Kling–Gupta efficiency explained

The Kling–Gupta efficiency (KGE) is a goodness-of-fit indicator widely used in the hydrologic sciences for comparing simulations to observations. It was created by hydrologic scientists Harald Kling and Hoshin Vijai Gupta.^[1] Its creators intended for it to improve upon widely used metrics such as the coefficient of determination and the Nash–Sutcliffe model efficiency coefficient.

KGE =1- \sqrt{(r -1 ) ^{2+(\alpha-1)} ^{2+(\beta-1)} ^{2}}

where:

• $r$ is the Pearson correlation coefficient,
• $\backslash alpha$ is a term representing the variability of prediction errors,
• $\backslash beta$ is a bias term.

The terms $\backslash alpha$ and $\backslash beta$ are defined as follows:

\beta = (\mu _{s})/(\mu _{o})

where:

• $\backslash mu\_s$ is the mean of the simulated time series (e.g.: flows predicted by the model)
• $\backslash mu\_o$ is the mean of the observed time series

and

\alpha = (\sigma _{s})/(\sigma _{o})

where:

• $\backslash sigma\_s^2$ is the variance of the simulated time series, so $\backslash sigma\_s$ is estimated by the standard deviation of simulated data.
• $\backslash sigma\_o^2$ is the variance of the observed time series

A modified version, KGE', was proposed by Kling et al. in 2012.^[2]

Notes and References

1. 10.1029/2011WR010962. 1944-7973. 47. 10. Gupta. Hoshin Vijai. Kling. Harald. On typical range, sensitivity, and normalization of Mean Squared Error and Nash–Sutcliffe Efficiency type metrics. Water Resources Research. 2023-08-24. 2011. 2011WRR....4710601G. 119636876.
2. 10.1016/j.jhydrol.2012.01.011. 424. 264–277. Kling. Harald. Fuchs. Martin. Paulin. Maria. Runoff conditions in the upper Danube basin under an ensemble of climate change scenarios. Journal of Hydrology. 2012. 2012JHyd..424..264K.