Dix inversion constrained by L1-norm optimization

Introduction

Dix formula (Dix, 1952) estimates interval velocities from picked stacking velocities. The conventional result of constrained least-squares Dix inversion (Harlan, 1999; Clapp, 2001; Koren and Ravve, 2006) is always a smooth velocity model, because the regularization is imposed in the $L2$ sense. However, to represent a geological environment with sharp velocity contrasts, e.g., carbonate layers, salt bodies and strong faulting, we may need a blocky velocity model rather than a smooth velocity model. Valenciano et al. (2003) proposed to use edge-preserving regularization with Dix inversion in order to get sharp edges in interval velocity. However, one of the solvers they used, IRLS (Iterative Re-weighted Least Squares) is cumbersome to use because users must specify numerical parameters with unclear physical meaning.

$L1$-norm optimization is known to be a robust estimator to yield sparse models. Many works (Guitton, 2005; Claerbout and Muir, 1973; Darche, 1989; Nichols, 1994) has shown that $L1$-norm is not sensitive to outliers, while it penalizes the small residuals down to zero. In theory, when the model space is sparse and the data are noisy, regressions produced by $L1$ optimization always outperform those produced by $L2$ norms.

In this study, we analyze, improve and test different methods on a simple synthetic problem as well as a field-data problem. We aim to develop robust and efficient solvers to perform regressions of an $L1$ nature. We initially improve the traditional IRLS method, explore the conjugate direction $L1$ method, and finally test an $L1/L2$ hybrid method. The inversion results of a 1-D, synthetic, 2-step, interval-velocity model and a 1-D field data example are given at the end of the paper.

Dix inversion constrained by L1-norm optimization