The general linear inverse problem: Implication of surface waves and free oscillations for Earth structure

Abstract

The discrete general linear inverse problem reduces to a set of m equations in n unknowns. There is generally no unique solution, but we can find k linear combinations of parameters for which restraints are determined. The parameter combinations are given by the eigenvectors of the coefficient matrix. The number k is determined by the ratio of the standard deviations of the observations to the allowable standard deviations in the resulting solution. Various linear combinations of the eigenvectors can be used to determine parameter resolution and information distribution among the observations. Thus we can determine where information comes from among the observations and exactly how it constrains the set of possible models. The application of such analyses to surface‐wave and free‐oscillation observations indicates that (1) phase, group, and amplitude observations for any particular mode provide basically the same type of information about the model; (2) observations of overtones can enhance the resolution considerably; (3) the degree of resolution has generally been overestimated for many model determinations made from surface waves; and (4) computation of parameter and information resolution is such a simple extension of any inversion procedure based on perturbation parameters that such inversion studies are incomplete without considering resolution.