Exploring multidimensional landscapes without a map

A discussion of methodologies for nonlinear geophysical inverse problems is presented. Geophysical inverse problems are often posed as optimization problems in a finite-dimensional parameter space. An Earth model is usually described by a set of parameters representing one or more geophysical properties (e.g. the speed with which seismic waves travel through the Earth's interior). Earth models are sought by minimizing the discrepancies between observation and predictions from the model, possibly, together with some regularizing constraint. The resulting optimization problem is usually nonlinear and often highly so, which may lead to multiple minima in the misfit landscape. Global (stochastic) optimization methods have become popular in the past decade. A discussion of simulated annealing, genetic algorithms and evolutionary programming methods is presented in the geophysical context. Less attention has been paid to assessing how well constrained, or resolved, individual parameters are. Often this problem is poorly posed. A new class of method is presented which offers potential in both the optimization and the `error analysis' stage of the inversion. This approach uses concepts from the field of computational geometry. The search algorithm described here does not appear to be practical in problems with dimension much greater than 10.