Rapid inversion of multi dimensional magnetotelluric data

This dissertation addresses the problem of computationally efficient ways to recover the distribution of the Earth's electrical conductivity as a function of position from magnetotelluric (MT) data, for one-, two-, and three-dimensional conductivity distributions. A method (the "rapid method") is developed for inversion of two-dimensional (2-D) data that is roughly 70 times faster per interation than standard methods of inversion for small sized problems, with increased savings for larger problems. Approximate partial derivatives of MT data with respect to the conductivity directly beneath it are obtained from a perturbation argument in which only cross terms involving both horizontal gradients and variations in conductivity (or variations in the fields arising from variations in conductivity) are neglected. At each iteration of the method the partial derivatives are used in single site inversions at each site, the conductivity profiles below the various sites interpolated to obtain a 2-D model, and 2-D forward modeling performed to obtain a new set of residuals. The inversions minimize a weighted Laplacian of the logarithm of the conductivity. The rapid inversion method is demonstrated in several inversions of synthetic data. The rapid method is extended to inversion of 3-D data in two different manners; one using single site inversions and the other using multiple site inversions. Estimated operation counts are down by factors of 500 and 80 respectively from 3-D inversions formulated in a conventional manner (but using efficient sparse matrix techniques), making three-dimensional inversion affordable on current day computers. Our implementation of the rapid method makes use of a general purpose linear inversion that can minimize weighted norms of a model or its first or second derivatives and may be used in conventional inversions of one- and two-dimensional MT data. Step size control is particularly important for assuring the improvement of an objective function when the general purpose inversion is used in an iterative nonlinear inversion. Nonlinear 1-D MT inversions are also discussed briefly.