Regularization and Backus-Gilbert estimation in nonlinear inverse problems: Application to magnetotellurics and surface waves

Existing methods for solving inverse problems in earth structure determination have not fully exploited the Backus-Gilbert approach to handling nonuniqueness. I develop a theory of linear inference which combines Backus-Gilbert estimation and Tikhonov regularization. I show that, for linear inverse problems, a "regularized least squares" criterion for finding smooth, data-compatible models is equivalent to a Backus-Gilbert criterion for finding optimal estimates of linear spatial averages of earth structure. I attempt to extend the theory to nonlinear problems and show that, under appropriate assumptions, the equivalence of regularized least squares and Backus-Gilbert estimation holds as a first order approximation.;I apply linear inverse methods to the inversion of surface wave dispersion data for one-dimensional shear velocity and the inversion of magnetotelluric data for one- and two-dimensional variations of electrical conductivity. In the surface wave examples, I use a regularized least squares method to invert both synthetic and real data. The inversion method achieves the goals of the Backus-Gilbert approach by finding smooth shear velocity models providing good fits to the data, together with variance and spatial resolution functions which provide a quantitative measure of the uniqueness of the models.;In the one-dimensional magnetotelluric problem, numerical experiments indicate that the effects of nonlinearity on Backus-Gilbert estimates of conductivity are not severe. A theoretical analysis shows that magnetotelluric data are highly nonlinear and discontinuous functionals of two- and three-dimensional conductivity.;I test a regularized least squares method with synthetic and real two-dimensional magnetotelluric data sets. The experiments are successful in that acceptable models are obtained much more readily than with trial-and-error modeling. However, computational constraints severely limited the number of data which could be inverted and the number of iteration steps conducted to find a solution. Further, the regularization functional (roughness measure) was engineered to control oscillations in the iteration sequence, and it is not known what measure of spatial resolution it implies. For these reasons, the models obtained, while quite satisfactory, cannot be considered optimal and lack quantitative measures of uniqueness.