Fractal convergence properties of geophysical inversion

Abstract

Geophysical data, such as measurements of the Earth's gravitational or magnetic field, are routinely collected and studied to get information on the structure of the subsurface geology. The standard method of data analysis involves 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 to converge to local minima – that is they reach a set of model parameters whose geophysical response is a better fit to the observed data than the starting 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 to a particular minima of the misfit surface is studied here for some magnetic 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 inverted. The fractal dimension of the set is shown to be inversely proportional to the damping of the inversion process.