Low-frequency layer-induced anisotropy versus heterogeneity

Abstract

Anisotropy is an important property of the modern seismic models, and can be recognized as intrinsic and induced. The standard way to treat both in seismic upscaling is to use the classic Backus averaging technique which is the static method based on the zero-frequency limit for effective medium. Recently, we (Roganov and Stovas, 2012) propose a method to extend the Backus method to low frequencies which is more suitable for real seismic experiment. By analyzing the solution of the problem, we show that the effective medium properties computed for a given frequency do not correspond to any physically realized medium, despite of the fact that the effective medium possess the vertical axis symmetry. In order to preserve the symmetry axis direction for an effective medium, we approximate the frequency dependent eigen-values computed from the corresponding system matrix by the vertical slowness expansions taken from different modes: qP (quasi P), qSV (quasi SV) and qSH (quasi SH), in a transversely isotropic medium with vertical symmetry axis. By fitting the coefficients of these expansions, we obtain the frequency-dependent anisotropy parameters. This approximation guarantees kinematical equivalence for quasi-vertical wave propagation. Being averaged with the spectrum of seismic wavelet, the anisotropy parameters can be called the seismic data driven and used for anisotropic velocity model building. When the contrast in elastic properties of the finely layered medium is large, the effective slowness surface becomes discontinuous. This requires application of the least-square method to fit the discontinuous effective slowness surface in order to evaluate the anisotropy parameters.