Conclusions

We have presented a set of three nested scalar anisotropic approximations; beyond isotropy there is standard elliptical anisotropy, then the first anelliptic approximation, and then the second anelliptic approximation. Each subsequent approximation adds one more velocity parameter and includes the previous as a special case.

There are two velocity parameters for elliptical anisotropy, horizontal and vertical. The first anelliptic approximation adds to these a third, horizontal NMO velocity (the standard moveout velocity recorded by surface surveys). The second anelliptic approximation adds a fourth, vertical NMO velocity (the moveout velocity measured in a cross-borehole survey).

All members of the family share elliptical anisotropy's convenient unity of form and ease of transformation between the group-velocity and phase-slowness domains. Although the anelliptic representations in the two domains are not exactly consistent, the disparity should be of about the same magnitude as the mismatch between the anelliptic approximation and the TI wave mode being approximated. Trouble can occur for extremely anisotropic media because the single-valued anelliptic approximations cannot follow triplications. In this case the approximations should not be used in the group-velocity domain.

The approximations can be successfully used in the phase-slowness domain even in such extreme cases, as demonstrated by the qSV examples in Figures and . These results suggest that the anelliptic approximations should find wide use replacing TI dispersion relations in modeling, migration, and inversion programs.

There are several reasons to use the anelliptic approximations instead of exact transverse isotropy. For one, there are fewer parameters to find values for, because the equations are more simple. Most importantly, the anelliptic velocity parameters should be more natural and robust, because the velocity parameters used in the anelliptic approximations (horizontal and vertical true velocities and moveout velocities) are the most important ones in the natural independent coordinate sets for the sorts of surface-to-surface and cross-borehole geometries favored in geophysics.

We have derived the theoretical basis for a first-anelliptic extension of Dix's equations. How well our anelliptic analysis can work in practice remains to be determined.