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For about ten years, iterative algorithms called IRLS (Iterative Reweighted
Least-Squares) algorithms have been developed to solve *L*^{p}-norm minimization
problems, for , so lying between the least-absolute-values
problem and the classical least-squares problem. Their main advantage is to
provide an easy way to compute a *L*^{1} solution; previously, *L*^{1} solutions
were found by linear programming techniques (Taylor et al., 1979), which need
a large quantity of computer memory. *L*^{1} solutions are known to be more
``robust'' than *L*^{2} solutions, being less sensitive to spiky high-amplitude
noise (Claerbout and Muir, 1973).
The applications of IRLS algorithms to geophysical problems have
been limited up to now to travel-time tomography (Scales et al., 1988), 1-D
acoustic inversion (Gersztenkorn et al.,1986). A simple example of deconvolution
on a synthetic trace was given by Yarlagadda et al. (1985).

Each iteration of the IRLS algorithm consists of a weighted least-squares
minimization, which is usually solved by a conjugate-gradient method. In many
cases, such as tomography, this method is very efficient; if the linear systems
involved are sparse, this method is also computationally fast.

However, deconvolution processes don't offer the same guarantees of
computational efficiency, because they are *by essence* ill-conditioned
and require many computations. Consequently, this algorithm demands some care
before it can be implemented for deconvolution problems.

I will show the efficiency of the *L*^{1}-norm algorithm in presence of spiky
noise using a simple synthetic example. This algorithm will in fact sharpen the
*L*^{2} residuals, as I will illustrate for predictive deconvolution; however, in
that case, it does not improve the estimation of the seismic wavelet. This
illustration will be completed by an example on a marine CMP gather.

Finally, I suggest the extension of this algorithm to use a mixed
*L*^{1}-*L*^{2} norm, for more general noise problems.

** Next:** L^{p} MINIMIZATION WITH THE
** Up:** Gilles Darche: L Deconvolution
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Stanford Exploration Project

1/13/1998