Reflections in $\left(\tau,\xi\right)$

In this Appendix we derive the relationships between the ray parameters of the $\tau$-rays reflected from a planar dipping reflector, and the ray parameters of the incident rays. We start by the equivalent relationship for z-rays. The ray parameters of the reflected z-ray  $\left(p_{x\Uparrow},p_{x\Uparrow}\right)$are related to the ray parameters of the incident z-rays  $\left(p_{x\Downarrow},p_{x\Downarrow}\right)$as follows:

$$\begin{eqnarray} p_{x\Uparrow} & = & p_{x\Downarrow} \cos 2\alpha_z + p_{z\Downa... ... & p_{x\Downarrow} \sin 2\alpha_z - p_{z\Downarrow} \cos 2\alpha_z\end{eqnarray}$$ (21)

where $\alpha_z$ is the dip angle of the reflector.

The ray parameters of the $\tau$-rays are related to the ray parameters of the z-rays by

$$\begin{eqnarray} p_{x} & = & p_{\xi} + p_{\tau} \sigma_f \nonumber \\ p_{z} & = & p_{\tau} \frac{2}{V}.\end{eqnarray}$$ (22)

Substituting equation 22 into equation 21 we get

$$\begin{eqnarray} p_{\tau\Uparrow} &=& p_{\xi\Downarrow} \frac{V}{2} \sin 2\alpha... ...\frac{\sigma_f^{2} V}{2} \right) \sin 2\alpha_z \right]. \nonumber\end{eqnarray}$$ (23)

The dip $\alpha_z$ of a reflector in depth is related to the time-dip angle $\alpha_\tau$ by

$$\tan\alpha_z = \frac{V}{2} \left( \tan\alpha_\tau - \sigma_f \right).$$ (24)