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## Split-Step Fourier Approximation

The extrapolation wavenumber defined in equations 14 and 15 cannot be implemented purely in the Fourier domain due to the presence of mixed-domain fields (i.e. a function of both and simultaneously). This can be addressed using an extended version of the split-step Fourier approximation Stoffa et al. (1990), a popular approach that uses Taylor expansions to separate into two parts: . Wavenumbers and represent a pure Fourier domain phase-shift and a mixed domain split-step correction, respectively.

The phase-shift term is given by,
 (16)
where are reference values of . The split-step approximation is developed by performing a Taylor expansion about each coefficient ai and evaluating the results at stationary reference values bi. The stationary values of and are assumed to be zero. This leads to a split-step correction of,
 (17)
where '' denotes "with respect to a reference medium". The partial differential expressions in equation 17 are,
 (18)
resulting in the following split-step Fourier correction,
 (19)
Note that we could use many reference media followed by interpolation similar to the phase-shift plus interpolation (PSPI) technique of Gazdag and Sguazzero (1984).

Importantly, even though there are additional ai coefficients in the dispersion relationship, these can be made smooth through mesh regularization such that fewer sets of reference parameters are needed to accurately represent wavenumber . In addition, situations exist where some coefficients are zero or negligible. For example, the coefficients for a weakly non-orthogonal coordinate system (i.e. ) within a kinematic approximation reduce to,
 (20)
Additional special cases are presented in Appendix A.

Next: Example 1 - 2-D Up: Acoustic wave-equation in 3D Previous: One-way wavefield extrapolation
Stanford Exploration Project
4/5/2006