Theoretical and Applied Climatology https://doi.org/10.1007/s00704-018-2390-z

ORIGINAL PAPER

Temperature-based modeling of reference evapotranspiration using several artificial intelligence models: application of different modeling scenarios Hadi Sanikhani 1 & Ozgur Kisi 2 & Eisa Maroufpoor 1 & Zaher Mundher Yaseen 3 Received: 17 July 2017 / Accepted: 18 January 2018 # Springer-Verlag GmbH Austria, part of Springer Nature 2018

Abstract The establishment of an accurate computational model for predicting reference evapotranspiration (ET0) process is highly essential for several agricultural and hydrological applications, especially for the rural water resource systems, water use allocations, utilization and demand assessments, and the management of irrigation systems. In this research, six artificial intelligence (AI) models were investigated for modeling ET0 using a small number of climatic data generated from the minimum and maximum temperatures of the air and extraterrestrial radiation. The investigated models were multilayer perceptron (MLP), generalized regression neural networks (GRNN), radial basis neural networks (RBNN), integrated adaptive neuro-fuzzy inference systems with grid partitioning and subtractive clustering (ANFIS-GP and ANFIS-SC), and gene expression programming (GEP). The implemented monthly time scale data set was collected at the Antalya and Isparta stations which are located in the Mediterranean Region of Turkey. The Hargreaves–Samani (HS) equation and its calibrated version (CHS) were used to perform a verification analysis of the established AI models. The accuracy of validation was focused on multiple quantitative metrics, including root mean squared error (RMSE), mean absolute error (MAE), correlation coefficient (R2), coefficient of residual mass (CRM), and Nash–Sutcliffe efficiency coefficient (NS). The results of the conducted models were highly practical and reliable for the investigated case studies. At the Antalya station, the performance of the GEP and GRNN models was better than the other investigated models, while the performance of the RBNN and ANFIS-SC models was best compared to the other models at the Isparta station. Except for the MLP model, all the other investigated models presented a better performance accuracy compared to the HS and CHS empirical models when applied in a cross-station scenario. A cross-station scenario examination implies the prediction of the ET0 of any station using the input data of the nearby station. The performance of the CHS models in the modeling the ET0 was better in all the cases when compared to that of the original HS.

1 Introduction One of the major components of the hydrological cycle is the evapotranspiration process which is widely used for agricultural and irrigation management, water resources planning, * Hadi Sanikhani [email protected] 1

Water Eng. Department, Faculty of Agriculture, University of Kurdistan, Sanandaj, Iran

2

Faculty of Natural Sciences and Engineering, Ilia State University, Tbilisi, Georgia

3

Sustainable Developments in Civil Engineering Research Group, Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam

and other watershed sustainability (Kisi 2007; Rana and Katerji 2000a; Trajkovic 2005). The agriculture sector is the main consumer of water; increasing the efficiency and productivity of water in this sector has a significant contribution in its maintenance (Kisi et al. 2015a; Tabari et al. 2013). The prediction of ET0 in the global and local scales has always been crucial. ET0 can be predicted using different methods such as mass transfer method, water budget, and radiation temperaturebased methods (Gocic and Trajkovic 2010). From another aspect, several techniques like direct configuration (i.e., lysimeter) and indirect configuration such as artificial intelligence (AI) models are used remarkably for the prediction of ET0 (Fahimi et al. 2016). In the previous studies, FAO-56 Penman–Monteith (FAO56-PM) equation has been applied as a reference

H. Sanikhani et al.

model for the calculation of ET 0 (Penman 1948). The American Society of Civil Engineers (ASCE) predicted the monthly irrigation water requirement in 20 diverse approaches for different regions and compared the outcome with the results obtained from lysimeters and concluded that the PM method had the best estimation (Jensen et al. 1990). The main problem with this equation is that it requires a large number of meteorological parameters and scale (Trajkovic 2005). However, many countries either do not have the equipment required for keeping an accurate recording of these parameters or data are not recorded on a regular basis and have some deficiencies. Hence, the establishment of the sophisticated soft computing models is necessary for this kind of climatic problem (Rana and Katerji 2000b; Zhao et al. 2013). Over the past two decades, AI models, including neural network (NN), fuzzy logic (FL) (e.g., adaptive neuro-fuzzy inference system (ANFIS)), and evolutionary computing (e.g., gene expression programming (GEP)), have shown an outstanding application in modeling ET 0 (Abyaneh et al. 2011; Chauhan and Shrivastava 2009; Kisi 2008; Kisi and Cengiz 2013; Kumar et al. 2011; Landeras et al. 2009; Parasuraman et al. 2007). Many types of research have proven the successfulness of NN, ANFIS, and GEP in providing a reliable predictive system for hydrological processes (Zhao et al. 2013). Artificial neural networks (ANNs) are suitable tools due to their ability in the simulation of the nonlinear behavior of complex phenomena such as ET0. Many applications of ANNs have been reported on the modeling of ET0 prediction (Goci et al. 2015; Kisi et al. 2015b; Kumar et al. 2002; Pandorfi et al. 2016). One of the earliest researches was conducted by Trajkovic (2005). In this study, the performance of different temperature-based methods such as radial basis function (RBF) algorithm, Thornthwaite, and Hargreaves–Samani (HS) equation was evaluated by comparing them with a standard method (PM). The results indicated that the efficiency of RBF was better than the others. Zanetti et al. (2007) applied ANNs for the computing of ET0 using the minimum and maximum air temperatures for Campos dos Goytacazes county, State of Rio de Janeiro, for the data period 1996–2002. Kumar et al. (2011) presented a complete review of ANN application in evapotranspiration modeling and evidenced their predictability. Another attempt was conducted on the modeling of ET0 in the northern arid region of China using ANNs (Huo et al. 2012). Most recently, Traore et al. (2016) applied various types of ANNs for a shortterm prediction of evapotranspiration. They found that multilayer perceptron network provided a better prediction compared to the other methods. There have been many research attentions in the field of ET0 prediction due to the benefits for practical implementation (Abyaneh et al. 2011; Baba et al. 2013; Cobaner 2011;

Karimaldini et al. 2012; Kisi et al. 2015b; Ladlani et al. 2014; Petković et al. 2015a; Tabari et al. 2012). Using other version of soft computing model, the prediction of the daily ET0 with atmospheric parameters using the co-active ANFIS has been proposed by Aytek (2009). The performance accuracy of two types of ANFIS models such as adaptive neuro-fuzzy inference systems with grid partitioning (ANFIS-GP) and subtractive clustering (ANFISSC) on the estimation of ET0 has been compared by Cobaner (2011). The results indicated the superiority of ANFIS-SC over ANFIS-GP with fewer computations. The determination of the parameters that influence the estimation of ET0 using ANFIS has been investigated by Petković et al. (2015b). They reported that the most important variables for the ET0 prediction were the hours of sunshine, the minimum air temperature, and the actual vapor pressure. In general, the FL method performed noticeably in modeling ET0 over the other conceptual predictive models. On the other hand, GEP provided an explicate relation between the input and output variables as a powerful method for application in different fields of engineering (Kisi et al. 2015b; Mehdizadeh et al. 2017; Shiri et al. 2012a, b, 2014; Yassin et al. 2016). Some of the applications of genetic programming (GP) and GEP for ET0 prediction are presented here. The performance of the GP model has been verified and compared to the classical NN model and PM formulation for ET0 prediction (Parasuraman et al. 2007). The results indicated that GP can successfully simulate the dynamics of ET0 process compared to the other models. Another study has evaluated the effectiveness of the GEP model in the prediction of ET0 through the utilization of climatic information (Traore and Guven 2013) and found GEP as a powerful tool for a successful ET0 computing. Most recently, Yassin et al. (2016) used GEP and ANNs for determining the value of ET0 process in an arid climate of Saudi Arabia. The authors reported that the performance of ANNs in ET0 estimation was better compared to the GEP model. Studies on the comparison of different soft computing skills such as the NN models, ANFIS, GEP, hybrid models, and other empirical methods such as the Penman– Monteith, HS, and CHS on the ET0 prediction are still limited. This study aims to investigate the ET0 prediction accuracy of multilayer perceptron (MLP), generalized regression neural network (GRNN), radial basis function neural network (RBNN), ANFIS-GP, ANFIS-SC, GEP, HS, and CHS models using the monthly data from two stations (Antalya and Isparta) located in Turkey. The study focused on the temperature-based modeling for the ET0 prediction. This is necessary for our study because such a developing nation and ungauged basins are often faced with limited data availability. As a second objective, cross-station data scenario is inspected between the implemented meteorological stations.

Temperature-based modeling of reference evapotranspiration using several artificial intelligence models:...

2 Hydrological data description The data of two meteorological stations situated in the Mediterranean Region of Turkey (Antalya 36°53′N, 30°40′E, elevation = 47 m and Isparta 37°47′N, 30°34′E, elevation = 997 m) over the time period 1982–2006 were selected as a case study (Fig. 1). The Mediterranean zone is characterized by hot and dry during the summer season and rainy during the winter season. In general, the climate has an annual mean temperature of 18 °C and a precipitation value approximately 1100 mm, of which the majority occurs between October and April (Canakci et al. 2005). The climate variables included in the prediction modeling are solar radiation (SR), relative humidity (RH), minimum and maximum air temperatures (T min and T max ), and wind speed (W) over a 25-year period on a monthly scale. A summary of the statistical measures of the used data is tabulated in Table 1. Apparently, the correlation between the Tmax (or Tmin) and ET0 of the two stations was high (0.865 (0.787) and 0.892 (0.834)) for Antalya and Isparta stations, respectively. The statistical properties of the data from Antalya and Isparta stations differed from each other. The SR of the stations had the highest correlation with ET0 while W showed the least correlation. The correlation of each variable with ET0 was higher for the Isparta station compared to the Antalya station. The value of the ET0 ranged from 1.16 to 10.41 mm/day with a mean value of 5.63 mm/day for the Antalya station, and 0.69–6.80 mm/ day with a mean value of 13.52 mm/day for the Isparta station.

3 Predictive model overview 3.1 Penman–Monteith method The Penman–Monteith (PM) equation is derived based on the Penman classic equation proposed in 1948 for the ET0 estimation using various meteorological variables such as air

temperature, W, SR, and RH (Penman 1948). The mathematical formula of PM is given as:
 900 u2 ðes −ea Þ 0:408 Δ ðRn −GÞ þ γ ðT þ 273Þ ET0−PM ¼ ð1Þ Δ þ γ ð1 þ 0:34 u2 Þ where ET0 − PM is the reference evapotranspiration rate in mm/ day, G is the soil heat flux density (MJ m−2 day−1), Rn is the net radiation (MJ m−2 day−1), T is the mean air temperature (°C), Δ is the slope of the saturation vapor pressure versus air temperature curve (kPa °C−1), γ is the psychrometric constant (kPa °C−1), u2 is the average 24-h wind speed over the earth surface by 2 m (m s−1), and es and ea are the saturation and actual vapor pressure (kPa), respectively (Allen et al. 1998).

3.2 Hargreaves–Samani and its modified (calibrated) version The ET0 can be predicted using the Hargreaves–Samani (HS) equation given as (Hargreaves and Samani 1985): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ET0−HS ¼ 0:0023 Ra ðT mean þ 17:8Þ ðT max −T min Þ ð2Þ where Ra indicates the water equivalence of the extraterrestrial radiation (mm/day) and Tmax and Tmin are the maximum and minimum values of temperature (°C), respectively. Trajkovic (2005) demonstrated that the temperature-based approaches (i.e., HS model) either underestimate or overestimate the ET0 computed by the PM model. To overcome this issue, the empirical method can be calibrated (Allen et al. 1998). In the present study, the following equation was obtained for each station by calibrating a, b, and c coefficients: ET0−CHS ¼ a⋅Ra ðT mean þ bÞðT max −T min Þc

ð3Þ

where a, b, and c are the empirical calibration coefficients and ET0 − CHS is ET0 computed by the calibrated HS. It should be noted that c value is 0.5 in the HS equation given in Eq. (2).

3.3 Artificial neural network

Turkey

Isparta Antalya

Fig. 1 The location of the investigated meteorological stations (Antalya and Isparta) in Mediterranean Region of Turkey

The framework of the human brain is the basic knowledge of ANN model. The ANNs have been used for different applications in science, such as in pattern recognition, clustering, and simulation. The ANNs can be learned intelligently to map a set of input/output data and find the function approximators. Different algorithms have been constructed from the ANNs based on different structures. The MLP as a static ANN is the most predominant algorithm used in various engineering fields. The RBNN and GRNN are also other common statistical neural networks that have been developed based on statistical estimation.

H. Sanikhani et al. Table 1 The statistical parameters of data set for the stations. Xmean, Xmin, Xmax, Sx, Cv, and Csx indicate the mean, minimum, maximum, standard deviation, coefficient of variation, and skewness, respectively

Station

Antalya

Data set

Unit

Xmean

Xmin

Xmax

Sx

Cv

Csx

Correlation with ET0-PM

Tmax

°C °C

30.40 8.87

16.20 − 3.00

44.20 21.60

7.96 6.90

0.26 0.78

− 0.05 0.18

0.865 0.787

m/s Langley

2.64 401.68

0.90 119.60

4.90 679.20

0.66 153.41

0.25 0.38

0.17 − 0.07

0.002 0.924

RH ET0 Tmax Tmin W SR

% mm/day °C °C

56.18 5.63 24.75 − 0.09

45.50 1.16 7.00 − 18.50

68.50 10.41 37.80 14.20

4.20 2.05 8.25 7.65

0.07 0.36 0.33 − 85.63

0.18 0.17 − 0.27 − 0.01

− 0.379 1.000 0.892 0.834

m/s Langley

1.76 325.35

0.60 112.30

3.60 657.10

0.51 122.69

0.29 0.38

0.45 0.13

0.001 0.941

RH

% mm/day

60.44 3.52

46.00 0.69

72.50 6.80

4.97 1.52

0.08 0.43

− 0.38 0.01

− 0.724 1.000

Tmin W SR

Isparta

ET0

3.3.1 Multilayer perceptron The MLP is one of the common training approaches implemented for ANN models and has been effectively applied for several engineering purposes. The MLP has three distinctive properties: firstly, in the network, the model of each neuron consists of a nonlinear activation function; secondly, the network includes one or more layers of hidden neurons that help the network to learn complex tasks; and thirdly, the network exhibits a high degree of connectivity. Different training algorithms can be used in the MLP method. In this research, the Levenberg–Marquardt algorithm was applied. For more details on the MLP approach, please refer to the survey by Haykin (1999). 3.3.2 Generalized regression neural network In 1991, the theoretical aspect of GRNN, which is a model in the group of the probabilistic neural networks, was proposed by Specht. The ability of the probabilistic neural networks to converge the underlying data function with limited training data made them attractive to researchers. There are four layers of the GRNN, which are the input, pattern, summation, and output layers. The input number is usually the same as the total number of parameters. Every unit in the pattern layer is a training pattern whose output determines the input distance from the stored pattern. The pattern layers are connected to the summation layer by two neurons—S-summation and Dsummation. More theoretical details have been provided by Specht (1991). 3.3.3 Radial basis function neural network The RBNN was originally formulated in 1988 by Lowe and Broomhead. The radial basis functions (RBF) of the RBNN serve as the activation function (Lowe and

Broomhead 1988). The output of the network is generated by the linear combination of the RBF input and neuron parameters. The responsibility of the RBF is to interpolate data sets in a multi-dimensional space, and it is their main advantage over the others. The RBNN, in practical terms, is made up of three layers: the first layer where the input feature vectors are fed into the network by the input neuron; the hidden layer where the RBF neurons compute the outcome of the basis function; and the output layer where the output neurons compute the linear combination of the basis function. Bishop (1995) provided more insight into the theoretical aspect of the RBNN.

3.4 Adaptive neuro-fuzzy inference system Jang developed the ANFIS technique in 1991 as a powerful tool for natural operation modeling. Due to its imperfections, it has not been recognized as an authentic tool yet (Kurtulus and Razack 2010). It operates on the ANN and FL rules together and, therefore, has the capability of boosting the combined advantages of the two models that made it up in a single structure. There are fuzzy IF-THEN rules in the architecture of the ANFIS that can approximate nonlinear functions. To identify the parameters of the Sugeno-type fuzzy inference systems, the ANFIS adopts a hybrid learning algorithm such as the least square and backpropagation gradient descent methods. Different kinds of membership functions (MFs) can be applied to perform the ANFIS model. The MFs are the curves that determine how each pattern in the input variables is traced to a degree of membership between 0 and 1. Among various MFs, triangular and Gaussian MFs are prevalent and suggested for practical usage (Russel and Campbell 1996). The model would work based on trial and error with two or three MFs. The schematic structure of ANFIS with two inputs is shown in Fig. 2. In this modeling application, two

Temperature-based modeling of reference evapotranspiration using several artificial intelligence models:...

ANFIS-GP, there must be a maximum of six input variables. This paper applied three input variables for the estimation of the ET0 values, thereby fulfilling the right model condition requirement. 3.4.2 ANFIS-SC

Fig. 2 The architecture of an ANFIS with 2 input variables and 5 layers. Layer 1 (input fuzzy rules); Layer 2 (input MFs); Layer 3 (fuzzy neurons); Layer 4 (output MFs); Layer 5 (summation and weights). w = weights; x, y = metrological inputs; P1 and P2 = fuzzy rules; Q1, Q2 = fuzzy rules; x, y (with arrows = input-target in training phase); ET0 = target output

different methods (GP and SC) were used for the tuning of the ANFIS model. 3.4.1 ANFIS-GP The ANFIS-GP is an integration of the ANFIS model with a grid partition method. The input space, by the application of GP, can be categorized into rectangular subspaces using a series of local fuzzy regions. With this, the type and the number of predefined MFs in each dimension have a major role to play. The optimization process of the MF locations can be facilitated by the partitioning of the fuzzy grids (Abonyi et al. 1999). This study applied different MFs such as the trapezoidal, Gaussian, pi-shaped, and generalized bell subjected to 100 iterations for the ANFIS-GP models. The best MF number was selected after an evaluation of the minimum root mean squared error value of the different MFs. To use the Table 2 Test results of the applied models in modeling ET0—Antalya station

The ANFIS-SC is a hybrid model formulated from the combination of the ANFIS and the subtractive clustering (SC) approach. Each data point in this model is a potential cluster center. The influential radius determines the number of clusters; a small radius is synonymous to small cluster in the data space, therefore requiring additional rules (Chiu 1997). The generalization of the fuzzy rules using ANFIS-SC requires obtaining the right cluster radius. This radius, which can be determined through trial and error methods based on minimum root mean squared error (RMSE) values, is in the range of 0 and 1. Like the ANFIS-GP, the ANFIS-SC epoch number in this study was set at 100.

3.5 Gene expression programming The genetic algorithm (GA), GP, and GEP originated from biological evolution and basically have a similar architecture. In the GEP, the problem is transformed into codes of fixed length linear chromosomes. GEP uses parse tree to find the solution to problems (Ferreira 2001; Hardy and Steeb 2002). GEP has some advantages compared to other intelligent models such as ANN and ANFIS. The main advantage of GEP is the representation of an explicit equation between the input variables and the target parameter. However, the

Models

Parameters

RMSE mm/day

MAE mm/day

R2

CRM

NS

MLP GRNN RBNN ANFIS-GP ANFIS-SC GEP HS CHS

(3,1,1)******* Spread = 0.05****** (3,9,0.4,1)***** Gaussian (2,2,2)**** Radii = 0.95*** – (0.0023, 0.5, 17.78)** (0.000241, 0.6376898, 87.15)*

0.498 0.510 0.524 0.574 0.494 0.498 8.018 0.878

0.397 0.381 0.383 0.425 0.390 0.397 6.869 0.766

0.897 0.906 0.905 0.903 0.901 0.905 0.892 0.878

− 0.005 − 0.004 − 0.008 − 0.009 − 0.015 0.007 1.220 − 0.037

0.913 0.923 0.920 0.915 0.898 0.924 − 18.89 0.819

*

The numbers represent the optimal values of a, b, and c of CHS equation given in Eq. (3)

**

The numbers represent the constant values of the HS equation given in Eq. (2)

***

0.95 indicates the radii value of the ANFIS-SC model

****

Gaussian (2,2,2) refers to an ANFIS-GP model comprising 2 Gaussian membership functions for the inputs of Tmin, Tmax, and Ra

*****

(3,9,0.4,1) indicates a RBNN model having 3 inputs, 9 hidden nodes, 0.4 spread constant, and 1 output

******

0.05 indicates the spread constant of the GRNN model

*******

(3,1,1) indicates a MLP model consisting 3 inputs and 1 hidden and 1 output nodes

H. Sanikhani et al.

method. More details on the application of GEP in modeling can be found in Sanikhani et al. (2015).

3.6 Performance skill metrics The prediction skills of the conducted AI models and the empirical methods were validated using several statistical metrics including RMSE, mean absolute error (MAE), correlation

Estimates-MLP, mm/day

25 20 15 10

y = 3.5127x + 0.1108 R² = 0.8919

5 0

4 3 2 1 0 0

1 2 3 4 5 6 PM-FAO 56 ET0, mm/day y = 0.8655x + 0.4817 R² = 0.9037

5 4 3 2 1 0 0

1 2 3 4 5 6 PM-FAO 56 ET0, mm/day

7

y = 0.8748x + 0.4692 R² = 0.9051

5 4 3 2 1 0 0

3 2 1 0

1 2 3 4 5 6 PM-FAO 56 ET0, mm/day

7

1 2 3 4 5 6 PM-FAO 56 ET0, mm/day

7

7 y = 0.887x + 0.4157 R² = 0.9051

6 5 4 3 2 1 0

1 2 3 4 5 6 PM-FAO 56 ET0, mm/day

7

7 y = 0.8665x + 0.4802 R² = 0.9006

6 5 4 3 2 1 0 0

7 6

4

0

7 6

5

7

Estimates-ANFIS-SC, mm/day

Estimates-ANFIS-GP, mm/day

Estimates-RBNN, mm/day

y = 0.8812x + 0.4403 R² = 0.9067

5

y = 0.8609x + 0.4738 R² = 0.8969

6

0

7 6

7

1 2 3 4 5 6 7 PM-FAO 56 ET0, mm/day

Estimates-CHS, mm/day

Estimates-GRNN, mm/day

0

Estimates-GEP, mm/day

Fig. 3 The scatterplots between PM-FAO 56 ET0 and estimated values using the MLP, RBNN, GRNN, GEP, ANFIS-GP, ANFIS-SC, HS, and CHS models in testing phase at the Antalya station (dotted line indicates the best fit line)

Estimates-HS, mm/day

disadvantage of GEP is its increase in the parse tree depth which results in raising the nested functions in some cases (Ryan and Hibler 2011; Yaseen et al. 2015). The procedure for modeling ET0 using GEP model includes choosing the fitness function, selecting a set of terminals and functions, selecting the structure of the chromosomes, and selecting the linking function and genetic operators. In this study, the Genexpro software was utilized to model ET0 via the GEP

10

1 2 3 4 5 6 PM-FAO 56 ET0, mm/day

7

y = 1.0796x - 0.6548 R² = 0.8782

8 6 4 2 0

0

2 4 6 8 PM-FAO 56 ET0, mm/day

10

Temperature-based modeling of reference evapotranspiration using several artificial intelligence models:...

coefficient (R2), coefficient of residual mass (CRM), and Nash–Sutcliffe efficiency coefficient (NS). Mathematically, the performance metrics can be described as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 ∑Ni¼1 ETPM;i −ET M;i ð4Þ RMSE ¼ N   ETPM;i −ETM;i  ð5Þ MAE ¼ N    ∑Ni¼1 ETPM;i −ET0 ETM;i −ETM 2 2 R ¼ ð6Þ  2  2 ∑Ni ETPM;i −ET0  ∑Ni ETM;i −ETM 

∑N ETPM;i −∑Ni¼1 ETM;i CRM ¼ i¼1 N ∑i¼1 ETPM;i



 2 ∑Ni¼1 ETPM;i −ETM;i NS ¼ 1−  2 ∑Ni¼1 ETPM;i −ET0

ð7Þ

4.1 Scenario I In this sub-section, the comparison of the six different AI models with the HS and CHS empirical methods for the estimation of ET0 at Antalya and Isparta stations was focused. Different control parameter values were strained for each of the applied predictive models. Table 2 showed the optimal results of the prediction accuracies at the Antalya station. It can be seen based on the presented results that the accurate MLP model was attained when three input corresponding variables were included (i.e., Tmin, Tmax, and Ra) with 1 hidden and 1 output nodes, respectively. The RBNN model performed its best application through (3,9,0.4,1), having three input variables corresponding to Tmin, Tmax, and Ra, 9 hidden nodes, 0.4

ð8Þ

where, ETPM, i is the calculated ET0 by the Penman–Monteith method (mm/day), ETM, i is the predicted ET0 values by the different models (mm/day), ET0 is the mean value of PM ET0 values (mm/day), ETM is the mean value predicted by the various models (mm/day), and N is the number of data set.

4 Application, results, and discussion In the hydrological field, obtaining an informative value of the hydrology cycle processes is significant for decisionmakers and planners. Particularly, knowing the future trend of the ET0 is extremely valuable for climatological, agricultural, and water resource engineering. In this research, intelligent models were developed and applied for temperaturebased modeling of ET0. The obtained results of the AI models were compared with empirical formulas of PM, HS, and calibrated HS (CHS). The established modeling was based on two different scenarios: (i) comparing the six different AI models and HS and CHS empirical methods for the estimation of the ET0 of Antalya and Isparta stations, separately; and (ii) investigating the same predictive model and comparing their performance with the empirical formulation based on a cross-station application. The cross-station application scenario was built based on constructing the input variables of the Antalya station, while the target ET0 value was at the Isparta station. The second scenario is highly beneficial for the watershed that lacks meteorological information or monitoring stations; hence, the diagnosis of the applicability of the intelligent models is practically efficient. Both scenarios were verified and evaluated using several statistical metrics defined in section 3.6.

Fig. 4 The expression tree of the GEP model for the Antalya station (d0, d1, and d2 indicate the Tmin, Tmax, and Ra and co and c1 are the coefficients; refer to Eq. 9)

H. Sanikhani et al.

spread constant, and 1 output. Gaussian (2,2,2) refers to an ANFIS-GP model comprising of 2 Gaussian membership functions for the inputs of Tmin, Tmax, and Ra. The optimal spread and radii numbers were 0.05 and 0.95 for GRNN and ANFIS-SC models, respectively. New coefficients of the calibrated Hargreaves–Samani equation are provided in Table 2, where 0.000241, 0.6376898, and 87.15 represent the optimal values of a, b, and c given in Eq. (3). Apparently, the statistical prediction skills showed that GEP performed better than the predictive models, particularly with respect to CRM, NS, and R2 measures. There was a slight difference between GEP and GRNN models with respect to NS, while the RMSE and MAE values of the GEP, MLP, and ANFIS-SC were close to each other. This is expected because these statistics provided different forms of information on the accuracy of applied models. NS is a normalized measure that identifies the relative magnitude of the residual variance (Bnoise^) compared to the variance of the observed data (Nash and Sutcliffe 1970), while RMSE and MAE calculate the square and absolute differences and provide information on the exactness of the applied models in catching high and average values, respectively. The accuracy rank of the applied models was GEP, GRNN, RBNN, ANFIS-GP, MLP, ANFIS-SC, HS, and CHS with respect to NS. It is apparent from the table that the calibration of the coefficients considerably improved the accuracy of the HS model. CHS increased the accuracy of the HS by 89, 89, 2, 103, and 104% with respect to RMSE, MAE, R2, CRM, and NS, respectively. Furthermore, the RMSE, MAE, and NS accuracies of the CHS were increased by 43, 48, and 119% Table 3 Test results of the applied models in modeling ET0—Isparta station

Models

Structure

MLP

(3,1,1)*******

GRNN RBNN ANFIS-GP ANFIS-SC GEP HS CHS *

******

Spread = 0.1 (3,13,2,1)***** Triangular (2,2,2)**** Radii = 0.99*** – (0.0023, 0.5, 17.78)** (0.00000243, 2.36118, 7.2668)*

respectively, using GEP. Figure 3 displayed the graphical presentation of the scatter plot of the applied AI predictive models for the Antalya station. There is no much difference between the AI models. The HS model considerably overestimated the PM-FAO 56 ET0 values, while the CHS performed better than the HS. The optimum GEP tree modeling was presented in Fig. 4. It was evident from the figure that the GEP model has only three sub trees, implying a simple equation. The derived GEP equation for the Antalya station was: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

 T min ET0 ¼ T min þ sin cosðT max Þ þ −3:699 0 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s

 !3 T min A þ @cos Ra þ −7:8678 7:4248 þ sin

pffiffiffiffiffi 3 Ra þ sinðRa Þ þ atanðT max Þ

ð9Þ

On the other hand, Table 3 tabulated the test phase results of the Isparta station. Clearly, the performance of the RBNN and ANFIS-SC models attained almost a similar level of prediction accuracies with respect to RMSE, MAE, R2, and CRM, and both models performed more accurately than the other predictive models in modeling the monthly ET0. The ranking of the prediction skills of the applied AI models was RBNN, ANFIS-SC, ANFIS-GP, MLP, GEP, GRNN, CHS, and HS with respect to RMSE and MAE measures. Similarly, for the Antalya station, the calibration of the HS

RMSE mm/ day

MAE mm/ day

R2

CRM

NS

0.447 0.474 0.430 0.444 0.431 0.469 8.126 1.021

0.351 0.381 0.332 0.335 0.326 0.357 6.862 0.903

0.923 0.910 0.929 0.924 0.927 0.913 0.892 0.923

− 0.153 0.013 0.006 0.009 0.003 0.009 1.947 0.013

0.575 0.919 0.908 0.925 0.920 0.924 − 25.90 0.910

The numbers represent the optimal values of a, b, and c of CHS equation given in Eq. (3)

**

The numbers represent the constant values of the HS equation given in Eq. (2)

***

0.99 indicates the radii value of the ANFIS-SC model

****

Triangular (2,2,2) refers to an ANFIS-GP model comprising 2 triangular membership functions for the inputs of Tmin, Tmax, and Ra

*****

(3,13,2,1) indicates a RBNN model having 3 inputs, 13 hidden nodes, 2 spread constant, and 1 output

******

0.1 indicates the spread constant of the GRNN model

*******

(3,1,1) indicates a MLP model consisting 3 inputs and 1 hidden and 1 output nodes

Temperature-based modeling of reference evapotranspiration using several artificial intelligence models:...

y = 3.5782x - 2.2268 R² = 0.9228

15 10 5 0

0

7 y = 0.8707x + 0.4767 R² = 0.9102

6 5 4 3 2 1 0

y = 0.863x + 0.495 R² = 0.9237

6 5 4 3 2 1 0 0

7

1 2 3 4 5 6 7 PM-FAO56 ET0, mm/day y = 0.8712x + 0.4995 R² = 0.9131

6 5 4 3 2 1 0 0

1 2 3 4 5 6 7 PM-FAO56 ET0, mm/day

Estimates-ANFIS-SC, mm/day

7

y = 0.8653x + 0.5213 R² = 0.9232

6 5 4 3 2 1 0

0

7

1 2 3 4 5 6 7 PM-FAO56 ET0, mm/day y = 0.8721x + 0.4815 R² = 0.9285

6 5 4 3 2 1 0

1 2 3 4 5 6 7 PM-FAO56 ET0, mm/day

0

7

1 2 3 4 5 6 7 PM-FAO56 ET0, mm/day y = 0.8827x + 0.446 R² = 0.9268

6 5 4 3 2 1 0 0

10

Estimates-CHS, mm/day

Estimates-ANFIS-GP, mm/day

0

7

1 2 3 4 5 6 7 PM-FAO56 ET0, mm/day Estimates-RBNN, mm/day

20

Estimates-GRNN, mm/day

Estimates-HS, mm/day

25

Estimates-GEP, mm/day

Fig. 5 The scatterplots between PM-FAO 56 ET0 and estimated values using the MLP, RBNN, GRNN, GEP, ANFIS-GP, ANFIS-SC, HS, and CHS models in testing phase at the Isparta station (dotted line indicates the best fit line)

the HS model was considerably improved by calibration. CHS increased the accuracy of the HS by 87, 87, 3, 99, and 104% with respect to RMSE, MAE, R2, CRM, and NS. Furthermore, the RMSE, MAE, and CRM accuracies of the CHS were increased by 54, 60, and 31% respectively, using GEP. It

Estimates-MLP, mm/day

equation considerably improved its accuracy. Figure 5 exhibited the scatter plot of the estimation of the MLP, GRNN, RBNN, ANFIS-GP, ANFIS-SC, GEP, HS, and CHS models. Like the previous application, there was no much difference between the performances of the AI models. The accuracy of

1 2 3 4 5 6 7 PM-FAO56 ET0, mm/day y = 1.1676x - 1.129 R² = 0.8302

8 6 4 2 0

0

2 4 6 8 10 PM-FAO 56 ET0, mm/day

H. Sanikhani et al.

difficulties because of the higher PM-FAO 56 ET0 values in the test period compared to the training period. The other reason for this may be the fact that the data number of high ET0 values was not enough for an adequate learning of the data-driven methods compared to the average ones. The optimum GEP tree modeling is presented in Fig. 6. In this station, the model also has only three subtrees. The GEP equation derived for the Isparta station was:

ET0 ¼

sinð9:5442−T min Þ−4:9587Ra Ra qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 3 þ Ra þ T min :e−1:05978 þ ð9:9871T min − ðRa −8:6857ÞÞ1=3 −logðT min Þ ð10Þ

Fig. 6 The expression tree for the GEP model for the Isparta station (d0, d1, and d2 indicate the Tmin, Tmax, and Ra and co and c1 are the coefficients; refer to Eq. 10)

should be noted that all the AI models generally underestimated the peak ET0 values in both Antalya and Isparta stations (see Figs. 3 and 5). The difference between the training and testing data ranges may be the reason for this performance. The models encountered extrapolation

Table 4 Test results of the applied models in modeling ET0 of the Isparta station using input data of the Antalya station crossstation application

To validate the current research with the literature studies, Rahimikhoob (2010) used MLP to estimate ET0 based on the maximum and minimum air temperatures and extraterrestrial radiation. He found RMSE values of 0.41, 0.46, 0.38, and 0.37 mm/day and R2 of 0.93, 0.94, 0.96, and 0.95 for the Noshahr, Babolsar, Ramsar, and Sari stations respectively, located in Iran. Wang et al. (2011) modeled ET0 with MLP using maximum and minimum air temperatures and extraterrestrial radiation inputs and found RMSE of 0.4, 0.4, and 0.3 mm/day and R 2 of 0.80, 0.81, and 0.92 for Dori, Bogande, and Fada N’Gourma respectively, in Burkina Faso. It is clear from Tables 2 and 3 that the applied AI models generally provided accurate results in modeling ET0 from the RMSE and R2 viewpoints.

Models

Structure

RMSE mm/day

MAE mm/day

R2

CRM

NS

MLP

(3,1,1)*******

GRNN RBNN ANFIS-GP ANFIS-SC GEP HS CHS

Spread = 0.125****** (3,7,1,1)***** Triangular (2,2,3)**** Radii = 0.87*** – (0.0023, 0.5, 17.78)** (0.000251, 0.61095, 50.218)*

0.507 0.481 0.484 0.490 0.497 0.485 8.126 0.554

0.395 0.383 0.384 0.384 0.385 0.367 6.862 0.416

0.897 0.907 0.901 0.904 0.901 0.905 0.892 0.879

− 0.153 0.013 0.006 0.009 0.003 0.009 1.947 0.013

0.575 0.919 0.908 0.925 0.920 0.924 − 25.90 0.910

*

The numbers represent the optimal values of a, b, and c of CHS equation given in Eq. (3)

**

The numbers represent the constant values of the HS equation given in Eq. (2)

***

0.87 indicates the radii value of the ANFIS-SC model

****

Triangular (2,2,3) refers to an ANFIS-GP model comprising 2 triangular membership functions for the inputs of Tmin, Tmax, and Ra

*****

(3,7,1,1) indicates a RBNN model having 3 inputs, 7 hidden nodes, 1 spread constant, and 1 output

******

0.125 indicates the spread constant of the GRNN model

*******

(3,1,1) indicates a MLP model consisting 3 inputs and 1 hidden and 1 output nodes

Temperature-based modeling of reference evapotranspiration using several artificial intelligence models:...

20 15 10 5 0

0

1 2 3 4 5 6 7 PM-FAO56 ET0, mm/day

7 6

y = 0.8812x + 0.4403 R² = 0.9067

5 4 3 2 1 0

0

1 2 3 4 5 6 7 PM-FAO56 ET0, mm/day

7

y = 0.8655x + 0.4817 R² = 0.9037

6 5 4 3 2 1 0

7 6 5 4 3 2 1 0

0

1 2 3 4 5 6 7 PM-FAO56 ET0, mm/day y = 0.8748x + 0.4692 R² = 0.9051

0

1 2 3 4 5 6 7 PM-FAO56 ET0, mm/day

Estimates-MLP, mm/day

y = 3.5127x + 0.1108 R² = 0.8919

Estimates-RBNN, mm/day

25

Estimates-ANFIS-SC, mm/day

Estimates-GRNN, mm/day Estimates-ANFIS-GP, mm/day Estimated-GEP, mm/day

Fig. 7 The scatterplots between PM-FAO 56 ET0 and estimated values using the MLP, RBNN, GRNN, GEP, ANFIS-GP, ANFIS-SC, HS, and CHS models in testing phase at the Isparta station using the metrological information of the Antalya station as input variables (dotted line indicates the best fit line)

Estimates-HS, mm/day

In this scenario, a cross-station prediction based on Isparta and Antalya was conducted. This scenario is essential because in

some stations, especially in developing countries such as Turkey, some meteorological data are missing and should be estimated using data from nearby stations. The accuracy of the temperature-based models (MLP, GRNN, RBNN, ANFIS-

Estimates-CHS, mm/day

4.2 Scenario II

7 6 5 4 3 2 1 0

y = 0.8609x + 0.4738 R² = 0.8969

0

1 2 3 4 5 6 7 PM-FAO56 ET0, mm/day

7

y = 0.8665x + 0.4802 R² = 0.9006

6 5 4 3 2 1 0

0

1 2 3 4 5 6 7 PM-FAO56 ET0, mm/day

7

y = 0.8665x + 0.4802 R² = 0.9006

6 5 4 3 2 1 0

7 6 5 4 3 2 1 0

0

1 2 3 4 5 6 7 PM-FAO56 ET0, mm/day y = 1.0459x - 0.088 R² = 0.8786

0

1 2 3 4 5 6 7 PM-FAO56 ET0, mm/day

H. Sanikhani et al.

GP, ANFIS-SC, and GEP) was estimated for the determination of the ET0 of the Isparta meteorological station in Turkey using the input data of the Antalya station (cross-stationed). The outcome of the study with respect to the statistical measures for all the models was shown in Table 4. The visualization of the predictive models evidenced that GRNN outperformed the other models with respect to the prediction metrics. A slight difference appeared between the GRNN and GEP (or RBNN) with respect to the RMSE. From the MAE viewpoint, however, GEP performed a better prediction than GRNN due to the differences in the measures as AI models perform differently from one case to another. Another remarkable observation, which is a decrease in the accuracy of the applied AI models in the absence of local input data, was evidenced by a comparative analysis of Tables 3 and 4. The RMSE values of the MLP, GRNN, RBNN, ANFIS-GP, ANFIS-SC, and GEP were increased by 13, 1.4, 13, 10, 15, and 3.4% when using input data from other stations. Figure 7 illustrated the estimates of the applied models. There was no much difference in the performance of the GEP, GRNN, RBNN, ANFIS-GP, ANFIS-SC, and MLP models. The ET0 values were highly overestimated by the HS model, while a better performance was obtained with the calibrated HS model. Figure 8 showed the expression trees of the optimal GEP

model in a cross-station application. The GEP equation derived for the Isparta station without local climatic inputs was: ET0 ¼

Ra 17:3948 þ atanð−4:170715Þ−T min ðTmin þ sinðT max ÞÞ 22:86676



2 Ra −atanðRa Þ þ sin log 6:80719 þ

ð11Þ

The main advantage of the GEP approach compared to the GRNN, RBNN, ANFIS-GP, MLP, and ANFIS-SC approaches is the provision of more insight on the temperature-based ET0 modeling. The reason for this is that GEP explicitly provides the form of the function identified (Savic et al. 1999). Because of this advantage, GEP models can be preferred in modeling ET0 using limited climatic data when the other methods may be inappropriate. It has a simple formulation and can be used as a module in hydrologic modeling studies. The calibrated HS equation also had a good accuracy in the second scenario and can be used as a good alternative to the data-driven approaches in modeling ET0 without local input data.

5 Conclusions and remarks

Fig. 8 The expression tree for the optimal GEP model in cross-station application (estimation of Isparta ET0 using Antalya inputs; d0, d1, and d2 indicate the Tmin, Tmax, and Ra and co and c1 are the coefficients; refer to Eq. 11)

A temperature-based modeling of reference evapotranspiration was conducted using different AI models (MLP, GRNN, RBFNN, ANFIS-GP, ANFIS-SC, and GEP). The predictive models were constructed using maximum and minimum air temperatures and extraterrestrial radiation as input variables. The climatic data of two stations (Antalya and Isparta) located in Turkey were used to perform the prediction. The results of the inspected AI models were compared with each other and to those of the Hargreaves–Samani equation and its calibrated version. For the Antalya station, the GEP and GRNN models gave similar results and generally provided a better accuracy compared to the other models. In quantitative measurements, the GEP model increased the absolute error metrics, RMSE and MAE, and the best fit metric NS accuracies over the CHS by 43, 48, and 19%, respectively. On the other hand, in the Isparta station, the best accuracy was obtained from the GRNN and ANFIS-SC models. Increments of enhancement for the Isparta station in terms of the RMSE, MAE, and NS accuracies over the CHS were obtained as 54, 60, and 31% using GEP, respectively. The GEP model considerably increased the absolute error (RMSE and MAE) and best fit (NS) metric accuracies of the CHS in both stations. The applied models were also compared in the temperature-based estimation of ET0 using the input data of nearby stations. All the AI models and CHS empirical method provided good accuracies and performed better than

Temperature-based modeling of reference evapotranspiration using several artificial intelligence models:...

the HS. In both stations, the intelligent models performed better compared to the HS and its calibrated version. Calibration process significantly increased the accuracy of the HS in modeling ET0. The main limitation of this study was using two stations located in the same climatic region. In the future studies, more stations including longer period data and more empirical methods may be used and compared with different AI methods in estimating ET0. Some different scenarios such as using both local data and information from nearby stations may also be produced in the future studies.

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