Least-squares imaging and deconvolution using the hybrid norm conjugate-direction solver

Introduction

L1 norm optimization in many situations are more desirable than the conventional least squares (L2) optimization. However currently widely used methods like IRLS (Iterative Reweighted Least Squares) or weighted-median (Guitton, 2005; Claerbout and Muir, 1973; Darche, 1989) require the users to fine tune extra solver parameters in order to obtain a pleasing result. The sensitivity of such parameters make the solvers cumbersome to use since the users have to do trial-and-error. We developed a robust and efficient L1-type solver (Claerbout, 2009b) that uses a hybrid norm function to approximate the L1 norm, and implemented a generalized conjugate-direction (CD) method by using Taylor's expansion (Maysami and Mussa, 2009).

This solver is convenient to apply, because the function interface is almost the same as the traditional least-squares (L2) solver in the SEP library. The user must specify one additional parameter: the residual quantile. Fortunately this parameter has a clear physical meaning (Claerbout, 2009b). Users should assign this parameter according to prior observation or expectation of the model's spikiness/blockness.

In this paper we show the usefulness of the hybrid solver by applying it on the LSI imaging and deconvolution problems. The L1 inversion of LSI imaging (Least Squares Inverse) problem is preferable to L2 inversion, because it better perseves the spikiness/sparseness that are commonly encountered in reflectivity models. When the model regularization is posed with the L2 norm, it is hard to honor spikness/sparseness, because the L2 norm cannot tolerate large values in the model. In contrast, the L1 type norm fits our regularization goal very well.

A similar motivation applies to the deconvolution problem; conventional deconvolution assumes the reflectivity series to be random (white spectrum), whereas we argue that a sparse reflectivity series is more appropriate (and often more desirable) in practice.

Least-squares imaging and deconvolution using the hybrid norm conjugate-direction solver