Understanding Inverse Theory

Abstract

Much of our knowledge of the Earth's interior is perforce based on the interpretation of measurements made at the surface, rather than sampling of the material in the interior. In the past few years there have been great advances in the mathematical aspects of this problem, and the topic has come to be called geo­ physical inverse theory. To apply these ideas, there must be a valid mathematical model of the physics of the system under study, so that one would be able to calculate the values of observations made on an exactly known structure: the calculation of the behavior of a specified system is the solution of the or direct problem. Frequently it is the forward problem that presents a difficult challenge to the theoretical geophysicist. Illustrations include the mechanism of earthquake rupture or the generation of the Earth's magnetic field; in problems like these, inverse theory is normally quite inappropriate. When the forward problem has been completely solved, there are of course unknown parameters in the mathematical model representing physical properties of the Earth such as Lame parameters, density or electrical conductivity. The goal of inverse theory is to determine the parameters from the observations or, in the face of the inevitable limitations of actual measurement, to find out as much as possible about them. The quality that distinguishes inverse theory from the parameter estimation problem of statistics (Bard 1974, Rao 1973) is that the unknowns are junctions, not merely a handful of real numbers. This' means that the solution contains in principle an infinite number of variables, and therefore with real data the problem is as under­ determined as it can be. Naturally, there are geophysical problems containing a relatively small number of free parameters: for example, in describing the relative instantaneous motion of N lithospheric plates, we find that the assumption of internal rigidity reduces the number of unknowns to 3N 3 for the N 1 relative angular velocity vectors (McKenzie & Parker 1974). Sometimes, however, unknown structures are conceived in terms of small numbers of homogeneous layers for reasons of computational simplicity rather than on any convincing geophysical or