Ensemble inference in geophysical inverse problems

Abstract

Non‐invasive studies of the shallow Earth are an essential element in a broad range of earth science disciplines. Such studies are usually accomplished using geophysical techniques, which means that a geophysical inverse problem must be solved. Unfortunately, non‐uniqueness is a basic property of almost all solutions to geophysical inverse problems. In this paper we describe a method of dealing with this non‐uniqueness which we believe is more direct, more complete, and more informative than previous attempts in this area. Instead of using regularization methods to constrain the non‐uniqueness of geophysical inverse problems, we confront directly the non‐uniqueness and attempt to describe it. The basic technique is to generate and describe a collection of models that fit the data within acceptable limits based upon observational errors. This collection of models, called an ensemble, is analysed with statistical methods in order to characterize the uncertainty in the solution and also make inferences about properties that are shared by all acceptable models. A by‐product of this approach is that it produces basic information about the degree of linearity contained in the problem and maps trade‐offs between various parameters. The method proposed is quite general and should be applicable to a broad range of geophysical inverse problems. We illustrate the basic method by applying it to two typical geophysical problems. The first application is to a set of cross‐borehole traveltime observations gathered in Kesterson, California. The small deviations of the observed traveltime residuals from those predicted by a uniform half‐space indicates that the velocity contrasts are not large. The set of acceptable models appears to form a non‐degenerate hyperellipsoid in the model space. The mean of the velocity models converges quite rapidly to a smoothed version of a linearized singular‐value decomposition solution. The velocity variations agree very well with the results of a hydrological tracer test conducted prior to the seismic experiment. The second application is to the problem of simultaneously estimating the shape and density of a constant density body using surface gravity measurements. In this case a linear analysis is found to be inadequate. A quadratic formulation is necessary to accurately represent the trade‐off between the magnitude of the density perturbation and the depth extent of the body. Copyright © 1993, Wiley Blackwell. All rights reserved