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A Kirchhoff perspective for PS-ADCIG transformation

  In this appendix, I obtain the relation to transform subsurface offset-domain common-image gathers into angle-domain common-image gathers for converted waves. To perform this derivation, I use the geometry in Figure [*] in order to obtain the parametric equations for migration on a constant velocity medium.

Following the derivation of Fomel (1996) and Fomel and Prucha (1999), and applying simple trigonometry and geometry to Figure [*], I obtain parametric equations for migrating an impulse recorded at time tD, data-midpoint mD, and data surface-offset hD as follows:

Figure 1
Parametric formulation of the impulse response.

z_\xi&=& (L_s+L_r)\frac{\cos{\beta_r} \cos{\beta_s}}{\cos{\beta...

where the total path length is:

t_D &=& S_sL_s+S_rL_r, \nonumber\ z_s - z_r &=& L_s\cos{\beta_s}-L_r\cos{\beta_r}.\end{eqnarray}

From that system of equations, Biondi (2005) shows that the total path length is

L=\frac{t_D}{2}\frac{\cos{\beta_r}+\cos{\beta_s}}{S_s\cos{\beta_r}+S_r\cos{\beta_s}}.\end{displaymath} (81)

I can rewrite system [*] as:

 z_\xi&=& \frac{(L_s+L_r)}{2} \frac{\cos^2{\alpha}-\sin^2{\gamm...
 ...\xi&=& m_D - \frac{(L_s+L_r)}{2}\frac{\sin{\alpha}}{\cos{\gamma}}.\end{eqnarray}

where $\alpha$ and $\gamma$ follow the same definition as in equation [*]. The total path length, L, in terms of the angles $\alpha$ and $\beta$ is:

L(\alpha,\beta)=\frac{t_D}{(S_r+S_s)+(S_r-S_s)\tan{\alpha}\tan{\gamma}}\end{displaymath} (83)