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A classical approach for attenuating multiples consists of building
a multiple model, e.g., as described by Verschuur et al., 1992 or
Berkhout and Verschuur 1997, and adaptively subtracting
this model from the multiple infested-data by estimating shaping filters
Dragoset (1995); Liu et al. (2000); Rickett et al. (2001).
The estimation of the shaping filters is usually done in a least-squares
sense making these filters relatively easy to compute.
Implicitly, by using the norm, the resulting signal, after the
filter estimation step, is assumed to be orthogonal to the noise and has minimum
energy. These assumptions might not hold and other methods, such as
pattern-based approaches Guitton et al. (2001); Spitz (1999)
have been proposed to circumvent these limitations. For instance, when
a strong primary is surrounded by weaker multiples, the multiple model
will match the noise (multiples) as well as the signal (primaries)
such that the difference between the data and the filtered multiple
model is minimum in a least-squares sense. Consequently, some primary energy
might leak in the estimated multiples and *vice versa*.
Therefore, a new criterion or norm for the filter estimation step is needed.
In this paper, I propose estimating the shaping filters with the
norm instead of the norm, thus relaxing the need for
the signal to have minimum energy. This choice is driven by the simple
fact that the norm is robust to ``outliers'' Claerbout and Muir (1973)
and large amplitude anomalies. Because the norm is
singular where any residual component vanishes, the Huber norm
with L-BFGS solver of Chapter is chosen instead.
This method gives an excellent approximation of the norm.

In this first section following the introduction, some
limitations of the least-squares criterion on a simple 1-D
problem are illustrated. I then introduce the proposed approach based on the
norm to improve the multiple attenuation results. In a second
synthetic example, internal multiples are attenuated with
the and norms.
Finally, I utilize shaping filters on a multiple contaminated
gather from a seismic survey showing that the norm
leads to a substantial attenuation of the multiples.

** Next:** Principles of norm and
** Up:** Adaptive subtraction of multiples
** Previous:** Summary
Stanford Exploration Project

5/5/2005