Geophysical applications of a novel and robust L1 solver

Introduction

L1 norm optimization is known to be a powerful estimator when the data are noisy or the model is sparse (Guitton, 2005; Claerbout and Muir, 1973; Darche, 1989; Nichols, 0994). However, the most widely used L1 solver-Iterated Reweighted Least-Squares (IRLS)-is cumbersome to use because users must specify numerical parameters with unclear physical meanings. To develop a robust, efficient L1 solver, Claerbout (2009) proposed a hybrid norm function to approximate the L1 norm (absolute value function), and he generalized the conjugate direction (CD) method using Taylor's expansion to guide the plane search in the minimization.

The parametrization of this proposed hybrid norm is straightforward. Users specify thresholds for the data residual and model residual, respectively. These thresholds determine the transition point from L2 to L1. The threshold for the data residual ($R_d$ ) can be chosen according to the signal-to-noise ratio in data space; the threshold for the model residual ($R_m$ ) can be specified by the desired blockiness in the model space.

In theory, the convergence of this hybrid norm solver is guaranteed, because the objective function is strictly convex. Nevertheless, difficulties may occur as the hybrid-norm approaches the L1 limit.

To test the performance and analyze the stability of this hybrid solver, we apply the solver to a 1-D field RMS velocity inversion, a 2-D regularized Kirchhoff migration inversion, and a 2-D velocity analysis problem. The inversion results show that this hybrid solver is robust and simple to use.

Geophysical applications of a novel and robust L1 solver