Calculation method for ray velocity in the general TI medium using Thomsen's parameter model

The group velocity in a general Transversely Isotropic (TI) medium is a function of the phase slowness direction angle. However, in practical calculations, it is often necessary to compute the group velocity along a specific ray direction. Thus, the corresponding phase slowness direction angle must first be derived from the ray direction, and then the group velocity in that direction can be calculated. Traditional ray velocity calculation methods for general TI media typically discretize the phase slowness angle with small angular intervals, compute the ray direction angle and ray velocity for each discrete angle, and then use interpolation based on the ray direction angle-ray velocity database to obtain the ray velocity for the actual ray direction. Nevertheless, this approach suffers from time-consuming computations and insufficient accuracy. This paper proposes a calculation method for ray velocity in the context of the general TI medium Thomsen parameter model. Firstly, the phase slowness vector is represented using the phase slowness direction vector and phase velocity. Subsequently, a system of equations is directly established between the ray direction and the phase slowness direction angle. Finally, the phase slowness direction angle obtained from solving the system of equations is substituted into the group velocity formula to derive the ray velocity for that ray direction. In the numerical simulations section, this ray velocity calculation method was used to obtain correct traveltime isosurfaces for homogeneous Vertical Transverse Isotropy (VTI) and Tilted Transverse Isotropy (TTI) medium models. Additionally, when applying the ray velocity formula to the traveltime calculation of an overthrust TTI model, the traveltime results were identical to those obtained from an elastic parameter model. Furthermore, by introducing an interpolation method to solve the phase slowness direction angle along the ray direction and applying it to the shortest path ray tracing method, the corresponding seismic wave traveltime field was obtained, providing a reliable comparison for the traveltime results of qP-waves and qSV-waves in the overthrust TTI model. Comparative tests show that the ray velocity calculation method for the Thomsen parameter model of general TI media proposed in this paper is correct, and compared with traditional interpolation methods, this method significantly improves the computational efficiency.