Dispersion relation and wave equation for attenuative elliptically anisotropic media

TL;DR

This paper derives the dispersion relation and wave equation for attenuative elliptically anisotropic media, providing conditions for elliptical P-wave surfaces and validating them through numerical examples.

Contribution

It extends elliptical anisotropy models to include attenuation, deriving new dispersion relations and wave equations for such media.

Findings

Elliptical conditions ensure P-wave surfaces satisfy elliptical equations.

Derived viscoacoustic wave equation for elliptically anisotropic media.

Numerical examples confirm the validity of the elliptical conditions.

Abstract

The P-wave slowness and group-velocity surfaces in elliptically anisotropic media are ellipsoids. Elliptical anisotropy is convenient to use as the reference medium in perturbation methods designed to solve P-wave wave-propagation problems for transverse isotropy (TI). Here, we make the elliptically anisotropic TI model attenuative and discuss the corresponding P-wave dispersion relation and the wave equation. Our analysis leads to two conditions in terms of the Thomsen-type parameters, which guarantee that the P-wave slowness surface and the dispersion relation satisfy elliptical equations. We also obtain the viscoacoustic wave equation for such elliptically anisotropic media and solve it for point-source radiation using the correspondence principle. Numerical examples validate the proposed elliptical conditions and illustrate the behavior of the P-wavefield in attenuative elliptical TI models.