An acoustic wave equation for anisotropic media

Tariq Alkhalifah

tariq@sep.stanford.edu

ABSTRACT

A wave equation, derived under the acoustic medium assumption for P-waves in transversely isotropic media with a vertical symmetry axis (VTI media), though physically impossible, yields good kinematic approximation to the familiar elastic wave equation for VTI media. The VTI acoustic wave equation is fourth-order and has two sets of complex conjugate solutions. One set of solutions is just perturbations of the familiar acoustic wavefield solutions in isotropic media for incoming and outgoing waves. The second set describes a wave type that propagates at speeds slower than the P-wave for the positive anisotropy parameter, $\eta$, and grows exponentially, becoming unstable, for negative values of $\eta$. Luckily, most $\eta$values corresponding to anisotropies in the subsurface have positive values. Placing the source in an isotropic layer, a common occurrence in marine surveys where the water layer is isotropic, eliminates most of the energy of this additional wave type. Numerical examples prove the usefulness of this acoustic equation in simulating wave propagation in complex models. From this acoustic wave equation, the eikonal and transport equations that describe the ray theoretical aspects of wave propagation are derived. These equations, based on the acoustic assumption (shear wave velocity equals zero), are much simpler than their elastic counterparts, and yet yield exceptionally accurate description of traveltime and geometrical amplitude, or wavefront spreading.