Prestack Stolt residual migration

In general, residual migration improves the quality of an image without re-migration of the original data; instead, a transformation is applied to the current migrated image.

In prestack Stolt residual migration, we attempt to correct the effects of migrating with an inaccurate reference velocity by applying a transformation to images transformed to the Fourier domain. Supposing that the initial migration was done with the velocity v₀, and that the correct velocity is v, we can use dsr-sg to derive ${\kzz}_0$, the vertical wavenumber for the reference velocity, and $\kzz$, the vertical wavenumber for the correct velocity.

Mathematically, the goal of prestack Stolt residual migration is to obtain $\kzz$ from ${\kzz}_0$. If we elliminate $\omega$ from ${\kzz}_0$ and $\kzz$,and make the notation $\rho=\frac{v_0}{v}$,we obtain the residual migration equation for full 3D prestack seismic images:  

$$\bea{r} \kzz= \frac{1}{2} \sqrt{ \rho^2 \frac{ \lb 4{\kzz}_0... ...f k}_s\vert\rp^2\rb}{16{\kzz}_0^2}-\vert{\bf k}_s\vert^2}, \eea$$ (33)

which can also be represented in midpoint-offset coordinates using the change of variables in changevar. If we make the change of variables

$$\mu^2 = \frac{ \lb 4 {{\kzz}_0}^2 + \lp {\bf k}_r- {\bf k}_s... ...}_0}^2 + \lp {\bf k}_r+ {\bf k}_s\rp^2 \rb } {16{{\kzz}_0}^2}.$$ (34)

we obtain a simplified version of myresmig-3d-pr:

$$\kzz = \frac{1}{2}\sqrt{ \rho^2 \mu^2 - \vert{\bf k}_s\vert^2} + \frac{1}{2}\sqrt{ \rho^2 \mu^2 - \vert{\bf k}_r\vert^2} \;.$$ (35)

In the 3D post-stack case, when ${\bf k}_h={\bf 0}$, myresmig-3d-pr becomes:  

$$\kzz= \sqrt{ \rho^2 \lb {\kzz}_0^2+ \vert{\bf k}_m\vert^2 \rb -\vert{\bf k}_m\vert^2}.$$ (36)

In the 2D prestack case (k_my=0 and k_mx=0), we can write myresmig-3d-pr as:  

$$\bea{r} \kzz=\frac{1}{2} \sqrt{ \rho^2 \frac{ \lb {\kzz}_0^2... ...zz}_0^2+{k_m}_x^2 \rb} {{\kzz}_0^2}-({k_m}_x-{k_h}_x)^2}. \eea$$ (37)

For 2D post-stack data, myresmig-3d-po and myresmig-2d-pr become  

$$\kzz= \sqrt{ \rho^2\lb {\kzz}_0^2+{k_m}_x^2\rb-{k_m}_x^2},$$ (38)

which can also be written in the familiar form (102):  

$$\omega = \sqrt{\omega_0^2+{k_m}_x^2 \lp v_0^2-v^2\rp},$$ (39)

where, by definition, $\ww_0= {\kzz}_0 v_0$, and $\ww = \kzz v$.