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© 2020 Conscientia Beam. All Rights Reserved. 

PENMAN AND THORNWAIT EQUATIONS FOR ESTIMATING REFERENCE 
EVAPOTRANSPIRATION UNDER SEMI-ARID ENVIRONMENT   

 

 

 Muhammad 
Hafeez1+ 
Allah Bakhsh2  
Abdul Basit3 
Zia Ahmad Chattha4  
Alamgir Akhtar Khan5    
Muhammad Adnan 
Majeed6  
Fatima Tahira7 

 

1Department of Agricultural Engineering, Faculty of Agricultural Sciences 
and Technology, Bahauddin Zakariya University, Multan, Pakistan. 

 
2,3Department of Botany, Ghazi University, Dera Ghazi Khan, Pakistan. 

 
4Faculty of Agricultural Engineering, University of Agriculture, Faisalabad, 
Pakistan. 

 
5Department of Agricultural Engineering, MNSUA, Multan, Pakistan. 

 
6University of Waikato, New Zealand. 

 
7Department of Mathematics, Institute of Southern Punjab, Multan, 
Pakistan. 

 

 
(+ Corresponding author) 

 
 
 
 

 

 ABSTRACT 
 
Article History 
Received: 17 January 2020 
Revised: 19 February 2020 
Accepted: 24 March 2020 
Published: 22 April 2020  
 

Keywords 
Penman 
Thornwait 
Penman-Monteith 
Reference evapotranspiration 
Semi-arid 
Environment. 

 
The estimation of reference evapotranspiration (ETo) is required for effective 
development and management of agriculture water systems. In order to define the most 
accurate method to estimate ETo in semi-arid climatic environment of Faisalabad, 
Lahore and Peshawar. Penman and Thornwait ETo methods are compared with 
standard Penman-Monteith (PM) ETo method. The statistical results show that 
Penman ETo method overestimates PM ETo method in all the semi-arid climatic 
regions of Faisalabad, Lahore and Peshawar by 34.91%, 39.51% and 30.75%, 
respectively. The R2 were 0.98, 0.98 and 0.99 at Faisalabad, Lahore and Peshawar 
weather stations, respectively. The RMSE were 2.47 mm/day, 2.64 mm/day and 2.19 
mm/day at Faisalabad, Lahore and Peshawar weather station, respectively. The MBE 
of -2.41 mm/day, -2.58 mm/day and -2.13 mm/day were noted at Faisalabad, Lahore 
and Peshawar weather stations, respectively. The statistical result of Thornwait (Th) 
ETo method as compared with PM ETo method indicates underestimation of ETo in 
winter season and overestimation of ETo in summer season by 13.81%, 22.43% and 
14.54% at Faisalabad, Lahore and Peshawar stations, respectively. The R2 of 0.92, 0.89 
and 0.95 were noted at Faisalabad, Lahore and Peshawar weather stations, respectively. 
The RMSE were 2.14 mm/day, 2.36 mm/day and 1.16 mm/day at Faisalabad, Lahore 
and Peshawar weather stations, respectively. The MBE were -0.68 mm/day, -1.12 
mm/day and 0.61 mm/day at Faisalabad, Lahore and Peshawar weather stations, 
respectively. Overall, Thornwait method gave better estimation of ETo than Penman 
ETo method at all the weather stations.  
 

Contribution/Originality: The main objective of this research is to compare the performance of Penman and 

Thornwait ETo methods against standard PM ETo method under semi-arid climatic conditions of Lahore, 

Faisalabad and Peshawar, Pakistan. 

 

 

Current Research in Agricultural Sciences 
2020 Vol. 7, No. 1, pp. 6-14. 
ISSN(e): 2312-6418 
ISSN(p): 2313-3716 
DOI: 10.18488/journal.68.2020.71.6.14 
© 2020 Conscientia Beam. All Rights Reserved. 

 
 
 

 
 
 
 

 

 
 
 
 

https://orcid.org/0000-0002-3262-7014
https://www.doi.org/10.18488/journal.68.2020.71.6.14


Current Research in Agricultural Sciences, 2020, 7(1): 6-14 

 

 
7 

© 2020 Conscientia Beam. All Rights Reserved. 

1. INTRODUCTION 

Pakistan lies in arid to semi-arid region where average annual rainfall is 254 to 356 mm against a potential 

demand (of water for maximum crop production) of 1778 mm. This gap between the demands and supplies is met 

through applying irrigation. Moreover, the country is facing threat of rapidly increasing population with the annual 

growth rate of 2.05 percent. It has been observed that water availability for agriculture is expected to decline 

globally to 62 percent by 2020 as was available (72%) in 1995 and from 87% to 73% in developing countries  [1]. 

Reference evapotranspiration (ETo) is one of the most significant factor to design and manage water reservoirs [2] 

scheme of irrigation structures [3] effective irrigation management [4] and hydrological and meteorological 

investigations [5]. Types of crop and land use affect the evapotranspiration process [6]. The most accurate ETo 

method for the estimation of ETo is lysimeter [7, 8]. Since lysimeters manufacturing is very expenses, 

experimental ETo methods are generally applied to estimate ETo. Numerous researchers have argued that 

Penmen–Monteith (PM) ETo method can be applied as a reference ETo method as compared to the other 

experimental ETo methods [9-12]. The PM ETo method requires large number of weather parameters i-e air 

temperature, humidity, solar radiation, wind speed etc. But, availability of these weather parameters is not 

accessible at all the weather stations of the world especially in developing country like Pakistan.  

Therefore, it appears reasonably to substitute it by other ETo methods which require small number of weather 

parameters [13]. The accuracy of a particular ETo method depends greatly on the climatic situations of the 

research area [14]. For humid subtropical weather climatic conditions PM ETo method is commonly suggested 

[15, 16].  

Many researchers including [17-20] revealed that temperature and radiation dependent ETo methods lean 

towards the highest and pan-coefficient dependent ETo methods give lowest ETo values. It is concluded that in dry 

and semi-dry climatic conditions solar radiation-dependent ETo methods give poor results [21].  

However, application of regionally modified radiation-dependent ETo methods can give more accurate results 

than air temperature dependent ETo methods and even complex ETo methods [22, 23].  As the accuracy of 

estimated values of ETo by different ETo methods is significant for water resources design and management, 

proper irrigation timing, control and agricultural efficiency; it has given rise to many researchers that were carried 

out in various regions of the globe to determine the most accurate ETo method which is appropriate for estimation 

of ETo in such regions [24]. A study is carried out to compare the various ETo methods including Turc [25] 

Blaney and Criddle [26], Haith and Shoemaker [27], Thornthwaite [28] and Priestley and Taylor [29] ETo 

methods against  standard Penman-Monteith [30] ETo method for the estimation of ETo by applying weather 

parameters of 12 various weather stations. The results of the study indicated that the Turc and PM ETo methods 

showed the most accurate results [31].  

Another research is conducted to evaluate the accuracy of 9 ETo methods against PM ETo method to estimate 

ETo. The conclusion of research showed that the Blaney-Criddle (BC) Eto method indicated the most accurate ETo 

estimation and the Thornthwaite ETo method indicated the poor results of ETo estimation [32]. The main 

objective of this research is to compare the performance of Penman and Thornwait ETo methods against standard 

PM ETo method under semi-arid climatic conditions of Lahore, Faisalabad and Peshawar, Pakistan. 

 

2. MATERIALS AND METHODS 

2.1. Geographical Area and Weather Data Set 

The mean monthly weather data of three weather stations of semi-arid regions (Lahore, Faisalabad and 

Peshawar) is used to estimate reference evapotranspiration (ETo) by Penman and Thornwait ETo methods. The 

GPS (Global Positioning system) coordinates of Lahore are 31.33o N and 74.20o E and height of 214 m from the 

ocean. Lahore sorts  semi-dry climatic conditions. The GPS (Global Positioning System) coordinates of Faisalabad 

are 31.26o N and 73.08o E and elevation of 185.6 meters. The weather of Faisalabad sorts semi-arid climatic 



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© 2020 Conscientia Beam. All Rights Reserved. 

conditions with very warm and moist midsummers and arid cold wintertime. The GPS (Global Positioning System) 

coordinates of Peshawar are 34.02o'N, 71.56o E and elevation of 327 m from the sea. It has warm semi-arid weather 

conditions with very thirsty summers and slight winters-time. The mean monthly weather data period, climate 

conditions and Global Positioning System (GPS) of weather stations used in the study are given in Table 1. 

 
Table-1. Global Positioning System and climate of weather stations of study regions. 

Station Latitude Longitude Elevation (m) Data Period Climate 

Lahore 31.33o N 74.20o E 214.0 2000-2009 hot semi-arid 
Faisalabad 31.26o N 73.08o E 185.6 2001-2010 hot semi-arid 
Peshawar 34.02o N 71.56o E 327.0 2000-2007 hot semi-arid 

 

 

2.2. Methods for Estimation of ETo 

2.2.1. Penman-Monteith (PM) ETo Method 

In this research paper, the Penman-Monteith (PM) ETo method [30] is recommended as the reference ETo 

method for estimation ETo. The accuracy of this ETo method has been proved by many researcher under various 

weather conditions [33-35]. The Penman-Monteith (PM) ETo method presented by Allen, et al. [30] is given as: 

                                                (1) 

Where, ETo is reference crop evapotranspiration (mm/day); Δ is slope of the saturation vapor pressure 

function (kPa (°C) -1 ); Rn is net solar radiations (MJ m-2 day-1); G is earth heat flux thickness (MJ m-2 day-1); T is 

average atmospheric temperature (°C); U2 is the mean 24-hour air velocity at 2m elevation (ms-1); (es-ea) is the vapor 

pressure deficit (kPa); and γ is psychometric constant (kPa (°C)-1) The estimation of all weather data essential for 

Equation 1 for estimation of ETo followed the method of Allen, et al. [30]. 

 

2.3. Thornthwaite Method 

The Thornthwaite ETo method had been developed in 1948 by Thornthwaite [28]. This ETo method is given 

as: 

                                                                                      (2) 

                                                                                           (3)                                                                                           

                                                                                         (4)                                                                                            

Where, N is the maximum number of sunny hours in function of the month latitude; dm is the number of day 

per month; ETgr is the gross evapotranspiration; Tm is the mean temperature (0C); I is the monthly heat index in 

Equations 2,3,4 and 5. 

        (5) 

2.4. Penman Method 

The Penman [36] ETo method is given as: 



Current Research in Agricultural Sciences, 2020, 7(1): 6-14 

 

 
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© 2020 Conscientia Beam. All Rights Reserved. 

                                                   (6) 

Where, ETo is the reference evapotranspiration (mm/day); Δ is slope of the saturation vapor pressure function 

(kPa (°C) -1); Rn is net solar radiations (MJ m-2 day-1); G is earth heat flux thickness (MJ m-2 day-1);  u2 is the mean 

24-hour air velocity at 2m elevation (ms-1); (es-ea) is the vapor pressure deficit (kPa); γ is psychometric constant (kPa 

(°C)-1) and  is  the latent heat of vaporization in MJ kg-1 (  = 2.45 MJ kg-1 at a temperature of 20 oC) in Equation 

6. 

 

2.5. Evaluation Criteria 

In this study, the root mean square error (RMSE) (7), percentage error of estimate (PE) (8), mean bias error 

(MBE) (9) and coefficient of determination (R2) (10) are used for the evaluation of the ETo methods. The RMSE, 

PE, MBE and R2 are defined as: 

                                                                (7) 

                                                                        (8)   

          

                                                                           (9) 

                                     (10) 

Where, Pi are the projected values and Oi are observed values.   is the mean of Pi    and  is the mean of Oi, 

and n is the whole number of values. 

 

3. RESULTS AND DISCUSSION 

The Penman ETo method  and Thornwait ETo method that are temperature dependent ETo methods are 

compared with standard Penman-Monteith  ETo method in different semi-arid climatic regions of Lahore, 

Faisalabad and Peshawar. According to the statistical analysis applied between Penman and PM ETo methods, the 

Penman ETo method indicated overestimation of ETo by 34.91% at Faisalabad weather station as concluded by 

Djaman, et al. [37] as shown in Figure 1 (a) and Table 2. The difference of variation between Penman and PM ETo 

methods has coefficient of determination (R2) of 0.98 with root mean square error (RMSE) of 2.47 mm/day and 

mean bias error (MBE) of -2.41 mm/day at Faisalabad weather station. The statistical results between Thornwait 

ETo method and PM ETo method show that the Thornwait ETo method indicated underestimation in winter and 

overestimation in summer by 13.81% at Faisalabad station as concluded by Pereira and Pruitt [38]; Trajkovic, et al. 

[39] as shown in 1 (b) and Table 2. The difference of variation between Thornwait ETo method and PM ETo 

method has coefficient of determination (R2) of 0.92 with root mean square error (RMSE) of 2.14 mm/day and mean 

bias error (MBE) of -0.68 mm/day.  

 

 



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Table-2. Statistical analysis of ETo calculated by Penman and Thornwait ETo methods compared with PM ETo method at Faisalabad station. 

Method RMSE R2 MBE % Error 

Penman 2.47 0.98 -2.41 34.91 
Thornwait 2.14 0.92 -0.68 13.81 

 

 

The monthly comparison of ETo estimated by Penman and PM ETo method at Lahore weather station 

indicate that the ETo estimated by Penman ETo method overestimated the PM ETo method by 39.51% as 

concluded by Hussein [40] as shown in the Figure 2 (a) and Table 3. The difference of variation between Penman 

ETo method and PM ETo method has coefficient of determination (R2) of 0.98 with root mean square. 

 

 
Figure-1. Comparison of ETo by (a) Penman and (b) Thornwait ETo methods with PM ETo method at Faisalabad station. 

 

RMSE of 2.64 mm/day and MBE of -2.58 mm/day at Lahore weather station as shown in Table 3. The 

Thornwait ETo method indicate underestimation of ETo in first 3 and last months (January, February, March and 

December) and overestimated ETo in the remaining months of the year by 22.43% as concluded by Trajkovic, et al. 

[41] as shown in the Figure 2 (b) and in Table 3. The difference of variation between Penman ETo method and PM 

ETo method has R2 of 0.89 with RMSE of 2.36 mm/day and MBE of -1.12 mm/day at Lahore weather station as 

shown in the Table 3. 

 
Table-3.  Statistical analysis of e to calculated by Penman and Thornwait ETo methods compared with PM ETo method at Lahore station. 

Method RMSE R2 MBE % Error 

Penman 2.64 0.98 -2.58 39.51 
Thornwait 2.36 0.89 -1.12 22.43 

 

 



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© 2020 Conscientia Beam. All Rights Reserved. 

 
Figure-2. Comparison of ETo (a) Penman (b) Thornwait ETo methods with PM ETo method at Lahore station. 

 

The statistical analysis between Penman ETo method and PM ETo method at Peshawar weather station 

indicate that Penman ETo method show overestimation of ETo by 30.75% as compared to the PM ETo method as 

concluded by Lang, et al. [42] as shown in Figure 3 (a) and in Table 4. The difference of variation among Penman 

ETo method and PM ETo method has R2 of 0.99 with RMSE of 2.19 mm/day and MBE of -2.13 mm/day. The 

mean monthly comparison between Thornwait ETo method and PM ETo method at Peshawar weather station 

indicate that Thornwait ETo method overestimated in 3 months of summer (June, July and August) and 

underestimated in the remaining months of the year by 14.54% as concluded by Lakatos, et al. [43] shown in the 

Figure 3 (B) and Table 4. The variation difference between Thornwait ETo method and PM ETo method has R2 of 

0.95 with RMSE of 1.16 mm/day and MBE of 0.61 mm/day. 

 
Table-4. Statistical analysis of ETo calculated by Penman and Thornwait ETo methods compared with PM ETo method at Peshawar station. 

Method RMSE R2 MBE % Error 

Penman 2.19 0.99 -2.13 30.75 
Thornwait 1.16 0.95 0.61 14.54 

 

 

 
Figure-3. Comparison of ETo by (a) Penman (b) Thornwait ETo methods with PM ETo method at Peshawar station. 

 

4. CONCLUSION 

This study compared the Penman and Thornwait ETo methods with PM ETo method to estimate ETo in 

different semi-arid climatic regions. The PM ETo method has been taken as reference ETo method as stated by 

many researchers including [44, 45]. The statistical results show that the Penman ETo method overestimated PM 

ETo method for estimation of ETo at all the weather stations (Faisalabad, Lahore and Peshawar) of semi-arid 

climatic conditions. The Thornwait ETo method underestimated PM ETo method in winter season and 

overestimated PM ETo method in summer season in semi-arid climatic conditions of Faisalabad, Lahore and 



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Peshawar weather stations as concluded by Moeletsi, et al. [46]. Overall, Thornwait ETo method gave better 

estimation ETo than Penman ETo method at all the weather stations.  

 

Funding: This study received no specific financial support.    
Competing Interests: The authors declare that they have no competing interests.  
Acknowledgement: The authors would like to thank Pakistan Metrological Department, 
Lahore and Peshawar for providing the climatic data records used in this research. 

 

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