The use of a priori data to resolve non-uniqueness in linear inversion

The recent, but by now classical method for dealing with non-uniqueness in geophysical inverse problems is to construct linear averages of the unknown function whose values are uniquely defined by empirical data (Backus & Gilbert). However, the usefulness of such linear averages for making geophysical inferences depends on the good behaviour of the unknown function in the region in which it is averaged. The assumption of good behaviour, which is implicit in the acceptance of a given average property, is equivalent to the use of a priori information about the unknown function. There are many cases in which such a priori information may be expressed quantitatively and incorporated in the analysis from the very beginning. In these cases, the classical least-squares method may be used both to estimate the unknown function and to provide meaningful error estimates. In this paper I develop methods for exploring the resolving power in such cases. For those problems in which a continuous unknown function is represented by a finite number of‘layer averages’, the ultimately achievable resolving width is simply the layer thickness, and perfectly rectangular resolving kernels of greater width are achievable. The method is applied to synthetic data for the inverse‘gravitational edge effect’problem where yi are data, f(z) is an unknown function, and ei are random errors. Results are compared with those of Parker, who studied the same problem using the Backus-Gilbert approach.