Mapping of PS-ADCIGs

  This appendix presents the derivation for the equations that allow to map the existing PS-ADCIGs that are a function of the half-aperture angle into two angle-domain common-image gathers. The first angle gather is a function of the P-incidence angle, the second angle gather depends on the S-reflection angle. The first element to obtain these mapping equations is Snell's law:

 

$$\frac{\sin\phi}{v_p} = \frac{\sin\sigma}{v_s}.$$ (72)

The angles $\phi$ and $\sigma$ are the incidence and reflection angles, respectively. From the definition of the full-aperture angle, $\theta$,(equation ), I obtain the following:

$$\begin{eqnarray} \sigma & = & 2\theta - \phi, \ \phi & = & 2\theta - \sigma.\end{eqnarray}$$ (73) (74)

Introducing equations  and  into equation , and using the P-to-S velocity ratio ($\gamma$), I obtain:

$$\begin{eqnarray} \sin\phi & = & \gamma \sin(2\theta - \phi), \ \sin\sigma & = & \gamma^{-1} \sin(2\theta - \sigma).\end{eqnarray}$$ (75) (76)

Using simple trigonometric relations and basic algebra, from equations  and , I get, respectively,

$$\begin{eqnarray} (1+\gamma \cos 2\theta) \tan\phi & = & \gamma \sin 2\theta, \ (\gamma + \cos 2\theta) \tan\sigma & = & \sin 2\theta.\end{eqnarray}$$ (77) (78)

Equations  and  translate into equations  and , in Chapter 3.