Unravelling the contribution of potential evaporation formulation to uncertainty under climate change

The increasing air temperature in a changing climate will impact actual evaporation and have consequences for water resources management in energy-limited regions. In many hydrological models, evaporation is assessed by a preliminary computation of potential evaporation (PE) representing the evaporative demand of the atmosphere. Therefore, in impact studies the quantification of uncertainties related to PE estimation, which can arise from different sources, is crucial. Indeed, a myriad of PE formulations exist and the uncertainties related to climate variables cascade into PE computation. So far, no consensus 5 has emerged on the main source of uncertainty in the PE modelling chain for hydrological studies. In this study, we address this issue by setting up a multi-model and multi-scenario approach. We used seven different PE formulations and a set of 30 climate projections to calculate changes in PE. To estimate the uncertainties related to each step of the PE calculation process (namely Representative Concentration Pathways, General Circulation Models, Regional Climate Models and PE formulations), an analysis of variance decomposition (ANOVA) was used. Results show that PE would increase across France by the end of 10 the century, from +40 to +130 mm/year. In ascending order, uncertainty contributions by the end of the century are explained by: PE formulations (below 10%), then RCPs (above 20%), RCMs (30-40%) and GCMs (30-40%). Finally, all PE formulations show similar future trends since climatic variables are co-dependent to temperature. While no PE formulation stands out from the others, in hydrological impact studies the Penman-Monteith formulation may be preferred as it is representative of the PE formulations ensemble mean and allows accounting for climate and environmental drivers co-evolution. 15

studies have focused on the most pessimistic greenhouse gas emission scenario associated with the greatest global warming.
For instance, Milly and Dunne (2017) only explored the CMIP5 RCP 8.5 scenario, and before them Kay and Davies (2008) only used SRES A2. Third, results are apparently linked to the study spatial scale, depending on whether the authors are interested in climate impact at the global scale (e.g. Dunne, 2016, 2017), the regional scale (e.g. Kay and Davies, 2008), or the local scale (e.g. Koedyk and Kingston, 2016). Vidal et al. (2013) only focused on mountainous areas, where all PE formulations are questionable. Finally, studies differ on the variable of interest on which the total uncertainty is computed: it can be either the PE estimate itself or streamflow simulated by an hydrological model. Assessing the PE uncertainty on the resulting streamflow simulations is legitimate for case studies where water resources are assessed but it largely complicates the analysis since the sensitivity of streamflow simulations to PE inputs is conditioned both by hydrological model parameterization and by the climatic settings of the studied area (Koedyk and Kingston, 2016).
In this study, we use a comprehensive framework, including a large variety of seven PE formulations under several scenarios 70 and using a large set of thirty CMIP5 GCM/RCM outputs. The sensitivity of impact models to PE is not addressed in this study, we focus our analysis on PE and assess the contribution of the formulations to the total PE uncertainty over a large domain, France. First, the analysis will focus on the future change in potential evaporation under climate change. Second, projection uncertainty will be analysed to characterise the influence of the PE formulations on projections compared to the other impact modelling steps. Three different RCPs (RCP 2.6, RCP 4.5 and RCP 8.5) were used to account for the uncertainty on the unknown future greenhouse gas emissions trajectories and climate variables from thirty GCM/RCM couples from EURO-CORDEX (Jacob et al., 2014) were collected ( Table 1). The use of several models at each step allowed for a more robust quantification of the 80 uncertainties stemming from each step. The reference period used with climate projections to compute PE anomalies is 1976-2005 and projections were analysed 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. All data were available at the daily time step.

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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 since they are derived from the energy budget coupled with aerodynamic considerations. While the Penman formulation is recommended for open water 90 Table 1. The available climate projection data. The numbers (2.6, 4.5 and 8.5) refer to the RCPs used by the GCM (rows)/RCM (columns) pairs. Empty boxes or missing RCPs show the absence of data.

Quantifying and partitioning projected PE uncertainties
A Bayesian data augmentation technique, the QUALYPSO method (Evin et al., 2019; Evin, 2020), 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 1). This framework was successfully applied by Lemaitre-Basset et al. (2021)   were calculated by averaging the seven PE formulations (a) and also the different GCM/RCM couples (b). As mentioned in Table 1, the RCP 2.6 time series is obtained by averaging 8*7 time series (8 GCM/RCM couples and 7 PE formulations), the RCP 4.5 time series is obtained by averaging 10*7 time series (10 GCM/RCM couples and 7 PE formulations) and the RCP 8.5 time series is obtained by averaging 12*7 time series (12 GCM/RCM couples and 7 PE formulations).

Behavioural differences between PE formulations and links to climate variables
The PE formulations show large differences in terms of magnitude, whatever the time period considered (Figure 2a). This is in line with previous studies that compared PE amounts depending on the PE formulation (Federer et al., 1996;Kingston et al., 2009). 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.  By 2085, GCMs and RCMs provide the major sources of uncertainty in the modelling chain, with a contribution of 30% each. The contribution of RCPs to the total uncertainty is particularly higher than for other periods (especially in the south part of the domain), so that in 2085, the divergence between RCP scenarios explains 20% of the variances in PE estimates. Finally, differences between PE formulations, although they lead to largely different PE changes, remain a minor source of uncertainty 175 compared to other factors, with a contribution to the total uncertainty below 10%. which is essential to correctly assess the contribution of each modelling step and to avoid inducing a biased estimation of the variance explained by a specific modelling step over or under represented. The PE increase for future lead times, computed 185 from the inferred complete matrix with the augmentation process does not modify the spatial distribution of the original data as shown in Figure 1. The south-north gradient is still well represented for PE anomalies, only mountainous areas are no longer standing out.