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 I 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):  

$$V=\sqrt{\widetilde{V}^2+\left(\frac{d\widetilde{V}}{d\widetilde{\theta}}\right)},$$ (30)

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

$$\tan \theta= \frac {\tan \widetilde{\theta}+ \frac{1}{\wide... ...d \widetilde{V}}{d\widetilde{\theta}} \tan \widetilde{\theta}}.$$ (31)

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

$$\widetilde{S}=\sqrt{S^2+\left(\frac{dS}{d\theta}\right)},$$ (32)

where $\widetilde{S}$ and S are respectively the phase slowness and the group slowness, and  

$$\tan \widetilde{\theta}= \frac {\tan \theta+ \frac{1}{S}\frac{d S}{d\theta}} {1- \frac{1}{S}\frac{d S}{d\theta} \tan \theta}.$$ (33)

I use the heuristic relation in equation 33 to derive some of the analytical results presented in this paper. Furthermore, I use all the above relationships to compute the kinematic numerical results presented in this paper. B