UC San Diego

This dissertation is a work on the development of mathematical tools for uncertainty quantification in environmental flow and transport models. In hydrology, data scarcity and insufficient site characterization are the two ubiquitous factors that render modeling of physical processes uncertain. Spatio-temporal variability (heterogeneity) poses significantly impact on predictions of system states. Standard practices are to compute (analytically or numerically) the first two statistical moments of system states, using their ensemble means as predictors of a system's behavior and variances (or standard deviations) as a measure of predictive uncertainty. However, such approaches become inadequate for risk assessment where one is typically interested in the probability of rare events. In other words, full statistical descriptions of system states in terms of probabilistic density functions (PDFs) or cumulative density functions (CDFs), must be sought. This is challenging because not only parameters, forcings and initial and boundary conditions are uncertain, but the governing equations are also highly nonlinear. One way to circumvent these problems is to develop simple but realistic models that are easier to analyze. In chapter 3, we introduce such reduced-complexity approaches, based on Green-Ampt and Parlange infiltration models, to provide probabilistic forecasts of infiltration into heterogeneous media with uncertain hydraulic parameters. Another approach is to derive deterministic equations for the statistics of random system states. A general framework to obtain the cumulative density function (CDF) of channel- flow rate from a kinematic-wave equation is developed in the third part of this work. Superior to conventional probabilistic density function (PDF) procedure, the new CDFs method removes ambiguity in formulations of boundary conditions for the CDF equation. Having developed tools for uncertainty quantification of both subsurface and surface flows, we apply those results in final part of this dissertation to perform probabilistic forecasting of algae growth in an enclosed aquatic system