Transformation into independent angles

From equation (3) we established a relation between the propagation angles for the down-going and up-going plane-waves, $\beta_s$ and $\beta_r$, respectively. Now, from Figure  it is easy to see that the propagation angles are related to: 1) the incidence angle of the down-going plane wave into the reflector ($\gamma_i$); 2) the reflection angle of the up-going plane wave ($\gamma_r$); and the structural dip ($\alpha_x$). The relation among all the angles is  

$$\beta_s= \alpha_x- \gamma_i, \;\;\;\;\;\; {\rm and} \;\;\;\;\;\; \beta_r= \alpha_x+ \gamma_r.$$ (11)

Combining equation (3) and (11), we can see the direct relation between the angles that we compute with relations (7) and/or (10) and the real structural dip, the incidence angle, and the reflection angle. That is:

$$\begin{eqnarray} 2\gamma &=& \gamma_r+ \gamma_i, \nonumber\\ 2\alpha &=& 2\alpha_x+ (\gamma_r- \gamma_i).\end{eqnarray}$$ (12)

It is easy to note that the opening angle $\gamma$ is the reflection angle and $\alpha$ is the geological dip when $\gamma_i=\gamma_r$, which is only valid for the single-mode case.

With these equations and Snell's law, we can convert the full-aperture angle ($\gamma$)obtained with equation (7) or (10) into the incidence angle ($\gamma_i$) or the reflection angle ($\gamma_r$):

$$\begin{eqnarray} \tan{\gamma_i} &=& \frac{\phi \sin{2\gamma}}{1+\phi \cos{2\gamm... ...r\\ \tan{\gamma_r} &=& \frac{\sin{2\gamma}}{\phi + \cos{2\gamma}}.\end{eqnarray}$$ (13)

Appendix A presents a full derivation of the same equations but with the perspective of the Kirchhoff approach. The reader is encourage to follow that demonstration.