Ray-direction velocities in VTI media

Two-point ray tracing in anisotropic media requires the group and phase velocities to be calculated along ray directions available at each step of a ray bending algorithm. This computation, usually done iteratively or through velocity tables, becomes exceedingly involved for shear-waves that have multivalued group-velocity surfaces, such as in the presence of triplications on the SV wavefronts in vertically transversely isotropic (VTI) media. The difficulties encountered in computing the SV-wave velocities for a given ray direction can be circumvented by solving a polynomial equation whose real-valued roots provide the phase directions of the P- and either one or three SV-waves propagating along a selected ray; those phase directions then allow the group and phase velocities to be computed in a standard fashion. I construct the polynomial and supply computer codes implementing its solution, the codes that can be used in two-point ray-tracing software to improve its performance.