Application of Formal and Informal Bayesian Methods for Water Distribution Hydraulic Model Calibration

Water distribution system model parameter calibration is an important step to obtain a representative system model, such that it may be applied to understand system operational performance, often in real time. However, few approaches have attempted to quantify uncertainty in calibrated parameters, model predictions, and consider the sensitivity of model predictions to uncertain parameters. A probabilistic Bayesian approach is applied to calibrate and quantify uncertainty in the pipe roughness groups of an Epanet2 hydraulic model of a real-life water distribution network. Within the applied Bayesian framework, the relative performance of formal and informal Bayesian likelihoods in implicitly quantifying parameter and predictive uncertainty is considered. Both approaches quantify posterior parameter uncertainty with similar posterior distributions for parameter values (mean and standard deviation). However, the uncertainty intervals identified with the informal likelihood are too narrow, regardless of the behavioral threshold applied to derive these bounds. In contrast, the formal Bayesian approach produces more realistic 95% prediction intervals based on their statistical coverage of the observations. This results as the error model standard deviation is jointly inferred during calibration, which also helps to avoid potential overconditioning of the posterior parameter distribution. However, posterior diagnostic checks reveal that the prediction intervals are not valid at percentiles other than the 95% interval as the assumptions of normality, residual homoscedasticity, and noncorrelation, often assumed in hydraulic model calibration, do no hold. More robust calibration requires the development of error models better suited to the nature of residual errors found in water distribution system models.