A comparison of automatic techniques for estimating the regularization parameter in non-linear inverse problems

Two automatic ways of estimating the regularization parameter in underdetermined, minimum-structure-type solutions to non-linear inverse problems are compared: the generalized cross-validation and L-curve criteria. Both criteria provide a means of estimating the regularization parameter when only the relative sizes of the measurement uncertainties in a set of observations are known. The criteria, which are established components of linear inverse theory, are applied to the linearized inverse problem at each iteration in a typical iterative, linearized solution to the non-linear problem. The particular inverse problem considered here is the simultaneous inversion of electromagnetic loop-loop data for 1-D models of both electrical conductivity and magnetic susceptibility. The performance of each criteria is illustrated with inversions of a variety of synthetic and field data sets. In the great majority of examples tested, both criteria successfully determined suitable values of the regularization parameter, and hence credible models of the subsurface.