Theory Review

Rosales et al. (2001) presented residual pre-stack Stolt migration for converted waves, this new operator is asymmetric and this asymmetry property can be used for imaging under salt edges, by keeping unchanged one travel-time leg and modifying the other.

The asymmetric pre-stack Stolt residual operator can be written as:

 

$$\begin{array} {r} k_{z_m}=\frac{1}{2} \sqrt{ \rho_s \overlin... ...mma_0}^2 \overline{\overline{{\kappa_0}^2}} - k_g^2}\end{array}$$ (1)

where

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

$\rho_s =\frac{v_{s0}}{v_{sm}}$, $\rho_g=\frac{v_{g0}}{v_{gm}}$, and $\gamma_0=\frac{v_{s0}}{v_{g0}}$;the subscripts s and g denote for source and geophone, respectively.

The definition of the sources and receivers position is important since it explains the meaning of the three parameters used for the residual migration. If we define the source position along the fastest path, the value of $\gamma_0$ will be bigger than one. The contrary would happen if we define the geophone position along the fastest path.