** Next:** Example 1 - 2-D
** Up:** Acoustic wave-equation in 3D
** Previous:** One-way wavefield extrapolation

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 *a*_{i} and evaluating the results
at stationary reference values *b*_{i}. 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 *a*_{i} 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