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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 *c*_{j}.
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 *s*_{0} and *c*_{j}_{0} 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 (*s*_{0},*c*_{j}_{0}).
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