02 Nov 2021
02 Nov 2021
Status: a revised version of this preprint is currently under review for the journal HESS.

Pitfalls and a feasible solution for using KGE as an informal likelihood function in MCMC methods: DREAM(ZS) as an example

Yan Liu1, Jaime Fernández-Ortega2, Matías Mudarra2, and Andreas Hartmann1,3 Yan Liu et al.
  • 1Chair of Hydrological Modeling and Water Resources, University of Freiburg, 79098 Freiburg, Germany
  • 2Department of Geology and Centre of Hydrogeology, University of Málaga (CEHIUMA), 29071 Málaga, Spain
  • 3Department of Civil Engineering, University of Bristol, Bristol, UK

Abstract. The Kling-Gupta Efficiency (KGE) is a widely used performance measure because of its advantages in orthogonally considering bias, correlation and variability. However, in most Markov chain Monte Carlo (MCMC) algorithms, error-based formal likelihood functions are commonly applied. Due to its statistically informal characteristics, using the original KGE in MCMC methods leads to problems in posterior density ratios due to negative KGE values and high proposal acceptance rates resulting in less identifiable parameters. In this study we propose adapting the original KGE using a gamma distribution to solve these problems and to apply KGE as an informal likelihood function in the DiffeRential Evolution Adaptive Metropolis DREAM(ZS), which is an advanced MCMC algorithm. We compare our results with the formal likelihood function to show whether our approach is robust and plausible to explore posterior distributions of model parameters and to reproduce the discharge behaviors. For that, we set three case studies that contain different uncertainties. Our results show that model parameters cannot be identified and the uncertainty of discharge simulations is large when directly using the original KGE. Our approach finds similar posterior distributions of model parameters compared to the formal likelihood function. Even though the acceptance rate of the adapted KGE is lower than the formal likelihood function for some systems, the convergence rate (efficiency) is similar between the two approaches for the calibration of real hydrological systems showing generally acceptable performances. We also show that both the adapted KGE and the formal likelihood function provide low performances for low flows, with the larger overestimations obtained from using the formal likelihood function. Furthermore, the adapted KGE approach behaves closely to the formal likelihood function in terms of the correlation between simulations and observations. Thus, our study provides a feasible way to use KGE as an informal likelihood in the MCMC algorithm and provides possibilities to combine multiple data for better and more realistic model calibrations.

Yan Liu et al.

Status: final response (author comments only)

Comment types: AC – author | RC – referee | CC – community | EC – editor | CEC – chief editor | : Report abuse
  • RC1: 'Comment on hess-2021-514', Anonymous Referee #1, 22 Nov 2021
    • AC1: 'Reply on RC1', Yan Liu, 21 Jan 2022
  • RC2: 'Comment on hess-2021-514', Anonymous Referee #2, 17 Dec 2021
    • AC2: 'Reply on RC2', Yan Liu, 21 Jan 2022

Yan Liu et al.


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Short summary
We adapt the informal performance measure Kling-Gupta Efficiency (KGE) with a gamma distribution to apply it as an informal likelihood function in the DiffeRential Evolution Adaptive Metropolis DREAM(ZS) method. Our results show that the adapted KGE performs as good as the formal likelihood function for exploring posterior distributions of model parameters. The adapted KGE even has a higher general performance and a smaller bias overestimation of low flows than the formal likelihood function.