Derivation of the equation for the angle-domain transformation

This appendix presents the derivation of the main equation for this paper, that is, the transformation from the subsurface offset domain into the angle domain for converted-wave data. Rosales and Biondi (2005) present the derivation for the angle-domain transformation. Biondi (2005) and Shragge et al. (2005) present similar equations for different applications, for anisotropic case and for the forward-scatter case, respectively. Throughout this Appendix we will use the following defintions:

$$\begin{eqnarray} \tan{\theta_0}&\equiv& -\frac{\partial z_\xi}{\partial h_\xi} \... ...ber\\ \d &\equiv& -\frac{\partial z_\xi}{\partial m_\xi} \nonumber\end{eqnarray}$$

In this definitions, $\tan{\theta_0}$ is the pseudo-reflection angle, and $\d$ is field for the local image dips. The bases for this definition resides in the conventional PP case. For that case, the pseudo-reflection angle represents the reflection angle, and the field $\d$ represents the geological dip Fomel (1996). Based on this definition, the angle equations from Biondi (2005); Rosales and Biondi (2005); Shragge et al. (2005) can be rewritten as:

$$\begin{eqnarray} \tan{\theta_0}&=& \frac{ \tan{\theta} + \mathcal{S}\tan{\alpha}... ...cal{S}\tan{\theta} } { 1 - \mathcal{S}\tan{\theta} \tan{\alpha}. }\end{eqnarray}$$ (6) (7)

Following basic algebra, equation 7 can be rewritten as:  

$$\tan{\alpha} = \frac { \d - \mathcal{S}\tan{\theta} } { 1 - \mathcal{S}\d \tan{\theta} }$$ (8)

Substituing equation 8 into equation 6, and following basic algebraic manipulations, we obtain equation 4 in the paper.