Introduction

Interval velocity estimation is an fundamental problem in reflection seismology. An accurate velocity model is essential to creating an interpretable image from seismic data. There are many techniques for estimating velocity Clapp (2001); Sava (2004) in complex geological settings, but these are often very expensive due to, not only, the operator but also the non-linear nature of the problem and coherent noise that can lead the linear problem to local minima. In this paper an inversion technique is presented for $\ell_1$ regularized problems that could potentially decrease the computation time for velocity estimation.

Grid based techniques have an additional drawback, in that they tend to create smooth models even where sharp contrast exists. When considering velocity inversion problems, $\ell_1$ regularization can be used to create a sparse solution, resulting in more ``blocky'' velocity models. The $\ell_1$ regularization preserves sharp geologic boundaries, such as channel margins, salt bodies, or carbonate layers. Recently, a specialized interior point method has been presented for efficiently solving $\ell_1$ regularized least squares problems Kim et al. (submitted).

A modified version of that algorithm is presented here. To exemplify its utility it will be used to solve the least squares Super Dix equations, originally presented by Clapp et al. (1998). Expanding on this work,Valenciano et al. (2003) introduced $\ell_1$ regularization to the problem formulation using a nonlinear iterative approach that approximated an $\ell_1$ regularization. Witten and Grant (2006) solved the same problem using a MATLAB based convex optimization solver. MATLAB, however, was pushed to its limits to solve even this small problem.

In this paper, the algorithm will be described. The method is first applied to a simple synthetic model. Then it is applied to a real data set from the Gulf of Mexico. It initial results compared to previous methods for this same dataset.