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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:

 
 \begin{displaymath}
\frac{\sin\phi}{v_p} = \frac{\sin\sigma}{v_s}.\end{displaymath} (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.


next up previous print clean
Next: A Kirchhoff perspective for Up: Imaging of converted-wave Ocean-bottom Previous: PS angle-domain transformation
Stanford Exploration Project
12/14/2006