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I applied *L*^{1} deconvolution on a marine CMP gather from BP Alaska.
This gather, recorded in South California, is composed of 36 traces;
the near-offset is 65.7 m, the intertrace distance 33.3 m.
The time window I used is 0.7-3.0 sec; the time sampling is 4 msec.
The CMP gather is presented in Figure , along with the
residuals of the *L*^{2} deconvolution.
For graphics convenience, I only present a time window between 0.7 and 2
sec. The residuals were computed with
the same filter for all the traces, composed of 50 samples; I used a
small prewhitening factor (5 %), in accordance to the small level of noise
before the first arrival. It appears on the first arrival that the seismic
wavelet has a precursor, making it clearly non minimum-phase; the source
was composed of two water-guns.
I applied *L*^{1} deconvolution on this gather, with the *L*^{2} predictive
filter as initial estimate; I computed one 50-sample filter for all traces. The
residuals of this *L*^{1} deconvolution are presented in Figure ,
with also the difference between the *L*^{2} residuals and the *L*^{1} residuals.
As I expected, the *L*^{1} residuals are a sharpened version of the *L*^{2}
residuals: the largest amplitudes have been increased, the smallest have been
decreased. This is not really obvious on the sections, but it appeared clearly
when I computed the kurtosis of the traces in the central part of the gather
(between 1.5 and 2.5 sec, to avoid the strong water-bottom reflections):
the kurtosis of the *L*^{1} residuals were larger than those of the *L*^{2}
residuals. It also appears on the difference section when we look at the
water-bottom reflection. However, the estimation of the phase of the seismic
wavelet has not been improved, and especially the precursor has not been
removed, but accentuated.

This sharpening of the residuals presents the advantage of increasing the
contrast between reflectors, and stressing the main reflectors. Moreover,
as this gather presents some lateral variations of amplitude, these
variations will be enhanced by the sharpening of the residuals.

I also tried a *L*^{p} deconvolution with *p*=0.1, though the objective function
is no longer a norm. However, with the *L*^{2} filter as initial estimate, the
algorithm converged. As expected, the result was once again a sharpening of
the *L*^{2} residuals, but it was very similar to the result of the *L*^{1}
deconvolution.

** Next:** MIXED L-L NORM MINIMIZATIONS
** Up:** Gilles Darche: L Deconvolution
** Previous:** L norm and non
Stanford Exploration Project

1/13/1998