Stolt Migration

Prestack Stolt migration (PrSM) was summarized as Claerbout (1985)

$$p(t,y,h) \rightarrow P(\omega,k_y,k_h) \rightarrow P'(k_z,k_y,k_h) \rightarrow p'(z,y,h).$$

An important component of PrSM is the remapping from the $(\omega,k_y,k_h)$domain to the (k_z,k_y,k_h) domain, where $\omega$, k_z represent, respectively, the frequency and the vertical wavenumber, and k_y,k_h represent the midpoint and offset wavenumbers.

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

$$k_z=\frac{1}{2} \left ( \sqrt{\frac{\omega^2}{v^2}-k_g^2}+\sqrt{\frac{\omega^2}{v^2}-k_s^2} \right ),$$ (1)

where k_g and k_s stand for, respectively, the geophone and the source wavenumbers. It is desirable to implement the remapping as a pull operator, to avoid numerical problems in the inverse Fourier transform. A detailed discussion on the advantages and disadvantages of the different mappings is done by Levin 1994. We can, therefore, express $\omega$ as a function of k_z from Equation (1) as:  

$$\omega^2 = \frac{v^2}{16 k_z^2} \left[ 4 k_z^2+(k_g-k_s)^2\right] \left[ 4 k_z^2+(k_g+k_s)^2\right]$$ (2)

or  

$$\omega^2 = \frac{v^2}{k_z^2} \left[ k_z^2+k_h^2\right] \left[ k_z^2+k_y^2\right].$$ (3)