Regularization and trade-off associated with nonlinear geophysical inverse problems: penalty homotopies

Abstract

The presence of local extrema complicates the construction of trade-off curves for nonlinear geophysical inverse problems. Techniques based on path-following or operator homotopies are of use in determining trade-off and in estimating how to weight regularization penalty terms. A homotopy approach has several advantages over linearization about a particular solution. First, it is non-local in the sense that the solution can change significantly as the regularization is varied. Second, nearby distinct local minima can often be found by following branches from the primary homotopy path. The homotopy method tracks a single extremum as the regularization weight is varied. Therefore, it produces a smooth trade-off curve, something that is difficult to accomplish by repeated minimization with different regularization parameters. The homotopy approach is used to invert a set of seismic first arrival times for near-surface velocity structure near Yucca Mountain, Nevada. A trade-off curve of travel time misfit versus model norm is monotonic and provides an estimate of regularization weighting. The velocity model contains two low-velocity zones which correlate with known fault intersections.