In the early stages of prospect evaluation, an inexpensive interval velocity estimate is often desired. The Dix equation Dix (1952) analytically inverts root-mean-square (RMS) velocity for interval velocity as a function of time. In addition to many physical shortcomings (assumption of a stratified v(z) earth), Dix inversion suffers from numerical problems that lead to poor velocity estimates. Dix inversion is unstable when RMS velocities vary rapidly, and may produce interval velocities with unreasonably large and rapid variations. For this reason, the problem is often cast as a least-squares problem, with is regularized in time with a differential operator to penalize such rapid variations and to produce a smooth result Clapp et al. (1998).
While temporal smoothness may often be justified from a geologic stand point (v(z)), in some cases velocity can change abruptly (e.g., carbonate layers, salt bodies, strong faulting). In these cases we desire a regularization technique that gives smooth velocity in most places, but which preserves sharp ``geologic'' interval velocity contrasts when they are present, without requiring pre-defined edges to be supplied.
In this paper we present two automatic edge-preserving regularization methodologies for the least-squares implementation of Dix formula. The first uses IRLS to effectively change the norm of the problem to permit a ``spiky'' or ``sparse'' model residual, which leads to a ``blocky'' velocity model. The second uses an isotropic edge detector, the gradient magnitude, in a nonlinear scheme to compute a measure of the edges of the model. This edge measure is then used as a model residual weight, along the lines of Clapp et al. (1998) and Lizarralde and Swift (1999).