## 1. Introduction

[2] It is generally more suitable to characterize strongly heterogeneous subsurface environments using a stochastic, rather than deterministic, approach. This means that we estimate parameters describing the geostatistical nature of the heterogeneity, rather than a detailed distribution of material properties. To this end, estimation of the geostatistical properties of subsurface velocity heterogeneity from surface-based seismic and ground-penetrating radar (GPR) reflection images has been a long-standing problem of significant interest [e.g., *Holliger et al.*, 1992; *Hurich*, 1996; *Pullammanappallil et al.*, 1997; *Rea and Knight*, 1998; *Bean et al.*, 1999; *Poppeliers and Levander*, 2004; *Carpentier and Roy-Chowdhury*, 2007; *Knight et al.*, 2007]. Of particular interest has been the estimation of the *lateral* statistics of a subsurface velocity field from those of the corresponding reflection image, as this information cannot be obtained from borehole log or core analysis. A primary motivation for this work has been the prospect of having an effective means of obtaining realistic geostatistical models of inaccessible subsurface regions, which could greatly facilitate the understanding, characterization, and modeling of such diverse environments as the crystalline crust, hydrocarbon reservoirs, groundwater aquifers, the vadose zone, and mining prospects.

[3] Previous work on the problem of estimating subsurface geostatistical properties from seismic or GPR reflection data has shown that the second-order spatial statistics of the data are indeed related to those of the underlying heterogeneous subsurface velocity distribution. However, such work has either been largely empirical or methodologically inadequate, and a rigorous means of linking the statistical properties of depth-imaged reflection data to the geostatistical properties of velocity has only recently been presented. *Irving et al.* [2009, 2010] describe the corresponding methodology and provide a comprehensive review of previous work on this topic. With the availability of this method and the development of an effective inversion approach, however, have come some surprising and somewhat enigmatic observations. Most notably, there is evidence to suggest that the second-order spatial statistics of seismic or GPR reflection images are sensitive only to the structural aspect ratio of the velocity heterogeneity, and not to the horizontal and vertical correlation lengths individually. Consequently, it may not be possible to resolve separately the horizontal and vertical correlation lengths of the underlying velocity structure from a reflection image. Another important, and as of yet unresolved, question concerns the sensitivity with regard to the decay of the power spectrum characterizing the fine-scale details of the velocity heterogeneity.

[4] In this paper, we explore these questions using an analytical approach that is based on the realistic assumption that the subsurface velocity heterogeneity obeys some generic band-limited scaling laws and hence can be characterized by a van-Karman-type power spectrum. Our goal is to concretely demonstrate just what, and what not, we may ideally hope to recover regarding the stochastic properties of velocity from those of the corresponding seismic or GPR reflection image. We first review the methodological background for our approach, and then analytically assess the sensitivity of the 2-D power spectrum of a geophysical reflection image to the parameters describing the underlying von-Karman-type heterogeneity. We then test and verify our analytical work with a series of numerical examples.