Stable Iterative Methods for the Inversion of Geophysical Data

Summary Interpretation of earth electrical measurements can often be assisted by inversion, which is a non-linear model-fitting problem in these cases. Iterative methods are normally used, and the solution is defined by ' best fit ' in the sense of generalized least-squares. The inverse problems we describe are ill-posed. That is, small changes in the data can lead to large changes in both the solution and in the iterative process that finds the solution. Through an analysis of the problem, based on local linearization, we define a class of methods that stabilize the iteration, and provide a robust solution. These methods are seen as generalizations of the well-known Singular Value Truncation and Marquardt Methods of iterative inversion. Here, and in a companion paper, we give examples illustrating the successful application of the method to ill-posed problems relating to the resistivity of the Earth. In this paper we present an analysis of the solution to a number of geophysical inverse problems. We also provide a reference for the companion paper (Joint Inversion of Geophysical Data, Vozoff & Jupp 1975), where the results are applied to some specific examples. Solutions to geophysical inverse problems are generally non-unique (Backus & Gilbert 1967, 1968, 1970), and it is usual to reduce the non-uniqueness by restricting the complexity of the Earth models. The mathematical problem that arises is commonly ill-posed (unstable) in the sense that small changes in the data lead to large changes in the solution. The solution methods must take careful account of this inherent problem. In the companion paper, and the example given in Section 3 we have data in the form of apparent resistivity measurements for both magnetotelluric (MT), and Direct Current (DC) survey methods. The restricted class of earth models consists of horizontally layered, isotropic media, with constant resistivity in each layer. The simplified inverse problem is, in this case, to find the layer resistivities and thicknesses that best fit the observed data. The analysis of the problem is not, however, restricted to layered models, but applies to any geophysical inverse problem in which the partial derivatives of the (predicted) data with respect to the (unknown) model parameters can be calculated.