## 1. Introduction

[2] Geophysical tomography (seismic, electromagnetic, electrical, and radar) is commonly used in petroleum engineering, global seismology, and hydrogeologic studies. Tomograms provide valuable, qualitative information about geologic, aquifer and reservoir structure and processes, as reported by abundant examples in the literature. Contrasts in electromagnetic (EM), electrical, or seismic properties provide information about lithology, rock fractures, depth to bedrock, depth to water table, and other structures and properties of geologic relevance. In the environmental and petroleum fields, tomograms are used increasingly for more quantitative estimation of hydraulic properties. Petrophysical formulas are used to convert geophysical images into two-dimensional (2-D) cross sections or three-dimensional (3-D) volumes of quantities such as saturation, concentration, porosity, or permeability [e.g., *Hubbard et al.*, 2001; *Slater et al.*, 2002; *Berthold et al.*, 2004]. In some cases, site-specific conversions are used [e.g., *Alumbaugh et al.*, 2002], whereas in others, general petrophysical relations such as Archie's law [*Archie*, 1942] are applied [e.g., *Slater et al.*, 2002]. To account for uncertain and nonunique relations between geophysical properties and the hydrologic parameter of interest, some have considered conditional simulation, Bayesian, and co-kriging frameworks [e.g., *McKenna and Poeter*, 1995; *Cassiani et al.*, 1998; *Hubbard et al.*, 2001]. In some studies, models of statistical spatial structure (i.e., the variogram or covariance) have been inferred from tomograms. Such work implicitly assumes that tomographic estimates approximate point-scale measurements. Unfortunately, for many practical field situations tomographic estimates should be viewed as weighted local averages because tomograms bear the imprint of prior information and regularization criteria, survey geometry, measurement physics, and measurement error.

[3] Although qualitative insights into subsurface architecture are readily made from geophysical data, quantitative use of geophysical images to estimate values of hydrologic parameters suffers from limitations arising from imperfect and variable tomographic resolution. Geotomography tends to overpredict the extent and underpredict the magnitude of geophysical targets; moreover, the spatial structure of tomograms may only weakly reflect the true spatial structure of the subsurface. In this paper, we refer to this loss of information as “correlation loss.” This problem was recognized by *McKenna and Poeter* [1995], who noted weak correlation between tomographic estimates of seismic velocity and colocated measurements of hydraulic conductivity compared to the correlation seen for higher resolution sonic logs; they derived a correction based on regression and applied the correction uniformly over the tomogram to compensate for the correlation loss. Similarly, *Cassiani et al.* [1998] noted correlation loss between tomographic estimates of seismic velocity and hydraulic conductivity in poorly resolved regions of tomograms. In an effort to monitor tracer experiments with electrical-resistivity tomography (ERT), *Singha and Gorelick* [2005] noted the impact of regularization and inversion artifacts on the estimated spatial moments of the tracer plume. In an ERT study to monitor a fluid tracer in the vadose zone, *Binley et al.* [2002] applied locally derived petrophysical relations to convert resistivity tomograms to changes in moisture content; their analysis revealed a 50% mass balance error that was attributed to the poor sensitivity in the center of the image volume where the tracer was applied.

[4] Quantifying tomographic resolution and geophysical measurement support is a long-standing and active area of geophysical research [e.g., *Dahlen*, 2004; *Sheng and Schuster*, 2003; *Friedel*, 2003; *Alumbaugh and Newman*, 2000; *Oldenburg and Li*, 1999; *Schuster*, 1996; *Rector and Washbourne*, 1994; *Ramirez et al.*, 1993; *Menke*, 1984; *Backus and Gilbert*, 1968]. Resolution depends on the measurement physics; physical approximations in the forward model and (or) sensitivity calculation; survey design; measurement error; regularization criteria and inversion approach; and parameterization. The correlation between point-scale measurements of hydrologic and geophysical properties is commonly degraded by the inversion process, which produces images that represent blurry, blunted, and artifact-prone versions of reality.

[5] Recently, *Day-Lewis and Lane* [2004] investigated the implications of correlation loss and variable model resolution for geostatistical utilization of tomograms. Analytical methods were developed to model the correlation loss as a function of measurement physics, survey geometry, measurement error, spatial correlation structure of the subsurface, and regularization. For the simplified case of linear, ray theoretic radar tomography and linear correlation between radar slowness (1/velocity) and the natural logarithm of permeability, they derived formulas to predict (1) how the inversion process degrades the correlation between imaged geophysical properties and related hydrologic parameters, compared to point measurements, (2) how the variance of the estimated geophysical parameter compares to the variance of the true property, and (3) how the inversion alters the spatial covariance of the estimated parameter. Using synthetic examples for radar travel time tomography (RTT), they demonstrated that tomograms of qualitative value may hold limited quantitative utility for standard geostatistical applications.

[6] In this paper, we extend the methodology of *Day-Lewis and Lane* [2004] to consider nonlinear tomographic inversion and nonlinear petrophysical relations; furthermore, we compare correlation loss, variance reduction, and tomogram spatial structure for RTT and ERT. In the present treatment of radar tomography, we use a more physically based forward model and sensitivity calculation that accounts for refraction and finite frequency limitations on measurement sensitivity and tomographic resolution. The goals of the present study are (1) to raise awareness of the limitations of geophysical data, (2) to provide a framework to improve survey design and assess tomograms for hydrologic estimation, and (3) to develop insights into the different patterns of correlation loss for electrical resistivity and radar tomography. Although the examples used to illustrate our approach are based on near-surface radar and electrical tomography to characterize variations in water content, our approach is applicable to other forms of geotomography, and our results have implications for quantitative use of tomograms across geoscience subdisciplines.