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

Research output: Contribution to journalResearch articleContributedpeer-review


  • Yan Liu - , University of Freiburg (Author)
  • Jaime Fernández-Ortega - , University of Málaga (Author)
  • Matías Mudarra - , TUD Dresden University of Technology (Author)
  • Andreas Hartmann - , Chair of Groundwater Systems, University of Freiburg (Author)


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 system behaviors. For that we use three case studies that contain different uncertainties and different types of observational data. Our results show that model parameters cannot be identified and the uncertainty of discharge simulations is large when directly using the original KGE. The adapted KGE finds similar posterior distributions of model parameters derived from 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 formal and the adapted KGE 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, while the adapted KGE has a balanced performance for both low and high flows. Furthermore, the adapted KGE shows a generally better performance for calibrations of solute concentrations. 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.


Original languageEnglish
Pages (from-to)5341-5355
Number of pages15
JournalHydrology and earth system sciences
Issue number20
Publication statusPublished - 27 Oct 2022

External IDs

Scopus 85141939038
Mendeley c16405b1-e104-3fc2-afba-6b9db327d4df
unpaywall 10.5194/hess-26-5341-2022