## 1. Introduction

[2] The DC resistivity survey is an inexpensive and widely used technique for investigation of near-surface resistivity anomalies. It recently has become popular for the investigation of subsurface pollution problems [*National Research Council* (*NRC*), 2000]. In principle, it measures the voltage generated by a transmission of current between electrodes implanted at the ground surface. Apparent (bulk or effective) electrical resistivity is then calculated and used to interpret subsurface anomalies.

[3] Classical formulas for determining apparent electrical resistivity assume homogeneity, and the potential field is smooth because of its highly diffusive nature. Consequently, conventional interpretations of electrical resistivity survey data have been virtually ineffective for environmental applications, where electrical resistivity anomalies are subtle, complex, and multiscale. To overcome these difficulties, a contemporary electrical resistivity survey has been designed to collect extensive electric current and electric potential data sets in multi-dimensions. Without assuming subsurface homogeneity, a mathematical computer model is employed to invert the data sets to estimate the resistivity field, using a regularized optimization approach [e.g., *Daily et al.*, 1992; *Ellis and Oldenburg*, 1994; *Li and Oldenburg*, 1994; *Zhang et al.*, 1995]. However, the general uniqueness and resolution of the three-dimensional electrical resistivity inversion have not been investigated sufficiently thus far [*Carle et al.*, 1999; *NRC*, 2000].

[4] While the physical process is different, the governing equation for electric currents and potential fields created in the electrical resistivity survey is analogous to that for steady flow in saturated porous media. The mathematical solution to the inversion of an electrical resistivity survey is therefore similar to that of a groundwater hydrological survey. Groundwater hydrologists and reservoir engineers have attempted to solve the inverse problem of flow through multidimensional, heterogeneous porous media for the last few decades [e.g., *Gavalas et al.*, 1976]. Extensive reviews on the inverse problem of subsurface hydrology and various solution techniques are given by *Yeh* [1986], *Sun* [1994], and *McLaughlin and Townley* [1996]. They concluded that prior information on geological structure, and some point measurements of parameters to be estimated are necessary to better constrain the solution of the inverse problem. A similar finding was also reported by *Oldenburg and Li* [1999] and *Li and Oldenburg* [2000] for the inverse problems in geophysics.

[5] Groundwater hydrologists also have used a multicomponent linear estimator (cokriging) to estimate the hydraulic conductivity field from scattered measurements of pressure head and hydraulic conductivity in saturated flow problems [*Kitanidis and Vomvoris*, 1983; *Hoeksema and Kitanidis*, 1984]. The popularity of cokriging is attributed to its ability to incorporate spatial statistics, point measurements of hydraulic conductivity, and hydraulic head into the estimation and to yield conditional mean estimates. Cokriging is also capable of quantifying the uncertainty associated with its estimate due to limited information and heterogeneity. *Kitanidis* [1997] articulated the differences between cokriging and the classical inverse methods in subsurface hydrology. Nevertheless, cokriging is a linear estimator and it is limited to mildly nonlinear systems, such as aquifers of mild heterogeneity, where the variance of the natural logarithm of hydraulic conductivity, σ_{lnK}^{2}, is less than 0.1. When the degree of aquifer heterogeneity is large (σ_{lnK}^{2} > 1), the linear assumption becomes inadequate. Therefore cokriging cannot take full advantage of the hydraulic head information to obtain a good estimate of hydraulic properties [*Yeh et al.*, 1996].

[6] To overcome this shortcoming, *Yeh et al.* [1995, 1996], *Gutjahr et al.* [1994], and *Zhang and Yeh* [1997] developed an iterative geostatistical technique, referred to as a successive linear estimator (SLE). In this technique, a linear estimator was used successively to incorporate the nonlinear relation between hydraulic properties and the hydraulic head. This method also employs a conditional covariance concept to quantify reductions in uncertainty due to the incorporation of subsequent information. *Yeh et al.* [1995, 1996] and *Zhang and Yeh* [1997] demonstrated that with the same amount of information, the SLE method revealed a more detailed hydraulic conductivity field than cokriging. *Hughson and Yeh* [1998, 2000] extended the SLE method to the inverse problem in three-dimensional, variably saturated, heterogeneous porous media. On the basis of the SLE algorithm, *Yeh and Liu* [2000] developed a sequential SLE technique for hydraulic tomography to process the large amount of data created by the tomography, and subsequently characterize aquifer heterogeneity. They investigated the effect of monitoring intervals, pumping intervals, and the number of pumping locations on the final estimate of hydraulic conductivity, and they established guidelines for a design of a hydraulic tomography test.

[7] In section 2 of this paper, we introduce the concept of stochastic representation of electrical resistivity tomography (ERT) inverse problems. In sections 3 and 4 we describe the development of a geostatistically based sequential SLE methodology for ERT inversion problems. Section 5 offers numerical examples that illustrate the usefulness of the new inversion approach, and describes the effects of geological structures on the layout of electrical resistivity surveys. The relation between the electrical resistivity and the moisture content of 25 soil cores were measured and analyzed for spatial variability in section 6. Impacts of spatial variability on the estimated changes in moisture content in the vadose zone, using ERT surveys were explored and discussed in section 7.