The TTI slowness surface approximation

Abstract

The relation between the vertical and horizontal slownesses, better known as the dispersion relation, for a transversely isotropic media with titled symmetry axis (TTI) requires solving a quartic polynomial, which does not admit a practical explicit solution to be used, for example, in downward continuation. Using a combination of perturbation theory with respect to the anelliptic parameter and Shanks transform to improve the accuracy of the expansion, we develop an explicit formula for the dispersion relation that is highly accurate for all practical purposes. It also reveals some insights into the anisotropy parameter dependency of the dispersion relation including the low impact that the anelliptic parameter has on the vertical placement of reflectors for small tilt in the symmetry angle.