Inverse methods applied to continuation problems in geophysics

Abstract

This paper focuses on the application of inverse methods to continuation problems in geophysics. It is divided in three sections. The first introductory section lists a number of continuation problems which are of interest to geo physicists and gives a brief review of earlier work on the subject. The second section is devoted to the solution we have developed in three papers from 1973 to 1975, which we have further generalized and termed a global inverse method. This method is equally applicable to two- and three-dimensional problems. A particular solution of the problem is found as a linear combination of continuation kernels expressed at the observation points (they generate a vector space En); this solution is acceptable i.e. consistent with the data. The general solution of the problem requires that solutions from the null space of the kernels be added to the particular solution belonging to En. The properties of the Gram matrix built from the continuation kernels are investigated. one important result provided by the global inverse method is that an analytic expression of the elements of this matrix is easily obtained when the concept of images is introduced. We also show how an approximate Green's function for the irregular surface over which the data points are distributed can be found. In the third section we compare the global inverse method with generalized inverse matrix theory. When the latter theory is generalized to the case when one dimension becomes infinite the two formalisms are equivalent. We show how the concepts of resolution and information density are related in the two methods and investigate the properties of the base of eigenfunctions of the problem. Expanding the solution over this base leads to the concept of ranking and winnowing (Gilbert, 1971) and allows the computation of various covariance matrices for model components. Also, in the case of continuation problems, the relationship between the two sets of eigenvectors which appear in the theory of generalized inverse matrices is made clear and is quite enlightening.