Log-transformed discharge is often used to calculate performance criteria to
better focus on low flows. This prior transformation limits the
heteroscedasticity of model residuals and was largely applied in criteria
based on squared residuals, like the Nash–Sutcliffe efficiency (NSE). In the
recent years, NSE has been shown to have mathematical limitations and the
Kling–Gupta efficiency (KGE) was proposed as an alternative to provide more
balance between the expected qualities of a model (namely representing the
water balance, flow variability and correlation). As in the case of NSE,
several authors used the KGE criterion (or its improved version KGE

In the context of rainfall–runoff modelling, evaluating the quality of the
models' outputs is essential. Deterministic simulations are commonly
evaluated using efficiency criteria such as the Nash–Sutcliffe efficiency

However, the Nash–Sutcliffe criterion was shown to have limitations. Indeed,
using a decomposition of NSE based on the correlation, bias and ratio of
variances,

Given that this criterion tends to be sensitive to large errors, some users
chose to apply prior transformations on flows before computing KGE, e.g. to
put more weight on low flows, as done with NSE. For example,

In this technical note we show that the use of a logarithmic transformation
when computing KGE or KGE

The KGE and KGE

The KGE

Because the three terms

KGE

Consequently, because the conversion term becomes additive when applying the
logarithmic transformation, the

When using a logarithmic (or an inverse) transformation, the case of null
flows, which may exist in the case of intermittent or ephemeral streams, prevents
proper calculation. To avoid this, different techniques may be set up in the
case of NSE:

The first involves discarding the zero-flow values from the series, i.e. considering them as gaps

The second involves adding a small constant to all flow values

The third involves using a Box–Cox transformation to reproduce the effects of the logarithmic transformation without
the zero-flow issue

To illustrate these numerical issues and their potential impacts, several
tests were carried out in a wide range of catchments, using the GR4J rainfall–runoff
model

A daily data set of 240 catchments across France (Fig.

Location of the 240 flow gauging stations in France used for the tests and their associated catchments.

The tests were performed with the daily lumped conceptual GR4J
model

Values of KGE

Figure

Values of KGE

The KGE

Comparison between KGE

In this technical note, the impact of a near-zero standard deviation of log-transformed flows is not presented because it is rarer than near-zero mean values. The standard deviations of flows in the catchments studied are indeed all significantly higher than zero.

The dependence of KGE

Dependence on flow units of the KGE

The higher model performance when using litres per second than when using
square metres per second can be explained analytically. Considering
Eq. (

Because

Sensitivity of NSE and KGE

As presented in Sect.

Values of KGE

The Box–Cox transformation is also dependent on the units
(Fig.

Using this equation, the KGE

Using Eq. (

Dependence on flow units of the KGE

Furthermore, because the zero of the modified Box–Cox function is not

Values of KGE

The modified Box–Cox transformation (Eq.

Comparison between KGE

Pros (

Given the previous results, we can argue that using log-transformed flows to
calculate the KGE or the KGE

Instead of KGE

Note that many studies use NSE on log-transformed flows

the underestimation of variability,

the low weight of water balance errors for catchments with highly variable flows,

the poor benchmark represented by the mean flows for catchments with highly variable flows.

Two additional remarks should be taken into account on this topic. First, as noted by Harald Kling in a personal communication, 2018, prior transformations on flows in KGE (or in NSE) lead to a misinterpretation in the estimation of the water balance. The other components of the KGE also lose their initial physical meaning. KGE on transformed flows can give more information on low flows, but the physical interpretation of the criterion is not as simple as in the case of untransformed flows.

Secondly, even if it did not occur in our experiment, the issue described in this technical note may lead to problems during the calibration process. Indeed, it can create a strongly negative zone in the objective function hyperspace, which may negatively impact the performance of local calibration algorithms.

The daily flow data can be downloaded from the Banque HYDRO
website (

LS made the technical development and the analysis. The paper was written by him, GT and CP.

The authors declare that they have no conflict of interest.

The authors thank Météo France for providing the data used in this work. We also wish to thank Alban De Lavenne, Laure Lebecherel, Maria-Helena Ramos and Cedric Rebolho for the discussions on the different aspects of the issues using the logarithmic transformation with KGE. We thank Andrea Ficchí for his work on the database and Linda Northrup for her correction of the English language of an earlier version of the paper. Finally, we extend our thanks to Harald Kling for discussions on this issue.

We thank the topical editor, Bettina Schaefli, for her careful reading of the
paper, her suggestion on the modified Box–Cox transformation and the
following discussions. We also thank the two reviewers, Lieke Melsen and
Björn Guse, for taking the time to read our paper and for their
remarks that helped us to make the paper and the figures more
understandable. We thank Sivarajah Mylevaganam for the discussions that
helped us to be more precise in the KGE and KGE