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Abstract

Predictive uncertainty in hydrological modelling is quantified by using post-processing methods or Bayesian statistical models. The former methods are not straightforward because they combine two models of different nature while the latter methods are not distribution-free. Moreover, calibration of hydrological models is largely based on the squarer error scoring function, or related skill scores and efficiency metrics (e.g. the Nash-Sutcliffe (NSE) and the Kling-Gupta (KGE) efficiencies) that are appropriate when one aims to issue predictions of the mean functional of the probability distribution of the model’s response. To remedy preceding limitations, we propose to move beyond the mean by calibrating and assessing hydrological models with consistent scoring functions. The proposed method is non-parametric, thus the specification of probability distributions in Bayesian settings is not necessary. Furthermore, predictive uncertainty can be estimated directly by calibrating the hydrological model using quantile (or expectile) scoring functions, consequently post-processing residual errors with statistical models is not required. By varying the quantile (or expectile) level of the quantile (expectile) scoring function one can directly simulate pre-specified quantiles (expectiles) of the predictive distribution of the hydrological model’s response. We apply our method to three airGR hydrological models at 511 river basins in the contiguous US. We illustrate the predictive quantiles and expectiles and we show how an honest assessment of the predictive performance of the hydrological models can be made by using consistent scoring functions.