by Kwangwon Park, PhD

Modeling Uncertainty in metric space is a relatively new approach to Bayesian modeling with non-Gaussian prior and computational complexity in the forward model. In this work, we provide an alternative approach to the traditional McMC framework for sampling the posterior, a framework that is not practical computation-wise for most Earth Science problems. We reformulate the non-linear inverse problem by defining distances between model outcomes to be equal to the difference in forward model response. A Karhoene-Loeve expansion is developed that allows creating new models from the prior that have zero distance to the field data (exact matching) or a specified likely distance (the traditional likelihood in Bayesian modeling). The problem of drawing models from the prior which have certain distance to other models is in image analysis termed the "pre-image" problem and in this work we develop practical solution for high-dimensional problem via multi-dimensional scaling and prior-consistent optimization.Ample examples demonstrate that the method has sampling properties similar to the rejection sampler but at a fraction of the computational cost.

The PDF can be downloaded here.