AVO-A response of an anisotropic half-space bounded by a dipping surface for P–P, P–SV and P–SH data

Abstract

We analyze amplitude variations with offsets and azimuths (AVO-A) of an anisotropic half-space bounded by a dipping surface. By analyzing the response of a dipping reflector instead of a horizontal one, we integrate the fundamental problem of lateral heterogeneity vs. anisotropy into our study. This analysis is limited to the three scattering modes that dominate ocean bottom seismic (OBS) data: P–P, P–SV and P–SH. When the overburden is assumed isotropic, the AVO-A of each of these three scattering modes can be cast in terms of a Fourier series of azimuths, φ, in general form, R avoa (φ)=F 0+ ∑ n=1 4 [F n cos(nφ)+G n sin(nφ)], where F 0, F n and G n are the functions that describe the seismic amplitude variations with offsets (AVO) for a given azimuth. The forms of AVO functions are similar to those of classical AVO formulae; for instance, the AVO functions corresponding to the P–P scattering mode can be interpreted in terms of the intercept and gradient, although the resulting numerical values can differ significantly from those of isotropic cases or horizontal reflectors. One of the benefits of describing the AVO-A as a Fourier series is that the contribution of amplitude variations with azimuths (AVAZ) is distinguishable from that of AVO. The AVAZ is characterized by the functions {1, cos φ, sin φ, cos2 φ, sin2 φ, cos3 φ, sin3 φ, cos4 φ, sin4 φ}, that are mutually orthogonal. Thus, the AVO–A inversion can be formulated as a series of AVO inversions where the AVO behaviors are represented by the functions F 0, F n and G n . When the coordinate system of seismic acquisition geometry coincides with the symmetry planes of the rock formations, the series corresponding to P–P and P–SV simplify even further; they reduce to F 0 for azimuthally isotropic symmetry and to F 0, F 2, F 4, G 2 and G 4 for orthorhombic symmetry. The series corresponding to P–SH scattering is reduced to G 2 and G 4 for these two symmetries. Unfortunately, the coordinate system of seismic acqusition geometry rarely coincides with the symmetry planes of the rock formations; therefore, the other terms are rarely zero. In particular, the functions F 1, F 3, G 1 and G 3 become important for large dips and are actually largely dependent on the angle of the dipping reflector. For P–P scattering, these functions are zero if the reflector is horizontal, irrespective of the anisotropic behavior. For P–SV and P–SH scattering, these functions are not necessarily zero for horizontal reflectors because they are affected by the asymmetry of the P–S reflection in addition to the effect of dip.