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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
stolt
view

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 vm, we can then write  
 \begin{displaymath}
\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. \end{displaymath}   

The goal of RPrSM is to obtain kzm from kz0. If we use the first equation of (4) to substitute $\omega$in the second equation of (4), we obtain  
 \begin{displaymath}
\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}\end{displaymath} (4)
or  
 \begin{displaymath}
\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}\end{displaymath} (5)

Equation (6) represents the RPrSM equation in two dimensions. For post-stack data, the same equation takes the familiar form
\begin{displaymath}
k_{z_m}= \sqrt{ \frac{v_0^2}{v_m^2}\left[k_{z_0}^2+k_y^2\right]-k_y^2}.\end{displaymath} (6)


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Next: 3-D Residual Stolt Migration Up: Sava: Residual migration Previous: Stolt Migration
Stanford Exploration Project
6/30/1999