INTRODUCTION
Grass reference evapotranspiration (ET_{o}) is an important information
in irrigation systems planning. ET_{o} is defined as the rate of evaporation
from an extensive surface of 8 to 15 cm tall, green grass cover of uniform height,
actively growing, completely shading the ground and with adequate water (Allen
et al., 1998). The grass reference ET_{o} is used to calculate
crop water requirements by multiplying with crop coefficient (Kc) for planning
irrigation systems. Generally, the ET_{o} is acceptably calculated by
FAO PenmanMonteith equation (Doorenbos and Pruitt, 1977;
Jensen et al., 1990). This computation of ET_{o
}is based on sound theoretical reasoning and is obtained by a combination
of the energybalance and masstransfer approach. The PenmanMonteith’s
equation requires necessary climatological data such as solar radiation, vapour
pressure, duration of bright sunshine, temperature, relative humidity and wind
velocity etc. (Irmak et al., 2008; Allen
et al., 2006; Chow et al., 2003). Therefore,
the lack of necessary climatological data is difficult to obtain the reliable
ET_{o}. In addition, some parameters are difficult to measure and record
in small meteorological station. However, these basic hydrological parameters
temperature, wind speed and humidity are easily to measure and collect by small
meteorology station especially in poor countries. It seem to be more benefit
for poor countries, if can use only a few basic hydrological data to estimate
the ET_{o} by new technique.
A fuzzy set is mathematical theory for describing the interested variables
from uncertain factors or variables like daily climatological data such as temperature,
wind speed and humidity. The relationship between input and output variables
is defined from fuzzy rule, according to human processes in thinking and decision.
In addition, fuzzy rules are relatively easy to explain and understand. Recently,
the fuzzy model was accepted to describe the relationship of the uncertain variables
(Ross, 1995; Shrestha et al.,
1996; Jairaj and Vedula, 2000; Panigrahi
and Mujumda, 2000; Kangrang and Chaleeraktrakoon, 2007;
Lu et al., 2007). Often, the calibration processes
of the fuzzy model were performed by manual adjusting (trial and error) the
membership functions and rule bases. Therefore, depending on the result of the
human adjustment, it does not guarantee to yield the optimal solution.
A Genetic algorithm (GAs) is search and optimization techniques based on the
principles of national selection and genetics. GAs is a robust method for searching
for the optimum solution of a complex problem. It can provide the near global
optimal solution. The GAs was applied to solve the optimal solution of water
resource problems (Goldberg, 1989; Wardlaw
and Sharif, 1999; Chang et al., 2003; Kangrang
and Chleeraktrakoon, 2008; Hormwichian et al.,
2009). The best part of GAs is that they can handle any type of objective
function describing decision variables.
This study thus proposes the fuzzy set model for estimating the ET_{o}
using only a few basic hydrological parameters such as temperature, humidity
and wind speed. The genetic algorithm technique is applied to calibrate the
membership of the fuzzy model. The FAO PenmanMonteith equation was adopted
to compute the ET_{o} for evaluating the proposed model.
MODEL FORMULATION
The fuzzy sets model and its rulebased system are applied to estimate the ET_{o}. The input parameters include temperature, wind speed and humidity. The output is ET_{o}. The steps for working are described as following.
Firstly, the input variables are transformed to the fuzzy variable through
the membership function. The number and type of membership functions are constructed
based on statistical data and experience of engineers, generally upon the considering
problem (Saruwatari and Yomota, 1995; Jang
et al., 1997). The fuzzy sets with triangular membership functions
are used to describe all parameters due to their easy computation.
Secondly, the fuzzy rule bases are created using daily historical data and fuzzy operator. For the historical data of will be presented in the next section. These fuzzy operators including AND and OR are applied to combine the input variables.
Thirdly, the input membership functions and the rule bases are applied to obtain the output membership functions. This step is done by the implication method which obtaining a fuzzy set of output when given a single number of each inputs. Then the output membership functions of each rule are jointed to one output fuzzy set.
Finally, a fuzzy set of output is converted into a single crisp value using the centroid method.
The calculated ET_{o} from PenmanMonteith equation are used to evaluate the obtained ET_{o} of the fuzzy model. This formula is described as:
where, ET_{o} is the grass reference evapotranspiration (mm day^{1}), γ is psychrometric constant (kPa°C^{1}), Δ is slope vapour pressure curve (kPa°C^{1}), R_{n} is net radiation (MJ m^{2} day^{1}), G is soil heat flux (MJ m^{2} day^{1}), T is mean daily temperature (°C), U_{2} is mean wind speed measurement at 2 m. height (m sec^{1}) and e_{a}e_{d} is vapour pressure deficit (kPa).
The adequacy of the fuzzy model is evaluated by considering the coefficient of determination (R^{2}) which defined, based on the estimated ET_{o} as:
where, Em_{i} is the estimated grass reference evapotranspiration using
the Fuzzy model of day i, Ea_{1} is the calculated grass reference evapotranspiration
using the PenmanMonteith equation of day I,
and
are, respectively the average of above mentions and n is the number of daily
data. The fuzzy model is calibrated by adjusting the membership functions and
rule bases using the genetic algorithms technique, these performances will be
stopped when the results obtained the highest coefficient of determination (closed
to 1.0).
The calibration processes using the genetic algorithm were described as follows. The genetic algorithm requires encoding schemes that transform the decision variables into chromosome. The decision variables were the typical membership function of each type. Figure 1 and Eq. 3 show the typical membership function of triangular type. They present that the decision variables of each membership function for 1 group are a, b and c. These variables were transferred into the chromosome for searching in the process of genetic algorithm:
where, μ_{r} is membership function of value r for input or output variable (i.e., temperature, humidity and wind speed), μ is membership value of the variable, r is the value of input or output variable.
Then, the genetic operations (reproduction, crossover and mutation) were performed.
These genetic operations generated new sets of chromosomes. The objective function
of the search was to maximize the coefficient of determination (R^{2}).
This study used population size = 80, crossover probability = 0.9, mutation
probability = 0.01 (Goldberg, 1989).
The search will be stopped when obtained the highest coefficient of determination,
hence the optimal value of a, b and c was met.
This study considered the numbers of membership function of each parameter from 2 to 4 groups based on the distribution of historical data.
ILLUSTRATIVE APPLICATION
These daily climatological data such as solar radiation, vapour pressure, duration
of bright sunshine, temperature, relative humidity and wind velocity etc., of
five locations were used in the study. The meteorological stations were Nakhon
Sawan, Chiang Rai, Chiang Mai, Phitsanulok and Phetchabun (in the Central and
Northern regions of Thailand). These data were used to compute the Et_{o}
by the PenmanMonteith equation, only temperature, relative humidity and wind
velocity were used in the proposed model.

Fig. 2: 
Temperature and ET_{o} of PenmanMonteith 

Fig. 3: 
Humidity and ET_{o} of PenmanMonteith 

Fig. 4: 
Wind speed and ET_{o} of PenmanMonteith 
Figure 2 shows relationship between ET_{o} of PenmanMonteith equation and temperature for five locations during 19931995. Figure 3 shows relationship between ET_{o} of PenmanMonteith equation and humidity. Figure 4 shows relationship between ET_{o} of PenmanMonteith equation and wind speed. They indicate that the temperature during 1535°C affects to the ET_{o} and high temperature influences to high ET_{o}. While the humidity about 5060% persuades to high ET_{o}. For the wind speed values are varying during 0.52.5 m sec^{1} and high wind speed affects to high ET_{o}. These data are used to set the initial membership function and rule base of fuzzy model.
Table 1 shows an example of fuzzy rule bases using AND and OR operators such as If the medium temperature AND the low humidity AND high wind speed THEN the ET_{o} is high, If the medium temperature AND the low humidity AND low wind speed THEN the ET_{o} is low. There are eighteen rule bases. These rule bases were used to construct the relationship between input and output parameters.
RESULTS AND DISCUSSION
Table 2 presents the coefficient of determination of several
membership functions using the GAs calibration which the number of temperature
group of 2 and the number of other parameters are 2, 3 and 4. It indicated that
the range of R^{2} is 0.803 for 2232 to 0.932 for 2434. Table
3 presents the coefficient of determination of several membership functions
using the GAs calibration which the number of temperature group of 3 and the
number of other parameters are 2, 3 and 4.
Table 1: 
Example of fuzzy rule bases for estimating ET_{o} 

Table 2: 
Membership function numbers and R^{2} (the number
of Temperature group = 2) 

Table 3: 
Membership function numbers and R^{2} (the number
of Temperature group = 3) 

*The optimal value of the search using GA 
They present that the lowest of R^{2} is 0.826 for 3222 and the
highest of R^{2} is 0.992 for 3333 according the previous study (Kangrang
and Chaleeraktrakoon, 2007).
Table 4: 
Membership function numbers and R^{2} (the number
of Temperature group = 4) 

Table 4 presents the coefficient of determination of several
membership functions using the GAs calibration which the number of temperature
group of 4. It indicated that the range of R^{2} is 0.832 to 0.976.
These results found that the optimal number and shape of membership functions
give the highest coefficient of determination. Hence, this condition is accepted
to calculate the ET_{o} according to the concept of Goldberg
(1989). However, this accepted model is necessary to validate with other
data for evaluating the performance of the model.
The accepted model was further validated using climatological data which were
not considered (2002) in calibrating process. These data of five stations were
used to calculate the ET_{o} by the PenmanMonteith equation and the
proposed model. Figure 5 shows the calculation of ET_{o}
computed by PenmanMonteith equation compared against the daily ET_{o}
calculations using the fuzzyGAs model of Nakhon Sawan, Chiang Rai, Chiang Mai,
Phitsanulok and Phetchabun respectively. They indicate that calculations by
two methods are linear with each another, with R^{2} higher than 0.9117.
Figure 6 presents the calculation of 120day Et_{o}
computed by the PenmanMonteith equation and the fuzzyGAs model. It indicates
that the ET_{o} of both method are similar with the standard error of
the estimate (SEE) 0.154 mm day^{1}. In summary, the proposed fuzzyGAs
model can use to calculate the ET_{o} that closed to the ET_{o}
of PenmanMonteith equation (Irmak et al., 2008;
Allen et al., 2006; Chow et
al., 2003).

Fig. 5: 
Calculation of ET_{o} computed by PenmanMonteith
equation compared against the daily ET_{o} calculations using the
fuzzyGAs model 

Fig. 6: 
Calculation of 120day ET_{o} computed by the PenmanMonteith
equation and the fuzzyGAs model 

Fig. 7: 
The optimal Triangular membership functions of the input and
output variables for the number 3333 
Figure 7 shows the optimal Triangular membership functions of the input and output variables for the number 3333 using GAs technique. It indicates that triangular membership function is suitable to apply in this study.
CONCLUSIONS
This study applied a fuzzy model for estimating the ET_{o}. Genetic
algorithm technique was used to calibrate membership function condition of fuzzy
sets model. The applied model using only a few basic hydrological parameters
including temperature, humidity and wind speed. The model was applied to determine
the ET_{o} of five Meteorological Stations (in the Central and the Northern
region of Thailand). The daily data (19931995 and 2002) of the Stations were
used in the study. Results showed that the fuzzyGAs model can be used to estimate
the ET_{o}, given only the basic hydrological parameters; temperature,
humidity and wind speed. The obtained ET_{o} of the proposed model is
close to the ET_{o} of the PenmanMonteith equation. Furthermore, the
results presented that the Genetic algorithm calibration provided the optimal
condition of membership function. In summary, a few basic hydrological data
can provide the ET_{o} in some area which insufficient climatological
data.
ACKNOWLEDGMENTS
The authors would like to acknowledge the financial support by the National Research Council of Thailand, NRCT. Thanks are also due to Dr. Preeyaphorn Kosa for supporting data.