Given a probability space, we analyze the uncertainty, that is, the amount of information of a finite system, by studying the entropy of the system. We also extend the concept of entropy to a dynamical system by introducing a measure preserving transformation on a probability space. After showing some theorems and applications of entropy theory, we study the concept of ergodicity, which helps us to further analyze the information of the system.

Streamflow forecasting is a fundamental component of various water resources management systems, ranging from flood control and mitigation to long-term planning of irrigation and hydropower systems. In the context of floods, a probabilistic forecasting system is required for proper and effective decision-making. Therefore, the primary goal of this research is the development of an advanced ensemble-based streamflow forecasting framework to better quantify the predictive uncertainty and generate enhanced probabilistic forecasts. This research started by comprehensively evaluating the performances of various lumped conceptual models in data-poor watersheds and comparing various Bayesian Model Averaging (BMA) modifications for probabilistic streamflow simulation. Then, using the concept of BMA, two novel probabilistic post-processing approaches were developed to enhance streamflow forecasting performance. The combination of the entropy theory and the BMA method leads to an entropy-based Bayesian Model Averaging (En-BMA) approach for enhanced probabilistic streamflow and precipitation forecasting. Also, the integration of the Hydrologic Uncertainty Processor (HUP) and the BMA methods is proposed for probabilistic post-processing of multi-model streamflow forecasts. Results indicated that the MACHBV and GR4J models are highly competent in simulating hydrological processes within data-scarce watersheds, however, the presence of the lower skill hydrologic models is still beneficial for ensemble-based streamflow forecasting. The comprehensive verification of the BMA approach in terms of streamflow predictions has identified the merits of implementing some of the previously recommended modifications and showed the importance of possessing a mutually exclusive and collectively exhaustive ensemble. By targeting the remaining limitation of the BMA approach, the proposed En-BMA method can improve probabilistic streamflow forecasting, especially under high flow conditions. Also, the proposed HUP-BMA approach has taken advantage of both HUP and BMA methods to better quantify the hydrologic uncertainty. Moreover, the applicability of the modified En-BMA as a more robust post-processing approach for precipitation forecasting, compared to BMA, has been demonstrated. / Thesis / Doctor of Philosophy (PhD) / Possessing a reliable streamflow forecasting framework is of special importance in various fields of operational water resources management, non-structural flood mitigation in particular. Accurate and reliable streamflow forecasts lead to the best possible in-advanced flood control decisions which can significantly reduce its consequent loss of lives and properties. The main objective of this research is to develop an enhanced ensemble-based probabilistic streamflow forecasting approach through proper quantification of predictive uncertainty using an ensemble of streamflow forecasts. The key contributions are: (1) implementing multiple diverse forecasts with full coverage of future possibilities in the Bayesian ensemble-based forecasting method to produce more accurate and reliable forecasts; and (2) developing an ensemble-based Bayesian post-processing approach to enhance the hydrologic uncertainty quantification by taking the advantages of multiple forecasts and initial flow observation. The findings of this study are expected to benefit streamflow forecasting, flood control and mitigation, and water resources management and planning.