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Stolt Migration

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

\begin{displaymath}
p(t,y,h) \rightarrow
P(\omega,k_y,k_h) \rightarrow
P'(k_z,k_y,k_h) \rightarrow
p'(z,y,h).\end{displaymath}

An important component of PrSM is the remapping from the $(\omega,k_y,k_h)$domain to the (kz,ky,kh) domain, where $\omega$, kz represent, respectively, the frequency and the vertical wavenumber, and ky,kh 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  
 \begin{displaymath}
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 ),\end{displaymath} (1)
where kg and ks 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 kz from Equation (1) as:  
 \begin{displaymath}
\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]\end{displaymath} (2)
or  
 \begin{displaymath}
\omega^2 = \frac{v^2}{k_z^2} 
\left[ k_z^2+k_h^2\right]
\left[ k_z^2+k_y^2\right].\end{displaymath} (3)


next up previous print clean
Next: 2-D Residual Stolt Migration Up: Sava: Residual migration Previous: Introduction
Stanford Exploration Project
6/30/1999