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We can use equation (12) to construct a numerical solution
to the oneway wave equation in the mixed
, domain.
The extrapolation wavenumber described in equation (12)
is, in general, a function which depends on several quantities
 
(14) 
where is the spacevariable slowness, and
are
coefficients which are computed numerically from the definition
of the coordinate system, as indicated by
equations (8).
For any given coordinate system, can be regarded as known.
Next, we write the extrapolation wavenumber as a
firstorder Taylor expansion relative to a reference medium:
 
(15) 
where and represent the spatially variable slowness and coordinate
system parameters, and and are the
constant reference values in every
extrapolation ``slab'' Sava (2000).
As usual, the first part of equation (15),
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
 

 
 (17) 
Equation (16) is motivated by
a wavefront normal propagation approximation.
By ``0'', we denote the reference medium (.We could also use many reference media, followed by
interpolation, similarly to the technique of
Gazdag and Sguazzero (1984).
For the particular case of Cartesian coordinates
(),
equation (16) reduces to
 
(18) 
which corresponds to the popular SplitStep Fourier (SSF)
extrapolation method Stoffa et al. (1990).
Next: Finitedifference solutions to the
Up: Sava and Fomel: Riemannian
Previous: Oneway waveequation in 3D
Stanford Exploration Project
10/14/2003