ORIGINAL PAPER

Performance evaluation of modified versions of Hargreavesequation across a wide range of Iranian climates

Parisa Hosseinzadeh Talaee

Received: 1 September 2012 / Accepted: 28 April 2014

� Springer-Verlag Wien 2014

Abstract Reference evapotranspiration (ETo) is signifi-

cant for water resources planning and environmental

studies. Many equations have been developed for ETo

estimation in various geographic and climatic conditions,

of which, the Penman–Monteith FAO 56 (PMF-56) equa-

tion was accepted as reference method. A major compli-

cation in estimating ETo by the PMF-56 model is the

requirement for meteorological data that may not be readily

available from typical weather stations in many areas of the

globe. This restriction necessitates use of simpler models

which require less input data. In this study, the original and

five modified versions of the Hargreaves equation that

require only temperature and rainfall were evaluated in

humid, semi-humid, semi-arid and arid climates in Iran.

The results showed that the original and modified versions

of the Hargreaves equation had the poorest performance in

semi-humid climate and the best performance in windy

humid environment. Further, the ETo estimations with the

Hargreaves equations having rainfall parameter were poor

as compared to those from the PMF-56 method under

majority of the climatic situations studied.

1 Introduction

The accurate calculation of ETo plays an important role in

regional management, water saving agriculture and

efficient use of agricultural water resources (Aguilar and

Polo 2011; Xing et al. 2012). With increasing pressure on

water resources from competing users particularly in arid

and semi-arid environments, large emphasis has been

placed on water use efficiency in irrigated fields (Deh-

ghanisanij et al. 2004).

Evapotranspiration can be directly measured by high-

cost techniques or indirectly estimated with using weather

data (Marti et al. 2011). A large number of equations have

been developed for estimating ETo based on meteorologi-

cal data. The United Nations Food and Agriculture Orga-

nization (FAO) adopted the Penman–Monteith method

FAO 56 (PMF-56) as a global standard for estimating ETo

from four meteorological parameters (temperature, wind

speed, radiation and relative humidity). The main limita-

tion to generalized application of this methodology in

irrigation practice is the time and cost involved in acqui-

sition and processing of the necessary meteorological data.

Additionally, over many areas of the globe the number of

meteorological stations where all these parameters are

observed is limited (Shahidian et al. 2012).

The lack of requisite meteorological data for application

of the PMF-56 model motivated Hargreaves et al. (1985) to

develop less-demanding models in terms of input data. The

reduced data requirement of the Hargreaves equation is

advantageous in the regions where solar radiation,

humidity and wind data are lacking or are of low or

unreliable quality (Hargreaves and Allen 2003). The Har-

greaves equation has been examined based on high-quality

lysimeter data under various climatological conditions

(Hargreaves 1994). The results have showed this equation

was nearly as accurate as PMF-56 ETo on a weekly or

longer time step and was therefore recommended in cases

where reliable data were lacking (Droogers and Allen

2002). The Hargreaves method can be considered as a

Responsible editor: L. Gimeno

P. Hosseinzadeh Talaee (&)

Young Researchers and Elite Club, Hamedan Branch,

Islamic Azad University, Hamedan, Iran

e-mail: p.hosseinzadeh@iauh.ac.ir

123

Meteorol Atmos Phys

DOI 10.1007/s00703-014-0333-5

semi-empirical approximation as it incorporates extrater-

restrial radiation in combination with temperature as indi-

cators of global radiation, and the daily temperature range

as an indicator of humidity and cloudiness (Aguilar and

Polo 2011). Allen et al. (1998) suggested that when suffi-

cient or reliable data to use the PMF-56 method are not

available, the Hargreaves equation can be used.

In the last decade, considerable attention has been

focused globally on evaluation of simple ETo models (e.g.,

Chen et al. 2005; Trajkovic 2007; Trajkovic and Kolakovic

2009; Sabziparvar et al. 2010; Martinez and Thepadia

2010; Tabari and Hosseinzadeh Talaee 2011). In Iran,

Rahimikhoob (2008) studied the ETo estimates obtained

from the Hargreaves equation in the very dry south region.

The results indicated that the Hargreaves equation fails to

calculate ETo values above 9 day-1, whereas the PMF-56

model reaches values of more than 13 mm day-1. Tabari

(2010) evaluated four simpler models based on monthly

performance for various climates in Iran and noted that the

Makkink and Priestley–Taylor equations estimated ETo

values less accurately than Turc and Hargreaves models for

all the climates. Razzaghi and Sepaskhah (2010) examined

the Penman–FAO, PMF-56, Hargreaves–Samani, Jensen–

Haise, Turc, Priestley–Taylor, FAO–Blaney–Criddle,

FAO-Radiation and Pan Evaporation equations in a semi-

arid environment in Iran. The results indicated that the

FAO-Radiation and Hargreaves–Samani were the most

appropriate methods and the Priestley–Taylor method was

the least appropriate. Sabziparvar and Tabari (2010) eval-

uated the performance of the Makkink, Priestley–Taylor,

and Hargreaves models compared to that of the PMF-56

method for arid and semi-arid regions in northeastern Iran.

They reported that the Hargreaves model had the best

performance in estimating monthly ETo values. Foolad-

mand (2011) tested the Hargreaves, Thornthwaite and

Blaney–Criddle equations in Fars Province in Iran. The

results showed that there was no specific relationship

between the climate of the station and the best equation for

estimating ETo. Tabari et al. (2013) evaluated 31 ETo

methods under humid conditions and reported the Blaney–

Criddle model as the best temperature-based equation.

The main aim of this study was to investigate the per-

formance of the original and five modified versions of the

Hargreaves equation in humid, semi-humid, semi-arid and

arid climates of Iran. The ETo values estimated by the

Hargreaves equations were compared with estimates by the

standard PMF-56 method.

2 Materials and methods

In the current study, ETo values were calculated by the

Hargreaves equations in four climates of Iran. The data set

for the period 1966–2005 were obtained from Babolsar,

Gorgan, Zanjan and Birjand synoptic stations, having

humid, semi-humid, semi-arid and arid climate, respec-

tively. Station maintenance and data archiving are under

the supervision of the Islamic Republic of Iran Meteoro-

logical Organization (IRIMO). The climatic and geo-

graphic characteristics of the stations are presented in

Table 1.

This study uses the ETo values estimated by the PMF-56

model for examining the efficiency of the Hargreaves

equations. The PMF-56 model given by Allen et al. (1998)

is as follows:

ETo ¼0:408DðRn � GÞ þ c 900

Taþ273U2 es � eað Þ

Dþ c 1þ 0:34U2ð Þ ; ð1Þ

where ETo is the reference crop evapotranspiration

(mm day-1), Rn is the net radiation (MJ m-2 day-1), G is

the soil heat flux (MJ m-2 day-1), c is the psychrometric

constant (kPa �C-1), es is the saturation vapor pressure

(kPa), ea is the actual vapor pressure (kPa), and D is the

slope of the saturation vapor pressure–temperature curve

(kPa �C-1), Ta is the daily mean air temperature (�C), and

U2 is the mean daily wind speed at 2 m height (m s-1). The

computation of all data required for calculating ETo fol-

lowed the method and procedure given in Chapter 3 of

FAO-56 (Allen et al. 1998).

The original Hargreaves equation (Hargreaves et al.

1985) can be written as:

ETo ¼ 0:0023� Ta þ 17:8ð Þ � ðTmax � TminÞ0:5 � Ra: ð2Þ

Allen (1993) modified the Hargreaves equation by fit-

ting coefficients based on monthly calculations of ETo by

the PMF-56 method using the FAO Climwat data set

Table 1 Climatic and geographic characteristics of the weather stations

Station Latitude

(N)

Longitude

(E)

Elevation

(m a.s.l.)

Air temperature

(�C)

Rainfall

(mm/year)

ETo

(mm/year)

Climate

(Tabari et al. 2014)

Babolsar 52�390 36�430 -21 17.9 962.7 1003.5 Humid

Gorgan 36�510 54�160 13 18.1 562.9 1,097.9 Semi-humid

Zanjan 36�41 48�290 1,663 11.4 269.8 1,488.7 Semi-arid

Birjand 32�520 59�120 1,491 16.7 149.9 1,691.3 Arid

P. Hosseinzadeh Talaee

123

(Smith 1993) consisting of 3,200 stations and using

lysimeter measurements of ETo from Davis, California,

resulting in:

ETo ¼ 0:408� 0:0030� ðTa þ 20Þ � ðTmax � TminÞ0:4 � Ra:

ð3Þ

Subsequently, Droogers and Allen (2002) in an attempt

to improve the agreement of the Hargreaves equation with

the PMF-56 method used the IWMI (International Water

Management Institute) Climate Atlas data grids. Compar-

isons around the globe using the grid were used to adjust

two parameters in the original Hargreaves equation. The

result was the following forms for the Hargreaves equation:

ETo ¼ 0:408� 0:0025� ðTa þ 16:8Þ � ðTmax � TminÞ0:5 � Ra;

ð4Þ

ETo ¼ 0:408� 0:0013� ðTa þ 17Þ � ðTmax � Tmin

� 0:0123PÞ0:76 � Ra: ð5Þ

Following Eq. (5) above, Fooladmand (2008) modified

the Hargreaves equation using meteorological data from

the south of Iran as follows:

ETo ¼ 0:408� 0:0045ðTa þ 46:2Þ � ðTmax � Tmin

� 0:0156PÞ0:11 � Ra: ð6Þ

In another study, Trajkovic (2007) adjusted the Har-

greaves equation using data from Western Balkans region

as follows:

ETo ¼ 0:0023� Ta þ 17:8ð Þ � ðTmax � TminÞ0:424 � Ra;

ð7Þ

where ETo is in mm day-1; P is monthly rainfall (mm); Ra

is the water equivalent of the extraterrestrial radiation

(mm day-1) computed according to Allen et al. (1998);

Tmax, Tmin and T are the maximum, minimum and mean air

temperatures (�C), respectively. The coefficient of 0.408 is

for converting MJ m-2 day-1 into mm day-1 (Allen et al.

1998). The Eqs. (2)–(7) are defined hereafter as H, MH1,

MH2, MH3, MH4 and MH5, respectively.

The coefficient of determination (R2), mean absolute

error (MAE), root mean square error (RMSE), and mean

bias error (MBE) criteria were used to evaluate the per-

formance of the Hargreaves models. These criteria were

worked out using Eqs. (8)–(11) given below:

R2 ¼Pn

i¼1 Xi � �Xð Þ Yi � �Yð Þ� �2

Pni¼1 Xi � �Xð Þ2

Pni¼1 Yi � �Yð Þ2

; ð8Þ

MAE ¼Pn

i¼1 Xi � Yij jn

; ð9Þ

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn

i¼1 Xi � Yið Þ2

n

s

; ð10Þ

MBE ¼Pn

i¼1 ðXi � YiÞn

; ð11Þ

where Xi and Yi are the ith observed and estimated values,

respectively; �X and �Y are the average of Xi and Yi, and n is

the total numbers of data.

3 Results and discussion

3.1 Humid climate

The results from the ETo estimates obtained from the

Hargreaves equations were compared with the ETo results

obtained using the PMF-56 equation. The evaluation cri-

teria (MAE, MBE, RMSE and R2) for the Hargreaves

equations in humid climate are presented in Table 2. The

Hargreaves models except for the MH4 equation performed

best in humid climate among the climates studied. Good

coefficients of determination were obtained for all equa-

tions, with values above 0.94 thus indicating a good linear

relation between the Hargreaves equations and the PMF-56

method under humid climatic conditions. The H equation

Table 2 Statistical performance of the Hargreaves equations vs. the

PMF-56 model for ETo estimation under four climates

Climate Model R2 MAE

(mm/day)

RMSE

(mm/day)

MBE

(mm/day)

Humid H 0.983 0.163 0.213 0.004

MH1 0.985 0.367 0.408 -0.366

MH2 0.983 0.198 0.243 -0.155

MH3 0.949 0.458 0.551 0.435

MH4 0.959 1.568 1.638 -1.568

MH5 0.984 0.412 0.515 0.384

Semi-humid H 0.896 1.246 1.447 -1.246

MH1 0.886 1.646 1.861 -1.646

MH2 0.898 1.494 1.712 -1.494

MH3 0.913 1.143 1.351 -1.141

MH4 0.822 2.840 3.078 -2.840

MH5 0.891 0.623 0.795 -0.558

Semi-arid H 0.927 0.500 0.620 0.218

MH1 0.931 0.442 0.579 0.016

MH2 0.926 0.515 0.663 0.037

MH3 0.913 0.725 0.921 -0.165

MH4 0.933 0.758 0.869 -0.708

MH5 0.929 0.859 0.999 0.857

Arid H 0.910 0.642 0.955 0.379

MH1 0.915 0.593 0.867 0.188

MH2 0.910 0.577 0.830 0.135

MH3 0.888 0.655 0.889 -0.216

MH4 0.918 0.843 0.983 -0.328

MH5 0.915 1.207 1.633 1.194

Evaluation of modified versions of Hargreaves equation

123

with the MAE, RMSE and MBE values of 0.163, 0.213 and

0.004 mm/day, respectively, was the best suited model for

humid climate. In this climate, the modified versions of the

Hargreaves equations that require rainfall and temperature

parameters provided worse ETo estimates compared with

those with only temperature parameter as input (H, MH1,

MH2 and MH5). Figure 1 illustrates that the ETo values

obtained by the original Hargreaves equation were close to

those estimated by the PMF-56 method. The trend of the

ETo values estimated by the modified Hargreaves equations

with exception of the MH4 equation was similar to that of

PMF-56 ETo.

3.2 Semi-humid climate

The results of this study for semi-humid climate are given

in Table 2. The modified versions of the Hargreaves

equation had a poor performance under semi-humid con-

ditions. The Hargreaves equations overestimated the ETo

as compared to PMF-56 model and the highest overesti-

mation was found in summer season (Fig. 2). This equation

generally overestimates ETo at humid locations (Jensen

et al. 1990; Amatya et al. 1995; Itenfisu et al. 2003; Te-

mesgen et al. 2005; Trajkovic 2005) and underestimates in

arid locations (Jensen et al. 1990; Tabari et al. 2012). Ta-

bari (2010) reported poor performance of the original

Hargreaves equation in humid climate of Iran. The

underestimations of the Hargreaves equations at the semi-

humid location are higher than those at the humid location.

This may be due to the fact that wind speed in the humid

site (4.10 m/s) is about 34 % higher than that in the semi-

humid site (3.05 m/s), and conditions with high wind speed

may result in the underestimation of ETo by the Hargreaves

equation (Temesgen et al. 1999; Tabari 2010).

The obtained results showed that the MH5 equation had

the lowest error (MAE = 0.623 mm/day, RMSE = 0.759

mm/day and MBE = -0.558 mm/day) for semi-humid

climate. It was interesting that the MH4 equation which has

similar meteorological parameter input as that of MH3

gave the worst ETo estimates. The better performance of

the MH3 equation compared with the MH4 can be due to

the meteorological data by which the equation was devel-

oped. The MH3 equation was developed using the IWMI

Climate Atlas data grids, while the MH4 equation was

developed based on the meteorological data from Fars

province located in south Iran.

3.3 Semi-arid climate

The results of the analysis for semi-arid climate are pre-

sented in Table 2. As the results in Fig. 3 indicate that the

MH1 equation was the more precise for this climate

(R2 = 0.931, MAE = 0.442 mm/day, RMSE = 579 mm/

day and MBE = 0.016 mm/day). Inclusion of rainfall

parameter in the MH3 and MH4 equations increased the

estimation errors. Further, it was noted that the best per-

formance of the MH4 equation was obtained in semi-arid

climate possibly be because of the reason that the model

was developed using the weather data from the stations

having semi-arid climate. The worst performance was

obtained by the MH5 equation for semi-arid climate.

Fig. 1 Comparison of the ETo values calculated from the PMF-56

model and the Hargreaves equations in humid climateFig. 2 Comparison of the ETo values calculated from the PMF-56

model and the Hargreaves equations in semi-humid climate

Fig. 3 Comparison of the ETo values calculated from the PMF-56

model and the Hargreaves equations in semi-arid climate

P. Hosseinzadeh Talaee

123

Moreover, the H equation presented acceptable results in

this climate. Tabari (2010) also reported the good perfor-

mance of the original Hargreaves equation in semi-arid

northwestern regions of Iran. Jensen et al. (1997) recom-

mended the Hargreaves equation as one of the most simple

and accurate empirical methods for the regions where not

all the variables required in the standard PMF-56 model are

measured or when measurements have errors, especially

concerning relative humidity data.

3.4 Arid climate

The statistical measures for the ETo values estimated by the

original and modified versions of the Hargreaves equation

are presented in Table 2. Next to semi-arid climate, the

modified Hargreaves equations except for the MH5 equa-

tion presented the best performance in the arid climate.

Figure 4 shows the comparison between the monthly

means of the estimated ETo values by the Hargreaves

equations and those obtained from the PMF-56 model. The

Hargreaves equation tends to overestimate ETo at lower

ETo rates and underestimate at higher ETo rates (Droogers

and Allen 2002; Xu and Singh 2002). Samani (2000) also

reported the underestimation of the Hargreaves equation

under arid climate. Since the thermal range does not

completely consider the aerodynamic terms, the applica-

tion of the Hargreaves formula in arid zones must be done

with caution (Garcia et al. 2004). The Hargreaves equa-

tions which require only air temperature showed common

behavior of the method in arid regions (i.e., overestima-

tion). However, the results of this study indicated that

inclusion of rainfall in the Hargreaves equation led to

overestimation of ETo. A closer look at the MH3 and MH4

equations having rainfall parameter indicates that adding

rainfall parameter to the equations should lead to decrease

ETo values, and so overestimation of these equations is not

due to rainfall parameter and can be related to the cali-

brated coefficients of the equations.

4 Conclusions

In the current study, the original and five modified versions

of the Hargreaves equation were evaluated against the

PMF-56 model in arid, semi-arid, semi-humid and humid

climates of Iran. The poorest performance of the Har-

greaves equations was found in semi-humid climate, while

the equations yielded better ETo estimations in windy

humid climate as compared to the PMF-56 model. The best

suited model for humid climate was the original Har-

greaves equation. Similarly, the modified versions of the

Hargreaves equation with the exception of the MH4 had

higher precision under humid climate. For arid climate, the

ETo estimates obtained by use of the MH1 and MH2

equations were closer to PMF-56 ETo than those by the H,

MH3, MH4 and MH5 equations. This indicates that site

climate has significant effect on the performance of the

Hargreaves equations.

Overall, the lowest errors (MAE = 0.163 mm/day,

RMSE = 0.213 mm/day and MBE = 0.004 mm/day)

were obtained while using the original Hargreaves equation

in windy humid climate, whereas the highest errors

(MAE = 2.840 mm/day, RMSE = 3.078 mm/day and

MBE = -2.84 mm/day) and lowest correlation

(R2 = 0.822) were obtained using the MH4 equation in

semi-humid climate. The good performance of the Har-

greaves equations in semi-arid and windy humid climates

must be emphasized, since this being very simple equation

that requires only the air temperature and rainfall mea-

surements. Results from this study recommend the use of

Hargreaves equation for semi-arid and windy humid

regions where full weather data for the application of the

PMF-56 model are missing or the quality of data is

questionable.

Acknowledgments The authors wish to express their gratitude to

the Islamic Republic of Iran Meteorological Organization (IRIMO)

for access to the weather station data. The author is also grateful to

anonymous reviewers whose suggestions and remarks have greatly

helped to improve the quality of the paper.

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