Robust Uncertainty Quantification and Scalable Computation for Computer Models with Massive Output

Uncertainty quantification (UQ) is both an old and new concept. The current novelty lies in the interactions and synthesis of mathematical models, computer experiments, statistics, field/real experiments, and probability theory, with a particular emphasize on the large-scale simulations by computer models. The challenges not only come from the complication of scientific questions, but also from the size of the information. It is the focus in this thesis to provide statistical models that are scalable to massive data produced in computer experiments and real experiments, through fast and robust statistical inference.

Chapter 2 provides a practical approach for simultaneously emulating/approximating massive number of functions, with the application on hazard quantification of Soufri\`{e}re Hills volcano in Montserrate island. Chapter 3 discusses another problem with massive data, in which the number of observations of a function is large. An exact algorithm that is linear in time is developed for the problem of interpolation of Methylation levels. Chapter 4 and Chapter 5 are both about the robust inference of the models. Chapter 4 provides a new criteria robustness parameter estimation criteria and several ways of inference have been shown to satisfy such criteria. Chapter 5 develops a new prior that satisfies some more criteria and is thus proposed to use in practice.

Influence of Increased Human Presence in the Mills River Basin on Water Availability and Drought

Periods of drought and low streamflow can have profound impacts on both human and natural systems. People depend on a reliable source of water for numerous reasons including potable water supply and to produce economic value through agriculture or energy production. Aquatic ecosystems depend on water in addition to the economic benefits they provide to society through ecosystem services. Given that periods of low streamflow may become more extreme and frequent in the future, it is important to study the factors that control water availability during these times. In the absence of precipitation the slower hydrological response of groundwater systems will play an amplified role in water supply. Understanding the variability of the fraction of streamflow contribution from baseflow or groundwater during periods of drought provides insight into what future water availability may look like and how it can best be managed. The Mills River Basin in North Carolina is chosen as a case-study to test this understanding. First, obtaining a physically meaningful estimation of baseflow from USGS streamflow data via computerized hydrograph analysis techniques is carried out. Then applying a method of time series analysis including wavelet analysis can highlight signals of non-stationarity and evaluate the changes in variance required to better understand the natural variability of baseflow and low flows. In addition to natural variability, human influence must be taken into account in order to accurately assess how the combined system reacts to periods of low flow. Defining a combined demand that consists of both natural and human demand allows us to be more rigorous in assessing the level of sustainable use of a shared resource, in this case water. The analysis of baseflow variability can differ based on regional location and local hydrogeology, but it was found that baseflow varies from multiyear scales such as those associated with ENSO (3.5, 7 years) up to multi decadal time scales, but with most of the contributing variance coming from decadal or multiyear scales. It was also found that the behavior of baseflow and subsequently water availability depends a great deal on overall precipitation, the tracks of hurricanes or tropical storms and associated climate indices, as well as physiography and hydrogeology. Evaluating and utilizing the Duke Combined Hydrology Model (DCHM), reasonably accurate estimates of streamflow during periods of low flow were obtained in part due to the model’s ability to capture subsurface processes. Being able to accurately simulate streamflow levels and subsurface interactions during periods of drought can be very valuable to water suppliers, decision makers, and ultimately impact citizens. Knowledge of future droughts and periods of low flow in addition to tracking customer demand will allow for better management practices on the part of water suppliers such as knowing when they should withdraw more water during a surplus so that the level of stress on the system is minimized when there is not ample water supply.