Since the construction of wastewater treatment plants combined sewer overflows have become an increasingly important limitation to the quality of the surrounding surface waters. Over the years urban water resources have often been so modified by anthropogenic activity that water quality management requires an integrated approach both at an evaluation and an investment level. Effects of acute pollutants, such as oxygen depletion caused by excessive organic material, should be evaluated on the basis of their return periods. Models of the relevant water systems are then used both to calculate the present return periods and to predict those of proposed amelioration projects. In traditional deterministic modelling of combined sewer systems there has been a tendency to continuously add new processes to the model structure in an attempt to improve the quality of the predictions made. This has often resulted in many model parameters with unknown values and the inclusion of processes much less significant than others that are not known or well understood. An alternative approach is to describe only the most essential processes and to include stochastic terms to describe the remaining variation. The present study has focused on comparing and contrasting deterministic and stochastic approaches to modelling of urban runoff pollution and water quality in general. Methodologies surrounding the application of models in return period analysis and its uncertainty have also been studied.A multivariate analysis was made on event mean concentrations data sets from three Dutch and two Danish combined sewer catchments (Chapter 3). This was done to examine the underlying structure of variations in event mean concentrations. Results confirmed expectations that the most pronounced common variations relate to the groups of particulate pollutants and dissolved pollutants. The distribution of the principal factors clearly reconfirmed the bimodal or mixed distribution that have earlier been reported for event mean concentrations of particulate substances.Non-linear event lumped models were developed to predict combined sewer discharged volume and event mean concentrations as a function of rainfall variables (Chapter 4). The aim was to combine basic understanding of the physical system with information held in the data. The discharged volume was well described with a wetness dependent runoff coefficient. Seasonality revealed initially by the data and then described using an empirical "cut-off" sinusoidal expression exhibited a remarkable agreement with average monthly open water evaporation data. Using open water evaporation as an input variable to the model improved the prediction whilst at the same time reducing the number of model parameters. The event lumped rainfall variables were only able to explain very little of the variations in the event mean concentrations of the combined sewer overflow and subsequently some of the water quality variables were characterised by their probability distributions alone.An analysis of the underlying assumptions made during mathematical modelling of water systems in time has resulted in a new portrayal of the essential differences between deterministic and stochastic modelling (Chapter 5). The implicit assumption made during deterministic modelling is that our model gives a perfect description of reality and that all deviation between modelled and observed values is a result of observation error. During stochastic modelling the implicit assumption is that the model only gives a partial description of reality and that deviation between modelled and observed values results from unexplained random behaviour of the system being modelled as well. Having isolated to the core differences between deterministic and stochastic modelling allows for more interchange of methods and approaches, thus enhancing the quality of water and water quality modelling. Knowledge of the dominating physical, chemical and biological processes of our system can be built into the traditionally empirical stochastic models. Parameter statistics, experimental design, empirical elements and concepts of identifiability can be applied to deterministic models. Quantitative a priori knowledge of given model parameter values can be incorporated into the estimation procedure. In the long term it is the aim that the selected approach will depend more on the appropriateness of the assumptions made (viewed also in relation to the available resources and the possible consequences of a poor model) than on background of the modeller, as is often the case today. Parameters of a combined sewer rainfall-runoff model have been estimated both in a deterministic and in a stochastic model to study and illustrate the main points of the chapter.Using a stochastic differential equations approach water quantity and quality models for a combined sewer system were formulated and their parameters estimated (Chapter 6). The aim was to evaluate the potentials and limitations of this approach where the sewer system is defined by a set of differential equations that is solved stochastically in continuous time. Parameter estimation was possible for the water quantity model and a very small observation error confirmed the relevance of a stochastic modelling approach. Results from the water quality modelling suggest that more work is needed in order to fully appreciate potentials and limitations of the approach.A non-linear random coefficient model to describe suspended chemical oxygen demand in a combined sewer system was identified and its parameters were estimated (Chapter 7). In random coefficient modelling certain selected parameters are assumed to vary from event to event and a value for these parameters is estimated for each event. In the present study a critical soft threshold flow at which resuspension begins is assumed to be a random coefficient. Although there is a lack of data in the period before overflow begins, the results suggest that there is a high potential for random coefficient modelling in urban runoff pollution both as an alternative to and in combination with stochastic modelling. The recipient water quality model used in the uncertainty analysis of Chapter 9 was also estimated using this approach.Methods and approaches studied in the preceding chapters have been discussed in a broader perspective whilst drawing attention to some interesting developments within the field of water and water quality modelling (Chapter 8). Structuring our physical, chemical and biological theory in stochastic state space models we acknowledge that the deviation between "what we model" and "what we see" is the result of both unexplained random behaviour of the system being modelled and observation error. This acknowledgement will reduce bias in parameter estimates and therefore improve the models' abilities to predict and extrapolate in time and to new circumstances. Although stochastic state space modelling using the Kalman filter had its main entry into hydrology and water quality modelling in the late 1970s, this was mostly with empirical formulations based entirely on observed data and therefore of little use to the engineer wishing to examine and compare alternative scenarios. To avoid over-parameterised models with highly interchangeable parameters it is important that model structure is identifiable on the basis of data being used to estimate the model parameters. The a posteriori estimation criteria incorporating quantitative a priori knowledge present an interesting formalised method of introducing the engineer's intuition and experience into the parameter estimation procedure.A new methodology for evaluating the uncertainty of a return period analysis is presented and exemplified in an integrated approach to urban runoff pollution involving models of both the combined sewer and the receiving water (Chapter 9). The underlying hypothesis of the presented methodology is that a distinction has to be made between inherent variation and uncertainty resulting from a lack of knowledge. This distinction is attained through embedded error propagation, which was here implemented as Embedded Monte Carlo Simulations. It is argued that pooling uncertainty with inherent variation systematically increases the frequency of extreme events resulting in return period curves with little or no engineering value. The study also demonstrates that efforts are needed to implement faster alternatives to the crude Monte Carlo simulations to reduce computation time, which would be necessary for use in practice.A review of methodologies surrounding return period analysis in urban runoff pollution and its uncertainty was carried out with the aim of viewing the new methodology presented in Chapter 9 in its broader perspective (Chapter 10). Three principally different methods of calculating return periods of given effects have been described: direct fitting, moments transformation and analytical or numerical integration. Combining these methods with the different types of models (in terms of input and output being time series or event lumped variables) results in a framework encompassing most approaches to return period analysis. Uncertainty in engineering work becomes particularly relevant when design criteria are based on return periods of very rare events. Because they are rare the precision with which they are described is poor and cannot be ignored when large investments and consequences are at stake. A distinction should be made between inherent variation and uncertainty due to a lack of knowledge. Furthermore, an effort should be made to use stochastic models in return period analysis to reduce bias resulting from inappropriate assumptions during parameter estimation and to avoid underestimation of the frequencies of extreme occurrences due to the exclusion of certain inherent random behaviour.

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