3-D Residual Stolt Migration

The RPrSM equations in three dimensions can be written similarly to those in two dimensions if we consider the relationship between source-geophone and midpoint-offset coordinates as follows:

$$\vec{k_g}= \vec{k_y}+ \vec{k_h}$$

$$\vec{k_s}= \vec{k_y}- \vec{k_h}.$$

The equivalent prestack residual migration equation in source-geophone coordinates thus becomes

$$\begin{array} {r} k_{z_m}=\frac{1}{2} \sqrt{ \frac{v_0^2}{v_... ...t)^2\right]} {16k_{z_0}^2}-\vert\vec{k_s}\vert^2} .\end{array}$$ (7)

In midpoint-offset coordinates, the same equation becomes

$$\begin{array} {r} k_{z_m}=\frac{1}{2} \sqrt{ \frac{v_0^2}{v_... ...]} {16k_{z_0}^2}-\vert\vec{k_y}-\vec{k_h}\vert^2} ,\end{array}$$ (8)

which, for the post-stack case, takes the form

$$k_{z_m}= \sqrt{ \frac{v_0^2}{v_m^2}\left[k_{z_0}^2+ \vert\vec{k_y}\vert^2\right]-\vert\vec{k_y}\vert^2}.$$ (9)