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*V*(*x*,*y*,*z*) and non-stationary inverse convolution

Implicit extrapolation with the helical factorization discussed in the
previous chapter can be easily extended to smoothly-variable
velocities.
Stationary filtering and inverse filtering can be replaced by their
non-stationary counterparts, and the spectral factorization becomes a
problem of *LU* decomposition.
I develop a solution the cost of which remains proportional to the
number of grid nodes [*O*(*N*)]. This solution is also exactly
equivalent to the constant velocity factorization in the
smooth-velocity limit.
Unfortunately, however, I will show that the Godfrey/Muir
1979 bulletproofing cannot be simply applied to the
helical factorization, and so the method is susceptible to stability
problems.
Numerical examples confirm the accuracy of the method in models with
smoothly-varying velocity; however, I observe instability in models
with strong lateral velocity variations.

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** Up:** Spectral factorization of wavefields
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Stanford Exploration Project

5/27/2001