Entropy and geophysical inverse problems

Abstract

Geophysical surveys often result in the inference of Earth properties based on the data collected from organised experiments. Such problems are known as inverse problems. The mathematical theory relating the data and the Earth properties is generally non-linear which, combined with ill-posed and underdetermined attributes, leads to ambiguous solutions. A single solution needs to be chosen for the geophysical experiment to be of benefit. Of the range of possible solutions it appears that some are better than others ? simply by imposing common sense. The entropy principle consists of selecting the solution that has the maximum entropy value. Such a solution will be consistent with all experimental data and maximally non­committal with regard to unavailable data. This solution contains structure only if it is needed to satisfy the given data. The entropy function is a simple convex function of logarithmic form. The convex nature of the function is enough to provide most of the benefits associated with maximum entropy solutions. However, a simple demonstrative example shows that the logarithmic form is essential to ensure that the solution is maximally non-committal to unavailable data. The entropy principle also provides a natural mechanism for using prior information. Prior information is that obtained from experiences prior to the execution of the geophysical experiment. Such information is generally not of the form of a hard constraint and is frequently difficult to use. The entropy values can be measured relative to a prior estimate of the model parameters. This allows prior information to be used as default values and as a guide without restricting the inversion. The addition of useful prior information models can be beneficial to the outcomes of geophysical inversions. A synthetic cross-hole tomography example shows the benefits of maximum entropy solutions when compared to the results of the popular SIRT inversion algorithm. The most obvious of these are the lack of spurious background structure and increased resolution of the anomalies.