On anelliptic approximations for $qP$ velocities in transversally isotropic media

Conclusions

I have developed a general approach for approximating both phase and group velocities in a VTI medium. Suggested approximations use three elastic parameters as opposed to the four parameters in the exact phase velocity expression. The phase velocity approximation coincides with the acoustic approximation of Alkhalifah (1998,2000a) but is derived differently. The group velocity approximation has an analogous form and similar superior approximation properties. It is important to stress that the two approximations do not correspond exactly to each other. The exact group velocity corresponding to the acoustic approximation is different from the approximation derived in this paper and can be too complicated for practical use (Alkhalifah, 1999). The suggested phase and group approximations match each other in the sense that they have analogous approximation accuracy in the dual domains.

The group velocity approximation is useful for approximating normal moveout and diffraction traveltimes in applications to non-hyperbolic velocity analysis and prestack time migration. It is also useful for traveltime computations that require ray tracing in locally homogeneous cells. I have shown examples of such computations utilizing an anisotropic extension of the fast marching finite-difference eikonal solver.

On anelliptic approximations for $qP$ velocities in transversally isotropic media