Introduction

For about ten years, iterative algorithms called IRLS (Iterative Reweighted Least-Squares) algorithms have been developed to solve L^p-norm minimization problems, for $1\leq\!p\!\leq 2$, 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 L¹ solution; previously, L¹ solutions were found by linear programming techniques (Taylor et al., 1979), which need a large quantity of computer memory. L¹ solutions are known to be more ``robust'' than L² 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 L¹-norm algorithm in presence of spiky noise using a simple synthetic example. This algorithm will in fact sharpen the L² 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 L¹-L² norm, for more general noise problems.