Marginal Solutions to Quasi-Linear Inverse Problems in Geophysics: The Edgehog Method

The complete solution of an inverse problem generally involves two tasks: (1) finding a solution, and (2) representing in a meaningful way the degree of nonuniqueness of the solution. In this paper I shall deal only with the second task, presuming at the outset that at least one satisfactory solution has been found which is in agreement with available geophysical data and any necessary a priori constraints. I begin with the premise that all geophysical inverse problems involve some degree of non-uniqueness, because geophysical data are always finite in number and/or extent, because they contain random errors, and more fundamentally, because some arbitrariness must always enter the problem when a real physical situation is ' modelled ' by a mathematical formulation. It is thus important to ask which features of the solution are in fact required in order to fit the data within their uncertainty. In many cases an adequate answer to the question requires graphical representation of a multior infinitely-dimensioned family of possible solutions; this problem may be more challenging than the mathematical problem of defining the solution space. For linear inverse problems, the ' resolving power ' approach of Backus & Gilbert (1968, 1970) provides an excellent tool for characterizing non-uniqueness. Their technique is also applicable to non-linear problems which may be linearized by perturbation around some approximate solution, provided the range of linearity is large enough. One limitation of the Backus-Gilbert technique is that it provides no convenient test of the validity of the linearity assumption. Another limitation is the rather weak conclusions which may be drawn concerning features which are just at the limit of resolvability. For non-linear inverse problems, the investigation of non-uniqueness has generally involved searching for, testing, and displaying representative solutions. Two methods in common use in geophysics are Monte Carlo search, in which solutions are sought at random within a preset range, and Hedgehog search (Keilis-Borok & Yanovskaja 1967), in which the search is conducted in an organized manner around some satisfactory initial solution. The greatest limitation of these techniques is the difficulty of assuring that the solutions found are representative of all possibilities. For instance, the presence of a common feature in any finite number of models does not guarantee its presence in all possible models, and it may be quite difficult to prove that the search is not biased in some way. A further limitation is that some arbitrary grid size and search range must be set before searching; poor choice may result in failure to discover important solutions and/or astronomical computation costs. Finally, the graphical representation problem becomes severe when many successful solutions are found. A response to these problems which is gaining in popularity is to seek solutions which are extreme in some sense, yet which satisfy the data (e.g. Wiggins, McMechan & Toksoz 1972). In the present paper I shall use this approach to extend the resolving power approach of Backus & Gilbert to non-linear problems. I shall use linear theory