Stolt migration

Prestack Stolt migration can be summarized (24) as a succession of transformations from seismic data to seismic images as follows:  

$$\mathcal{d} ( t ,{\bf m}, {\bf h}) \rightarrow \mathcal{D} (... ..._m, {\bf k}_h) \rightarrow \mathcal{r} ( \zz,{\bf m}, {\bf h}).$$ (29)

where $\mathcal{d}$ and $\mathcal{r}$ are representations of the data and image in the space domain, while $\DD$ and $\RR$ are the equivalent representations in the Fourier domain. In stoltmain, t stands for time, ${\bf m}$ for midpoint location, ${\bf h}$ for half-offset, and $\zz$ for depth.

The central component of prestack Stolt migration is the re-mapping from the $(\omega,{\bf k}_m,{\bf k}_h)$domain to the $(\kzz,{\bf k}_m,{\bf k}_h)$ domain, where $\omega$ and $\kzz$ represent, the temporal frequency and the depth wavenumber respectively, and ${\bf k}_m=({k_m}_x,{k_m}_y)$ and ${\bf k}_h=({k_h}_x,{k_h}_y)$ represent the midpoint and offset wavenumbers.

If we consider the representation of the input data in shot-receiver coordinates, the mapping takes the form  

$$\kzz= \frac{1}{2}\sqrt{\frac{\omega^2}{v^2}-\vert{\bf k}_r\v... ... +\frac{1}{2}\sqrt{\frac{\omega^2}{v^2}-\vert{\bf k}_s\vert^2},$$ (30)

where ${\bf k}_r$ and ${\bf k}_s$ stand for, respectively, the receiver and the source wavenumbers. From dsr-sg we can express $\omega$ as a function of $\kzz$: 

$$\omega^2 = v^2 \frac{\lb 4 \kzz^2+\lp\vert{\bf k}_r\vert-\ve... ...p\vert{\bf k}_r\vert+\vert{\bf k}_s\vert\rp^2 \rb} {16 \kzz^2}.$$ (31)

We can obtain an equation equivalent to () in midpoint-offset coordinates, if we make the usual change of variables:  

$$\bea{l} {\bf k}_r= {\bf k}_m+ {\bf k}_h\\ {\bf k}_s= {\bf k}_m- {\bf k}_h. \eea$$ (32)