The use of constraints in geophysical tomographic reconstruction

Abstract

A significant problem with geophysical tomography is that the image reconstruction problem does not usually have a unique solution, given the measured data. A number of tomograms can be consistent with the measured data because of the limited access available for taking measurements. It is necessary to introduce additional information into the reconstruction process in order to find a unique solution. A common approach is to choose the solution that optimises some objective function. However, such criteria (eg, minimum norm and maximum entropy) are not necessarily consistent with the geology. Other prior information is usually available which is consistent with the geology, much of which can be expressed in the form of constraints which define sets of images, called property sets. The intersection of the property sets contains feasible solutions. Set theoretic estimation finds solutions in this feasibility set which are consistent with the measured data and all prior information. A number of tomography algorithms used in geophysics fit into the category of set theoretic estimation techniques. This paper gives an introduction to some of the concepts that are common to these techniques and compares several reconstruction algorithms, using straight-ray synthetic data.