Determination of slowness vector from ray directions for qP- And qSV-wave in TTI media

In anisotropic media, the group velocity often differs from the phase velocity and always deviate from the ray direction. As a result, calculating the group velocity from ray direction is critical problem in anisotropic field. The phase velocity direction which is the same with the slowness vector, is always along the wave normal, while the group velocity describes the energy transportation of the ray. Vavrycuk (2006) developed a method to calculate the slowness vector from the ray vector. But there are two main problems for this method: (1) it leads to multiple solutions for the P- and SV-wave slowness vector, and even some complex solutions; (2) it mixes the solutions of P- and SV-wave, so it is hard to distinguish these two wave modes from the solutions. Zhou and Greenhalgh (2004) designed another way to calculate the group velocity in a general anisotropic medium. By combining these two methods, a new way of individually finding the slowness vector from the ray vector is developed. This method solved the two problems of the original method: (1) it splits P- and SV-wave by separate the original two equations into four new equations; (2) and it also succeeds to find the only solution for P- and SV-wave slowness vector by simplifying the original 2D problem into a new 1D problem, which reduce the unknown components in the equations. As a result, the only solution for the slowness vector for each wave mode can be find. And this method is applicable for both VTI and TTI media.