We used data from 325 catchments around France in order to (i) generalize the conclusions drawn from this study and (ii) explore possible links between catchment characteristics and specific behaviours of transformations. These catchments were chosen for the low human impact on the precipitation–streamflow relationship and for the low rate of missing streamflow data (<0.5%) over the period of interest. Moreover, the catchments are spread throughout France (Fig. ), thus representing a wide variety of meteorological and hydrological conditions.

Precipitation and temperature data were retrieved from the Météo-France SAFRAN reanalysis . Streamflow data were retrieved from the French HydroPortail database . Daily meteorological and hydrological data from the period August 1985–July 2005 were used, with August 1985–July 1995 as the calibration period and August 1995–July 2005 as the independent evaluation period. The main characteristics of the 325 catchments are summarized in Table , illustrating the large diversity of the catchment characteristics encountered, with small to large catchments, various precipitation and temperature conditions, rain-fed and snow-fed catchments, and catchments with low- to high-baseflow components. Moreover, the large range of land cover, slope, and hydraulic length strengthens the diversity of possible catchment responses. This table also shows that the climatic and hydrological conditions are similar between the two periods, with the evaluation period being only slightly warmer and wetter than the calibration period and the other indicators showing only slight variations.

The GR4J model is a lumped conceptual daily rainfall-runoff model . In this model, the effective precipitation is derived from the reduction in total precipitation by vegetation interception and by evapotranspiration from a soil-moisture-accounting production store. The effective precipitation is then routed through two unit hydrographs and one routing store. Groundwater exchange can occur from or to neighbouring catchments. A complete description of the model's equations is provided by .

This model contains four free parameters to calibrate against streamflow observations: the maximum capacity of the production store (X1, mm), the groundwater potential exchange (X2, mmd-1), the 1 d ahead routing store capacity (X3, mm), and the time characteristics of the unit hydrographs (X4, d).

For the catchments with a proportion of solid precipitation (considered here to be precipitation occurring with negative air temperatures) greater than 10 % of the total precipitation, a snow model (CemaNeige) was used. This model is based on a degree-day approach and comprises two parameters to calibrate: the melt rate coefficient (Kf, mm°C-1d-1) and a parameter regulating the energy of the snowpack (cT, dimensionless). In order to consider intra-catchment variability, CemaNeige was applied to five elevation bands of equal area, which makes it possible to account for temperature and precipitation gradients seefor more details.

In this work, we also use the GR6J model to assess the transferability of the conclusions drawn. GR6J adds two parameters to GR4J: X5 [–], which enables an inversion of the direction of the groundwater exchange throughout the year, and X6 [mm], which is the maximum capacity of an additional exponential store, whose purpose is to improve low-flow simulations.

All the calculations are made with the airGR R package . The built-in optimization algorithm, an initial parameter grid screening followed by the steepest-gradient approach, is chosen due to its known satisfactory performance with GR models . All optimization criteria and streamflow transformations used in this work are embedded in airGR.

In order to assess the impact of transformations, the hydrological models are calibrated with several objective functions over the 1985–1995 period. However, in order to estimate how transformations impact the simulated time series, the 1995–2005 independent evaluation period is also used. In both cases, a 1-year spin-up period preceding the aforementioned periods is used.

Three objective functions are chosen for their wide-ranging use in calibrating hydrological models: the well-known Nash–Sutcliffe efficiency NSE; see, the Kling–Gupta efficiency KGE; see, and the modified Kling–Gupta efficiency KGE^′; see. The NSE concentrates most of the analyses of this work, and the KGE and KGE^′ objective functions are used to assess the generality of the results. These three criteria are detailed in Eqs. (), (), and (). 1ENSE=1-∑t=1N(Qts-Qto)2∑t=1N(Q‾o-Qto)22EKGE=1-(r-1)2+(Qs‾Q‾o-1)2+(sd(Qs)sd(Qo)-1)23EKGE′=1-(r-1)2+(Qs‾Q‾o-1)2+(CV(Qs)CV(Qo)-1)2 N is the total number of days of the test period. Qts and Qto are the simulated and observed streamflows, respectively, at time step t. Q‾o (Q‾s) is the average observed (simulated) streamflow over the period. r is the correlation coefficient, sd is the standard deviation, and CV is the coefficient of variation.

The hydrological models are calibrated by applying different transformations to streamflow values in the calculation of objective functions. Nine to 11 transformations are used (Table ), together with three objective functions. In addition to the transformations mentioned in the Introduction section, four additional transformations are used. The squared (Q2) transformation is applied, as this can be used for focusing on floods , and its inverse (Q-2) is applied, as it focuses on low flows. Furthermore, two composite criteria, f(Q)+f(Q-1)2 and f(Q)+f(log(Q))2 with f standing for NSE, KGE, or KGE^′;, are added since they can be used as a compromise between criteria focusing on ranges of streamflows that are too specific. The two transformations containing the use of a logarithm are not applied to KGE and KGE^′, as they cause numerical instabilities and unit dependence, as shown by . Regarding the Box–Cox transformation, Eq. (10) in is used to avoid the same issues as for the logarithmic transformation with a λ value equal to 0.25, as suggested by .

In order to evaluate the impact of the transformations on model calibration, we use a common analysis framework that aims at analysing the behaviour of transformations at every simulation time step. The general methodology, which is applied for each catchment and objective function, is detailed here and in Fig.  (for illustrative purposes for only two transformations):

The hydrological model is calibrated against observed streamflows for a catchment and with a given objective function successively with different transformations (see Fig. a for two transformations only).

The use of ranks to classify the proximity between model simulations and streamflow observations could be criticized, since it gives the same importance to large and small errors. This option was preferred to the use of direct (normalized) errors. Indeed, ranks make it possible to consider together various flow ranges, where the magnitudes of errors can be very different. The impact of this methodological choice will be discussed in Sect. .

The methodology above is applied catchment by catchment. Then, to aggregate the results over the 325 catchments, we either identify the transformation with the most number-1 ranks or average the ranks over the 325 catchments.

When modellers choose an objective function (or, if relevant, a transformation), the main objective is to have a model fit for purpose, e.g. to be the best for low flows if the target is low flows. In the following, we evaluate the link between the objective function and transformation selected and the accuracy of the model using the 200 flow intervals described above. We successively performed this analysis on

the calibration period over a single catchment with GR4J calibrated on NSE,

We also tried to link the results to the characteristics of the 325 catchments over the evaluation period with GR4J calibrated on NSE, using the Spearman correlation. Lastly, we questioned the methodology used to compare transformations for the calibration period over the 325 catchments with GR4J calibrated on NSE.

Figure  illustrates an example application of the methodology for a single catchment, the Fecht River at Wintzenheim, for the GR4J model calibrated with the NSE objective function and for 11 different transformations. Here we show which transformation leads to the most number-1 ranks for each of the 200 intervals, for low flows (on the left) to high flows (on the right). It appears that some transformations are often ranked first (such as the -2, -1, 1, and 2 transformations). Conversely, some transformations are rarely or never ranked first (such as QlogQ or 0.2). In this figure, the transformations are represented in an order from a pre-supposed good representation of high flows (top row) to a pre-supposed representation of low flows (bottom row), because, except for composite transformations, the aforementioned transformations are presented in order of decreasing power. We can see that the logic is respected quite well, with transformations 2 and 1 being very well represented regarding intervals corresponding to high flows and transformations -2 and -1 being very well represented regarding intervals corresponding to low flows. This does not preclude some transformations from being identified as the best ones (or equally the best ones, as ties are represented in Fig. ) for unexpected intervals, such as transformation 2, which shows good results for some low-flow categories.

A second representation is given in Fig.  with the average rank of the transformations (i.e. the average of the ranks of transformations over all the time steps of the interval concerned) for each of the 200 intervals. In this figure, we see that the transformations remain rather close together for low flows, with an average rank between 5 and 7. By contrast, the spread is larger for high streamflows, with average ranks between 4 and 9. Specifically, several transformations share the best average rank values for low flows, such as the -1, log, and -0.5 transformations. Interestingly, the -2 transformation, which is supposedly the transformation giving the highest weight to low flows and identified as the transformation with the most number-1 ranks for a high number of intervals in Fig. , only shows the best average rank for the very first interval and then quickly shows a much worse average rank. This might indicate that the -2 transformation gives a high weight to errors over a limited number of time steps with the lowest streamflows (see Fig.  in the Appendix for further analyses).

Regarding the middle range of streamflows, a couple of transformations show the best average rank, such as the -1, log, and -0.5 transformations but, also progressively, as the streamflows get higher, the 0.2, QinvQ, and Box–Cox transformations. Interestingly, this indicates that, while being quite average most of the time (analysis not shown here), these transformations still have better average ranks than transformations with more occurrences of rank 1. Finally, regarding high flows, the 2, 1, QlogQ, and 0.5 transformations take the lead.

We tried to identify links between catchment characteristics and the performance of transformations to better understand the behaviour of the different transformations and to potentially help prescribe transformations, knowing the catchment characteristics. To do so, we used the Spearman correlation to analyse how the interval-averaged ranks of transformations for each catchment could relate to the catchment characteristics listed in Table . The GR4J model calibrated with NSE was used here. In order to maximize the possibility of identifying strong links, only the catchment characteristics and interval-averaged ranks over the calibration period were used.

The most important correlations were found between the BFI values and the QlogQ (correlation equal to 0.56), boxcox (0.51), 0.2 (0.50), and log (0.38) transformations. Negative correlations were found between the BFI and the -1 (-0.32) and -0.5 (-0.30) transformations. This means that, the lower the BFI, the better the use of transformations giving intermediate weights between high and low flows. Conversely, the higher the BFI, the better the use of transformations that give a large weight to low flows. We observed that the central slope of the flow duration curve shows correlations of the same order of magnitude but opposite to the one shown by the BFI, with exactly the same transformations. This suggests a strong correlation between these indicators. All the other catchment characteristics show no correlation over 0.30 or under -0.30, signifying a weak explanation of the performance of transformations with these characteristics. Using the stats::cor.test() function in R, we found that the correlations lower than -0.11 and higher than 0.11 were all significant (i.e. p values < 0.05), whereas none of the others was significant. Regarding the BFI and the central slope of the flow duration curve, although the above-mentioned correlation values are interesting, they are not sufficient to permit any prescription to be made regarding the choice of transformation for a specific kind of catchment. To deepen this analysis, a co-inertia analysis was undertaken of two principal component analyses made for the table of catchment characteristics and the table of transformations; however, it could not show any further informative interdependence between catchment characteristics and transformations.

In this work, the transformations have been compared using ranks. The reasons behind choosing to work with ranks instead of direct (normalized) errors were (i) being less impacted by different orders of error magnitudes between catchments or ranges of streamflows and (ii) answering the question of which transformations are the best ones, rather than how good the transformations are. Assigning ranks can, however, have the effect that a rank difference of 1 can signify a small error or a larger one. Nevertheless, ranks are accounted for time step by time step, meaning that, if differences between two simulations are very low, there can quite easily be changes in the order, which then result in similar average ranks over the intervals.

The impact of using absolute differences rather than ranks to compare transformations was assessed in . The general shapes of curves obtained with this alternative method were found to be similar, although some discrepancies were observed. The large errors of some outlier transformations made it difficult to identify differences between the other transformations. While this could lead to the conclusion that these transformations can be used interchangeably because they seem to lead to very similar errors, it is important to note that the very high normalized mean squared error (MSE) of one or two transformations leads to smaller differences for other transformations. In other words, the results using the MSE are impacted by the large error of some transformations and lead to less informative results. We therefore believe that the methodology based on ranks used in this study provides more readable results.

The results of this study confirmed some common hypotheses in the literature about efficiency criteria but also provided specific insights that should be helpful for modellers. That is, we confirmed that, as mentioned by various authors, transformations can be used “to remove the bias towards high flows” , “to fit low flow periods” , or “to put more weight on low flow” . In addition, we showed that the log or 0.2 transformations could lead to a better representation of low flows, as stated by and .

However, we also added some knowledge to the existing literature. First, we showed that using no transformation of streamflows leads to a performance that is among the worst for a large range of streamflows (from low to medium ranges). As using no transformation is still the most widespread way of calculating criteria, this could impact how modellers calibrate models or evaluate them. In addition, we showed that using the most extreme transformations for calibration, for both high (2) and low (-2, -1) flows, leads to a narrow range of streamflows for which the simulations seem satisfactory. This reinforces the idea of trying to define well the purpose of the modelling chains developed and choosing the adapted transformation for model calibration. In addition, the fact that it is difficult to identify a transformation leading to the best simulations overall or simply sufficiently high performance for the whole range of streamflows indicates that developing generic models fitting all purposes is still a challenging task for modellers.