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The numerical scheme described by equation ()
can probably handle some amount of lateral velocity variations,
but it would be inaccurate for more complex velocity
functions.
In the derivation presented in the previous section
there is no assumption
of mild lateral velocity variations
up to equation ().
The problem with equation () is that it
is fourth order in time.
In addition to the desired
solution it has another solution that can generate artifacts
and cause instability.
Therefore, a direct solution by finite-differences
would encounter problems with the spurious solution.
() describes a similar problem
when solving an acoustic wave equation for anisotropic media.
However,
it is fairly straightforward to derive
an approximation to equation () that is more accurate than
equation ().
Equation () can be easily solved for 299#299because it contains only the even powers of 2#2.We can then approximate
the square root that appears in the formal solution for
299#299 as
The useful solution of
equation () can then be approximated as
Equation () degenerates to
equation () when 309#309 is zero.
It is more accurate than
Equation () for 309#309 different than zero,
but it is likely to break down for strong lateral velocity
variations.
Equation () shares
with equation ()
the fundamental problem of instability for horizontally
propagating waves.
Therefore, I have not implemented a numerical scheme to solve
equation () yet.
However,
it is possible to define a mixed implicit-explicit
method to solve
equation (),
similar to the one proposed by
()
to solve Alkhalifah's
acoustic wave equation for anisotropic media.
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Up: From downward continuation to
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Stanford Exploration Project
6/7/2002