Introduction

Interval velocity estimation is a fundamental problem in seismology. The simplest technique for finding interval velocity is the Dix equation Dix (1952) which analytically inverts the root-mean-square (RMS) velocity for interval velocity. The Dix equation has many flaws including the assumption of a vertically stratified earth and numerical problems that can cause the inversion to become unstable for rapidly varying velocities. To better constrain the solution, the Dix equation is often cast as a least-squares problem, which is regularized in time with a differential operator that penalizes rapid variations to produce a smooth result Clapp et al. (1998).

Valenciano et al. (2003) expanded on this work to use both $\ell_2$ and $\ell_1$ regularization. The $\ell_2$ regularization is justified when the expected velocity model is smooth. When geological expectations dictate abrupt changes in interval velocity $\ell_1$regularization can be utilized to preserve sharp boundaries when they are present, yet allows for smooth velocity elsewhere.

Since least-squares problems are already convex, this is a perfect problem to test the utility of convex optimization. Here we utilize convex optimization to solve the problem of interval velocity estimation using the same examples as those present by Valenciano et al. (2003) for the $\ell_2$ and $\ell_1$ regularization, which used conjugate gradient methods. As well, bounds will be enforced on the solution to further constrain the Dix inversion to a more geologically sensible answer.