# Multi-Algorithm Hybrid Optimization of Back Propagation (BP) Neural Networks for Reference Crop Evapotranspiration Prediction Models

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{0}) statistic is useful for estimating agricultural system water requirements and managing irrigation. In dry areas, the accurate calculation of ET

_{0}is crucial for optimal agricultural water resource utilization. By investigating the relationship between meteorological information and ET

_{0}in Shihezi City, four prediction models were developed: a BP neural network prediction model, a BP neural network prediction model improved by genetic algorithm (GA-BP), a BP neural network prediction model improved by particle swarm algorithm (PSO-BP), as well as an improved hybrid BP neural network prediction model (GA-PSO-BP). The Pearson correlation analysis found that the key parameters influencing ET

_{0}were temperature (T

_{max}, T

_{ave}, T

_{min}), hours of sunshine (N), relative humidity (RH), wind speed (U), as well as average pressure (AP). Based on the analysis results, different combinations of meteorological input factors were established for modeling, and the results showed that when the input factors were temperature (T

_{max}, T

_{ave}, T

_{min}), hours of sunshine (N), as well as relative humidity (RH), the overall effect of the ET

_{0}prediction model was better than the other input combinations, and the GA-PSO-BP prediction model was the best, which could provide some guidance for the deployment and use of water resources. This may assist in the allocation and utilization of agricultural water resources in Shihezi.

## 1. Introduction

_{a}), which is calculated by multiplying the reference crop evapotranspiration (ET

_{0}) by the crop factor (K

_{c}) [5,6]. As a result, correct calculation of ET

_{0}will effectively alleviate problems like excess irrigation in the irrigation process [7]. The ET

_{0}is a notion that dates back to the 1970s and was properly defined and codified in 1985. In the 1990s, a variety of equations for estimating ET

_{0}, including Penman’s combinatorial equations, were widely utilized, and studies were performed to enhance these equations. Finally, in the Expert Consultation Report on the Procedure for Revision of the FAO Guidelines for the Prediction of Water Requirements of Crops, the Food and Agriculture Organization of the United Nations (FAO) advised using the Penman–Monteith equation to estimate ET

_{0}[8,9].

_{0}using machine learning, such as those based on temperature, light intensity, wind speed, and so on [10]. Kumar et al. employed an artificial neural network (ANN) to create a model for calculating the ET

_{0}of grassland, and the findings indicated that their ANN model for forecasting was more precise than the previous method [11]. Feng et al. developed ET

_{0}prediction models based on generalized regression neural network (GRNN) and random forest (RF) algorithms, and in each case, the findings showed that the random forest (RF) model performed marginally better compared to the GRNN model [12]. Zhang et al. built a spatially dispersed ET

_{0}model using remote sensing data and machine learning methods, then examined its adaptability, which revealed that the method was better at estimating ET

_{0}[13]. Bellido et al. used a number of techniques and different combinations of input variables to develop a prediction model for reference crop evapotranspiration (ET

_{0}), and the performance of the models built was comparable [14].

_{0}is critical for achieving optimal deployment and exploitation of agricultural irrigation water resources. In this paper, an ET

_{0}prediction model based on a genetic algorithm (GA), particle swarm optimization (PSO), and BP neural network is developed by investigating the impacts of various climatic parameters on ET

_{0}with various input combinations. The following are the specific research objectives: (1) Determine the consequences of various combinations of input elements on each day ET

_{0}estimation in order to identify appropriate input components for modeling. (2) Create an ET

_{0}prediction model built around a BP neural network and optimize it using a genetic algorithm (GA), a particle swarm algorithm (PSO), as well as a hybrid approach (GA-PSO). (3) The four ET

_{0}prediction models were simulated and analyzed using MATLAB to determine the best model for agricultural productivity and water resource management in the Shihezi city.

## 2. Materials and Methods

#### 2.1. Study Area and Sources of Data

_{max}, T

_{ave}, and T

_{min}, °C), wind speed (U, m/s), hours of sunshine (N, h), relative humidity (RH, %), average air pressure (AP, hap), and precipitation (P, mm).

_{0}in cotton fields exist, and therefore the Penman–Monteith formula is typically employed to calculate ET

_{0}[15,16].

^{−2}·d

^{−1}); G denotes soil heat flux (MJ·m

^{−2}·d

^{−1}); T denotes the average daily air temperature (°C); γ denotes the wet and dry table constants; U

_{2}denotes wind speed at a distance of 2 m (m/s); e

_{s}denotes saturated water vapor pressure; and e

_{a}represents real water vapor pressure (kPa).

#### 2.2. BP Neural Network

**Step 1:**Identify the number of nodes a, b, c in the BP neural network’s input, hidden, and output layers, initialize the connection weights W

_{ij}and W

_{jl}within the neurons in each layer, given the learning rate and neuron excitation function, and initialize the hidden and output layer thresholds m, n.

**Step 2:**From the input quantity X and the hidden layer link weights W

_{ij}and threshold m, compute the hidden layer output. The following is the formula:

**Step 3:**Determine the output layer O

_{l}by implicitly producing H

_{j}and connecting the weights W

_{jl}and the threshold n. The following is the formula:

**Step 4:**Using the following formula, calculate the output error E

_{l}between the expected output O

_{l}and the intended output T

_{l}:

**Step 5:**Use the output error E

_{l}to correct the output layer and implied layer weights as follows:

**Step 6:**Adjust the threshold based on the output error E

_{l}, which is determined as follows:

**Step 7:**Check to see if the network has met the end conditions; if not, continue training.

#### 2.3. BP Neural Networks’ Genetic Algorithm Optimization

**Step 1:**Generate N chromosomes at random, with each chromosome representing the weights and thresholds between the BP neural network’s input, hidden, and output layers.

**Step 2:**The function that best describes the fitness fit was chosen to be the MSE Equation (17), and the fit of the fitness function was utilized to calculate the fitness value for each chromosome.

**Step 3:**Using the roulette approach, choose the chromosome with the highest applicability, and the selection probability is determined as follows:

_{c}denotes the fitness value of individual c; and P

_{c}denotes the probability of individual c being selected.

**Step 4:**The crossover operation is performed on the chromosomes using the genuine crossover approach to produce new chromosomes utilizing the crossover probability process described below:

_{i}and x

_{j}denote the two paternal chromosomes; x′

_{i}and x′

_{j}denote the two child chromosomes; and α takes the value range of [0, 1].

**Step 5:**Mutational operations on the staining are carried out utilizing uniform mutation to generate new chromosomes distinct from the parent, and the mutational operations are as follows:

_{max}and x

_{min}are the highest and lowest limits of x

_{i}; g is the right now number of iterations; G

_{max}is the highest possible number of evolutions; and γ and r have values in the range [0, 1].

**Step 6:**Repeat steps 3–5 until the individual with the greatest fitness level is found.

#### 2.4. Particle Swarm Optimization for BP Neural Networks

**Step 1:**Set the velocity v

_{i}, position x

_{i}, population size N, individual highest value pbest, as well as global extreme value zbest for the particle swarm.

**Step 2:**MSE is chosen as the fitness function to calculate the particle swarm’s initial fitness value.

**Step 3:**If the individual fitness value calculated in the second step is a better value, then the individual’s current position will be used as the individual’s historical optimal placement, that is, the individual extreme value pbest; otherwise, continue to maintain the current individual extreme value pbest until a better individual appears in the update.

**Step 4:**Change the global extreme value zbest; compare the fitness scores of pbest and zbest; if pbest’s fitness value is better, the individual optimal position will be used as the population’s historical optimal position, that is, the global extreme value; otherwise, the present global extreme value will remain in place until an improved severe value appears.

**Step 5:**Change the particle’s velocity v and location x using the particle swarm algorithm velocity and position update equations, which are as follows:

_{ij}(t) represents the j-dimensional velocity component of particle i evolving to generation t; x

_{ij}(t) is the j-dimensional position component of particle i evolving to generation t; pbest

_{ij}(t) represents the pbest

_{i}component of the best position of the j-dimensional individual of particle i evolving to generation t; and zbest

_{ij}(t) represents the j-dimensional best position of the whole particle swarm evolving to generation t zbest

_{i}component.

_{1}and c

_{2}constitute learning variables with a value range of (0, 2); r

_{1}and r

_{2}are taking the value range of (0, 1) range. The magnitude of the inertia factor ω directly impacts the particle swarm algorithm’s optimization ability. When the value of ω is bigger, the value now displays a better global search ability, while a smaller value shows a better local convergence performance, which can be calculated as follows:

_{max}and ω

_{min}are the highest and lowest values of the inertia gravity factor, respectively; R is the at present number of iterations; and E is the overall number of iterations.

**Step 6:**End condition judgment of the particle swarm algorithm, based on the set end condition of the greatest number of repetitions or go after fitness value, to determine whether the finish scenario is reached; if the finish scenario is not reached, come back to the second step; alternatively, output zbest, which is the global optimal solution.

#### 2.5. Hybrid Optimization of BP Neural Network

**Step 1:**Determine the number of nodes in the input, hidden, and output layers as well as initialize the neural network according to the model’s input and output data.

**Step 2:**The particle characteristics and number of particles were determined based on the network’s structure, and the speed as well as position of the particles were encoded in binary.

**Step 3:**MSE is implemented as the function of fitness to calculate each particle’s fitness value, and the results are utilized to determine whether the target conditions are fulfilled. If the desired circumstances are fulfilled, the results are outputted; otherwise, the particles’ individual and global optimizations are changed.

**Step 4:**The particle swarm crossover operation, utilizing the betting wheel approach to select better adapted particles, selected better adapted particles put into the particle swarm in the next iteration, based on the set probability of the position and speed of the particles to crossover.

**Step 5:**For the particle swarm mutation operation, select some of the particles from the particle swarm with low fitness values, use the velocity mutation operator and position mutation operator to mutate the velocity as well as the position of the particles based on the set probability, and reintroduce the mutated particles into the original particle swarm.

**Step 6:**The fitness function is used to determine the fitness value, update the individual particle polarity pbest, and update the particle swarm’s global polarity zbest.

**Step 7:**Determine whether the particle swarm’s fitness value meets the target value or whether the particles’ evolutionary generation is up to the greatest evolutionary generation; when the above conditions are fulfilled, then output the best possible result zbest; otherwise, proceed to the third step to finish the iteration.

**Step 8:**After the iteration is finished, decode the best possible result and replace the initial weights and thresholds in the specified BP neural network.

#### 2.6. Criteria for Evaluation

^{2}) are introduced for assessing the efficiency of the BP, GA-BP, PSO-BP, and GA-PSO-BP models.

^{2}), which has a range of [0, 1], reflects the correlation among the expected outcome and the actual result; when R2 is close to 1, the model has a stronger predictive capacity. Where Y

_{i}′ is the predicted value, Y

_{i}is the real value, y

_{i}′ is the predicted value’s average, y

_{i}represents the real value’s average, and n represents the total amount of the real value.

## 3. Results

#### 3.1. Correlation Analysis between ET_{0} and Meteorological Factors

_{0}) must be considered when researching the relationship between various meteorological factors and ET

_{0}. Excessive consideration of climatic parameters raises computational costs, diminishes model computational efficiency, and may reduce prediction accuracy. To determine the association between climatic parameters and ET

_{0}, a Pearson correlation analysis was utilized. Pearson correlation analysis is a statistical method for determining the degree of linearity between two variables that are continuous, and the correlation factor ranges from −1 to 1. When the absolute value approaches one, the correlation is stronger, and the positive and negative signs reflect the direction of the correlation. The presence of a positive correlation indicates that the two variables have an advantageous linear link, an unfavorable correlation indicates that the two variables have a linear connection that is unfavorable, and a correlation value close to 0 indicates that the two variables have essentially no linear connection between them.

_{0}, with maximum temperature (T

_{max}), average temperature (T

_{ave}), and minimum temperature (T

_{min}) having correlation coefficients of 0.51, 0.78, and 0.61, respectively, while hours of sunshine (N), wind speed (U), average pressure (AP), and relative humidity (RH) having correlation coefficients of 0.51, 0.3, −0.12 and −0.44. Temperature (T

_{max}, T

_{ave}, T

_{min}), wind speed (U), and hours of sunshine (N) are all positively connected with ET

_{0}, with a substantial association, although average air pressure (AP) and relative humidity (RH) are negatively correlated. However, mean air pressure (AP) has a weak association and is not used in the model’s input component selection.

#### 3.2. Simulation Analysis of ET_{0} Prediction Model

_{0}, seven distinct combinations of meteorological input factor combinations and models were chosen in this article. Temperature (T

_{max}, T

_{ave}, T

_{min}) was chosen as the first three input elements for the prediction model since it has the strongest association with ET

_{0}. Modeling simulation analysis was performed using MATLAB 2017b to assess the performance of the BP neural network prediction model, GA-BP prediction model, PSO-BP prediction model, and GA-PSO-BP prediction model under seven combinations of inputs, and the results are given in Table 2.

_{max}, T

_{ave}, T

_{min}), the MAR, RMSE, and R

^{2}of the model are in the ranges of 0.209–0.411 mm/day, 0.258–0.435 mm/day, and 0.793–0.893, respectively, and when the input factors are increased to 4, the MAR, RMSE, and R

^{2}of the model are in the ranges of 0.174–0.393 mm/day, 0.212–0.401 mm/day and 0.802–0.901. When the number of input components is raised to 5, the model’s MAR, RMSE, and R

^{2}range from 0.145 to 0.341 mm/day, 0.163 to 0.352 mm/day, and 0.843 to 0.952.

#### 3.3. Analysis of Results

_{1}, X

_{2}, X

_{3}and X

_{4}, it can be seen that the model’s effect is improved with the increase in input factors, and the superposition of sunshine hours (N), relative humidity (RH), wind speed (U) and temperature (T

_{max}, T

_{ave}, T

_{min}) all help to improve the model performance. Further comparison of combinations X

_{5}, X

_{6}and X

_{7}shows that the superposition of N, RH and T (T

_{max}, T

_{ave}, T

_{min}) is more effective in improving the model performance, and the model effect of combination X

_{7}is optimal compared to other groups. The comparison of the four prediction models is shown in Figure 4.

## 4. Discussion

_{0}) is critical in determining the selection of input factors for prediction models, and many studies show that temperature crop input factors are better suited for ET

_{0}prediction modeling in the Shihezi region [27]. This is consistent with Rachid et al.’s findings that temperature is a crucial factor regulating ET

_{0}[28]. However, in the case of limited input factors, the performance of the prediction model built solely on the neural network is poor; having said that, the performance of the model can be improved by introducing other optimization algorithms, and it has been demonstrated that the accuracy of the optimized BP neural network is significantly higher than that of the unoptimized prediction model [29,30,31]. These findings are congruent with the findings of our investigation.

_{max}, T

_{ave}, T

_{min}), hours of sunlight (N), and relative humidity (RH) were input factors.

## 5. Conclusions

_{0}) prediction model: the genetic algorithm (GA), particle swarm algorithm (PSO), and hybrid optimization. The influencing elements having the highest association with ET

_{0}were found using correlation analysis, and the ET

_{0}prediction model was constructed and simulated using various combinations of meteorological factors. It is summarized below:

- Temperature (T
_{max}, T_{ave}, T_{min}), hours of sunlight (N), relative humidity (RH), wind speed (U), and average air pressure (AP) all had an effect on reference crop evapotranspiration (ET_{0}). And when the input factors include temperature (T_{max}, T_{ave}, T_{min}), daylight hours (N), and relative humidity (RH), the model performance is better than other input combinations, indicating that this input combination is optimum for building the model. - When the four ET
_{0}prediction models are compared under the combination of X_{7}input factors, the GA-PSO-BP prediction model outperforms the other three prediction models, with optimal MAE, RMSE, and R^{2}values of 0.145 mm/day, 0.163 mm/day, and 0.952, respectively. - Analyzing seven sets of meteorological factors input combinations reveals that the hybrid algorithm (GA-PSO) provides the best performance boost to the BP neural network, and the prediction impact of the GA-PSO-BP model is optimal under each input combination. As a result, when meteorological circumstances are constrained, the use of the GA-PSO-BP model to estimate ET
_{0}for water resource allocation has a significant reference value.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Neissi, L.; Albaji, M.; Nasab, S.B. Site Selection of Different Irrigation Systems Using an Analytical Hierarchy Process Integrated with GIS in a Semi-Arid Region. Water Resour. Manag.
**2019**, 33, 4955–4967. [Google Scholar] [CrossRef] - Marsal, J.; Girona, J.; Casadesus, J.; Lopez, G.; Stöckle, C.O. Crop coefficient (Kc) for apple: Comparison between measurements by a weighing lysimeter and prediction by CropSyst. Irrig. Sci.
**2013**, 31, 455–463. [Google Scholar] [CrossRef] - Mokhtar, A.; Al-Ansari, N.; El-Ssawy, W.; Graf, R.; Aghelpour, P.; He, H.M.; Hafez, S.M.; Abuarab, M. Prediction of Irrigation Water Requirements for Green Beans-Based Machine Learning Algorithm Models in Arid Region. Water Resour. Manag.
**2023**, 37, 1557–1580. [Google Scholar] [CrossRef] - Shen, J.L.; Zhao, Y.K.; Song, J.F. Analysis of the regional differences in agricultural water poverty in China: Based on a new agricultural water poverty index. Agric. Water Manag.
**2022**, 270, 107745. [Google Scholar] [CrossRef] - Xiang, K.Y.; Li, Y.; Horton, R.; Feng, H. Similarity and difference of potential evapotranspiration and reference crop evapotranspiration—A review. Agric. Water Manag.
**2020**, 232, 106043. [Google Scholar] [CrossRef] - Fan, J.L.; Wu, L.F.; Zhang, F.C.; Xiang, Y.Z.; Zheng, J. Climate change effects on reference crop evapotranspiration across different climatic zones of China during 1956–2015. J. Hydrol.
**2016**, 542, 923–937. [Google Scholar] [CrossRef] - Subedi, A.; Chávez, J. Crop Evapotranspiration (ET) Estimation Models: A Review and Discussion of the Applicability and Limitations of ET Methods. J. Agric. Sci.
**2015**, 7, 6. [Google Scholar] [CrossRef] - Pereira, L.S.; Allen, R.G.; Smith, M.; Raes, D. Crop evapotranspiration estimation with FAO56: Past and future. Agric. Water Manag.
**2015**, 147, 4–20. [Google Scholar] [CrossRef] - Anderson, R.G.; French, A.N. Crop Evapotranspiration. Agronomy
**2019**, 9, 614. [Google Scholar] [CrossRef] - Rodrigues, G.C.; Braga, R.P. Estimation of Reference Evapotranspiration during the Irrigation Season Using Nine Temperature-Based Methods in a Hot-Summer Mediterranean Climate. Agriculture
**2021**, 11, 124. [Google Scholar] [CrossRef] - Kumar, M.; Raghuwanshi, N.S.; Singh, R. Artificial neural networks approach in evapotranspiration modeling: A review. Irrig. Sci.
**2011**, 29, 11–25. [Google Scholar] [CrossRef] - Feng, Y.; Cui, N.B.; Gong, D.Z.; Zhang, Q.W.; Zhao, L. Evaluation of random forests and generalized regression neural networks for daily reference evapotranspiration modelling. Agric. Water Manag.
**2017**, 193, 163–173. [Google Scholar] [CrossRef] - Zhang, Z.X.; Gong, Y.C.; Wang, Z.J. Accessible remote sensing data based reference evapotranspiration estimation modelling. Agric. Water Manag.
**2018**, 210, 59–69. [Google Scholar] [CrossRef] - Bellido-Jimenez, J.A.; Estevez, J.; Garcia-Marin, A.P. New machine learning approaches to improve reference evapotranspiration estimates using intra-daily temperature-based variables in a semi-arid region of Spain. Agric. Water Manag.
**2021**, 245, 106588. [Google Scholar] [CrossRef] - Shiri, J.; Nazemi, A.H.; Sadraddini, A.A.; Landeras, G.; Kisi, O.; Fard, A.F.; Marti, P. Comparison of heuristic and empirical approaches for estimating reference evapotranspiration from limited inputs in Iran. Comput. Electron. Agric.
**2014**, 108, 230–241. [Google Scholar] [CrossRef] - Feng, Y.; Jia, Y.; Cui, N.B.; Zhao, L.; Li, C.; Gong, D.Z. Calibration of Hargreaves model for reference evapotranspiration estimation in Sichuan basin of southwest China. Agric. Water Manag.
**2017**, 181, 1–9. [Google Scholar] [CrossRef] - Qi, W.; Zhang, Z.Y.; Wang, C.; Huang, M.Y. Prediction of infiltration behaviors and evaluation of irrigation efficiency in clay loam soil under Moistube (R) irrigation. Agric. Water Manag.
**2021**, 248, 106756. [Google Scholar] [CrossRef] - Chen, R.Y.; Song, J.J.; Xu, M.B.; Wang, X.L.; Yin, Z.; Liu, T.Q.; Luo, N. Prediction of the corrosion depth of oil well cement corroded by carbon dioxide using GA-BP neural network. Constr. Build. Mater.
**2023**, 394, 132127. [Google Scholar] [CrossRef] - Yu, A.X.; Liu, Y.K.; Li, X.; Yang, Y.L.; Zhou, Z.W.; Liu, H.R. Modeling and Optimizing of NH
_{4}^{+}Removal from Stormwater by Coal-Based Granular Activated Carbon Using RSM and ANN Coupled with GA. Water**2021**, 13, 608. [Google Scholar] [CrossRef] - Zhu, F.L.; Zhang, L.X.; Hu, X.; Zhao, J.W.; Meng, Z.H.; Zheng, Y. Research and Design of Hybrid Optimized Backpropagation (BP) Neural Network PID Algorithm for Integrated Water and Fertilizer Precision Fertilization Control System for Field Crops. Agronomy
**2023**, 13, 1423. [Google Scholar] [CrossRef] - Singh, N.K.; Singh, Y.; Kumar, S.; Upadhyay, R. Integration of GA and neuro-fuzzy approaches for the predictive analysis of gas-assisted EDM responses. Appl. Sci.
**2019**, 2, 137. [Google Scholar] [CrossRef] - Wang, T.; Fang, G.H.; Xie, X.M.; Liu, Y.; Ma, Z.Z. A Multi-Dimensional Equilibrium Allocation Model of Water Resources Based on a Groundwater Multiple Loop Iteration Technique. Water
**2017**, 9, 718. [Google Scholar] [CrossRef] - Yi, W.Z. Forecast of agricultural water resources demand based on particle swarm algorithm. Acta Agric. Scand. Sect. B-Soil Plant Sci.
**2022**, 72, 30–42. [Google Scholar] [CrossRef] - Lu, G.Y.; Xu, D.; Meng, Y. Dynamic Evolution Analysis of Desertification Images Based on BP Neural Network. Comput. Intell. Neurosci.
**2022**, 2022, 5645535. [Google Scholar] [CrossRef] [PubMed] - Yan, J.Z.; Xu, Z.B.; Yu, Y.C.; Xu, H.X.; Gao, K.L. Application of a Hybrid Optimized BP Network Model to Estimate Water Quality Parameters of Beihai Lake in Beijing. Appl. Sci.
**2019**, 9, 1863. [Google Scholar] [CrossRef] - Jahandideh-Tehrani, M.; Bozorg-Haddad, O.; Loaiciga, H.A. Application of particle swarm optimization to water management: An introduction and overview. Environ. Monit. Assess.
**2020**, 192, 281. [Google Scholar] [CrossRef] - Jiao, P.; Hu, S.-J. Optimal Alternative for Quantifying Reference Evapotranspiration in Northern Xinjiang. Water
**2022**, 14, 1. [Google Scholar] [CrossRef] - Hadria, R.; Benabdelouhab, T.; Lionboui, H.; Salhi, A. Comparative assessment of different reference evapotranspiration models towards a fit calibration for arid and semi-arid areas. J. Arid Environ.
**2021**, 184, 104318. [Google Scholar] [CrossRef] - Qin, A.; Fan, Z.; Zhang, L. Hybrid Genetic Algorithm−Based BP Neural Network Models Optimize Estimation Performance of Reference Crop Evapotranspiration in China. Appl. Sci.
**2022**, 12, 10689. [Google Scholar] [CrossRef] - Kim, S.; Kim, H. Neural networks and genetic algorithm approach for nonlinear evaporation and evapotranspiration modeling. J. Hydrol.
**2008**, 351, 299–317. [Google Scholar] [CrossRef] - Zhang, Z.-C.; Zeng, X.-M.; Li, G.; Lu, B.; Xiao, M.-Z.; Wang, B.-Z. Summer Precipitation Forecast Using an Optimized Artificial Neural Network with a Genetic Algorithm for Yangtze-Huaihe River Basin, China. Atmosphere
**2022**, 13, 929. [Google Scholar] [CrossRef]

**Figure 3.**Correlation analysis of meteorological factors with ET

_{0}are presented; T

_{max}, T

_{ave}, and T

_{min}denote maximum, average, and minimum temperatures (°C); RH denotes relative humidity (%); U denotes wind speed (m/s); AP denotes average air pressure (hap); and N denotes hours of sunshine (h).

**Figure 4.**Box plots of the overall accuracy of the prediction model are presented: (

**a**) mean absolute error (MAE) box plot, (

**b**) root mean square error (RMSE) box plot, and (

**c**) coefficient of determination (R

^{2}) box plot.

Month | T_{max} (°C) | T_{ave} (°C) | T_{min} (°C) | U (m/s) | N (h) | RH (%) | AP (hap) |
---|---|---|---|---|---|---|---|

Jan | −2.0 | −12.8 | −23.5 | 0.9 | 290.0 | 82 | 974 |

Feb | 4.3 | −12.6 | −25.2 | 0.8 | 294 | 78 | 977 |

Mar | 16 | 4.2 | −13.1 | 1.3 | 370.3 | 74 | 968 |

Apr | 30.2 | 15.5 | −1.5 | 1.5 | 430.6 | 35 | 966 |

May | 34.6 | 23.5 | 10.8 | 2.0 | 457.0 | 40 | 959 |

Jun | 39.9 | 26.3 | 12.3 | 1.6 | 462.0 | 43 | 956 |

Jul | 37.1 | 25.9 | 13.1 | 1.6 | 467.4 | 46 | 955 |

Aug | 35.6 | 23.4 | 7.1 | 1.4 | 432.5 | 51 | 959 |

Sep | 39.5 | 19.7 | 4.0 | 1.2 | 375.2 | 47 | 962 |

Oct | 22.8 | 8.7 | −2.7 | 1.0 | 341.2 | 62 | 971 |

Nov | 13.0 | −0.7 | −27.7 | 1.0 | 290.5 | 80 | 973 |

Dec | −5.0 | −16.0 | −26.0 | 0.7 | 278.5 | 81 | 983 |

Input | MAE (mm/day) | RMSE (mm/day) | R^{2} | ||
---|---|---|---|---|---|

X_{1} | T | BP | 0.411 | 0.435 | 0.793 |

GA-BP | 0.306 | 0.331 | 0.842 | ||

PSO-BP | 0.303 | 0.326 | 0.849 | ||

GA-PSO-BP | 0.209 | 0.258 | 0.893 | ||

X_{2} | T, U | BP | 0.393 | 0.401 | 0.802 |

GA-BP | 0.289 | 0.318 | 0.851 | ||

PSO-BP | 0.283 | 0.312 | 0.858 | ||

GA-PSO-BP | 0.198 | 0.245 | 0.901 | ||

X_{3} | T, RH | BP | 0.361 | 0.374 | 0.813 |

GA-BP | 0.261 | 0.295 | 0.869 | ||

PSO-BP | 0.255 | 0.286 | 0.877 | ||

GA-PSO-BP | 0.183 | 0.231 | 0.912 | ||

X_{4} | T, N | BP | 0.349 | 0.365 | 0.835 |

GA-BP | 0.258 | 0.283 | 0.876 | ||

PSO-BP | 0.248 | 0.277 | 0.882 | ||

GA-PSO-BP | 0.174 | 0.212 | 0.921 | ||

X_{5} | T, RH, U | BP | 0.341 | 0.352 | 0.843 |

GA-BP | 0.241 | 0.256 | 0.889 | ||

PSO-BP | 0.237 | 0.251 | 0.893 | ||

GA-PSO-BP | 0.165 | 0.201 | 0.933 | ||

X_{6} | T, N, U | BP | 0.319 | 0.331 | 0.857 |

GA-BP | 0.233 | 0.245 | 0.898 | ||

PSO-BP | 0.228 | 0.237 | 0.902 | ||

GA-PSO-BP | 0.153 | 0.181 | 0.945 | ||

X_{7} | T, N, RH | BP | 0.295 | 0.313 | 0.871 |

GA-BP | 0.214 | 0.229 | 0.907 | ||

PSO-BP | 0.211 | 0.224 | 0.911 | ||

GA-PSO-BP | 0.145 | 0.163 | 0.952 |

_{max}), average temperature (T

_{ave}), and minimum temperature (T

_{min}); RH denotes relative humidity (%); U denotes wind speed (m/s); AP denotes average air pressure (hap); and N denotes hours of sunshine (h).

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zheng, Y.; Zhang, L.; Hu, X.; Zhao, J.; Dong, W.; Zhu, F.; Wang, H.
Multi-Algorithm Hybrid Optimization of Back Propagation (BP) Neural Networks for Reference Crop Evapotranspiration Prediction Models. *Water* **2023**, *15*, 3718.
https://doi.org/10.3390/w15213718

**AMA Style**

Zheng Y, Zhang L, Hu X, Zhao J, Dong W, Zhu F, Wang H.
Multi-Algorithm Hybrid Optimization of Back Propagation (BP) Neural Networks for Reference Crop Evapotranspiration Prediction Models. *Water*. 2023; 15(21):3718.
https://doi.org/10.3390/w15213718

**Chicago/Turabian Style**

Zheng, Yu, Lixin Zhang, Xue Hu, Jiawei Zhao, Wancheng Dong, Fenglei Zhu, and Hao Wang.
2023. "Multi-Algorithm Hybrid Optimization of Back Propagation (BP) Neural Networks for Reference Crop Evapotranspiration Prediction Models" *Water* 15, no. 21: 3718.
https://doi.org/10.3390/w15213718