Conclusions

Though physically impossible, the acoustic wave equation for P-waves in transversely isotropic media with a vertical symmetry axis (VTI media) that I have derived yields good kinematic approximations to the familiar elastic wave equation for VTI media. The fourth-order nature of this acoustic equation results in two sets of complex conjugate solutions. One set of solutions are 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 $\eta$. Most $\eta$values corresponding to anisotropies in the subsurface are likely to have positive values. Placing the source or receivers in an isotropic layer, a common occurrence in marine surveys where the water layer is isotropic, will eliminate most of the energy of this additional wave type. Numerical examples, provided in this paper, prove the usefulness of this acoustic equation in simulating wave propagation in VTI media.

A ray theoretical (high-frequency) approximation is used to derive the eikonal and transport equations that describe the traveltime and amplitude behavior, respectively. These equations are also simpler than those we have grown accustomed to in anisotropic media.