Theory Review

() describe three possible ways to perform residual migration. The most precise method is an exact derivation which involves the combination of the $\rho_p$, $\rho_s$ and $\gamma_0$parameters. This method attempts to simultaneously correct the effect of two inaccurate velocity fields.

Assuming that the initial migration was done with the velocities v_0p and v_0s, and that the correct velocities are v_mp and v_ms, we can then write  

$$\left\{\begin{array} {l} k_{z_0}=\frac{1}{2} \left ( \sqrt{\... ...2}{v_{ms}^2}-k_g^2} \right ) . \ \nonumber\end{array} \right.$$

Solving for $\omega^2$ in the first equation of () and substituting it in the second equation of (), we obtain the expression for prestack Stolt depth residual migration for converted waves:  

$$\begin{array} {r} k_{z_m}=\frac{1}{2} \sqrt{ \rho_p^2 {\kapp... ... \sqrt{ \rho_s^2 {\gamma_0}^2 {\kappa_0}^2 - k_g^2},\end{array}$$ (115)

where ${\kappa_0}^2$ is the transformation kernel defined as:

$${\kappa_0}^2= \frac{4({\gamma_0}^2 +1){k_{z_0}}^2 + ({\gamma... ...2 - {k_g}^2)+4{\gamma_0}^2 {k_{z_0}}^2}}{({\gamma_0}^2 - 1)^2},$$

and $\rho_p=\frac{v_{0p}}{v_{mp}}$, $\rho_s=\frac{v_{0s}}{v_{ms}}$, and $\gamma_0=\frac{v_0p}{v_0s}$.

This formulation depends only on velocity ratios. This fact implies that it is a valid formulation for non-constant velocity media, as suggested by ().