Abstract

In all areas related to water resources engineering and other sciences, precise evapotranspiration estimation is crucial for water management sustainability and efficient irrigation process. This research includes a simplified procedure for predicting the reference evapotranspiration (ETo) via the FAO-56 Penman-Monteith Procedure, as the original procedure is complicated and requires many steps to compute ETo. A set of empirical equations were introduced to calculate the main parameters of ETo with almost the same accuracy. CLIMWAT was considered to attain climate data for number of countries. These data were fed to CROPWAT to calculate ETo values. The calculated values of ETo were used to estimate the coefficients of the proposed equations. The coefficient of determination, R², was 0.9558 for the proposed model. A trial was conducted for improving the performance of the suggested model by modifying the coefficients of the original FAO-56 ETo equation, so the R² became 0.9634. To evaluate the performance of the proposed procedure, CLIMWAT was also used to obtain climate data for several countries other than those used for derivation, and a good agreement was obtained (R² = 0.9506). The proposed procedure requires only seven steps to calculate ETo, while the original procedure requires 22 steps, so the calculations were reduced by 68%, which can be considered very useful, especially for academic purposes.

1 Introduction

Reference evapotranspiration (ETo) is among the most imperative aspects in designing and managing irrigation systems (Berti et al. 2014), and water reservoirs (Zarei et al. 2015), irrigation structures’ scheme (Fooladmand 2012), and meteorological and hydrological surveys (Kisi 2014). The evapotranspiration process is affected by land use, crop type, climatic factors, plant traits, and management practices including irrigation and fertilization (Majeed et al. 2017; Jerin et al. 2021; Wang et al. 2022). Evapotranspiration could be estimated via several methods including direct measurements, such as lysimeters and evaporation pans (Xu and Chen 2005; Rana and Katerji 2009; Valipour 2014), empirical equations (Fisher and Pringle 2013), physically-based approaches (Zeleke and Wade 2012), neural network approaches (Kisi 2007; Aytek et al. 2008), and machine learning algorithms (Yang et al. 2021). Due to the costly manufacturing of lysimeters, experimental ETo approaches are generally preferred. As stated by many researchers, the Penman-Monteith (PM) ETo procedure can be considered as a reference technique against other experimental ETo approaches (Azhar and Perera 2011; da Silva et al. 2011; Tabari et al. 2013; Awchi and Hasan 2013; DeJonge et al. 2015; Altalib et al. 2021; Sholagberu et al. 2022). Regarding design and management of water resources, control and irrigation efficiency, and proper irrigation timing, the precision of predicted values of ETo methods is important, which led researchers to assess the capability of various ETo methods in different regions in the world to identify suitable ETo methods in certain regions (Gocić et al. 2015). Hafeez et al. (2020) assessed the accuracy of Penman and Thornthwaite ETo methods against the standard Penman-Monteith ETo method in the semi-arid region of Faisalabad, Lahore, and Peshawar. According to some statistical indices considered, they found that the Penman ETo method overestimated ETo in comparison with the PM ETo method in all the sites. On the other hand, they concluded that the Thornthwaite ETo method underestimated ETo in the winter season and overestimated ETo in the summer season of the studied sites compared to the PM ETo method. They also noted that, in general, the Thornthwaite ETo method resulted in better predictions of ETo than the Penman ETo method at all the considered sites.

Yasin and Abdullah (2020) presented sets of empirical equations to estimate ETo for Mosul city in Iraq. These sets of equations included ETo as a dependent variable, and at least two climatic factors or more as dependent variables. They used climatic data of Mosul from 1980–2008, which included relative humidity, temperature, wind speed, and sunshine to estimate ETo using the ETo calculator. Then, they used nonlinear regression to develop seven empirical equations to predict ETo with high values of coefficient of determination (R²). They concluded that air temperature is a significant factor and must be included in the empirical equations because the only equation that showed slightly lower agreement was the equation of dependent variables free of air temperature.

Jerin et al. (2021) studied the spatiotemporal changes in ETo depending on the number of techniques such as empirical Bayesian Kriging model, cross-wavelet transform (XWT), Morlet wavelet analysis (MWA), and the Mann-Kendall test. The partial correlation coefficient (PCC) and stepwise linear regression analysis were considered to identify the driving factors of ETo changes. They used daily meteorological data for the period 1980–2017 for 18 climatological stations in Bangladesh. They utilized the FAO-56 Penman-Monteith method to compute ETo owing to its suitability for a variety of cases of climate, geographic position, and elevation without calibration and validation. They concluded that there was a reduction in yearly and seasonal ETo in Bangladesh, which is the same as too many countries in the world. This decrease was mainly owing to the reduction in sunshine duration and wind speed, although there was an increase in mean temperature. From the spatial analysis, it was found that the lowest annual values of ETo were in the northwest, while the highest ones were in the southwest. The chief factors responsible for ETo variations were relative humidity, mean temperature, and sunshine duration.

Da Silva et al. (2021) examined the performance of five empirical approaches (Hargreaves-Samani, Camargo, Thornthwaite, Priestley-Taylor, and Jensen-Haise) of calculating ETo in the semi-arid region of Petroquina, northeastern Brazil, in comparison with the standard Penman-Monteith method (PM-FAO-56) using daily meteorological data for the period 2004–2013. They concluded that empirical methods cannot completely replace the Penman-Monteith method, but they provide effective alternatives in the case of data shortages, making them valuable tools for agricultural water management sustainability in semi-arid environments.

Hadria et al. (2021) conducted a study to compare the performance of five simple temperature-dependent reference evapotranspiration (ETo) models with the standard FAO Penman-Monteith model in arid and semi-arid regions of Morocco, using climate data from 22 meteorological stations. The results exhibited that solar radiation, mean, and maximum temperatures are the main factors influencing ETo. Among the considered models, the Dorji model was the most accurate and stable compared to the other models and was improved through advanced calibration of its parameters to suit Moroccan climatic conditions, resulting in the development of a new version called ETo-Hadria. The improved model revealed high accuracy in lowland and flat areas, while the original Dorji model was more suitable for mountainous areas. The improved model aimed to support improved water resources management in arid and semi-arid regions.

Huerta et al. (2022) presented a new FAO-56 Penman-Monteith ETo gridded dataset named “PISCOeo_pm”. To create PISCOeo_pm, the FAO-56 Penman-Montieth technique and the years 1981–2016 were taken into consideration, with a spatial resolution of roughly 1 km (0.01°) for the entire continental Peruvian region. Various steps were followed to manage the sub-variables (sunshine duration, minimum and maximum air temperatures, wind speed, and dew point temperature) of ETo viz (1) quality control, (2) gap-filling, (3) homogenization, and (4) spatial interpolation. Because PISCOeo_pm had a greater spatial resolution and an estimative capacity above the mean shown using daily and monthly data, it was determined to be more accurate than the gridded products CRU_TS, TerraClimate, and ERAS-Land.

Duan et al. (2023) combined the effect of land surface soil moisture on evaporation and root zone soil moisture on transpiration, as well as climatological and energy factors when applying the Penman-Monteith-Leuning ET model for better simulations of global ET for the period 2001–2015. Thirteen flux locations around the world were used to validate the proposed approach, which showed significant enhancements in agreement with the measured values. They determined that their study was just a starting point, and that further development of the model was possible.

It is evident based on the abovementioned review that almost all the studies related to evapotranspiration depend directly or indirectly on the FAO-56 Penman-Montieth procedure (FPM), whether to estimate ETo or to compare its performance with other methods. Therefore, a trial was conducted in this research to simplify the original FPM procedure, as it is complicated and requires many steps to calculate ETo, which can save both time and resources. A total of 22 steps is required to calculate ETo, based on the original FAO-56 procedure, while the enhanced proposed procedure requires only 7 steps. CLIMWAT was considered to obtain climate data from different countries to use CROPWAT and the original FPM procedure to derive empirical equations to estimate the factors included in ETo calculations. Climate data, other than those used in derivation, were also extracted from CLIMWAT to test the proposed empirical equations to show excellent agreement between the proposed equations and the original procedure.

2 Materials and Methods

2.1 Climatological data

The climatological data considered in the current study were extracted from CLIMWAT 2.0. The climate database CLIMWAT is proposed to be utilized in combination with the computer application CROPWAT 8.0. These software packages were developed by the Food and Agriculture Organization (FAO). CLIMWAT gives CROPWAT access to meteorological data from more than 5000 climate stations across the globe, enabling it to determine crop water needs, irrigation supply, and irrigation scheduling for different crops. Mean relative humidity, mean sunshine hours or solar radiation, mean daily minimum temperature, mean daily maximum temperature, mean wind speed, and monthly total and effective rainfall are all climatic parameters that CLIMWAT provides for the stations in its database. These values are essential for calculating potential evapotranspiration using the Penman-Monteith method. Apart from potential evapotranspiration, every variable is a direct observation or a conversion of an observation. A trial was performed to include data from 1971 to 2000 in the compilation; however, in cases where data for this time frame was unavailable, any recent series with at least 15 years of data and ending after 1975 was included. Despite having at least 15 years of data, some of the series are "broken" (e.g., 1961–70 and 1992–2000). Therefore, CLIMWAT was used to obtain climatological data from seven counties around the world and different stations from each country were selected, so a total of 95 stations were considered (Table 1). These stations covered various climate conditions, elevations, latitudes, longitudes, temperatures, humidities, and wind speeds. Consequently, any proposed equation that can represent this wide range of variety, would be a good equation. To evaluate the performance of the suggested equations, another set of climatological data from other countries and different stations were considered. This set of data consists of eight countries and a total of 67 stations within these countries (Table 2).

Table 1  Stations considered for climatological data from CLIMWAT, used to develop the proposed equations.

No.	Country	Station	No.	Country	Station
1	Australia	Barrow-Island	50	Malaysia	Bintulu
2	Australia	Bidyadanga	51	Malaysia	Cameron-Highlands
3	Australia	Canberra-Airport	52	Malaysia	Kangar
4	Australia	Coonabarabran	53	Malaysia	Kota-Bharu
5	Australia	Cunnamulla	54	Malaysia	Kota-Kinabalu
6	Australia	Darwin-Airport	55	Malaysia	Kuala-Kangsar
7	Australia	Eucla	56	Malaysia	Kuala-Trengganu
8	Australia	Liverpool	57	Malaysia	Kuantan
9	Australia	Longreach-Airport	58	Malaysia	Kuching
10	Australia	Macquarie-Island	59	Malaysia	Malacca
11	Australia	Monto-P_O_	60	Malaysia	Port-Dickson
12	Australia	Oenpelli	61	Malaysia	Sandakan
13	Australia	Ongerup	62	Malaysia	Sitiawan
14	Australia	Port-Lincoln	63	Malaysia	Tawau
15	Australia	Smoky-Cape	64	Slovenia	Ljubljana-Bezigrad
16	Australia	Swansea	65	Slovenia	Maribor-Mesto
17	Australia	Tarcoola	66	Slovenia	Murska-Sobota
18	Australia	Willis-Island	67	Slovenia	Nova-Gorica
19	Australia	Yuendumu	68	Slovenia	Novo-Mesto
20	Iraq	Amarah	69	Slovenia	Ratece-Planica
21	Iraq	Baghdad	70	Turkey	Adiyaman
22	Iraq	Basrah	71	Turkey	Afyon
23	Iraq	Habbaniyah-Lake	72	Turkey	Ankara-Central
24	Iraq	Kanaqin	73	Turkey	Ankara-Esenboga
25	Iraq	Mosul	74	Turkey	Bingol
26	Iraq	Nukaib	75	Turkey	Dogubeyazit
27	Iraq	Rutbah	76	Turkey	Edirne
28	Iraq	Salahaddin	77	Turkey	Hakkari
29	Iraq	Shaibah	78	Turkey	Inebolu
30	Iraq	Sinjar	79	Turkey	Iskenderun
31	Iraq	Sulaimaniya	80	Turkey	Kartal
32	Japan	Chichijima	81	Turkey	Kayseri-Erkilet
33	Japan	Fukuyama	82	Turkey	Mardin
34	Japan	Hiroo	83	Turkey	Menemen
35	Japan	Izuhara	84	Turkey	Milas
36	Japan	Minamitorishima	85	Turkey	Polatli
37	Japan	Miyakejima	86	Turkey	Sinop
38	Japan	Mutsu	87	Turkey	Tortum
39	Japan	Nara	88	Turkey	Yozgat
40	Japan	Nemuro	89	Yemen	Aden-Khormaksar
41	Japan	Saga	90	Yemen	Amran-Minjida
42	Japan	Saigo	91	Yemen	El-Kod-Research-C_
43	Japan	Shizuoka	92	Yemen	Jumeisha
44	Japan	Suttsu	93	Yemen	Riyan
45	Japan	Utsunomiya	94	Yemen	Sana_A
46	Japan	Wajima	95	Yemen	Wadi-Zabid
47	Japan	Wakkanai
48	Japan	Yamagata
49	Japan	Yonagunijima

Table 2  Stations considered for climatological data from CLIMWAT, used to assess the proposed equations.

No.	Country	Station	No.	Country	Station
1	Egypt	Alexandria-Nouzha	35	Mexico	Merida-Aero
2	Egypt	Asswan	36	Mexico	Piedras-Negras-Coah_
3	Egypt	Baharia	37	Mexico	Puerto-Cortes
4	Egypt	Dakhla	38	Mexico	San-Luis-Potosi-S_L_
5	Egypt	Hurguada	39	Mexico	Soto-La-Marina-Tamps
6	Egypt	Mallawi	40	Mexico	Tapachula-Chis
7	Egypt	Mersa-Matruh	41	Mexico	Univ_-De-Chihuahua-C
8	Egypt	Port-Said-El-Gamil	42	Moroco	Casablanca
9	Egypt	Salloum	43	Moroco	Kasba-Tadla
10	Egypt	Shandaweel	44	Moroco	Meknes
11	Egypt	Siwa	45	Moroco	Ouarzazate
12	France	Auxerre	46	Moroco	Oujda
13	France	Biarritz	47	Moroco	Tan-Tan
14	France	Boulogne	48	Netherlands	De-Bilt
15	France	Bourges	49	Netherlands	De-Kooy
16	France	Brest	50	Netherlands	Eindhoven
17	France	Cazaux	51	Netherlands	Groningen-Ap-Eelde
18	France	Chateauroux	52	Netherlands	Maastricht-Ap-Zuid-L
19	France	Grenoble-St-Geoirs	53	Netherlands	Twenthe
20	France	Le-Mans	54	Netherlands	Vlissingen
21	France	Limoges	55	Pakistan	Astore
22	France	Marseille-Marignane	56	Pakistan	Chaman
23	France	Metz-Frescaty	57	Pakistan	Islamabad-Airport
24	France	Nice	58	Pakistan	Karachi-Manora
25	France	Nimes-Courbessac	59	Pakistan	Nawabshah
26	France	Strasbourg	60	Pakistan	Nokkundi
27	France	Toulon	61	Pakistan	Peshawar
28	Mexico	Acapulco-Gro_	62	Pakistan	Zhob
29	Mexico	Cozumel-Intl-Arpt	63	Singapore	Singapore-Changi-Air
30	Mexico	Durango-Dgo_	64	Singapore	Singapore-Paya-Lebar
31	Mexico	Ensenada	65	Sweden	Basel-Binningen
32	Mexico	Guanajuato-Gto_	66	Sweden	Geneve-Cointrin
33	Mexico	Hermosillo-Son_	67	Sweden	Zurich-Kloten
34	Mexico	Lagos-In-Jalisco

2.2 Derivation of the proposed procedure

FAO-56 Penman-Monteith Procedure for estimating ETo

The monthly reference evapotranspiration as stated by FAO-56 (Allen et al. 1998) can be computed via Equation 1:

(1)

Where:

ETo	=	reference evapotranspiration (mm.day-1),
Rn	=	crop surface net radiation (MJ.m-2.day-1),
G	=	density of soil heat flux (MJ.m-2.day-1),
T​	=	average daily air temperature at a height of two metres (°C),
u2	=	wind speed at 2 metres height (m.s-1),
es	=	saturation vapor pressure (kPa),
ea	=	actual vapor pressure (kPa),
∆	=	slope of vapor pressure curve (kPa/°C), and
γ	=	psychrometric constant (kPa/°C).

To determine ETo, a series of equations must be followed. As a starting parameter, the atmospheric pressure (P in kPa) can be determined from Equation 2:

(2)

Where:

z

=

elevation above mean sea level (m).

The psychrometric constant (γ) can be calculated from Equation 3:

(3)

Combining Equations 2 and 3 resulted in Equation 4:

(4)

The slope of the saturation vapor pressure curve (∆ in kPa/°C) can be determined from Equation 5:

(5)

Another important parameter is the mean saturation vapor pressure (e_s in kPa) which can be calculated via Equation 6:

(6)

Where:

Tmax and Tmin

=

maximum and minimum air temperatures (°C).

Actual vapor pressure (e_a in kPa) is among the parameters that must be determined. This pressure can be determined via number of equations based on the available climatic data. According to the available data in the current research, Equation 7 is the most suitable one:

(7)

Where:

RH

=

relative humidity (%).

Therefore, the vapor pressure deficit (e_s – e_a) can be computed using Equation 8 (after combining Equations 6 and 7):

(8)

The mean daily air temperature at 2 m height (T, °C) can simply be computed by Equation 9:

(9)

The soil heat flux density (G, MJ/m²/day) can be calculated via Equation 10:

(10)

Where:

Tmonth i and Tmonth i-1

=

mean air temperature of the current month and the previous month, respectively (°C).

The net radiation at the crop surface (R_n in MJ/m²/day) is the most complex and difficult factor to compute. It requires many steps as illustrated below:

Step 1: Find the number of the day in the year (J in days). For monthly calculations, J at the middle of the month is approximately given by Equation 11:

(11)

Where:

M

=

month.

For example, for January M = 1, for February M = 2, and so on. By applying Equation 11 to the months of a year, the following values of J can be obtained (Table 3):

Table 3  Values of the number of the day in the year (J).

Month no. (M)	Month	J
1	January	15
2	February	46
3	March	76
4	April	107
5	May	137
6	June	167
7	July	198
8	August	228
9	September	259
10	October	289
11	November	319
12	December	350

Step 2: Calculate the inverse relative distance Earth–Sun (d_r in rad). This can be determined via Equation 12:

(12)

Step 3: Determine the solar declination (δ in rad). This can be calculated with Equation 13:

(13)

Step 4: Convert the latitude angle (φ in rad) from degree to radian using Equation 14:

(14)

Step 5: Find the sunset hour angle (ω_s in rad) from Equation 15:

(15)

Step 6: Compute the extraterrestrial radiation (R_a in MJ/m²/day) by Equation 16:

(16)

Where:

Gsc

=

solar constant = 0.082 MJ/m2/min.

Step 7: Determine the daylight hours (N in h/day) through Equation 17:

(17)

Step 8: Calculate the solar or shortwave radiation (R_s in MJ/m²/day) by Equation 18:

(18)

Where:

as and bs	=	fraction of extraterrestrial radiation reaching the earth on clear days, their recommended values are 0.25 and 0.5, respectively, and
n	=	actual duration of sunshine in a day (h).

Step 9: Find the net solar or net shortwave radiation (R_ns in MJ/m²/day) via Equation 19:

(19)

Where:

α

=

canopy or albedo reflection coefficient, its value is 0.23.

Step 10: Compute clear-sky solar radiation (R_so in MJ/m²/day) via Equation 20:

(20)

Step 11: Determine net outgoing longwave radiation (R_nl in MJ/m²/day) via Equation 21:

(21)

Where

σ	=	fraction of extraterrestrial radiation reaching the earth on clear days, their recommended values are 0.25 and 0.5, respectively, and
Tmax, K, and Tmin, K	=	maximum and minimum absolute temperatures in Kelvin (K = C° + 273.15).

Step 12: Finally, the net radiation is given by Equation 22:

(22)

Proposed procedure

In this section, a series of simpler equations for calculating different parameters of the FAO-56 Penman-Monteith equation were proposed as follows:

The psychrometric constant (γ): this constant can be calculated from Equation 23 instead of Equation 4:

(23)

Where:

a1 and a2

=

empirical coefficients.

The slope of the saturation vapor pressure curve (∆ in kPa/°C) can be computed via Equation 24 instead of Equation 5:

(24)

Where:

b1 and b2

=

empirical coefficients.

The vapor pressure deficit (e_s – e_a) can be calculated using Equation 25 instead of Equations 6, 7, and 8:

(25)

Where:

c1 – c5

=

empirical coefficients.

The net radiation at the crop surface (R_n in MJ/m²/day) is the most complicated parameter among the other parameters. It requires 12 steps, or 12 equations (Equations 11–22), to be calculated. However, the following equation can be used to estimate this parameter with one step only:

(26)

Where:

d1 – d14

=

empirical coefficients, and the latitude φ is in degrees.

To estimate the coefficients of the proposed equations, an optimization tool called “Microsoft Excel Solver” (available in Excel) was used (Mailapalli et al. 2008). The coefficients (a₁ and a₂) of Equation 23 (the psychrometric constant, γ) were estimated based on values of elevations range from 0–4000 m, with 1 m intervals. The coefficients b₁ and b₂ of Equation 24 (the slope of the saturation vapor pressure curve, Δ) were estimated based on temperature values ranging from -15–50°C, with 0.5°C intervals. The coefficients c₁ – a₅ of Equation 25 (vapor pressure deficit, e_s – e_a), and the coefficients (d₁ – d₁₄) of Equation 26 (net radiation at the crop surface, R_n) were predicted based on the data available in Table 1.

Performance evaluation of the proposed equations

To assess the performance of the suggested equations, some statistical criteria were considered, i.e., root mean square error (RMSE), mean absolute error (MAE), percent bias (PBIAS), and coefficient of determination (R²). These statistical criteria were calculated based on Equations 27–30 (Horn 1993; Willmott 2012):

(27)

(28)

(29)

(30)

Where:

Ei	=	predicted values,
Oi	=	calculated values,
k	=	number of values, and
	=	mean observed value.

3 Results and Discussion

Table 4 illustrates the coefficients of all the proposed equations and the values of statistical indicators. It is obvious from Table 4 that, in general, all the proposed equations performed well due to the high values of the coefficients of determination and the good values of other statistical criteria. It can also be noted that all the available digits in Excel were included in Table 4 to achieve a high accuracy in calculating the parameters when they are required to be calculated by others. For further demonstration, the calculated and predicted values of the parameters of ETo were plotted against the 1:1 line as shown in Figure 1.

Table 4  Values of the coefficients of the proposed equations and the statistical indicators.

Proposed Equation	Coefficients	Coefficient Values	Statistical Indicators
Equation 23	a1	0.0675692467808435	RMSE	8.92E-05
	a2	-0.000121790599803648	MAE	7.68E-05
			PBIAS	1.38E-05
			R2	0.9999
Equation 24	b1	0.0535849374354397	RMSE	0.0077
	b2	0.0493559668438695	MAE	0.0069
			PBIAS	0.0116
			R2	0.9979
Equation 25	c1	0.0100062914197956	RMSE	0.0686
	c2	-0.0364066089985376	MAE	0.0001
	c3	0.0474809272162379	PBIAS	-0.0437
	c4	-0.49478099193758	R2	0.9938
	c5	0.794212219342794
Equation 26	d1	-17.5701012898306	RMSE	1.8917
	d2	0.000459230734374548	MAE	0.0011
	d3	46.0000976625741	PBIAS	1.52E-05
	d4	-0.321686086895198	R2	0.7919
	d5	0.396891774576683
	d6	0.315579030040339
	d7	0.0116303061564504
	d8	0.128739758971131
	d9	55.5111575387937
	d10	2.09726264235546
	d11	0.000914175162392261
	d12	0.00523089954765904
	d13	0.00282250179635253
	d14	0.00890654067136002

Figure 1  Relationship between the calculated and predicted parameters of ETo 
NOTE: c denotes calculated, and p is for predicted.

A high agreement can be noted between the calculated parameters from the original equations in FAO-56 and the predicted parameters from the suggested equations for the psychrometric constant, γ (Figure 1a), the slope of the saturation vapor pressure curve, ∆ (Figure 1b), and the vapor pressure deficit (e_s – e_a) (Figure 1c). However, lower agreement was noted in predicting the net radiation at the crop surface, R_n (Figure 1d) as this parameter requires 12 steps to be calculated from the original procedure, while in the current study, only one step is required to calculate this parameter. Although the proposed equation is complicated and considers all the variables included in the 12 steps, it is still of relatively low performance.

For further assessment of the proposed equations, the reference evapotranspiration ETo was calculated, and their values were compared with the values resulting from the original FAO-56 procedure. The sum of square errors (SSE) between ETo values was calculated and added to the same statistical indicators to evaluate the ETo values resulting from both methods. Table 5 shows the statistical indicators of ETo calculation based on the proposed equations.

Table 5  Statistical indicator values of ETo calculation based on the proposed equations.

SSE	206.071
RMSE	0.4252
MAE	0.3534
PBIAS	-0.0710
R2	0.9559

The high value of the coefficient of determination R², and the good values of other indicators, reflect the good performance of the suggested model. Figure 2 illustrates the relationship between the values of calculated ETo based on FAO-56 procedure and those estimated based on the suggested procedure.

Figure 2  Relationship between values of calculated ETo based on FAO-56 procedure (ETo) and those estimated based on the suggested procedure (ETop). 
NOTE: Plots are based on locations noted in Table 1.

Figure 2 confirms the high performance of the proposed procedure, as all the point are close to the 1:1 line and they are uniformly distributed around it, and there are almost no extreme values. It is likely that the relatively low performance of the proposed equation for estimating the net radiation on crop surface (R_n) is the main cause of the relative disagreement of ETo that is represented by some points that are not so close to the 1:1 line (Figure 2). Therefore, a trial was suggested to enhance the estimated values of ETo and improve the performance of the proposed procedure. This trial includes modifying the constants of the FAO-56 Penman Montieth equation (Equation 1) to be as follows:

(31)

Where:

f1 – f6

=

empirical coefficients.

By doing this, the disagreements in estimating R_n and ETo may be improved. The coefficients were estimated using the same previously mentioned technique. Table 6 illustrates the coefficients f₁ – f₆ and the statistical indicators. Equation 31 will be called the enhanced model of the FAO-56 Penman Montieth equation in this study.

Table 6  Coefficient values of the enhanced model of FAO-56 Penman Montieth and the statistical indicators.

Proposed Equation	Coefficients	Coefficient Values	Statistical Indicators
Equation 31	f1	0.443225165784686	SSE	171.119
	f2	831.192725329481	RMSE	0.3874
	f3	1.02346070456977	MAE	0.3132
	f4	0.944070485096514	PBIAS	0.0656
	f5	1.20270402034287	R2	0.9634
	f6	0.251468741269277

It is clear from Table 6 that the performance of the enhanced model of ETo calculation is better than the previous one, as all the statistical indicators were improved. Figure 3 illustrates the relationship between values of calculated ETo based on the FAO-56 procedure and those estimated based on the suggested enhanced procedure.

Figure 3  Relationship between calculated ETo values based on the FAO-56 procedure (ETo) and those estimated based on the suggested enhanced procedure (ETope).
NOTE: Plots are based on locations noted in Table 1.

The last evaluation of the proposed procedure was done using climate data for other countries, i.e., locations that were not included in developing the proposed procedure. These locations are shown in Table 2. This is the best test of the proposed procedure. Table 7 shows the statistical criteria of the suggested model for only ETo values, and Figure 4 illustrates the relationship between the calculated values ETo based on the FAO-56 procedure and those estimated based on the suggested enhanced procedure.

Table 7  Statistical indicator values of ETo calculation based on the proposed enhanced procedure for different locations that were not included when developing the procedure.

SSE	167.999
RMSE	0.4571
MAE	0.3570
PBIAS	0.0688
R2	0.9506

Figure 4  Relationship between the calculated ETo values based on the FAO-56 procedure (ETo) and those estimated based on the suggested enhanced procedure (ETope) for the climate data of the locations listed in Table 2.

It can be seen that there is an excellent agreement between the calculated and estimated values of ETo. This is due to the good values of the statistical indicators shown in Table 7 and the majority of the points that are close to the 1:1 line illustrated in Figure 4. Based on the coefficient of determination listed in Table 7 which is 0.9506, a very slight error may occur in the calculations of irrigation requirements when depending on the proposed model, which may not exceed 5%.

Although there are some free and user-friendly software for calculating ETo, such as ETo Calculator and CROPWAT, it is vital to know and understand the original procedure, which is lengthy, complicated, and time consuming. Therefore, it is crucial to find a simpler procedure to calculate ETo, like the procedure developed in the current study which can save both time and resources. A total of 22 steps is required to calculate ETo based on the original procedure of FAO-56, while the proposed enhanced procedure requires only seven steps. Thus, for academic purposes, it is easier to decrease the required steps to calculate ETo with insignificant errors that are achieved by considering the proposed procedure, which can reduce the calculations by about 68% compared to the original procedure.

4 Conclusions

Accurate assessment of evapotranspiration is essential in many aspects of water resources engineering and other sciences. As part of this study, a simplified approach was developed for calculating the reference evapotranspiration (ETo) using the FAO-56 Penman-Monteith process because the initial method is intricate and necessitates numerous steps to calculate ETo. To determine the primary ETo parameters with almost the same accuracy, a set of empirical equations was presented. CLIMWAT was used to obtain climatic data for a variety of nations. To determine ETo values, these data were supplied to CROPWAT. The coefficients of the suggested equations were estimated using the computed values of ETo. For the suggested model, the R² (coefficient of determination) was 0.9558. A trial was carried out to enhance the performance of the suggested model by altering the original FAO-56 ETo equation's coefficients, resulting in an R² of 0.9634. CLIMWAT was also utilized to gather climate data for countries other than those used for derivation to evaluate the effectiveness of the suggested technique, and good agreement was found (R² = 0.9506). In comparison to the prior approach, which required 22 steps to calculate ETo, the new procedure only takes 7 steps, resulting in a 68% reduction in calculations. For further research, it is suggested to develop a new empirical formula for calculating ETo by directly relating the basic climatic data (temperature, wind speed, sunshine, and humidity) and location elements (altitude, longitude, and latitude) in one equation without needing to calculate the basic parameters of the FAO-56 Penman-Monteith procedure.

Acknowledgments

This research was conducted at the University of Mosul, College of Engineering, Dams and Water Resources Engineering Department. The author would like to thank all those who provided insight and expertise that greatly assisted the research. The author also appreciates the effort presented by academic staff of the Dams and Water Resources Engineering Department in providing all the requirements for doing the research. The helpful comments of the anonymous reviewers are also very appreciated.

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