A General Probabilistic Approach for Inference of Gaussian Model Parameters from Noisy Data of Point and Volume Support

Abstract

Methods that rely on Gaussian statistics require a choice of a mean and covariance to describe a Gaussian probability distribution. This is the case using for example kriging, sequential Gaussian simulation, least-squares collocation, and least-squares-based inversion, to name a few examples. Here, an approach is presented that provides a general description of a likelihood function that describes the probability that a set of, possibly noisy, data of both point and/or volume support is a realization from a Gaussian probability distribution with a specific set of Gaussian model parameters. Using this likelihood function, the problem of inferring the parameters of a Gaussian model is posed as a non-linear inverse problem using a general probabilistic formulation. The solution to the inverse problem is then the a posteriori probability distribution over the parameters describing a Gaussian model, from which a sample can be obtained using, e.g., the extended Metropolis algorithm. This approach allows detailed uncertainty and resolution analysis of the Gaussian model parameters. The method is tested on noisy data of both point and volume support, mimicking data from remote sensing and cross-hole tomography.