ABSTRACT

The low reliability of evaporation estimates in reservoirs challenges the management of water resources, especially when drought occurs. The evaporation rates measured in buried tanks (20 m² evaporimeters) are close to that of lakes. However, there are few studies in Brazil using long data sets with this type of tank. In this context, this study aimed to evaluate evaporation estimates of free water surface using the method of Penman (1948)Penman, H. L. Natural evaporation from open water, bare soil and grass. Proceedings of the Royal Society A, v.193, p.120-145, 1948. http://dx.doi.org/10.1098/rspa.1948.0037
http://dx.doi.org/10.1098/rspa.1948.0037... and other five empirical equations in comparison with data from a 20 m² tank observed during 10 years. The annual relationship between rainfall and evaporation was 1.0, showing variations in dry years (0.7) and rainy years (1.3). Among all evaluated methods, Linacre (1993)Linacre, E. T. Data-sparse estimation of lake evaporation using a simplified Penman equation. Agricultural and Forest Meteorology, v.64, p.237-256, 1993. http://dx.doi.org/10.1016/0168-1923(93)90031-C
http://dx.doi.org/10.1016/0168-1923(93)9... shows the best performance. Camargo et al. (1999)Camargo, A. P.; Marin, F. R.; Sentelhas, P. C.; Picini, A. G. Ajuste da equação de Thornthwaite para estimar a evapotranspiração potencial em climas áridos e superúmidos, com base na amplitude térmica diária. Revista Brasileira de Agrometeorologia, v.7, p.252-257, 1999. has very good estimates using air temperature as input variable and is a simple and reliable alternative when data are missing.

Key words:
water resources; evaporimeters; free water surface; Penman

RESUMO

A baixa confiabilidade da quantificação da evaporação nos reservatórios desafia a gestão de recursos hídricos, principalmente em situações de escassez. A evaporação medida em tanques enterrados (evaporímetro de 20 m²) se aproxima à de lagos, porém existem poucos estudos no Brasil utilizando longas séries de dados com este tipo de tanque. Objetivou-se, neste trabalho, avaliar a estimativa de evaporação de superfície livre pelo método de Penman (1948)Penman, H. L. Natural evaporation from open water, bare soil and grass. Proceedings of the Royal Society A, v.193, p.120-145, 1948. http://dx.doi.org/10.1098/rspa.1948.0037
http://dx.doi.org/10.1098/rspa.1948.0037... e outros cinco modelos (empíricos) em comparação com valores observados em um tanque de 20 m² durante dez anos. A relação anual entre a precipitação e a evaporação medida foi igual a 1,0, apresentando variações em anos secos (0,7) e chuvosos (1,3). Dentre os métodos de estimativa avaliados o método de Linacre (1993)Linacre, E. T. Data-sparse estimation of lake evaporation using a simplified Penman equation. Agricultural and Forest Meteorology, v.64, p.237-256, 1993. http://dx.doi.org/10.1016/0168-1923(93)90031-C
http://dx.doi.org/10.1016/0168-1923(93)9... apresenta o melhor desempenho enquanto o método de Camargo et al. (1999)Camargo, A. P.; Marin, F. R.; Sentelhas, P. C.; Picini, A. G. Ajuste da equação de Thornthwaite para estimar a evapotranspiração potencial em climas áridos e superúmidos, com base na amplitude térmica diária. Revista Brasileira de Agrometeorologia, v.7, p.252-257, 1999. apresenta desempenho muito bom e utiliza a temperatura do ar como variável de entrada sendo alternativa eficaz na ausência de dados.

Palavras-chave:
recursos hídricos; evaporímetros; superfície de água livre; Penman

Introduction

Water evaporation under natural conditions (E) is an important component in the hydrological cycle, since it represents approximately 63% of the total rainfall in the continents (Trenberth et al., 2011Trenberth, K. E.; Fasullo, J. T.; Mackaro, J. Atmospheric moisture transports from ocean to land and global energy flows in reanalyses. Journal of Climate, v.24, p.4907-4924, 2011. http://dx.doi.org/10.1175/2011JCLI4171.1
http://dx.doi.org/10.1175/2011JCLI4171.1... ). Water surfaces significantly contribute to the return of the water to the atmosphere. In the Brazilian semi-arid region, studies quantifying evaporation in reservoirs are common, due to the situations of drought (Pereira et al., 2009Pereira, S. B.; Pruski, F. F.; Silva, D. D. da; Ramos, M. M. Evaporação líquida no lago de Sobradinho e impactos no escoamento devido à construção do reservatório. Revista Brasileira de Engenharia Agrícola e Ambiental, v.13, p.346-352, 2009. http://dx.doi.org/10.1590/S1415-43662009000300018
http://dx.doi.org/10.1590/S1415-43662009... ; Leão et al., 2013Leão, R. A. de O.; Soares, A. A.; Teixeira, A. dos S.; Silva, D. D. Estimation of evaporation in the Banabuiú dam, in the state of Ceará, Brazil, by different combined methods, derived from the Penman equation. Engenharia Agrícola, v.33, p.129-144, 2013. http://dx.doi.org/10.1590/S0100-69162013000100014
http://dx.doi.org/10.1590/S0100-69162013... ).

In the period from 2012 to 2014, per year, 56 billion m³ of water were not replenished in the Southeast and 49 billion m³ in the Northeast (Getirana, 2015Getirana, A. C. V. Extreme water deficit in Brazil detected from space. Journal of Hydrometeorology, v.17, p.591-599, 2015. http://dx.doi.org/10.1175/JHM-D-15-0096.1
http://dx.doi.org/10.1175/JHM-D-15-0096.... ). Details of the drought in the Southeast region can be seen in Porto et al. (2014)Porto, R. L.; Porto, M. F. A.; Palermo, M. A ressurreição do volume morto do Sistema Cantareira na Quaresma. Revista DAE, v.197, p.18-25, 2014. http://dx.doi.org/10.4322/dae.2014.131
http://dx.doi.org/10.4322/dae.2014.131... and Coutinho et al. (2015)Coutinho, R. M.; Kraenkel, R. A.; Prado, P. I. Catastrophic regime shift in water reservoirs and São Paulo water supply crisis. PLoS ONE, v.10, p.e0138278, 2015. http://dx.doi.org/10.1371/journal.pone.0138278
http://dx.doi.org/10.1371/journal.pone.0... . Studies quantifying evaporation are essential to Brazil. Evaporation is measured using tanks due to their easy handling (Kohler & Parmele, 1967Kohler, M. A.; Parmele, L. H. Generalized estimates of free-water evaporation. Water Resources Research, v.3, p.997-1005, 1967. http://dx.doi.org/10.1029/WR003i004p00997
http://dx.doi.org/10.1029/WR003i004p0099... ). Evaporation in buried and large-surface tanks is close to that of lakes (Hounam, 1973Hounam, C. E. Comparison between pan and lake evaporation. Geneva: World Meterological Organization, 1973. 52p.).

In Brazil, there are six 20 m² evaporimeters; besides the tank of this research, two are located in São Paulo, two in Paraíba and one in Pernambuco (Leitão et al., 2007Leitão, M. de. M. V. B. R.; Oliveira, G. M. de; Leitão, T. J. V. Avaliação do desempenho de diferentes métodos de estimativa da evaporação para duas regiões da Paraíba. Revista Brasileira de Engenharia Agrícola e Ambiental, v.11, p.585-593, 2007. http://dx.doi.org/10.1590/S1415-43662007000600006
http://dx.doi.org/10.1590/S1415-43662007... ; Oliveira 2009Oliveira, G. X. S. Relações entre medidas de evaporação de superfícies de água livre por evaporímetros e estimativas por métodos meteorológicos em duas regiões do Estado de São Paulo. Piracicaba: ESALQ/USP, 2009. 108p. Tese Doutorado. http://dx.doi.org/10.11606/t.11.2009.tde-04082009-111344
http://dx.doi.org/10.11606/t.11.2009.tde... ). Studies with long data series using these tanks are rare and serve to assess or calibrate models and smaller evaporimeters (such as the Class A pan).

When there are no data of tanks, models that combine energy and aerodynamic balance, such as that of Penman (1948)Penman, H. L. Natural evaporation from open water, bare soil and grass. Proceedings of the Royal Society A, v.193, p.120-145, 1948. http://dx.doi.org/10.1098/rspa.1948.0037
http://dx.doi.org/10.1098/rspa.1948.0037... and its simplifications, show good results for evaporation from data obtained above lawns in weather stations (Camargo & Sentelhas, 1997Camargo, A. P; Sentelhas, P. C. Avaliação do desempenho de diferentes métodos de estimativas da evapotranspiração potencial no Estado de São Paulo, Brasil. Revista Brasileira de Agrometeorologia, v.5, p.89-97, 1997.; Camargo & Camargo, 2000Camargo, Â. P. D.; Camargo, M. B. P. D. Uma revisão analítica da evapotranspiração potencial. Bragantia, v.59, p.125-137, 2000. http://dx.doi.org/10.1590/S0006-87052000000200002
http://dx.doi.org/10.1590/S0006-87052000... ; Leitão et al., 2007Leitão, M. de. M. V. B. R.; Oliveira, G. M. de; Leitão, T. J. V. Avaliação do desempenho de diferentes métodos de estimativa da evaporação para duas regiões da Paraíba. Revista Brasileira de Engenharia Agrícola e Ambiental, v.11, p.585-593, 2007. http://dx.doi.org/10.1590/S1415-43662007000600006
http://dx.doi.org/10.1590/S1415-43662007... ; Oliveira, 2009Oliveira, G. X. S. Relações entre medidas de evaporação de superfícies de água livre por evaporímetros e estimativas por métodos meteorológicos em duas regiões do Estado de São Paulo. Piracicaba: ESALQ/USP, 2009. 108p. Tese Doutorado. http://dx.doi.org/10.11606/t.11.2009.tde-04082009-111344
http://dx.doi.org/10.11606/t.11.2009.tde... ; Pereira et al., 2009Pereira, S. B.; Pruski, F. F.; Silva, D. D. da; Ramos, M. M. Evaporação líquida no lago de Sobradinho e impactos no escoamento devido à construção do reservatório. Revista Brasileira de Engenharia Agrícola e Ambiental, v.13, p.346-352, 2009. http://dx.doi.org/10.1590/S1415-43662009000300018
http://dx.doi.org/10.1590/S1415-43662009... ; Leão et al., 2013Leão, R. A. de O.; Soares, A. A.; Teixeira, A. dos S.; Silva, D. D. Estimation of evaporation in the Banabuiú dam, in the state of Ceará, Brazil, by different combined methods, derived from the Penman equation. Engenharia Agrícola, v.33, p.129-144, 2013. http://dx.doi.org/10.1590/S0100-69162013000100014
http://dx.doi.org/10.1590/S0100-69162013... ).

An analysis between the data obtained in the 20 m² tank with the method of Penman (1948)Penman, H. L. Natural evaporation from open water, bare soil and grass. Proceedings of the Royal Society A, v.193, p.120-145, 1948. http://dx.doi.org/10.1098/rspa.1948.0037
http://dx.doi.org/10.1098/rspa.1948.0037... and its variations is necessary to identify which formulation is the most adequate in the management of water resources. The aim of this study was to evaluate the performance of the model of Camargo et al. (1999)Camargo, A. P.; Marin, F. R.; Sentelhas, P. C.; Picini, A. G. Ajuste da equação de Thornthwaite para estimar a evapotranspiração potencial em climas áridos e superúmidos, com base na amplitude térmica diária. Revista Brasileira de Agrometeorologia, v.7, p.252-257, 1999., methods of Penman (1948)Penman, H. L. Natural evaporation from open water, bare soil and grass. Proceedings of the Royal Society A, v.193, p.120-145, 1948. http://dx.doi.org/10.1098/rspa.1948.0037
http://dx.doi.org/10.1098/rspa.1948.0037... and simplifications, compared with a standard tank (20 m²), in the high central region of São Paulo (Itirapina, SP) in a period of ten years (2003-2014).

Material and Methods

The study was carried out in Itirapina, SP, Brazil, at the Climatological Station of the Center of Water Resources and Applied Ecology (CRHEA) of the School of Engineering of São Carlos – University of São Paulo (22º 01’ 22” S; 43º 57’ 38” W; 733 m). The region has mean temperature and relative air humidity of 21.6 ^oC and 71%, respectively; the mean annual rainfall between 1979 and 2014 is 1486 mm. The data of evaporation were collected in a buried tank, with diameter of 5 m and depth of 2 m (20 m²).

The water level in the tank was measured with micrometric screw. The daily evaporation in the tank (E₂₀) results from the difference between subsequent readings corrected with the rainfall measured with a Ville de Paris pluviometer. Failures occurred in 16% of the days, due to the overflow of the tank. Thus, the failures were filled with the mean evaporation of the previous and subsequent days, according to Collischonn & Tucci (2014)Collischonn, B.; Tucci, C. E. M. Relações regionais entre precipitação e evapotranspiração mensais. Revista Brasileira de Recursos Hídricos, v.19, p.205-214, 2014. http://dx.doi.org/10.21168/rbrh.v19n3.p205-2014
http://dx.doi.org/10.21168/rbrh.v19n3.p2... .

The utilized meteorological data were: air temperature (t_a), relative humidity (RH), hours of insolation (n) and wind speed (U₂), collected at a height of 1.5 to 2.0 m above the grassed surface (Table 1).

For temperature, the monthly mean was calculated with the daily values; for E₂₀ and rainfall, monthly accumulations were also calculated.

Methods of Penman (1948)Penman, H. L. Natural evaporation from open water, bare soil and grass. Proceedings of the Royal Society A, v.193, p.120-145, 1948. http://dx.doi.org/10.1098/rspa.1948.0037
http://dx.doi.org/10.1098/rspa.1948.0037... and Kohler & Parmele (1967)Kohler, M. A.; Parmele, L. H. Generalized estimates of free-water evaporation. Water Resources Research, v.3, p.997-1005, 1967. http://dx.doi.org/10.1029/WR003i004p00997
http://dx.doi.org/10.1029/WR003i004p0099...

Using the relationship between the flux of sensible and latent heat and the evaporating power of the air, Penman (1948)Penman, H. L. Natural evaporation from open water, bare soil and grass. Proceedings of the Royal Society A, v.193, p.120-145, 1948. http://dx.doi.org/10.1098/rspa.1948.0037
http://dx.doi.org/10.1098/rspa.1948.0037... elaborated the method without the need for measurements at two levels above the soil, of temperature and humidity. These measurements at two levels do not exist in weather stations and require high-resolution sensors (Pereira et al., 2013Pereira, A. R.; Villa Nova, N. A.; Sediyama, G. C. Evapotranspiração. Campinas: Fundag, 2013. 318p.). This method (Kohler & Parmele) is based on the modification of the psychrometric coefficient and of the empirical function of the wind. The method uses an anemometer at height of 4 m; in this study, the anemometer is installed at height of 2 m above the soil, requiring the multiplication by the coefficient 1.15.

Methods of Priestley & Taylor (1972)Priestley, C. H. B.; Taylor, R. J. On the assessment of surface heat flux and evaporation using large-scale parameters. Monthly Weather Review, v.100, p.81-92, 1972. http://dx.doi.org/10.1175/1520-0493(1972)100%3C0081:OTAOSH%3E2.3.CO;2
http://dx.doi.org/10.1175/1520-0493(1972... and Stewart & Rouse (1976)Stewart, R. B.; Rouse, W. R. A simple method for determining the evaporation from shallow lakes and ponds. Water Resources Research, v.12, p.623-628, 1976. http://dx.doi.org/10.1029/WR012i004p00623
http://dx.doi.org/10.1029/WR012i004p0062...

The method of Penman (1948)Penman, H. L. Natural evaporation from open water, bare soil and grass. Proceedings of the Royal Society A, v.193, p.120-145, 1948. http://dx.doi.org/10.1098/rspa.1948.0037
http://dx.doi.org/10.1098/rspa.1948.0037... comprises two terms, diabatic and adiabatic. Priestley & Taylor (1972)Priestley, C. H. B.; Taylor, R. J. On the assessment of surface heat flux and evaporation using large-scale parameters. Monthly Weather Review, v.100, p.81-92, 1972. http://dx.doi.org/10.1175/1520-0493(1972)100%3C0081:OTAOSH%3E2.3.CO;2
http://dx.doi.org/10.1175/1520-0493(1972... eliminated the adiabatic term (aerodynamic) and corrected the diabatic term with a coefficient, α = 1.26 (Pereira et al., 2013Pereira, A. R.; Villa Nova, N. A.; Sediyama, G. C. Evapotranspiração. Campinas: Fundag, 2013. 318p.).

Stewart & Rouse (1976)Stewart, R. B.; Rouse, W. R. A simple method for determining the evaporation from shallow lakes and ponds. Water Resources Research, v.12, p.623-628, 1976. http://dx.doi.org/10.1029/WR012i004p00623
http://dx.doi.org/10.1029/WR012i004p0062... simplified the estimate of the net radiation (R_n) of the model of Priestley & Taylor (1972)Priestley, C. H. B.; Taylor, R. J. On the assessment of surface heat flux and evaporation using large-scale parameters. Monthly Weather Review, v.100, p.81-92, 1972. http://dx.doi.org/10.1175/1520-0493(1972)100%3C0081:OTAOSH%3E2.3.CO;2
http://dx.doi.org/10.1175/1520-0493(1972... with a linear regression from data of global solar irradiance, whose method was tested in Canada, in a shallow lake.

Methods of Linacre (1993)Linacre, E. T. Data-sparse estimation of lake evaporation using a simplified Penman equation. Agricultural and Forest Meteorology, v.64, p.237-256, 1993. http://dx.doi.org/10.1016/0168-1923(93)90031-C
http://dx.doi.org/10.1016/0168-1923(93)9... and Camargo et al. (1999)Camargo, A. P.; Marin, F. R.; Sentelhas, P. C.; Picini, A. G. Ajuste da equação de Thornthwaite para estimar a evapotranspiração potencial em climas áridos e superúmidos, com base na amplitude térmica diária. Revista Brasileira de Agrometeorologia, v.7, p.252-257, 1999.

The utilization of empirical relationships of net radiation and the evaporating power of the air, which showed satisfactory performance in various climates, are the basis of this methodology of Linacre (1993)Linacre, E. T. Data-sparse estimation of lake evaporation using a simplified Penman equation. Agricultural and Forest Meteorology, v.64, p.237-256, 1993. http://dx.doi.org/10.1016/0168-1923(93)90031-C
http://dx.doi.org/10.1016/0168-1923(93)9... , in which the author substituted the aerodynamic function of the wind by 2.5U₂.

Camargo et al. (1999)Camargo, A. P.; Marin, F. R.; Sentelhas, P. C.; Picini, A. G. Ajuste da equação de Thornthwaite para estimar a evapotranspiração potencial em climas áridos e superúmidos, com base na amplitude térmica diária. Revista Brasileira de Agrometeorologia, v.7, p.252-257, 1999. aimed to simplify the estimation of reference evapotranspiration (ETo) and proposed the effective temperature (T_ef = 1.08tmax-0.36tmin); thus, the estimated ETo will be considered as equal to the evaporation. Camargo & Camargo (2000)Camargo, Â. P. D.; Camargo, M. B. P. D. Uma revisão analítica da evapotranspiração potencial. Bragantia, v.59, p.125-137, 2000. http://dx.doi.org/10.1590/S0006-87052000000200002
http://dx.doi.org/10.1590/S0006-87052000... pointed out that in a humid area the 20 m² tank with buffer area presents optimal data of ETo.

The following variables were analyzed: residual errors, the difference between estimated and observed values, efficiency and the characterization of the prediction between overestimation and underestimation. Hence, the following statistical parameters were used: RMSE – root-mean-square error; CRM – coefficient of residual mass; ME – maximum error; BIAS – mean difference; NSE - Nash-Sutcliffe efficiency; R² - coefficient of determination; D – coefficient of agreement; C – coefficient of performance (C = RD) (Camargo & Sentelhas, 1997Camargo, A. P; Sentelhas, P. C. Avaliação do desempenho de diferentes métodos de estimativas da evapotranspiração potencial no Estado de São Paulo, Brasil. Revista Brasileira de Agrometeorologia, v.5, p.89-97, 1997.; Gupta et al., 2009Gupta, H. V.; Kling, H.; Yilmaz, K. K.; Martinez, G. F. Decomposition of the mean squared error and NSE performance criteria: Implications for improving hydrological modelling. Journal of Hydrology, v.377, p.80-91, 2009. http://dx.doi.org/10.1016/j.jhydrol.2009.08.003
http://dx.doi.org/10.1016/j.jhydrol.2009... ).

Results and Discussion

The monthly accumulations of rainfall and evaporation in the 20 m² tank are shown in Figure 1. The t_a and R_s are also indicated during the months. The mean annual E₂₀ was equal to 4.0 mm d^-1. Allen et al. (1998)Allen, R. G.; Pereira, L. S.; Raes, D.; Smith, M. Crop evapotranspiration: Guidelines for computing crop water requirements. Rome: FAO, 1998. 174p. Irrigation and drainage paper 56 cite that, in a humid tropical climate with mean temperature close to 20 ^oC, the mean ETo remains between 3 and 5 mm d^-1. The equation of Penman-Monteith is considered as the standard method to estimate ETo (Pereira et al., 2015Pereira, L. S.; Allen, R. G.; Smith, M.; Raes, D. Crop evapotranspiration estimation with FAO56: Past and future. Agricultural Water Management, v.147, p.4-20, 2015. http://dx.doi.org/10.1016/j.agwat.2014.07.031
http://dx.doi.org/10.1016/j.agwat.2014.0... ). Values of ETo become close to E₂₀ as reported by Camargo & Camargo (2000)Camargo, Â. P. D.; Camargo, M. B. P. D. Uma revisão analítica da evapotranspiração potencial. Bragantia, v.59, p.125-137, 2000. http://dx.doi.org/10.1590/S0006-87052000000200002
http://dx.doi.org/10.1590/S0006-87052000... . The monthly means during the entire period (2003-2014) did not exceed these values.

In Figure 1, the lowest E₂₀ rates are observed in June (minimum accumulated value was 57 mm in June/2013) due to the lower availability of R_s for the net radiation on the liquid surface. Hence, June is the month with the lowest mean of R_s (13 MJ (m² d)^-1), which, along with a RH of 72%, has a mean E₂₀ of 72 mm month^-1. The peaks of R_s that occur in the summer months, November and February, show a monthly mean R_s of 21 MJ (m² d)^-1, doubling E₂₀ to 144 mm month^-1. The highest monthly mean of R_s (26 MJ (m² d)^-1) was observed in January 2014, which showed the second highest accumulated value of E₂₀ (184 mm), due to an anomalous dry spell. The maximum accumulation of E₂₀ was 196 mm in October 2014; this month had R_s (24 MJ (m² d)^-1) lower than that of January/2014, but with lower RH (54%).

The highest accumulations of rainfall occurred in December, January and February, characterizing a humid, rainy summer, while the first month of the year had the lowest n (167 h month^-1), resulting, in general, in R_s lower (19 MJ (m² d)^-1) than that in December and February. On the other hand, August exhibits, in general, peaks of insolation with 263 h month^-1, and the lowest rates of RH, with 64%, indicate the dry winter of the region, with mean R_s of 19 MJ (m² d)^-1. With the decrease in nebulosity, mean R_s equal to that of November and for being the third month of the summer, February has the highest t_a (24.3 ^oC).

Each year was divided into two periods, dry and rainy. The rainy period comprehends the months of October to December and January to March. The dry period refers to the months of April to September. The lowest E₂₀ was equal to 1325 mm in 2013 and the highest one to 1613 mm in 2014, a difference of almost 300 mm between both years. The mean E₂₀ was 1446 mm year^-1. The mean annual relationship between rainfall and E₂₀ was equal to 1.0 during the analyzed period. In 2011, due to the excess of rainfall, especially in the months of January, March and December, this relationship increased to 1.3; two years later, with the dry period in the region of São Paulo, in 2014, this relationship reached the lowest value (0.7).

In some months with low rainfall in 2014, the atmospheric demand for water vapor increased, intensifying evaporation, due to the inexistence of buffer area around the 20 m² tank. Thus, the energy balance above the tank is no longer only vertical (Oasis Evaporation), a fact that is not taken into account by the models. Table 2 shows statistical indices for the evaluation of the models.

Figure 2 shows the comparison between E₂₀ and modeled values separated as dry and rainy periods; the index c is also indicated. Figure 3 shows the same comparison for the period from 2003 to 2014. The evaporation estimation quality of the methods is evaluated by the coefficient of determination (R²) and the accuracy is represented by the coefficient of agreement (D), both presented in Figure 3.

E_PN has the best indices for the dry period. The indices CRM and BIAS suggest that, during the dry period, only E_KP overestimated, while the other methods underestimated E. E_PN and E_SR during the dry period exhibit acceptable results; however, in the rainy period, the estimates of these methods showed the lowest C values. In this humid period, E_PN and E_SR showed high evaporation, above the observed value. E_PT shows the best accuracy during the rainy period; E_LN has values closer to 1 in the rainy period and in the entire period. Except for E_LN and E_PT, the other methods showed C coefficient lower than 0.7 during the rainy period.

The methods that exhibited similar behavior under both analyzed conditions (rainy and dry) were: Camargo et al. (1999)Camargo, A. P.; Marin, F. R.; Sentelhas, P. C.; Picini, A. G. Ajuste da equação de Thornthwaite para estimar a evapotranspiração potencial em climas áridos e superúmidos, com base na amplitude térmica diária. Revista Brasileira de Agrometeorologia, v.7, p.252-257, 1999. and Linacre (1993)Linacre, E. T. Data-sparse estimation of lake evaporation using a simplified Penman equation. Agricultural and Forest Meteorology, v.64, p.237-256, 1993. http://dx.doi.org/10.1016/0168-1923(93)90031-C
http://dx.doi.org/10.1016/0168-1923(93)9... , with underestimation, and Kohler & Parmele (1967)Kohler, M. A.; Parmele, L. H. Generalized estimates of free-water evaporation. Water Resources Research, v.3, p.997-1005, 1967. http://dx.doi.org/10.1029/WR003i004p00997
http://dx.doi.org/10.1029/WR003i004p0099... , with overestimation. When the statistical index NSE has negative value, it is advisable not to use the model; instead, the observed mean should be used as estimate (Gupta et al., 2009Gupta, H. V.; Kling, H.; Yilmaz, K. K.; Martinez, G. F. Decomposition of the mean squared error and NSE performance criteria: Implications for improving hydrological modelling. Journal of Hydrology, v.377, p.80-91, 2009. http://dx.doi.org/10.1016/j.jhydrol.2009.08.003
http://dx.doi.org/10.1016/j.jhydrol.2009... ).

Ultimately, the estimation quality of the methods is evaluated by the coefficient (C) presented in Table 3. The best statistical fit and accuracy are obtained by the methods E_LN and E_PN. E_LN has the best accuracy, demonstrated by the indices RMSE and D.

In Piracicaba, SP, and Jaboticabal, SP, E_LN showed good results with the 20 m² tank with the coefficient C varying from very good to optimal on the monthly scale (Oliveira, 2009Oliveira, G. X. S. Relações entre medidas de evaporação de superfícies de água livre por evaporímetros e estimativas por métodos meteorológicos em duas regiões do Estado de São Paulo. Piracicaba: ESALQ/USP, 2009. 108p. Tese Doutorado. http://dx.doi.org/10.11606/t.11.2009.tde-04082009-111344
http://dx.doi.org/10.11606/t.11.2009.tde... ). For Boqueirão, PB, and Patos, PB, it showed very good and good C factor, respectively (Leitão et al., 2007Leitão, M. de. M. V. B. R.; Oliveira, G. M. de; Leitão, T. J. V. Avaliação do desempenho de diferentes métodos de estimativa da evaporação para duas regiões da Paraíba. Revista Brasileira de Engenharia Agrícola e Ambiental, v.11, p.585-593, 2007. http://dx.doi.org/10.1590/S1415-43662007000600006
http://dx.doi.org/10.1590/S1415-43662007... ). Pereira et al. (2009)Pereira, S. B.; Pruski, F. F.; Silva, D. D. da; Ramos, M. M. Evaporação líquida no lago de Sobradinho e impactos no escoamento devido à construção do reservatório. Revista Brasileira de Engenharia Agrícola e Ambiental, v.13, p.346-352, 2009. http://dx.doi.org/10.1590/S1415-43662009000300018
http://dx.doi.org/10.1590/S1415-43662009... concluded that E_LN overestimated the evaporation for the reservoir of Sobradinho in comparison to the measurements in the Class A pan corrected by the coefficient of 0.6. However, in this study, Q_g was not estimated with data of insolation, making the accuracy of the model worse.

When Penman (1948)Penman, H. L. Natural evaporation from open water, bare soil and grass. Proceedings of the Royal Society A, v.193, p.120-145, 1948. http://dx.doi.org/10.1098/rspa.1948.0037
http://dx.doi.org/10.1098/rspa.1948.0037... developed the method with lysimetric and evaporimetric data, he did not have a net radiometer; thus, he elaborated an equation that is adopted by FAO 56 and is the basis to estimate net radiation (Rn) of the estimates of E_PN, E_KP and E_PT. Under the conditions of the present study, the evaporation measured in the 20 m² tank indicates that the equation used in the models overestimates the value of Rn.

In the simplification made by Priestley & Taylor (1972)Priestley, C. H. B.; Taylor, R. J. On the assessment of surface heat flux and evaporation using large-scale parameters. Monthly Weather Review, v.100, p.81-92, 1972. http://dx.doi.org/10.1175/1520-0493(1972)100%3C0081:OTAOSH%3E2.3.CO;2
http://dx.doi.org/10.1175/1520-0493(1972... , the adiabatic term was suppressed and incorporated to the diabatic term through the multiplication by a coefficient α = 1.26; thus, a good estimation of evaporation depends even more on the correct estimation of Rn. Oliveira (2009)Oliveira, G. X. S. Relações entre medidas de evaporação de superfícies de água livre por evaporímetros e estimativas por métodos meteorológicos em duas regiões do Estado de São Paulo. Piracicaba: ESALQ/USP, 2009. 108p. Tese Doutorado. http://dx.doi.org/10.11606/t.11.2009.tde-04082009-111344
http://dx.doi.org/10.11606/t.11.2009.tde... concluded that E_PT showed results consistent with those of the 20 m² tank in Piracicaba-SP; however, in this study the author measured Rn above a lake.

Stewart & Rouse (1976)Stewart, R. B.; Rouse, W. R. A simple method for determining the evaporation from shallow lakes and ponds. Water Resources Research, v.12, p.623-628, 1976. http://dx.doi.org/10.1029/WR012i004p00623
http://dx.doi.org/10.1029/WR012i004p0062... utilized the approach of Priestley & Taylor (1972)Priestley, C. H. B.; Taylor, R. J. On the assessment of surface heat flux and evaporation using large-scale parameters. Monthly Weather Review, v.100, p.81-92, 1972. http://dx.doi.org/10.1175/1520-0493(1972)100%3C0081:OTAOSH%3E2.3.CO;2
http://dx.doi.org/10.1175/1520-0493(1972... modifying the form of estimating Rn, by using the equation 0.38+0.30Rs, which was developed in temperate climate and does not show good results for the estimation of evaporation in the tropical climate.

The form of determination of net radiation through the method of Kohler & Parmele (1967)Kohler, M. A.; Parmele, L. H. Generalized estimates of free-water evaporation. Water Resources Research, v.3, p.997-1005, 1967. http://dx.doi.org/10.1029/WR003i004p00997
http://dx.doi.org/10.1029/WR003i004p0099... produced values underestimated by, on average, 2.38 MJ m^-2 d^-1 (~1.0 mm) during the entire period, in comparison to those estimated by the method of Penman (1948)Penman, H. L. Natural evaporation from open water, bare soil and grass. Proceedings of the Royal Society A, v.193, p.120-145, 1948. http://dx.doi.org/10.1098/rspa.1948.0037
http://dx.doi.org/10.1098/rspa.1948.0037... . Thus, the estimation of the method showed unsatisfactory performance.

The difference between the height of the anemometer (2.0 m) and the recommendation of the authors (4.0 m) does not influence the estimation of evaporation, due to the correction performed by the logarithmic profile. According to Woodhead (1972)Woodhead, T. Mapping potential evaporation for tropical East Africa: The accuracy of Penman estimates derived from indirect assessments of radiation and windspeed. In: Symposium on World Water Balance, 1972, Madrid. Proceedings... Madrid: WMO, 1972. p.232-241., this occurs because the uncertainties in U₄ measurements alter in at most 5% the calculated values of evaporation.

Local data, such as the aerodynamic resistance of the surface or the empirical function of the wind expressed as a linear function f(u), are rare. Pereira et al. (2013)Pereira, A. R.; Villa Nova, N. A.; Sediyama, G. C. Evapotranspiração. Campinas: Fundag, 2013. 318p. point out that these functions must be determined locally to improve the efficiency of the method. Thus, given the various and increasing options found in the literature to estimate net radiation and establish f(u), the model proposed by Linacre (1993)Linacre, E. T. Data-sparse estimation of lake evaporation using a simplified Penman equation. Agricultural and Forest Meteorology, v.64, p.237-256, 1993. http://dx.doi.org/10.1016/0168-1923(93)90031-C
http://dx.doi.org/10.1016/0168-1923(93)9... simplifies these choices and does not require specific measurements.

If there are no data of insolation or R_s and UR, the method of Camargo et al. (1999)Camargo, A. P.; Marin, F. R.; Sentelhas, P. C.; Picini, A. G. Ajuste da equação de Thornthwaite para estimar a evapotranspiração potencial em climas áridos e superúmidos, com base na amplitude térmica diária. Revista Brasileira de Agrometeorologia, v.7, p.252-257, 1999., based only on the effective temperature, has a very good performance and should be used in the estimation of evaporation of the 20 m² tank. Between competing hypotheses, the one with lowest number of assumptions must be selected (Ockham’s razor). Temperature and rainfall are the variables most easily measured in the stations and with higher reliabilities. Therefore, the expansion of platforms of environmental data collection (ANA, INMET, CPRM and INPE) in Brazil increases the possibility of application of this method.

Conclusions

1. The method of Linacre (1993) is recommended to estimate open-water evaporation.

2. In the rainy period, the method of Priestley & Taylor (1972) provides evaporation estimates as good as those of the method of Linacre (1993).

3. The method of Camargo et al. (1999) is a simple alternative for the estimation of evaporation from the 20 m² tank. The original method of Penman (1948) overestimates it.

4. The mean annual relationship between rainfall and the evaporation observed in the 20 m² tank is unitary, varying between 0.7 for a dry year (2014) and 1.3 for a rainy year (2011).

Literature Cited

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