Determination of parameter uncertainty and evaluation of the predictive significance of an urban hydrological model
Computer based simulation models are widely used in urban drainage design and management. For instance, these models are often applied in order to predict system behaviour for different planning conditions. The knowledge of basic physical processes is still limited, so models are always a simplified representation of the regarded system. For this reason model results should be considered as estimates or predictions with uncertainty as inherent attribute. The sources of model predictive uncertainty are diverse. They can be assorted to external disturbances outside the modelled domain, system state, errors in observed input variables and output responses, model structure and model parameters.
Decisions of planners and engineers are based on uncertain model results. A quantification of model predictive uncertainty increases the information content of model results and model credibility. It will also help decision makers to better estimate risks. The objective of modelling should not only be determining single values but also probability distribution functions, the main statistical moments or at least the potential range of modeled system variables. Confidence limits for model results can be indicated based on this additional information.
In modelling practice the different sources of uncertainty are not considered for systematically. Common modelling approaches are only partly physically based. They introduce conceptual parameters and parameters with physical meaning that cannot be directly measured. The values of these parameters are determined by variation in order to fit the curve of calculated versus - if available - observed values of the considered system response. In this manner, presence of external disturbances and lack of knowledge of initial system state are ignored and it is implicitly assumed that errors due to model structure are negligible and input data and observations of system response are error free. The estimated parameters are likely to be ambiguous or biased because they indirectly account for other sources of uncertainty. So in the strict sense, the set of estimated parameters is only valid for applications similar to calibration conditions. Consequently the usefulness and applicability of the model for predictive purposes are reduced. The suitability of the model for predictive purposes has to be proved by validation.
Subject of the presented work is to examine and compare the implication of single sources of uncertainty on uniqueness of estimated parameters and model predictive significance. In detail, uncertainties due to input data, model parameters and output data are analysed.
Effects of input data uncertainty are studied by perturbing ‘true’ rainfall input by assumed errors.
In order to quantify uncertainties of model parameters, estimates are determined for several events. Presuming that the set of events constitutes a representative sample of sufficient size the obtained values are used as a sample for defining model parameter uncertainties. The characteristics of the events, e.g. variability, length, succession etc., influences the universality of parameter estimates.
Multiple objective functions are used for parameter estimation in order to reduce parameter uncertainty by extracting maximum information from available data.
Errors in output data are represented inherently by the type of objective functions used for parameter estimation.
For the presented work a typical conceptual urban rainfall runoff model is used to determine the discharge hydrograph from urban areas. The model concept for paved areas is based on time variable losses for determination of effective rainfall and constant dicharge coefficient for formation of runoff. Runoff from unpaved areas is determined by the SCS-method. Parallel linear reservoir cascades are used for concentration of runoff.
Sensitive parameters are: degree of imperviousness, curve number, flow partitioning factor between reservoir cascades, storage coefficient, number of reservoirs per cascade.
The model is linked to a global optimisation algorithm based on evolutionary algorithms for automated estimation of parameters.
Within this framework, best parameter estimates for different ‘uncertainty scenarios’ are determined. Then the predictive significance of the obtained parameter sets is compared using a validation event.
First, a single objective parameter estimation based on the least squares criterion is carried out. Parameter estimates are obtained for different events and uncertainty ranges are derived.
Then input data are pertubated by errors and parameter estimation for different events is conducted.
The procedure is repeated for a multi objective parameter estimation using least squares criterion and overall volume error.
Discharge hydrographs and variation bands are determined for a validation rainfall event using different parameterised models via Monte Carlo simulation. Output reliability is measured by the observed hydrograph falling within the confidence interval. This comparison reveals the suitability for predictions of the respective model and the amount of variance contributed by different sources of uncertainty. On this basis indications on improvement of model parts and data basis can be derived. Also different approaches to minimise model predictive uncertainty can be evaluated.