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: [email protected]
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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