Slowness vector and ray velocity magnitude from ray direction in TTI media

Abstract

Two-point ray bending problems in anisotropic elastic media require finding the (non-unique) slowness vectors and the ray (group) velocity magnitudes from a given ray direction at each node of the ray trajectory. In this study, we consider polar anisotropic elastic media (transverse isotropy with tilted axis of symmetry – TTI), where the solutions for the coupled qP and qSV waves consist of a single slowness vector of a qP wave and single or triple slowness vectors of qSV waves. We provide an original and efficient solution for this challenging problem, formulated as a single polynomial equation of degree six, whose coefficients depend on the medium properties and the ray angle, where the unknown parameter is the non-unique phase angle between the slowness vectors and the tilted symmetry axis. Additionally, we provide a direct indication of the existence of a qSV triplication without the need to solve the polynomial equation. For SH waves, the solution is unique and straightforward.