Chapter 1 - Probabilistic Inverse Theory

Abstract

Geophysical investigations which commenced thousands of years ago in China from observations of the Earth shaking caused by large earthquakes (Lee et al., 2003) have gone a long way in their development from an initial, intuitive stage to a modern science employing the newest technological and theoretical achievements. In spite of this enormous development, geophysical research still faces the same basic limitation. The only available information about the Earth comes from measurement at its surface or from space. Only very limited information can be acquired by direct measurements. It is not surprising, therefore, that geophysicists have contributed significantly to the development of the inverse theory—the theory of inference about sought parameters from indirect measurements. For a long time this inference was understood as the task of estimating parameters used to describe the Earth's structure or processes within it, like earthquake ruptures. The problem was traditionally solved by using optimization techniques following the least absolute value and least squares criteria formulated by Laplace and Gauss. Today the inverse theory faces a new challenge in its development. In many geophysical and related applications, obtaining the model “best fitting” a given set of data according to a selected optimization criterion is not sufficient any more. We need to know how plausible the obtained model is or, in other words, how large the uncertainties are in the final solutions. This task can hardly be addressed in the framework of the classical optimization approach. The probabilistic inverse theory incorporates a statistical point of view, according to which all available information, including observational data, theoretical predictions and a priori knowledge, can be represented by probability distributions. According to this reasoning, the solution of the inverse problem is not a single, optimum model, but rather the a posteriori probability distribution over the model space which describes the probability of a given model being the true one. This path of development of the inverse theory follows a pragmatic need for a reliable and efficient method of interpreting observational data. The aim of this chapter is to bring together two elements of the probabilistic inverse theory. The first one is a presentation of the theoretical background of the theory enhanced by basic elements of the Monte Carlo computational technique. The second part provides a review of the solid earth applications of the probabilistic inverse theory.