Probabilistic modelling in urban drainage – two approaches that explicitly account for temporal variation of model errors

This article presents and compares two approaches that consider temporal variations of model errors during stochastic modelling and uncertainty analysis. Time-dynamic error variations should be considered especially in urban drainage modelling because of model structure deficits and the sources of input uncertainties observed in the models. The explicit inclusion of such variations in the modelling process will lead to a better fulfilment of the assumptions made in formal statistical frameworks, thus reducing the need to resolve to informal methods. The two approaches presented here are the external bias description (EBD) and the internal noise description (IND, also known as stochastic grey-box model). The former approach can add a bias with time-varying mean and variance to the output of any model, while the latter approach uses stochastic model equations and continuously updates the model to observations. After a brief discussion of the assumptions made for likelihood-based parameter inference, we illustrated the basic principles of both approaches on the example of sewer flow modelling with a conceptual rainfallrunoff model. The results from a real-world case study suggested that both approaches can yield reliable simulations and forecasts. The EBD approach had performed stronger in simulation but was computationally more expensive while the IND approach was suitable for online applications.