Three different RCPs (RCP2.6, RCP4.5 and RCP8.5) were used to account for the uncertainty in future greenhouse gas emission trajectories and climate variables from 30 GCM–RCM couples (Table ) from EURO-CORDEX . The use of several models at each step allowed for a more robust quantification of the uncertainties stemming from each step. The modeling chain describes a pathway from different RCPs to an impact model (here PE formulations), using a succession of models whose simulation outputs feed the next model. Figure 1 represents the modeling chain used for this study, showing each modeling step, namely RCPs, GCMs, RCMs and PE formulations. The reference period used for climate projections to compute PE anomalies is 1976–2005, and projections were analyzed over 1976–2099. In practice, 30-year periods are used for climate impact studies, and the EURO-CORDEX simulations using the climate change scenarios cover the 2006–2100 time period. For all RCP–GCM–RCMs combinations, only a realization is available. This hinders the quantification of the internal variability as an uncertainty source. However, as anomalies are computed over long time slices, we can assume that natural climate variability has a limited impact. All data were available at the daily time step.

Seven PE formulations were selected in this study (see Table 2). This selection was made to represent diverse ways of estimating PE, including physically based methods derived from the energy balance and empirical methods. In this study, all formulations were applied at a daily time step.

The Penman and Penman–Monteith formulations are often referred to as combinational methods, as they are derived from the energy budget coupled with aerodynamic considerations. While the Penman formulation is recommended for open water evaporation estimation, the Penman–Monteith formulation was proposed to estimate the potential evaporation from a reference crop. These two formulations are widely used in crop water requirement and hydrological models, as they make full use of currently measurable climate variables, under a physically derived framework. For climate change impact studies, these formulations allow one to account for possible interactions and feedbacks between climate variables. The Priestley–Taylor formulation is a simplification of the Penman equation that allows for the estimation of PE with only radiative climate variables. This formulation does not make use of aerodynamic climate variables, which are highly uncertain in climate change impact studies. The Morton formulation was recommended by for hydrological modeling. Based on the Priestley–Taylor formulation, it includes an iterative estimation of a so-called equilibrium temperature, which better represents the surface temperature. The Oudin, Hamon and Hargreaves formulations use air temperature only. Proxies of solar radiation are implicitly included in these formulations, either via extraterrestrial radiation estimation or by using empirically derived equations that relate radiation to the mean or amplitude of daily air temperature. For climate change impact studies, these formulations are interesting, as air temperature is probably the least uncertain variable in climate projections. However, the absence of other climate variables is questionable under climate change because feedbacks between climate variables exist, such as the dimming effect causing radiation to decrease while the air temperature increases. pointed out higher future PE amounts with the temperature-based formulations, compared with other formulations, over five predominantly hardwood forest sites at differing latitudes in the eastern United States.

A Bayesian data augmentation technique, the QUALYPSO (Quasi-ergodic AnaLYsis of climate ProjectionS using data augmentatiOn) method , was applied to deal with the lack of balance in terms of representation within the combinations of climate models (GCMs–RCMs) and RCPs (see gaps in Table ). The data set available for this study is composed of 30 RCP–GCM–RCM chains multiplied by seven PE formulations. The Bayesian data augmentation process of the QUALYPSO framework fills a complete and balanced matrix of climate projections composed of 162 members (3 RCPs × 6 GCMs × 9 RCMs) multiplied by 7 PE formulations. This process results in a balanced data set, which is essential to correctly assess the contribution of each modeling step and to avoid inducing a biased estimation of the variance explained by a specific modeling step being over- or underrepresented. This framework was successfully applied by to analyze projected hydrological uncertainties with an incomplete ensemble of projections. Analysis of variance (ANOVA) methods are frequently used to quantify the contribution of different models to total uncertainty. They rely on the respective variance contributions of the different modeling chain steps to the total variance. An ANOVA can be performed with a time series approach, as mentioned by , which considers the quasi-ergodicity of climate variables for the long term. This approach was also used by and , who added the downscaling and hydrological modeling steps to the modeling chain in the evaluation of uncertainties. In the present study, to quantify and partition the total uncertainty with respect to the projected changes among the different modeling steps, the quasi-ergodic analysis of variance (QE-ANOVA) framework was chosen . The QE-ANOVA method allows for the decomposition of the total variance of the projected potential evaporation estimates. The total uncertainty partitioning is composed of the sum of the specific variance of each modeling chain step (namely RCPs, GCMs, RCMs and the PE formulations) as well as a residual term, representing the interaction between models. Moreover, a 30-year rolling mean is applied on projected variables that are available from 1976 to 2099 in order to reduce the impact of internal variability. Within the QE-ANOVA analysis, the trend signal analyzed by a variance decomposition is a trend model fitted to rolling mean projections. This statistical analysis allows for the assessment of the importance of the choice of PE formulation, according to the timescale and geographical-area targets. Finally, a signal-to-noise ratio is used to determine the strength of PE changes over France. The ensemble mean PE expected changes represents the signal, and the noise is represented by the standard deviation between simulations.

Figure a shows the mean annual PE, from a multi-scenario and multi-model average, over the reference period from 1976 to 2005 based on climate projection data. Across France, a north–south gradient is visible, with the highest values obtained over the Mediterranean region, and the lowest values obtained over the northern part of the country and the mountainous areas (the Alps and Pyrenees).

Annual PE is shown to increase over the 1976–2005 period (Fig. b). A mean increase from about 730 mm y-1 at the beginning of the century to about 840 mm y-1 by the end of the century is simulated for all RCPs on average (Fig. b). However, this PE increase highly depends on RCPs, as PE reaches around 775, 820 and 920 mm y-1 for RCP2.6, RCP4.5 and RCP8.5, respectively. This means that, for the majority of climate models and PE formulations, the projected increase in greenhouse gas concentrations clearly leads to an increase in PE estimates, and the trend in PE is closely related to the RCP considered: an exponential increase for RCP8.5, a rather linear increase for RCP4.5 and a plateau reached by 2040 for RCP2.6.

The PE formulations show large differences in terms of magnitude, regardless of the time period considered (Fig. a). This is in line with previous studies that compared PE amounts depending on the PE formulation . The differences between formulations reach about 400 mm y-1, which is much higher than the expected PE changes over the 1976–2099 period for a given formulation.

Figure b shows gradual positive anomalies for each formulation. By the end of the century, the annual PE changes over the 1976–2099 period are between +30 mm y-1 (on average over 30-year periods) with the Morton formulation and +130 mm y-1 with the Hamon formulation, whose relatively high sensitivity to air temperature increase has already been demonstrated for other locations . However, interestingly, the growth rate does not necessarily depend on the selected forcing variables nor on the type of equation, as Penman, Penman–Monteith and Oudin present similar trend slopes while probably being most different in terms of formulation and the selected forcing variables.

To better understand this point, we analyzed the link between the PE and climate variables anomalies, regardless of their use in PE formulations. We show that, irrespective of the PE formulation, an increase in air temperature or radiation or a decrease in relative humidity leads to an increase in PE (Fig. ). The relationships exhibit rather linear behaviors, with higher slopes obtained for the Hamon PE formulation. This suggests that the covariance between climate variables from GCM–RCM couples of models allows for PE formulations that do not make use of all available climate variables to still show consistent evolutions of annual PE. Surprisingly, no clear relationship is observed between the PE and wind speed anomalies. A large discrepancy exists between the GCM–RCM climate models regarding the sign of the evolution of wind speed, suggesting large uncertainties for this variable.

In the previous section, we showed that differences between PE evolutions are large in the future. In this section, we aim to decipher the real contribution of PE formulations to the total uncertainty, relative to other uncertainty contributors (RCPs, GCMs and RCMs). Figure  shows the contribution of each modeling step (RCPs, GCMs, RCMs and PE formulations) to the total uncertainty of the future PE changes, i.e., the proportion of the variance explained by each step in the modeling chain for three different 30-year periods centered around 2030, 2050 and 2085, over the entire study domain (France), obtained by applying the QUALYPSO method.

By 2030, GCMs are the main source of uncertainty, with a contribution ranging from 30 % (over the southeastern part of the domain) to 50 % (over the northwest part of the domain). RCMs are the second largest source of uncertainty, with a contribution ranging from 20 % to 30 %. The other steps in the modeling chain exhibit a much lower contribution: PE formulations account for less than 5 % to the total uncertainty, except over the southeastern part of the domain where it reaches 10 %; the RCPs represent less than 10 % of the total uncertainty; and the residual is quite low, with a proportion below 10 %.

By 2050, GCMs are still the largest source of uncertainty over a large part of the territory, with a contribution of about 40 % to the total uncertainty. However, in the southern part of the territory, this contribution is still lower (30 %). The lower contribution of GCMs in this part of the territory can be explained by the increase in the contribution of RCPs to the total uncertainty, by up to 10 % on average. The contributions of RCMs show a small increase in the center of the French territory, from 30 % to 40 % on average. The contributions of other steps remain similar to those for 2030.

By 2085, GCMs and RCMs provide the major sources of uncertainty in the modeling chain, with a respective contribution of 30 %. The contribution of RCPs to the total uncertainty is notably higher than for other periods (especially in the southern part of the domain); thus, in 2085, the divergence between RCP scenarios explains 20 % of the variance in PE estimates. Finally, differences between PE formulations, although they lead to largely different PE changes, remain a minor source of uncertainty compared with other factors, with a contribution to the total uncertainty below 10 %.

The total variance of PE change is the total uncertainty of the PE response to climate change. Interestingly, the total uncertainty increases with time, especially for the southern regions, where the contribution of RCPs is relatively high.

The contribution of PE formulations to the PE projection uncertainty might be underestimated due to the combination of RCPs in the previous analysis. As a consequence, the contribution of PE formulations to the PE projection uncertainty is assessed here for RCP8.5 only. Figure  presents the results of the uncertainty analysis for a single scenario: RCP8.5 by 2085. According to the results presented in Fig. , the contribution of RCPs becomes higher for the end of the century compared with earlier periods, i.e., when RCPs convey more divergent climate signals with respect to RCP8.5, the total variance (Fig. ) is moderately smaller than in the multi-scenario analysis (Fig. ). Therefore, considering multiple RCPs introduces more variability in the long term. Figure 6 shows that the relative contribution of the PE formulations is higher for the analysis considering only RCP8.5 than for the multi-scenario analysis. This is especially the case for the southern locations, where the PE contribution reaches almost 50 % of the variance. GCM and RCM contributions remain stable between the single- and multi-scenario analyses. The results of the uncertainty analysis for a single scenario highlight the important contribution of the PE formulations over the total uncertainty (RCPs excluded), and they even rank the PE formulations as the first uncertainty contributor for the south of France.

The top row of Fig.  shows the mean yearly PE increase (mm y-1) for RCP8.5 computed with the complete matrix of data, i.e., completed with the QUALYPSO framework, for three different 30-year periods centered around 2030, 2050 and 2085. The PE increase for future lead times, computed from the inferred complete matrix with the augmentation process, does not modify the spatial distribution of the original data as shown in Fig. . The south–north gradient is still well represented for PE anomalies, although mountainous areas no longer stand out.

Finally, the signal-to-noise ratio due to climate change, based on the method of , is presented in Fig.  (bottom row). This indicator allows one to assess if the change signal is more important than the uncertainty associated with the change trend (i.e., the noise of the ensemble). When the signal-to-noise ratio is above 1, the change is more important than the uncertainty, and an emergence time can be associated with the change trend. The expected increase in PE becomes greater than the variability for the southern areas of the domain, indicating a signal emergence. For northern locations, a large uncertainty is associated with the estimation of future PE.

We performed a descriptive analysis to evaluate the distribution of PE changes, more specifically the distributions of PE anomalies averaged over the entire domain (France) according to each scenario and model. Figure  shows the variability of the mean PE anomaly projected over France for each model and scenario used in the study, namely RCPs, GCMs, RCMs and PE formulations, for three different 30-year periods centered around 2030, 2050 and 2085. The figure allows one to identify which scenario or model in the ensemble is likely to convey more uncertainty to the projections. For example, a model showing greater variability than other models or a significantly different projection can be identified as contributing a significant amount to the ensemble uncertainty range. At first glance, Fig. 8 shows an increase in the variability of the projections with time, i.e., a growing uncertainty associated with the PE projections from horizon 2030 to horizon 2085. This increase is relevant to previously shown results, such as the divergence between RCPs (Fig. ), the divergence between PE formulations (Fig. ) or the increase in total uncertainty (Fig. ).

Regarding the RCPs, the divergence between the three pathways increases with time. At the 2085 horizon, PE projections clearly differ according to the emission scenario, which explains the results observed in Fig.  (increased proportion of uncertainty due to RCPs). As expected, RCP8.5 shows the largest uncertainty range and mean value increase, followed by RCP4.5 and RCP2.6. Given that the signal-to-noise ratio (in Fig. ) is calculated from the mean of the total set of projections, we can assume that if only RCP8.5 were considered to estimate future PE, the signal-to-noise ratio would be greater and would be more likely to be above unity for some areas and under some horizons (as shown in Fig. 6), whereas the signal-to-noise ratio would be lower when considering RCP2.6 only.

Regarding the GCMs, the divergence of PE projections between each model increases over time in terms of the median change and distributions of change. Moreover, the uncertainty spread also increases with time, which adds up to the total uncertainty, despite the fact that the relative contribution of this factor to the total uncertainty decreases with time (see Fig. ). Two GCMs stand out from the rest: ICHEC-EC-EARTH projects notably higher PE than the others, whereas MPI-ESM-LR shows significantly lower PE values.

The divergence between RCMs increases over time as well. It must be highlighted that not all RCMs are used with the same GCMs. For example, CLMcom-CCLM4-8-17 appears in Fig.  as the RCM with the most variability; however, this RCM only uses the outputs of two different GCMs, the MPI-ESM-LR model that projects the lowest PE increase and the MOHC-HadGEM2-ES model that projects one of the highest PE increases. Thus, the projected higher uncertainty derived from this RCM can be explained by significantly divergent GCM inputs to this model compared with the other RCMs.

Finally, Fig.  also shows changes in the PE projection uncertainty level and spread derived from the PE formulations. For each time horizon, the uncertainty level and spread show similar values across the seven PE models. However, for each formula, the variability and, hence, the uncertainty to PE projections increases over time. In addition, it is clear that, except for the Hamon formulation, all PE projection anomalies remain within similar value ranges. This important result confirms that PE formulations are not the main factor of uncertainty in PE projections, compared with RCPs, GCMs and RCMs.

Two related questions arise when using PE data as input for climate change impact models:

Do we need to consider multiple PE formulations or does a single PE formulation suffice?

As for the first question, our results suggest that PE formulations may have different future trends and that assessing these differences using a multi-formulation approach is undoubtedly a relevant practice. The PE anomalies obtained in this study are of the same order of magnitude as those of previous studies, and they rank PE formulations in a similar way . However, uncertainty analysis results depend on the framework of the climate impact study: combining multiple scenarios or treating each scenario independently. For the multi-scenario analysis, considering the other sources of uncertainty conveyed by RCPs, GCMs and RCMs, we found that PE formulations account for only around 10 % of the total uncertainty in PE projections. This result contrasts with previous studies that have highlighted the relatively important role of PE formulation in the impact modeling chain . However, we must note that many of these studies focused on the RCP8.5 scenario, which is associated with the highest greenhouse gas emissions and, therefore, the highest radiative forcing. The results of the uncertainty analysis performed for RCP8.5 are more consistent with other studies. This leads to greater divergence between the formulations than in the multi-scenario approach, due to the higher future air temperature gradient. We attribute the differences between our results and those from previous studies to the inclusion of several climate model simulations under a large range of emission scenarios and to the approach used to partition uncertainty (relying on a Bayesian method to complete our unbalanced set of projections). However, another hypothesis that could explain the discrepancy between our results and those from other studies is the use (or not) of bias-corrected data. Using bias-corrected data reduces the contribution of GCMs/RCMs to total uncertainty (as their distributions are forced to match an observed data set), in contrast to using raw data. In this study, we prefer not to include bias-corrected data, as most bias-corrected climate projections with statistical methods perform poorly with respect to retaining the properties of the covarying variables needed for the PE calculation . In addition, in contrast to these studies, we did not use an impact model (such as a hydrological model or a crop water requirement model), as we aimed to make this study as general as possible. Due to the minor contribution of PE formulations to the total uncertainty in PE projections, when multiple scenarios are considered, it does not seem necessary to consider several formulations in multi-scenario climate impact studies. However, for a single-scenario analysis (treating each scenario independently), the contribution of the PE formulation becomes higher, and it seems necessary to quantify this source of uncertainty as well as the GCM and RCM contributions. Furthermore, if several impact models are used, due to the differences in the magnitude of PE and its changes between formulations, the same formulation should be conserved for all modeling chains. These recommendations do not apply if only one climate projection is used, which should be avoided anyway.

As for the second question, it is difficult to provide clear guidelines. In theory, the choice of a PE formulation for an impact study needs to consider both the uncertainties with respect to the projected climate variables used by the formulations and the physical consistency handled by these formulations. In practice, as the choice of a PE formulation is often driven by the personal experience of the modeler, the relatively low contribution of PE formulations in multi-scenario analysis, relative to other sources of uncertainty, tends to confirm this practice. We showed that the behaviors of the PE formulations along projected climatic gradients are generally consistent (Fig. ), even for very different formulations in terms of the climate variables used and physical consistency. This results from the fact that other climate variables covary with temperature, and these relationships are maintained in climate projections. If one wants to take other aspects into account, such as CO2, we would recommend choosing a Penman–Monteith-type formula that explicitly allows for this. We found that Penman–Monteith PE trends are close to the average of the PE formulations tested. Moreover, the physical soundness of this method allows one to more explicitly account for interactions between climate drivers than other formulations (even though we showed that those interactions tend to be implicitly taken into account in all formulations). In addition, its structure theoretically allows one to consider other environmental changes, such as land use or plant behavior under elevated atmospheric CO2, in a more explicit way . Accounting for these changes probably represents a greater challenge than identifying the “best” PE formulation with fixed vegetation parameters.

This paper has tackled the issue of uncertainty in PE projections, but PE is only a modeling result. Here, we discuss how PE uncertainty is transferred to actual evaporation (AE) uncertainty. Generally, AE is computed as a fraction of PE, typically constrained by soil moisture. How uncertainty in PE propagates to AE depends on the region. In water-limited regions (such as in the Mediterranean region in our study), the impact of PE uncertainty on the AE estimate is negligible. However, the precipitation projections have a large uncertainty, which affects the estimated AE in such regions. Indeed, an increasing (decreasing) trend of precipitation results in increased (decreased) soil moisture and, thus, AE. In energy-limited regions, the uncertainty of PE is more important than the uncertainty of precipitation to estimate long-term AE.

This leads one to question the sensitivity of impact models to PE variability. Numerous rainfall–runoff models employed for impact studies use PE as input. The representation of the evaporation process in the model may have consequences for the transfer of uncertainties. For example, the HBV (Hydrologiska Byråns Vattenbalansavdelning) model considers AE to be equal to PE if soil moisture is sufficient; if not, AE is reduced to satisfy the previous condition, which could reduce the PE uncertainty . Calibrated hydrological models can accommodate errors in PE estimates but are much more sensitive if the errors are not constant over time . The optimization of any hydrological model parameters with observed streamflow data might compensate for errors and uncertainty with respect to PE. This becomes an issue for the extrapolation period, where the risk is to project excessive dry or wet future conditions under climate change. While the equations described in this paper are used for rainfall–runoff models and, thus, for hydrological climate impact studies, they are not necessarily appropriate for crop water requirements, as crop-specific parameters might be considered. For such a purpose, the set of PE formulations should represent vegetation parameters, which could lead to different evolutions than the set of formulations chosen in this study. Crop-specific parameters are indeed susceptible to be sensitive to aspects other than the climatic variables tested in our study, which might modify the uncertainty contribution.