting the original inverseproblem to an equivalent problem where the number of

Figure 5. Posterior concentrations after 120 time steps in case model errors are neglected.Concentration measurement locations are denoted by asterisks. Solid lines denote the true concentrationcontours. Contours denote 0.001, 0.01, and 0.1 g L�1.

Figure 4. (a) Generated (‘‘true’’) solute concentrations, (b) prior concentrations, (c) posteriorconcentrations, and (d) histogram of the normalized concentration residuals after 120 time steps.Concentration measurement locations are denoted by asterisks in Figures 4a–4c. Solid lines in Figures 4band 4c denote the true concentration contours from Figure 4a. Contours in Figures 4a–4c denote 0.001,0.01, and 0.1 g L�1.

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independent unknowns is proportional to the number ofmeasurements. The solution obtained with the representer-based inverse method is equal to the original large-scaleinverse problem which has a number of unknown parame-ters proportional to the number of grid nodes. This isespecially advantageous in groundwater problems, wherethe total number of measurements is often relatively small.[44] The explicit incorporation of model errors allows

the algorithm to properly balance information from mea-surements and model predictions. If the measurement errorvariances are small compared to the model error variancesthe measurements will be given more weight. If theopposite is true the model predictions will be given moreweight. The balanced approach to errors taken in therepresenter formulation highlights the need to quantifythe accuracy of both model and data in inverse applica-tions. In practice it is unrealistic to assume that eithermodel predictions or measurements are perfect. Otherinversion procedures commonly lack the flexibility tobalance the uncertainty of parameters, model errors, andmeasurement errors.[45] Although our representer-based inverse algorithm is

complicated to derive it is relatively easy to implement. Therepresenters and adjoint variables are obtained by solvingequations similar in structure to the original flow andtransport equations. In most cases the matrix derivativescan be derived in closed form and the iterative solutionalgorithm converges quickly. This may not always be thecase, especially if the original problem is highly nonlinear.Overall, the algorithm’s performance and computationalefficiency are good enough to encourage applications tofield problems of realistic size. Such applications willundoubtedly provide better understanding of the method’sadvantages and limitations.

Appendix A: Derivation of theEuler-Lagrange Equations

[46] The Euler-Lagrange equations are derived by multi-plying the flow equation (3) with 2 times the head adjointvector lh and multiply the transport equation (4) with2 times the concentration adjoint vector lc

t and take thesummation of the last product over all time steps and addthem to the objective function (7):

J ¼ z�M h; cð Þ½ �TP�1v z�M h; cð Þ½ � þ a� �að ÞTP�1

a a� �að Þ

þ wTh P

�1whwh þ

XNt

t¼1

�tTP�1� �t þ 2lT

h Ah� q� wh½ �

þ 2XNt

t¼1

ltT

c Bct � Dct�1 � ut � Y ct�1� �

�t� �

ðA1Þ

[47] Taking the variation of this equation yields:

1

2DJ ¼

XNt�1

t¼1

"� zn �Mn h; cð Þ½ � P�1

v

� �np

@Mp h; cð Þ@ctj

þ ltciBij � ltþ1

ciDij�ltþ1

ci

@Yim ctð Þ@ctj

�tþ1m

#Dctj

þ"� zn �Mn h; cð Þ½ � P�1

v

� �np:@Mp h; cð Þ

@cNt

j

þ lNt

ciBij

#DcNt

j

þ lhf Afg þXNt

t¼1

ltci

@Bij

@hgctj �

@Dij

@hgct�1j

�"� zn �Mn h; cð Þ½ �

� P�1v

� �np

@Mp h; cð Þ@hg

#Dhg þ

�al � �alð Þ P�1

a

� �lk

þXNt

t¼1

ltci

@Bij

@ak

ctj �@Dij

@ak

ct�1j

� þlhf

@Afg

@ak

hg

�Dak

þ whf P�1wh

h ifd�lhd

� �Dwhd

þXNt

t¼1

�tr P�1�

� �rm�lt

ciYim ct�1

� �h iD�tm ðA2Þ

[48] In the minimum of the objective function the varia-tion of the objective function is zero for any variation of therandom variables. Forcing this constraint, equation (A2)yields the Euler-Lagrange equations:[49] For the variations of the concentrations Dc, it yields

the adjoint system for concentrations:

Bjiltci¼ Djiltþ1

ciþ @Mp h; cð Þ

@ctjP�1v

� �pn

zn �Mn h; cð Þ½ �

þ �tþ1m

@Ymi ctð Þ

@ctltþ1ci

; t ¼ Nt � 1; . . . ; 1 ðA3Þ

BjilNt

ci¼ @Mp h; cð Þ

@cNt

j

P�1v

� �pn

zn �Mn h; cð Þ½ � ðA4Þ

[50] For the variations of the heads Dh, it yields theadjoint system for heads:

Agf lhf ¼@Mp h; cð Þ

@hgP�1v

� �pn

zn �Mn h; cð Þ½ �

�XNt

t¼1

ctj@Bji

@hg� ct�1

j

@Dji

@hg

� �ltci

ðA5Þ

[51] For the variations of the parameters Dak, it yields theparameter equation:

al ¼ �al � Palkhg

@Agf

@ak

lhf þXNt

t¼1

ctj@Bji

@ak

� ct�1j

@Dji

@ak

� ltci

" #

ðA6Þ

[52] For the variation of the model errors of the transportequation D�t, it yields:

�tr ¼ P�½ �rmYmi ct�1

� �ltci; t ¼ Nt � 1; . . . ; 1 ðA7Þ

[53] For the variation of the model errors of the flowequation D whd, it yields:

whf ¼ Pwh½ �fdlhd ðA8Þ

[54] The flow and transport equations are equal to theoriginal equations (3) and (4).

Appendix B: Derivation ofRepresenter Equations

[55] In equations (17)–(21), the representer definitionswere introduced in order to solve the Euler-Lagrange

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equations (8)–(16). By inserting these representer defini-tions in the Euler-Lagrange equations, explicit expressionsfor all representers and their coefficients and correctionterms are obtained. However, these expressions still dependon the optimal estimates for the parameters and statevariables, which are unknown initially and have to be founditeratively.

B1. Concentration Adjoint Representer

[56] Inserting the representer definitions (17)–(21) in theconcentration adjoint equations (8) and (9) yields:

BjiFtipbp ¼ DjiFtþ1

ip bp þ@Mp h; cð Þ

@ctjP�1v

� �pn

� zn �Mn hF þ hcorr þ Xb; cF þ ccorr þ Wbð Þ½ �

þ �tþ1m

@Ymi ctð Þ

@ctFtþ1ip bp; t ¼ Nt � 1; . . . ; 1 ðB1Þ

BjiFNt

ip bp ¼@Mp h; cð Þ

@cNt

j

P�1v

� �pn

� zn �Mn hF þ hcorr þ Xb; cF þ ccorr þ Wbð Þ½ � ðB2Þ

[57] So far the adjoint representers F and the coefficientsb are both unknown. In order to get a unique expression forthe adjoint representer F, the coefficients b for all pmeasurements are defined as:

bp ¼ P�1v

� �pn

zn �Mn hF þ hcorr þ Xb; cF þ ccorr þ Wbð Þ½ � ðB3Þ

[58] Inserting this definition in equations (B1) and (B2),these equations can only be fulfilled for nonzero b when:

BjiFtip ¼ DjiFtþ1

ip þ @Mp h; cð Þ@ctj

þ �tþ1m

@Ymi ctð Þ

@ctFtþ1ip ;

t ¼ Nt � 1; . . . ; 1 ðB4Þ

BjiFNt

ip ¼ @Mp h; cð Þ@cNt

j

ðB5Þ

B2. Head Adjoint Representer

[59] Inserting the representer definitions (17)–(21) in thehead adjoint equation (10) yields an expression for the headadjoint representer:

Agf Gfpbp ¼@Mp h; cð Þ

@hgP�1v

� �pn

zn �Mn hF þ hcorr þ Xb; cF þ ccorrð½

þ WbÞ� �XNt

t¼1

ctj@Bji

@hg� ct�1

j

@Dji

@hg

� �Ftipbp ðB6Þ

[60] Inserting the definition of representer coefficients(B3), equation (B6) can only be fulfilled for nonzero b if:

Agf Gfp ¼@Mp h; cð Þ

@hg�XNt

t¼1

ctj@Bji

@hg� ct�1

j

@Dji

@hg

� �Ftip ðB7Þ

B3. Parameter Representers

[61] Inserting the representer definitions (17)–(21) in theparameter equation (11) yields:

�al þ�lpbp ¼ �al � Palk

"hg

@Agf

@ak

Gfpbp

þXNt

t¼1

ctj@Bji

@ak

� ct�1j

@Dji

@ak

� Ftipbp

#ðB8Þ

[62] For any nonzero b, this equation can only be fulfilledif for each p:

�lp ¼ �Palk

"hg

@Agf

@ak

GfpþXNt

t¼1

ctj@Bji

@ak

� ct�1j

@Dji

@ak

� �Ftip

#ðB9Þ

B4. Head Representers

[63] We would like to define the head representers in sucha way that during each iteration the head representers are theexact linearization around the head estimate of the previousiteration. Afterward, the head correction term will be chosenin such a way that the flow equation (12) will be fulfilled.First, we perturb the flow equation around the estimate ofthe previous iteration:

Afg aþ�pdbp� �

hg þ Xgpdbp� �

¼ qf þ whf þ Pwh½ �fdGdpdbp

ðB10Þ

where:�pdbp perturbation of parametersXgpd p perturbation of heads[Pwh

]fdGdpdbp perturbation of model errors for flow equation.[64] Linearizing this equation yields:

AfgXgpdbp ¼ qf � Afg þ@Afg

@ak

�kpdbp

� �hg þ whf þ Pwh

½ �fdGdpdbp

ðB11Þ

[65] When we take the expectation of this equation andsubtract it from the same equation, it yields:

AfgXgpdbp ¼ � @Afg

@ak

�kpdbphg þ Pwh½ �fdGdpdbp ðB12Þ

[66] Divide this equation by dbp yields the head repre-senter equation we wanted to acquire:

AfgXgp ¼ � @Afg

@ak

�kphg þ Pwh½ �fdGdp ðB13Þ

[67] Now we will choose the head correction term in sucha way that the forward flow equation (12) will be fulfilled.First, we insert the representer definitions (17)–(21) in theflow equation (12). This yields:

Afg hFgþ hcorrg þ Xgpbp

� �¼ qf þ Pwh

½ �fdGdpbp ðB14Þ

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W05116

[68] Now we multiply equation (B13) by bp, sum it overall measurements, subtract it from equation (B14) and userepresenter definition (19). It yields:

Afghcorrg ¼ qf þ@Afg

@ak

ak � �akð Þhg � AfghFgðB15Þ

B5. Concentration Representer and Correction Term

[69] The derivation of the concentration representers andconcentration correction term are done by the same steps aswe used in the derivation of the head representers and thehead correction term.

B6. Determination of Representer Coefficients

[70] The coefficient equationwas defined in equation (B3).Rearranging yields:

Pvnp þMn Xp;Wp

� �� �bp ¼ zn �Mn hF þ hcorr; cF þ ccorrð Þ½ �

ðB16Þ

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����������������������������D. B. McLaughlin, Massachusetts Institute of Technology, Ralph

M. Parsons Laboratory, 15 Vassar Street, Cambridge, MA 02139, USA.

C. B. M. te Stroet, J. R. Valstar, and F. C. van Geer, Netherlands Instituteof Applied Geoscience TNO–National Geological Survey, PO Box 80015,3508 TA Utrecht, Netherlands. (j.valstar@nitg.tno.nl)

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