Weak-anisotropy moveout approximations for P-waves in homogeneous layers of monoclinic or higher anisotropy symmetries

Abstract

We have used so-called weak anisotropy (WA) parameterization as an alternative to the parameterization of generally anisotropic media by a stiffness tensor. WA parameters consist of linear combinations of normalized stiffness-tensor elements controlling various seismic signatures; hence, they are theoretically extractable from seismic data. They are dimensionless and can be designed to have the same order of magnitude. WA parameters, similarly to Thomsen-type parameters, have a clear physical interpretation. They are, however, applicable to anisotropy of any symmetry, strength, and orientation. They are defined in coordinate systems independent of the symmetry elements of the studied media. Expressions using WA parameters naturally simplify as the anisotropy becomes weaker or as the anisotropy symmetry increases. We expect that, due to these useful properties, WA parameterization can potentially provide a framework for seismic data processing in generally anisotropic media. Using the WA parameterization, we have derived and tested approximate P-wave moveout formulas for a homogeneous layer of up-to-monoclinic symmetry, underlain by a horizontal reflector coinciding with a symmetry plane. The derived traveltime formulas represent an expansion of the traveltime with respect to (small) WA parameters. For the comparison with standard moveout formulas, we expressed ours in the common form of nonhyperbolic moveout, containing normal moveout velocity and a quartic coefficient as functions of the WA parameters. The accuracy of our formulas depends strongly on the deviation of ray- and phase-velocity directions (controlled by the deviation of the ray and phase velocities). The errors do not generally increase with increasing offset, nor do they increase with decreasing anisotropy symmetry. The accuracy of our formulas is comparable with, or better than, the accuracy of commonly used formulas.