The Backup-Gilbert method and its application to the electrical conductivity problem

Abstract

The theory of Backus and Gilbert gives a technique for solving the general linear inverse problem. Observational error and lack of data are shown to reduce the reliability of the solution in different ways: the former introduces statistical uncertainties in the model, while the latter smooths out the detail. Precision can be improved by sacrificing resolving power, and vice versa, so that some compromise may be made between the two in choosing the best model. Nonlinear inverse problems can be brought into the domain of the theory by linearizing about a typical solution. The inverse problem of electrical conductivity in the mantle is used to illustrate the Backus-Gilbert technique; an example of the tradeoff diagram is given.