Alternative reference evapotranspiration methods for the main climate types of the state of Paraná, Brazil

The objective of this work was to define the best alternative methods for estimating the reference evapotranspiration (ETo) in the main climatic types (Cfa and Cfb) of the state of Paraná, Brazil. The methods tested were Budyko, Camargo, Hargreaves-Samani, Linacre, and Thornthwaite, which were compared to the ETo calculated with the Penman-Monteith ASCE (ETo_PM) method, between 1986 and 2015, in eight meteorological stations. The performance of the alternative methods was obtained from the coefficient of determination (R²), index “d” of agreement, index “c” of performance, and root mean square error (RMSE). The Hargreaves-Samani method has a better performance in estimating the ETo for the main climatic types in the state of Paraná. The Camargo method allows smaller errors between the standard values of ETo, obtained with the Penman-Monteith method, and the estimated values. The methods of Thornthwaite, Linacre, and Budyko are not adequate to estimate the ETo in any climatic type of the state of Paraná, Brazil.

Index terms:
climate variability; irrigation management; missing climate data; Penman Monteith; simplified models; water balance

Authorship

Bruno César Gurski

Universidade Federal do Paraná, Rua dos Funcionários, no 1.540, CEP 80035-050 Curitiba, PR, Brazil. E-mail: brunogurski@ufpr.br, jmoretti@ufpr.brUniversidade Federal do ParanáBrazilCuritiba, PR, BrazilUniversidade Federal do Paraná, Rua dos Funcionários, no 1.540, CEP 80035-050 Curitiba, PR, Brazil. E-mail: brunogurski@ufpr.br, jmoretti@ufpr.br

Daniela Jerszurki

Jacob Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, 84990, Negev, Israel. E-mail: djerszurki@ufpr.brBen-Gurion University of the NegevIsraelNegev, IsraelJacob Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, 84990, Negev, Israel. E-mail: djerszurki@ufpr.br

Jorge Luiz Moretti de Souza

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables | Formulas

Figures (1)
Tables (3)
Formulas (5)

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Figure 1.
Location of weather stations in the state of Paraná, Brazil, in the main climate types of the state. Source: Adapted from INMET (2016)INMET. Instituto Nacional de Meteorologia. BDMEP - Banco de dados meteorológicos para ensino e pesquisa. 2016. Available at: <Available at: http://www.bdmpe.inmet.br/ >. Accessed on: Jun. 22 2016.
http://www.bdmpe.inmet.br/... .

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Table 1.
Interpretation criteria of the alternative methods performance, according to c index proposed by Camargo & Sentelhas (1997)CAMARGO, A.P. de; SENTELHAS, P.C. Avaliação do desempenho de diferentes métodos de estimativa da evapotranspiração potencial no Estado de São Paulo, Brasil. Revista Brasileira de Agrometeorologia, v.5, p.89-97, 1997..

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Table 2.
Parameters used to compare the performance of the alternative methods for determination of reference evapotranspiration (ETo), in relation to Penman-Monteith standard method (ETo_PM), for the climate type Cfa, in state of Paraná, Brazil⁽¹⁾.

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Table 3.
Parameters used to compare the performance of the alternative methods for determination of reference evapotranspiration (ETo), in relation to Penman-Monteith standard method (ETo_PM), for the climate type Cfb, in state of Paraná, Brazil⁽¹⁾.

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Figure 1. Location of weather stations in the state of Paraná, Brazil, in the main climate types of the state. Source: Adapted from INMET (2016)INMET. Instituto Nacional de Meteorologia. BDMEP - Banco de dados meteorológicos para ensino e pesquisa. 2016. Available at: <Available at: http://www.bdmpe.inmet.br/ >. Accessed on: Jun. 22 2016.
http://www.bdmpe.inmet.br/... .

Table 1. Interpretation criteria of the alternative methods performance, according to c index proposed by Camargo & Sentelhas (1997)CAMARGO, A.P. de; SENTELHAS, P.C. Avaliação do desempenho de diferentes métodos de estimativa da evapotranspiração potencial no Estado de São Paulo, Brasil. Revista Brasileira de Agrometeorologia, v.5, p.89-97, 1997..

C value	Performance
c > 0.85	Great
0.85 ≤ c < 0.75	Very good
0.75 ≤ c < 0.65	Good
0.65 ≤ c < 0.60	Average
0.60 ≤ c < 0.50	Tolerable
0.50 ≤ c < 0.40	Bad
c ≤ 0.40	Very Bad

Table 2. Parameters used to compare the performance of the alternative methods for determination of reference evapotranspiration (ETo), in relation to Penman-Monteith standard method (ETo_PM), for the climate type Cfa, in state of Paraná, Brazil⁽¹⁾.

Parameter	Penman-Monteith	Budyko	Camargo	Hargreaves & Samani	Linacre	Thornthwaite
Campo Mourão
Average	3.14	4.07	2.84	4.08	2.69	1.50
Linear coefficient		-1.37	0.45	-0.24	-0.25	1.56
Angular coefficient		1.11	0.95	0.83	1.26	1.05
R²		0.45	0.57	0.78	0.63	0.77
r		0.67	0.75	0.88	0.79	0.88
Index “d”		0.88	0.93	0.92	0.92	0.76
Index “c”		0.59	0.70	0.82	0.72	0.66
Performance		Tolerable	Good	Very good	Good	Good
RMSE		1.35	0.91	1.15	0.94	1.76
Londrina
Average	2.23	4.30	3.00	4.17	3.33	1.82
Linear coefficient		-1.33	0.27	-0.16	0.56	1.71
Angular coefficient		0.88	0.73	0.62	0.65	0.82
R²		0.26	0.35	0.43	0.29	0.64
r		0.51	0.59	0.65	0.54	0.80
Index “d”		0.74	0.63	0.60	0.66	0.62
Index “c”		0.38	0.37	0.39	0.35	0.50
Performance		Very bad	Very bad	Very bad	Very bad	Bad
RMSE		2.17	1.23	2.06	1.33	1.57
Maringá
Average	3.06	4.44	3.10	4.05	3.84	2.01
Linear coefficient		-1.63	0.44	-0.01	0.86	1.92
Angular coefficient		1.10	0.91	0.81	0.63	0.77
R²		0.43	0.56	0.70	0.40	0.65
r		0.66	0.75	0.84	0.63	0.80
Index “d”		0.88	0.69	0.72	0.68	0.62
Index “c”		0.58	0.52	0.60	0.43	0.50
Performance		Tolerable	Tolerable	Tolerable	Bad	Bad
RMSE		1.52	0.86	1.08	1.24	1.65
Paranaguá
Average	1.95	4.32	3.00	3.41	2.45	1.31
Linear coefficient		-1.63	0.16	0.07	-0.10	1.62
Angular coefficient		0.90	0.70	0.65	0.96	0.87
R²		0.41	0.53	0.60	0.43	0.67
r		0.64	0.73	0.78	0.66	0.82
Index “d”		0.74	0.76	0.74	0.74	0.58
Index “c”		0.47	0.55	0.58	0.49	0.47
Performance		Bad	Tolerable	Tolerable	Bad	Bad
RMSE		2.20	1.07	1.39	0.80	1.60

⁽¹⁾
Data from 1986 to 2015.

Table 3. Parameters used to compare the performance of the alternative methods for determination of reference evapotranspiration (ETo), in relation to Penman-Monteith standard method (ETo_PM), for the climate type Cfb, in state of Paraná, Brazil⁽¹⁾.

Parameter	Penman-Monteith	Budyko	Camargo	Hargreaves & Samani	Linacre	Thornthwaite
Castro
Average	2.86	3.41	2.36	3.63	2.27	0.93
Linear coefficient		-0.33	0.73	0.04	0.09	2.15
Angular coefficient		0.95	0.92	0.79	1.24	1.11
R²		0.32	0.45	0.58	0.42	0.32
r		0.56	0.67	0.76	0.65	0.57
Index “d”		0.76	0.72	0.78	0.68	0.51
Index “c”		0.43	0.48	0.59	0.44	0.29
Performance		Bad	Bad	Tolerable	Bad	Very bad
RMSE		1.21	1.13	1.17	1.21	2.49
Curitiba
Average	2.67	3.48	2.42	3.46	2.50	1.06
Linear coefficient		-0.28	0.65	0.53	0.66	1.62
Angular coefficient		0.85	0.84	0.62	0.81	1.09
R²		0.40	0.59	0.57	0.38	0.59
r		0.64	0.77	0.75	0.61	0.77
Index “d”		0.88	0.63	0.62	0.55	0.48
Index “c”		0.56	0.49	0.47	0.33	0.37
Performance		Tolerable	Bad	Bad	Very bad	Very bad
RMSE		1.16	0.73	1.16	0.87	1.85
Irati
Average	2.30	3.46	2.43	3.60	2.45	1.09
Linear coefficient		-0.47	0.58	0.02	-0.21	1.56
Angular coefficient		0.93	0.89	0.75	1.17	1.22
R²		0.38	0.53	0.70	0.63	0.58
r		0.61	0.73	0.84	0.79	0.76
Index “d”		0.87	0.72	0.77	0.66	0.55
Index “c”		0.53	0.52	0.64	0.52	0.42
Performance		Tolerable	Tolerable	Median	Tolerable	Bad
RMSE		1.21	0.90	1.15	0.75	1.96
Ivaí
Average	3.12	3.65	2.55	3.82	2.60	1.27
Linear coefficient		-0.52	0.76	0.06	0.16	1.93
Angular coefficient		1.00	0.93	0.81	1.15	1.07
R²		0.37	0.51	0.71	0.51	0.52
r		0.61	0.71	0.84	0.71	0.72
Index “d”		0.74	0.71	0.78	0.64	0.56
Index “c”		0.45	0.51	0.66	0.46	0.40
Performance		Bad	Tolerable	Good	Bad	Very bad
RMSE		1.17	1.10	1.02	1.09	2.21

⁽¹⁾
Data from 1986 to 2015.

E T o_{P M} = \frac{0.408 \times Δ \times (R_{n} - G) + γ_{p s y} \times \frac{C_{n}}{(T + 273)} \times u_{2} \times (e_{s} - e_{a})}{Δ + γ_{p s y} \times (1 + C_{d} \times u_{2})}

Q o_{i (m m d a y^{- 1})} = R_{a} / 2.45

E T o_{L i} = \frac{700 \times \frac{T + 0.006 z}{100 - φ} + 15 (T - T d_{i})}{80 - T}

\begin{array}{l} E T o_{H i} = \frac{N_{i}}{12} \times \frac{1}{30} \times 16 \times {(\frac{10 \times T}{I})}^{a}, for: T > 0 º C \\ a = 6.75 x 10^{- 7} \times I^{3} - 7.71 x 10^{- 5} \times I^{2} + 1.7912 x 10^{- 2} \times I + 0.49239 \\ I = {\sum_{i = 1}^{12} (0.2 \times T_{a})}^{1.514}, f o r : T_{a} > 0 º C \end{array}

\begin{array}{l} d = 1 - [\frac{\sum_{i = 1}^{n} {(E_{i} - O_{i})}^{2}}{\sum_{i = 1}^{n} {(| E_{i} - {\bar{O}}_{i} | + | O_{i} - {\bar{O}}_{i} |)}^{2}}] \\ c = | R \times d | \\ R M S E = \sqrt{\frac{1}{n} \times \sum_{i = 1}^{n} {(E_{i} - O_{i})}^{2}} \end{array}

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Alternative reference evapotranspiration methods for the main climate types of the state of Paraná, Brazil

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[TFN1] ⁽¹⁾
Data from 1986 to 2015.