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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1. Introduction

Mathematical modeling of subsurface ow and mass transportis needed, for instance, for groundwater resources management

Corresponding author at: Center for Petroleum and Geosystems EngineeringResearch, University of Texas at Austin, Austin, TX 78712, USA.

E-mail addresses: haiyanzhou@utexas.edu (H. Zhou), jaime@dihma.upv.es

Advances in Water Resources 63 (2014) 2237

Contents lists availab

Advances in Wa(J.J. Gmez-Hernndez), liangpingli@utexas.edu (L. Li).4. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4. Parameterizing the conductivity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.5. Assessing the uncertainty of prediction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0309-1708/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.advwatres.2013.10.014. . . . 32

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. . . . 332.6. Preserve prior structure or not? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.7. MultiGaussian or not?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.8. Evolution of inverse approaches to date . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3. Recent trends of inverse methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1. Integrating multiple sources of information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2. Combining high order moments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3. Handling multiscale problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Contents

1. Introduction . . . . . . . . . . . . . . . . .1.1. The forward problem and1.2. Why is the inverse problem1.3. Why is the inverse problem1.4. Outline of the paper . . . . .

2. Evolution of inverse methods . . .2.1. Direct or indirect approach2.2. Linearization or not? . . . . .2.3. Deterministic estimation o2.4. Minimization or sampling?2.5. Real time integration or no. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22rse problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23sary? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23ult? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25astic simulation? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Groundwater modelingata assimilation discussed. 2013 Elsevier Ltd. All rights reserved.le online 7 November 2013

ds:geneityter identication

This paper does not attempt to be another overview paper on inverse models, but rather to analyze andtrack the evolution of the inverse methods over the last decades, mostly within the realm of hydrogeol-ogy, revealing their transformation, motivation and recent trends. Issues confronted by the inverse prob-lem, such as dealing with multiGaussianity and whether or not to preserve the prior statistics areccepted 28 October 2013vailab

from trial-and-error manual calibration to the current complex automatic data assimilation algorithms.for Petroleum and Geosystems Engineering Research, University of Texas at Austin, Austin, TX 78712, USA

i c l e i n f o

history:d 4 July 2013d in revised form 24 October 2013

a b s t r a c t

Parameter identication is an essential step in constructing a groundwater model. The process of recog-nizing model parameter values by conditioning on observed data of the state variable is referred to as theinverse problem. A series of inverse methods has been proposed to solve the inverse problem, rangingResearch Institute of Water and Environmental Engineering, Universitat Politcnica de Valncia, 46022 Valencia, SpainCenterrse methods in hydrogeology: Evolution and recent trends

an Zhou a,b,, J. Jaime Gmez-Hernndez a, Liangping Li a,beviewjournal homepage: www.e lsev ier .com/ locate/advwatresle at ScienceDirect

ter Resources

ateror for contaminant remediation. The forward model requires spec-ication of a variety of parameters, such as, hydraulic conductivity,storativity and sources or sinks together with initial and boundaryconditions. However, in practice, it is impossible to characterizethe model exhaustively from sparse data because of the complexhydrogeological environment; for this reason, inverse modeling isa valuable tool to improve characterization. Inverse models areused to identify input parameters at unsampled locations by incor-porating observed model responses, e.g., hydraulic conductivitiesare derived based on hydraulic head and/or solute concentrationdata. Deriving model parameters from model state observationsis common in many other disciplines, such as petroleum engineer-ing, meteorology and oceanography. This work mostly focuses oninverse methods used in hydrogeology.

1.1. The forward problem and the inverse problem

The forward problem involves predicting model states, e.g.,hydraulic head, drawdown and solute concentration, based on aprior model parameterization. Combining mass conservation andDarcys laws, the forward groundwater ow model in an incom-pressible or slightly compressible saturated aquifer can be writtenas [1]

r Krh Ss @h@t

Q 1

subject to initial and boundary conditions, where r is the diver-gence operator @

@x @@y @@z

, r is the gradient operator@@x ;

@@y ;

@@z

T; K is hydraulic conductivity (L T1), h is hydraulic head

(L), Ss is specic storage (L1), t is time (T), and Q is source or sink(T1). The differential equation governing non-reactive transportin the subsurface is:

/@C@t

r qC r /DrC 2

subject to initial and boundary conditions, where C is the concen-tration of solute in the liquid phase (M L3), / is porosity (), D isthe local hydrodynamic dispersion tensor (L2 T1) usually denedas Di aijqj Dm where ai refers to the longitudinal and transversedispersivities (L) and Dm is the molecular diffusion coefcient(L2 T1), and q is the Darcy velocity (L T1) given by Darcys lawas q Krh.

The inverse problem aims at determining the unknown modelparameters by making use of the observed state data. In the earlydays of groundwater modeling, it was common to start with a priorguess of the model parameters, run the forward model to obtainthe simulated states, and then enter in a manual loop iterativelymodifying the parameters, and then running the forward model,until observed and simulated values were close enough so as to ac-cept the model parameter distribution as a good representation ofthe aquifer. This trial and error method falls into the scope ofindirect methods as opposed to the direct methods which donot require multiple runs of the forward model to derive the modelparameters [2] as will be discussed below.

1.2. Why is the inverse problem necessary?

Sagar et al. [3] classied the inverse problem into ve typesaccording to the unknowns, i.e., model parameters, initial condi-tions, boundary conditions, sources or sinks and a mixture of theabove. Most documented inverse methods fall into the rst type,

H. Zhou et al. / Advances in Wthat is, they try to identify model parameters, which contribute lar-gely to the model uncertainty due to the inherent heterogeneity ofaquifer properties. Parameter identication is of importanceconsidering the fact that no reliable predictions can be acquiredwithout a good characterization of model parameters. Parameteridentication is a broad concept here including not only the prop-erty values within facies but the facies distribution, or in otherwords, geologic features. The effect of geologic uncertainty ingroundwater modeling is examined, for instance, by He et al. [4]in a real case study. Furthermore, data scarcity deteriorates thecharacterization of the model parameters and raises the uncer-tainty. Besides estimating aquifer parameters, the inverse methodsalso play a critical role in assessment of uncertainty for the predic-tions. Furthermore, the inverse problem might be used as a guidefor data collection and the design of an observation network. Thereader is referred to Poeter and Hill [5] who discussed the benetsof inverse modeling in depth. In this work we are mainly con-cerned with the uncertainty introduced by the unknown modelparameters and thus the inverse methods that are used to charac-terize these parameters.

1.3. Why is the inverse problem difcult?

A problem is properly posed if the solution exists uniquely andvaries continuously as the input data changes smoothly. However,most of the inverse problems in hydrogeology are ill-posed andthey cannot be solved unless certain assumptions and constraintsare specied. Ill-posedness may give rise to three problems: non-uniqueness, non-existence and non-steadiness of the solutions,among which non-uniqueness is the most common. Non-uniqueness primarily stems from the fact that the number ofparameters to be estimated exceeds that of the available observa-tion data. Another reason is that the observations are sometimesnot sensitive to the parameters to be identied; in other words,the information content of the observations is very limited. For in-stance, hydraulic heads close to the prescribed head boundaries aremore inuenced by the boundaries than by the nearby hydraulicconductivities (i.e., the hydraulic heads are not so sensitive to theconductivities), and on the contrary, the hydraulic heads close tothe prescribed ux boundaries are determined to a large extentby the hydraulic conductivities nearby [6].

A series of suggestions have been proposed to alleviate the ill-posedness:

1. Reduce the number of unknown parameters, e.g., using zona-tion, or collect more observation data so that the numbers ofdata and unknowns are balanced.

2. Consider the prior information or some other type of constraintto restrict the space within which parameters may vary.

3. Impose regularization terms to reduce uctuations during theoptimization iterations [2].

4. Maximize the sensitivity of observations to model parameters,for instance, by designing properly the observation network.

5. Minimize the nonlinearity in the model equation. Carrera andNeuman [6] argued that working with the logarithm of hydrau-lic conductivity reduces the degree of non-convexity duringoptimization. An alternative is to infer hydraulic conductivityusing uxes rather than heads as done by Ferraresi et al. [7],since the relationship between hydraulic conductivity and uxis linear (Darcys law) while the relationship between hydraulicconductivity and head is nonlinear.

Detailed discussions on this subject can be found in [2,6,8,9]among others.

Besides the ill-posedness problem, computational burden is thesecond main hurdle for inverse problems [10]. There are several

Resources 63 (2014) 2237 23reasons for the high CPU time requirement. Since many inversemodels are iterative, the forward model has to be run many timesuntil an acceptable parameter distribution is obtained. The time

that model parameters be known over the entire domain. In such1

that piezometer heads have been measured at all nodes of the dis-

aterneeded to run the forward model grows exponentially with the de-gree of discretization and the level of heterogeneity. When multi-ple realizations are sought, CPU demand grows with the numberof realizations. For those methods that use gradient minimization,the sensitivity matrices of model variables on parameters are com-puted, which is very time consuming. A few measures to reducecomputational demand have been proposed, for instance, (a) cer-tain kernel techniques render it possible to select representativerealizations from a large ensemble so that the size of the ensemblecan be reduced (e.g., [11,12]); (b) upscaling can be performed priorto any forward simulation in order to reduce solution time (e.g.,[1315]); (c) use of parallel/distributed computing technologies(e.g., [16,17]); (d) use of efcient surrogate models to reduce thenumber of unknowns that must be computed at each time step,i.e., reduced-order ow modeling (e.g., [1820]).

The problem of scales is the third difculty to be confronted inthe application of the inverse method. Measurements from bore-holes (made in situ or in the laboratory), local pumping tests,and regional aquifer estimates are the three common scales[21,22] at which information is handled in aquifer modeling. AsEmsellem and De Marsily [8] pointed out, permeability is aparameter with no punctual value but with an average value in aregion of a given size. The support of the permeability measuredfrom the eld is normally smaller than the cell size of the numer-ical model. In practice, permeability should be upscaled to a scaleconsistent with that of the numerical model discretization, other-wise the forward model would be computationally very expensive.A variety of approaches to calculate the upscaled permeability orhydraulic conductivity are available (e.g., [2327]). Besides, thescale inconsistency between eld measurement support andnumerical model discretization extends also to the observations,which can be obtained at different supports, too.

1.4. Outline of the paper

Despite all sorts of difculties, many inverse methods have beenproposed to solve the inverse problem. In the present paper we donot intend to review all current inverse methods, since several oth-ers have reviewed the topic from different points of view (e.g.,[9,2831]). But rather, we would like to analyze the evolution ofthe inverse models, from the simple trial-and-error approaches ofyesterday to the sophisticated ensemble Kalman lters of today,pointing out the incremental improvements that happened in theway.

In the remainder of this paper we mainly focus on seven keytopics, as follows:

Section 2.1 discusses the direct method. Then, in the followingsections, we focus on the indirect approaches.

Section 2.2 shows a linear inverse method in which the ground-water ow model is solved by linearizing the partial differentialequation under certain assumptions. Its shortcoming motivatesdevelopment of the nonlinear inverse methods in which the for-ward problem is solved numerically. The inverse methods in theremaining sections all belong to the nonlinear type.

Section 2.3 highlights the importance of considering uncer-tainty and introduces the inverse method based on Monte Carlosimulation in which multiple plausible realizations are used torepresent the real system.

Section 2.4 discusses sampling the posterior distribution ratherthan minimizing an objective function as the solution to theinverse problem.

Section 2.5 focuses on integrating new observations sequen-

24 H. Zhou et al. / Advances in Wtially without the need to reformulate the problem. Section 2.6 discusses whether the prior statistical structure ofthe model should be preserved through the inversion algorithm.cretized domain, and for it to yield stable results, head measure-ments are needed everywhere for several orthogonal boundaryconditions in the sense explained by Ponzini and Lozej [35].

It is worthwhile mentioning that, recently, Brouwer et al. [38]proposed a direct approach called the double constraint methodto determine permeability. Although it is not strictly a direct meth-od, since it does not require having observations extensively overa case, the inverse problem can be simply formulated as K F(h), provided, that h is known exhaustively. (The boundary condi-tions, source and sink terms could also be identied if necessary[3].)

Although the theory is straightforward, it is virtually impossibleto obtain a realistic solution by solving the algebraic equationsresulting after the discretization of the inverse equation due tothe serious ill-posedness and the singularity of the matrices in-volved in the numerical formulation [34]. Somemodications wereproposed to cope with these difculties, such as, considering moreequations than there are unknowns to build an over-determinedsystem so that the effect of measurement errors is reduced [35];or, imposing a constraint on the objective function, which convertsthe inverse problem into a linear programming problem [2,36] or aminimization problem [37].

The main shortcoming of the direct method is that it requires Section 2.7 addresses the issue of multiGaussianity in inversemodeling and the difculties to get away from it.

In each section, we will introduce a typical inverse methodexplaining its principle, implementation details, advantages andshortcomings. Recent trends of the inverse modeling are summa-rized in Section 3. The paper ends with some conclusions.

2. Evolution of inverse methods

Many approaches have been proposed to solve the inverseproblem. Several comparison studies have been carried out to eval-uate their performances, among them Zimmerman et al. [32] andHendricks Franssen [33] both compared seven different inversemethods. The former work focused on the transmissivity estima-tion and subsequent forecast of transport at the Waste Isolation Pi-lot Plant (WIPP). The latter used a different set of seven inversemethods to characterize well catchments by groundwater owmodeling. Some methods, such as the maximum likelihood meth-od, the self-calibration method and the pilot point method wereanalyzed in both works. In this section we would like to trackthe evolution of inverse methods along with the objectives thatcope with the pitfalls of the inverse methods.

2.1. Direct or indirect approach?

Inverse methods fall into two groups: direct ones and indirectones [2]. Nowadays, only indirect methods are considered; how-ever, it was the direct method that, somehow, gave rise to the indi-rect one. The move from direct methods to indirect ones would bethe rst major evolution in solving the inverse problem, and forthis reason we start this analysis by recalling the direct methodand why it had to be discarded in favor of the indirect one.

The forward problem of groundwater ow can be expressed asFK h, where F is an equation such as Eq. (1) relating the sys-tem parameters (e.g., hydraulic conductivity K) to the model re-sponses (e.g., hydraulic head h). The forward problem requires

Resources 63 (2014) 2237the entire domain, the nal step of the method, computing thepermeabilities, uses a direct approach. The method assumes thatpairs of pressure/ow rates are available at a number of points in

aterthe domain. A guess of the spatial distribution of the permeabilitiesis made and two forward runs are performed, the rst one consid-ers the measured pressures as prescribed boundary conditions(disregarding the ow rate data), the second one uses the owrates as prescribed boundary conditions (disregarding the pressuredata), then the prior permeability guess is forgotten and new per-meabilities are computed directly at each block interface usingthe pressure gradients derived from the rst run and the ow ratesderived from the second one. The process is repeated, and eventu-ally the spatial distribution of permeabilities converges to a stableone. The method was compared in a synthetic example with the re-sults obtained by the ensemble Kalman lter yielding good results.Another direct inverse method was proposed recently [39], inwhich the hydraulic conductivity is determined using steady-stateow measurements with unknown boundary conditions. In thismethod the groundwater ow equation is solved analytically usinga Trefftz-based approximation, then a collocation technique en-forces the global ow solution. The method is mainly applied onhomogenous aquifers and also on heterogeneous aquifers with aknown prior distribution of hydraulic conductivities.

The virtual impossibility of having state observation data overthe entire domain gave rise to the indirect methods capable of han-dling limited numbers of observations. In the following, only indi-rect methods are discussed.

2.2. Linearization or not?

Kitanidis and Vomvoris [22] proposed the geostatistical ap-proach (GA) as a very clever way to reduce the number of unknownvalues and subsequently mitigate the ill-posedness problem. Con-ductivities are not the subject of the identication problem, butrather the parameters of the variogram that describe the spatialcorrelation of the conductivities. Once the variogram has beenidentied, conductivities are interpolated by kriging onto the mod-el cells.

The procedure of the GA can be summarized into two mainparts: structure analysis and linear estimation. Structure analysisconsists of three steps as follows:

1. Select a geostatistical model, e.g., decide a variogram functionand whether the model is stationary or not. The model structureis selected based on all available information.

2. Estimate the parameters characterizing the model structuresuch as trend (if any), variance and correlation ranges. The jointprobability function of log-permeability and head is assumedmultiGaussian, and the hydraulic heads are expressed as a lin-ear function of log-permeability, after linearizing Eq. (1). Then,the parameters of the geostatistical model (generally no morethan ve) are estimated through maximum likelihood. TheGaussNewton method is used to solve the iterative maximiza-tion of the likelihood function.

3. Examine the validity of model. The estimated structure is eitheraccepted or modied during the test (i.e., the variogram func-tion is changed, or anisotropy is introduced).

As soon as the geostatistical model is accepted, the log-perme-ability eld is estimated by cokriging, a best linear unbiased esti-mation algorithm, which is capable of providing estimations withminimum error variance. Later, Dagan [40] and Rubin and Dagan[41] proposed an extension of the GA using Bayesian conditionalprobabilities.

The advantages of the GA reside in two main aspects. First, it re-duces the number of the effective parameters to be estimated by

H. Zhou et al. / Advances in Wintroducing the concept of random function into the inverse prob-lem. In this way, the ill-posedness is alleviated since (a) the num-ber of unknown values is far less than the number of observationsand (b) the estimated parameters are independent of grid discret-ization. Second, this method is computationally efcient arisingfrom two facts: hydraulic head is obtained by a rst-order approx-imation of the ow equation instead of numerically solving it; alinear estimation (cokriging) is applied as soon as the geostatisticalstructure is identied with no iterative optimization involved, sav-ing CPU time to a large extent. The method was rst veried on aone-dimensional test and found stable and well-behaved [22].

Despite the several advantages of this method, we have to men-tion some shortcomings. First, the approximation of the hydraulichead after linearizing the ow equation is only valid if the log-con-ductivity exhibits a small variance. Hoeksema and Kitanidis [42]alleviated this problem by applying the method to a two-dimen-sional isotropic conned aquifer in which hydraulic heads are ob-tained by solving numerically the partial differential equation, andthen Kitanidis [43] further generalized it onto a quasi-linear ap-proach, which was applied to condition on concentration data ina steady-state ow by Schwede and Cirpka [44]. Second, the nalconductivity map is obtained by kriging, this has two implications:rst, and most important, the nal maps are smooth since theyrepresent an ensemble expected value of the random functionmodel, and therefore cannot represent a real aquifer, and second,since kriging only uses the covariances for the estimation, as soonas the heads cannot be approximated as a linear combination ofthe conductivities, the solution of the ow equation in the nalconductivity maps will not honor the measured piezometric heads.

The need to apply the inverse model to hydraulic conductivityspatial distributions with large heterogeneity forced moving fromthe linearized approaches to other approaches in which this linear-ization is not necessary. The inverse methods involved in the fol-lowing sections do not apply linearization of the forward problem.

2.3. Deterministic estimation or stochastic simulation?

A typical example of nonlinear inverse approaches is the maxi-mum likelihood method (MLM) developed by Carrera and Neuman[45]. The method is able to estimate simultaneously such parame-ters as hydraulic conductivity, specic storage, porosity, dispersiv-ity, recharge, leakage and boundary conditions by incorporatinghead and concentration measurements as well as prior information[46]. Zonation is employed to reduce the number of parameters tobe estimated, that is, hydraulic conductivities are assumed con-stant or vary gradually over large patches of the aquifer, thus,the number of unknowns is proportional to the number of patches.Then, the groundwater problem (Eq. (1), or (2) or both) is solvednumerically subject to initial and boundary conditions. Let x be avector of all the unknown parameters, yobsi be a vector of availablemeasurements of type i and yobs be a vector with all measurementsof all types, a set of optimum parameter estimates is obtained bymaximizing the likelihood Lxjyobs. Under the hypothesis that allthe data could be transformed to jointly follow a multiGaussiandistribution, the likelihood function can be expressed as follows:

Lxjyobs / exp 12

XNmi1

yi yobsi TC1y;i yi yobsi

( ); 3

where yi is a vector of computed model states (e.g., head and con-centration), Cy;i is the corresponding covariance of observation er-rors and Nm is the number of types of measurements. MaximizingLxjyobs is equivalent to minimizing 2 lnL, and the optimizationproblem turns to minimizing the objective function:

J XNmi1

yi yobsi TC1y;i yi yobsi : 4

Resources 63 (2014) 2237 25Iterative minimization algorithms are applied on the objective func-tion until certain convergence criteria are met. The uncertainty of

aterparameter estimates is evaluated by a lower bound of the covari-ance matrix. Note that a regularization term can be included inthe objective function above in order to ensure stability of the opti-mization problem [46]. The objective function becomes then

J XNmi1

yi yobsi TC1y;i yi yobsi x xpri

TC1x x xpri; 5

where xpri is the prior model parameter vector and Cx is the covari-ance of x.

One of the important features of using zonation is that the num-ber of unknown parameters is reduced signicantly so that the po-tential ill-posedness problem is circumvented to some extent.Furthermore, the MLM might be used as a conceptual model iden-tication tool, i.e., to identify the best model structure among sev-eral alternatives; in this respect, Carrera [47] argued that thecriterion KIC proposed by Kashyap [48] is the most effective. Re-cently Riva et al. [49] demonstrated the discriminatory power ofKIC numerically concluding that the Bayesian model quality crite-rion KIC is well suited for the estimation of statistical data- andmodel-parameters in the context of nonlinear maximum-likeli-hood geostatistical inverse problems.

However, some limitations of the MLM are apparent. Althoughthe zonation scheme does help to reduce the number of the un-knowns, it causes over-smoothness, i.e., inherently heterogeneousgeologic properties are represented by a few patches of homoge-neous conductivities, and at the same time the zonation schememay introduce unacceptable discontinuities between zones.Although it is also true that the maximum likelihood formulationcould also be used to estimate a posterior covariancemap that couldbe used, together with the estimated map to generated stochasticrealizations. Furthermore, some zone discretizations may causebias, depending on the number of zones and measurements [50].Also, as stated earlier, the objective function is constructed underthe assumptionof amultinormaldistribution, i.e., theprior residualsand estimates follow amultiGaussian distribution. The implicationsof this assumption will be discussed further in Section 2.7.

The MLM was probably the rst widely successful inversemethod, it could incorporate many types of observations, it in-cluded a regularization term to prevent wide uctuations duringthe optimization phase, and, because it did not use any approxima-tion for the relationship between the state variables and the aqui-fer parameters, it yielded a zoned map of hydraulic conductivitiesthat reproduced very well the observed data. However, the MLMmethod produced a single map, too smooth to really describe theheterogeneity observed in nature.

Small scale variability had already being identied as one of theimportant factors controlling aquifer response. Recognizing this,De Marsily et al. [51] developed the pilot point method (PiPM) asa procedure to introduce more variability into an aquifer model ob-tained by kriging interpolation. Starting from a kriging map of thehydraulic conductivities, smooth as all kriging estimation mapsare, De Marsily et al. [51] position a ctitious datum where noobservation exists, and assigns to it a value so that when krigingis performed again with this new datum, the new kriging map pro-vides a better approximation to the observed piezometric headdata. New pilot points are introduced sequentially into the modeluntil there is enough heterogeneity in the model so as to reproducethe observed head data. This procedure had several advantages:the aquifer could be discretized at any scale, since after each iter-ation, (block) kriging was performed on the entire aquifer; the het-erogeneity was consistently treated throughout the process, sincethe same variogram model was used for all kriging estimations;

26 H. Zhou et al. / Advances in Wand, because of the way the procedure is implemented, the cti-tious data introduced at the pilot points were always within thelocal limits of variability of the variable as induced by the underly-ing random function model. The PiPM was successfully applied tomodel the Culebra formation overlying the Waste Isolation PilotPlan (WIPP) [5255]. The main problem of the PiPM was still that,at the end, there was only a single representation of the aquifer, asingle kriging map that, although more heterogeneous than thekriging map computed from the conductivity data alone, still wastoo smooth.

Moreover, the PiPM has been criticized recently by Rubin et al.[56] stating that rst it uses prior as a constraint rather than astarting point of parameter identication; second, it causes insta-bility and artifacts of the generated eld. Alternatively Rubinet al. [56] presented the anchored distribution method, and subse-quently Murakami et al. [57] and Chen et al. [58] applied thismethod for three-dimensional aquifer characterization at the Han-ford 300 Area.

An estimated map, let it be obtained by kriging, maximum like-lihood or Bayesian conditional probabilities, is an average map, andthe average does not necessarily represent reality. The same waythat the average outcome of throwing a dice (3.5) does not corre-spond to any of the dice face pips, the average of an ensemble ofpotential realizations, does not correspond with any realization.The smooth elds obtained by these methods fail to reect the lo-cal spatial variability and will necessarily fail in properly predictingmass transport (e.g., [59]).

It is then when the self-calibrated method is proposed (SCM)and the concept of inverse stochastic modeling starts being consid-ered. The idea is not to seek a single smooth representation of thespatial variability of hydraulic conductivity capable of reproducingthe observed piezometric head and/or concentration data, but togenerate multiple realizations, all of which display realistic pat-terns of short scale variability, all of which reproducing the ob-served piezometric head and/or concentration data. In the scopeof the stochastic inverse methods, non-uniqueness is not anymorea problem but an advantage, since all alternative solutions to theinverse problem are considered likely realizations of the aquiferheterogeneity, and all solutions are treated as an ensemble ofrealizations that must be further analyzed to make uncertainty-qualied predictions.

The concept of the SCMwasrst outlinedby Sahuquillo et al. [60]and then elaborated by Gmez-Hernndez [61] accompanied withtwo applications and an implementation program [6264]. TheSCM is based on the PiPM with the following rationale: instead ofstarting from a kriging map and introducing local perturbations byadding ctitious pilot points, lets start from multiple realizationsgenerated by a conditional simulation algorithm; and instead ofidentifying the optimal location of the next pilot point and introduc-ing them sequentially, locatewhat Gmez-Hernndez et al. [61] callmaster points all at once (as many as two or three per correlationlength) and determine the values of the perturbations in a singleoptimization step. Master point locations can be randomly selectedand vary at different iterations. To understand the SCM, one has torecall that a conditional realization is the sum of a conditionalmean(krigingmap) plus a correlated residual map [65], what SCM does isto apply the pilot pointmethodwithmultiple points at once tomod-ify the conditionalmeanwith the objective that the new conditionalmean plus the correlated residuals would match the observationdata. By applying this optimization approach to all the realizationsof an ensemble, the SCM is capable of generating multiple realiza-tions, all of which are conditional to the state observation. Thus, itcomes the term inverse stochastic modeling.

With such an ensemble of realizations, it was possible to maketransport predictions in each of the realizations and collect all of

Resources 63 (2014) 2237these predictions to build a model of prediction uncertainty basedon realistic realizations.

MPiP

e esh pithe

aterIt was soon realized that the concept of the SCM could be imple-mented in the original PiPM, and it has been applied multiple timessince then (e.g., [6668]).

MLM

logKupdate

logK

logKprior

x

x

x

0

Measurement

Fig. 1. Schematic illustration of MLM, PiPM and SCM. Suppose the parameter to bperturbation, logKupdate logKprior DlogK. The PiPM adds a perturbation around eacSCM, which is modied by adding a perturbation that is computed by interpolating

H. Zhou et al. / Advances in WGmez-Hernndez [69] reviewed stochastic conditional inversemodeling showing the strengths of SCM. The SCM was extended toincorporate concentration data [70,71] for the characterization ofhydraulic conductivity, and also to incorporate breakthrough dataof both reactive and nonreactive data to characterize the spatialvariability of the sorption coefcient [72]. Recently, genetic algo-rithms have been coupled with the SCM to search for the optimallocations of the master points as well as the optimal perturbationat these locations [73,74]; Hendricks Franssen [75] applied SCM tointegrate the pattern information from remote sensing images;Heidari et al. [76] coupled SCM with ensemble Kalman lter toassimilate the dynamic data in real-time; Li et al. [77] appliedthe master points concept of the SCM into the ensemble patternmatching method that have a capability to preserve the geologicstructures as well as the static and dynamic data [78].

The MLM, the PiPM and the SCM are closely related in that theyfollow a very similar perturbation and updating scheme. The up-date process for the PiPM is illustrated in Fig. 1 of [54] and inFig. 1 of [66]. For the sake of completeness and comparison, wegraphically show a sketch of the updating algorithm for all thethree methods (Fig. 1). In the MLM, it is as if there is a pilot pointin each zone, and the perturbation in the pilot point extends uni-formly constant over the entire zone, in the PiPM, the pilot pointperturbation dies out with the correlation length of the randomfunction model, and in the SCM, the perturbation of the entire eldis obtained by kriging the perturbations in each master point. Allthree methods seek nding those perturbations that added to theinitial guess will result in a new eld that is conditional to the ob-served state data.

It is worth to point out that the widely used program PEST[79], a model-independent non-linear parameter estimationprogram, which is also based on the minimization of an objectivefunction, could be framed in the family of PiPM-based methods.In this approach, the regularized inversion is commonly used whenthe model has many different parameters or when a number ofmodels are simultaneously simulated (e.g., [8082]). In the pilot

MCS

x x

x x

x x

timated is the log hydraulic conductivity. The prior guess is updated by adding alot point sequentially. A seed logK eld of the realization ensemble is shown for theperturbations at the master points.

Resources 63 (2014) 2237 27point implementation of the PEST code, parameters are estimatedat the predened pilot point locations and then spatial interpola-tion is used to complete the eld; then, a regularization term is in-cluded in the penalty function to control the stability anduniqueness of the solution of a highly parameterized model [80].The truncated singular-value decomposition and Tikhonov regular-ization (i.e., hybrid subspace) schemes [83] have been proposedand integrated into this approach, which was subsequently ex-tended to account for the uncertainty of estimated parameterwithin a Monte Carlo framework in the so-called subspace MonteCarlo technique [84,85]. Note that, Moore and Doherty [86,87]show that the regularized inversion method produces elds rela-tively smoother or simpler than the true conductivity eld. Clearly,this regularized inversion method, like the PiPM, only optimizes afew parameters (simple model) in the groundwater ow model,which is commonly insufcient for transport predictions such astravel times that require a higher level of system detail, as demon-strated in the case study by Moore and Doherty [86]. As a conse-quence, optimization of the whole aquifer, with millions ofdegrees of freedom (complex model), in a stochastic frameworkis warranted. The issue of simplicity versus complexity in modelconceptualization has already been subject of debate betweenhydrogeologists (e.g., [80,8893]).

An alternative to the Monte Carlo approach within the frame-work of maximum likelihood is a moment equation based inversealgorithm as proposed by Hernandez et al. [94,95]. Optimum unbi-ased estimates of model states such as hydraulic head and ux areobtained by their rst ensemble moments. Additionally the ap-proach is able to provide the second order moments which canbe used to measure the estimate uncertainty of model parametersand states. The moment equation inverse algorithm is applied atthe Montalto Uffugo research site (Italy) by Janetti et al. [96]. Themethod has been extended from steady state ow to transient ow

probaSup

ateraussian distribution with mean l and variance Cx; x Nl;Cx,with a prior probability density given by

px / exp 12x lTC1x x l

: 6

Assuming that the discrepancies between observed state variablesyobs and their corresponding model simulations ysim Fx is alsomultiGaussian with error covariance Cy, the joint probability distri-bution of yobs given x is,

pyobsjx / exp 12yobs FxTC1y yobs Fx

: 7

Using Bayes theorem, the posterior distribution of x given theobservations yobs can be written as

pxjyobs1cpx pyobsjx

/ exp 12xlTC1x xl

12yobsFxTC1y yobsFx

8

with c being a normalization constant. The Markov chain MonteCarlo method (McMC) [98100] is suitable for drawing samplesfrom the posterior conditional distribution pxjyobs. If a suf-ciently large chain is generated following the procedure describedbelow, the chain will converge so that its members will be drawnfrom the posterior conditional distribution. The procedure ofMcMC is the following [101].

1. Initialize the rst realization x.2. Update x according to the MetropolisHastings rule:

Propose a candidate realization x conditional on the lastrealization by drawing from the transition kernelx qxjx.

Accept the candidate x with probability minf1;ag and

a pxjy

pxjy qxjxqxjx : 9

3. Loop back to the second step.

The two critical points on the McMCmethod are the selection ofthe transition kernel qxjx and how to compute the acceptanceprobability a. The rst attempt to apply the McMC in hydrogeologyis by Oliver et al. [100] who generated permeability realizationsconditioned to variogram and pressure data using a local transitionkernel. This local kernel only modies a single cell in the realizationof x when making the transition from x to x. Such a localized per-turbation, specially if the aquifer discretization is large, makes themethod quite slow, since after each proposal there is a need to eval-uate the state equation (groundwater ow model) and to decide

whethglobalthe aqoptimization problem, but rather to sampling a multivariatebility distribution.pose that model parameter x is characterized by a multiG-[97] and from model state prediction to model parameter identi-cation [49]. Compared with the SCM, the moment equation meth-od is computationally efcient but it only provides a single smoothestimate of the random functions in that it utilizes the conditionalmean as the estimate.

2.4. Minimization or sampling?

Up to here all inverse methods discussed are based on the min-imization of an objective function that measures the mismatch be-tween the simulated state and the observed values. However, thereare alternative ways to achieve the same results without resortingto an

28 H. Zhou et al. / Advances in Wer the proposal is accepted or not. If the transition kernel is, producing a new realization which changes in every cell ofuifer, the probability that the candidate is rejected is quitehigh. An alternative is to use a block kernel in which the proposedrealizations differs from the previous one only over a certain regioninside the aquifer [102]. The so-called blocking McMC gives betterresults than either the local or global transition kernels. If, in addi-tion, the evaluation of the state of the system, for the purposes ofcomputing the acceptance probability, is made on a coarse scalewith the aid of upscaling, and only the high acceptance probabilitymembers are submitted to the ne scale evaluation, the conver-gence rate of the chain will be improved.

As mentioned, the McMC is not an optimization algorithm, itaims at generating multiple independent realizations by samplingfrom the posterior parameter distribution conditioned on theobservations. It is also important to notice that, since the posteriordistribution is built from the prior parameter distribution, the real-izations generated are consistent with the prior model. We will re-turn later to the issue of whether it is important or not to preservethe prior model structure throughout the inversion process inSection 2.6.

The original McMC is computationally demanding since eachproposed realization is subject to forward simulation that involvessolving a partial differential equation over a large domain and, pos-sibly, a long time period. Furthermore, these proposed realizationsare not necessarily accepted, which generally requires that a largenumber of candidate realizations have to be generated and evalu-ated. A lot of work has been devoted to increase the efciency ofMcMC by means of increasing the acceptance rates or reducingthe dimensionality of the forward simulation. A limited-memoryalgorithm has been used to accelerate sampling using low-dimen-sional jump distributions [103]. A two-stage McMC has been pro-posed to improve the efciency of McMC [104,105], in which nescale simulations are performed only if the proposal at the coarsescale is accepted. Another drawback of McMC is related with thelow mixing speed of the chain, in other words, the McMC shouldsample from the entire posterior distribution, but it takes quite along chain until this happens [102,106,107].

2.5. Real time integration or not?

The trajectory of inverse modeling up to here shows quite alarge evolution from the initial methods. At this stage, there arestill two main problems. The rst, and most important one, isCPU requirements; the second is the need to reformulate the prob-lem from the beginning if new data in space or time are collected.Inverse stochastic modeling supposed a big leap in aquifer charac-terization, but, in essence, it was equivalent (from a computationalpoint of view) to performing inverse modeling seeking a single bestestimate, but as many times as realizations were needed. It wasnecessary to nd an alternative capable of generating multipleconditioned realizations of conductivity in a more efcient man-ner. If this could be achieved, it would be interesting that as newdata are collected, as it happens in any monitoring network, thenewly collected piezometric heads or concentrations could beincorporated into the inverse model naturally without any modi-cation of the algorithm. The ensemble Kalman lter (EnKF) is anexample of such methods capable of it [108,109].

The EnKF is based on the Kalman lter, a data assimilation algo-rithm for systems in which the relation between model parametersand states is linear [110]. This linearity renders an exact propaga-tion of the covariance with time. However, the equations depictinggroundwater and solute transport model are nonlinear (Eqs. (1)and (2)), which prevents the computation of the covariance evolu-tion in time. As a solution to this problem, the extended Kalmanlter was proposed, the nonlinear function is approximated line-

Resources 63 (2014) 2237arly by a Taylor-series expansion and this linearization is usedfor covariance propagation. The problem with the extended Kal-man lter is that the covariance approximation deteriorates as

The inverse methods based on optimization attempt to mini-mize the deviation between predicted states and observation data,disregarding, sometimes, the prior model used to generate the ini-tial guess elds from which the optimization starts. Kitanidis [135]stated that the degree of data reproduction is a poor indicator ofthe accuracy of estimates questioning whether seeking the bestreproduction of the observed data justies any departure from

atertime passes, especially in highly heterogeneous elds for which theTaylor-series approximation is poor. The method is also very time-consuming when the aquifer is nely discretized [111].

The EnKF circumvents the problem of covariance propagation intime by working with an ensemble of realizations. The forwardproblem is solved on each realization, and the ensemble of result-ing states is used to compute the covariance explicitly. This is oneof the reasons that the EnKF is computationally efcient. Anotherreason resides in that the EnKF is capable of incorporating theobservations sequentially in time without the need to store all pre-vious states nor the need to restart groundwater simulation fromthe very beginning.

The theory and numerical implementation of the EnKF is de-scribed extensively in [111,112]. Here we will only recall the basicsof the EnKF. The EnKF deals with dynamic systems, for which ob-served data are obtained as a function of time and used to sequen-tially update the model. The joint state vector xi, for realization i,including both the parameters controlling the transfer functionand the state variables, is given by:

xi A

B

i

a1; a2; . . . ; aNT

b1; b2; . . . ; bNT !

i

; 10

where A is the vector of model parameters such as hydraulic con-ductivities and porosities, and B is the vector with the state vari-ables such as hydraulic heads and concentrations. The size of thestate vector ensemble x is determined by the number of grid cellsin which the domain has been discretized (N) and the number ofrealizations in the ensemble (Nr), i.e., x x1; x2; . . . ; xNr . Note thatthe boldface indicates the vector ensemble.

The EnKF consists of two main steps: forecast and update. Theforecast step involves the transition of the state vector from timet 1 to time t, i.e., xt Fxt1, where F is the transfer function.Normally this transfer function leaves the model parameters un-changed and forecasts the state variables to the next time stepusing the groundwater model (Eqs. (1) and/or (2)). After observa-tion data are collected the state vector is updated by Kalmanltering:

xut xft Gtyobst eHxft ; 11where xut is the joint vector ensemble with the updated states attime t and xft is the vector ensemble with the forecasted states;yobst is the observation at time t; e is an observation error with zeromean and covariance R;Gt is the Kalman gain, derived after theminimization of a posterior error covariance,

Gt PftHTHPftHT R1

; 12it multiplies the residuals between observed and forecasted valuesto provide an update to the latter; H is the observation matrix; Pft isthe ensemble covariance matrix of the state xft computed through

Pft 1

Nr 1 xft xft xft xft

T; 13

where xft is the ensemble mean, xft 1Nr

PNri1x

ft;i; and x

ft;i is a realiza-

tion of the ensemble of state vectors. It is worth noting that we donot have to compute the whole covariance matrix explicitly becausewe can compute directly PftH

T and HPftHT taking advantage of the

fact that most of the entries of H are zeroes.Most inverse methods need to store the previous states when

conditioning to new observed data, and the forward simulationhas to be restarted from the initial time until the time when thenew observation data are collected. On the contrary, the EnKF is

H. Zhou et al. / Advances in Wable to assimilate the real time observation data storing only thelatest state. Once the updating is done, the forward model is runfrom the last time step, and the assimilated observation data canbe discarded. A new forecast is performed, new data collected,and updated repeated. This is one of the reasons why the EnKF iscomputationally efcient. The sequential data assimilation schemeof the EnKF is shown in Fig. 2. The EnKF has been widely applied asa data assimilation tool in diverse disciplines such as oceanogra-phy, meteorology and hydrology (e.g., [113118]).

Notice that the EnKF is neither an optimization method nor asampling one, it is a data assimilation lter based on the minimiza-tion of a posterior covariance. For this reason, the EnKF is optimalwhen parameter and states are linearly related and follow a mul-tiGaussian distribution [119]. However, in hydrogeology, hydraulicconductivity is likely not to be properly modeled as multiGaussian,and even if it were, the states (heads and concentrations) wouldnever be linearly related to the parameters or propagate in timewith a linear transfer function. Some work has been done attempt-ing to circumvent the problem of nonGaussianity, but this issuewill be discussed later in Section 2.7.

Besides the issue of multiGaussianity, some disadvantages re-lated with the EnKF include: (a) underestimation of model variabil-ity, especially when the ensemble size is small and the parametereld is highly heterogeneous; (b) non-physical and spurious up-date of state vectors. These disadvantages can lead to lterinbreeding or divergence. To address this problem, several regular-ization strategies are proposed such as covariance ination [120],distance-based covariance localization [121,122], adaptive localiz-tion with wavelets [123], the use of damping factors [124] and theapplication of the conrming approach [118]. Wen and Chen [125]addressed some practical issues in applying EnKF, such as non-lin-earity via iteration, the selection of initial models, and nonGaussianity.

The EnKF has been successfully applied for building geologicalmodels conditioned on piezometric head data (e.g.,[17,114,124,126129]). With regard to conditioning on concentra-tion data, Huang et al. [130] applied the EnKF to update a conduc-tivity eld by assimilating both steady-state piezometric headand concentration data; Liu et al. [131] simulated multipleparameters (e.g., hydraulic conductivity, dispersivities, mobile/immobile porosity) for the MADE site; Li et al. [132] applied EnKFto jointly calibrate hydraulic conductivity and porosity by condi-tioning on both piezometric head and concentration data in afully transient ow; Schniger et al. [133] assimilated the nor-mal-score transformed concentration data to calibrate conductiv-ities. Jafarpour and Tarrahi [134] assessed the performance ofEnKF in subsurface characterization accounting for uncertaintyin the variogram.

2.6. Preserve prior structure or not?

xut-1 xft xut

yobst

......xft+1

yobst+1

......

Fig. 2. Workow of the EnKF.

Resources 63 (2014) 2237 29the prior model. There has been some debate on whether the priormodel structure should be preserved through the inverse modelingprocess, or, on the contrary, the optimization process may allow

the nmodel as driven by the need to reproduce the observed states. The

tributions in order to achieve the match to the observed data. Re-call tmodiprobabilities is performed by means of a single parameter rD that is

aterbest option, probably, lies in between; the prior model should betaken into account and should be used as a regularization element,while the new data should allow to introduce some deviations onthe prior model when this is the only way to approximate them. Inthis respect, Neuman [2] proposed the multiple-objective algo-rithm in which the model parameters are constrained not onlyby the minimization of the reproduction error but also by a phys-ical plausibility criterion. A similar strategy is used by Carrera andNeuman [45] and Medina and Carrera [46] in the MLM method, inwhich prior information is combined into the likelihood functions,or by Alcolea et al. [66] in the PiPM, in which a regularization termis added to the objective function.

There are methods which, by construction, will produce realiza-tions consistent with the prior model structure, such as the McMC,in which the prior model is implicitly built into the denition ofthe posterior conditional probability distribution from which thechain of realizations is drawn.

There are two other methods that, by construction, will pre-serve the prior model during the inversion process, thus ensuringthat the parameter distributions are physically plausible at theend of the inversion process: the gradual deformation method(GDM) and the probability perturbation method (PrPM).

The GDM method, as initially proposed by Hu [136], is based inthe successive linear combination of pairs of realizations. A singleparameter controls this linear combination, the value of which iscomputed by a simple optimization procedure. The optimal valueproduces a linear combination that improves the reproduction ofthe observed state. In addition, if both realizations are multiGaus-sian with the same mean and covariance it is easy to show that,after the linear combination, the resulting realization will also bemultiGaussian with the same mean and covariance. This pairwisecombination is repeated until an acceptable match to the observeddata is attained. This simple, but extremely effective, approach,which only worked for multiGaussian elds was soon extendedto work with other random function elds in conjunction withsequential simulation algorithms. In sequential simulation algo-rithms, each node of the realization is visited sequentially, a ran-dom number is drawn and a nodal value is obtained from thelocal conditional distribution, which has been computed accord-ingly to the random function model. There are sequential simula-tion algorithms to generate multiGaussian realizations (e.g.,[137]), realizations with arbitrary indicator covariance functions(e.g., [138]) or realizations based on the multiple-point patternsderived from a training image (e.g., [139]). In all cases, it all reducesto mapping a set of independent uniform random numbers into arealization of correlated values using as a transfer function the lo-cal conditional probabilities computed according to Bayes rule.When the GDM is applied to generate realizations of independentuniform random numbers, the resulting realizations will always beconsistent with the prior random function model, which was usedto compute the local conditional probabilities.

The attractiveness of GDM is that each iteration is a simple opti-mization step, and that it preserves the prior spatial structure. TheGDM algorithm can be summarized as follows:

1. Generate two independent Gaussian white noises, z1 and z2,with zero mean and unit variance. The two noises are combinedto form a new Gaussian white noise vector z with zero meanand unit variance according to

z z1 sinqp z2 cosqp; 14whq al set of realizations to depart (drastically) from the original30 H. Zhou et al. / Advances in Were q is a deformation parameter ranging from 1 to 1. If0; z is the same as z2 and if q 1=2; z is the same as z1.subject to optimization. This parameter can be interpreted as thedegree of perturbation needed to apply to the seed realization inorder to match the state data, if rD is close to zero, the actual real-ization gives a good reproduction of the state data and there is noneed to change anything, if rD is close to one, the current realiza-tion is far from matching the observation data and there is a needto generate another realization independent of the previous one,any value in between would generate a realization that representsa transition between these two independent realizations. Caers[145] compared the performances of the GDM and the PrPM in sev-eral simple examples from such aspects as preservation of prior

struct

Thebationhat the GDM freezes the probability distributions andes the random numbers. The perturbation of the conditionalNote that more than two noises could be combined to increasethe convergence rate ([140,141]).

2. The random vector z, is transformed into a uniform white noisevector u G1z, with G being the Gaussian cumulative dis-tribution function, and u is used with a geostatistical sequentialsimulation algorithm to yield a realization x; x Sz.

3. Run the forward model F (e.g., Eq. 1) on the generated prop-erty realization to obtain the simulated model responses, suchas ow rates and hydraulic heads.

4. An objective function measuring the mismatch between thesimulated model response and the observed data is built as

Jq Xni1xiFxqi Fxqobsi

2; 15

where xi is a weight and n is the number of observed data.5. Minimize the objective function to obtain the optimal q, and

update the random vector z.6. The updated random vector z will replace the previous z1 and

will be combined with a newly generated vector z2 to constructa new random vector. Then loop back to the second step untilall the observed data are matched up to some tolerance.

The drawback of the GDM is related to the convergence rate. LeRavalec-Dupin and Noetinger [141] found that the convergencerate is strongly inuenced by the number of realizations whichare combined in each iteration. Caers [142] proposed an efcientgradual deformation algorithm by coupling the traditional GDM,multiple point geostatistics and a fast streamline-based historymatching method with the aim to reduce CPU demand for param-eter identication in highly heterogeneous reservoir. Another crit-icism to GDM has been whether it generates realizations spanningthe entire space of variability coherent with the observed data; inthis respect, Liu and Oliver [143] assessed the performance of theGDM through a one-dimensional experiment; after comparisonof the GDM with other inverse methods, such as the McMC, theyconcluded that the GDM is able to produce reasonable distribu-tions of permeability and porosity. Wen et al. [144] comparedthe SCM and a specic variant of GDM that they call geomorphingapplied to a binary aquifer (sand/shale).

Another method that attempts to preserve the prior structuralmodel is the probability perturbation method [142]. The probabil-ity perturbation is also based on the sequential simulation algo-rithm, therefore, it will preserve the random function model thatis implicit to the algorithm used. Given a xed random path to visitthe nodes of the aquifer, a xed set of random numbers, and a xedset of conditional probability distributions, the PrPM will freezethe random numbers and perturb the conditional probability dis-

Resources 63 (2014) 2237ure and accuracy of posterior sampler.PrPM method has been extended to allow different pertur-s in different geological zones of the aquifer, that is, rD is

aterallowed to vary piecewise within the aquifer according to pre-de-ned zones [146]. The PrPM was initially applied to categoricalvariables, and later it was extended to continuous ones [147]. Hoff-man et al. [148] applied PrPM in a real oil eld. The PrPM has beenmostly used in petroleum engineering although recently it hasbeen applied in groundwater modeling, e.g., combined with a mul-tiple-point geostatistical method to locate high permeability zonesin an aquifer [149].

Before Caers proposed the PrPM, the idea of perturbing proba-bilities had already been proposed by Capilla et al. [150] althoughin a slightly different context. They used the concept of the SCMmethod, but instead of working with the conductivity values di-rectly, they transformed these conductivity values onto cumulativeprobabilities using the local conditional probability distributionsobtained by kriging (the probability eld of [151]). The type of kri-ging used could be indicator kriging and could incorporate softconditioning data, and therefore, the spatial structure associatedto such type of kriging would be preserved through the optimiza-tion process. Once the probabilities had been computed, the SCMmethod would be applied to seek the best spatial distribution ofprobabilities that when backtransformed onto conductivitieswould result in the best match to the observed state data. Later,to improve its efciency, the optimization step of the probabilityelds was combined with the GDM by Capilla and Llopis-Albert[152].

The problem associated to these latter methods is: what if theprior model is not correct? What if the prior model implies an iso-tropic spatial correlation, but, in reality conductivities are highlyanisotropic with channels of high permeability and quasi imper-meable barriers? Some studies have analyzed the impact of awrong a priori model choice, for instance, Kerrou et al. [153] ap-plied the SCM method on a uvial sediment aquifer in a steady-state ow with a wrong prior model (i.e., multiGaussian insteadof non-multiGaussian) and concluded that the channel structurescannot be retrieved, even when a large number of direct and indi-rect data are used for conditioning; Freni and Mannina [154] ana-lyzed the impact of different a priori hypotheses and found thatimproper assumptions could lead to very poor parameter estima-tions; Li et al. [155] and Xu et al. [129] assessed the performanceof normal-score EnKF (NS-EnKF) in a transient ow in non-multiG-aussian media and they argued that, if the monitoring net was de-signed properly, the localized NS-EnKF was able to identify thechannel structure even when an erroneous prior random functionmodel was used. A possible solution to account for the prior modeluncertainty is to use multiple prior models as done by Suzuki andCaers [156] although at a very large computational demand.

2.7. MultiGaussian or not?

Ever since the publication of the seminal paper on stochastichydrogeology by Freeze [157], hydraulic conductivity has beenassumed to follow a univariate lognormal distribution. Hisassumption was based on experimental data, and it was latercorroborated by Hoeksema and Kitanidis [158] who analyzed thehistogram and covariance of hydraulic conductivity data from 31regional aquifers to conclude that, indeed, hydraulic conductivityis best modeled by a logGaussian histogram. However, there arestill many cases, such as aquifers in uvial deposits, in which sev-eral highly contrasting facies coexist and in which conductivitiesare better characterized by a multimodal distribution. Multimodaldistribution also applies when one lumps data in a single sampleand tries to homogenize a composite medium [159]. But, the mostimportant issue is not whether the histogram of the conductivities

H. Zhou et al. / Advances in Wis normal or not, after all, it is always possible to apply a normal-score transformation to the data so that the transformed data fol-lows a Gaussian histogram; the most important issue is whetherthe best model to characterize the spatial continuity of hydraulicconductivity is the multiGaussian model or not. Applying a nor-mal-score transform to the data will render them univariate nor-mal, but it does not imply that the higher-order moments (theones controlling the continuity of the extreme values, or the curvi-linear arrangements of some conductivity values) should follow amultiGaussian model.

The nonGaussian models have been explored for some time(e.g., [153,160165]), and the dangers of using a multiGaussianmodel in aquifers with high continuity of extreme values were al-ready exposed by Journel and Deutsch [162] and Gmez-Hernn-dez and Wen [161].

Of all the methods discussed, all of those which are based on theminimization of the sum of square deviations have a tendency togenerate multiGaussian realizations, even if the seed realizationsprior to the start of the inverse procedure are nonGaussian. Basi-cally, all methods that use only moments up to the order two(covariance) in their formulation will behave in this way as a con-sequence that the multiGaussian distribution is the only one fullycharacterized by a mean value and a two-point covariance func-tion. This is the case of the GA, the SCM, the PiPM, or the MLM.Even the EnKF, which only uses the ensemble derived covarianceto update the realizations after each data collection stage, willend up with realizations with a multiGaussian avor even if theinitial ensemble is not.

Some of the methods discussed can handle non-multiGaussianpatterns of variability, such as the McMC, the GDM or the PrPM.It is apparent that those methods that can benet from techniquessuch as multiple point geostatistical simulation, which can gener-ate realizations of conductivity with realistic patterns according toa training image [166], are very promising. There have been someattempts to modify the EnKF to handle non multiGaussian ensem-bles. For example, Zhou et al. [167,168] proposed to transform boththe parameter and state variables from non-multiGaussian toGaussian ones through normal-score transformation, and then per-form the updating of transformed variables using EnKF; Sun et al.[169] coupled EnKF with Gaussian mix models to handle the nonG-aussian conductivity realizations; Sarma and Chen [170] developeda kernel EnKF to handle the connectivity of conductivity based onmultiple point statistics; Jafarpour and Khodabakhshi [171] sug-gested to update the mean of ensemble conductivity realizationsand then use this mean as soft data to regenerate the updatedmodel using multiple point statistics; Hu et al. [172] proposed toupdate the uniform-score random number that is used to drawthe local conditional probability in the multiple point statistics,using the EnKF; Zhou et al. [173] and Li et al. [78] presented anensemble pattern matching method, which is an update of EnKFmethod by using the multiple point statistics (i.e., pattern) toquantify the correlation between parameter and state variables in-stead of the two-point statistics (i.e., mean and covariance), andthus it has a capability to handle dynamic data integration in thecomplex formations such as the uvial depositions; Li et al. [77]further improved the computational efciency by coupling ensem-ble pattern matching method with the master points concept ofSCM method.

2.8. Evolution of inverse approaches to date

Table 1 summarizes the inverse methods discussed so far. Astime has passed, inverse models have, sequentially, gotten awayfrom the linearization of the state equation, become stochastic, at-tempted to preserve the prior structure (or at least introduce somecontrols which will give the prior structure information some

Resources 63 (2014) 2237 31weight during the inverse modeling process), and become capableof handling non-multiGaussian realizations. In our opinion the bestinverse model should be the one that is stochastic, is capable to

method are reported based on its most popular implementation.

ption M or Sa Referencesb

M Navarro [37]

M Kitanidis and Vomvoris [22] and Kitanidis [43]M Carrera and Neuman [45]M De Marsily et al. [51], RamaRao et al. [54] and Alcolea et al. [66]M Gmez-Hernndez et al. [61] and Hendricks Franssen [210]S Oliver et al. [100]M Hu [136]M Caers [211] and Hoffman and Caers [146] Evensen [108,112]

ntonal

ater Resources 63 (2014) 22373.1. Integrating multiple sources of information

Direct measurements of parameters of interest (usually knownas hard data) represent the rst constraint that the model mustmeet. These data are easily handled by standard geostatistic meth-ods [175]. Then, there are soft data, that is, indirect measurementsof the parameters. Recent developments in physical and geophys-ical techniques provide indirect measurements that are non-line-deal with multiple sources of state data governed by a complexstate equation, is not limited to multiGaussian realizations, canweight in prior information, and can generate multiple realizationsin an efcient manner. At the same time we have to mention thatsome preliminary methods in the evolution of the inverse model-ing are revisited and modied to propose new approaches, for in-stance, a variant of the direct method [39] and a zonation-krigingmethod [174].

3. Recent trends of inverse methods

The methods discussed so far have already been thoroughlytested and their advantages and pitfalls are well known. In the lastfew years, new issues have been brought into the inverse modelformulation that we would like to mention next.

Table 1Details of the inverse methods mentioned in the paper. The attributes of each inverse

Linearization Stochastic Structure preservation Gaussian assum

Direct methodYes No No

Indirect methodGA Yes No No YesMLM No No No YesPiPM No Noc No d

SCM No Yes No d

McMC No Yes Yes NoGDM No Yes Yes NoPrPM No Yes Yes NoEnKF No Yes No Yes

a Minimization of an objective function (M) or Sampling from a distribution (S).b References to original work and to main improvements.c In its inception PiPM pursued a single aquifer map, but it was later converted od Although implicitly there is no multiGaussian assumption in its formulation, the

formulated.

32 H. Zhou et al. / Advances in Warly related to the parameter of interest and that should be usedalso to constrain our aquifer model. Examples of these techniquesare ground-penetrating radar [176,177], time-lapse electrical resis-tivity [178], 4D-seismic [179], spatial altimetry [180], and remotesensing [181,182]. It is worth noting that certain techniques (e.g.,remote sensing) are able to provide information over a large arearather than at quasi-point scale. Besides the hard and soft (geo)physical measurements, state data other than hydraulic heads orow rates should be used to infer aquifer parameters, for instancepeak concentration arrival times [183] or groundwater ages [184].

Two other types of data have been used to identify aquiferstructure, i.e., water table uctuations due to tides, and connectiv-ity data. Tidal induced water table uctuations carry informationabout the aquifer properties in coastal aquifers. Head uctuationshave been used to identify possible preferential ow paths be-tween the sea and the coastal aquifer [185187]. The other rarelyapplied constraint is connectivity, which is found to play a criticalrole in transport modeling. In a synthetic example, Zinn andHarvey [164] demonstrated how the same conductivity valueswhen rearranged in space to induce different connectivity patternshave very different ow and transport behavior. Some indicatorshas been proposed to measure the connectivity (e.g., [188190]),and some attempts to include connectivity information in inversemodeling have been carried out [191,192].

3.2. Combining high order moments

To include curvilinear features in the spatial distribution of thehydraulic conductivities amounts to go beyond the two pointcovariance (a second-order moment) and to account for high ordermoments. A possible approach is with multiple point geostatistics[139,193]. We have already discussed how the GDM and the PrPMtake advantage of the sequential simulation methods based onmultiple point geostatistics to generate inverse conditional realiza-tions following the patterns extracted from a training image (suchas the one in Fig. 3). An alternative avenue is the use of spatialcumulants [194]. Other examples include those by Alcolea and Re-nard [191] who proposed a block moving window algorithm takingadvantage of multiple point simulations, Mariethoz et al. [195]who proposed an iterative resampling algorithm, and Zhou et al.[173] and Li et al. [78] who presented an ensemble pattern match-ing inverse method.

3.3. Handling multiscale problem

Observation data may be available at different support scales. If

a stochastic approach.realizations tend to become multiGaussian given the way the optimization model isthe aquifer model is discretized very ne to characterize the ex-treme heterogeneity of the geological parameters, it can be compu-tationally impossible to solve a stochastic inverse problem as westated in the Section 1.3. On the contrary if the model discretization

East

North

0 3000

240

-3.0

-2.0

-1.0

.0

1.0

2.0

3.0

Fig. 3. Conceptual model of a uvial deposited aquifer used as a training image bythe multiple point based simulation algorithms.

might lose some small scale but important features of the aquifer.

aterA scale balance must be reached. Some authors have combinedupscaling and inverse modeling to address the scale problem. Forinstance, upscaling is introduced into the McMC and forms a blockMcMC method [102,196]; upscaling is combined with EnKF toassimilation coarse scale observations and at the same time honorthe small scale properties [14,197]; a multiresolution algorithm isproposed by applying regression in the space of wavelet transfor-mation [20].

3.4. Parameterizing the conductivity eld

Reducing the number of unknowns of a heterogeneous conduc-tivity eld (i.e., parametrization) remains an active research area ininverse modeling. This approach usually consists of two steps: (a)reconstruct the hydraulic conductivity eld using a relatively smallnumber of parameters, (b) and then apply an optimization methodin the new parameterized space. The purpose of parameterizingthe conductivity eld is apparent: to mitigate the ill-posedness inthe inverse modeling.

The covariance-based principal component analysis (PCA) orthe KarhunenLoeve expansion are commonly used to representthe geological model in a reduced space for the multiGaussianmedia, (e.g., [18,198]). Jafarpour and McLaughlin [199] introduceda discrete cosine transformation to parameterize the conductivityeld. Sarma et al. [200] developed a kernel PCA to parameterize,not the covariance, but multiple point statistics of the conductivityeld. In this approach, the high-dimensional data is transformedinto a feature space by dening a kernel transformation function,and the parametrization is accomplished in the corresponding ker-nel feature space; a model is obtained by transforming back fromthe feature space to the model space (i.e., pre-image problem),which raises another non-linear optimization problem in this pro-cess (e.g., [201203]). Li and Jafarpour [204] proposed to use dis-crete wavelet transform to reconstruct the conductivity eld.Bhark et al. [205] developed a generalized grid-connectivity basedparametrization method to integrate dynamic data into geologicalmodels. More recently, there is a set of papers dealing with thereconstruction of conductivity eld by using sparse representativesand dictionaries (e.g., [206208]).

3.5. Assessing the uncertainty of prediction

Ensemble based inverse methods have a capability to assistassessing the uncertainty of prediction that is routinely requiredby the decision maker. Any method based on Bayes theorem def-initely is able to model a posterior probability. Examples of thismethod include traditional reject sampling [209], iterative spatialresampling [195] and McMC-based methods [100,102,191]. Allthe methods mentioned above could be formulated as resamplingmethods and are extremely computational expensive due to theiterative evaluation of the forward simulation on thousands of con-ductivity realizations. On the contrary, minimization-based meth-ods, such as the EnKF and the pattern-search based method (e.g.,[78,173]) are computational efcient and are able to assess theuncertainty of predictions using updated ensemble conductivities.However, both methods work with ensembles of realizations whatimplies that a large ensemble size is commonly required, resultingin a high computational cost, too.

4. Conclusionsis coarse in order to integrate the coarse scale observation data, it

H. Zhou et al. / Advances in WWe have given an overview of the evolution of inverse methodsin hydrogeology, i.e., how the algorithms have evolved during thelast decades to solve the inverse problem, from direct solutionsto indirect methods, from linearization to non-linearization ofthe transfer function, and from single estimate to stochastic MonteCarlo simulation. Furthermore, we consider a few issues involvedin solving the inverse problem, e.g., whether the multiGaussianassumption is appropriate and whether the prior structure shouldbe honored. Inverse models have gone a long way since theirinception, and they keep evolving with the aim of improving aqui-fer characterization while, at the same time, respecting all theinformation and data available.

Overall, the development of inverse methods shows somefeatures:

The goal of inverse problem is not just parameter identication,but also improved predictions.

Stochastic inverse methods are becoming the trend for the gen-eration of multiple realizations, which will serve to build amodel of uncertainty on both parameters and states.

There is a need for methods that are capable to generate geolog-ical models as diverse as possible while matching observed datato ensure that the uncertainty in the predictions is properlycaptured.

Multiple sources of observations are integrated in the inversemodeling, multiple scale problems are handled and multiplenew algorithms are introduced into the inverse modeling, forinstance, multiple point geostatistics and wavelet transform.

Acknowledgments

The authors gratefully acknowledge the nancial support by theSpanish Ministry of Science and Innovation through projectCGL2011-23295. We would like to thank Dr. Alberto Guadagnini(Politecnico di Milano, Italy) for his comments during the review-ing process, which helped improving the nal paper.

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