Articles | Volume 23, issue 4
Hydrol. Earth Syst. Sci., 23, 2147–2172, 2019
Hydrol. Earth Syst. Sci., 23, 2147–2172, 2019

Research article 30 Apr 2019

Research article | 30 Apr 2019

A likelihood framework for deterministic hydrological models and the importance of non-stationary autocorrelation

Lorenz Ammann1,2, Fabrizio Fenicia1, and Peter Reichert1,2 Lorenz Ammann et al.
  • 1Swiss Federal Institute of Aquatic Science and Technology (Eawag), Dubendorf, Switzerland
  • 2Department of Environmental Systems Science, ETH Zurich, Zurich, Switzerland

Abstract. The widespread application of deterministic hydrological models in research and practice calls for suitable methods to describe their uncertainty. The errors of those models are often heteroscedastic, non-Gaussian and correlated due to the memory effect of errors in state variables. Still, residual error models are usually highly simplified, often neglecting some of the mentioned characteristics. This is partly because general approaches to account for all of those characteristics are lacking, and partly because the benefits of more complex error models in terms of achieving better predictions are unclear. For example, the joint inference of autocorrelation of errors and hydrological model parameters has been shown to lead to poor predictions. This study presents a framework for likelihood functions for deterministic hydrological models that considers correlated errors and allows for an arbitrary probability distribution of observed streamflow. The choice of this distribution reflects prior knowledge about non-normality of the errors. The framework was used to evaluate increasingly complex error models with data of varying temporal resolution (daily to hourly) in two catchments. We found that (1) the joint inference of hydrological and error model parameters leads to poor predictions when conventional error models with stationary correlation are used, which confirms previous studies; (2) the quality of these predictions worsens with higher temporal resolution of the data; (3) accounting for a non-stationary autocorrelation of the errors, i.e. allowing it to vary between wet and dry periods, largely alleviates the observed problems; and (4) accounting for autocorrelation leads to more realistic model output, as shown by signatures such as the flashiness index. Overall, this study contributes to a better description of residual errors of deterministic hydrological models.

Short summary
The uncertainty of hydrological models can be substantial, and its quantification and realistic description are often difficult. We propose a new flexible probabilistic framework to describe and quantify this uncertainty. It is show that the correlation of the errors can be non-stationary, and that accounting for temporal changes in correlation can lead to strongly improved probabilistic predictions. This is a promising avenue for improving uncertainty estimation in hydrological modelling.