Construction of non-negative resolving kernels in Backus--Gilbert theory

Abstract

In linear Backus—Gilbert theory, resolving kernel approximations to delta distributions are linear combinations of data kernels which sometimes exhibit undesirable negative side lobes. For some problems, optimum delta-like data kernel combinations can be constructed which are everywhere non-negative. The square root of the resolving kernel is expanded in terms of a complete set of basis functions. Then, a standard measure of delta-ness (here the quadratic criterion) is minimized to find real expansion coefficients, subject to constraints of unimodularity and approximate equality of the square of the expansion to a linear combination of data kernels. The resulting non-linear Lagrange multipier problem is solved by a multidimensional Newton iteration. As an example, the method is applied to the gravitational edge-effect problem.