Phase and group angles and velocities

In anisotropic wave propagation the phase angles and velocities are different from the group angles and velocities. In this appendix we briefly review the concepts of phase and group angles and velocities and the relationships between these physical quantities.

The transformation from phase velocity $\widetilde{V}$to group velocity V is conventionally defined as the following Tsvankin (2001):  

$$V=\sqrt{\widetilde{V}^2+\left(\frac{d\widetilde{V}}{d\widetilde{\theta}}\right)},$$ (10)

where $\widetilde{\theta}$ is the phase propagation angle. The associated transformation from phase angles to group angles $\theta$ is defined as:  

$$\tan \theta= \frac {\tan \widetilde{\theta}+ \frac{1}{\wide... ...d \widetilde{V}}{d\widetilde{\theta}} \tan \widetilde{\theta}}.$$ (11)

Dellinger and Muir (1985) propose, and heuristically motivate, the following symmetric relations for the inverse transforms:  

$$\widetilde{S}=\sqrt{S^2+\left(\frac{dS}{d\theta}\right)},$$ (12)

where $\widetilde{S}$ and S are respectively the phase slowness and the group slowness, and   We use the heuristic relation in equation 13 to derive some of the analytical results presented in this paper. Furthermore, we use all the above relationships to compute the kinematic numerical results presented in this paper.