THORNTHWAITE AND MATHER SOIL WATER BALANCE MODEL ADAPTED FOR ESTIMATION OF REAL EVAPOTRANSPIRATION OF THE PASTURE

1 Federal Fluminense University, Department of Agricultural and Environmental Engineering, Niterói, Rio de Janeiro, Brazil 2 Federal Rural University of Rio de Janeiro, Department of Agricultural and Environmental Engineering, Seropédica, Rio de Janeiro, Brazil 3 Federal Rural University of Rio de Janeiro, Department of Environmental Sciences, Seropédica, Rio de Janeiro, Brazil 4 State University of Maringá, Department of Agronomy, Umuarama, Paraná, Brazil 5 Federal University of Alagoas, Center of Agrarian Sciences, Rio Largo, Alagoas, Brazil


INTRODUCTION
According to the prediction made by the Food and Agriculture Organization of the United Nations (FAO), it is estimated that by 2050, the world population will be approximately 9.7 billion (FAO, 2018). Thus, it is essential to increase agricultural productivity, associated with the rational and optimized use of natural resources, in order to supply the demand for food sustainably.
In this context, pastures are decisive in the Brazilian agricultural sector, as they are characterized as the basis for feeding the Brazilian herds (RETORE et al., 2019). Thus, the correct management of forage becomes essential, as it directly provides an increase in the productivity of livestock activity (AGUIAR et al., 2017).
Irrigation is a practice that can be used to optimize crop management, as it allows the reduction of losses caused by water deficit and, as a consequence, increased productivity (ANTONIEL et al., 2016;SANCHES et al., 2017b). However, it is among agricultural activities with the highest water consumption, requiring the control of water supply and demand, optimizing decision-making (BOSI et al., 2020). Therefore, for the practice to reach its potential, it is necessary to know the real water demand of the crops.
Evapotranspiration (ET) represents the transfer of water from the surface to the atmosphere through the simultaneous occurrence of evaporation and transpiration processes (CRUZ et al., 2017). As a result, ET is an essential factor in estimating the crop water demand. Several approaches have been proposed to obtain it, among them, direct methods (lysimeters), empirical models (e.g., Thornthwaite, Camargo, Hargreaves-Samani, Jensen-Haise, Makkink) or physical-physiological (Penman-Monteith), micrometeorological (e.g., Bowen ratio -energy balance, eddy correlation, aerodynamic method) and through the soil water balance (GOMES et la., 2015a;SILVA et al., 2016).
The Bowen ratio -energy balance (BREB) is a micrometeorological method that has been widely used as a standard in determining the flux of latent heat (LH) and sensitive heat (H) of several crops and allows to obtain the evapotranspiration (ET) (SILVA et al., 2018;WALLS et al., 2020).
However, BREB requires measurements of micrometeorological elements, resulting in its limitation, especially for small and medium farmers. Alternatively, the soil water balance (SWB) allows estimating ET by quantifying the inflows and outflows of water in the soil, as it is based on the mass conservation principle (O'REILLY et al., 2020). As it only requires soil and weather (rain and evapotranspiration) physical-hydric parameters, SWB is characterized by being a more accessible approach.
Thus, the aim of this study was to evaluate the crop coefficient and the performance of the Thornthwaite and Mather BH model adapted to estimate ETR of a pasture area, in relation to the estimates obtained by the BREB method, from July 2018 to June 2019.

Study area and micrometeorological/meteorological measurements
The experiment was carried out on a pasture area located in the municipality of Cachoeiras de Macacu, Rio de Janeiro State (RJ) (22º 27' S; 42º 45' W and 30 m altitude). The area covered 30 hectares cultivated with Brachiaria and fortnightly occupied, on average by 160 Nellore animals. The climate in the region, according to Köppen's classification, is Aw -a Humid Tropical Megathermal, with a dry season in winter (ALVARES et al., 2013).
For soil classification, six trenches were opened around the micrometeorological mast (MM). The profiles were described morphologically according to the methodology of Santos et al. (2018) and the soil was classified as typical Tb Melanic Gleisol with a texture between loam sandy and sand (Table  1). Such types of soil are characterized by occurring in flat terrains of floodplains and presenting a high level of the water table.
For physical-water analyses, three points were randomly selected around the MM. Three undisturbed samples were collected at each point at the depths of 0.10, 0.30, and 0.60 m, and later, the samples were taken to the Soil and Water Quality Laboratory at the Escola Superior de Agricultura "Luiz de Queiroz", ESALQ. Moisture at the field capacity (0.42 m³ m -3 ) and permanent wilting point (0.19 m³ m -3 ) was obtained through the soil water retention curve. Analyses of hydraulic conductivity and infiltration rate were also carried out.
Micrometeorological measurements were performed in a 4 m high MM installed on the pasture area. The micrometeorological station was composed of several instruments, described in Table 2. Rainfall data from an automatic meteorological station, installed 1 km from the experimental area were also used. For this experiment, it was evaluated the period from June 2018 to July 2019, which covered winter, spring, summer, and autumn.

Bowen ratio -Energy balance (BREB)
The energy balance consists of the energy conservation law. It is made up of the radiation balance, heat flux in the soil, sensitive heat flux and latent heat flux, which corresponds to ET (Equation 1). To obtain the energy balance, data from the MM were used, which were stored every 10 seconds and, subsequently, an average was performed every 30 minutes. The collected data were analyzed according to the methodology (PEREZ et al., 1999) proposed for disposal and selection (Equation 1).
Where, R n = net radiation, W.m -2 ; H = sensitive heat flux, W.m -2 ; LE = latent heat flux, W.m -2 ; and G = heat flux in the soil, W.m -2 Bowen (1926) incorporated the ratio between the fluxes of sensitive and latent heat. This relationship was denominated Bowen's ratio (β), which can also be determined by the relationship between the vertical gradients of temperature (∆T, o C) and air humidity (∆e, kPa) (Equation 2).
By replacing equation 2 into the 1 and reorganizing it, the ET a -BREB (ETa β ) was obtained: Where, λ = latent heat of vaporization, 2,45 MJ.kg -1 ; 1800 =mean time in seconds, from which data were obtained.

Selection of the observations
To ensure consistent estimates, the criterion established by Perez et al., (1999) was used. Thus, data collected at night and in the early morning were discarded. In addition, to avoid the effects of horizontal gradient fluxes, the sensors were installed within the constant or equilibrium limit layer (MONTEITH; UNSWORTK,1990), with a border of approximately 300 m, determined according to PEREIRA, 2007).

Soil water balance -Thornthwaite and Mather
The soil water balance is based on the law of mass conservation, represented by the variation of soil water storage (Equation 5). The real evapotranspiration (ETa-ThM) was obtained through the modified WB (LYRA et al., 2020).
Where, ALT = change in water storage in the soil, mm; SW = soil water storage, mm; P = precipitation, mm; ETa = actual evapotranspiration, mm; and EXC = water surplus, mm.
Storage was calculated using Equation 6, and according to the methodology, it is penalized according to water availability, denominated accumulated negative (ACN) (Equation 7).
Where, ET c = crop evapotranspiration.
Once the K c was obtained, ET c was estimated using the single crop coefficient method suggested in FAO 56 (Equation 9).

Single crop coefficient (K c )
For the calculation of ETc, the crop coefficient was used, which was obtained based on experimental data. Therefore, it was selected days considered without water deficit, characterized on the basis of the fraction of water available in the soil (FDA) (Equation 12).
The days without water deficit were those that met the criterion 0.8 ≤ fDA ≤ 1.1. In these conditions, ET c = ET a was characterized (PEREIRA; SEDIYAMA; NOVA, 2013). Next, a linear regression model was used between ET o (independent variable) and ET c (dependent variable) forced to pass through the origin, so that K c was determined by the angular coefficient of the line (Equation 13).
Data normality was assessed through the Kolmogorov-Smirnov test (p < 0.05), and subsequently the significance of the regression was analyzed using the analysis of variance (ANOVA). The quality of the model adjustment was observed using the coefficient of determination (R²) and the standard error of estimate (SES).

Statistical analysis
The accuracy of the ETa-ThM and ETa β estimates were assessed based on the modified Willmott concordance index (d m ) (WILLMOTT; ROBESON;MATSURA, 2012), (Equation 14). The simple linear regression analysis forced to pass at the origin between ETa β (standard) and ETa-ThM was also applied as well as the coefficient of determination (R²) (Equation 15) and the root mean squared error (RMSE) (Equation 16).
Where, d m = index of agreement or adjustment; P i = values of predictable ETa_ThM, mm.d -1 ; O i = values of observed ETa β, mm.d -1 ; Ō = means of the ETa β, mm.d -1 values; and N = number of observations.
Error indices were also used to assess whether the error was related to the model, through systematic error (MSEs) (Equation 17), or whether it was conditioned to external factors, through random error (MSEu), (Equation 18).

Climatic conditions and crop coefficient (K c )
Total rainfall of 245.6 mm was recorded in winter, distributed over 44 days; the average ET o for the period was 2.76 (± 0.97 mm.d -1 ). Regarding spring, a greater total of rainfall of 767.6 mm was observed in 56 days and the average ET o for the period was 4.03 (± 1.84 mm.d -1 ). The total rainfall for the summer was 693.4 mm, which occurred for 59 days. In the same period, an average ET o of 4.49 (± 1.52 mm.d -1 ) was obtained. Autumn had a total rainfall of 582.2 mm, distributed over 55 days, and an average ET o of 2.78 (± 0.85 mm.d -1 ). The highest mean magnitude per event was observed in spring, followed by summer, while the lowest occurred in winter (Table 3 and Figure 1).
Also, according to the accumulated total for each season, it is observed that rainfall showed greater values than ET o and ETa β and this difference was more accentuated in the spring. As for the accumulated ET o , ETa β, and ETa_ThM, they presented higher values in the summer, 408.5 (± 1.52 mm.d -1 ), 377.9 (± 1.39 mm.d -1 ) and 411.5 (± 1.71 mm.d -1 ), respectively.
Regarding the ET o variation during the analyzed period, a trend of higher values was observed in the spring and summer, and a decrease in the values in the winter and autumn seasons. This fact indicated that ET o showed a relationship with the availability of solar radiation, as expected. A similar pattern was observed by Bueno et al. (2019), when evaluating the effect of irrigation on two grass species during the period from May/2018 to June/2019. The authors found that ET o was influenced by incident solar radiation and air temperature. Alencar et al. (2015) also found that solar radiation was the parameter that influenced ET o estimates the most.  The fit of the linear regression used to determine the K c showed high precision, being able to explain 97% of the data variability with EPE of 0.0052 (Figure 2). When assessing the relationship between ET c and ET o , Graham et al., (2016) found that ET c explained 97% of ET o . Santos et al. (2017) evaluated the regression between ET c and ET o for three of the phenological stages of Moringa, and observed an R² of 0.99 for all stages.
According to the ANOVA, the ET o , ET c data showed a significant relationship (p < 0.01). The K c value of 1.04 corroborates the K c values for pastures shown in the literature (ANTONIEL et al., 2016;BARNOSA;OLIVEIRA;DE FIGUEIREDO, 2015;SANCHES et al., 2017a;SANTANA et al., 2016). Sanches et al. (2017b) when evaluating the average K c of Mombasa grass, also found values similar to those in this study (1.07). In addition, the authors observed that the K c values were not influenced by seasonality.

Performance of the Thornthwaite and Mather model
In general, ET a estimates were less dispersed in autumn and spring, which presented R² of 0.84 and 0.83, respectively. The season with the largest dispersion was winter, followed by summer, with R² of 0.43 and 0.56, respectively. Except for the spring, it was observed that the ET a estimated by the ThM model was underestimated concerning the ET a determined by the BREB (Figures 3 and 4). For winter and summer, an average underestimation of 11% was observed, while in the autumn period, the underestimation was 16%. Silva et al. (2016) found an R² of 0.83 in the winter period when evaluating the ET estimate calculated using Penman-Monteith and Bowen's ratio in an area cultivated with Bahia grass. Gomes et al. (2015b), when evaluating ET estimates of brachiaria decumbens performed using a simulation model of energy flux transfer, in relation to ET determined by Bowen's method, found a higher R² value, of 0.91. In relation to the accuracy of the ET a ThM estimates, the highest d m value was observed in the spring (0.82), followed by summer (0.73). Autumn and winter showed less accuracy, with values of 0.68 and 0.66, respectively (Table 4). As for the error, the months corresponding to spring presented lower RMSE% and the lowest error related to the model and its parameters, that is, systematic error, MSEs%. In contrast, the period with the highest RMSE% was winter.
The highest values of systematic error were observed in autumn and winter. It is worth mentioning that both periods are characterized by a lower incidence of solar radiation and air temperature, which provides a slower pasture growth in relation to spring and summer, and reflects in a smaller leaf area of the canopy and height of the crop (ZHUANG et al., 2020). Thus, the higher MSEs% values are probably the result of the use of a constant K c , considering that it is intrinsically related to the ratio between the total area of the canopy and the occupied area of the soil, defined as the leaf area index (LAI) and the height of the crop. Bueno et al. (2019) found that after eight cuts of two pasture species, the largest dry and fresh matter yield occurred in the period from November to January, corresponding to summer, which indicates that seasonality influences the forage.
Although summer had a higher total rainfall than autumn, the greater dispersion of the ET a estimates observed in the summer can be explained by the lower frequency between the occurrence of the rainfall. This fact may have corroborated the greater ETR penalty estimated by the BH method, due to the low storage of water in the soil, which also explains the greater dispersion obtained in winter, which was the period of lowest total rainfall observed.
The method proposed by Thornthwaite and Mather when adapted to the crops penalizes the ET a more rigorously than other models, such as, for example, the dual Kc Furtado (2017), considering that the removal of water from the soil is caused by the exponential and the initial penalty immediately when soil water storage is less than the available water capacity, while the replacement is carried out directly, through the sum between the storage value and the positive balance between precipitation and ET (PEREIRA; SEDIYAMA; NOVA, 2013).
Another possible factor responsible for the underestimation of the values from the ET a estimated by SWB is due to the type of soil of the study site, considering that it has a high level of groundwater. This characteristic can contribute to the occurrence of capillary rise, a phenomenon not quantified by the SWB proposed by Thornthwaite and Mather.

CONCLUSIONS
• The Thornthwaite and Mather soil water balance model adapted for pasture presents satisfactory estimates of the real evapotranspiration, particularly in periods with higher rainfall frequency; • In dry periods and or with less rainfall frequency, the model presents less performance, due to the intrinsic penalty to soil water balance; • The use of a single crop coefficient constant throughout the year does not represent seasonal variations in leaf area index and crop height; and • The model is more accurate, especially in the spring and autumn.