The Anatomy of Inverse Problems

Abstract

A major task of geophysics is to make quantitative statements about the interior of the earth. For this reason, inverse problems are an important area of geophysical research and industrial application. Figure 1 shows how many texts present inverse problems. The earth model is an element of a mathematical space that contains all allowable parameterizations of the earth’s properties (or at least those properties relevant to a given experiment); this space is referred to as model space . The physics of the problem determines which data d correspond to a given model m . The problem of computing the model response (synthetic “data”) given a model is called the forward problem . The corresponding data reside in a mathematical space that is called data space . In many applications, one records the data, and the goal is to find the corresponding model. The task is called the inverse problem , as shown in Figure 1. FIG. 1. The conventional view of inverse problems: find the model that predicts the measurements. Unfortunately, Figure 1 is wrong. There is a simple reason for this. In general the model that one seeks is a continuous function of the space variables with infinitely many degrees of freedom. For example, the 3-D velocity structure in the earth has infinitely many degrees of freedom. On the other hand, the data space is always of finite dimension because any real experiment can only result in a finite number of measurements. A simple count of variables shows that the mapping from the data to a model cannot be unique; or equivalently, there must be elements of the model space that have no influence on the data. This lack of uniqueness is apparent even for problems involving idealized, noise-free measurements. The problem only becomes worse when the uncertainties of real …