Dynamic modelling of predictive uncertainty by regression on absolute errors

Uncertainty of hydrological forecasts represents valuable information for water managers and hydrologists. This explains the popularity of probabilistic models, which provide the entire distribution of the hydrological forecast. Nevertheless, many existing hydrological models are deterministic and provide point estimates of the variable of interest. Often, the model residual error is assumed to be homoscedastic; however, practical evidence shows that the hypothesis usually does not hold. In this paper we propose a simple and effective method to quantify predictive uncertainty of deterministic hydrological models affected by heteroscedastic residual errors. It considers the error variance as a hydrological process separate from that of the hydrological forecast and therefore predictable by an independent model. The variance model is built up using time series of model residuals, and under some conditions on the same residuals, it is applicable to any deterministic model. Tools for regression analysis applied to the time series of residual errors, or better their absolute values, combined with physical considerations of the hydrological features of the system can help to identify the most suitable input to the variance model and the most parsimonious model structure, including dynamic structure if needed. The approach has been called dynamic uncertainty modeling by regression on absolute errors and is demonstrated by application to two test cases, both affected by heteroscedasticity but with very different dynamics of uncertainty. Modeling results and comparison with other approaches, i.e., a constant, a cyclostationary, and a static model of the variance, confirm the validity of the proposed method.