## 1. Introduction

[2] A fundamental Earth sciences problem is to determine the properties of an object that we cannot directly observe. A variety of geophysical tomography techniques have been developed to provide detailed subsurface information. One such technique, electrical resistance tomography (ERT), is a relatively recent geophysical imaging scheme that provides two-dimensional (2-D) and 3-D images of resistivity that are consistent with measurements made on an array of electrodes. With the increasing availability of computer controlled multielectrode instruments and robust data inversion tools, ERT is becoming widely available. The value of ERT for monitoring dynamic subsurface processes has promoted new applications in a wide range of environments [e.g., *Daily et al.*, 1987; *Park and Van*, 1991; *Daily et al.*, 1992; *Ellis and Oldenburg*, 1994; *Sasaki*, 1994; *LaBrecque et al.*, 1996, 1999; *Binley et al.*, 1996; *Morelli and LaBrecque*, 1996; *Park*, 1998; *Kemna et al.*, 2000; *Slater et al.*, 2000].

[3] The goal of any ERT inversion method is to calculate the subsurface distribution of electrical resistivity from a large number of resistance measurements made from electrodes. A deterministic inversion procedure searches for a model (i.e., a spatially varying distribution of resistivity) that gives an acceptable fit to the data and satisfies any other prescribed constraints. A common solution minimizes an objective function consisting of a regularized, weighted least squares formulation. Typically, the search is conducted using iterative, gradient-based methods [e.g., *Park and Van*, 1991; *Ellis and Oldenburg*, 1994; *LaBrecque et al.*, 1996].

[4] The ERT inversion problem is typically complicated by a nonlinear relationship between data and the inverted parameters, state-space dimensionality, under/over determined systems, noisy and dependent data, etc. Hence an exact inversion is rarely possible. It is common to use unrealistic simplifying assumptions to mitigate the severity of these problems when using classical optimization algorithms.

### 1.1. Previous Work: Stochastic Methods

[5] One alternative to the classical ERT inverse methodologies uses stochastic techniques that search for electrical resistivity models that best fit the collected electrical resistance measurements. The literature describes a variety of these methods and their application to geophysical problems. *Zhang et al.* [1995] suggest an inversion method that seeks to maximize a specified a posteriori probability density function of model parameters. In this case, maximizing the a posteriori density function is equivalent to minimizing the objective function in the classical inverse approach. *Yang and LaBrecque* [1999] proposed an alternate solution that extends Zhang et al.'s approach by allowing a more efficient estimate of the parameter covariance matrix.

[6] *Mosegaard and Tarantola* [1995] and *Mosegaard and Sambridge* [2002] describe an alternative stochastic inversion approach. The technique utilizes Markov Chain Monte Carlo (MCMC) methods, a class of importance sampling techniques that search for models that are most consistent with available data. In this approach, the inverse problem is formulated as a Bayesian inference problem. An importance sampling search algorithm is used to generate an empirical estimate of the a posteriori probability distribution based on available observations. Specifically, solutions are sampled at rates proportional to their posterior probabilities. This implies that models consistent with a priori information as well as observed data are sampled most often, while models that are incompatible with either prior information and/or observations are rarely sampled. This is the key difference between traditional Monte Carlo and MCMC: the former samples the space of possible models completely at random, while the latter moves through the models according to their posterior probabilities. This method yields an efficient sampling scheme that affords the user the flexibility to employ complex a priori information and data with non-Gaussian noise. Mosegaard and Tarantola show that this approach can be used to jointly invert disparate data types such as seismic and gravity data.

[7] *Kaipio et al.* [2000] describe the application of the MCMC approach to medical imaging problems using electrical impedance tomography (EIT, for the purpose of this paper, ERT and EIT are synonymous). They considered a variety of nondifferentiable priors including a minimum total variation prior, a second-order smoothness prior and an “impulse” prior that penalizes the L^{1} norm of the resistivity. Their approach estimates the posterior distribution of the unknown impedances conditioned on measurement data. From the posterior density, various estimates of the resistivity distribution and associated uncertainties are calculated.

[8] *Andersen et al.* [2001a] describe an MCMC geophysical approach for the Bayesian inversion of electrical resistivity data. They used random, polygonal models to represent the layered composition of the Earth, and demonstrate the performance of the method using field data. They analyze the posterior distribution by looking for the resistivity model that is most consistent with the data and comparing it to the estimated posterior mean model. They also estimate the variability of the transitions between Earth materials by comparing the standard deviation for each image pixel to its corresponding mean. *Andersen et al.* [2001b] describe another MCMC application aimed at the detection of cracks in electrically conductive media. Their approach assumes that the cracks are linear, nonintersecting and perfectly insulating. Using synthetic data, they demonstrate an updating scheme that assumes that the number of cracks is known a priori. Their approach estimates the posterior distribution of crack configurations and their associated variances.

[9] *Yeh et al.* [2002] describe a sequential, geostatistical ERT approach that allows inclusion of prior knowledge of general geological structures through the use of spatial covariance. Their method also uses point electrical resistivity measurements (well logs) to further constrain the solution. They use the successive linear estimator approach to find an optimal model that consists of the “conditional effective electrical conductivity” (CEEC). They define CEEC as the parameter field that agrees with electrical resistivity measurements at core sample locations and that honors electrical potential measurements. They also compute conditional variances to estimate the uncertainty associated with their optimal CEEC model.