## 1. Introduction

[2] Geophysical tomography has been used to monitor mass transport associated with tracer experiments [e.g., *Day-Lewis et al.*, 2003; *Singha and Gorelick*, 2005, 2006], infiltration tests [e.g., *Binley et al.*, 2002], engineered remediation [e.g., *Lane et al.*, 2004, 2006], and other transport processes. Tomograms serve as time-lapse snapshots of geophysical properties (e.g., electrical resistivity or radar attenuation) that can be related to hydrologic parameters of interest (e.g., tracer concentration); hence they provide valuable information for construction and calibration of numerical models of flow and transport. In several studies [*Hubbard et al.*, 2001; *Singha and Gorelick*, 2005; *Binley et al.*, 2002], plume moments were inferred from time-lapse tomograms. Moment inference provides direct insight into transport processes and controlling parameters. The reliability of moment inference from tomograms, however, remains poorly understood.

[3] In using moments calculated from tomograms, it is assumed that the tomogram provides an adequate approximation of the solute plume; the resolution of tomographic estimates, however, is known to be a function of survey geometry, measurement physics, measurement error, regularization, and parameterization [e.g., *Day-Lewis and Lane*, 2004; *Day-Lewis et al.*, 2005]. The model resolution matrix, **R**, is commonly used to quantify tomographic resolving power:

where

**J**Jacobian of predicted measurements with respect to model parameters,

*m*;**W**diagonal matrix where

*W*_{ii}is the reciprocal of the standard error for measurement*i*;**D**Tikhonov regularization matrix (e.g., second-derivative filter); and

*β*tradeoff parameter between the regularization and measurement fitting.

[4] The model resolution matrix is a powerful tool for understanding how well a given survey will resolve a hypothetical target. For linear problems, = **Rm**, where is the vector of tomographic model estimates and **m** is the vector of true model parameters; for nonlinear problems, **R** is commonly calculated for the final iteration of the inversion, and ≈ **Rm**, where the approximation degrades with increasing nonlinearity. For the case of perfect resolution, **R** is an identity matrix and parameter estimates are independent. More commonly, resolution is imperfect, the geophysical parameter estimates represent local averages, and tomograms are smooth, blunted versions of reality. In such cases, moments calculated on tomograms may be unreliable. The objective of this research is to quantify how well the moments calculated from tomograms reflect the moments of the true image, and how the quality of inferred moments depends on regularization criteria, measurement error, and survey geometry. Toward this end, we derive a formulation for a new resolution matrix for moment inference from tomograms.

**J**Jacobian of predicted measurements with respect to model parameters,

*m*;**W**diagonal matrix where

*W*_{ii}is the reciprocal of the standard error for measurement*i*;**D**Tikhonov regularization matrix (e.g., second-derivative filter); and

*β*tradeoff parameter between the regularization and measurement fitting.