Regularization of geophysical ill-posed problems by iteratively re-weighted and refined least squares

Abstract

The iteratively re-weighted least squares (IRLS) is a commonly used algorithm which has received significant attention in geophysics and other fields of scientific computing for regularization of discrete ill-posed problems. The IRLS replaces a difficult optimization problem by a sequence of weighted linear systems. The optimum solution of the original problem is usually determined by computing the solution for various regularization parameters λ, each needing several re-weighted iterations (usually 10–15). In this paper, in order to decrease the required computation time (iterations) while maintaining good properties of the algorithm such as edge-preserving, the IRLS is augmented with a refinement strategy and the value of λ is progressively updated in a geometrical form during the iterations. The new algorithm, called iteratively re-weighted and refined least squares (IRRLS), can be interpreted as a Landweber iteration with a non-stationary shaping matrix which is updated based on the solution obtained from previous iteration. Two main properties of IRRLS are (1) the regularization parameter is the stopping iteration and (2) it is equipped with a tuning parameter which makes it flexible for recovering models with different smoothness. We show numerically that both the residual and regularization norms are monotone functions of iteration and hence well behaved for automatic determination of stopping parameter. The Stain’s unbiased risk estimate (SURE), generalized cross validation (GCV), L-curve analysis, and discrepancy principle (DCP) techniques are employed for automatic determination of optimum iteration. Experimental results from seismic deconvolution and seismic tomography are included showing that the proposed methodology outperforms the conventional IRLS with significantly lower computational burden.