Modelling uncertainty in flood forecasting systems; Dissertation, UNESCO-IHE Institute for Water Education , Delft.

Uncertainty is a common factorof everyday life. In almostall circumstances we find ourselves in a state of uncertainty. The subject of this research is about uncertainty in flood forecasting systems. Like all natural hazards flooding is a complex and inherently uncertain phenomenon. The uncertainty in flood forecasting comes from the uncertainty in the precipitation forecast and other inputs, the model parameters, model structure and so forth. Despite increasing advances in the development of flood forecasting models and techniques, the uncertainty in actual flood forecasts remains unavoidable. It is therefore important that the existence of uncertainty in flood forecasts be admitted and properly appraised. Hiding uncertainty may create the illusion of certainty, the consequences of which can be very significant. The major benefits of estimating uncertainty in flood forecasting are that itprovides a rational basis for flood warning (risk-based warning) and it offers potential by providing economic benefits from flood forecasting and warning systems. Even with the lack of risk-based flood warning procedures, quantifying uncertainty provides additional information about the forecasts and helps decision makers to use their own judgement more appropriately. The aim of this research is to develop a framework and tools and techniques for modelling uncertainty in flood forecasting systems. This research applies uncertainty assessment methods based on probability theory and fuzzy set theory to the problem of flood forecasting. The outcomes of this research can be summarised as follows: 1. Review of different types of flood forecasting models and their appraisal with respect to uncertainty assessment. 2. Review of uncertainty representation and modelling theories and methods, particularly in regard to flood forecasting. 3. Development of a methodology using temporal disaggregation for uncertainty assessment in model outputs due to the uncertainty in time series inputs. 4. Development of an Improved First-Order Second Moment (IFOSM) method for uncertainty assessment using a second-order reconstruction of the model function. 5. Development of a qualitative uncertaintyscale using the best-case and worstcase scenarios for the interpretation ofthe uncertainty analysis results generated by the expert judgement-based qualitative method. 6. Exploration of hybrid techniques of uncertainty modelling and probabilitypossibility transformations. Different types of models for flood forecasting are reviewed. Broadly they can be classified as (i) physically-based, (ii) conceptual and empirical, and (iii) data-driven. Uncertainty analysis is a well accepted procedure in the first two categories of models, although many operational flood forecasting systems may not have components for uncertainty analysis. Normally, increasing the complexity of the model decreases the uncertainty in the model output due to the model structure. In this sense, model uncertainty should in principle decrease as we move from empirical to conceptual to physically based models. But model uncertainty is only a part of the total uncertainty. If there exists a large uncertainty in the inputs and if the parameters cannot be estimated with high precision, using complex models does not guarantee less uncertain model results. Some of the data-driven techniques, such as fuzzy rule- based systems and fuzzy regression, workwith imprecise data and implicitly incorporate the uncertainty concept in modelling. These models however, do not have the flexibility of using uncertainty methods based on other theories (e.g. the most popular probability theory). In the case of the model based on artificial neural network (ANN) (the most popular so far of the data-driven techniques), model performance is, by and large, expressed in a form based on the difference between the observed and model predicted results using measures such as Root Mean Square Error (RMSE). More study is required for the application of the more commonly used uncertainty analysis techniques, such as those based on probability and fuzzy set theories, to ANN-based models. The literature review on uncertainty suggests that probability theory and fuzzy set theory (including fuzzy measures and possibility theory) are the two most widely used theories for uncertainty representation. Probability theory assumes uncertainty mainly due to randomness, whereas fuzzy set theory assumes it due to vagueness (or fuzziness) and imprecision. In flood forecasting, however, the application of theories other than probability has so far been insignificant. It is argued that these two theories must be treated as complementary rather than competitive. In this research the Monte Carlo method and the First-Order Second Moment (FOSM) method are used as the standard techniques for uncertainty propagation in the probabilistic approach. At the same time, this research extends the application of fuzzy set theory for uncertainty modelling in flood forecasting. In particulartwo methods based on fuzzy set theory are explored: (i) the fuzzy Extension Principle, and (ii) the expert judgement based qualitative method. The development of the methodology using temporal disaggregation for uncertainty assessment in model results due to the uncertainty in time series inputs is an important contribution of this research. This methodology can be implemented in the frameworks of both the Monte Carlo method and the fuzzy Extension Principle, and can be applied to any type of deterministic rainfall-runoff model. This methodology explicitly requires the uncertainty in the time series inputs to be represented by probability distributions for the probabilistic approach or by membership functions for the fuzzy approach. The latter approach is particularly useful when the precipitation forecasts are nonprobabilistic. The fact that the popular FOSM method usesthe linearisation of the function gives rise to some limitations of the method. As part of this research, an Improved FOSM method is proposed using the second-degreereconstruction of the function to be modelled. The IFOSM method retains the simplicity and the smaller computational requirement of the FOSM method and it has a particular advantage when the average value of the input variable corresponds tomaximum/minimum of model function or to values in regions where the slope of the function is very mild compared to the effects of curvature (non-linearity). This research also demonstrates the application of a qualitative method of uncertainty assessment in flood forecasting. The qualitative method is based on expert judgement and fuzzy set theory. The advantage of this method is that it allows the incorporation of uncertainty due to all recognisable sources without a significant increase in the computation time. This method has the same mathematical structure as that of the FOSM method but the evaluations of the qualityand importance(similar to the variance and sensitivity, respectively, of the FOSM method) are fully based on expert judgement. Consequently it is a very approximate yet more holistic method. One of the issues in using this method concerns the interpretation of the results. In the course of this research, a Qualitative Uncertainty Scale was developed on which the results from the expert judgement-based method can be classified qualitatively using linguistic variables such as small uncertainty, moderate uncertainty, large uncertainty, etc. The derivation of the uncertainty scale is based on the concept of best-case and worst-case scenarios. This research also investigates the possibility of a hybrid technique of uncertainty modelling, whereby the combined application of both probabilistic and possibilistic or fuzzy approaches are used. There are at least two types of situation where the hybrid technique can be useful, which are referred in this thesis as Type I and Type II Problems. The Type I Problem is the situation where an uncertain variable possesses components from randomness and fuzziness. The Type II Problem refers to the situation where there exist two sets of uncertainty parameters: one represented by probability distributions and the other represented by possibility distributions or membership functions. Whereas the concepts, such as fuzzy probability and fuzzyrandom variable can be used to deal with the Type I Problem, a solution to the Type II Problem is also possible if an acceptable transformation between probability and possibility can be established. The major obstacle in modelling uncertaintyof the Type II Problem comes from the differences of operation between random-random (R-R) and fuzzy-fuzzy (F-F) variables. Part of this research is also devoted to the exploration of the differences and similarities in operation between R-R and F-F variables, and to investigate the derivation of a probability-possibility (or fuzzy) transformation that takes these differences into account. This research shows that the arithmeticoperations between two fuzzy variables by the fuzzy Extension Principle using a method called α-cut are similar to corresponding operations between two functionally dependent random variables for some specific conditions. It also provides an alternative method for the evaluation of the Extension Principle for a monotonic function without using the α-cut method. Although the direct implication ofthis finding is very limited, it is certainly an important basis for further research in probability-possibility transformations and hybrid approaches in uncertainty modelling. The application examples used in this research are flood forecasting models for the Klodzko catchment in Poland and the Loire River in France. The uncertainty assessment methodology using temporal disaggregation is applied to the Klodzko model to assess the uncertainty in the forecasted floods due to the uncertainty in the precipitation time series. It is implemented in the framework of the fuzzy Extension Principle and assisted by genetic algorithms (GAs) for the determination of the minimum and maximum of the model function. The results are illustrated using two versions of GAs, namely a normal (or conventional) GA and a micro GA. The Loire model is used to apply the FOSM, IFOSM and expert judgement-based qualitative methods for uncertainty assessment in the model forecasts due to the uncertainty in different parameters. In this application the Monte Carlo method is used as a standard method to compare the results from the FOSM and IFOSM methods