The Anatomy Of Inverse Problems

GEOPHYSICS, VOL. 65, NO. 6 (NOVEMBER-DECEMBER 2000); P. 1708–1710, 4 FIGS.

The Anatomy of Inverse Problems

John A. Scales∗ and Roel Snieder∗
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 inverseproblems. 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 iscalled 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. Unfortunately, Figure 1 is wrong. There is a simple reason for this. In general the model that one seeks is a continuous function of thespace 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; orequivalently, 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 measurements are taken into account. Although the uniqueness question is a hotly debated issue in the mathematical literature on inverse problems, it islargely irrelevant for practical inverse problems because they invariably have a nonunique solution (if by solution we mean an earth model). It is this nonuniqueness that makes Figure 1 deceptive, because the arrow pointing from data space to model space suggests that a unique model corresponds to every data set. A more realistic scheme of inverse problems is shown in Figure 2. Given a model m, thephysics of the problem determines the predicted data d; this is called the forward problem. For a given data set, one determines an estimated model m. We ˆ refer to this as the model estimation problem. (Later, we will consider the generalization to the problem of estimating properties of models, rather than the models themselves.) Note that there may be many reasonable model estimates for a givendata set and that the estimation procedure may be nonlinear even when the forward problem is linear. Thus, the mean of a set of numbers is a linear function of the numbers, whereas the median is a nonlinear function. Yet both the median and the mean may be reasonable estimators of the “center” of the set of numbers. Part of the art of solving inverse problems comes from the need to define what itmeans for an estimate to be reasonable. Since the mapping from data space to model space is nonunique, the estimated model may also depend on the details of the algorithm that one has used for the estimation problem as well as on the regularization and model parameterization that has been used. In general, the estimated model m differs ˆ from the true model m. For example, in seismic inversion theestimated model may be a blurred version of the true model. In addition, the data are always contaminated with errors; these errors represent an additional source of discrepancy between the estimated model and the true model. One is not finished when the estimated model is constructed; it is essential to somehow quantify the error between the estimated model and the true model. This is called the...