Percolation: Inference and Applications in Hydrology

Abstract

Percolation theory is a branch of probability theory describing connectedness in a stochastic network. The connectedness of a percolation process is governed by a few, typically one or two, parameters. A central theme in this thesis is to draw inference about the parameters of a percolation process based on information whether particular points are connected or not. Special attention is paid to issues of consistency as the number of points whose connectedness is revealed tends to infinity. A positive result concerns Bayesian consistency for a bond percolation process on the square lattice $\mathbb{L}^2$ - a process obtained by independently removing each edge of $\mathbb{L}^2$ with probability $1-p$. Another result on Bayesian consistency relates to a continuum percolation model which is obtained by placing discs of fixed radii at each point of a Poisson process in the plane, $\mathbb{R}^2$. Another type of results concerns the computation of relevant quantities for the inference related to percolation processes. Convergence of MCMC algorithms for the computation of the posterior, for bond percolation on a subset of $\mathbb{L}^2$, and the continuum percolation, on a subset of $\mathbb{R}^2$, is proved. The issue of convergence of a stochastic version of the EM algorithm for the computation of the maximum likelihood estimate for a bond percolation problem is also considered.

Finally, the theory is applied to hydrology. A model of a heterogeneous fracture amenable for a percolation theory analysis is suggested and the fracture's ability to transmit water is related to the fractures median aperture.