A Bayesian criterion for simplicity in inverse problem parametrization

To solve a geophysical inverse problem in practice means using noisy measurements to estimate a finite number of parameters. These parameters in turn describe a continuous spatial distribution of physical properties. (For example, the continuous solution may be expressed as the linear combination of a number of orthogonal functions; the parameters are the coefficients multiplying each of these functions.) As the solution is non-unique, estimating the parameters of interest also requires a measure of their uncertainty for the given data. In a Bayesian approach, this uncertainty is quantified by the posterior probability density function (pdf) of the parameters. This 'parameter estimation', however, can only be carried out for a given way of parametrizing the problem; choosing a parametrization is the 'model selection' problem. The purpose of this paper is to illustrate a Bayesian model selection criterion that ranks different parametrizations from their posterior probability for the given set of geophysical measurements. This posterior probability is computed using Bayes' rule and is higher for parametrizations that better fit the data, which are simple in that they have fewer free parameters, and which result in a posterior pdf that departs the least from what is expected apriori. Bayesian model selection is illustrated for a gravitational edge effect inverse problem, where the variation of density contrast with depth is to be inferred from gravity gradient measurements at the surface. Two parametrizations of the density contrast are examined, where the parameters are the coefficients of an orthogonal function expansion or the values at the nodes of a cubic spline interpolation. Bayesian model selection allows one to decide how many free parameters should be used in either parametrization and which of the two parametrizations is preferred given the data. While the illustration used here is for a simple linear problem, the proposed Bayesian criterion can be extended to the general non-linear case.