Alternative Methods of Reference Evapotranspiration for Brazilian Climate Types

Métodos Alternativos de Evapotranspiração de Referência para Tipos Climáticos Brasileiros

Abstract

The choice of consistent alternative methods is essential for the improvement of reference evapotranspiration (ETo) estimation for different climatic regions. Due to a critical gap in knowledge concerning the most adequate alternative ETo methods for the climatic conditions in Paraná, Brazil, this study aimed to test and to evaluate the main estimation alternative methods (Thornthwaite - ETo_TH; Camargo - ETo_C; Hargreaves and Samani - ETo_HS; Linacre - ETo_L; and, Budyko - ETo_B) for the subtropical (Cfb) and semi-arid (Bsh) climate types in Brazil. We compared our results with standard ETo_PM (Penman-Monteith) estimated between 1970 and 2015, using the minimum and maximum air temperature (T), sunshine hours (n), relative humidity (RH) and wind speed (U₂). Least square regression analysis of ETo estimated by alternative methods vs ETo_PM were used to calibrate the methods for each analyzed climate type. The performance of calibrated and noncalibrated methods was evaluated by index of agreement “d” and performance “c”, root mean square error (RMSE) and mean error (ME). Our results showed the importance of calibration process of alternative methods for the improvement of ETo estimations in Brazil. The Hargreaves and Samani and Linacre calibrated methods showed better performance in the subtropical and semi-arid climates, respectively. Also, the Linacre and Budyko calibrated methods were particularly robust in subtropical and semi-arid climates, outlining the importance of continuous measurements of T used in the ETo_L and ETo_B modeling effort. The results presented here showed the importance to calibrate the alternative methods on ETo estimations and outlined the need for improvement and proposition of new ETo methods based on a limited number of climatic variables commonly available in subtropical and semi-arid climates in Brazil.

Keywords:
modeling; Penman Monteith ASCE; semi-arid; subtropical zones

Resumo

A escolha de métodos alternativos consistentes é essencial para a melhoria das estimativas da evapotranspiração de referência (ETo) em diferentes zonas climáticas. Devido ao desconhecimento dos métodos alternativos mais adequados para as condições climáticas paranaenses, teve-se por objetivo no presente trabalho testar e avaliar os principais métodos alternativos de estimativa da ETo (Thornthwaite - ETo_TH; Camargo - ETo_C; Hargreaves e Samani - ETo_HS; Linacre - ETo_L; and, Budyko - ETo_B) para os tipos climáticos subtropical (Cfb) e semi-árido (Bsh) no Brasil. As estimativas alternativas foram comparadas com a ETo estimada como o método de Penman-Monteith ASCE (ETo_PM) entre 1970 e 2015, a partir da temperatura máxima e mínima do ar (T), horas de brilho solar (n), umidade relativa média (UR) e velocidade do vento a dois metros de altura (U₂). Análises de regressão linear simples entre a ETo estimada com os métodos alternativos vs ETo_PM para cada tipo climático analisado, foram utilizadas para a calibração dos métodos. O desempenho dos métodos calibrados e não calibrados foi obtido a partir do índice “d” de concordância, “c” de desempenho, raiz quadrada do erro médio (RMSE) e erro médio (EM). Os resultados demonstraram a importância da calibração dos métodos alternativos para a melhoria das estimativas nas condições climáticas brasileiras. Os melhores desempenhos foram observados para os métodos calibrados Hargreaves and Samani e Linacre para os tipos climáticos subtropical e semi-árido, respectivamente. Além disso, os métodos calibrados Linacre e Budyko mostraram-se particularmente robustos nos tipos climáticos subtropical e semi-árido, respectivamente, destacando a importância de medidas contínuas da variável T, utilizada na modelagem de ETo_L e ETo_B. Os resultados obtidos indicaram a necessidade da melhoria das estimativas alternativas da ETo, a partir da utilização de pequeno conjunto de variáveis disponíveis para os tipos climáticos subtropical e semi-árido no Brasil.

Palavras-chave:
modelagem; Penman Monteith ASCE; semi-árido; zonas subtropicais

1. Introduction

The reference evapotranspiration (ETo) has fundamental role in the study and determination of water relations in the activities of rural engineering. Besides that, it is one of the most important hydrological variables for crop evapotranspiration, estimation and interpretation of agricultural water balances and irrigation management (Blaney and Criddle, 1950; Xu and Singh, 2005). The ETo can be measured directly from lysimeters or evapotranspirometers (Doorenbos and Pruitt, 1977) or estimated from theoretical methods based on climate variables such as temperature (Thorthwaite, 1948; Doorenbos and Pruitt, 1977) and solar radiation (Hargreaves and Samani, 1985). Due to the dissemination of weather stations and the high cost of direct measurement, the estimated ETo has been used with satisfactory performance around the world (Pandey et al., 2016). Traditionally, the combined method of Penman-Monteith parameterized by the Food and Agriculture Organization of the United Nations - FAO (Allen et al., 1998) and the American Society of Civil Engineers - ASCE (ASCE-EWRI, 2005) is recognized as a standard method to estimate ETo (Chauhan and Shrivastava, 2009). However, all the weather data needed to solve the method are often incomplete or not available in many of the Brazilian regions, which limits their use (Souza et al., 2014; Alencar et al., 2015). Even considering the different climate database of meteorological data around world, as World Climate Database (Hijmans et al., 2005), the complete set of data needed to estimate ETo by Penman-Monteith method is not available, because key variables such as wind speed, solar radiation, daily insolation and relative humidity are still missing or not consistent.

Therefore, alternative methods developed to estimate the reference evapotranspiration, with a small number of climate variables of easy measurement, are promising for the alternative estimation of ETo (Chauhan and Shrivastava, 2009). Over the last 50 years, many alternative models have been developed (Penman 1948; Thornthwaite 1948; Campbell 1971; Budyko 1974; Linacre 1977; Hargreaves and Samani, 1985), but the literature only reports a comparative performance to the Penman-Monteith standard method (Borges and Mediondo, 2007; Oliveira et al., 2008; Trajkovic and Kolakovic, 2009), with few studies analyzing the adjustment of ETo methods under different climatic types (Todorovic et al., 2013.). Therefore, given the existing climate variability in Brazil, such studies are important, because they allow to identify trends and limitations of the alternative methods and to choose the best suited method for each region.

Accordingly, here we test, adjust and evaluate the main alternative methods of reference evapotranspiration for different climate types in Brazil.

2. Material and Methods

Analyzes were carried out in a set of 45 years of daily historical data (January 1970 to December 2015) of maximum, minimum and average air temperature (°C), relative humidity (%), daily sunshine hours (MJ m^-2 d^-1) and wind speed at ten meters height (m s^-1), available by the National Institute of Meteorology (INMET, 2016) from two automatic weather stations located in the cities of Petrolina and Curitiba (Table 1), under the climate types Bsh and Cfb, respectively. According to Koppen (1936), the Bsh climate is characterized as dry semi-arid, occurring in low latitude and altitude locations with high average annual temperatures above 26 °C and average precipitation of 522.3 mm year^-1. The Cfb climate is characterized as a subtropical without dry season, with temperate summers and average precipitation of 1500 mm year^-1 (Alvares et al., 2013).

Table 1
Climatic classification, location and assessed weather stations.

Climatic classification	State	Station	Latitude (degrees)	Longitude (degrees)	Altitude (m)
Bsh	Pernambuco	Petrolina	-9.38	-40.48	370.46
Cfb	Paraná	Curitiba	-25.43	-49.26	923.50

Daily reference evapotranspiration was estimated by Penman-Monteith method, parameterized by American Society of Civil Engineers (ASCE) (ASCE EWRI, 2005).

(1) $$ET{o}_{PM}=\frac{0.408·\mathrm{\Delta }·\left(R_n-G\right)+{\text{γ}}_{psy}·\frac{C_n}{\left(T+273\right)}·u_2·\left(e_s-e_a\right)}{\mathrm{\Delta }·{\text{γ}}_{psy}·\left(1+C_d·u_2\right)}$$

where ETo_PM - reference evapotranspiration (mm d^-1); Δ - slope of the saturated water-vapor-pressure curve (kPa °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 °C^-1); T - daily average air temperature (°C); U₂ - wind speed at 2 m height (m s^-1); e_s - saturation 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) was estimated by difference between saturated and actual vapor pressure. Saturated vapor pressure was calculated using air temperature based on the Tetens formula. Actual vapor pressure was obtained by saturated vapor pressure multiplied by fractional humidity. Daily net radiation (Rn) was estimated by the difference between net longwave and shortwave radiation. The net longwave radiation (Rnl) was obtained by relative shortwave radiation (Rs/Rso), air temperature and actual vapor pressure. The net shortwave radiation (Rns) was obtained by solar radiation (Rs), which was estimated by relation between extraterrestrial radiation (Ra) and relative sunshine duration (n/N) (Pereira, 2015). The soil heat flux (G) was calculated using air temperature (Pereira et al., 1997). The wind speed measurements were transformed to wind speed at 2 m height by the wind profile relationship (Allen et al., 1998).

Alternative ETo methods consisted of the following empirical models: Thornthwaite (1948), Camargo (1971), Budyko (1974), Linacre (1977) and Hargreaves and Samani (1985).

a) Thornthwaite (1948):

Thornthwaite (1948) method uses daily average air temperature (T), considering a month of 30 days and 12 hours of photoperiod:

(2) $$ET{o}_{THi}=\frac{N_i}{12}·\frac{1}{30}·16·\left(\frac{10·T}{I}\right),\text{for}:T>0^{°}C$$

where ETo_THi - reference evapotranspiration by Thornthwaite (1948) method for the i-th day (mm day^-1); N_i - photoperiod in i-th day (h); T - daily average air temperature (°C).

(3) $$\begin{array}{l}a=6.75·{10}^{-7}·I^3-7.71·{10}^{-5}·I^2\\ +1.7912·{10}^{-2}·I+0.49239\end{array}$$

where a - cubic function of the heat index (I) of the region (dimensionless); I - heat index in the region (dimensionless).

(4) $$I={\displaystyle \sum _{i=1}^{12}{\left(0.2·T_a\right)}^{1.514},\text{for}}{T}_a>0$$

where T_a - average normal temperature of the m-th month of the year (°C).

b) Camargo (1971):

(5) $$ET{o}_{Ci}=Q{o}_i·F·T$$

where ETo_Ci - reference evapotranspiration by Camargo (1971) method for the i-th day (mm day^-1); Qo_i - extraterrestrial radiation of the i-th day, calculated as evaporation (mm day^-1); F - adjustment factor varying according to the site of the annual average temperature; T - daily average air temperature (°C).

The extraterrestrial radiation of the i-th day in equivalent evaporation (Qoi - mm day^-1) was transformed from the latent heat of evaporation (λ = 2.45 MJ kg^-1):

(6) $$Q{o}_{I\left(\text{mm}{\text{.day}}^{-1}\right)}=\frac{R_a}{245}$$

where Qo_{I(mm day^-1)} - extraterrestrial radiation of the i-th day in equivalent of evaporation (mm day^-1); R_a - extraterrestrial radiation of the i-th day (MJ m^-2 day^-1).

c) Hargreaves and Samani (1985):

(7) $$ET{o}_{HSi}=0.0023·Q{o}_i·\left(T+17.8\right)·{\left(T_{MAXi}-T_{MINi}\right)}^{0.5}$$

where ETo_HSi - reference evapotranspiration by Hargreaves and Samani (1985) method for the i-th day (mm day^-1); Qo_i - extraterrestrial radiation of the i-th day, calculated as evaporation (mm day^-1); T - daily average air temperature (°C); T_MAXi - maximum air temperature in the i-th day (°C); T_MINi - minimum air temperature in the i-th day (°C).

The transformation of the extraterrestrial radiation of the i-th day in equivalent of evaporation (Qo_i - mm day^-1) was from the latent heat of evaporation (λ = 2.45 MJ kg^-1), as proposed by Camargo (1971).

d) Linacre (1977):

(8) $$ET{o}_{Li}=\frac{700·\frac{T+0.006z}{100-\text{φ}}+15·\left(T-T{d}_i\right)}{80-T}$$

where ETo_Li - reference evapotranspiration by Linacre method for the i-th day (mm dia^-1); T - daily average air temperature (°C); z - local altitude (m); φ - site latitude (degrees); Td_i - dew point temperature in the i-th day (°C).

e) Budyko (1974):

(9) $$ET{o}_{Bi}=0.20·T$$

where ETo_Bi - reference evapotranspiration by Budyko (1985) method for the i-th day (mm day^-1); T - daily average air temperature in the i-th day (°C).

The daily values of ETo estimated by alternative methods were adjusted from linear regression analyzes in a monthly and annual basis, between 01/01/1970 and 12/31/2005 (35 years). The linear regression analysis to verify the performance and association between ETo_PM and calibrated ETo_alternative were held for the last ten years of data series (01/01/2006 to 12/31/2015). The calibrated ETo_alternative values were obtained using the adjustment coefficients “a” and “b” obtained from the relation ETo_alternative and ETo_PM (1970 to 2005).

(10) $$ET{o}_{alternativei}=a+b·ET{o}_{PMi}$$

where: ETo_{alternative i} is the reference evapotranspiration estimated by calibrated alternative methods at each i-day (mm d^-1); ETo_PMi - reference evapotranspiration by Penman-Monteith method in the i-th day (mm day^-1); a is the linear coefficient (mm d^-1); b is the angular coefficient (dimensionless).

Daily reference evapotranspiration obtained with alternative methods and standard Penman-Monteith method were compared by linear regression analysis using R² as an index of precision, agreement index “d” (Willmott et al., 1985) as an index of accuracy and confidence index “c” ( Camargo and Sentelhas, 1997) as a general index to evaluate both precision and accuracy. 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 relative data to the 1:1 line.

The interpretation criteria of “c” performance, was classified by great (“c” > 0.85); very good (0.75 < “c” ≤ 0.85); good (0.65 < “c” ≤ 0.75); average (0.60 < “c” ≤ 0.65); tolerable (0.50 < “c” ≤ 0.60); bad (0.40 < “c” ≤ 0.50); and, very bad (“c” ≤ 0.40).

For further comparison, root mean squared error (RMSE) and mean error (ME) were used to evaluate the reference evapotranspiration estimated by alternative methods:

(9) $$d=1\left[\frac{{\displaystyle \sum _{i=1}^n{\left(E_i-O_i\right)}^2}}{{\displaystyle \sum _{i=1}^n{\left(\left|E_i-{\overline{O}}_i\right|+\left|O_i-{\overline{O}}_i\right|\right)}^2}}\right]$$

(10) $$c=\left|R·d\right|$$

(11) $$RMSE=\sqrt{\frac{1}{n}·{\displaystyle \sum _{i=1}^n{\left(E_i-O_i\right)}^2}}$$

(12) $$ME=\frac{E_i-O_i}{O_i}·100$$

where d - agreement index of Willmott et al. (1985) (dimensionless); Ei - ETo estimated by alternative methods in the i-th day (mm day^-1); O_i - ETo estimated by Penman-Monteith method in the i-th day (mm day^-1); O_i - mean of the ETo estimated by Penman-Monteith method (mm day^-1); n - number of observations (dimensionless); R - correlation coefficient (dimensionless); RMSE - root mean square error (mm day^-1); ME - mean error (%).

3. Results and Discussion

Contrasting performances were observed between methods for all climate types (Tables 2 and 3). The calibration process resulted in improved estimations in the annual period for some methods, showing better performances in the monthly period.

Table 2
Coefficient of determination (R²), correlation (R), root mean square error (RMSE), mean error (ME), indexes “d” and “c” and performance of alternative methods, calibrated (C) or not (NC), compared to standard (ETo_PM) for climate type “Cfb”, in monthly and annual periods, between 1970 and 2005.

Month	R 2		R		“d”		“c”		RMSE (mm day-1)		ME (%)		a (mm day-1)		b (dimensionless)
	NC	C	NC	C	NC	C	NC	C	NC	C	NC	C	NC	C	NC	C
Thornthwaite (1948)
Annual	0.55	0.55	0.74	0.74	0.81	0.84	0.60	0.62	0.90	0.77	-14.60	9.99	0.79	1.34	0.87	0.55
January	0.20	0.32	0.45	0.56	0.79	0.88	0.35	0.50	0.93	0.90	-8.60	15.17	1.48	3.38	0.70	0.20
February	0.18	0.41	0.42	0.64	0.76	0.87	0.32	0.56	0.83	0.86	-5.32	12.87	1.22	3.09	0.73	0.24
March	0.21	0.35	0.46	0.59	0.64	0.73	0.29	0.43	0.71	0.82	-6.20	15.21	1.32	2.84	0.63	0.19
April	0.13	0.14	0.36	0.38	0.76	0.62	0.27	0.23	0.69	0.59	-7.23	15.13	1.63	2.41	0.37	0.10
May	0.10	0.07	0.31	0.26	0.93	0.94	0.29	0.24	0.62	0.52	-10.44	17.96	1.33	1.87	0.30	0.06
June	0.07	0.45	0.26	0.67	0.96	0.97	0.25	0.64	0.57	0.47	-9.38	13.68	1.23	1.62	0.23	0.04
July	0.23	0.22	0.48	0.47	0.94	0.93	0.45	0.43	0.67	0.53	-19.82	24.52	1.01	1.74	0.53	0.20
August	0.32	0.39	0.57	0.63	0.84	0.72	0.48	0.45	0.90	0.60	-27.71	16.05	1.18	1.96	0.66	0.29
September	0.29	0.33	0.54	0.57	0.72	0.61	0.39	0.35	1.01	0.80	-25.88	17.44	1.39	2.31	0.63	0.25
October	0.20	0.23	0.45	0.48	0.50	0.69	0.22	0.33	1.10	0.89	-21.81	16.29	1.77	2.88	0.56	0.17
November	0.17	0.13	0.41	0.36	0.50	0.78	0.21	0.28	1.29	1.03	-21.75	21.55	2.11	3.53	0.56	0.13
December	0.14	0.18	0.38	0.43	0.69	0.87	0.26	0.37	1.10	0.97	-13.83	15.80	2.05	3.63	0.57	0.15
Camargo (1971)
Annual	0.63	0.58	0.79	0.76	0.86	0.84	0.68	0.64	0.76	0.81	-8.96	-7.47	0.56	0.68	0.91	0.65
January	0.19	0.30	0.44	0.55	0.86	0.88	0.38	0.48	0.81	0.84	0.21	8.46	-0.13	3.09	1.08	0.22
February	0.17	0.42	0.42	0.65	0.81	0.86	0.34	0.56	0.76	0.81	1.02	5.18	-0.26	2.73	1.11	0.26
March	0.21	0.36	0.46	0.60	0.62	0.70	0.28	0.42	0.65	0.77	-2.99	6.17	0.47	2.49	0.91	0.22
April	0.15	0.17	0.39	0.41	0.79	0.77	0.31	0.32	0.63	0.54	-8.64	7.04	1.13	2.13	0.61	0.14
May	0.11	0.09	0.33	0.30	0.95	0.95	0.31	0.29	0.57	0.48	-13.53	7.45	1.05	1.66	0.50	0.08
June	0.05	0.03	0.23	0.18	0.97	0.97	0.22	0.17	0.53	0.47	-15.71	4.48	1.10	1.49	0.35	0.03
July	0.23	0.22	0.48	0.47	0.94	0.96	0.45	0.45	0.66	0.45	-20.74	4.81	0.60	1.37	0.88	0.22
August	0.31	0.40	0.56	0.63	0.85	0.82	0.47	0.52	0.82	0.59	-22.08	-1.01	0.67	1.54	0.94	0.31
September	0.26	0.26	0.51	0.51	0.73	0.61	0.37	0.31	0.81	0.81	-12.28	3.66	0.74	2.02	0.86	0.23
October	0.20	0.23	0.44	0.48	0.57	0.64	0.25	0.31	0.81	0.85	-5.96	6.94	0.97	2.55	0.77	0.19
November	0.18	0.13	0.42	0.36	0.71	0.77	0.30	0.28	0.97	0.95	-7.81	14.21	1.03	3.25	0.82	0.14
December	0.14	0.18	0.38	0.42	0.83	0.86	0.31	0.36	0.88	0.91	-2.77	10.11	0.82	3.39	0.85	0.16
Hargreaves and Samani (1985)
Annual	0.72	0.71	0.85	0.85	0.81	0.84	0.69	0.71	1.09	1.01	36.54	31.18	0.10	0.93	0.74	0.93
January	0.41	0.55	0.64	0.74	0.86	0.91	0.55	0.68	1.34	0.69	34.01	2.10	0.32	2.16	0.71	0.42
February	0.41	0.58	0.64	0.76	0.83	0.90	0.54	0.69	1.22	0.70	33.17	-1.25	0.13	1.75	0.75	0.48
March	0.45	0.60	0.67	0.77	0.74	0.82	0.50	0.64	1.01	0.63	30.49	0.64	0.49	1.72	0.66	0.43
April	0.38	0.39	0.62	0.62	0.54	0.87	0.33	0.54	0.81	0.46	29.80	3.34	0.53	1.51	0.62	0.38
May	0.27	0.34	0.52	0.58	0.81	0.97	0.42	0.56	0.74	0.42	38.20	1.86	0.57	1.22	0.52	0.29
June	0.12	0.17	0.35	0.41	0.88	0.97	0.31	0.40	0.73	0.45	44.84	1.34	0.80	1.32	0.35	0.12
July	0.28	0.43	0.53	0.65	0.84	0.97	0.45	0.63	0.75	0.40	41.41	-1.19	0.37	0.99	0.59	0.39
August	0.36	0.46	0.60	0.68	0.62	0.85	0.37	0.57	0.94	0.59	39.22	-5.37	0.49	1.26	0.59	0.39
September	0.46	0.48	0.68	0.69	0.58	0.75	0.40	0.52	1.14	0.71	41.27	-2.00	0.53	1.50	0.58	0.38
October	0.47	0.44	0.69	0.66	0.70	0.78	0.48	0.52	1.30	0.73	40.25	1.88	0.57	1.77	0.60	0.41
November	0.40	0.46	0.63	0.68	0.84	0.87	0.53	0.59	1.32	0.73	34.16	5.11	0.52	2.03	0.67	0.42
December	0.42	0.40	0.65	0.63	0.86	0.90	0.56	0.57	1.37	0.77	34.20	3.11	0.46	2.32	0.68	0.39
Linacre (1977)
Annual	0.46	0.53	0.68	0.73	0.76	0.81	0.52	0.59	0.87	0.81	-3.67	0.84	0.29	1.19	1.02	0.51
January	0.35	0.47	0.59	0.69	0.62	0.92	0.37	0.63	1.12	0.74	-20.13	9.37	1.08	2.37	0.93	0.43
February	0.30	0.53	0.55	0.72	0.64	0.91	0.35	0.66	0.94	0.72	-15.08	6.59	1.04	2.23	0.86	0.42
March	0.28	0.48	0.53	0.70	0.61	0.82	0.32	0.57	0.71	0.68	-9.27	9.06	0.97	2.09	0.78	0.38
April	0.26	0.31	0.51	0.55	0.80	0.81	0.41	0.45	0.54	0.50	3.83	7.61	1.06	1.90	0.57	0.24
May	0.25	0.38	0.50	0.62	0.92	0.97	0.46	0.60	0.57	0.41	19.30	8.63	0.91	1.36	0.43	0.27
June	0.28	0.36	0.53	0.60	0.92	0.98	0.49	0.59	0.64	0.39	31.10	5.47	0.82	1.15	0.36	0.27
July	0.48	0.52	0.69	0.72	0.93	0.98	0.65	0.71	0.60	0.35	21.87	4.47	0.65	0.90	0.53	0.50
August	0.52	0.53	0.72	0.73	0.90	0.88	0.64	0.64	0.63	0.51	3.06	-0.93	0.95	1.04	0.56	0.53
September	0.44	0.51	0.66	0.72	0.82	0.81	0.54	0.58	0.74	0.65	-9.09	3.46	1.16	1.38	0.61	0.47
October	0.29	0.35	0.53	0.59	0.60	0.73	0.32	0.43	0.95	0.78	-17.07	5.84	1.46	2.15	0.65	0.31
November	0.33	0.41	0.58	0.64	0.56	0.86	0.32	0.55	1.21	0.79	-22.81	12.06	1.50	2.49	0.79	0.35
December	0.24	0.42	0.49	0.65	0.54	0.91	0.26	0.59	1.25	0.79	-22.21	10.67	1.70	2.59	0.75	0.38
Budyko (1974)
Annual	0.44	0.46	0.67	0.68	0.71	0.70	0.47	0.47	1.08	1.14	39.81	42.06	-0.58	2.38	0.96	0.44
January	0.19	0.31	0.44	0.56	0.89	0.88	0.39	0.49	0.88	0.84	15.49	8.38	-0.08	3.08	0.93	0.22
February	0.17	0.39	0.41	0.62	0.84	0.86	0.35	0.53	0.99	0.83	24.04	5.38	-0.39	2.82	0.94	0.24
March	0.19	0.31	0.44	0.55	0.67	0.68	0.29	0.37	1.11	0.79	35.51	6.39	0.12	2.59	0.74	0.19
April	0.11	0.13	0.33	0.36	0.21	0.76	0.07	0.27	1.31	0.55	54.18	7.19	1.15	2.23	0.36	0.10
May	0.08	0.05	0.28	0.22	0.26	0.95	0.07	0.21	1.36	0.49	78.17	7.64	1.09	1.71	0.23	0.05
June	0.05	0.03	0.23	0.18	0.39	0.97	0.09	0.17	1.42	0.47	94.34	4.50	1.10	1.49	0.15	0.03
July	0.20	0.18	0.44	0.42	0.52	0.96	0.23	0.41	1.21	0.46	73.62	5.19	0.65	1.43	0.38	0.19
August	0.27	0.34	0.52	0.59	0.56	0.81	0.29	0.47	0.99	0.61	42.60	-0.49	0.64	1.61	0.53	0.28
September	0.26	0.29	0.51	0.54	0.58	0.63	0.30	0.34	0.92	0.79	31.32	3.91	0.65	1.97	0.60	0.25
October	0.18	0.21	0.43	0.46	0.71	0.63	0.30	0.29	0.87	0.86	20.43	7.27	0.92	2.61	0.61	0.17
November	0.17	0.13	0.41	0.36	0.85	0.77	0.35	0.28	0.87	0.95	8.20	14.29	1.05	3.26	0.70	0.14
December	0.14	0.18	0.38	0.42	0.89	0.86	0.34	0.36	0.86	0.91	10.46	10.14	0.80	3.40	0.75	0.16

Table 3
Coefficient of determination (R²), correlation (R), root mean square error (RMSE), mean error (ME), indexes “d” and “c” and performance of alternative methods, calibrated (C) or not (NC), compared to standard (ETo_PM) for climate type “Bsh”, in monthly and annual periods, between 1970 and 2005.

Month	R 2		R		“d”		“c”		RMSE (mm day-1)		ME (%)		a (mm day-1)		b (dimensionless)
	NC	C	NC	C	NC	C	NC	C	NC	C	NC	C	NC	C	NC	C
Thornthwaite (1948)
Annual	0.45	0.49	0.67	0.70	0.72	0.79	0.48	0.55	1.55	1.28	-17.46	-10.13	2.75	0.14	0.65	0.87
January	0.49	0.32	0.70	0.56	0.79	0.84	0.55	0.48	1.15	1.04	-6.60	9.41	1.31	3.38	0.86	0.20
February	0.57	0.41	0.75	0.64	0.81	0.88	0.61	0.57	1.05	0.90	-8.26	10.98	1.29	3.09	0.87	0.24
March	0.61	0.35	0.78	0.59	0.88	0.84	0.69	0.49	0.90	0.92	-5.53	9.62	1.03	2.84	0.88	0.19
April	0.50	0.14	0.71	0.38	0.88	0.80	0.62	0.30	0.94	0.90	-9.67	11.56	1.21	2.41	0.86	0.10
May	0.44	0.07	0.67	0.26	0.88	0.88	0.59	0.23	1.14	0.71	-15.97	9.02	1.92	1.87	0.73	0.06
June	0.36	0.45	0.60	0.67	0.83	0.88	0.50	0.59	1.56	0.77	-28.78	11.95	2.01	1.62	0.82	0.04
July	0.22	0.22	0.47	0.47	0.75	0.86	0.35	0.40	1.97	0.72	-36.23	10.67	2.78	1.74	0.68	0.20
August	0.24	0.39	0.49	0.63	0.38	0.51	0.19	0.32	2.46	0.76	-39.69	9.76	3.66	1.96	0.62	0.29
September	0.19	0.33	0.43	0.57	0.22	0.87	0.10	0.50	2.30	0.74	-30.95	6.66	4.75	2.31	0.41	0.25
October	0.26	0.23	0.51	0.48	0.56	0.89	0.28	0.42	1.64	0.92	-16.72	7.24	4.08	2.88	0.50	0.17
November	0.44	0.13	0.67	0.36	0.77	0.84	0.52	0.31	1.33	1.20	-8.31	9.78	2.08	3.53	0.77	0.13
December	0.44	0.18	0.66	0.43	0.78	0.84	0.52	0.36	1.17	1.06	-6.61	12.96	2.41	3.63	0.68	0.15
Camargo (1971)
Annual	0.45	0.47	0.67	0.69	0.52	0.56	0.35	0.38	2.18	1.92	-29.60	-27.29	2.75	1.99	0.65	0.35
January	0.49	0.37	0.70	0.61	0.46	0.20	0.33	0.13	1.99	1.49	-21.45	-13.21	1.31	4.67	0.86	0.09
February	0.57	0.70	0.75	0.84	0.50	0.44	0.37	0.37	1.85	1.29	-21.17	-6.71	1.29	4.26	0.87	0.16
March	0.61	0.59	0.78	0.77	0.66	0.54	0.51	0.42	1.65	1.32	-18.72	-10.56	1.03	3.97	0.88	0.15
April	0.50	0.44	0.71	0.66	0.73	0.76	0.52	0.50	1.64	1.08	-23.66	-6.27	1.21	3.84	0.86	0.14
May	0.44	0.30	0.67	0.55	0.78	0.90	0.52	0.49	1.75	0.77	-28.45	-3.38	1.92	3.89	0.73	0.10
June	0.36	0.30	0.60	0.55	0.77	0.92	0.46	0.51	1.92	0.68	-35.46	3.80	2.01	4.09	0.82	0.09
July	0.22	0.30	0.47	0.55	0.73	0.88	0.34	0.48	2.05	0.69	-37.07	8.46	2.78	4.49	0.68	0.08
August	0.24	0.30	0.49	0.55	0.35	0.26	0.17	0.14	2.55	0.73	-40.92	8.55	3.66	5.18	0.62	0.11
September	0.19	0.31	0.43	0.56	0.13	0.83	0.05	0.46	2.77	0.72	-38.87	2.28	4.75	5.83	0.41	0.09
October	0.26	0.20	0.51	0.45	0.12	0.77	0.06	0.34	2.71	0.97	-34.59	-2.80	4.08	5.92	0.50	0.06
November	0.44	0.26	0.67	0.51	0.27	0.23	0.18	0.12	2.50	1.39	-28.22	-12.10	2.08	5.02	0.77	0.09
December	0.44	0.59	0.66	0.77	0.30	0.28	0.20	0.22	2.20	1.36	-26.50	-4.34	2.41	4.86	0.68	0.10
Hargreaves and Samani (1985)
Annual	0.45	0.49	0.67	0.70	0.70	0.78	0.47	0.55	1.37	1.09	-15.90	-12.92	2.75	0.14	0.65	0.87
January	0.49	0.32	0.70	0.56	0.65	0.80	0.46	0.45	1.24	0.87	-8.07	-0.78	1.31	3.38	0.86	0.20
February	0.57	0.41	0.75	0.64	0.65	0.72	0.49	0.46	1.19	0.97	-8.88	5.71	1.29	3.09	0.87	0.24
March	0.61	0.35	0.78	0.59	0.79	0.73	0.62	0.43	1.07	0.87	-7.27	-0.14	1.03	2.84	0.88	0.19
April	0.50	0.14	0.71	0.38	0.83	0.86	0.59	0.32	1.13	0.74	-14.19	1.67	1.21	2.41	0.86	0.10
May	0.44	0.07	0.67	0.26	0.87	0.93	0.58	0.24	1.20	0.59	-18.20	3.82	1.92	1.87	0.73	0.06
June	0.36	0.45	0.60	0.67	0.86	0.88	0.51	0.59	1.32	0.74	-22.99	14.00	2.01	1.62	0.82	0.04
July	0.22	0.22	0.47	0.47	0.83	0.67	0.39	0.31	1.36	0.92	-22.87	18.73	2.78	1.74	0.68	0.20
August	0.24	0.39	0.49	0.63	0.51	0.34	0.25	0.21	1.58	1.11	-24.37	19.56	3.66	1.96	0.62	0.29
September	0.19	0.33	0.43	0.57	0.22	0.86	0.10	0.49	1.68	0.84	-21.59	10.07	4.75	2.31	0.41	0.25
October	0.26	0.23	0.51	0.48	0.34	0.90	0.17	0.43	1.60	0.81	-18.45	5.91	4.08	2.88	0.50	0.17
November	0.44	0.13	0.67	0.36	0.53	0.86	0.35	0.31	1.53	0.86	-12.87	0.66	2.08	3.53	0.77	0.13
December	0.44	0.18	0.66	0.43	0.53	0.75	0.35	0.32	1.32	1.00	-12.05	5.24	2.41	3.63	0.68	0.15
Linacre (1977)
Annual	0.36	0.21	0.60	0.46	0.31	0.32	0.19	0.15	4.09	3.80	-61.09	-59.05	10.18	2.89	-2.12	-0.13
January	0.56	0.39	0.75	0.62	0.28	0.82	0.21	0.51	4.13	1.03	-58.62	-1.90	12.33	2.24	-2.85	0.61
February	0.51	0.47	0.71	0.69	0.29	0.80	0.21	0.55	3.88	0.94	-56.99	2.65	12.32	3.03	-2.80	0.48
March	0.54	0.47	0.74	0.68	0.43	0.80	0.32	0.55	3.43	0.93	-52.14	-2.25	12.42	2.24	-2.94	0.56
April	0.51	0.34	0.71	0.59	0.53	0.82	0.38	0.48	3.10	0.88	-51.55	1.28	10.56	2.90	-2.32	0.41
May	0.46	0.26	0.68	0.51	0.63	0.91	0.43	0.46	2.93	0.72	-51.73	1.43	9.75	2.70	-2.27	0.41
June	0.40	0.33	0.63	0.57	0.65	0.93	0.41	0.53	2.92	0.64	-55.61	5.89	8.65	2.87	-1.93	0.40
July	0.40	0.29	0.63	0.54	0.60	0.92	0.38	0.50	3.17	0.61	-59.27	4.95	8.49	2.95	-1.92	0.39
August	0.33	0.23	0.57	0.48	0.23	0.59	0.13	0.28	4.17	0.68	-68.28	6.39	8.78	4.19	-1.67	0.27
September	0.42	0.17	0.65	0.42	0.07	0.85	0.04	0.35	4.94	0.74	-71.85	2.99	10.17	4.63	-2.03	0.29
October	0.42	0.40	0.65	0.63	0.06	0.92	0.04	0.58	5.27	0.74	-71.96	1.74	10.91	3.52	-2.17	0.47
November	0.58	0.42	0.76	0.65	0.15	0.88	0.11	0.57	5.00	0.93	-66.64	-2.16	12.27	1.96	-2.84	0.67
December	0.48	0.41	0.69	0.64	0.17	0.81	0.12	0.52	4.55	1.05	-64.08	1.13	11.67	2.94	-2.51	0.49
Budyko (1974)
Annual	0.36	0.21	0.60	0.46	0.33	0.35	0.20	0.16	2.88	2.55	-33.56	-30.32	10.91	4.74	-1.44	-0.20
January	0.56	0.38	0.75	0.62	0.27	0.82	0.20	0.51	2.88	1.02	-30.19	-1.75	13.57	2.27	-2.01	0.60
February	0.50	0.48	0.71	0.69	0.31	0.80	0.22	0.55	2.57	0.94	-27.72	2.66	13.71	3.04	-2.02	0.47
March	0.54	0.47	0.74	0.68	0.46	0.80	0.34	0.55	2.18	0.93	-19.97	-2.17	14.00	2.25	-2.14	0.56
April	0.51	0.35	0.71	0.59	0.61	0.82	0.43	0.48	1.84	0.88	-18.64	1.37	11.71	2.91	-1.67	0.41
May	0.45	0.27	0.67	0.52	0.72	0.91	0.49	0.47	1.74	0.71	-17.91	1.43	10.68	2.73	-1.58	0.41
June	0.40	0.33	0.63	0.57	0.77	0.93	0.48	0.54	1.69	0.63	-23.46	5.93	9.27	2.89	-1.30	0.39
July	0.39	0.29	0.63	0.54	0.70	0.92	0.44	0.50	1.96	0.61	-29.24	4.96	8.99	2.99	-1.25	0.38
August	0.32	0.23	0.57	0.48	0.27	0.59	0.15	0.28	2.94	0.67	-44.51	6.32	9.13	4.22	-1.07	0.27
September	0.41	0.17	0.64	0.41	0.07	0.85	0.04	0.35	3.69	0.74	-50.81	2.90	10.65	4.65	-1.31	0.28
October	0.41	0.39	0.64	0.62	0.06	0.91	0.04	0.57	4.01	0.75	-51.28	1.83	11.44	3.67	-1.41	0.45
November	0.57	0.41	0.76	0.64	0.13	0.88	0.10	0.56	3.79	0.94	-42.88	-2.10	13.14	2.07	-1.90	0.65
December	0.48	0.39	0.69	0.62	0.16	0.80	0.11	0.50	3.25	1.08	-38.81	1.71	12.67	2.96	-1.74	0.50

Comparatively, several authors have been demonstrating the influence of climate type on the estimated ETo, as well as on the performance of alternative methods (Lemos Filho et al., 2010; Silva et al., 2011). In this sense, the results showed that the evaluation of performance of alternative methods estimating ETo for different climate types is essential when climatic data required for ETo_PM are unavailable or unreliable.

The ETo_HS method showed the best performance and lower estimation errors for the climate Cfb in the monthly and annual periods. The ETo_L showed similar trends to ETo_HS in a monthly period (Table 2). Similar results were obtained by Syperreck et al. (2008), when the ETo_HS was a better suited method for subtropical climate (R = 0.86, d = 0.85, c = 0.73).

Camargo and Sentelhas (1997), comparing 20 alternative methods with ET obtained by evapotranspirometers in different Brazilian climate types, observed better estimates for ETo_C and ETo_TH for the subtropical and dry tropical climate types, respectively. The ETo_HS, ETo_C and ETo_TH use solar radiation or daily insolation as input variables of the model, which can improve the estimative of ETo, mainly under warm and dry climates (Gardiman et al., 2012).

The use of solar radiation in alternative methods to estimate ETo is well-recognized in the literature, providing good results in the analyzed regions (Irmak et al., 2006). According to Hupet and Vanclooster (2001), solar radiation has great influence on ETo_PM in extremely cold and wet climates, due to lower magnitude and influence of other climatic variables. According to Camargo and Sentelhas (1997), the easiest use of ETo_C and ETo_TH stands out among other alternative methods. However, ETo_TH method presents particularities in the estimative, as the need for the normal monthly average temperature of the year, which is considered a difficult information to access because it is not provided by weather stations, which makes it even more difficult spreading the method.

According to Souza et al. (2014), the use of solar radiation in alternative methods results in consistent estimates of ETo for both warm and dry as to cold and wet climates. For these authors, ETo_HS tends to perform better estimations in warm climates with high average temperatures throughout the year, since the method was based on the average temperature and maximum temperature of the day. Opposite trend was observed in this study, with the worst performance observed for ETo_HS in Bsh climate (Table 3).

The ETo_L method is a simplification of the Penman (1948) model, which uses air temperature functions, such as the difference between the average temperature and dew point temperature, being proposed in Africa and South America (Souza et al., 2014). In this sense, the best performance of the method occurred for Bsh, given the similarity with the climatic characteristics of the regions where the method was adjusted and validated (Irmak et al., 2006). Compared to our results, Mendonça et al. (2003), Choi Junior et al. (2011) and Todorovic et al. (2013) also observed better adjustment of methods based only on air temperature in warm and dry climates, justifying its use when there is unavailability of climate data.

Thus, it is observed the link between the performance of each alternative method with climatic conditions of the region where it is being used, highlighting the importance of detailed study of tendency of ETo and input variables in climate models.

Furthermore, the best performance of the calibrated alternative methods emphasizes the importance of calibration of models to the region of interest (Garcia et al., 2007; Pandey et al., 2016).

It was found similar trend of estimated ETo by alternative methods in relation to the standard method of Penman-Monteith for Cfb climate (Fig. 1a), especially after the calibration of the methods (Fig. 1b). The highest values observed occurred in the summer, reflecting the greater availability of energy in the soil-plant-atmosphere system at this period (Syperreck et al., 2008).

Figure 1
Monthly average reference evapotranspiration (ETo) estimated by the standard method (ETo_PM) for climate type “Cfb” between 1970 and 2005, compared to alternative methods uncalibrated (a) and calibrated (b).

Similar results were observed for the Bsh climate type (Fig. 2), but with less seasonality of ETo throughout the year, characteristic of this climate type. However, the smaller values of ETo remained in the winter period. Prior to calibration, all alternative methods underestimated ETo_PM, and it was observed great variation between estimations (Fig. 2a). The same trend was observed by Cai et al. (2007) in warm and dry regions, due to overestimation of actual vapor pressure, resulting in inconsistent estimations of some alternative methods.

Figure 2
Monthly average reference evapotranspiration (ETo) estimated by the standard method (ETo_PM) for climate type “Bsh” between 1970 and 2005, compared to alternative methods uncalibrated (a) and calibrated (b).

Although confirmed the possibility of using alternative methods to estimate ETo in conditions where climatic variables required for the standard method are not available, the results obtained in this study demonstrate the need for obtaining alternative consistent estimations for all Brazilian climatic conditions, from the simplified generation of alternative methods. In this sense, it stands out the understanding of the real influence of climate variables on the trend of ETo, which will form the basis for the simplification of the estimation. Therefore, given the territorial extension of Brazil and the great variability of latitude and altitude over regions, which result in different climate types, the sensitivity study of ETo is interesting and necessary. The study enables trends identification, constraints, to develop simplifications of the standard method and to propose alternative methods with a possible decrease of the number of input variables needed for an accurate estimation.

4. Conclusions

1. In order of importance, the Hargreaves and Samani, and Linacre methods showed a strong linear association with standard ETo_PM in the subtropical and semi-arid climate, respectively.

2. 2. The Linacre and Budyko methods were particularly robust in subtropical and semi-arid climates, outlining the importance of continuous measurements of the air temperature used in the ETo_L and ETo_B modeling effort.

3. 3. The results presented here showed the importance to calibrate the alternative methods on evapotranspiration estimative and outlined the need for improvement and proposition of new ETo methods based on a limited number of climatic variables commonly available in subtropical and semi-arid climates in Brazil.

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Publication Dates

• Publication in this collection
  Jul-Sep 2018

History

• Received
  18 Oct 2016
• Accepted
  13 May 2018