Probabilistic Approach to Gramian Inversion

Abstract

The meaning of Gramian and its role in the joint inversion are explained using a probabilistic approach to inverse problem solution. We introduce a Hilbert space of random variables with the metric defined by the covariance matrix between random variables, representing different model parameters. Using this Hilbert space, the probabilistic Gramian is represented as the determinant of the covariance matrix between the different model parameters and their attributes. By minimizing the probabilistic Gramian, we enforce the linear correlation between various inverse models produced by joint inversion. Several gradient-type techniques are considered for solving this minimization problem.