## 1. Introduction

[2] It has become common to treat spatially varying subsurface flow parameters as autocorrelated random fields [*Kitanidis*, 1997]. This, together with uncertainty in forcing terms (initial conditions, boundary conditions and sources), renders the groundwater flow equations stochastic [*Dagan and Neuman*, 1997]. The solution of such equations consists of the joint, multivariate probability distribution of its dependent variables or, equivalently, the corresponding ensemble moments [*Dagan*, 1989; *Gelhar*, 1993]. It is advantageous to condition the solution on measured values of the parameters at discrete point in space-time [*Rubin*, 2003]. The equations can then be solved by conditional Monte Carlo simulation or by approximation. A typical solution includes the first two conditional moments (mean and variance-covariance) of head and flux [*Zhang*, 2001]. The first moments constitute optimum unbiased predictors of these random quantities and the second moments are measures of the associated prediction errors [*Neuman*, 1997].

[3] The Monte Carlo approach is conceptually straightforward but has several drawbacks. It requires generating many independent realizations of the random input variables (parameter fields and forcing terms including initial conditions, boundary conditions and source terms) and solving the flow problem for each. It requires specifying the joint multivariate probability distribution of all random input variables which is difficult to do unless the variables (or their log transform) are assumed to be Gaussian. Conditioning the input variables on measured values of the state variables (which are related to each other in a nonlinear fashion) through linearization or quasi-linearization may cause uncertainty to be underestimated by orders of magnitude [*Carrera and Glorioso*, 1991]. Another alternative is to generate conditioned random head fields directly through inversion (calibration) of each realization of the input variables [e.g., *Gómez-Hernández et al.*, 1997]. Unfortunately, it remains unclear whether such realizations are in fact statistically independent and, more importantly, whether they span the entire range of admissible parameter distributions. Moreover, the computational effort required for the inversion of numerous (typically thousands) realizations may be very large. The number of realizations needed for convergence of the Monte Carlo approach is difficult to ascertain a priori so that one is generally not sure about the computational feasibility of the process till it has been actually completed. In summary, Monte Carlo methods tend to be computationally intensive and difficult to condition properly on observed system behavior [*Carrera et al.*, 2005].

[4] Difficulties with the Monte Carlo approach have stimulated an interest in alternative methods of solution, which allow one to compute the first two conditional moments directly by solving a corresponding system of moment equations [*Zhang*, 2001]. Our interest centers on nonlocal (integro-differential) formulations of such moment equations such as those originally introduced by *Neuman and Orr* [1993] and *Tartakovsky and Neuman* [1998]. These authors developed exact equations for the first two conditional moments of head and flux under steady state and transients flows in bounded, randomly heterogeneous saturated domains. To render these nonlocal moment equations workable, *Tartakovsky and Neuman* [1998] and *Guadagnini and Neuman* [1999a] approximated them recursively through expansion in powers of σ_{Y}, a measure of the conditional standard deviation of (natural) log hydraulic conductivity *Y* = ln *K*. Their recursive approximations are nominally limited either to mildly heterogeneous or to well-conditioned strongly heterogeneous media.

[5] *Guadagnini and Neuman* [1999a, 1999b] and *Ye et al.* [2004] developed recursive computational algorithms for the solution of nonlocal steady state and transient flow moment equations conditional on measured values of hydraulic conductivity. Notwithstanding their nominal limitation, these algorithms have yielded highly accurate solutions for complex flows in strongly heterogeneous media with unconditional log conductivity variances as high as 4. The transient algorithm of Ye et al. has outperformed the Monte Carlo method by a significant margin. It would therefore be of considerable interest to allow further conditioning of these stochastic moment solutions on measured values of state variables such as hydraulic head and flow rates. For this, however, it would be necessary to solve the stochastic moment equations in an inverse mode.

[6] Linearized stochastic inverse solutions based on cokriging were developed for steady state flow by *Kitanidis and Vomvoris* [1983], *Hoeksema and Kitanidis* [1984], *Dagan* [1985], *Rubin and Dagan* [1987], and *Gutjahr and Wilson* [1989]. As demonstrated by *Zimmerman et al.* [1998], these methods yield reliable parameter estimates for moderate variability but relatively poor estimates and unduly small estimation variances when variability or nonlinearity is pronounced. The difficulty stems in part from the failure of these approaches to (1) consider statistical moments containing products of randomly fluctuating quantities, (2) account for the nonlinear relationship between hydraulic conductivity and head, and (3) take advantage of a computational grid to resolve spatial variability. *Woodbury and Ulrych* [2000] introduced a linearized Bayesian geostatistical inverse approach coupled with a maximum entropy principle that eliminates the last deficiency (but not the first two) by resolving log transmissivity variations on a finite element grid, resulting in improved ability to deal with strongly heterogeneous media [*Jiang et al.*, 2004].

[7] One way to overcome all three shortcomings is to generate multiple realizations that honor measurements using approaches such as those described by *Sahuquillo et al.* [1992], *Gutjahr et al.* [1994], *LaVenue et al.* [1995], *RamaRao et al.* [1995], *Capilla et al.* [1997], *Gómez-Hernández et al.* [1997], *Oliver et al.* [1997], *Hanna and Yeh* [1998], *Hu* [2000], and *Franssen and Gómez-Hernández* [2002], some of which have been evaluated by *Zimmerman et al.* [1998]. These approaches require generating a large set of random inverse solutions via a Monte Carlo simulation process, which may consume a large amount of computer time and produce biased parameter estimates due to a possible lack of statistical independence between the solutions. Applying them to only a few realizations, as has been the practice to date, may yield plausible representations of reality which however are random and therefore nonunique.

[8] Another way to overcome all three of the aforementioned shortcomings, without resorting to Monte Carlo simulation, is to formulate a nonlinear inverse problem on the basis of fully consistent nonlocal ensemble moment equations on a computational grid as we propose in this paper. The approach we propose is additionally unique in providing not only estimates of unknown hydraulic as well as geostatistical (variogram) parameters and corresponding predictions of hydraulic head and flux (through their conditional first moments) but also of second head and flux moments, which constitute measure of predictive uncertainty. A brief preliminary description of our proposed approach has been published earlier by *Hernandez et al.* [2002, 2003]. In the present paper we (1) develop the underlying idea, theory and computational algorithms in much greater detail than has been done in our preliminary report, (2) explore the ability of our approach to deal with strongly heterogeneous media; (3) present a detailed statistical analysis of estimation errors, (4) compare our estimates with theoretical error bounds; and (5) show how to estimate variogram parameters (in addition to log transmissivities) by relying on formal model discrimination criteria.