Mapping of PS-ADCIGs

Following definition , and after explicitly computing the half-aperture angle with equation , I have almost all the tools to compute the P-incidence angle ($\phi$), and the S-reflection angle ($\sigma$). Snell's law, and the P-to-S velocity ratio are the final two components for this procedure. The final result of this process is the mapping of the PS-ADCIGs, that are function of the half-aperture angle, into two angle-gathers. The first one is a function of the P-incidence angle, I refer to this angle gather as P-ADCIG. The second angle-gather is a function of the S-reflection angle to form an S-ADCIG. After basic algebraic and trigonometric manipulations, the final two expressions for this mapping are (Appendix B):

$$\begin{eqnarray} \tan{\phi} &=& \frac{\gamma \sin{2\theta}}{1+\gamma \cos{2\thet... ... \ \tan{\sigma} &=& \frac{\sin{2\theta}}{\gamma + \cos{2\theta}}.\end{eqnarray}$$ (33) (34)

Expressions  and  clearly show a non-linear relation between the half-aperture angle and both the incident and reflection angles. The main purpose of this set of equations is to observe and analyze the converted-wave angle-gathers in two different domains each one corresponding to the incidence angle and the reflection angle. The analysis of these angle-gathers might help to obtain residual moveout equations for both the P-velocity and the S-velocity; therefore, individual updates for each of the velocity models.