Expanding the geography of evapotranspiration: An improved method to quantify land-to-air water fluxes in tropical and subtropical regions

The development of new reference evapotranspiration (ETo) methods hold significant promise for improving our quantitative understanding of climatic impacts on water loss from the land to the atmosphere. To address the challenge of estimating ETo in tropical and subtropical regions where direct measurements are scarce we tested a new method based on geographical patterns of extraterrestrial radiation (Ra) and atmospheric water potential (Ψair). Our approach consisted of generating daily estimates of ETo across several climate zones in Brazil–as a model system–which we compared with standard EToPM (Penman-Monteith) estimates. In contrast with EToPM, the simplified method (EToMJS) relies solely on Ψair calculated from widely available air temperature (oC) and relative humidity (%) data, which combined with Ra data resulted in reliable estimates of equivalent evaporation (Ee) and ETo. We used regression analyses of Ψair vs EToPM and Ee vs EToPM to calibrate the EToMJS(Ψair) and EToMJS estimates from 2004 to 2014 and between seasons and climatic zone. Finally, we evaluated the performance of the new method based on the coefficient of determination (R2) and correlation (R), index of agreement “d”, mean absolute error (MAE) and mean reason (MR). This evaluation confirmed the suitability of the EToMJS method for application in tropical and subtropical regions, where the climatic information needed for the standard EToPM calculation is absent.


Introduction
The amount of water that flows through the soil-plant-atmosphere continuum is a key factor to be considered in ecosystem conservation and management efforts. Estimates of water fluxes from land-to-air are needed, for example, for the introduction of new crops, prediction of migration of plant species, and improvement of soil and irrigation management under climate a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 to other gases, and climatic dynamism [60]. However, previous investigations of the water potential gradients along the soil-plant-atmosphere continuum have suggested that net water fluxes can be simplified to produce reliable ETo baselines that are important for management as well as conservation efforts aimed at mitigating the effects of climate change [57,[61][62][63]. Accordingly, here we propose an alternative ETo method based on atmospheric water potential and solar radiation, using a wide range of climate types in Brazil as a model system for improving tropical and subtropical land-to-air water flux estimates.

Theoretical considerations
The basic principle that surrounds the notion of atmospheric water potential as a driving force of evapotranspiration, regardless of plant cover and soil properties, is rooted in the first and second laws of thermodynamics [57,61]. Briefly, the balance of heat, mechanical work (W), and variation of internal energy (ΔU) of a system are considered to be in equilibrium at time zero: where: Q is heat added to the system; W is the mechanical work; and, ΔU is the change in internal energy U of the system. Considering changes in energy that trigger dynamic responses: where: dU is a differential function of U, depending only of initial and final state of a transformation; dQ is the differential of line function, representing the input and outputs of heat; and, dW is the differential of work, equal to dQ in adiabatic processes. dQ is equal to TdS, where S is the entropy. The definition of S from the initial equilibrium (A) to the dynamic (B) state is given by: Considering the second law of thermodynamics, which defines other energy functions of thermodynamic potential, the Gibbs free energy function is given by: The Gibbs free energy (G) is only dependent on the system state, i.e., the pressure (P), volume (V) and temperature (T). This function explains the available energy to make work, which is given by: where: G is a function of T and P. Representing the Eq (5) in a mass (m) basis, considering P = e a and g = C in an isothermal path (dT = 0) and assuming the water vapor acting as an ideal gas we have: Integrating the equation from standard condition (e s ) to actual condition (e a ) we have: Due to the difficult of measurement of the absolute C air , between the standard (C o ) and interest condition (C air ), we set C o = 0 and ΔC air = C air : where: C air is the atmospheric water potential (MPa); R is the gas constant (8.314 J mol −1 K −1 ); T is the absolute temperature (K); e a is the actual vapor pressure (MPa); e s is the saturated vapor pressure (MPa); and, Mv is the partial molar volume of water (18.10 −6 m 3 mol -1 ). In order to combine the effect of C air to extraterrestrial radiation (Ra) in the equivalent water evaporation, the C air is turned into a coefficient of proportionality K Cair , ranging from 0 to 1: where: K Cair is the coefficient of proportionality of C air (dimensionless); C air.i is the atmospheric water potential at the i-day (MPa); C air.max is the maximum atmospheric water potential at the analyzed period (MPa); C air.min is the minimum atmospheric water potential at the analyzed period (MPa). The equivalent water evaporation (E e −mm d -1 ) is obtained by transformation of Ra (MJ m -2 d -1 ) by use of the inverse constant of the latent heat of vaporization (1/λ) [6] multiplied by K Cair .

Ra
where: Ra is the extraterrestrial radiation (MJ m -2 d -1 ); G sc is the solar constant (G sc = 0.0820 MJ m -2 min -1 ); dr is the relative distance Earth-Sun (dimensionless); ω s is the hourly angle corresponding to sunset (rad); φ is the latitude (rad); δ is the inclination of the sun (rad); E e is the equivalent water evaporation obtained by solar radiation and weighted by atmospheric water potential at each i-day (mm d -1 ); K Cair is the coefficient of proportionality of atmospheric water potential (dimensionless); λ is the latent heat of vaporization (λ = 2.45 MJ kg -1 ). The estimated E e can then be converted into ETo as explained in the calibration step described below.

Climate data
To perform the calculations described above in Brazil as a case study we used a set of nine meteorological stations [64] (Table 1) distributed across the most representative climatic zones of that country (Table 2) [65]. We relied on daily observations of maximum, minimum, and average air temperature ( o C), relative humidity (%), daily sunshine hours (MJ m -2 d -1 ), and wind speed (m s -1 ) measured at ten meters above the ground level, from January 2004 to January 2014. Daily sunshine hours were measured by the heliograph Campbell-Stokes (model 240-1070-L) at hourly intervals. Daily wind speed was obtained by the anemometer Vaisala WT521. Wind speed measurements were transformed to wind speed at 2 m height by the wind profile relationship [6]. Daily air temperature and relative humidity were obtained by the thermometer Fluke 5699 and the humidity sensor Vaisala HMK15, respectively.

Penman-Monteith reference evapotranspiration (ETo PM )
The Penman-Monteith method estimates ETo as follows [9]: where: ETo PM −reference evapotranspiration (mm d -1 ); Δ-slope of the saturated water-vapor- pressure curve (kPa o C -1 ); R n −net radiation at the crop surface (MJ m -2 d -1 ); G-soil heat flux (MJ m -2 d -1 ); γ psy −psychrometric constant (kPa o C -1 ); T air −average daily air temperature ( o C); u 2 -wind speed at two meters height (m s -1 ); e s −saturated vapor pressure (kPa); e a −actual vapor pressure (kPa); C n −constant related to the reference type and calculation time step, considered equal to 900 for grass (dimensionless); C d −constant related to the reference type and calculation time step, considered equal to 0.34 for grass (dimensionless). Daily vapor pressure deficit (e s −e a ) is estimated by the difference between saturated and actual vapor pressure. Saturated vapor pressure is calculated using air temperature based on the Tetens formula [66]. Actual vapor pressure is obtained by saturated vapor pressure multiplied by fractional humidity. Daily net radiation (Rn) is estimated by the difference between net longwave and shortwave radiation. The net longwave radiation (Rnl) is obtained by relative shortwave radiation (Rs/Rso), air temperature and actual vapor pressure. The net shortwave radiation (Rns) is obtained from solar radiation (Rs) measurements, which are determined by the relation between extraterrestrial radiation (Ra) and relative sunshine duration (n/N) [6]. Finally, soil heat flux (G) is calculated using air temperature [67].

Alternative "Moretti-Jerszurki-Silva" method: ETo MJS(Ψair) and ETo MJS
The alternative "Moretti-Jerszurki-Silva" method is easily calibrated and used to estimate the ETo. The method is proposed based on C air (ETo MJS(Cair) ); and, on C air and Ra by estimation of E e (ETo MJS ). Daily values of C air (Eq 8) vs ETo PM and E e (Eq 11) vs ETo PM obtained from meteorological stations are adjusted from regression analysis in a monthly and annual basis, between 2004 and 2011. As a general practice in validation procedures, an independent dataset should be used to fit the model; accordingly, the performance assessment and validation of ETo MJS(Cair) and ETo MJS against ETo PM are determined based on regression analysis for the last two years of the time series (January of 2012 -January of 2014). ETo MJS(Cair) and ETo MJS values are obtained using coefficients "a" and "b" for C air vs ETo PM and E e vs ETo PM , respectively, between 2004 and 2011.
where: ETo MJS(Cair).i is the calibrated reference evapotranspiration estimated by atmospheric water potential at each i-day (mm d -1 ); C air.i is the atmospheric water potential at each i-day (MPa); ETo MJS.i is the calibrated reference evapotranspiration estimated by atmospheric water potential and solar radiation at each i-day (mm d -1 ); E e is the equivalent evaporation obtained by solar radiation and weighted by atmospheric water potential at each i-day (mm d -1 ); a is the linear coefficient (mm d -1 ); b is the angular coefficient (dimensionless).

Validation of ETo MJS(Ψair) and ETo MJS estimates using lysimetric measurements
In addition to the 10-years comparison with standard ETo PM in multiple climatic zones, described above, a seasonal validation of the new ETo MJS(Cair) and ETo MJS method was conducted in situ using available lysimetric measurements (ETo LIS ) located at a reference pasture plantation. This independent validation was performed at a site of typical semi-arid climate type Bsh (latitude 3 o 18'S, longitude 39 o 12'W at an altitude of 30 m above the sea level) using climatic data collected between 1997 and 1998 as well ETo LIS previously reported in the literature [68]. The validation could only be performed at this site and period due to the scarcity of co-located ETo LIS and reliable weather stations in other climatic regions. This is sufficient, however, to demonstrate that the proposed method holds in the analysis of both seasonal and multi-year ET patterns without the need for the detailed climatic data that is required for the standard ETo PM calculation. As described above, the new ETo MJS(Cair) (Eqs 8 and 13) and ETo MJS (Eqs 8-11 and 14) were estimated using only air temperature, relative humidity and altitude. The calibration of ETo MJS(Cair) (Eq 13) and ETo MJS (Eq 14) were carried out using the monthly coefficients "a" and "b" of the linear relation between C air vs ETo

Statistics
We used coefficients of variation (CV) to assess the variability of ETo PM in response to climatic data collected across sites between 2004 and 2014. We relied on multiple regression analyses to correlate the estimated ETo PM to climatic variables for each specific climatic zone (Table 1).
We then compared daily reference evapotranspiration obtained with the alternative method to standard daily ETo PM and/or ETo LIS (validation) using regression analysis. The goodness of fit of the alternative methods was obtained by use of R 2 and R as an index of precision and correlation, and agreement index "d" as an index of accuracy [69]. The agreement index is a measure of the effectiveness with which the alternative method estimates the Penman-Monteith reference evapotranspiration, considering the dispersion of the data relative to the 1:1 line: ðjETo alternativei À ETo j þ jETo i À ETo jÞ where: d is the agreement index (dimensionless); ETo alternative.i is the reference evapotranspiration estimated by alternative method at each i-day (mm d -1 ); ETo. i is the reference evapotranspiration estimated by Penman-Monteith method or measured in the lysimeters at each i-day (mm d -1 ); ETo is the average reference evapotranspiration estimated by Penman-Monteith method or measured in the lysimeters (mm d -1 ). For further comparison, the mean absolute error (MAE) and the mean ratio (MR) [70] were used to evaluate the reference evapotranspiration estimated by atmospheric water potential: ðjETo alternativei À ETo i jÞ ð16Þ where: ETo alternative.i is the reference evapotranspiration estimated by the alternative method at each i-day (mm d -1 ); ETo i is the reference evapotranspiration estimated by Penman-Monteith method or measured in the lysimeters at each i-day (mm d -1 ); n is the number of observations (dimensionless). Finally, MAE was used to measure the accuracy of the proposed method and MR was used as an index of under-or overestimation of the standard ETo PM , such that when standard and alternative data are similar, MAE is close to zero and MR is close to one, indicating a more accurate estimation.

Results
As expected, our observations showed large variability of all climatic parameters (T max , T min , RH, Rs, u 2 and VPD) across the different climate zones sampled throughout Brazil. The results described here span ETo trends in humid subtropical, tropical with dry summers, and semiarid regions (Table 3). In general, VPD was the most seasonally variable parameter in humid climatic zones, reaching its lowest values during wet summers. Across sites, high T max and T min and low RH in semi-arid climate resulted in the highest VPD and ETo PM . These results reflect the geographical influence-governed by variation in atmospheric water potential and Rs-on ETo throughout the country.

Adjustment of atmospheric water potential and performance of the alternative methods ETo MJS(Ψair) and ETo MJS
We identified a strong negative linear relationship between C air and ETo PM (P<0.05), with the coefficients "a" and "b" varying with climatic zone (S1 Fig). The linear coefficient "a" corresponds to other climatic variables that in addition to C air drive atmospheric water demand (VPD) and control ETo PM , while coefficient "b" corresponds to the rate of change in the ETo PM relative to C air . Even though C air is not used in the calculation of the standard ETo PM (Eq 12), it strongly affects its variability over time and space. The exception occurred at the humid subtropical site, which showed the smallest linear coefficients between C air and ETo PM , due to lower magnitudes of ETo PM . We also identified an ETo threshold (2 mm d -1 ) beyond which evapotranspiration is primarily controlled by solar radiation and wind speed (S1 Fig). The highest angular coefficients |"b"| were observed in the tropical climate with dry winter (Aw) and the lowest in the subtropical climate. Accordingly, stronger associations and lower errors of adjustment in ETo MJS(Cair) were observed for tropical and semi-arid climates (S2 Fig and Table 4). Across all climate zones, the lowest associations between ETo MJS(Cair) and ETo PM were observed during dry winter months (Fig 1). We also identified a linear relationship (P<0.05) between E e and ETo PM (S3 Fig), which demonstrates the suitability of using C air and Ra-sole drivers of E e −as the key parameters in the alternative method. The linear coefficients "a" were around: 3 mm day -1 for the semi-arid climate; 1.5 to 2.5 mm day -1 for tropical climates; and 1.0 to 2.5 mm day -1 for subtropical climates. The angular coefficients | "b" | ranged from 0.35 to 0.5 for subtropical climates; 0.4 for the semi-arid climate; and between 0.27 and 0.33 for tropical climates (S3 Fig). The calibration process, which involved the relation between E e and ETo PM , further improved the performance of the alternative method ETo MJS for subtropical climates (Fig 1, S4 Fig and Table 4).
The smallest adjustment errors (MAE and MR) of ETo MJS(Cair) and ETo MJS were observed in the tropical climate types ( Table 4). The smallest associations between ETo MJS vs ETo PM were observed in winter months, which is to be expected given the relatively low Ra and ETo PM typical of this season [71].
In an attempt to establish generic "a" and "b" coefficients for the different climate zones and seasons, we used the linear and angular coefficients determined at the three main climate groups: tropical, semi-arid and sub-tropical. This resulted in a predictable ET trend response to C air and E e . The highest significant ETo MJS(Cair) and ETo MJS associations with ETo PM were observed when using monthly average coefficients for the climate subgroups: humid tropical (Af, Am, As and Aw), semi-arid (Bsh), humid subtropical without dry season (Cfa and Cfb) and humid subtropical climates with dry summers (Cwa and Cwb) (Fig 2). Table 3

Validation of the alternative method "Moretti-Jerszurki-Silva": ETo MJS(Ψair) and ETo MJS
For the in situ validation, performed at the semiarid climate type Bsh, we identified a strong linear relationship (P<0.05) between C air and ETo LIS , which further supports the relationship identified between ETo and C air across all climatic zones (S1 Fig). We also identified strong agreement in the ETo MJS(Cair) and ETo MJS estimated with respect to their monthly coefficients "a" and "b" (Fig 3 and Table 5). The highest error of adjustment obtained for ETo MJS (MAE = 0.65 mm d -1 ) resulted in only 10% maximum overestimation of ETo LIS at the reference site.

Discussion
Our results demonstrate that extraterrestrial radiation and atmospheric water potential can be used to reliably estimate ETo in tropical and subtropical regions. The influence of other climatic variables needed for the standard ETo PM calculation, such as Rs and u 2 , was indirectly but sufficiently accounted for in the analysis of radiation and atmospheric water potential, as evidenced by the strong agreement identified between ETo PM and ETo MJS estimates. Previous studies have shown similar results in cold and wet climates [54][55]; however, in our analysis of subtropical climates, the response of ETo to changes in C air indicates lower sensitivity relative to that observed in warmer and drier climates. In colder and wetter climatic zones, where atmospheric water demand is low (Table 4), ETo estimates were most strongly associated with Rs and u 2 [72].
In tropical and semi-arid climates 1 MPa of variation in C air led to strong (up to 0.0861 mm d -1 ) ETo PM responses. Similarly, we observed high sensitivity of the ETo MJS(Cair) method in response to air temperature and RH, which are the key variables controlling C air and used in climate classification systems [65]. This result was also confirmed in the analysis of "a" and "b" coefficients in warm climates (S1 Fig). Thus, the new simplified methods showed its best performances in warm and dry climatic zones (Fig 1 and S2 Fig) (Table 4). Despite the high Rs observed in the semi-arid climate, the influence of VPD on ETo PM was the most important, which is expected given the predominant warm and dry conditions [73][74][75], which led to strong agreement between ETo MJS(Cair) and ETo PM at daily to seasonal scales.
In a previous study, the performance of 12 alternative ETo methods evaluated in 28 locations in central-western Brazil [46], the agreement ("d" index) of ETo obtained based on solar radiation (relative to the standard ETo PM ) ranged from 0.32 to 0.91. In contrast, the agreement obtained here with the proposed ETo MJS method ranged from 0.92 to 0.95 across the same climatic regions. Moreover, the ETo MJS agreement index was higher ("d" index = 0.92 to 0.95) than observed with other alternative methods ("d" index = 0.50 to 0.82) [47]. Therefore, the use of solar radiation in the alternative methods significantly improved ETo estimates. Notably, we also observed higher associations between ETo MJS and ETo PM (R = 0.84 to 0.89) than reported in previous studies (R = 0.76 to 0.83) in subtropical humid climate of southern Brazil [41,43]. In general, compared to the method based on atmospheric water potential (ETo MJS (Cair) ), the ETo MJS method was more sensitive to seasonal and regional climate heterogeneity.
Our validation results showed the higher association and agreement between ETo MJS(Cair) vs ETo LIS (R = 0.76 and "d" index = 0.80) and ETo MJS vs ETo LIS (R = 0.75 and "d" index = 0.75), using the coefficients "a" and "b" described above. Even the highest error of adjustment obtained for ETo MJS (MAE = 0.65 mm d -1 ) resulted in low overestimations based on direct ETo LIS measurements (<10%; Table 5). Since VPD is thought to has great influence on ETo in dry and warm conditions [36], the best performance of ETo MJS(Cair) is consistent with expected responses in semi-arid regions. As hypothesized, for such dry and warm climates, the best performance of ETo MJS(Cair) shows that C air and monthly average coefficients "a" and "b" grouped into climate subgroups, can be used to estimate ETo more confidently than in previous simplified models. Comparing the findings in this study with previous tests of alternative ETo methods in other climatic regions, our ETo MJS(Cair) method also resulted in higher association and agreement with the standard ETo PM . For example, in tropical climate with dry winters (Aw) we obtained stronger agreement than previous simplifications based on either RH and air temperature (R = 0.49 to 0.83; "d" index = 0.49 to 0.69) [76] or based on solar radiation alone (R = 0.83, "d" index = 0.75) [40]. In addition, the association between ETo PM and our alternative ETo estimates was stronger than those previously performed using RH and air temperature in other parts of the world [14,[73][74][75]. Methods based on solar radiation have been reported as a good alternative to the Penman-Monteith method either in humid [33,51,54,[77][78][79] and semi-arid climates [74,80]. As it stands, these models present some difficulty of measurement [1,5] due to scarce direct measurements and resulting large errors [31][32]. In our analysis, however, we show that even the sole use of C air in the alternative ETo MJS(Cair) method is sufficiently robust to estimate ETo in tropical and semi-arid climates. Moreover, the use of the solar radiation (Ra) in our method has proven to further improve ETo estimates, regardless of across all climatic zones studied here. Finally, the monthly average coefficients for climate subgroups improved the estimated ETo MJS , such that the results indicate the possibility of using of the monthly average "a" and "b" coefficients to expand the geography of estimates and assess the potential of land-to-air water losses across different climatic zones (Fig 2C and 2D).
Taking into account the well-known limitations of the existing alternative methods [81], such as applicability in regions of strong seasonality of across regions of that encompass multiple climates [10,12,81], another important improvement of the proposed method is its sensitivity to spatial variability of climate conditions. The use of the C air -based estimates allows for investigating the spatial variability of ETo, which necessary to both conserve limited water resources as well as maintain food and energy production under changing climates [82][83][84].

Conclusions
In this study, we present a new model to estimate reference evapotranspiration in tropical and subtropical regions, where the climatic information needed for the standard ETo calculation is  scarce or absent. We describe how geographical and seasonal variability in evapotranspiration can be accurately predicted based on radiation and atmospheric water potential estimates. The new simplified method is particularly robust in tropical and semi-arid climates, but can also be applied in subtropical and wet climates. In all cases, the new method has significant benefits with respect to accuracy and spatiotemporal scale of application relative to previous models. Continued measurements of air temperature and relative humidity (needed for C air modeling) across different land uses will improve the accuracy of land-to-air water flux estimates in future studies.