Multiresolution statistical modeling with application to modeling groundwater flow

The development of accurate mathematical models describing the flow of groundwater is an important problem due to the prevalence of contaminants in or near groundwater supplies. An important parameter of these models is hydraulic conductivity, which describes the ability of the subsurface geology to conduct water flow. Because hydraulic conductivity is a function of the earth's subsurface, direct measurements can be made only at a relatively small number of locations. Instead, one must rely on indirect measurement sources, which supply observations of conductivity at different locations and resolutions. An important problem is to estimate the hydraulic conductivity function from all available data, and to characterize the remaining uncertainty. The class of multiscale processes introduced in [Chou et al., 19941 appears to be well-suited to this problem, since these processes can be estimated efficiently at every scale at which they are modeled; the multiscale estimator also provides an uncertainty measure for the estimates. However, this multiscale framework has some limitations that must be overcome before it can be applied to general data fusion problems. First, because all of the previous applications have focused on the measurement and estimation of the finest-scale process, arbitrary nonlocal properties of interest (e.g., coarse-resolution measurements of hydraulic conductivity) have not been represented within the multiscale framework. Second, the class of stochastic processes that is well modeled by low-order tree models has not been fully characterized. To address the first limitation, this thesis (1) extends multiscale realization theory so that coarse-scale variables can represent particular non-local properties to be measured or estimated and (2) applies these realization algorithms to the estimation of hydraulic conductivity from sparse measurements made at different resolutions. To partially address the second limitation, the multiscale processes are shown to provide natural approximations of fractional Brownian motion. These approximations are based on the observation that the multiscale realization problem can be considerably simplified when modeling random processes that are statistically self-similar and/or have stationary increments.