Calculating slowness direction from the ray direction for the qP, qSV and SH waves in 2D TTI media with Newton’s Method

Summary Phase and group velocities along specific ray directions are needed for qP, qSV and SH wave traveltime computation in tilted transversely isotropic (TTI) media. The phase and group velocities are not unique functions of the ray direction, but of the slowness direction, so efficient computation of the slowness direction from a given ray direction becomes necessary. The eigenvalue method and the generalized method have been proposed to facilitate the computation by formulating governing equations for the phase angle, which requires efficient root-finding algorithms. In this work, we address this problem in two-dimensional (2D) TTI media by applying Newton’s method. For the eigenvalue method, a set of two nonlinear equations for the qP and qSV waves and one nonlinear equation for the SH wave must be solved. For the generalized method, only one nonlinear equation of the phase angle needs to be solved for the qP, qSV and SH waves. Numerical experiments, including phase and group velocity computation and their application in first-arrival traveltime calculation, are performed to verify the effectiveness of the proposed methods.