An acoustic wave equation for anisotropic media

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 for isotropic media for incoming and outgoing waves. The second set describes an unwanted wave type that propagates at speeds slower than the P-wave for the positive anisotropy parameter, ƞ, and grows exponentially, becoming unstable, for negative values of ƞ. Luckily, most ƞ values corresponding to anisotropies in the subsurface have positive values which is in the stability range of the acoustic equation. 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.