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A predictive *L*^{2} deconvolution produces a minimum-phase inverse filter,
independently of the phase of the wavelet *w*. Thus it would be interesting
to study the effect of the *L*^{1} deconvolution on non minimum-phase inputs.
I used the same input reflection sequence as before, but changed the input
wavelet into a constant phase (60^{o}) wavelet. I applied *L*^{2} and *L*^{1}
deconvolution on the new synthetic trace. I initialized the *L*^{1} algorithm
first with the *L*^{2} filter, then without the input filter. The results are
shown on Figure .
It appears that the *L*^{1} deconvolution does not improve the phase properties
of the output. In one case (starting from *L*^{2} filter), the filter is
from the start close to the *L*^{2} filter; in the other case, the first
iteration of the IRLS algorithm forces the filter to be close to the *L*^{2}
filter. In the first case, the *L*^{1} deconvolution results in a sharpening
of the *L*^{2} result, again because the residuals have an a priori exponential
distribution.

The *L*^{1} deconvolution can be related to
the variable norm deconvolution (Gray, 1979), in the particular case when
the processing tries to draw the statistical distribution of the time series
from a gaussian one (seismic trace) to an exponential one (residuals).
However, *L*^{1} and Gray's deconvolutions are slightly different, because the
IRLS algorithm does not suppose any a-priori distribution for the inputs; the
normal equations of the variable norm deconvolution are in fact a linear
combination of the *L*^{2} normal equations and of the *L*^{1} normal equations.

In conclusion, the IRLS algorithm in predictive deconvolution essentially
sharpens the residuals. If an initial estimate is used (Wiener filter, Burg
filter, or any minimum-entropy deconvolution filter), the residuals of
*this initial filter* will be sharpened; otherwise, the result will be a
sharpened version of the *L*^{2} residuals. It would be interesting in fact to
modify the first iteration of the algorithm, but would it still guarantee the
convergence?

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** Up:** SYNTHETIC EXAMPLE
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Stanford Exploration Project

1/13/1998