Open Journal of Geology, 2013, 3, 397-403
Published Online November 2013 (
Open Access OJG
Application of Soft Computing Methods
in Predicting Evapotranspiration
Afshin Honarbakhsh1, Mostafa Moradi Dashtpagerdi2*, Hassan Vagharfard3
1Faculty of Natural Resources and Earth Sciences, University of Shahrekord, Shahrekord, Iran
2Graduated Watershed Management, Organization of Natural Resources and Watershed Management,
Karoon Watershed Management Office (KWMO), Shahrekord, Iran
3Faculty of Natural Resources, Hormozgan University, Bandarabbas, Iran
Email: *
Received May 21, 2013; revised June 20, 2013; accepted June 28, 2013
Copyright © 2013 Afshin Honarbakhsh et al. This is an open access article distributed under the Creative Commons Attribution Li-
cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Exact prediction of evapotranspiration is necessary for study, design and management of irrigation systems. In this re-
search, the suitability of soft computing approaches namely, fuzzy rule base, fuzzy regression and artificial neural net-
works for estimation of daily evapotranspiration has been examined and the results are compared to real data measured
by lysimeter on the basis of reference crop (grass). Using daily climatic data from Haji Abad station in Hormozgan,
west of Iran, including maximum and minimum temperatures, maximum and minimum relative humidities, wind sp eed
and sunny hours, evapotranspiration was predicted by soft computing methods. The predicted evapotranspiration values
from fuzzy rule base, fuzzy linear regression and artificial neural networks show root mean square error (RMSE) of
0.75, 0.79 and 0.81 mm/day and coefficient of determination of (R2) of 0.90, 0.87 and 0.85, respectively. Therefore,
fuzzy rule base approach was found to be the most appropriate method employed for estimating evapotranspiration.
Keywords: Evapotranspiration; Fuzzy Rule Base; Fuzzy Regression; Artificial Neural Network
1. Introduction
Evapotranspiration is one of the most important factors
in agriculture and the hydrological cycle that can be in-
fluenced by global warming and climatic changes [1].
The process of evapotranspiration (ET) is an important
part of the water cycle and exactly estimating the value
of ET is necessary for designing irrigation systems and
water resources management. Accurate estimation of ET
is crucial in agriculture. This is due to the fact that its
over-estimation causes waste of valuable water resources
and its underestimation leads to the plant moisture stress
and decrease in the crop yield. Prediction methods de-
veloped over the past few decades range from simple
ones such as Blany-Criddle to complex ones which use
physical processes like Penman compound method [2].
Penman approach used parameters such as dynamic of
evaporation, intensity of net radiations and surface aero-
dynamic characteristics. Montieth et al. (1965) later im-
proved this method by considering the plant daily resis-
tance and Penman-Montieth equation [3]. Several re-
searchers studied validation of these equations [4,5].
Jenson et al. (1990) compared results of twenty of such
methods with the results of lysimeters in 11 stations lo-
cated in different parts of the world with various climates
and concluded that in all climates the Penman-Montieth
method gave the best results [6]. In recent years, soft
computing methods including fuzzy rule base model
(FRBM), artificial neural Networks (ANN) and also a
combination of them have been employed for estimating
Burton et al. (2000) used ANNs and estimated daily
evaporation from pan evaporation by 2044 data gathered
from various places all over the world from 1992 to 1996
[7]. Input data were precipitation, temperature, relative
humidity, solar irradiance and wind speed. Compared
with multiple linear regressions methods such as the one
proposed by Priestley-Taylor (1972), ANN provided the
minimum error of 1.11 mm/day in ET estimation [8].
Odhiambo et al. (2001) compared the results from
FRBM with those of Penman-Montieth [9] and Sha-
yannejad et al. (2007 ) used Fuzzy linear regression (FLR)
for ET estimation in Hamadan, Iran and demonstrated
*Corresponding a uthor.
that FLR gave higher determination coefficient (R2) with
less errors than Penman-Montieth method [10]. Also,
Hargreaves-Samani methods (1994, 1985) used two
fuzzy rule base models, in which solar irradiance and
relative humidity were the input data in the first model
(FRBM-1) and wind speed was also added in the second
model (FRBM-2) [11-13]. A Comparison with the
lysimeter data showed that the standard errors for
FRBM-1, FRBM-2 models, Penman-Montieth and Har-
greaves-Samani were 0.73, 0.54, 0.50 and 0.66 mm per
day respectively. It can be seen that FRBM-2 and Pen-
man-Montieth yield similar errors despite the fact that
the number of input parameters was less in FRBM-2.
2. Materials and Methods
The necessary climatic data for this research were pro-
vided from Haji Abad meteorological station, near Hor-
mozgan, west of Iran. This station has longitude 55˚ and
55'' Nort h, and latitude 28˚ and 19'' East, and elev ation of
870 m above sea level. The climate can be described as
arid and hot according to Kopen’s classification. Maxi-
mum and minimum daily air temperature is 49˚C and 5˚C
respectively. The average annual rainfall during the pe-
riod of 2000-2008 was 265 mm. A 1 m × 2.25 m × 1.2 m
lysimeter equipped with drainage is used to measure ETP
with grass reference crop. The soil characteristics could
be described as: alkaline, deep, medium to heavy texture,
electric conductivity of 0.55 to 0.85 deci siemens per
meter, specific gravity of 1.63 - 1.91 gram per cm3. A
layer of 27 cm thickness gravel consisting of various
sizes covered the slopped bottom of the lysimeter at the
station and soil was added in separate horizontal layers.
Daily ET was obtained using water balance model meas-
uring water input and output and soil humidity.
In this study, three soft computing approaches namely,
FRBM, FLR and ANN were used to estimate the poten-
tial evapotranspiration and they were evaluated using the
lysimeter data.
2.1. Fuzzy Rule Base
Fuzzy rule-based models developed by Lotfizadeh (1965)
for handling imprecise information, has found important
application in various fields including water based sys-
tems in the last five decades [14]. Introduction of Lin-
guistic Terms (LT) by Fontane et al. (1997) and applica-
tion of complex mathematical models by Bárdossy et al.
(1995), Pesti et al. (1996) have established this method-
ology as a reliable tool for predicting water resource pa-
rameters [15-17]. A FRBM contains membership func-
tions of fuzzy sets constructed on the range of all the
inputs to the model. The model matches the input and
output, which also contains membership functions, with
fuzzy rules. In this study, as suggested by Bárdossy and
Duckstein (1995), following a local search on the four
available membership functions of triangular, bell-
shaped, dome-shaped and inverted cycloid, the triangular
input membership function was selected based on the
lowest root square mean error (RSME) of 0.75 and high-
est R2 of 0.90 as shown in Tabl e 1 [18].
2.2. FRBM Design
In the design of the FRBM, six inputs containing min.
and max daily temperature (Tmin, Tma x), min. and max.
Daily air relative humidity (Rhmin, Rhmax), daily wind
speed (U), daily sunny hours (N), were considered and
ET was the model output. In order to establish the rule-
bases, 40 lines of the data containing inputs and outputs
were selected randomly.
Five FRBM models (FRBM-1 to FRBM-5) were de-
fined based on the quantity of linguistic terms and also,
the type and number of input parameters mentioned
above (see Table 2). Using 6 similar input parameters,
FRBM-1, FRBM-2 and FRBM-3 have been defined with
2, 3 and 5 LT respectively, and as suggested by Figures
1-6, FRBM-3 with 5 LT showed the least RMSE of 0.75.
FRBM-4 and FRBM-5 were hence defined using 5 LT
but different types and number of input parameters.
Based on the results demonstrated in Table 2, FRBM-3
with lowest RMSE, with input triangular membership
function and 5 LT was selected as the best FRBM for this
2.3. Artificial Neural Network Method
ANNs are mathematical models consisting of highly in-
terconnected processing nodes or elements (artificial
neurons) under a pre-specified topology (sequence of
layers or slabs with full or random connections between
the layers). In 1950s Rosenblatt built many variations of
a specific type of early neural computational models
called perceptron network and developed associated
learning rules which led to introduction of first practical
Table 1. Comparison of membership functions type used in
Number Membership
Function Type RMSE R2
1 TRI-MF 0.75 0.90
2 TRAP-MF 1.15 0.831
3 GBELL-MF 1.27 0.768
4 GAUSS1-MF 1.94 0.82
5 GAUSS2-MF 1.49 0.785
Membership Function Type: TRI: triangular, TRAP: Trapezoid, GBELL:
generaliz ed bell, GAUSS, GAUSS2-MF: Gaussian.
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Figure 1. Membership function, model FRBM-1, with 2
linguistic terms.
Figure 2. RMSE for model FRBM-1, with 2 linguistic terms.
Figure 3. Membership function, model FRBM-2, with 3
linguistic terms.
application of ANN. They have been used extensively
since 1980’s in a variety of diverse real world applica-
tions [19]. In this work, the multi-layer perceptron net-
work has one input layer (with three processing ele-
ments), one hidden layer (with two processing elements)
and one output layer (with one processing element).
2.4. Fuzzy Linear Regression
In regression analysis, the best mathematical expression
Figure 4. RMSE for model FRBM-2, with 3 linguistic terms.
Figure 5. Membership function, model FRBM-3, with 5
linguistic terms.
Figure 6. RMSE for model FRBM-3, with 5 linguistic terms.
describing the functional relationship between one re-
sponse and one or more independent variables are ob-
tained. Following the introduction of the fuzzy theory, by
Lotfizadeh, fuzzy regression model (FRM) was devel-
oped by Tanaka et al. (1982) in which fuzzy uncertain-
ties of dependent variables with the fuzziness of response
functions were explained [20]. Based on the conditions
of variables, there are 3 categories of FRM: 1) input and
output data are both non-fuzzy numbers, 2) input data is
non - f uzzy nu mber but output data is fuzzy number, and 3)
input and output are both non-fuzzy number [21,22] Es-
timation of fuzzy regression is the subject of continuous
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come this, Kao and Chyu (200 2) proposed a “two-stage”
approach for fitting fuzzy linear regression (FLR)
through fuzzy least square approach and showed superi-
ority over Diamond’s procedure [29]. This approach is
discussed by Singh et al. (2007) and relevant nonlinear
computer programs such as LINGO, have been devel-
oped to solve such cases [30]. As far as fuzzy nonlinear
regression is concerned, Buckley and Feuring (2000)
proposed “evolutionary algorithm solutions” in which for
a given fuzzy data, algorithm searches from a library of
fuzzy functions (including linear, polynomial, exponen-
tial and logarithmic) one which would fit the data [31]. In
this study, using HYDROGENERATOR and LINGO
softwares, a fuzzy possibilistic model was employed in
which coefficients are fuzzy, while inputs and outputs are
non-fuzzy observational. The model used may be repre-
sented by the following equation:
research, is often carried out by two techniques, e.g.:
fuzziness minimization by numerical method using linear
programming (as suggested by Tanaka, 1982) and devia-
tion minimization between the estimated and observed
outputs, sometimes referred to as fuzzy least square
method [23]. FLR has been used where response variable
is in intervals. By taking mean or mode, interval value
can be changed to crisp values but at a cost of losing
useful information about the spread. Hence, no proper
interpretation of the fuzzy regression interval can be
made (Wang and Tsaur, 2000) Tanaka’s approach [24],
referred to as possibilitic regression has also been criti-
cized for both not being based on sound statistical prin-
ciples [25], as well as creating computational difficulties
when large number of data points is encountered [26].
Peters (1994) complains about Tanaka’s model being
extremely sensitive to the outliers [27]. Kim et al. (199 6)
reported that fuzzy linear regression (FLR) may tend to
become multicollinear as more independ ent variables are
collected [28]. The drawback, on the other hand, with the
fuzzy least square method is the spread of estimated re-
sponse increases as the magnitude of explanatory re-
sponse increases, even though the spread of observed
responses are roughly constant or decreasing. To over-
AAxAxAxAx 
 
where, 01
 
are fuzzy coefficients and
xx x are observational input variables which
are normal numbers and
is the fuzzy output for each
variable n. Table 3 shows the object function and the
restrictions used for the FLR in this work.
Table 2. Characteristics of various FRBM’s de fined for this study.
Parameters FRBM-1 FRBM-2 FRBM-3 FRBM-4 FRBM-5
minimum temperature * * * *
Maximum temperature * * * *
minimum humidity * * * *
maximum humidity * * * *
wind speed * * * * *
sunny hour * * * * *
mean relative humidity *
mean temperature *
RMSE mm/day
0.893 0.859
0.75 1.18 1.06
Table 3. Sensitivity Analysis.
Input Parameters FRBM RMSE (mm/day) ANN RMSE (mm/day) FRM RMSE (mm/day)
Tmin, Tmax, RHmin , RHmax, U, n 0.75 0.81 0.79
Tmin, Tmax, RHmin , RHmax, n 0.97 0.81 0.96
Tmin, Tmax, RHmin , RHmax, U 0.88 0.89 0.97
Tmin, Tmax, RHmin , U, n 0.91 0.80 0.83
Tmin, Tmax, RHmax , U, n 1.08 0.91 0.95
Tmin, RHmin, RHma x, U, n 1.23 1.43 0.98
Tmax, RHmin, RHmax, U, n 1.64 1.22 1.09
Table 3: Liodel for solving linear near programming m
regression with non-fuzzy observations.
Fuzzy a linear regression:
AAxAx 
 
Ax Ax
 
Minimize: mcmn
 (2)
y (3)
y (4)
3. Results and Discussion
antis and Fuzzy regres-
as required to indicate which one
ft Computing Methods in predicting
iijiij j
ppx hccx
 
iijiij j
ppx hccx
 
For calculating ET in Penman-M
sion methods, Excel and MATLAB softwares were ap-
plied, respectively. It is noted that RMSE and R2 were
used for validation and approval of the results.
Sensitivity Analysis
A sensitivity analysis w
of the input parameters has more important role on de-
fining the ET in the models. This is carried out in two
following methods: addition of input parameters and re-
moval of input parameters. Accordingly, each parameter
its addition or removal causes the most reduction in
RMSE would be identified as the most sensitive parame-
ter. In this work, using the latter approach, one of the six
input parameters was removed at a time and the corre-
sponding RMSE was calculated as shown in Tab le 3.
Minimum temperature was therefore found to be the
most sensitive parameter in all methods used while, the
sunny hour showed the least sensitivity in FRBM, and
maximum relative humidity was the least sensitive for
ANN and FRM.
RMSE and R2 were used to select the best method to
determine ET amongst FRBM, ANN and FRM. As can
be seen from Table 4, the results indicate that R2 does
not vary much (0.80 to 0.90), while RMSE alters more so
that the least RMSE relates to FRBM model with five
linguistic terms (FRBM-3), followed by ANN, FRM,
FRBM-1, FRBM-2, FRBM-4 and FRBM-5, which
showed higher RMSE (RMSE altered in the range of
0.75 to 1.18).
4. Conclusion
In this study, So
evapotranspiration were reviewed in west of Iran. Con-
sidering Figures 7-9 in which the observed and esti-
mated ET is demonstrated using the three models FRM,
FRBM and ANN, fuzzy rule-based model proved to be
the best method and is proposed to be used for ET esti-
mation of the region. Ir an loses 70% of annu al precipita-
tion by ET. It is obvious that in this country where there
are many limitations to water resources management,
increase in ET could lead to more problems [1]. Also,
Water consumption was estimated 2200 m3 per person
Figure 7. Comparing observed and estimated ET using
FRBM-3 model.
Figure 8. Comparing observed and estimated ET using FR
Figure 9. Comparing observed and estimated ET using
Table 4. Comparison of RMSE andRM and FRBM.
parameter F-5 ANN FRM
ANN model.
R2 for ANN, F
RMy) SE (mm/da 0.893 0.859 0.75 1.18 1.06 0.81 0.79
R2 0.83 0.86 0.90 0.81 0.80 0.85 0.87
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will be 726 to 860 m3 [32]. Therefore, prediction of af-
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