Conversion from average angles to true aperture angles

In this appendix I present the expressions for evaluating the true reflection angles $\widetilde{\gamma}_s$ and $\widetilde{\gamma}_r$for the incident and reflected plane waves, from the `normalized slowness difference'' $\Delta_\widetilde{S}=(\widetilde{S}_r-\widetilde{S}_s)/(\widetilde{S}_r+\widetilde{S}_s)$and from the average aperture angles $\widetilde{\gamma}$ computed by solving equations 16 and 17.

Rosales and Biondi (2005) derived these relationships as follows:

$$\begin{eqnarray} \tan \widetilde{\gamma}_s & = & \frac { \frac{1+\Delta_\widetil... ...detilde{S}}{1-\Delta_\widetilde{S}} + \cos 2 \widetilde{\gamma}. }\end{eqnarray}$$ (30) (31)

It is easy to verify that when $\Delta_\widetilde{S}=0$ (isotropic case) we get, as expected, $\widetilde{\gamma}_s=\widetilde{\gamma}_r=\widetilde{\gamma}_r$.B