An introduction to linear inverse theory

Linear inverse theory provides a formalism by which many questions fundamental to signal processing may be entertained. Questions pertaining to the resolving power of the data, the types of models that will reproduce the observations, the importance of additional data, the determination of the optimum sampling rate, and the effects of observational inaccuracies can all be meaningfully attacked through inverse theory. This paper presents an elementary overview of some of the approaches used by geophysicists to extract information about a model from the observations. The paper is not intended to be a review of the numerous ways in which a solution of the linear inverse problem has been sought, but rather it concentrates upon the methods and approaches of Backus and Gilbert. The three essential aspects of inverse theory — model construction, appraisal, and inference — are outlined and applied to a single numerical example. It is my hope that, the applications of the techniques presented in this paper to specific problems in signal analysis will be apparent.