Fractal convergence properties of geophysical inversion

Geophysical data, such as measurements of the Earth's gravitational or magn etic field, are routinely collected and studied to get information on the s tructure of the subsurface geology. The standard method of data analysis in volves the least-squares inversion of the data with respect to various user -chosen model parameters (such as the geometry or density of the lithology) . One serious problem with most inversion schemes is that they are liable t o converge to local minima - that is they reach a set of model parameters w hose geophysical response is a better fit to the observed data than the sta rting model, but they do not reach the set of parameters that would provide the best fit possible. The set of initial model parameters that converge t o a particular minima of the misfit surface is studied here for some magnet ic models, and is found to be a fractal when there are two minima available for the model to converge to, and at least two model parameters are invert ed. The fractal dimension of the set is shown to be inversely proportional to the damping of the inversion process. (C) 2000 Elsevier Science Ltd. All rights reserved.