Optimal Solutions to a Linear Inverse Problem in Geophysics

This paper is concerned with the solution of the linear system obtained in the Backus-Gilbert formulation of the inverse problem for gross earth data. The theory of well-posed stochastic extensions to illposed linear problems, proposed by Franklin, is developed for this application. For given estimates of the statistical variance of the noise in the data, an optimal solution is obtained under the constraint that it be the output of a prescribed linear filter. Proper specification of this filter permits the introduction of information not contained in the data about the smoothness of an acceptable solution. As an example of the application of this theory, a preliminary model is presented for the density and shear velocity as a function of radius in the earth's interior.