## 1. Introduction

[2] There is growing interest in using geophysical methods, such as electrical resistivity imaging (ERI), to investigate systems where the evolution of subsurface properties through space and time is constrained by physical, chemical, or biologic processes, such as the infiltration of water or transport of solutes in porous media. Subsurface imaging using ERI, however, usually requires the solution of an ill-posed inverse problem. While there are a variety of approaches addressing this issue in the inverse theory literature, Tikhonov regularization [*Tikhonov and Arsenin*, 1977] is commonly applied to ERI as it readily allows spatial constraints, such as smoothly varying property variations, to be enforced in subsurface images [e.g., *Daily and Ramirez*, 1995; *LaBrecque and Yang*, 2001; *Kemna et al*., 2002]. The spatial constraints applied in these inversions are typically generalized filters selected independently from the underlying processes affecting the target resistivity distribution (Figure 1a). In the context of imaging solute transport, it is well known that this regularization can lead to imaging artifacts causing problems with mass recovery and poor spatial resolution [e.g., *Singha and Gorlick*, 2005].

[3] In contrast, an emerging approach known as coupled inversion explicitly takes advantage of the dependence of geophysical properties on subsurface processes by using geophysical measurements to calibrate the parameters of a hydrologic model [e.g., *Rucker and Ferré*, 2004; *Ferré et al*., 2009; *Hinnell et al*., 2010]. Figure 1b illustrates how the coupled hydrologic and geophysical models can be considered as a single model with hydrologic parameters as inputs and geophysical responses as outputs. An advantage of the technique is that the coupled process model may have only a few parameters that control the detailed spatial and temporal evolution of hydrologic state variables in the subsurface, which in turn control the geophysical response. Coupled inversion can therefore be viewed as an implicit form of regularization that enforces a physics-based constraint on the inversion through the physical process simulator, e.g., the flow and transport model. A disadvantage of the approach, however, is that poor results may be obtained if the hydrologic model is subject to conceptual or structural errors or the geophysical model fails to capture the influence of nonhydrologic factors, such as background variations in resistivity.

[4] We propose a new approach for physics-based regularization of inverse problems that is dependent on, but less restricted by, assumptions about subsurface processes compared to coupled inversion (Figure 1c). The approach uses proper orthogonal decomposition (POD) [e.g., *Banks et al*., 2000; *Kunisch and Volkwein*, 2003; *Rathinam and Petzold*, 2004; *Pinnau*, 2008] of a set of training data generated by Monte Carlo simulation of a hydrologic process to generate an optimal set of basis vectors for the imaging problem. These hydrologically “tuned” basis vectors are subsequently used within a basis-constrained inversion framework to obtain a resistivity image.

[5] To our knowledge, this work is the first use of POD to constrain geophysical inversions by physical process information. The use of training data to characterize spatially distributed patterns, however, is well established in a variety of fields in the geosciences. For example, the adoption of training images to infer spatial patterns in applications of multiple-point geostatistics [e.g., *Strebelle*, 2000] is increasingly common. *Moysey et al*. [2005] applied the training data concept by using geostatistically based Monte Carlo simulations of geophysical surveys to quantitatively capture and correct for spatially variable inversion artifacts associated with nonlinear imaging problems. Similarly, *Lehikoinen et al*. [2010] used Monte Carlo simulations of flow in a heterogeneous vadose zone to construct a statistical model of approximation errors resulting from the assumption of a homogenous medium, which they were subsequently able to utilize within a Kalman filter to improve resistivity imaging of water content changes. The use of simulations and training data to capture relevant information to constrain estimation and imaging problems is, therefore, already well-established in the literature. The key contribution of this work is establishing the use of POD to capture patterns from training data and efficiently integrate this information as a constraint within an inverse problem.

[6] In this paper, we compare the results of POD-based inversion to results obtained using standard Tikhonov regularization techniques and coupled inversion for a problem where ERI is used to image a solute plume. We investigate two distinct scenarios: one where the *a priori* understanding of flow and transport utilized in generating the training images for the inversions is correct and one where the training data are inconsistent with the actual processes.