Introduction

In the discussion of various methods in the previous chapters several examples of interpretation of geophysical field data have been given where ‘inverse’ modeling techniques were employed. Whilst the forward theory for computing predicted responses of various types of models has been discussed previously in varying detail, the inverse theory received only a cursory treatment. Also, with regard to tomographic investigations, very little has been said about the underlying theory. In this chapter we fill in these gaps.

There is a wide variety of inverse techniques for application to geophysical problems. The level of application of inverse theory may range from the simple straightline fitting of seismic refraction data to more sophisticated acoustic or electromagnetic tomography or multidimensional resistivity sounding interpretation. The field is too broad to be treated in a comprehensive way in a single chapter. Nonetheless, an attempt is made to provide the basic elements of some commonly used inversion techniques in the following sections. The later part of the chapter is devoted to tomographic imaging, which is treated as a special problem of the inverse theory.

The inverse problem and its formulation

In contrast to the ‘forward modeling’ approach, where the interpretation of field data involves comparison with theoretical responses of mathematical models with assumed sets of parameters, the basic approach used in ‘inverse modeling’ may be formulated as follows: Given some data on the values of some measured quantities (e.g., potential field or travel-time data), we use a theoretical relationship to derive the values of the set of parameters of a model that reproduces the given field observations.