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To handle the inherent non-stationarity of seismic data, I estimate
a bank of non-stationary filters using helical boundary
conditions Claerbout (1998); Mersereau and Dudgeon (1974). This approach has been successfully
utilized by Rickett et al. (2001) to attenuate surface-related multiples.
As described before [equation (
)],
I use the Huber norm to approximate the
norm and a standard conjugate-gradient
solver with the
norm. The filter coefficients vary smoothly
across the output space by introducing a regularization term inside
equation (
) Crawley (2000); Rickett et al. (2001).
The misfit function to minimize becomes
| ![\begin{displaymath}
e_1({\bf f})=\vert{\bf Mf}-{\bf d}\vert _{Huber}+\epsilon^2\Vert{\bf Rf}\Vert^2,\end{displaymath}](img41.gif) |
(13) |
interl2
Figure 7 (a) The estimated primaries
with the
norm. (b) The estimated internal multiples with the
norm. Ideally, (b) should look like Figure
b,
but it does not.
interl1
Figure 8 (a) The estimated primaries
with the
norm. (b) The estimated internal multiples with the
norm. Beside some edge-effects, (b) resembles
closely Figure
b. The adaptive subtraction worked
very well.
where
is the unknown vector of filter coefficients for the
non-stationary matching filters and
is a regularization
operator. The Helix derivative Claerbout (1998) is chosen for
.In the following results, the non-stationary filters
are 1-D. The same number of coefficients per filter are estimated with
both
and
norms.
Next: Adaptive subtraction results
Up: 2-D data example: attenuation
Previous: The synthetic data
Stanford Exploration Project
5/5/2005