Amplitude Verus Offset analysis (AVO) assuming isotropic media is a tool which has been known for many years Ostrander (1984). However little has been published so far about anisotropic treatment of AVO behaviour. While in some cases anisotropic AVO behaviour can be explained by an equivalent isotropic medium for a given azimuth; the azimuthal variation is a better indicator reflectivity changes. In some cases amplitude variations cannot be explained by an equivalent isotropic medium. We focus on a particular simple anisotropic case which is relevant in real data, since it can be produced by very finely layering isotropic material or by fracturing an isotropic rock. We use Austin Chalk as our practical example and derive material stiffnesses from data published by White 1982 and Wright 1987. We model reflections from Austin Chalk, where a transverse isotropic shale overlies chalk, with varied amounts of fracturing. We investigate wether we can practically determine fracture parameters and/or fracture content from azimuthal variations of reflectivity and find that we get measurable small indications when using converted wave modes or when going to very large offsets.
Using a macro model description of the fractured and finely layered medium, we can employ a modeling method which works efficiently on a large - the actual surface seismic - measurement scale. Thus we avoid calculating all the fine details which are averaged out during the wave propagation process and use a homogeneous average medium that includes the macroscopic effects of microscopic fluctuations.
We address issues of: (a) how do we include the effect of fractures on the material properties, and (b) how do we carry out the modeling procedure. We choose a model with a single interface: a block of transverse isotropic (TI) shale lies over a fractured chalk. The source is located 1 km above the interface and a surface seismic survey is modeled with a large aperture (<60 deg.). The chalk fracturing is calculated using Hudson's theory combined with Schoenberg & Muir group theory (S&M) Nichols (1989); Schoenberg and Muir (1989). The regime of validity of those theories limits us to low crack density and small crack aspect ratios.
Anisotropic Finite Differences (FD) are used to model wave propagation through an anisotropic medium. We chose to test whether FD modeling of reflection amplitudes is adequate by comparing it to an analytical solution. We use this example to calibrate our method before applying it to real data in arbitrary complex media. The reflection amplitudes produced by the FD modeling are in nearly perfect agreement with analytical solutions for the reflection amplitudes (i.e. Zoeppritz, for a single flat reflector).
Using S&M equivalent medium theory, we ``add'' fractures to the chalk rock matrix. The vertical cracks have a constant azimuth along the inline direction. We compare inline and crossline reflection amplitude effects produced by the fracturing. Amplitude analysis is carried out in the slant-stack domain. We use a least-squares ``optimum'' stack to get an aperture-compensated estimate of the modeled data. Travel time curves in the slant-stack domain of the TI medium can be calculated directly or after anisotropic velocity analysis. Reflected amplitudes are extracted within a centered window along that curve. Alternatively, a downward continuation in the slant- stack domain can replace the previous process.