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# Introduction

Modeling plane-wave reflection and transmission coefficients at a plane interface between two homogeneous half spaces is a fundamental problem in wave propagation with many geophysical applications. Zoeppritz's research 1919 was among the first theoretical work to study the solution for this problem in the case of two isotropic media. Ostrander 1984 was one of the first to use these results assuming isotropic media for doing amplitude versus offset analysis (AVO). In order to apply AVO analysis on azimuthally-anisotropic targets such as fracture reservoirs, anisotropy must be accounted for in modeling the reflection coefficients. In my opinion, while in some cases anisotropic AVO behavior for a given azimuth can be explained by an equivalent isotropic model, azimuthal variations are a better indicator of reflectivity changes.

It is important to study reflection amplitudes for anisotropic materials as a first step in modeling azimuthal anisotropic AVO signatures. To date only numerical modeling is available to fully model reflection amplitudes. Recently, Rüeger 1995 obtained approximate expressiones for P-wave reflection coefficients in transversely isotropic media; however, he assumes weak anisotropy, small discontinuities in elastic properties across boundaries, and the bulk of his analyses are confined to symmetry planes. Unfortunately, while symmetry planes are an understandable and tractable case, they are also a misleading special case. Anisotropic phenomena must be studied as a 3-D problem. While the precise behavior of such phenomena can be described mathematically, the analytical expressions are usually too complex to allow any comprehension, including the case of transverse isotropy, which is the next level up in complexity after isotropy.

In the quest to obtain an approximate expression of reflection amplitudes, valid for any propagation direction, any degree of anisotropy, and any degree of discontinuity in the elastic parameters across the boundaries, I look at the effects of horizontal transverse isotropy in P-wave reflection patterns on a simple anisotropic model, a finely fractured shale overlying an Austin Chalk with varied amounts of fracturing. I investigate whether it is feasible to determine the parameters, content, and orientation of fractures from azimuthal variations in P-wave reflectivity measurements and conclude that we get indications when plotting 2-D views even for pre-critical angles.

To check the validity of these modeling results with real applications, we must assume a medium where anisotropy has a much higher order effect than heterogeneity in wave propagation. We also must assume planar interfaces and that we are working in the far field approximation, in order to satisfy the hypothesis that plane waves expand the solution space of anisotropic wave propagation.

The first section of this paper reviews the theory on plane-wave propagation in a homogeneous medium. The second section follows along the lines of Dellinger's 1991 work on plane wave propagation for a horizontally transverse isotropy media. The third section focuses on plane wave propagation across horizontal interfaces for homogeneous half spaces following the work of Nichols 1991 on his two-way phase-shift modeling scheme. In the final section, I compute reflectivity maps with a modified version of Nichols's phase-shift modeling scheme.

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Stanford Exploration Project
11/12/1997