Theor Appl Climatol DOI 10.1007/s00704-013-0967-0

ORIGINAL PAPER

GANN models for reference evapotranspiration estimation developed with weather data from different climatic regions Zikui Wang & Pute Wu & Xining Zhao & Xinchun Cao & Ying Gao

Received: 30 November 2012 / Accepted: 3 June 2013 # Springer-Verlag Wien 2013

Abstract Accurate estimation of reference evapotranspiration (ET0) becomes imperative for better managing the more and more limited agricultural water resources. This study examined the feasibility of developing generalized artificial neural network (GANN) models for ET0 estimation using weather data from four locations representing different climatic patterns. Four GANN models with different combinations of meteorological variables as inputs were examined. The developed models were directly tested with climatic data from other four distinct stations. The results showed that the GANN model with five inputs, maximum temperature, minimum temperature, relative humidity, solar radiation, and wind speed, performed the best, while that considering only maximum temperature and minimum temperature resulted in the lowest accuracy. All the GANN models exhibited high accuracy under both arid and humid conditions. The GANN models were also compared with multivariate linear regression (MLR) models and three conventional methods: Hargreaves, Priestley–Taylor, and Penman equations. All the GANN models showed better performance than the corresponding MLR models. Although Hargreaves and Priestley–Taylor equations performed slightly better than the GANN models considering the same inputs at arid and semiarid stations, they showed worse performance at humid and subhumid stations, and GANN models performed better Z. Wang : P. Wu (*) : X. Zhao : X. Cao College of Water Resources and Architectural Engineering, Northwest A&F University, Yangling 712100, China e-mail: [email protected] Z. Wang : P. Wu : X. Zhao : X. Cao : Y. Gao Institute of Water Saving Agriculture in Arid Regions of China, Northwest A&F University, Yangling 712100, China P. Wu : X. Zhao : Y. Gao Institute of Soil and Water Conservation, Chinese Academy of Sciences and Ministry of Water Resources, Yangling 712100, China

on average. The results of this study demonstrated the great generalization potential of artificial neural techniques in ET0 modeling.

1 Introduction Evapotranspiration (ET), the sum of water evaporated from soil surface (evaporation) and plant stomata (transpiration), plays an essential role in irrigation and drainage practices. Accurate estimation of it is highly necessary, since water resources available for agriculture become more and more limited in many areas worldwide due to various reasons such as population growth, climate change, and water quality degradation (Farahani et al. 2007; Wu et al. 2010). A lysimeter can be used to measure actual evapotranspiration (ETc) from a well-watered agricultural crop directly and accurately, but its wide application is restricted because it is costly and time consuming. To estimate ETc, reference evapotranspiration (ET0) is usually calculated first. ET0 refers to ET rate from a hypothetical grass crop surface with specific characteristics, which represents the effect of the climate on the evapotranspiration process (Allen et al. 1998). ETc can then be obtained by multiplying ET0 with an empirical crop coefficient, which accounts for the difference between the standard surface and a specific crop. ET0 is usually estimated using mathematical models with measured meteorological parameters as independent variables. These mathematical approaches are divided into physical and empirical models. Jensen et al. (1990) concluded that the physical models which combined energy balance and aerodynamic equations gave the most satisfying results. As a physical model, the FAO-56 application of the Penman–Monteith equation is currently widely used and can be considered as a sort of standard (Allen et al. 1998). It is referred to hereafter as FAO-56 PM equation. The FAO-56 PM equation can be applied in a wide variety of environments and climate scenarios

Z. Wang et al.

because of its sound physical basis. However, it cannot be applied at the places where weather data are limited, since it requires varieties of inputs. Many empirical methods, such as Hargreaves (Hargreaves and Samani 1985) equation and Priestley–Taylor (Priestley and Taylor 1972) equation, were developed to estimate ET0 with limited dataset, but these methods are usually site-dependent and show poor accuracy in other places. The lack of physical understanding of ET0 process and unavailability of all relevant data results in inaccurate estimation of ET0. Over the past decade, considerable attention has been given to the application of artificial neural network (ANN) technique in modeling ET. Kumar et al. (2002) used multilayer perceptions (MLPs) for estimation of ET, and found that ET was better estimated using the ANN models than the FAO-56 PM method. Sudheer et al. (2003) established radial basis neural networks (RBNNs) in modeling ET0 using limited climatic data. Kisi (2006) developed generalized regression neural network (GRNNs) for estimating ET0, and concluded that GRNN models performed better than Penman and Hargreaves empirical models. Kisi (2008) investigated the potential of MLPs, RBNNs, and GRNNs in modeling of ET0. Rahimikhoob (2008, 2010) developed ANN model only with maximum and minimum temperature data, and proved that ANN model was superior to the Hargreaves method. Jain et al. (2008) examined the link weights for physical interpretation of ANN models and determined the relative importance of the input variables in ET modeling. More recently, Ozkan et al. (2011) developed neural networks with artificial bee colony algorithm to estimate ET0, Kisi (2011) tested the accuracy of evolutionary neural networks in ET0 modeling, and Hamid et al. (2011) successfully applied ANN models in estimating of garlic crop ET. Most of the ANN models in the previous studies were trained and tested using climate data from the same station and cannot be implemented to the other stations without local training. This is the main limitation of the ANN model compared to FAO-56 PM method, which can be used worldwide without local calibration (Kumar et al. 2011). To overcome this, several authors trained ANNs with data from multiple stations. Dai et al. (2009) used weather data from 26, 40, and 23 stations to train the ANNs for the arid, semiarid, and semihumid areas of the Inner Mongolia of China, respectively, and got satisfied results. Kumar et al. (2009) pooled the data of four distinct locations in ANN learning to develop generalized ANN models, and they found that the models performed satisfactory for arid location, but the performance deteriorated in humid location due to under representation of the humid region in ANN learning process. Kumar et al. (2011) suggested that increasing the range of variation in the input data set was a possible way to develop GANN model, although the

models may perform less accurately than those with local training. The objective of this study was to develop ANN models for ET0 estimation that have great generalization capability. To achieve this, we collected weather data from four weather stations located in different climatic regions (arid, semiarid, humid, and subhumid) and with far different geographical position to train the neural networks. After the networks were constructed, they were tested with weather data from other four distinct locations with different climates. All the GANN models were compared with multiple regression models and conventional equations to identify the generalization potential of artificial neural approach.

2 Artificial neural network overview ANN is a mathematical model that tries to simulate biological neural. The structure of ANN acts very similar to human brain and nervous system and is able to create a nonlinear relation, which connects input and output data. The basic structure of an ANN usually consists of three layers: (1) the input layer, where the data are introduced to the network; (2) the hidden layer, where data are processed; and (3) the output layer, where the results of given input are produced. Each of the neurons in any given layer was interconnected with each of the neurons in the next layer by weight, bias, and transfer function to demonstrate complex behavior of the ANN. Similar to human brain, ANN learns by examples and tries to improve the weights and biases to relate input and output comparing to the target. This process is called training of the network, and a specific algorithm is usually used to do this.

3 Model development 3.1 Selection of input variables Selecting input vector (dependent variables) is one of the most vital procedures in the construction of ANN model, which determines the model’s accuracy and applicability. Most of the researchers considered several combinations of climate factors when they developed ANN models for ET0 modeling. They all found that the combination of variables needed by FAO-56 PM method showed the highest accuracy, and the consideration of only temperature and solar radiation could also give acceptable results (Kisi 2006, 2007; Jain et al. 2008; Tabari et al. 2012). The weather parameters considered in this part of the study are maximum air temperature Tmax, minimum air

GANN models for reference evapotranspiration estimation

temperature Tmin, relative humidity RH, solar radiation SR, and wind speed at 2 m height U2. Four models were developed using different combinations of these variables as inputs: (1) GANN2 using Tmax and Tmin; (2) GANN3 using Tmax, Tmin, and SR; (3) GANN4 using Tmax, Tmin, SR, and RH; (4) GANN5 using Tmax, Tmin, SR, RH, and U2.

3.2 Database Daily meteorological data, including Tmax, Tmin, RH, U2, and sunshine hours (N), for the period of January, 1, 1991– December, 31, 2000 were collected from eight weather stations managed by the Chinese Bureau of Meteorology. These stations are of great distinction in spatial and altitudinal characteristics and cover a wide range of climatic zones of China (Fig. 1). Information regarding the sites, annual rainfall, and annual mean values of relevant weather variables for each station are given in Table 1. SR was calculated from N using the Angstrom’s equation, with coefficients applicable to China (Angstrom 1924; Zhou et al. 2005). The corresponding daily ET0 was calculated using the FAO56-PM method:
 900 U 2 ðes − ea Þ 0:408ΔðRn − GÞ þ g T þ 273 ET0 ¼ Δ þ g ð1 þ 0:34U 2 Þ ð1Þ

where ET0 is the reference evapotranspiration (mm day−1), Rn is the net radiation at the grass surface (in megajoule per square meter per day), G is the soil heat flux density (in megajoule per square meter per day), T is the mean daily air temperature (in Celsius), which was calculated as the average of Tmax and Tmin, U2 is the mean daily wind speed at 2-m height (in meters per second), es is the saturation vapor pressure (in kilopascal), ea is the actual vapor pressure (in kilopascal), Δ is slope vapor pressure curve (in kilopascal per degrees Celsius), and γ is the psychometric constant (in kilopascal per degrees Celsius). The computation of all data required for calculating ET0 followed the method and procedure given in Chap. 3 of FAO 56 irrigation and drainage paper (Allen et al. 1998). According to the new scheme of climate regionalization of China made by Zheng et al. (2010), LH and WW are located in arid zone, UM and LZ are located in semiarid zone, SQ and WG are located in subhumid zone, and GZ and YA are located in humid zone. Dryness (the ratio of annual ET0 to annual rainfall) was the main indexes for classification of the climatic zone in their study. The climatic data collected from LH, UM, SQ, and GZ were pooled together to train the models. These data were separated into training dataset (January 1, 1991–December 31, 1998) and cross validation dataset (January, 1, 1999–December, 31, 2000). The training set is for determining the weights and biases of the neural network, and the cross validation set is for evaluating the weights and biases and for deciding when to stop

Fig. 1 Spatial distribution of the eight meteorological stations used in the study. Data from the four stations marked with solid circle were used to train the networks, and those from the stations marked with empty circle were used in test process

Z. Wang et al. Table 1 Summary of weather station sites used in this study

Station

Code

Lat. (N)

Lon. (E)

Alt. (m)

Annual rainfall (mm)

T (°C)

RH (%)

U2 (m s−1)

N (hours)

Linhe Urumqi Shangqiu Ganzhou Wuwei Lanzhou Wugong Yaan

LH UM SQ GZ WW LZ WG YA

40.45 43.47 34.27 25.52 37.55 36.03 34.15 29.59

107.25 87.39 115.40 115.00 102.40 103.53 108.13 103.00

1039.3 935.0 50.1 137.5 1531.5 1517.2 447.8 627.6

145.6 286.3 681.1 1461.4 165.9 311.7 651.7 1692.5

8.1 6.9 14.1 19.6 7.9 9.8 14.7 16.2

48 58 71 76 53 56 65 79

2.1 2.4 2.2 1.6 1.8 0.9 1.3 1.4

8.48 6.91 5.87 4.87 7.87 6.64 5.19 2.70

training. Meteorological data from other four stations, namely, WW, LZ, WG, and YA, were used to test the models. The data during the period of January 1, 1993 to December 31, 1993 were used. These four stations also represent quite different climate patterns as shown in Table 1. All inputs to the neural networks are normalized to fall between 0 and 1 to minimize the influence of absolute scale. The normalization scheme is as follows: xnorm ¼

x0 −xmin xmax −xmin

ð2Þ

where xnorm, x0, xmin, and xmax are normalized value, real value, minimum value, and maximum value, respectively. 3.3 Selecting of ANN architecture There are multitudes of network types available for ANN applications, and its choice depends on the nature of problem and data availability. The feed forward back-propagation network is perhaps the most popular used for hydrologic modeling (ASCE Task Committee 2000) and applied by many authors in ET modeling (Kisi 2007, 2008; Leahy et al. 2008; Laaboudi et al. 2012; Tabari et al. 2012). A feed-forward network, with single hidden layer, is sufficient for the process of evapotranspiration (Kumar et al. 2002; Parasuraman et al. 2007) and was adopted in this study. Input Neurons

l

Figure 2 illustrated a three-layer feed forward neural network, whose input matrix is P, output matrix is T. IW1,1 and b1 are the interconnection weights and bias between the input layer and the hidden layer, and LW2,1 and b2 are the interconnection weights and bias between the hidden layer and the output layer. The sizes of the matrices are marked below them. The letters l, m, and n denote the number of neurons in the input layer, hidden layer, and output layer, respectively. H, the matrix of hidden neurons, is calculated as:   H ¼ f 1 IW 1;1  P þ b1 ð3Þ Where f1 is the activation function between the input layer and the hidden layer. The log-sigmoid function was used in this work. T is the output of the ANN, which is calculated as:   T ¼ f 2 LW 2;1  H þ b2 ð4Þ In which f2 is the transfer function between the hidden layer and the output layer. Linear transfer function was applied here. Determination of the number of hidden nodes for a particular problem is another important issue, as the network topology directly affects its computational complexity and its generalization capability. In most situations, there is no way to determine the optimal number of hidden nodes without training several networks and estimating the error of

Hidden Neurons

P

IW1,1

1

m l

H Transfer

m 1

Output Neurons

LW2,1 n

m

Function

n 1

Function

b1

b2

m 1

n 1

Fig. 2 Schema of a three-layer feed forward neural network architecture. IW1,1 and b1 are the interconnection weights and bias between the input layer and the hidden layer, and LW2,1 and b2 are the interconnection weights

T Transfer

and bias between the hidden layer and the output layer. The sizes of the matrices are marked below them. The letters l, m, and n denote the number of neurons in the input layer, hidden layer, and output layer, respectively

GANN models for reference evapotranspiration estimation

1 XN jO −Pi j i¼1 i N

each. A few hidden nodes result in a high training and generalization errors due to underfitting; whereas, too many hidden nodes result in low training error but give higher estimation error due to overfitting. Kumar et al. (2002) studied the effect of number of nodes in the hidden layer on the evapotranspiration estimating performance, and revealed that i+1 (where i is the number of nodes in the input layer) nodes was sufficient to model evapotranspiration for almost all conditions of data availability. We followed their suggestion in our research. So the structures of the ANN2, ANN3, ANN4, and ANN5 are 2-3-1, 3-4-1, 4-5-1, and 5-6-1, respectively.

MAE ¼

3.4 Training the networks

Neural network toolbox in MATLAB (version7, http:// www.mathworks.com), which has all necessary functions already set, was used to actualize the GANN models. Scatter plots of the predicted values by the ANN models versus observed data (FAO-56 PM estimated) for the test stations, WW, LZ, WG, and YA, are given in Fig. 3. The R2 statistics are also presented in the plots. It can be observed from the figure that the GANN2 model overestimated the low values of ET0 and underestimated the high values at WW, LZ, and YA stations, while overestimated almost all the values at WG station. GANN3 also overestimated most of the values at WG station but had the data spread evenly along both sides of the 1:1 line at the other three stations. No apparent overestimates or underestimates of ET0 can we found for GANN4 and GANN5 at each station, except that GANN4 overestimated most of the observed values at LZ station. From the viewpoint of R2, we can conclude that the more climatic parameters considered in the input dataset the stronger relationship between the simulated values and the observed ones. This observation is in agreement with Kisi (2006, 2007), Jain et al. (2008) and Huo et al. (2012), who developed ANN models with weather data from single station. It is worth mentioning that GANN4 model computed ET0 almost as accurately as GANN5 at WG and YA stations with the R2 values of 0.983 and 0.982. However, at WW and LZ stations, the performance of GANN4 is not as good as GANN5 with the R2 values of 0.963 and 0.949. Huo et al. (2012) analyzed relative importance of climate factors to ET0 predicting ANN performance at several weather stations and found that humidity was more important for humid climates. This may be the reason why GANN4 performed better under humid and subhumid climates in our study. The RMSE and MAE statistics of each ANN model for the test stations are listed in Table 2. The results presented in Table 2 confirmed that the GANN5 model was ranked the top among the GANN models, with RMSEs ranging from 0.137 to 0.249 mm day−1, averaging 0.193 mm day−1 and MAEs ranging from 0.109 to 0.204 mm day−1, averaging 0.158 mm day−1, while GANN2 performed the worst with RMSEs varying from 0.595 to 0.692 mm day−1, averaging 0.660 mm day−1 and MAEs varying from 0.473

During the training process of a feed forward network, the modeled outputs are compared with known outputs, and any error is back-propagated to determine the appropriate weight adjustments necessary to minimize the errors. The Levenberg– Marquardt algorithm, one of the most appropriate higher-order adaptive algorithms known for minimizing the MSE of a neural network, was used in the present study for adjusting the weights and biases of the feed-forward networks. One of the problems that threaten the training process is overfitting. It usually occurs when the network has memorized the training examples, but it has not learned to generalize to new situations. To minimize the effect of this problem, cross-validation dataset was used through an early stopping technique, in which the model performance on the cross-validation dataset was monitored, and training was stopped, when error on the cross-validation dataset began to rise (Neural Network ToolboxTM User’s Guide 2010; Izadifar and Elshorbagy 2010). 3.5 Model evaluation Once the network was trained, the generalization abilities of the network were evaluated using the testing dataset. The performances of GANN models were evaluated using three statistical criteria, coefficient of determination (R2), root mean square error (RMSE), and mean absolute error (MAE). The R2 measures the degree to which two variables are linearly related. The RMSE measures the goodness of fit relevant to high simulated values, whereas the MAE yields a more balanced perspective of the goodness of fit at moderate values. These indices were calculated as: hXN   i2 O −O P −P i i i¼1 XN   R2 ¼ XN  O −O P −P i i i¼1 i¼1

ð5Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Xn RMSE ¼ ðOi −Pi Þ2 i¼1 n

ð6Þ

ð7Þ

where Oi, Pi, P, and O are observed values, simulated values, mean of simulated, and mean of observed values, respectively. N is the number of instances in the dataset.

4 Results and discussion 4.1 The performance of GANN models

Z. Wang et al. 8

Modeled ET0 (mm)

7 6

GANN2 at WW

GANN3 at WW

R² = 0.8374

R² = 0.9110

GANN2 at LZ

GANN3 at LZ

GANN4 at WW

R² = 0.9634

GANN5 at WW R² = 0.9914

5 4 3 2 1

0 8 Modeled ET0 (mm)

7 6

R² = 0.8578

R² = 0.9325

GANN4 at LZ

R² = 0.9488

GANN5 at LZ R² = 0.9936

5 4 3 2 1 0 8

Modeled ET0 (mm)

7 6

GANN2 at WG R² = 0.8880

GANN3 at WG R² = 0.9643

GANN4 at WG

GANN5 at WG

R² = 0.9830

R² = 0.9936

GANN4 at YA

GANN5 at YA

5

4 3 2 1

Modeled ET0 (mm)

0 8 7 6

GANN2 at YA R² = 0.8479

GANN3 at YA R² = 0.9299

R² = 0.9822

R² = 0.9942

5 4 3

2 1

0 Observed ET0 (mm)

Observed ET0 (mm)

Observed ET0 (mm)

Observed ET0 (mm)

Fig. 3 Scatter plots of predicted values resulted from GANN models versus observed data for test stations

to 0.543 mm day−1, averaging 0.513 mm day−1. The GANN4 and GANN3 models took the second and third places in terms of these two indices. It can be obviously seen from Table 2 that all the models exhibit higher precision at WG and YA stations, which are located in subhumid and humid environments, respectively. The arbitrariness in constructing training dataset may be responsible to this. Four stations in different climatic regions were chosen to collect weather data from, and no other rules were obeyed. Although training dataset has wide data range and high variance for all variables, some data patterns of test stations still cannot be represented, especially those at WW

and LZ stations with high variance (see Table 3). Incorporating data from more stations representing different climatic conditions in model development may increase the representation of the ANN model; however, the accuracy of the model may decrease at the same time. 4.2 Comparison of GANN models with conventional equations Three conventional models, Hargreaves (Hargreaves and Samani 1985), Priestley–Taylor (Priestley and Taylor 1972), and Penman (Penman 1948) methods, were considered for the

GANN models for reference evapotranspiration estimation Table 2 Error statistics of the ANN, MLR, and conventional models for the test stations Model

WW

LZ

WG

YA

Average

RMSE MAE RMSE MAE RMSE MAE RMSE MAE RMSE MAE (mm day−1) (mm day−1) (mm day−1) (mm day−1) (mm day−1) (mm day−1) (mm day−1) (mm day−1) (mm day−1) (mm day−1) ANN models ANN2 0.682 ANN3 0.554 ANN4 0.325 ANN5 0.249 Conventional models ETHS 0.722 ETPT 0.520 ETP 1.065 MLR models MLR2 0.997 MLR3 0.964 MLR4 0.672 MLR5 0.659

0.520 0.428 0.252 0.204

0.695 0.486 0.464 0.235

0.543 0.374 0.352 0.198

0.668 0.442 0.207 0.151

0.514 0.352 0.154 0.120

0.595 0.357 0.190 0.137

0.473 0.251 0.140 0.109

0.660 0.460 0.296 0.193

0.513 0.351 0.224 0.158

0.519 0.365 0.962

0.690 0.462 1.232

0.522 0.343 1.139

0.813 0.557 0.812

0.651 0.440 0.728

0.786 0.468 0.632

0.629 0.404 0.580

0.753 0.502 0.935

0.580 0.388 0.852

0.778 0.751 0.548 0.522

0.931 0.900 0.877 0.680

0.748 0.722 0.724 0.564

0.923 0.891 0.658 0.721

0.751 0.727 0.539 0.599

0.819 0.791 0.580 0.560

0.679 0.655 0.487 0.462

0.918 0.886 0.697 0.655

0.739 0.714 0.574 0.537

RMSE root mean squared error, MAE mean absolute error

comparison in the current work. These methods are presented as follows:
 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T max þ T min þ 17:8 T max −T min ð8Þ ET HS ¼ 0:0023Ra 2
ET PT ¼ 1:26
ET P ¼

Δ Δþg

Δ Δþg





1 ðRn −GÞ λ

ð9Þ

Rn g 1 þ ½15:36ð1 þ 0:0062U 2 Þðes −ea Þ Δþg λ λ

ð10Þ Where ETHS, ETPT, and ETP denotes the reference evapotranspiration estimated by Hargreaves, Priestley–Taylor, and Penman methods, respectively. Other symbols have the same meaning as those in Eq. (1). The error statistics of the ET0 values computed with the conventional models are also

presented in Table 2. We can see from the table that when only Tmax and Tmin were considered, GANN2 showed slightly lower accuracy compared with HS method at WW and LZ stations in terms of MAE values, but it apparently outperformed HS method at WG and YA stations in terms of both MAE and RMSE values. Kisi (2006) and Rahimikhoob (2008) also stated that ANN model outperformed HS method when only Tmax and Tmin were available. As GANN2, GANN3 showed better performance than PT equation at WG and YA stations but had slightly worse performance than the PT equation at the other two stations. We can also see from Table 2 that in respect of four stations averaged RMSE and MAE values, both GANN2 and GANN3 exhibited higher accuracy than the corresponding conventional equations. GANN5 had far better performance than the original Penman method, which also had the most input variables. These outcomes imply that GANN models are more adaptable to different environment compared with conventional methods. Empirical equations are usually developed

Table 3 Data range and variance (Cv) of the training dataset and testing dataset Variable

Tmax (°C) Tmin (°C) SR (MJ m−2day−1) RH (%) U2 (m s−1)

Training Dataset

WW

LZ

WG

YA

Range

Cv

Range

Cv

Range

Cv

Range

Cv

Range

Cv

−19.5∼39.8 −27.0∼28.9 2.8∼30.2 7∼100 0∼10.8

0.75 2.13 0.47 0.33 0.59

−6.2∼34.6 −22.7∼21.2 3.9∼29.9 21∼96 0.3∼4.8

0.71 13.78 0.38 0.27 0.52

−4.8∼35.6 −14.2∼21.6 4.0∼29.1 20∼88 0∼3.3

0.58 1.74 0.45 0.27 0.59

0.6∼39.3 −8.2∼26.5 4.3∼29.5 26∼95 0∼4.5

0.50 1.06 0.46 0.21 0.38

3.6∼34.9 0.7∼25.4 6.5∼26.7 53∼99 0∼4.3

0.41 0.50 0.45 0.12 0.61

Z. Wang et al.

using local-related data, and the equations are site dependent. Allen et al. (1994) suggested that empirical methods be calibrated using the standard FAO-56 PM method. But no calibration was done for the equations used in this work because we supposed that there was no history data for local calibration for the test stations. Given that the relationship between climate factors and ET0 is nonlinear and highly complex, conventional equations cannot capture this relationship for every climatic pattern. However, trained GANN models can effectively simulate this complex relationship because of their capacity for nonlinear simulation. 4.3 Comparison of GANN models with MLR models Multivariate linear regression (MLR) is a method used to model the linear relationship between a dependent variable and one or more independent variables based on least squares. Since the climatic parameters are highly correlated with ET0, many authors adopted the linear models to estimate ET0. MLR models were also established in this study. The input variables of the MLR models correspond to those of the ANN models. The training dataset of the ANN models were used to develop the MLR models. Finally, four MLR models were constructed in the following forms: ET MLR2 ¼ −0:3028 þ 0:2234T max −0:0963T min ET MLR3 ¼ −0:3250 þ 0:2129T max −0:0889T min þ 0:0101SR ET MLR4 ¼ 3:6855 þ 0:0871T max þ 0:0407T min þ 0:0077SR−0:0433RH ET MLR5 ¼ 2:4553 þ 0:1031T max þ 0:0225T min þ 0:0078SR−0:0371RH þ 0:3650U 2

ð11Þ

ð12Þ

ð13Þ

ð14Þ

Where ETMLR2, ETMLR3, ETMLR4, and ETMLR5 denote the reference evapotranspiration calculated by MLR2, MLR3, MLR4, and MLR5, respectively. The performance of the MLR models in terms of RMSE and MAE values are also presented in Table 2. It is evident from Table 2 that the MLR models resulted in higher RMSEs and MAEs than the corresponding ANN models, especially more inputs were considered. This adds further justification to the choice of the neural networks. Many researchers also proved the superiority of ANN model over MLR model in modeling ET0 (Jain et al. 2008; Dai et al. 2009; Laaboudi et al. 2012). However, the ANN model is not that transparent compared to MLR model, from which some information about the

physics of the process can be extracted (Izadifar and Elshorbagy 2010). Some other techniques should be applied to identify the contribution of each variable, when the ANN approach is adopted.

5 Conclusions The great capability of the neural approach in modeling ET0 has been proved by many researchers. However, the neural network developed with climatic data from single station has the limitation of regional validity. So the present study tried to develop generalized ANN models using weather data from four weather stations located in different climatic regions. Weather data from other four distinct locations were collected to test the models. The results showed that the more the input variables, the better the performance of the GANN model. All the GANN models exhibited high accuracy under both arid and humid conditions and performed better than the conventional methods and MLR models with the same inputs. This is quite encouraging and demonstrates the potential of ANN models in simulating ET0 with weather data of large variance. However, the ANN models developed in this study resulted in slightly larger error in arid and semiarid areas. Longer time series weather data from more stations located at different climatic zones should be collected to develop more representative neural networks for ET0 modeling. Acknowledgments This work is jointly supported by the Special Foundation of National Science & Technology Supporting Plan (2011BAD29B09), the ‘111’ Project from the Ministry of Education and the State Administration of Foreign Experts Affairs (B12007), and the Supporting Plan of Young Elites and basic operational cost of research from Northwest A & F University. The anonymous reviewers are thanked for their constructive comments, which substantially improved the manuscript.

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