Three-Dimensional Bayesian Inversion With Application to Subsurface Sensing

A Bayesian formalism is considered for inverting for the parameters of a heterogeneity profile based on measured scattering data. It is shown that the typical use of regularization (e.g., Thikonov) corresponds to a maximum a posteriori point approximation to the full-posterior density function on the heterogeneity parameters, given the observed data. In the Bayesian framework considered here, the full posterior is approximated as a multidimensional Gaussian distribution. The mean of this distribution may be used as a point estimate of the heterogeneity profile, with the covariance matrix providing associated "error bars" (a measure of confidence in the inversion). In addition to providing an approximation to the full posterior of the heterogeneity profile, this formalism addresses the proper weighting to apply for inversion regularization. Specifically, an important limitation of previous regularization procedures is the need to place a weight on the importance of the regularization relative to the importance of fitting the data to the underlying model. In the Bayesian analysis outlined here, we also assign such a weight, but now the weight is treated as a random variable, with a statistical prior. The measured data are then used to determine a posterior distribution on the parameter, based on the measured data. We present here the basic Bayesian inversion framework, with several example results presented for subsurface-sensing problems