Environmental spatiotemporal mapping and groundwater flow modelling using the BME and ST methods

Modelling the natural processes that shape our environment is a difficult task, both numerically and theoretically. The numerical solution of physical laws describing natural processes, such as the flow and transport of pollutants in the subsurface, lead to solutions that are very expensive, even on modern day computers. Additionally natural processes exhibit high randomness in space and time and are described by various sources of knowledge (including sparse observations, uncertain measurements, empirical laws, etc.), but classical methods of geostatistics lack the theoretical underpinnings to accurately incorporate these types of knowledge. Therefore there is a need for novel approaches that are numerically efficient and theoretically sound. In this work two methods are implemented to respond to these challenges; the space transformation (ST) method used to solve efficiently the groundwater flow equation, and the Bayesian Maximum Entropy (BME) method for the spatiotemporal mapping of natural processes using uncertain information. The ST transforms the complicated set of PDEs describing three-dimensional flow into a set of one-dimensional equations (i.e. ordinary differential equations, or ODEs) which are much easier to solve. The method is implemented to solve the ground water flow equation for three dimensional flow domains, and numerical results show that accurate solutions are calculated and that an efficient parallel implementation is possible because the computational work is divided into independent tasks. The BME method offers a rigorous framework for spatiotemporal analysis and mapping. Due to its epistemological background and mathematical rigor, the BME approach offers a considerable flexibility in incorporating various sources of physical knowledge. In this work efficient formulations of the BME approach are derived, and the method is implemented for several mapping situations of practical interest. In particular the BME implementations presented can incorporate uncertain measurements of interval or probabilistic types. Numerical comparisons show that BME is substantially more accurate than classical kriging methods, because it accounts rigorously for uncertain information. Several case studies are presented, which demonstrate the practical usefulness of BME mapping in water resources and environmental health. Finally, a general framework is presented that permits to incorporate physical laws in the BME analysis. This framework unifies the goal of the ST method within the spatiotemporal mapping context, and leads to considerable gains in accuracy and flexibility.