Introduction

Robust error measures such as the l¹ norm have a number of uses in geophysics. As measures of data misfit, they show considerably less sensitivity to large measurement errors than does the least-squares (l²) measure. In a recent work, Guitton and Symes (1999) show that a possible approximation of the l¹ norm is viable using the Huber misfit function Huber (1973). In particular, their work on velocity inversion, using the hyperbolic Radon transform, suggests that (1) the Huber function gives far more robust model estimates than does least squares, (2) its minimization using a standard quasi-Newton method is comparable in computational cost to least-squares estimation using conjugate gradient iteration, and (3) the result of Huber data fitting is stable over a wide range of choices for the $l^2 \rightarrow l^1$ threshold and total number of quasi-Newton steps. For the same geophysical problem, Nichols (1994) and Hugonnet (1998) use a different implementation of the hybrid l²-l¹ norm, using an IRLS algorithm with a thoroughly chosen weighting function. Both strategies (IRLS and the Huber solver) show similar results, but at this stage, no direct comparisons between the two methods have been carried out.

This paper will compare the efficiency of the these two robust solvers with synthetic and real data. First, I will briefly review both methods and describe algorithmic issues. Then, I will compare the Huber solver and IRLS performing velocity inversion as suggested by Thorson and Claerbout (1985), Nichols (1994), and Lumley et al. (1994).