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Fourier finite-difference

In one of its most general forms Ristow and Ruhl (1994), we can write kz as  
 \begin{displaymath} k_z= {k_z}_o+ \omega\left[1- \frac{s}{s_o} \left(\sum\limits... ...hoose n} \S^{2n} \delta_n \right)\right] \left(s - s_o\right),\end{displaymath} (2)
where $\displaystyle{\frac{1}{2} \choose n}$ are binomial coefficients for integer numbers n, s represents the spatially variable slowness function at depth z, so is a constant approximation to s, and $\delta_n$ is a sum of terms derived from s and so. The FFD equation is obtained using two Taylor series expansions of the SSR equations written for kz and kzo. I give a full derivation of Equation (2) in Appendix A.

Higher accuracy can be achieved by replacing the Taylor expansion in Equation (2) with Muir's continuous fraction expansion Claerbout (1985). The equivalent form of the general Fourier finite-difference propagator is:  
 \begin{displaymath} k_z= {k_z}_o+ \omega\left[1- \frac{s}{s_o} \left(\sum\limits... ...\frac{\S^2}{a_n+b_n \S^2} \right)\right] \left(s - s_o\right),\end{displaymath} (3)
where an and bn are coefficients that, in general, depend on the medium and the constant reference slownesses, s and so. The coefficients an and bn are derived either by identification of terms between Equation (2) and Equation (3), after the approximation $\frac{1}{1-x^2}\approx 1+x^2$, or by an optimization problem as described by Ristow and Ruhl (1997).

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Next: Generalized screen Up: Mixed-domain migration theory Previous: Mixed-domain migration theory
Stanford Exploration Project
9/5/2000