1 Mathematical Statistics, Department of Informatics and Mathematical Modeling, Technical University of Denmark2 Department of Informatics and Mathematical Modeling, Technical University of Denmark3 Department of Applied Mathematics and Computer Science, Technical University of Denmark
With Focus on Uncertainties
The main topic of the thesis is grey box modelling of hydrologic systems, as well as formulation and assessment of their embedded uncertainties. Grey box model is a combination of a white box model, a physically-based model that is traditionally formulated using deterministic ordinary differential equations, and a black box model, which relates to models that are obtained statistically from input-output relations. Grey box model consists of a system description, defined by a finite set of stochastic differential equations, and an observation equation. Together, system and observation equations represent a stochastic state space model. In the grey box model the total noise is divided into a measurement noise and a process noise. The process noise is due to model approximations, undiscovered input and uncertainties in the input series. Estimates of the process noise can be used to highlight the lack of fit in state space formulation, and further support decisions for a model expansion. By using stochastic differential equations to formulate the dynamics of the hydrological system, either the complexity of the model can be increased by including the necessary hydrological processes in the model, or formulation of process noise can be considered so that it meets the physical limits of the hydrological system and give an adequate description of the embedded uncertainty in model structure. The thesis consists of two parts: a summary report and a part which contains six scientific papers. The summary report is divided into three distinct parts that introduce the main concepts and methods used in the following papers. The first part contains the basic concepts in hydrology and related hydrological models. The second part explains the grey box model by presenting stochastic differential equations and show how the equations can be linked to the available measurements. Moreover, impulse response function models are introduced as an alternative to stochastic differential equation basedmodels, but by exploiting known hydrological models as the impulse response function in this model makes this model framework partly physically-based. For estimating the parameters in the grey box models maximum likelihood method is used. The third important part of the summary report is predictions, and with focus on uncertainty of prediction intervals the corresponding performancemeasures have to include the intervals. The thesis illustrates three performance measures for this performance evaluations: reliability, sharpness and resolution. For decision making, a performance criterion is preferred that quantifies all of these measures in a single number, and for that the quantile skill score criterion is discussed in this thesis. The second part of the thesis, which contains the papers, is divided into two different subjects. First are four papers, which consider the grey box model approach to a well field with several operating pumps. The model foundation is the governing equation for groundwater flow, which can be simplified and represented a state space form that resembles the methods used in numerical methods for well field modelling. The objective in the first two papers is to demonstrate how a simple grey box model is formulated and, subsequently, extended in terms of parameter estimation using statistical methods. The simple models in these papers consider only part of the well field, but data analysis reveals that the wells in the well field are highly correlated. In the third paper, all wells pumping from the same aquifer are included in the state space formulation of the model, but instead, but instead of extending the physical description of the system, the uncertainty is formulated to handle the spatio-temporal variation in the output. The uncertainty in the model are then evaluated by using the quantile skill score criterion. In the fourth paper, the well field is formulated by considering the impulse response function models to describe water level variation in the wells, as a function of available pumping rates in the well field. The paper illustrates, through a case study, how the model can be used to define and solve the well field management problem. The second half of part II consists of two papers where the stochastic differential equation based model is used for sewer runoff from a drainage system. A simple model is used to describe a complex rainfall-runoff process in a catchment, but the stochastic part of the system is formulated to include the increasing uncertainty when rainwater flows through the system, as well as describe the lower limit of the uncertainty when the flow approaches zero. The first paper demonstrates in detail the grey box model and all related transformations required to obtain a feasible model for the sewer runoff. In the last paper this model is used to predict the runoff, and the performances of the prediction intervals are evaluated by the quantile skill score criterion.