Next: Finite-difference solutions to the Up: Riemannian wavefield extrapolation Previous: One-way wave-equation

# Mixed-domain solutions to the one-way wave-equation

We can use oneway.3d to construct a numerical solution to the one-way wave equation in the mixed , domain. The extrapolation wavenumber described in oneway.3d is, in general, a function dependent on several quantities
 (14)
where is slowness, and are coefficients computed numerically from the definition of the coordinate system, as indicated by coefs.3d. Specifying a coordinate system, implicitly defines all coefficients cj.

We can write the extrapolation wavenumber as a first-order Taylor expansion relative to a reference medium:
 (15)
where and represent the spatially variable slowness and coordinate system parameters, and s0 and cj0 are the constant reference values at every extrapolation step.

The first part of mixed.3d, corresponding to the extrapolation wavenumber in the reference medium ,is implemented in the Fourier () domain, while the second part, corresponding to the spatially variable medium coefficients, is implemented in the space () domain.

If we make the further simplifying assumptions that and , we can write
 (16)

where

. s|_0 &=& 2s_0 4_0s_0^2 - _0^2 ,
. |_0 &=& - i_0 2_0^2 + _0^2-2_0 s_0^2 2_0^2 4_0s_0^2 - _0^2 ,
. |_0 &=& i 2_0 - _0 2_0 4_0s_0^2 - _0^2 .

By 0'', I denote the reference medium (s0,cj0). In principle, we could also use many reference media, followed by interpolation, similarly to the phase-shift plus interpolation (PSPI) technique of (38).

For the particular case of Cartesian coordinates (), mixed.3d.explicit reduces to
 (17)
which corresponds to the popular split-step Fourier (SSF) extrapolation method (100).

Next: Finite-difference solutions to the Up: Riemannian wavefield extrapolation Previous: One-way wave-equation
Stanford Exploration Project
11/4/2004