Uncertainty estimation of a complex water quality model: GLUE vs Bayesian approach applied with Box – Cox transformation

In urban drainage modelling, uncertainty analysis is of undoubted necessity; however, several methodological aspects need to be clarified and deserve to be investigated in the future, especially in water quality modelling. The use of the Bayesian approach to uncertainty analysis has been stimulated by its rigorous theoretical framework and by the possibility of evaluating the impact of new knowledge on the modelling estimates. Nevertheless, the Bayesian approach relies on some restrictive hypotheses that are not present in less formal methods like GLUE. One crucial point in the application of Bayesian methods is the formulation of a likelihood function that is conditioned by the hypotheses made regarding model residuals. Statistical transformations, such as by the use of the Box–Cox equation, are generally used to ensure the homoscedasticity of residuals but this practice may affect the reliability of the analysis leading to a wrong estimation of the uncertainty. The present paper aims to study the impact of such a transformation considering five cases one of which is the “real” residuals distributions (drawn from available data). The analysis was applied to the Nocella experimental catchment (Italy) which is an agricultural and semi-urbanised basin where two sewer systems, two wastewater treatment plants and a river reach were monitored during both dry and wet weather periods. The results show that the uncertainty estimation is greatly affected by residual transformation and a wrong assumption may also affect the evaluation of model uncertainty. The use of less formal methods always provide an overestimation of modelling uncertainty with respect to Bayesian method but such effect is reduced if a wrong assumption is made regarding the residuals distribution. If residuals are not normally distributed, the uncertainty is over-estimated if Box-Cox transformation is not applied or non calibrated parameter is used.