# Modern Techniques to Modeling Reference Evapotranspiration in a Semiarid Area Based on ANN and GEP Models

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## Abstract

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_{0}) are either difficult or need a large number of inputs that are not always available from meteorological stations. Over a 6-year period (2006–2011), this study compares Feed Forward Neural Network (FFNN), Radial Basis Function Neural Network (RBFNN), and Gene Expression Programming (GEP) machine learning approaches for estimating daily ET

_{0}in a meteorological station in the Lower Cheliff Plain, northwest Algeria. ET

_{0}was estimated using the FAO-56 Penman–Monteith (FAO56PM) equation and observed meteorological data. The estimated ET

_{0}using FAO56PM was then used as the target output for the machine learning models, while the observed meteorological data were used as the model inputs. Based on the coefficient of determination (R

^{2}), root mean square error (RMSE), and Nash–Sutcliffe efficiency (EF), the RBFNN and GEP models showed promising performance. However, the FFNN model performed the best during training (R

^{2}= 0.9903, RMSE = 0.2332, and EF = 0.9902) and testing (R

^{2}= 0.9921, RMSE = 0.2342, and EF = 0.9902) phases in forecasting the Penman–Monteith evapotranspiration.

## 1. Introduction

_{0}) is calculated first. However, estimating ET

_{0}is known to be very complex. ET

_{0}is either measured directly (e.g., by lysimeter or pan setups), or complex physics-based experimentally validated equations are used. It is clear that direct measurements are very costly and time-consuming. Many commonly used physics-based equations [7,8,9,10,11], including the FAO-56 Penman–Monteith, involve multiple parameters which may not all be known from local observations [3]. Nevertheless, the FAO-56 Penman–Monteith method has been accepted as a standard and used by scientists in different climates. A high correlation is observed between the ET

_{0}values obtained from the FAO-56 Penman–Monteith method and direct measurements even in different climatic conditions. Therefore, scientists have considered the values computed using the FAO-56 Penman–Monteith method as the desired output of data-based artificial intelligence methods and different combinations of meteorological variables as inputs for such methods for accurately estimating ET

_{0}[12,13,14,15,16].

_{0}in California, USA, with Genetic Programming (GP). Rahimikhoob [24] predicted ET

_{0}values with Artificial Neural Networks (ANN) using temperature and relative humidity parameters in an eight-station region of Iran with a subtropical climate. Ozkan et al. [25] successfully estimated daily ET

_{0}amounts using ANN and bee colony hybrid method using the meteorological data of two stations in California, USA. Cobaner [26] estimated ET

_{0}amounts in the USA using wavelet regression (WR) and class A pan evaporation data. WR model results were found to give better results than the FAO-56 Penman–Monteith equation. Ladlani et al. [27] applied Adaptive Neuro-Fuzzy Inference System (ANFIS) and multiple linear regression models for daily ET

_{0}estimation in the north of Algeria. According to the results of the study, ANFIS yielded better results.

_{0}amounts using the Support Vector Machine (SVM) method in a region of China that was extremely arid. The authors utilized limited meteorological variables as model input. It was observed that modeling is sufficient in estimating daily ET

_{0}based on maximum and minimum temperature. Gocić et al. [29] used GP, ANN, SVM-firefly optimization algorithm, and SVM-wavelet models for ET

_{0}prediction in Serbia. This particular study took the FAO-56 Penman–Monteith equation to be the basic method. The results pointed to SVM-wavelet being the best performing methodology for the estimation of ET

_{0}under the given conditions. Petković et al. [30] estimated the amount of ET

_{0}in Serbia between 1980 and 2010 using Radial Basis Function Neural Network (RBFNN) coupled with particle swarm optimization and backpropagation RBFNN methods. Pandey et al. [31] estimated daily ET

_{0}by methods like ANN, support vector regression, and non-linear regression. In this study, limited climatic parameters were used as model input. Daily ET

_{0}values calculated from the FAO56PM method were compared to the model output. The results pointed to the acceptability of the ANN model estimations. Fan et al. [32] estimated the daily ET

_{0}amount using SVM, extreme learning machine models, and four tree-based ensemble methods in China’s different climatic conditions. The results pointed to the fact that tree-based ensemble methods can yield appropriate results in different climates. Wu et al. [33] used cross-station and synthetic climate data to estimate the amount of ET

_{0}. They also found that machine learning methods could perform successfully in the prediction process.

_{0}values using different combinations of climatic variables. The models are applied in a semi-arid farmland area, namely the Lower Cheliff Plain in northwest Algeria. This research made use of the well-known FAO-56 (PM56) equation as the basic method. In this article, the role of climatic parameters in ET

_{0}estimation in this semi-arid region was also determined.

## 2. Materials and Methods

#### 2.1. Study Area and Meteorological Data Acquisition

#### 2.2. Description of Data

_{max}, T

_{min}, and T

_{mean}), daily mean relative humidity (RH), wind speed (WS), sunshine duration (SD), and global radiation (GR). The days with data that proved to be inadequate were excluded from the patterns. The statistical parameters pertaining to the daily climatic data are given in Table 2, in which the X

_{mean}, X

_{max}, X

_{min}, S

_{x}, and CV stand for the mean, maximum, minimum, standard deviation, and coefficient of variation, respectively.

#### 2.3. Evapotranspiration Estimation Method

_{0}was implemented following the formulation in [3] as a function of daily mean net radiation, temperature, water vapor pressure, and wind speed. The procedure used was that outlined in Chapter 3 of FAO-56 [3].

^{−1}), ${R}_{n}$ is the net radiation (MJ m

^{−2}day

^{−1}), $G$ is the soil heat flux (MJ m

^{−2}day

^{−1}), $c$ is the psychrometric constant (kPaC

^{−1}), ${e}_{s}$ is the pressure of saturation vapor (kPa), ${e}_{a}$ is the pressure of the actual vapor (kPa), $D$ is the slope of the curve for saturation vapor pressure–temperature (kPaC

^{−1}), ${T}_{a}$ is the average daily air temperature (°C), and ${U}_{2}$ is the mean daily wind speed at 2 m (m s

^{−1}).

#### 2.4. Multilayer Perceptron Artificial Neural Network

_{ij}x

_{j}), or (for the first layer) of the network inputs and returns a non-linear transformation of this quantity.

_{ij}) are the parameters added to each source defining this linear combination and typically also include an intercept term called the activation threshold [35]. A non-linear activation function is then applied to the linear output combination (f (∑w

_{ij}x

_{j})). This activation function can be, for example, a sigmoid function, which constrains each neuron’s output values between two asymptotes. Once the activation function is applied, each neuron’s output feeds into the outputs of the next layer. The most frequently used architecture for an ANN consists of an input layer in which the data is introduced into the ANN, a hidden layer(s) in which the data undergoes processing, and the output layer in which the effects of the input generate a predicted output value(s) [35].

#### 2.5. Radial Basis Function

#### 2.6. Gene Expression Programming

#### 2.7. Evaluation Criteria

^{2}), root mean square error (RMSE), and Nash Sutcliffe efficiency coefficient (EF) [41,42,43]. The calculation of the three criteria was done according to Equations (2)–(4).

_{mean}is the mean of observed $ET$.

## 3. Results and Discussion

_{0}values were computed by the Penman–Monteith method using climatic data. Then the following equation was used to normalize the input (meteorological data) and output (calculated ET

_{0}by Penman–Monteith):

#### 3.1. Application of MLP

_{max}, T

_{mean}, (T

_{max}− T

_{min}), RH, I, WS, and GR, performed better than other FFNN-based input combination models with R

^{2}values as 0.9903, 0.9921, RMSE values as 0.2332, 0.2342, and E values as 0.9902, 0.9902 for both training and testing stages, respectively. Nineteen neurons were used in the hidden layer to achieve this ideal performance. The performance and agreement plot among actual and predicted values of the FFNN2 model for both the training and testing stage are mapped out in Figure 5, which shows that max values lie very close to the line of 450 and follow the same pattern as the actual values in both training and testing stages. If all the values lie on the line of 450 and follow the same path, the model is ideal and predicts values similar to actual ones.

_{mean}, RH, WS, and GR) is suitable for predicting ET with R

^{2}values as 0.9875, 0.9892, RMSE values as 0.2656, 0.2623, and E values as 0.9873, 0.9877 for both training and testing stages, respectively. The same number of neurons (19) is used in the single hidden layer for achieving this performance, similar to the FFNN2 model. The performance and agreement plot among actual and predicted values of the FFNN11 model for both the training and testing stage is shown in Figure 5, which points to the fact that max values lie very close to the line of perfect agreement and follow the same pattern as the actual values in both training and testing stages.

#### 3.2. Application of RBF

_{min}, T

_{max}, T

_{mean}, RH, I, WS, and GR, performs better than other input combination RBFNN based models with R

^{2}values as 0.9907, 0.9911, RMSE values as 0.2270, 0.2374, and E values as 0.9907, 0.9899 for both training and testing stages, respectively. The performance and agreement plot among actual and predicted values of the RBFNN5 model for both training and testing stages are shown in Figure 6, which shows that max values lie very close to the line of 450 and follow the same pattern as the actual values in both training and testing stages.

_{mean}, RH, WS, and GR) is suitable for predicting the ET with R

^{2}values as 0.9886, 0.9892, RMSE values as 0.2514, 0.2551, and E values as 0.9886, 0.9884 for both training and testing stages, respectively. A lower rate of spread (Table 5) was used in the development of this model than in the case of the RBFNN5 model. The performance and agreement plot among actual and predicted values of the RBFNN11 model for both training and testing stages are shown in Figure 6, which shows that max values lie very close to the line of perfect agreement and follow the same pattern as the actual values in both training and testing stages.

#### 3.3. Application of GEP

^{2}ranged between 0.6973 and 0.9664, RMSE ranged 0.4830–1.3112 mm day

^{−1}, and EF ranged 0.6895–0.9579. So, for the test phase, the R

^{2}ranged between 0.8057–0.9775, RMSE ranged 0.3701–1.1224 mm day

^{−1}, and E ranged 0.7744–0.9755 (Table 7). It is clear that the presence or absence of critical meteorological variables in the input combinations significantly affected GEP model performance. The results of Table 7 suggest that the GEP11 model, including T

_{mean}, RH, WS, and GR parameters in the input combination, performed better than other input combinations and GEP based models with R

^{2}values as 0.9606, 0.9775, RMSE values as 0.4830, 0.3701, and E values as 0.9579, 0.9755 for the training and testing stages, respectively. The performance and agreement plot among actual and predicted values of the GEP11 model for both training and testing stages are shown in Figure 7, which indicates that max values lie very close to the line of 450 and follow the same path as the actual values in both training and testing stages. Table 7 concludes that the GEP11 model is the best performing model with optimum input combinations.

#### 3.4. Inter Comparison among Best and Optimum Input Combination Based Models

_{0}. As such, T

_{max}, T

_{mean}, (T

_{max}− T

_{min}), RH, I, WS, and the GR input combination-based FFNN model could be used for the prediction of ET

_{0}. However, the results mapped out in Table 9 of single-factor ANOVA suggest that there is no significant difference between observed and predicted values using FFNN, RBFNN, and GEP best combination-based models.

_{0}by different best input combination-based models over the test period is depicted in Figure 10. It is clear that the representative points of all the applied models have nearly the same position. The FFNN2 model is located nearest to the observed point with the lower value of RMSE and SD and higher value of the coefficient of correlation, which picks out this model as the superior model.

_{0}, which suggests that T

_{mean}, RH, WS, and GR input combination-based RBFNN model could be used for the prediction of ET

_{0}. The results in Table 12 of single-factor ANOVA suggest that there is no significant difference between observed and predicted values using FFNN, RBFNN, and GEP optimum input combination-based models.

_{0}by different optimum input combination-based models over the test period is depicted in Figure 13. It is clear that the representative points of all the applied models have nearly the same position. The RBFNN11 model is located nearest to the observed point with the lower value of RMSE, SD, and higher value of the coefficient of correlation, making this model emerge as a superior model with the optimum number of input parameters.

## 4. Conclusions

_{0}obtained by the FAO PM method. They yielded reliable estimations for the semi-arid area in question. The study also found that modeling ET

_{0}utilizing the ANN technique leads to better estimates than the GEP model.

_{mean}, RH, WS, and GR parameters are the optimum parameters for the estimation of daily evapotranspiration in the semi-arid region of Algeria. The overall performance of all applied models is satisfactory, as there is no significant difference between actual and predicted values using the optimum number of input parameters in the models.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Gilbert, M.E.; Hernandez, M.I. How should crop water-use efficiency be analyzed? A warning about spurious correlations. Field Crops Res.
**2019**, 235, 59–67. [Google Scholar] [CrossRef] - Ghiat, I.; Mackey, H.R.; Al-Ansari, T. A review of evapotranspiration measurement models, techniques and methods for open and closed agricultural field applications. Water
**2021**, 13, 2523. [Google Scholar] [CrossRef] - Allen, R.; Peirera, L.S.; Raes, D.; Smith, M. Crop Evapotranspiration—Guidelines for Computing Crop Water Requirements; FAO—Food and Agriculture Organization of the United Nations: Rome, Italy, 1998; ISBN 92-5-104219-5. [Google Scholar]
- Sun, G.; Alstad, K.; Chen, J.; Chen, S.; Ford, C.R.; Lin, G.; Zhang, Z. A general predictive model for estimating monthly ecosystem evapotranspiration. Ecohydrology
**2011**, 4, 245–255. [Google Scholar] [CrossRef] - Amatya, D.M.; Muwamba, A.; Panda, S.; Callahan, T.J.; Harder, S.; Pellett, C.A. Assessment of spatial and temporal variation of potential evapotranspiration estimated by four methods for South Carolina, USA. JSCWR
**2018**, 5, 3–24. [Google Scholar] [CrossRef][Green Version] - Yu, H.; Cao, C.; Zhang, Q.; Bao, Y. Construction of an evapotranspiration model and analysis of spatiotemporal variation in Xilin River Basin, China. PLoS ONE
**2021**, 16, e0256981. [Google Scholar] [CrossRef] - Thornthwaite, C.W. An approach toward a national classification of climate. Geogr. Rev.
**1948**, 38, 55–94. [Google Scholar] [CrossRef] - Blaney, H.F.; Criddle, W.D. Determining Water Requirements in Irrigated Areas from Climatological and Irrigation Data; US Soil Conservation Service: Washington, DC, USA, 1950.
- Makkink, G.F. Testing the Penman formula by means of lysimeters. J. Inst. Water Eng.
**1957**, 11, 277–288. [Google Scholar] - Priestley, C.H.B.; Taylor, R.J. On the assessment of surface heat flux and evaporation using large-scale parameters. Mon. Weather Rev.
**1972**, 100, 81–92. [Google Scholar] [CrossRef] - Hargreaves, G.H.; Samani, Z.A. Reference crop evapotranspiration from temperature. Appl. Eng. Agric.
**1985**, 1, 96–99. [Google Scholar] [CrossRef] - Kumar, M.; Bandyopadhyay, A.; Raghuwanshi, N.S.; Singh, R. Comparative study of conventional and artificial neural network-based ET
_{0}estimation models. Irrig. Sci.**2008**, 26, 531–545. [Google Scholar] [CrossRef] - Mohawesh, O.E. Artificial neural network for estimating monthly reference evapotranspiration under arid and semi-arid environments. Arch. Agron. Soil Sci.
**2011**, 59, 105–117. [Google Scholar] [CrossRef] - Eslamian, S.S.; Gohari, S.A.; Zareian, M.J.; Firoozfar, A. Estimating Penman–Monteith reference evapotranspiration using artificial neural networks and genetic algorithm: A case study. Arab. J. Sci. Eng.
**2012**, 37, 935–944. [Google Scholar] [CrossRef] - Citakoglu, H.; Cobaner, M.; Haktanir, T.; Kisi, O. Estimation of monthly mean reference evapotranspiration in Turkey. Water Resour. Manag.
**2014**, 28, 99–113. [Google Scholar] [CrossRef] - Sattari, M.T.; Apaydin, H.; Shamshirband, S. Performance evaluation of deep learning-based gated recurrent units (GRUs) and tree-based models for estimating ET
_{0}by using limited meteorological variables. Mathematics**2020**, 8, 972. [Google Scholar] [CrossRef] - Sattari, M.T.; Mirabbasi, R.; Sushab, R.S.; Abraham, J. Prediction of level in Ardebil plain using support vector regression and M5 tree model. Groundwater
**2018**, 56, 636–646. [Google Scholar] [CrossRef] [PubMed] - Rouzegari, N.; Hassanzadeh, Y.; Sattari, M.T. Using the hybrid simulated annealing-M5 tree algorithms to extract the If-Then operation rules in a single reservoir. Water Resour. Manag.
**2019**, 33, 3655–3672. [Google Scholar] [CrossRef] - Apaydin, H.; Feizi, H.; Sattari, M.T.; Colak, M.S.; Shamshirband, S.; Chau, K.-W. Comparative analysis of recurrent neural network architectures for reservoir inflow forecasting. Water
**2020**, 12, 1500. [Google Scholar] [CrossRef] - Shabani, S.; Samadianfard, S.; Sattari, M.T.; Mosavi, A.; Shamshirband, S.; Kmet, T.; Várkonyi-Kóczy, A.R. Modeling pan evaporation using Gaussian process regression K-nearest neighbors random forest and support vector machines; Comparative analysis. Atmosphere
**2020**, 11, 66. [Google Scholar] [CrossRef][Green Version] - de Oliveira Ferreira Silva, C.; de Castro Teixeira, A.H.; Manzione, R.L. Agriwater: An R package for spatial modelling of energy balance and actual evapotranspiration using satellite images and agrometeorological data. Environ. Model. Softw.
**2019**, 120, 104497. [Google Scholar] [CrossRef] - Thorp, K.R.; Marek, G.W.; DeJonge, K.C.; Evett, S.R.; Lascano, R.J. Novel methodology to evaluate and compare evapotranspiration algorithms in an agroecosystem model. Environ. Model. Softw.
**2019**, 119, 214–227. [Google Scholar] [CrossRef] - Guven, A.; Aytek, A.; Yuce, M.I.; Aksoy, H. Genetic programming-based empirical model for daily reference evapotranspiration estimation. Clean Soil Air Water
**2008**, 36, 905–912. [Google Scholar] [CrossRef] - Rahimikhoob, A. Estimation of evapotranspiration based on only air temperature data using artificial neural networks for a subtropical climate in Iran. Theor. Appl. Climatol.
**2010**, 101, 83–91. [Google Scholar] [CrossRef] - Ozkan, C.; Kisi, O.; Akay, B. Neural networks with artificial bee colony algorithm for modeling daily reference evapotranspiration. Irrig. Sci.
**2011**, 29, 431–441. [Google Scholar] [CrossRef] - Cobaner, M. Reference evapotranspiration based on Class A pan evaporation via wavelet regression technique. Irrig. Sci.
**2013**, 31, 119–134. [Google Scholar] [CrossRef] - Ladlani, I.; Houichi, L.; Djemili, L.; Heddam, S.; Belouz, K. Estimation of daily reference evapotranspiration (ET
_{0}) in the North of Algeria using adaptive neuro-fuzzy inference system (ANFIS) and multiple linear regression (MLR) models: A comparative study. Arab. J. Sci. Eng.**2014**, 39, 5959–5969. [Google Scholar] [CrossRef] - Wen, X.; Si, J.; He, Z.; Wu, J.; Shao, H.; Yu, H. Support-vector-machine-based models for modeling daily reference evapotranspiration with limited climatic data in extreme arid regions. Water Resour. Manag.
**2015**, 29, 3195–3209. [Google Scholar] [CrossRef] - Gocić, M.; Motamedi, S.; Shamshirband, S.; Petković, D.; Ch, S.; Hashim, R.; Arif, M. Soft computing approaches for forecasting reference evapotranspiration. Comput. Electron. Agric.
**2015**, 113, 164–173. [Google Scholar] [CrossRef] - Petković, D.; Gocic, M.; Shamshirband, S.; Qasem, S.N.; Trajkovic, S. Particle swarm optimization-based radial basis function network for estimation of reference evapotranspiration. Theor. Appl. Climatol.
**2016**, 125, 555–563. [Google Scholar] [CrossRef] - Pandey, P.K.; Nyori, T.; Pandey, V. Estimation of reference evapotranspiration using data driven techniques under limited data conditions. Model. Earth Syst. Environ.
**2017**, 3, 1449–1461. [Google Scholar] [CrossRef] - Fan, J.; Yue, W.; Wu, L.; Zhang, F.; Cai, H.; Wang, X.; Lu, X.; Xiang, Y. Evaluation of SVM, ELM and four tree-based ensemble models for predicting daily reference evapotranspiration using limited meteorological data in different climates of China. Agric. For. Meteorol.
**2018**, 263, 225–241. [Google Scholar] [CrossRef] - Wu, L.; Peng, Y.; Fan, J.; Wang, Y. Machine learning models for the estimation of monthly mean daily reference evapotranspiration based on cross-station and synthetic data. Hydrol. Res.
**2019**, 50, 1730–1750. [Google Scholar] [CrossRef] - Douaoui, A.E.K.; Nicolas, H.; Walter, C. Detecting salinity hazards with in a semiarid context by means of combining soil and remote-sensing data. Geoderma
**2006**, 134, 217–230. [Google Scholar] [CrossRef] - Shiri, J.; Kisi, Ö.; Anderas, G.; López, J.J.; Nazemi, A.H.; Stuyt, L.C.P.M. Daily reference evapotranspiration modeling by using genetic programming approach in the Basque Country (Northern Spain). J. Hydrol.
**2012**, 414, 302–316. [Google Scholar] [CrossRef] - Leahy, P.; Kiely, G.; Corcoran, G. Structural optimization and input selection of an artificial neural network for river level prediction. J. Hydrol.
**2008**, 355, 192–201. [Google Scholar] [CrossRef] - Ioannou, K.; Myronidis, D.; Lefakis, P.; Stathis, D. The use of artificial neural networks (ANNs) for the forecast of precipitation levels of Lake Doirani (N. Greece). Fresenius Environ. Bull.
**2010**, 19, 1921–1927. [Google Scholar] - Haciismailoglu, M.C.; Kucuk, I.; Derebasi, N. Prediction of dynamic hysteresis loops of nano-crystalline cores. Expert Syst. Appl.
**2009**, 36, 2225–2227. [Google Scholar] [CrossRef] - Lin, F.; Chen, L.H. A non-linear rainfall-runoff model using radial basis function network. J. Hydrol.
**2004**, 289, 1–8. [Google Scholar] [CrossRef] - Wang, S.; Fu, Z.; Chen, H.; Nie, Y.; Wang, K. Modeling daily reference ET in the karst area of northwest Guangxi (China) using gene expression programming (GEP) and artificial neural network (ANN). Theor. Appl. Climatol.
**2015**, 126, 493–504. [Google Scholar] [CrossRef] - Dubovský, V.; Dlouhá, D.; Pospíšil, L. The calibration of evaporation models against the Penman–Monteith equation on Lake Most. Sustainability
**2020**, 13, 313. [Google Scholar] [CrossRef] - Myronidis, D.; Ivanova, E. Generating regional models for estimating the peak flows and environmental flows magnitude for the Bulgarian-Greek Rhodope mountain range torrential watersheds. Water
**2020**, 12, 784. [Google Scholar] [CrossRef][Green Version] - Han, X.; Wei, Z.; Zhang, B.; Li, Y.; Du, T.; Chen, H. Crop evapotranspiration prediction by considering dynamic change of crop coefficient and the precipitation effect in back-propagation neural network model. J. Hydrol.
**2021**, 596, 126104. [Google Scholar] [CrossRef]

**Figure 3.**RBF architecture [39].

**Figure 4.**General GEP model implementation and general structure [35].

**Figure 5.**Performance of the best performing (

**a**) FFNN M2 and (

**b**) FFNN M11 models for both training and testing stages.

**Figure 6.**Performance of the best performing (

**a**) RBFNN M5 and (

**b**) RBFNN M11 models for both training and testing stages.

**Figure 7.**Performance of the optimum combination of inputs/ best input combination based model GEP M11 model for both training and testing stages.

**Figure 8.**Scatter plot among observed and predicted values using best input combination-based models using testing data set.

**Figure 11.**Scatter plot among observed and predicted values using optimum input combination-based models using testing data set.

Name of Sensor | Measuring Unit |
---|---|

Psychrometer | % |

Heliograph | Minute |

Anemometer | 0.3 to 50 m/s |

Wind direction | 0° to 360° |

Pyranometer | 0…1400 W/m^{2} (Max 2000) |

Albedometer | −2000 to 2000 W/m^{2} |

Air temperature | −30 °C to 70 °C |

Soil temperature | −50 °C to 50 °C |

Evaporation pan | Mm of water |

Rain gauge | Mm of water (resolution 0.1 mm) |

Data Set | Unit | X_{min} | X_{max} | X_{mean} | S_{x} | CV (S_{x}/X_{mean}) |
---|---|---|---|---|---|---|

T_{min} | °C | −4.30 | 26.29 | 11.68 | 6.87 | 0.59 |

T_{max} | °C | 6.98 | 48.16 | 27.28 | 8.89 | 0.33 |

T_{mean} | °C | 3.87 | 37.23 | 19.48 | 7.53 | 0.39 |

RH | % | 21.50 | 95.66 | 59.69 | 14.39 | 0.24 |

WS | m/s | 0.00 | 28.94 | 6.66 | 3.81 | 0.57 |

SD | h | 0.00 | 14.10 | 7.21 | 4.14 | 0.57 |

GR | mm | 9.72 | 1791.04 | 969.52 | 446.08 | 0.46 |

Parameter | Value |
---|---|

Hidden layer transfer Function | Tangent sigmoid transfer function (tansig) |

Output layer transfer Function | Linear transfer function (purelin) |

Training function | Levenberg-Marquardt |

Maximum number of epochs to train | 1000 |

Maximum validation failures | 6 |

Minimum performance gradient | 1 × 10^{−7} |

Initial mu | 0.001 |

mu decrease factor | 0.1 |

mu increase factor | 10 |

Maximum mu | 1 × 10^{10} |

Maximum time to train in seconds | Inf |

**Table 4.**Statistical criteria for an estimation of ET

_{0}using different input variables for FFNN. The bold part shows that this model is superior to others.

Model | Input | Neurons | Training Phase | Testing Phase | ||||
---|---|---|---|---|---|---|---|---|

R^{2} | RMSE | EF | R^{2} | RMSE | EF | |||

FFNN1 | T_{min}, T_{max}, T_{mean}, (T_{max} − T_{min}), RH, I, WS, GR | 18 | 0.9903 | 0.2338 | 0.9901 | 0.9918 | 0.2389 | 0.9898 |

FFNN2 | T_{max}, T_{mean}, (T_{max} − T_{min}), RH, I, WS, GR | 19 | 0.9903 | 0.2332 | 0.9902 | 0.9921 | 0.2342 | 0.9902 |

FFNN3 | T_{min}, T_{mean}, (T_{max} − T_{min}), RH, I, WS, GR | 13 | 0.9905 | 0.2308 | 0.9904 | 0.9917 | 0.2368 | 0.9900 |

FFNN4 | T_{min}, T_{max}, (T_{max} − T_{min}), RH, I, WS, GR | 19 | 0.9903 | 0.2336 | 0.9901 | 0.9920 | 0.2378 | 0.9899 |

FFNN5 | T_{min}, T_{max}, T_{mean}, RH, I, WS, GR | 11 | 0.9899 | 0.2376 | 0.9898 | 0.9916 | 0.2393 | 0.9897 |

FFNN6 | T_{min}, T_{max}, T_{mean}, (T_{max} − T_{min}), I, WS, GR | 12 | 0.9782 | 0.3481 | 0.9781 | 0.9859 | 0.3102 | 0.9828 |

FFNN7 | T_{min}, T_{max}, T_{mean}, (T_{max} − T_{min}), RH, WS, GR | 19 | 0.9883 | 0.2566 | 0.9881 | 0.9900 | 0.2536 | 0.9885 |

FFNN8 | T_{min}, T_{max}, T_{mean}, (T_{max} − T_{min}), RH, I, GR | 14 | 0.9399 | 0.5799 | 0.9393 | 0.9603 | 0.5046 | 0.9544 |

FFNN9 | T_{min}, T_{max}, T_{mean}, (T_{max} − T_{min}), RH, I, WS | 14 | 0.9676 | 0.4245 | 0.9675 | 0.9748 | 0.4032 | 0.9709 |

FFNN10 | T_{mean}, RH, I, WS, GR | 16 | 0.9895 | 0.2435 | 0.9893 | 0.9907 | 0.2513 | 0.9887 |

FFNN11 | T_{mean}, RH, WS, GR | 19 | 0.9875 | 0.2656 | 0.9873 | 0.9892 | 0.2623 | 0.9877 |

FFNN12 | T_{mean}, RH, I, WS | 11 | 0.9672 | 0.4265 | 0.9671 | 0.9716 | 0.4124 | 0.9695 |

FFNN13 | RH, I, WS, GR | 8 | 0.9217 | 0.6593 | 0.9215 | 0.9465 | 0.5533 | 0.9452 |

FFNN14 | T_{mean}, RH, WS | 19 | 0.9172 | 0.6770 | 0.9172 | 0.9165 | 0.6845 | 0.9161 |

FFNN15 | T_{mean}, RH | 14 | 0.8528 | 0.9047 | 0.8522 | 0.8966 | 0.7745 | 0.8926 |

FFNN16 | T_{mean}, WS | 7 | 0.8326 | 0.9633 | 0.8324 | 0.8520 | 0.9224 | 0.8477 |

FFNN17 | RH, WS | 20 | 0.7771 | 1.1120 | 0.7767 | 0.8439 | 0.9488 | 0.8388 |

**Table 5.**Statistical criteria for an estimation of ET

_{0}using different input variables for RBF. The bold part shows that this model is superior to others.

Model | Input Combination | Training Phase | Testing Phase | |||||
---|---|---|---|---|---|---|---|---|

Spread | R^{2} | RMSE | EF | R^{2} | RMSE | EF | ||

RBF1 | T_{min}, T_{max}, T_{mean}, (T_{max} − T_{min}), RH, I, WS, GR | 1187.55 | 0.9911 | 0.2215 | 0.9911 | 0.9909 | 0.2406 | 0.9896 |

RBF2 | T_{max}, T_{mean}, (T_{max} − T_{min}), RH, I, WS, GR | 1187.55 | 0.9910 | 0.2238 | 0.9910 | 0.9910 | 0.2382 | 0.9898 |

RBF3 | T_{min}, T_{mean}, (T_{max} − T_{min}), RH, I, WS, GR | 1385.47 | 0.9906 | 0.2279 | 0.9906 | 0.9910 | 0.2377 | 0.9899 |

RBF4 | T_{min}, T_{max}, (T_{max} − T_{min}), RH, I, WS, GR | 1385.47 | 0.9907 | 0.2265 | 0.9907 | 0.9910 | 0.2378 | 0.9899 |

RBF5 | T_{min}, T_{max}, T_{mean}, RH, I, WS, GR | 1385.47 | 0.9907 | 0.2270 | 0.9907 | 0.9911 | 0.2374 | 0.9899 |

RBF6 | T_{min}, T_{max}, T_{mean}, (T_{max} − T_{min}), I, WS, GR | 791.70 | 0.9805 | 0.3284 | 0.9805 | 0.9842 | 0.3216 | 0.9815 |

RBF7 | T_{min}, T_{max}, T_{mean}, (T_{max} − T_{min}), RH, WS, GR | 1187.55 | 0.9890 | 0.2466 | 0.9890 | 0.9901 | 0.2445 | 0.9893 |

RBF8 | T_{min}, T_{max}, T_{mean}, (T_{max} − T_{min}), RH, I, GR | 593.77 | 0.9456 | 0.5489 | 0.9456 | 0.9549 | 0.5076 | 0.9539 |

RBF9 | T_{min}, T_{max}, T_{mean}, (T_{max} − T_{min}), RH, I, WS | 1781.32 | 0.9753 | 0.3696 | 0.9753 | 0.9616 | 0.4729 | 0.9600 |

RBF10 | T_{mean}, RH, I, WS, GR | 791.70 | 0.9907 | 0.2267 | 0.9907 | 0.9901 | 0.2530 | 0.9885 |

RBF11 | T_{mean}, RH, WS, GR | 593.77 | 0.9886 | 0.2514 | 0.9886 | 0.9892 | 0.2551 | 0.9884 |

RBF12 | T_{mean}, RH, I, WS | 1583.40 | 0.9704 | 0.4047 | 0.9704 | 0.9699 | 0.4298 | 0.9669 |

RBF13 | RH, I, WS, GR | 593.77 | 0.9300 | 0.6224 | 0.9300 | 0.9400 | 0.5873 | 0.9382 |

RBF14 | T_{mean}, RH, WS | 1385.47 | 0.9214 | 0.6599 | 0.9214 | 0.9140 | 0.6941 | 0.9137 |

RBF15 | T_{mean}, RH | 791.70 | 0.8569 | 0.8902 | 0.8569 | 0.8915 | 0.7834 | 0.8901 |

RBF16 | T_{mean}, WS | 791.70 | 0.8400 | 0.9413 | 0.8400 | 0.8480 | 0.9279 | 0.8458 |

RBF17 | RH, WS | 791.70 | 0.7779 | 1.1089 | 0.7779 | 0.8434 | 0.9441 | 0.8404 |

Parameter | Value |
---|---|

Number of chromosomes | 30 |

Head size | 8 |

Number of genes | 3 |

Linking function | Addition |

Fitness function error type | RMSE |

Mutation rate | 0.044 |

Inversion rate | 0.1 |

IS transposition | 0.1 |

RIS transposition | 0.1 |

One-point recombination rate | 0.3 |

wo-point recombination rate | 0.3 |

Gene recombination rate | 0.1 |

Gene transposition rate | 0.1 |

Model | Input Combination | Training Phase | Testing Phase | ||||
---|---|---|---|---|---|---|---|

R^{2} | RMSE | EF | R^{2} | RMSE | EF | ||

GEP1 | T_{min}, T_{max}, T_{mean}, (T_{max} − T_{min}), RH, I, WS, GR | 0.8959 | 0.7732 | 0.8920 | 0.9190 | 0.6945 | 0.9136 |

GEP2 | T_{max}, T_{mean}, (T_{max} − T_{min}), RH, I, WS, GR | 0.9075 | 0.7227 | 0.9057 | 0.9323 | 0.6251 | 0.9300 |

GEP3 | T_{min}, T_{mean}, (T_{max} − T_{min}), RH, I, WS, GR | 0.9026 | 0.7361 | 0.9021 | 0.9300 | 0.6652 | 0.9208 |

GEP4 | T_{min}, T_{max}, (T_{max} − T_{min}), RH, I, WS, GR | 0.8355 | 0.9692 | 0.8303 | 0.8627 | 0.9407 | 0.8416 |

GEP5 | T_{min}, T_{max}, T_{mean}, RH, I, WS, GR | 0.8415 | 0.9721 | 0.8294 | 0.9033 | 0.8151 | 0.8810 |

GEP6 | T_{min}, T_{max}, T_{mean}, (T_{max} − T_{min}), I, WS, GR | 0.9393 | 0.5804 | 0.9392 | 0.9629 | 0.4687 | 0.9607 |

GEP7 | T_{min}, T_{max}, T_{mean}, (T_{max} − T_{min}), RH, WS, GR | 0.9664 | 0.4323 | 0.9663 | 0.9762 | 0.3795 | 0.9742 |

GEP8 | T_{min}, T_{max}, T_{mean}, (T_{max} − T_{min}), RH, I, GR | 0.8636 | 0.8695 | 0.8635 | 0.9194 | 0.6925 | 0.9141 |

GEP9 | T_{min}, T_{max}, T_{mean}, (T_{max} − T_{min}), RH, I, WS | 0.9353 | 0.6081 | 0.9332 | 0.9537 | 0.5604 | 0.9438 |

GEP10 | T_{mean}, RH, I, WS, GR | 0.9085 | 0.7138 | 0.9080 | 0.9275 | 0.6435 | 0.9258 |

GEP11 | T_{mean}, RH, WS, GR | 0.9606 | 0.4830 | 0.9579 | 0.9775 | 0.3701 | 0.9755 |

GEP12 | T_{mean}, RH, I, WS | 0.9420 | 0.5885 | 0.9374 | 0.9597 | 0.5045 | 0.9544 |

GEP13 | RH, I, WS, GR | 0.8560 | 0.9004 | 0.8536 | 0.9236 | 0.7069 | 0.9105 |

GEP14 | T_{mean}, RH, WS | 0.8388 | 0.9563 | 0.8349 | 0.8797 | 0.8406 | 0.8735 |

GEP15 | T_{mean}, RH | 0.8136 | 1.0598 | 0.7972 | 0.8635 | 0.9331 | 0.8441 |

GEP16 | T_{mean}, WS | 0.7769 | 1.1312 | 0.7689 | 0.8057 | 1.0588 | 0.7993 |

GEP17 | RH, WS | 0.6973 | 1.3112 | 0.6895 | 0.8062 | 1.1224 | 0.7744 |

Model | Training Phase | Testing Phase | ||||
---|---|---|---|---|---|---|

R^{2} | RMSE | E | R^{2} | RMSE | E | |

FFNN2 | 0.9903 | 0.2332 | 0.9902 | 0.9921 | 0.2342 | 0.9902 |

RBFNN5 | 0.9907 | 0.2270 | 0.9907 | 0.9911 | 0.2374 | 0.9899 |

GEP11 | 0.9606 | 0.4830 | 0.9579 | 0.9775 | 0.3701 | 0.9755 |

Source of Variation | F | p-Value | F_{crit} | Variation among Groups |
---|---|---|---|---|

Actual-FFNN2 | 0.171751 | 0.678682 | 3.854264 | Insignificant |

Actual-RBFNN5 | 0.101036 | 0.750681 | 3.854264 | Insignificant |

Actual-GEP11 | 0.126406 | 0.72229 | 3.854264 | Insignificant |

Statistic | FFNN2 | RBFNN5 | GEP11 |
---|---|---|---|

Minimum | −0.8840 | −1.2231 | −0.8671 |

Maximum | 1.4199 | 1.5204 | 1.9343 |

1st quartile | −0.0681 | −0.0726 | −0.1503 |

Median | 0.0606 | 0.0250 | −0.0055 |

3rd quartile | 0.2091 | 0.1742 | 0.2176 |

Mean | 0.0713 | 0.0548 | 0.0630 |

Model | Training Phase | Testing Phase | ||||
---|---|---|---|---|---|---|

R^{2} | RMSE | E | R^{2} | RMSE | E | |

FFNN | 0.9875 | 0.2656 | 0.9873 | 0.9892 | 0.2623 | 0.9877 |

RBF | 0.9886 | 0.2514 | 0.9886 | 0.9892 | 0.2551 | 0.9884 |

GEP | 0.9606 | 0.4830 | 0.9579 | 0.9775 | 0.3701 | 0.9755 |

Source of Variation | F | p-Value | F_{crit} | Variation among Groups |
---|---|---|---|---|

Observed-FFNN11 | 0.101466 | 0.750169 | 3.854264 | Insignificant |

Observed-RBFNN11 | 0.119424 | 0.72976 | 3.854264 | Insignificant |

Observed-GEP11 | 0.126406 | 0.72229 | 3.854264 | Insignificant |

Statistic | FFNN11 | RBFNN11 | GEP11 |
---|---|---|---|

Minimum | −0.6918 | −0.7073 | −0.8671 |

Maximum | 1.5230 | 1.3700 | 1.9343 |

1st Quartile | −0.1227 | −0.0952 | −0.1503 |

Median | 0.0119 | 0.0221 | −0.0055 |

3rd Quartile | 0.2228 | 0.2111 | 0.2176 |

Mean | 0.0548 | 0.0599 | 0.0630 |

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**MDPI and ACS Style**

Achite, M.; Jehanzaib, M.; Sattari, M.T.; Toubal, A.K.; Elshaboury, N.; Wałęga, A.; Krakauer, N.; Yoo, J.-Y.; Kim, T.-W. Modern Techniques to Modeling Reference Evapotranspiration in a Semiarid Area Based on ANN and GEP Models. *Water* **2022**, *14*, 1210.
https://doi.org/10.3390/w14081210

**AMA Style**

Achite M, Jehanzaib M, Sattari MT, Toubal AK, Elshaboury N, Wałęga A, Krakauer N, Yoo J-Y, Kim T-W. Modern Techniques to Modeling Reference Evapotranspiration in a Semiarid Area Based on ANN and GEP Models. *Water*. 2022; 14(8):1210.
https://doi.org/10.3390/w14081210

**Chicago/Turabian Style**

Achite, Mohammed, Muhammad Jehanzaib, Mohammad Taghi Sattari, Abderrezak Kamel Toubal, Nehal Elshaboury, Andrzej Wałęga, Nir Krakauer, Ji-Young Yoo, and Tae-Woong Kim. 2022. "Modern Techniques to Modeling Reference Evapotranspiration in a Semiarid Area Based on ANN and GEP Models" *Water* 14, no. 8: 1210.
https://doi.org/10.3390/w14081210