Evapotranspiration with the Moretti-Jerszurki-Silva model for the Brazilian sub-tropical climate

Figures & data

Figure 1. Predominant climate types in Paraná State and location of weather stations. Source: Adapted from Paraná Agronomic Institute (IAPAR 2019); adapted from de Brazilian Institute of Geography and Statistics (IBGE 2010)

Figure 2. Seasonal average variation of the climatic variables of 15 and 10 weather stations in Cfa and Cfb climate types, respectively, in Paraná State, for the period 1 December 2016 to 8 November 2018: (a) air temperature (T; °C); (b) relative humidity (RH; %); (c) incident solar radiation (Rs; MJ m^–2 h^–1); (d) wind speed at 2 m height (u₂; m s^–1)

Table 1. Calibration of the MJS model for 25 weather stations located in Cfa and Cfb climate types, in Paraná State, for the period 1 December 2016 to 1 December 2017: linear coefficient (“a”); angular coefficient (“b”); correlation coefficient (r); and number of data in the analyses (n)

Weather station	a	b	r	n
	(mm h^–1)	(mm h^–1 MPa^–1)	(dimensionless)
Cfa climate
Campina da Lagoa	0.0328	−3.1307E-09	0.37	7487
Cidade Gaúcha	−0.0079	−2.9987E-09	0.57	7093
Diamante do Norte	0.0162	−3.0102E-09	0.55	8509
Dois Vizinhos	0.0278	−2.4630E-09	0.55	6524
Foz do Iguaçu	0.0098	−3.9421E-09	0.68	6422
Icaraíma	0.0197	−2.6140E-09	0.51	7651
Japirá	0.0153	−2.6234E-09	0.51	7496
Joaquim Távora	−0.0024	−3.2096E-09	0.58	8389
Marechal Cândido Rondon	0.0020	−3.5451E-09	0.69	7993
Maringá	0.0034	−2.7126E-09	0.56	8371
Morretes	0.0008	−6.9641E-09	0.77	4045
Nova Fátima	0.0085	−4.1062E-09	0.73	6209
Nova Tebas	0.0151	−3.0851E-09	0.57	8417
Paranapoema	−0.0133	−4.0179E-09	0.68	7182
Planalto	0.0208	−3.1454E-09	0.61	7462
Average for Cfa climate	0.0099	−3.4379E-09	0.60	7283
Cfb climate
Castro	−0.011	−3.486E-09	0.71	8614
Clevelândia	−0.004	−2.990E-09	0.69	4629
Colombo	−0.009	−4.580E-09	0.76	8378
Curitiba	−0.024	−3.595E-09	0.66	5697
General Carneiro	−0.012	−4.610E-09	0.78	8161
Inácio Martins	−0.002	−5.535E-09	0.73	5404
Ivaí	−0.011	−3.674E-09	0.68	8053
Laranjeiras do Sul	−0.009	−4.139E-09	0.73	7461
São Mateus do Sul	−0.001	−3.868E-09	0.73	7405
Ventania	0.026	−3.165E-09	0.58	7042
Average for Cfb climate	−0.006	−3.964E-09	0.71	7084

Figure 3. Spatialization of “a” and “b” coefficients of the Moretti-Jerszurki-Silva model, in Paraná State: (a) linear coefficient (“a”; mm h^–1); (b) angular coefficient (“b”; mm h^–1 MPa^–1)

Table 2. Seasonal averages^a of ETo_MJS.h and ETo_h (mm h^–1) of 25 weather stations in Paraná State, for the period 2 December 2017 to 8 November 2018

Weather station	Spring		Summer		Autumn		Winter		Annual average
	EToMSJ.h	EToh	EToMSJ.h	EToh	EToMSJ.h	EToh	EToMSJ.h	EToh	EToMSJ.h	EToh
	(mm h^–1)
Cfa climate
Campina da Lagoa	0.10	0.10	0.10	0.12	–	–	0.10	0.07	0.10	0.10
Cidade Gaúcha	0.15	0.14	0.09	0.14	–	–	0.20	0.11	0.15	0.13
Diamante do Norte	0.10	0.11	0.09	0.12	–	–	0.10	0.07	0.10	0.10
Dois Vizinhos	0.10	0.11	0.09	0.12	–	–	0.10	0.07	0.10	0.10
Foz do Iguaçu	0.12	0.11	0.11	0.12	–	–	0.12	0.07	0.12	0.10
Icaraíma	0.08	0.09	0.09	0.13	–	–	0.09	0.07	0.09	0.10
Japirá	0.09	0.11	0.08	0.12	–	–	0.09	0.07	0.09	0.10
Joaquim Távora	0.09	0.11	0.08	0.12	–	–	0.09	0.07	0.08	0.10
Marechal Cândido Rondon	0.10	0.11	0.09	0.12	–	–	0.10	0.07	0.10	0.10
Maringá	0.08	0.11	0.07	0.12	–	–	0.08	0.07	0.08	0.10
Morretes	0.19	0.11	0.17	0.12	–	–	0.19	0.07	0.19	0.10
Nova Fátima	0.12	0.11	0.11	0.12	–	–	0.12	0.07	0.12	0.10
Nova Tebas	0.12	0.11	0.11	0.12	–	–	0.12	0.07	0.12	0.10
Paranapoema	0.10	0.11	0.09	0.12	–	–	0.10	0.07	0.09	0.10
Planalto	0.20	0.22	0.17	0.22	–	–	0.25	0.17	0.20	0.20
Average for Cfa climate	0.12	0.11	0.10	0.13	–	–	0.12	0.08	0.11	0.11
Cfb climate
Castro	0.11	0.12	0.09	0.14	0.08	0.07	0.11	0.08	0.10	0.11
Clevelândia	0.08	0.02	0.07	0.03	–	–	0.08	0.02	0.08	0.02
Colombo	0.12	0.11	0.11	0.12	–	–	0.12	0.07	0.11	0.10
Curitiba	0.08	0.11	0.07	0.12	–	–	0.08	0.07	0.07	0.10
General Carneiro	0.12	0.11	0.11	0.12	–	–	0.12	0.07	0.11	0.10
Inácio Martins	0.15	0.11	0.14	0.12	–	–	0.15	0.07	0.14	0.10
Ivaí	0.12	0.11	0.11	0.12	–	–	0.12	0.07	0.11	0.10
Laranjeiras do Sul	0.11	0.11	0.10	0.12	–	–	0.11	0.07	0.10	0.10
São Mateus do Sul	0.11	0.11	0.10	0.13	–	–	0.11	0.08	0.10	0.10
Ventania	0.11	0.11	0.11	0.12	–	–	0.11	0.07	0.11	0.10
Average for Cfb climate	0.11	0.10	0.10	0.11	0.08	0.07	0.11	0.07	0.11	0.09

^aThe seasons were considered to occur in the following periods: summer begins on 21 December and ends on 20 March; autumn begins on 21 March and ends on 20 June; winter begins on 21 June and ends on 22 September; and spring begins on 23 September and ends on 20 December.

Figure 4. Averages of ETo_MJS.h and ETo_h, at the hourly scale, from weather stations according to the Cfa and Cfb climate types, in Paraná State, for the period 2 December 2017 to 8 November 2018: (a) seasonal average; (b) monthly average

Table 3. Seasonal^a and annual values of Nash-Sutcliffe efficiency (NSE; dimensionless), “d” index (dimensionless), root mean square error (RMSE; mm h^–1) and correlation coefficient (r; dimensionless) between ETo_MJS.h and ETo_h, of 15 and 10 weather stations in Cfa and Cfb climate types, respectively, in Paraná State, for the period 2 December 2017 to 8 November 2018

Weather station				Spring				Summer				Autumn				Winter				Annual average
			NSE	d	RMSE	r	NSE	d	RMSE	r	NSE	d	RMSE	r	NSE	d	RMSE	r	NSE	d	RMSE	r
					Cfa climate
Campina da Lagoa			0.73	0.90	0.03	0.88	0.52	0.81	0.04	0.89	–	–	–	–	−0.45	0.67	0.04	0.71	0.27	0.79	0.04	0.83
Cidade Gaúcha			0.49	0.89	0.05	0.81	0.11	0.76	0.07	0.84	–	–	–	–	−5.40	0.53	0.11	0.73	−1.60	0.73	0.08	0.73
Dois Vizinhos			0.24	0.77	0.06	0.72	−0.67	0.66	0.07	0.76	–	–	–	–	−8.87	0.38	0.06	0.74	−3.10	0.60	0.06	0.68
Diamante do Norte			0.82	0.94	0.02	0.93	0.55	0.85	0.04	0.92	–	–	–	–	0.06	0.79	0.03	0.81	0.48	0.86	0.03	0.83
Foz do Iguaçu			0.80	0.95	0.02	0.93	0.79	0.94	0.03	0.92	–	–	–	–	−1.31	0.68	0.05	0.81	0.09	0.86	0.03	0.83
Icaraíma			0.76	0.91	0.02	0.92	0.47	0.81	0.05	0.93	–	–	–	–	0.38	0.83	0.03	0.81	0.54	0.85	0.03	0.83
Joaquim Távora			0.72	0.92	0.03	0.93	0.33	0.80	0.05	0.92	–	–	–	–	0.44	0.87	0.03	0.81	0.50	0.86	0.04	0.83
Japirá			0.68	0.89	0.03	0.93	0.29	0.77	0.05	0.92	–	–	–	–	0.49	0.85	0.02	0.81	0.49	0.84	0.03	0.83
Marechal Cândido Rondon			0.84	0.95	0.02	0.92	0.56	0.87	0.04	0.92	–	–	–	–	−0.06	0.80	0.04	0.81	0.45	0.87	0.03	0.83
Maringá			0.55	0.86	0.04	0.93	0.06	0.73	0.06	0.92	–	–	–	–	0.06	0.89	0.02	0.81	0.23	0.83	0.04	0.83
Morretes			−2.29	0.68	0.10	0.93	0.42	0.80	0.07	0.92	–	–	–	–	−13.90	0.40	0.13	0.82	−5.26	0.63	0.10	0.84
Nova Fátima			0.75	0.94	0.03	0.93	0.83	0.95	0.02	0.92	–	–	–	–	−1.72	0.65	0.06	0.81	−0.05	0.85	0.04	0.83
Nova Tebas			0.21	0.71	0.14	0.71	−0.01	0.74	0.07	0.81	–	–	–	–	0.19	0.65	0.11	0.53	0.13	0.70	0.11	0.60
Paranapoema			0.80	0.95	0.02	0.92	0.40	0.85	0.04	0.89	–	–	–	–	−0.13	0.80	0.04	0.80	0.36	0.87	0.03	0.82
Planalto			0.68	0.91	0.05	0.86	−2.26	0.50	0.06	0.69	–	–	–	–	−1.51	0.67	0.10	0.79	−1.03	0.69	0.07	0.73
Average for Cfa climate			0.45	0.88	0.04	0.88	0.16	0.79	0.05	0.88	–	–	–	–	−2.12	0.70	0.06	0.77	−0.50	0.79	0.05	0.79
					Cfb climate
Castro			0.74	0.94	0.03	0.9	−0.07	0.73	0.06	0.91	0.71	0.03	0.03	0.65	−0.29	0.81	0.05	0.84	0.27	0.82	0.04	0.83
Clevelândia			−15.54	0.09	0.08	−0.34	−15.88	0.08	0.06	−0.37	–	–	–	–	−12.29	0.10	0.08	−0.16	−14.57	0.09	0.07	−0.29
Colombo			0.77	0.95	0.03	0.93	0.79	0.94	0.03	0.92	–	–	–	–	–1.42	0.69	0.05	0.81	0.05	0.86	0.04	0.89
Curitiba			0.55	0.88	0.04	0.93	0.02	0.76	0.06	0.92	–	–	–	–	0.49	0.89	0.02	0.81	0.36	0.84	0.04	0.89
General C.			0.76	0.95	0.03	0.93	0.8	0.95	0.03	0.93	–	–	–	–	−1.47	0.69	0.05	0.81	0.03	0.86	0.04	0.89
Inácio Martins			0.13	0.86	0.05	0.93	0.74	0.94	0.03	0.93	–	–	–	–	−5.22	0.54	0.09	0.83	−1.45	0.78	0.06	0.90
Ivaí			0.77	0.94	0.03	0.93	0.40	0.84	0.05	0.92	–	–	–	–	0.24	0.85	0.03	0.81	0.47	0.88	0.04	0.89
Laranjeiras do Sul			0.83	0.96	0.02	0.92	0.70	0.91	0.03	0.92	–	–	–	–	−0.05	0.76	0.04	0.81	0.49	0.88	0.03	0.88
São Mateus do Sul			0.86	0.97	0.02	0.93	0.63	0.90	0.04	0.92	–	–	–	–	−0.43	0.76	0.04	0.81	0.35	0.88	0.03	0.89
Ventania			0.84	0.95	0.02	0.93	0.71	0.90	0.03	0.91	–	–	–	–	−0.80	0.69	0.05	0.81	0.25	0.85	0.03	0.88
Average for Cfb climate			−0.93	0.85	0.04	0.87	−1.12	0.80	0.04	0.79	0.71	0.03	0.03	0.65	−2.12	0.68	0.05	0.72	−1.40	0.77	0.04	0.71

^aThe seasons were considered to occur in the following periods: summer begins on 21 December and ends on 20 March; autumn begins on 21 March and ends on 20 June; winter begins on 21 June and ends on 22 September; and spring begins on 23 September and ends on 20 December.

Figure 5. Variation and correlation between ETo_MJS.h and ETo_h, at the hourly scale, of 15 and 10 weather stations in Cfa and Cfb climate types, respectively, in Paraná State, for the period 2 December 2017 to 8 November 2018: (a) hourly ETo without delay adjustment, in Cfa climate; (b) hourly ETo with 2-h delay adjusted, in Cfa climate; (c) hourly ETo without delay adjustment, in Cfb climate; (d) hourly ETo with 2-h delay adjusted, in Cfb climate

Figure 6. Indexes and errors before and after the correction of the delay observed in the correlation between ETo_MJS.h and ETo_h, at the hourly scale, of 25 weather stations, 15 and 10 in climate types Cfa and Cfb, respectively, in Paraná State, for the period 2 December 2017 to 8 November 2018: (a) Nash-Sutcliffe efficiency (NSE; dimensionless); (b) root mean square error (RMSE; mm day^–1); (c) correlation (r; dimensionless)

Table 4. Average daily values of reference evapotranspiration (ETo; mm day^–1) of 24 weather stations in Paraná State, for the period 2 December 2017 to 8 November 2018

Weather station	ETo (mm day^–1)
	ASCE-PM	MJS	HS	PET	PMT
Cfa climate
Campina da Lagoa	2.38	2.14	2.08	2.58	2.10
Cidade Gaúcha	3.22	3.09	2.90	2.10	2.49
Diamante do Norte	2.48	2.21	1.23	1.97	1.36
Dois Vizinhos	4.23	3.77	1.44	4.12	1.49
Foz do Iguaçu	2.43	2.50	1.20	3.20	1.37
Icaraíma	2.27	2.09	1.15	1.89	1.40
Japirá	2.51	2.37	1.22	3.57	1.26
Joaquim Távora	2.49	2.14	1.22	4.55	1.34
Marechal Cândido Rondon	2.46	2.21	1.21	2.91	1.39
Maringá	2.42	2.27	2.14	1.30	2.32
Morretes	2.30	2.40	2.07	4.49	1.86
Nova Fátima	2.40	2.45	2.11	3.58	2.24
Nova Tebas	3.52	3.25	2.71	3.65	2.32
Paranapoema	2.36	2.12	2.11	1.31	2.38
Planalto	3.97	3.76	3.14	3.44	2.55
Average for Cfa climate	2.76	2.58	1.86	2.98	1.86
Cfb climate
Castro	2.59	2.19	2.22	3.23	1.94
Colombo	2.59	2.80	1.22	3.85	1.94
Curitiba	2.59	2.30	1.23	3.53	1.73
General Carneiro	1.84	1.92	0.97	3.75	1.18
Inácio Martins	2.41	2.67	1.20	4.49	1.20
Ivaí	2.50	2.71	1.21	5.15	1.17
Laranjeiras do Sul	2.44	2.27	1.19	2.24	1.28
São Mateus do Sul	2.38	2.31	2.06	4.47	1.29
Ventania	2.31	2.39	2.08	5.18	1.85
Average for Cfb climate	2.41	2.40	1.49	3.99	1.51

Figure 7. Daily reference evapotranspiration (ETo; mm day^–1) of 24 weather stations in Paraná State, obtained with the inverse distance weighting (IDW) method, for the following models: (a) ASCE-PM (ETo_d); (b) Moretti-Jerszurki-Silva (ETo_MJS.d); (c) Hargreaves and Samani (ETo_HS.d); (d) modified parametric (ETo_PET.d); and (e) Penman-Monteith temperature (ETo_PMT.d)

Table 5. Daily values of Nash-Sutcliffe efficiency (NSE), root mean square error (RMSE) and correlation coefficient (r; dimensionless) in the correlations between “ETo_MJS.d and ETo_d,” “ETo_HS.d and ETo_d,” “ETo_PET.d and ETo_d” and “ETo_PMT.d and ETo_d” of 15 and nine weather stations in Cfa and Cfb climate types, respectively, in Paraná State, for the period 2 December 2017 to 8 November 2018

Weather station	Interaction between the evaluated models (ETo; mm day^–1)
	EToMJS.d and ETod				EToHS.d and ETod				EToPET.d and ETod				EToPMT.d and ETod
	RMSE	d	r	NSE	RMSE	d	r	NSE	RMSE	d	r	NSE	RMSE	d	r	NSE
	(mm day^–1)	(dimensionless)			(mm day^–1)	(dimensionless)			(mm day^–1)	(dimensionless)			(mm day^–1)	(dimensionless)
Cfa climate
Campina da Lagoa	0.64	0.92	0.96	0.50	0.40	0.94	0.97	0.80	0.46	0.91	0.91	0.74	1.23	0.75	0.89	0.27
Cidade Gaúcha	0.78	0.87	0.80	0.62	0.61	0.93	0.92	0.77	1.39	0.71	0.76	−0.20	1.31	0.75	0.86	0.26
Diamante do Norte	0.47	0.92	0.92	0.74	1.38	0.56	0.96	−1.27	0.75	0.74	0.91	0.33	2.63	0.53	0.81	−1.09
Dois Vizinhos	0.77	0.83	0.81	0.46	2.89	0.41	0.88	−6.61	0.71	0.87	0.76	0.53	2.70	0.52	0.78	−1.42
Foz do Iguaçu	0.37	0.95	0.94	0.85	1.36	0.58	0.96	−1.09	0.86	0.80	0.91	0.17	2.40	0.54	0.79	−0.71
Icaraíma	0.35	0.92	0.94	0.77	1.20	0.56	0.95	−1.67	0.55	0.81	0.90	0.45	2.78	0.51	0.60	−0.76
Japirá	0.40	0.95	0.92	0.81	1.42	0.56	0.96	−1.29	1.14	0.71	0.91	−0.47	2.12	0.53	0.77	−1.06
Joaquim Távora	0.50	0.91	0.96	0.71	1.40	0.57	0.96	−1.26	2.12	0.53	0.91	−4.20	2.08	0.54	0.78	−1.15
Marechal C. Rondon	0.48	0.92	0.92	0.74	1.38	0.57	0.96	−1.18	0.60	0.87	0.91	0.59	2.45	0.54	0.82	−0.82
Maringá	0.45	0.94	0.90	0.78	0.37	0.96	0.98	0.85	1.32	0.56	0.89	−0.88	1.16	0.81	0.90	0.43
Morretes	0.32	0.97	0.94	0.87	0.35	0.96	0.97	0.85	2.24	0.51	0.92	−5.28	0.98	0.78	0.86	0.38
Nova Fátima	0.33	0.96	0.94	0.86	0.38	0.95	0.98	0.82	1.24	0.68	0.91	−0.95	1.35	0.75	0.88	0.36
Nova Tebas	0.44	0.94	0.94	0.79	0.90	0.78	0.94	0.09	0.43	0.94	0.90	0.79	1.54	0.69	0.86	−0.01
Paranapoema	0.43	0.93	0.92	0.76	0.37	0.95	0.97	0.82	1.22	0.57	0.91	−0.95	1.37	0.77	0.9	0.34
Planalto	0.37	0.96	0.95	0.85	0.84	0.83	0.95	0.22	0.61	0.89	0.90	0.59	1.49	0.71	0.87	0.01
Average	0.47	0.93	0.92	0.74	1.02	0.74	0.95	−0.61	1.04	0.74	0.89	−0.58	1.84	0.65	0.82	−0.33
Cfb climate
Castro	0.78	0.71	0.90	0.39	0.46	0.94	0.97	0.79	0.75	0.58	0.93	0.43	0.79	0.84	0.90	0.56
Colombo	0.57	0.92	0.88	0.74	1.55	0.56	0.96	−0.95	1.36	0.70	0.88	−0.51	1.63	0.55	0.39	0.15
Curitiba	0.48	0.92	0.94	0.76	1.49	0.57	0.97	−1.34	1.01	0.77	0.87	−0.08	1.74	0.56	0.85	−0.74
General Carneiro	0.28	0.98	0.98	0.94	1.37	0.63	0.99	−0.42	1.85	0.69	0.97	−1.58	1.56	0.59	0.89	−0.47
Inácio Martins	0.42	0.95	0.94	0.79	1.34	0.58	0.96	−1.06	2.13	0.54	0.91	−4.23	1.67	0.56	0.83	−0.40
Ivaí	0.43	0.95	0.92	0.80	1.43	0.57	0.96	−1.25	2.71	0.47	0.91	−7.08	1.89	0.55	0.85	−0.95
Laranjeiras do Sul	0.38	0.94	0.93	0.80	1.35	0.57	0.96	−1.50	0.44	0.91	0.92	0.74	1.96	0.56	0.86	−0.72
São Mateus do Sul	0.40	0.95	0.91	0.81	0.42	0.94	0.97	0.79	2.15	0.54	0.92	−4.46	3.26	0.52	0.63	−0.08
Ventania	0.35	0.95	0.93	0.84	0.34	0.96	0.97	0.85	2.92	0.43	0.91	−10.08	1.07	0.76	0.88	0.31
Average	0.45	0.92	0.93	0.76	1.08	0.70	0.97	−0.45	1.70	0.63	0.91	−2.98	1.73	0.61	0.79	−0.26

Figure 8. Simple linear regression analysis of the values of daily reference evapotranspiration (ETo; mm day^–1) of 24 weather stations in Paraná State, for the period 2 December 2017 to 8 November 2018, for the correlation between: (a) ETo_MJS.d and ETo_d; (b) ETo_HS.d and ETo_d; (c) ETo_PET.d and ETo_d; (d) ETo_PMT.d and ETo_d

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