Comparing Evapotranspiration Estimates from the GEOframe-Prospero Model with Penman–Monteith and Priestley-Taylor Approaches under Different Climate Conditions

: Evapotranspiration (ET) is a key variable in the hydrological cycle and it directly impacts the surface balance and its accurate assessment is essential for a correct water management. ET is difﬁcult to measure, since the existing methods for its direct estimate, such as the weighing lysimeter or the eddy-covariance system, are often expensive and require well-trained research personnel. To overcome this limit, different authors developed experimental models for indirect estimation of ET. However, since the accuracy of ET prediction is crucial from different points of view, the continuous search for more and more precise modeling approaches is encouraged. In light of this, the aim of the present work is to test the efﬁciency in predicting ET ﬂuxes in a newly introduced physical-based model, named Prospero, which is based on the ability to compute the ET using a multi-layer canopy model, solving the energy balance both for the sunlight and shadow vegetation, extending the recently developed Schymanski and Or method to canopy level. Additionally, Prospero is able to compute the actual ET using a Jarvis-like model. The model is integrated as a component in the hydrological modelling system GEOframe. Its estimates were validated against observed data from ﬁve Eddy covariance (EC) sites with different climatic conditions and the same vegetation cover. Then, its performances were compared with those of two already consolidated models, the Priestley–Taylor model and Penman FAO model, using four goodness-of-ﬁt indices. Subsequently a calibration of the three methods has been carried out using LUCA calibration within GEOframe, with the purpose of prediction errors. The results showed that Prospero is more accurate and precise with respect to the other two models, even if no calibrations were performed, with better performances in dry climatic conditions. In addition, Prospero model turned to be the least affected by the calibration procedure and, therefore, it can be effectively also used in a context of data scarcity. data acquired and shared by the FLUXNET community, including these networks: AmeriFlux, AfriFlux, AsiaFlux, CarboAfrica, CarboEuropeIP, CarboItaly, Car-boMont, ChinaFlux, Fluxnet-Canada, GreenGrass, ICOS, KoFlux, LBA, NECC, OzFlux-TERN, TCOS-Siberia, and USCCC. The ERA-Interim reanalysis data are provided by ECMWF and pro-cessed by LSCE. The FLUXNET eddy covariance data processing and harmonization was carried out by the European Fluxes Database Cluster, AmeriFlux Management Project, and Fluxdata project of FLUXNET, with the support of CDIAC and ICOS Ecosystem Thematic Center, and the OzFlux, ChinaFlux and AsiaFlux ofﬁces. This research was supported in part by the Ofﬁce of Science (BER), U.S. Department of Energy, Grant No. DE-FG02-03ER63638. This study uses the rainfall and


Introduction
The estimation of atmospheric turbulent fluxes (sensible and latent heat) at the land surface has long been recognized as the most important issue in the determination of the exchanges of energy and mass among hydrosphere, atmosphere and biosphere (e.g., [1][2][3][4][5][6][7][8]) and thus, it is of critical importance to the hydrological and energy cycles [9,10]. At global scale, the overland latent heat represents the 38% of the net radiation absorbed and the corresponding evapotraspiration (ET) amounts to 40% of total precipitation. Transpiration (T) The Prospero (PS) model has been tested in five different Eddy-covariance sites across the world, belonging to the FLUXNET network. The sites are: (i) AU-Dry in Northern Territory of Australia located at an elevation of 180 m a.s.l.; (ii) US-Cop in Utah, United States with an altitude of 1520 m; (iii) US-Var in California, United States at 129 m a.s.l; (iv) IT-Tor in Aosta Valley, Italy at the altitude of 2160 m a.s.l.; (v) GL-ZaH in Sermersooq, Greenland at 38 m. a.s.l. They differ in terms of climatic regime but not in terms of land cover. Indeed, the sites are located in tropical, arid, temperate, continental and polar climates which are the five main climate groups, and the dominant vegetation type is grassland. In detail, the grassland cover mainly consists of herbaceous and other understory systems in AU-Dry, perennial grasses in US-Cop, grasses and herbs in US-Var, herbaceous types in IT-Tor and patches of mosses in GL-ZaH. The grassland cover has been selected as it is one of the largest biomes on earth [36]. The accuracy of PS in predicting ET has been compared to those of the Priestley-Taylor and Penman FAO models, which are two of the most widely used methods in the scientific literature [37] in case of both calibrated and non-calibrated approaches. The location of each site is shown in Figure 1. The sites differ in terms of climatic conditions: they are located in (i) A (tropical), (ii) B (arid), (iii) C (temperate), (iv) D (continental), and (v) E (polar) climates according to the Koppen classification [38]. In terms of land cover, grassland is the dominant vegetation in each site and was selected as the reference ecosystem, since it is one of the most widespread among all the major biomes on Earth. In addition, it occurs under a wide range of climatic conditions [36]. As regards the climatic conditions, each of the five main groups selected for the present study is defined by temperature criteria, except for the B climate, which has a low amount of annual rainfall. Group A is characterized by the temperature of the coolest month, higher than or equal to 18 • C. Type C has the temperature of the coldest month, ranging between −3 • C and 18 • C, and the temperature of the warmest month, above 10 • C. Group D has the coldest month, with a temperature lower than −3 • C, while the hottest month featured an average temperature higher than 10 • C. Climate E has at least one month of the year with temperatures lower than 10 • C. The input datasets, required by the selected ET models, include measurements of precipitation, wind speed, relative air humidity, air temperature and solar radiation, at a sub-hourly time-step. The Dry River station (AU-Dry) [39][40][41] is characterized by Tropical savanna climate (identified by the symbol Aw according to Koppen classification). The flux tower of the site consists of a LI-7500A (LI-COR, Lincoln, NE, USA) infrared gas analyser and a CSAT3 sonic anemometer (Campbell Scientific, Logan, UT, USA). In addition, a CS702 tipping bucket rain gauge (Campbell Scientific, Logan, UT, USA), TCAV soil thermocouple probes, CN3 heat flux plates and a CNR1-net radiometer complete the equipment. The Corral Pocket site (US-Cop) [42][43][44] has a cold, semi-arid climate (BSk). The equipment of the study site consists of an eddy covariance (EC) station, a meteorological station and soil moisture and soil temperature sensors. The Eddy covariance station called Vaira Ranch "US-Var" [45,46] is located in an area with a hot-summer Mediterranean climate (Csa). The site includes a 2-meter-high eddy covariance tower with a triaxial sonic anemometer (Model 1352, Gill Instruments Ltd., Lymington, UK) and a CO 2 and water infrared gas analyser (IRGA, Li 7500, Li-Cor Inc., Lincoln, NE, USA). In addition, the site is equipped with a net radiometer, a pyranometer, a shielded and aspirated sensor and a reflectometry sensor. The experimental site of Torgnon (IT-Tor) [47,48] is characterized by a subarctic climate. The EC station is equipped with a CSAT3 three-dimensional sonic anemometer and a LI-7500 open-path infrared gas analyser. The Eddy covariance station called Zackenberg Heath (GL-ZaH) [49,50] is located in an area with Tundra climate. The flux measurements have been provided by an EC system consisting of an infrared gas analyzer LI-7000 (LI-COR, Lincoln, NE, USA) and a 3D sonic anemometer Gill R3 (Gill Instrument Ltd, Lymington; Hampshire, UK ). In addition, a meteorological station allows for additional weather measurements. The five aforementioned sites have been selected among a wide range of EC experimental locations belonging to the Fluxnet platform with the required climate and vegetation types, since they offer the longest flux time series and more complete datasets, including all the variables required in the present study, with few or no periods of gaps or missing data. The sites climatic characterizations, with the monthly patterns of precipitation and air temperature can be found in Figure 2. The rainfall and temperature data used for the climate characterization of the sites have been provided by climate-data.org, whose archives are based on European Centre for Medium-Range Weather Forecasts (ECMWF) data. The model uses more than 1.8 billion datapoints and and has a resolution of 0.1-0.25 grade. The rainfall/temperature data span moves from 1999 and 2019. The moisture index (I DM ) proposed by de Martonne [51] has been calculated for each investigated site so as to provide a more detailed climatic characterization. It can be estimated as

Case Studies
where P is the annual mean precipitation in mm and T is the annual mean air temperature in • C.

GEOframe Modelling System
GEOframe-Prospero was implemented as a Java component within GEOframe, an opensource, semi-distributed, component-based hydrological modeling system. It is developed in Java and based on the environmental modeling framework Object Modeling System V3 (OMS3) [31]. Within GEOframe, each part of the hydrological cycle is implemented in a self-contained building block, commonly called component [52]. Components can be joined together to obtain multiple modeling solutions that can accomplish from simple to very complicated tasks. Thanks to its solid informatic infrastructure, GEOframe proved a great flexibility and a great robustness in several applications (e.g., [35,53,54] Using the components for geomorphic and DEM analyses [55], the basin can be discretized into Hydrological Response Units (HRUs), i.e., hydrologically similar parts, such as a catchment or a hillslope or one of its parts. The meteorological forcing data can be spatially interpolated using a geostatistical approach, such as the Kriging technique [56]. Both shortwave and longwave radiation components are available for the estimation of the radiation budget [57,58]. ET can be estimated using three different formulations: the FAO ET model [32], the Priestley-Taylor model [4], and the Prospero model, tthe object of the present work. Snow melting and the snow water equivalent can also be simulated by three models, as described in [59]. Runoff production is performed by using the Embedded Reservoir Model (ERM) or a combination of its reservoirs [35]. The discharge generated at each hillslope is routed to the outlet using the Muskingum-Cunge method [35]. Travel time analysis of a generic pollutant within the catchment can be done using the approach proposed in [60,61]. Moreover, the GEOframe provides an integration of Richards 1D with and without temperature, decoupled and coupled with the Energy budget and integration of Richards Equation (2)D [62]. Model parameters can be calibrated using two algorithms and several objective functions: Let Us CAlibrate (LUCA) [63] and Particle Swarm Optimization (PSO) [64]. A graph-based structure, called NET3 [65], is employed for the management of process simulations. NET3 is designed using a river network/graph structure analogy, where each HRU is a node of the graph, and the channel links are the connections between the nodes. In any NET3 node, a different modeling solution can be implemented and nodes (HRUs or channels) can be connected or disconnected at run-time, through scripting. GEOframe is open source and helps the reproducibility and replicability of research [66]. Developers and users can easily collaborate, share documentation, and archive examples and data within the GEOframe community. All the links for the reproducibility of this work results are detailed in Appendix A. In this context, the modularity of the GEOframe modelling system offers the possibility to have multiple modelling solution for a better estimation of the ET fluxes.

The Prospero Model
The Prospero model (PS) was created e to be the core of a physically based, but computationally efficient, ecohydrological model. It is currently mainly thought to estimate ET, but it could be easily extended to the computation of photosynthesis or as the core of a lysimeter model, able to compute the water and energy exchanges between soil and atmosphere. The basic idea is that the ET is given by the sum of two different processes ( Figure 3): E from soils and T from the canopy, both from the fraction exposed to direct sunlight and from the one in shadow. E from soils is computed according to the FAO Penman-Monteith model, while the T is computed according to a modified version of the Schymanski and Or [33] model, which was upscaled in order to face the T at canopy level and the mass conservation when there is water stress. and where: The Prospero ET model is based on the the SO approach [33], but it has been further expanded to work at canopy level and to deal with plant s water stress: • Modifying the SO model to use the leaf area index to obtain the transpiring surface (A tr ); • Considering the stomata opening based on radiation stress [67]; • Adding a stress function, based on the soil moisture, in order to ensure the water balance conservation [32].
To extend this equation to the canopy, we decided to adopt a two big leaf approach [68][69][70][71], using the Sun/Shade model of [72], which allows for the computation of the fraction of canopy in sunlight and in shade and also the radiation absorbed by multiple canopy layers. In particular, Prospero adopts this two leaf, sun-shade approach, while soil is treated separately as a further layer. Moreover, air temperature, relative humidity, wind and longwave radiation were considered constant inside the canopy, allowing to consider it as a single big leaf that emits latent heat proportionally to the corresponding area and to the shortwave radiation. The sun-shade approach can be considered valid for any type of investigated canopy (from grasslands to forests).
The SO [33] approach overcomes some limitations of the Penman-Monteith equation, such as a wrong representation of transpiring area and of the leaf thermal capacity and its feedback on energy balance. In fact, an incorrect representation of the transpiring area of a leaf could impact on the whole energy balance. Therefore, the energy budget in Equation (2) can be rewritten in function of the area able to exchange fluxes and of the equilibrium leaf temperature, as: where: • a sH is the side of the surface exchanging sensible heat (1 for  where the longwave in computed as the T is computed as the sensible heat is computed as A tr is the transpiring surface for unit of ground surface [-], a sE are the sides of surface exchanging latent heat, equal to 1 for hypostomatous, 2 for amphistomatous [-]; a sH are the sides of surface exchanging sensible heat and longwave radiation, equal to 1 for soil, 2 for leaves [-]; P ws and P w are the saturation water vapour pressure and the water vapour pressure. Eventually, the leaf temperature (for each layer treated) T l is computed as where g s is the stomatal conductance [m s −1 ]. The authors decided to include a modified version of the stomatal conductance proposed in Schymanski and Or [33], based on [73], according to which where f (R PAR ) is the stress factor due to the fraction of absorbed photosynthetically-active radiation, f (T a ) is the stress factor due to the air temperature, f (VPD) is the stress factor due to the vapour pressure deficit and f (θ) is the stress factor due the soil water content.
A detailed description of the stress function in the Appendix C. Further information on the complete deployment of Prospero model can be found at Bottazzi [74].

Classical ET Model Descriptions
In addition to the Prospero model, two classical ET models were implemented as components in GEOframe and compared to the results provided by Prospero for each of the previously described sites. These models were selected, among many others belonging to the same classes, since they were widely used in previous studies and returned very acceptable and accurate results [75,76]. In addition, they require a small number of meteorological variables as input parameters, which are easily available from local weather stations. In particular, these models belong to different categories: the Priestley-Taylor (PT) model is a radiation-based method, where the principal weather parameter affecting ET is the radiation, while the FAO Penman-Monteith (FAO) model is a combination-type approach, where ET is affected by the combination of temperature, radiation and atmospheric drivers, including air humidity and wind speed. A brief description of the methods can be found below. Priestley-Taylor (PT) model can be expressed as where α PT represents the advection correction coefficient set at the value of 1.26 and is dimensionless, λ represents the latent heat of vaporization (MJ kg −1 ), it has been set at 2.45, as it varies only slightly over normal temperature ranges.
T mean is the average daily temperature ( • C). The variable γ represents the psychrometric constant (kPa • C −1 ).
The FAO-Penman-Monteith (FAO) equation can be written in the form and E A represents the drying power of the air, which is expressed as where u represents the wind speed (ms −1 ), e s represents the saturation vapor pressure (kPa), function of the maximum and minimum temperature and the parameter e a represents the vapor pressure (kPa) derived from the maximum and minimum relative humidity.

Calibration and Validation
Two different applications of the chosen three models (PS, FAO and PT) are proposed. In the first, the model parameters were set to te literature values and their results were directly compared against ET fluxes derived by EC observations, both at hourly and at daily time-steps. Four indices of goodness-of-fit (GOF) were computed to measure the agreement between the simulated and measured data: Root Mean Square Error (RMSE), Mean Absolute Error (MAE), coefficient of determination R 2 and the index of agreement (D). Further details about the GOFs are reported in Appendix B. This first application was primarily made since the authors want to demonstrate the goodness of the models when their parameters cannot be calibrated against measured data.
In the second application, the model parameters were first calibrated, at hourly timesteps, against a subset of measured data, using LUCA calibration within GEOframe and than validated against a different subset of measured data. Eventually, the performances of the models after calibration were also assessed at daily time-steps, keeping the same calibrated values. This second application is also interesting, since it shows the integration of the three models within the GEOframe framework, which allows for effortless parameter calibrations, when data are available.
The parameters to be calibrated for the PS model are the vapour pressure deficit (VPD 0 ); the lower and higher temperature (respectively, T l , T h ); the temperature at maximum conductance (T 0 ); the water content at wilting point (θ WP ) and the water content at field capacity (θ FC ), which allow for the calculatation of the stress coefficient used when moderate or intense water stress is imposed with the aim of switching from potential to actual ET; the slope (α RAD ) and shape (θ RAD ) parameters of the stress function f (R PAR ). In particular, VPD 0 is defined as the difference between the vapor pressure at the saturation condition and the actual water vapor pressure of the air. The parameters T l , T h , T 0 , respectively, refer to the lower and upper temperature of the range for which a positive stomatal conductance is predicted and the temperature at the maximum stomatal conductance. θ WP and θ FC represent the soil moisture corresponding, respectively, to the the minimum amount of water into the ground required to prevent plants withering and to the amount held in the soil after excess water has drained away. The parameters α RAD and θ RAD allow to quantify the total solar radiation stress. θ WP and θ FC are also calibrated in the Penman-Monteith approach, while, for the PT model, α PT is the selected calibration variable. The latter is a coefficient used to take the effect of the drying power of the air on ET into account. All the above-mentioned parameters are crucial to the ET assessment and they impact this process in different ways. With reference to VPD 0 , when it increases, the atmospheric demand for water and, consequently, the ET fluxes increase too. The lower, higher and optimal temperatures help to define the air temperature stress function, as well as (α RAD ) and (θ RAD ) do for the solar radiation stress function. Both these stress functions regulate the response of the stoma based on external forcings with associated effects on ET losses. The parameters θ WP and θ FC impact on the water stress coefficient, which describes the effect of water stress on the crop: higher values of the stress coefficient determine no soil-water-limiting conditions and higher ET fluxes. Finally, α PT allows to consider the reduction in ET, as the soil water content decreases. Further details about the calibration and validation periods, the calibrated parameters for each investigated site and the proposed model are shown in Table 2. Table 2. Calibration period, validation period and calibrated parameters for each Site and model. Note that VPD 0 , T l , T 0 , T h , θ WP , θ FC , α RAD , θ RAD , α PT , respectively, stand for vapour pressure deficit, lower, optimal and higher temperature, the water content at wilting point, the water content at field capacity, the slope and the shape parameters of the total solar radiation stress function and the PT coefficient.

Results and Discussion
In this Section, the results of the comparison among the performances of three models before and after the calibration procedures are shown and discussed. The calibrated and uncalibrated parameters, and the range of calibration, for each model are listed in Table 3. It is worth noticing that all the calibrated values differ from the literature ones, stressing the importance of using site-specific parameters, when it is possible to perform calibration versus observed data. What is also evident is that some calibrated values, e.g., the PT α, are coinciding with the lower value of the calibration range meaning that the calibration should always be supervised, since the range were chosen to be physically based [32,[77][78][79].
In fact, the choice to widen the ranges of calibration would have, probably, led to even better GOFs, but parameters with little meaning, from the process physics perspective.  This is further confirmed in Table 4 that reports the results, in terms of the four GOFs, for both cases (the calibrated case identifies with a "(C)").
It is immediately clear that the overall performances of both PS model and FAO are really good, with an RMSE and a MAE less than 0.2 mm h −1 , a R 2 and a D generally above the 0.5, at hourly time-steps. At the same time-step, Priestley-Taylor model shows worse performances, especially in terms of RMSE and MAE, which are often two or three times greater than the other two models. Similar considerations apply to the models' performances computed at daily time-steps. Indeed, the daily RMSE and MAE for PT model exceed, respectively, the values of 2 and 1.4 mm in most cases, while the errors corresponding to PS and FAO approaches, in a few cases, show values higher than 1 mm.
Looking at each sites' results in detail, the best GOFs are obtained for the Italian site Torgnon. For this station, all three models perform equally better, with a slight improvement after the parameter calibration. These results are confirmed if we have a look at Figures 4 and 5, where the annual cumulated, for non-calibrated models, and the time variation of ET, both calibrated and not, are shown, respectively. For this case, for the PS and PT models, the annual cumulated ET is comparable with the measured ET, while it is clear that a mean underestimation of around 200 mm occurs each year for the FAO model. Figure 5 confirms these results, both for the non-calibrated and calibrated cases. These results, including the strong FAO underestimation, can be explained by looking at Table 5, where the values of soil moisture (minimum, mean, maximum and 25th, 50th and 75th percentiles), for each investigated site, are reported. It is clear that the IT-Tor, together with GL-ZaH, presents the highest soil moisture values, which, in this case, do not represent a limiting factor for the ET fluxes.    The station with the worst indices is the AU-Dry, where the RMSE values are higher than in all the other stations for all the models. This could be for two reasons. First of all, the site has a temperature regime which strongly differs from the others. Indeed, the temperature is lower when no rain occurs, while it reaches its maximum value during the wettest months. In the other sites, temperature and precipitation are in opposite-phase ( Figure 2). This may have affected the models' perfomances. The other reason is to be found in the particularly low values of soil moisture, as reported in Table 5. It can be seen that, most of the time, the soil moisture is lower than the wilting point (Table 5); this has a strong impact on the stress function linked to the water content in the soil. In fact, it can be seen in Figure 4 that, on an annual scale, in the uncalibrated case, the quantities of ET are strongly underestimated. As an exception to this case, generally Prospero has better performances, due to the closure of the energy balance through the equilibrium temperature (Formula (8)), a correct representation of the transpiring surface given by the Sun/Shade canopy model, which can discriminate between the illuminated and shaded breathable surface. This separation allows us to write three different energy balances in Prospero, one for the sunlit surface, one for the shaded one and one for the soil. Moreover, the stress functions, and especially the one based on the soil water content, enhance the performance of Prospero, allowing the conservation of the water mass. This can be seen in the cases of US-Var, IT-Tor and GL-Zah, where the soil moisture values of the site are, on average, quite high (Table 5). In fact, in these cases, Prospero obtains excellent results, even not calibrated, both at hourly and at annual scale, as can be seen in Table 4 and Figure 4. Annual volumes, simulated by the PS model, are always comparable, with small (100 mm/year, generally less than 25%) to really negligible differences for all the stations. From a climatic perspective, it can be said that the PS model shows higher performances in the sites with wetter climatic regimes. Indeed, the sites IT-Tor and GL-Zah, which present the lowest errors, are featured by an Extremely Humid regime according to the moisture index (Table 1). Therefore, the prediction accuracy decreases, moving from very wet to arid sites. Looking at both the Figures 4 and 5, for the three stations, US-Cop, Us-Var and GL-Zah, despite the comparable GOFs, PT shows a great overestimation, both in the non-calibrated and calibrated cases. The opposite behaviour is shown for station US-Cop and Us-Var for the FAO model that shows a systematic underestimation of the annual volumes, which slightly improves for the calibrated case. For the GL-Zah station, the FAO model overestimates the annual volumes of around 200 mm each year.
The model which is most impacted by the calibration procedure is PT. Indeed, at hourly time-steps, the errors decrease to more than one half of the initial value. An emblematic case is the site GL-Zah where the RMSE moves from 0.11 mm h −1 to 0.05 mm h −1 while the MAE decreases from 0.07 mm h −1 to 0.03 mm h −1 with a reduction close to 60%. On the other hand, the values of α calibrated, for PT, return the same value in all sites and for all climates, i.e., the minimum of the calibration interval.
At the daily time scale (Table 6), the improvement due to the calibration process of PT appears substantial, even if less evident, with the exception of the site US-Cop, where the RMSE decreases by more than 100% and MAE of about 60%. The effect of the calibration process is lower for the other models where, in terms of RMSE, a reduction of, at most, 20% is reached, while in terms of MAE, an improvement no higher than 10% can be observed. Since the prediction accuracy of the PT model is so heavily impacted by the calibration process, in the context of data scarcity, where no or very few flux measurements are available, it results the least recommended for the prediction of the ET. So, in general, when measured values of ET are missing, the used model can be employed as originally proposed in literature, namely without site-specific coefficients with the consequent reduction of the predictive accuracy [75]. When further in-situ measurements are not available such as the net radiation or the soil heat flux, they can be derived using empirical formulations [32] which only require readily available temperature time series as input data. Moreover, for the calculation of Rn, there are several model components available in the GEOframe system [57]. Eventually, a wide literature available becomes available to derive soil hydraulic parameters from basic soil textural properties [80].

Conclusions
ET is one of the main components of the water cycle and it plays a key part in water resources' assessment. In light of this, an accurate prediction of this variable is essential, making the need for more and more accurate models an increasingly discussed issue. Within this framework, a new ET model, named Prospero (PS), has been introduced in the component-based hydrological modelling framework, GEOframe. Prospero extends the recently developed S0 model [33] at the canopy level, which overcomes some limitations of the PM model with respect to the representation of the leaf transpiring surfaces. The performances of this model have been compared with those of two of the most commonly used approaches in the scientific literature, which are Priestley-Taylor and Penman-Monteith FAO methods. The comparison has been performed in terms of RMSE, MAE, R 2 and D, using a high-quality dataset of selected EC towers at hourly and daily temporal resolutions. The selected stations are AU-Dry, US-Cop, US-Var, IT-Tor, GL-ZaH which are located in different climates but have the same vegetation cover, represented by grassland. Addi-tionally, the three method parameters were calibrated to the local conditions using LUCA calibration within GEOframe, in an attempt to reduce the prediction errors.
PS proved to be a robust model since it gave very good results for all the investigate sites, under different climatic conditions. In fact, the results show that, overall, PS returns higher performances than PT and PM, especially in the case of non-calibrated parameters. It follows that, in a context of data scarcity, where, due to the lack of measurements, a calibration procedure cannot be performed, the use of PS is recommended for ET assessment. Indeed, at hourly time steps, the RMSE and MAE reach, at most, 0.15 mm and 0.08 mm, respectively, while they approach the values of 0.17 mm and 0.10 for FAO and 0.22 mm and 0.12 mm for PT. In case of calibration, PS is always the most accurate model, but the gap with the other methods is narrowed. This implies that the PS model, besides having more parameters, is the least affected by the calibration procedure, while, for PT, the calibration process has the greatest impact. Besides this, Prospero returns temperature, the sensible heat fluxes, and the vapor pressure difference between the evaporating surface and the air, which, for the sake of topic and brevity, were not shown in the present work.
Future research directions of the present work are threefold. From one side, the PS model can be compared to meteorological data-based approaches for the prediction of actual ET, such as the Antecedent precipitation index model or the Advection aridity model [37,76], which are more effective at reproducing the ET fluxes than the potential ET models proposed in the present study. On the other side, since the PS model returns an ET overestimation during the energy-limited periods (particularly evident in GL-Zah site), a threshold mechanism could be applied in order to improve the overall prediction. The approach couples a potential ET model and an actual one (in this case PS) according to a threshold value of net radiation [75]. It would also be interesting to test the model in sites with different land uses, such as forest, cropland and wetland. This would allow for the exploration of all aspects, and an exploration of the behavior of PS approach and, consequently, its prediction accuracy.

Data Availability Statement:
The interested researcher can find the entire OMS project, containing input data, output, .sim files, jar files, and the Python script used for the plots, at the following link: https://zenodo.org/record/4643532, accessed on 27 April 2021. temperature data provided by climate-data.org (https://it.climate-data.org/, accessed on 26 April 2021) under the Attribution-NonCommercial 4.0 International license (CC BY-NC 4.0).

Appendix A. Code Availability
For the present work, version v.1.0 is the version of the codes of the GEOframe-Prospero package that we used, available at the following link: https://github.com/ geoframecomponents/ETP, accessed on 27 April 2021.

Appendix B. Indices of Goodness of Fit
• Root-Mean-Square Error The Root-Mean-Square Error (RMSE) is given by where M and S represent the measured and simulated time-series, respectively, and N is the number of components in the series. • Mean Absolute Error where M and S represent the measured and simulated time-series, respectively, and N is the number of components in the series. • Coefficient of determination where M and S represent the measured and simulated time-series, respectively, N is the number of components in the series and M i is the mean of measured values. • Index of agreement

Appendix C. Prospero Stress Functions
We used the model proposed by [73], but instead of using the stress factor given by the leaf water potential, we used the one based on the soil moisture proposed by FAO approach [81] where f (θ) is the normalised stress factor depending on available soil water.
where: Air temperature stress The air temperature stress factor can be computed as: where b and c are defined as: where: -T 0 is the temperature at maximum conductance [ • C]; -T l and T h the lower and upper temperature of the range for which a positive stomatal conductance is predicted [ • C].
If T l ≤ T air ≤ T h , f(T) = 0. We assigned the values for T l , T 0 and T h equal to 0 • C, 25 • C and 50 • C. These parameters can be set a priori or calibrated.
Total solar radiation stress The solar radiation stress can be computed as: where: α RAD and θ RAD are the slope and shape parameters of the stress function f (R SW ) and are set equal to 0.005 and 0.85 [-].
R SW is the total solar radiation expressed in µmol m −2 s −1 . If we want to express it in W m −2 we must include a conversion factor equal to ≈1/4.6.
Vapour pressure deficit stress The vapor deficit stress factor can be estimated as: