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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
| |
(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.

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** Up:** 2-D data example: attenuation
** Previous:** The synthetic data
Stanford Exploration Project

5/5/2005