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

Lorenz Ammann,Fabrizio Fenicia,Peter Reichert

doi:10.5194/hess-23-2147-2019

Abstract

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.

Highlights

Deterministic hydrological models are widely applied in research and decision-making processes
With case studies in two catchments, we investigate the following questions: (a) Can we confirm previous findings about the problems related to joint inference of hydrological and error model parameters?
How can the performance of empirical error models, such as those presented in this study, be quantified? We argue that the performance of an error model in joint inference with a hydrological model should be judged according to the following criteria: (a) good reproduction of observed dynamic fluctuations by individual model realisations, (b) good overall predictive marginal distribution of streamflow, and (c) small absolute deviance between model output and observations

Summary

Introduction

Deterministic hydrological models are widely applied in research and decision-making processes. The quantification of their associated uncertainties is an important task with high relevance for the scientific learning process, as well as for operational decisions with respect to water management. When performing inference, (iv) observation errors are an additional source of uncertainty, which arise for example due to errors in rating curves The sources (i–iv) usually result in residual errors of predicted streamflow observations with the following characteristics:. Model residuals are seldom well represented by a normal distribution with constant mean and variance. Residuals are typically heteroscedastic (increasing with streamflow), right-skewed due to non-negativity of streamflow and characterized by excess kurtosis (fat tails) Residuals are typically heteroscedastic (increasing with streamflow), right-skewed due to non-negativity of streamflow and characterized by excess kurtosis (fat tails) (e.g. Schoups and Vrugt, 2010)

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