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 t_D, data-midpoint m_D, and data surface-offset h_D as follows:

 

angles2
Figure 1 Parametric formulation of the impulse response.

$$\begin{eqnarray} z_\xi&=& (L_s+L_r)\frac{\cos{\beta_r} \cos{\beta_s}}{\cos{\beta... ...beta_r}+\sin{\beta_r}\cos{\beta_s}} {\cos{\beta_r}+\cos{\beta_s}}.\end{eqnarray}$$ (79)

where the total path length is:

$$\begin{eqnarray} t_D &=& S_sL_s+S_rL_r, \nonumber\ z_s - z_r &=& L_s\cos{\beta_s}-L_r\cos{\beta_r}.\end{eqnarray}$$ (80)

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

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

I can rewrite system as:

$$\begin{eqnarray} z_\xi&=& \frac{(L_s+L_r)}{2} \frac{\cos^2{\alpha}-\sin^2{\gamm... ...\xi&=& m_D - \frac{(L_s+L_r)}{2}\frac{\sin{\alpha}}{\cos{\gamma}}.\end{eqnarray}$$ (82)

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:

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

 

• Tangent to the impulse response