## 1. Introduction

[2] Large-scale inverse modeling has been drawing attention in the groundwater area mainly because of two facts. The first one is that engineering practices, such as remediation or deep injection projects, demand ever more accurate prediction of variables such as hydraulic head and solute concentration in groundwater systems. The other one is that current advances in technology allow the collection of even more measurements. Improvements in computational power hold the promise of a more detailed characterization of the physical properties of subsurface media. As an important category in interpretation of indirect measurements, various inverse modeling methods have been developed in the groundwater area since the 1960s, and they have been reviewed and generalized by *Yeh* [1986], *McLaughlin and Townley* [1997], *Zimmerman et al.* [1998], and *Carrera et al.* [2005]. Among all these methods, the Bayesian formulation followed by a quasi-linear solver proposed by *Kitanidis and Vomvoris* [1983], *Hoeksema and Kitanidis* [1984], *Dagan* [1985], and *Kitanidis* [1995] has been adopted in a number of applications. *Snodgrass and Kitanidis* [1997] used it to identity contaminant sources, *Michalak and Kitanidis* [2004] applied the method to estimate historical groundwater contaminant distributions, *Li et al.* [2005] estimated transmissivity and storativity distributions with transient hydraulic measurements, and *Cardiff et al.* [2009] employed the approach in steady state hydraulic tomography. In particular, *Nowak and Cirpka* [2006] utilized this method in a high-resolution hydraulic conductivity and dispersivity estimation problem with an efficient multiplication algorithm (complexity ) for Toeplitz matrices using fast Fourier transform (FFT). However, solving the Toeplitz system is still relatively computationally intensive, with complexity [*de Hoog*, 1987; *Ammar and Gragg*, 1987], and this fact affects in each iteration the efficiency of the line search, which on the basis of our experience, is a very important step to secure the global convergence.

[3] In the Bayesian formulation, inference is made through the posterior distribution of the parameters (i.e., conditioned on the measurements). The posterior distribution is composed of two parts, the likelihood of the measurements and the prior distribution of the parameters. The first part controls the extent of fitness to the data, and the second part controls the smoothness of the solution in the parameter space. The second part, the prior distribution, restricts the class of solutions that is considered possible. In this paper we reformulate the objective function so that we restrict the class of solutions through local variation of the parameters, and we pick out the solution that has the least local variation. This method is also a member of the Tikhonov regularization [*Tikhonov and Arsenin*, 1977], and it has been referred to as the Occam's inversion and has been widely used in geophysics [*Constable et al.*, 1987; *Gouveia and Scales*, 1997]. In our reformulation, a general framework for multidimensional aquifer characterization with consideration of structural anisotropy is proposed. In the end, the problem takes the form of a classical least squares (LS) problem with high sparsity that can be efficiently solved with a number of mature LS solvers, for example, with direct decomposition of the system or iterative methods such as LSQR [*Paige and Saunders*, 1982]. Compared with the classical setup where multiplication of the sensitivity of the observations with the covariance matrix raises the computational cost, this formulation is more efficient. We then apply this method to a laboratory-scale steady state hydraulic tomography problem, and the results are validated with various methods.