2-D Residual Stolt Migration

In general, residual migration represents a method of improving the quality of the image without having to remigrate the original data, but rather only applying a transformation to the current migration image.

 

stolt Figure 1 A sketch of Stolt residual migration

In residual prestack Stolt migration (RPrSM), we attempt to correct the effects of migrating with an inaccurate reference velocity by applying a transformation to the data that have been transformed to the Fourier domain (Figure 1). Supposing that the initial migration was done with the velocity v₀, and that the correct velocity is v_m, we can then write  

$$\left\{\begin{array} {l} k_{z_0}=\frac{1}{2} \left ( \sqrt{\... ...a^2}{v_m^2}-k_s^2} \right ) . \\ \nonumber\end{array} \right.$$

The goal of RPrSM is to obtain k_zm from k_z0. If we use the first equation of (4) to substitute $\omega$in the second equation of (4), we obtain  

$$\begin{array} {r} k_{z_m}=\frac{1}{2} \sqrt{ \frac{v_0^2}{v_... ...k_{z_0}^2+(k_g+k_s)^2\right]} {16 k_{z_0}^2}-k_s^2}\end{array}$$ (4)

or  

$$\begin{array} {r} k_{z_m}= \frac{1}{2} \sqrt{ \frac{v_0^2}{v... ...[k_{z_0}^2+k_y^2\right]} {k_{z_0}^2}-(k_y-k_h)^2} .\end{array}$$ (5)

Equation (6) represents the RPrSM equation in two dimensions. For post-stack data, the same equation takes the familiar form

$$k_{z_m}= \sqrt{ \frac{v_0^2}{v_m^2}\left[k_{z_0}^2+k_y^2\right]-k_y^2}.$$ (6)