# Optimal Sprinkler Application Rate of Water–Fertilizer Integration Machines Based on Radial Basis Function Neural Network

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## Abstract

**:**

## 1. Introduction

## 2. Neural Network Training Set Data Acquisition

#### 2.1. RBF Neural Network Basic Theory

#### 2.2. Spray Irrigation Soil Hydrology Model

#### 2.2.1. Soil Water Infiltration Process Equation

#### 2.2.2. Unsaturated Soil Hydraulic Properties Function

#### 2.2.3. Irrigation Water Interception Equation

#### 2.2.4. Soil Profile Parameter Setting

#### 2.3. Determination of Simulation Parameters

^{3}and 1.5 g/cm

^{3}. Therefore, in this paper, four parameters are selected from this interval and used as simulation parameters.

#### 2.4. Acquire and Analyze Data

#### 2.4.1. Simulation Data

#### 2.4.2. Data Analysis

## 3. Determination of Neural Network Input and Output Layers

## 4. Training Data Pre-Processing

_{min}is the minimum of all training samples, and X

_{max}is the maximum of all training samples. Partially normalized data for total irrigation water are presented in Table 4.

## 5. Basic Theory of Neural Networks

#### 5.1. BP Neural Network Basic Theory

#### 5.2. Generalized Regression Neural Network Theory

## 6. Evaluation Criteria for Neural Networks

## 7. Training and Comparison

#### 7.1. Neural Network Parameter Determination

**σ**. In this paper, the dichotomous method is used to perform continuous testing and to determine the value of the smoothness parameter based on the magnitude of the average relative error of the test set.

#### 7.2. Comparison of Neural Network Errors

## 8. Empirical Formula for Total Irrigation Water

## 9. Experimental Verification

#### 9.1. Spray Irrigation Soil Infiltration Experiment Site

#### 9.2. Experimental Soil Samples

#### 9.3. Construction of Experimental Platform

#### 9.3.1. Soil Infiltration Device

#### 9.3.2. Sprinkler Irrigation Equipment

#### 9.3.3. Soil Moisture Data Measurement and Transmission System

#### 9.4. Experimental Steps

- (1)
- As shown in Figure 23, the amount of water required to dry the soil to the moisture content in the table was calculated based on the initial soil moisture content listed in Table 5. The soil is then spread out and the water is sprayed evenly on top of the soil to ensure uniformity of soil moisture content.
- (2)
- The soil with the required initial moisture content was left to stand for 24 h and then filled into the soil sample bottles at 2 cm intervals according to the soil bulk density given in Table 5.
- (3)
- As shown in Figure 24, at the top end of the soil sample bottle, the soil surface is covered with another layer of grass to simulate crop interception.
- (4)
- (5)

#### 9.5. Comparison of Neural Network Prediction Results with the Actual Situation

## 10. Value Evaluation Using Optimal Spray Intensity

## 11. Conclusions

- (1)
- In this paper, the highest accuracy of the optimal sprinkler application rate predicted by RBF neural network is obtained by trial and error method when the smoothing parameter is 1.14. The average relative error of its prediction is 0.11 and the RBF neural network is compared with BP neural network and generalized regression neural network. It can be found that the RBF neural network has the highest prediction accuracy. This indicates that RBF neural network is more suitable for the prediction of sprinkler application rate than other neural networks.
- (2)
- Comparing the experimental data with the prediction results, the relative error of RBF neural network prediction is around 10%. This indicates that the RBF neural network has some practical value in the prediction of sprinkler application rate.
- (3)
- The RBF neural network not only predicts the best sprinkler application rate with high accuracy, but also its fast computing speed makes it ideal for placement into embedded devices.
- (4)
- Among the initial soil water content, bulk density and sprinkler application rate, the most complex effect on soil water accumulation time is the sprinkler application rate.
- (5)
- The finite element values simulated by the soil infiltration model established by the partial differential equation do not differ much from the actual values. Therefore, applying the finite element method to soil infiltration can greatly reduce the complexity of soil infiltration experiments.
- (6)
- The maximum application rate of sprinkler irrigation predicted by combining the RBF neural network with the total irrigation volume formula was compared with the conventional recommended application rate of sprinkler irrigation in actual sprinkler irrigation using the lawn as an example. When the optimal application rate of sprinkler irrigation proposed in this paper was used, the sprinkler time was greatly reduced. For an irrigation volume of 58 mm, for example, the optimal application rate of sprinkler irrigation saves 3.4 h and improves the overall efficiency by about 70%, which greatly reduces irrigation losses due to long sprinkler time and also reduces the operation and maintenance costs of the equipment.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Initial moisture content versus time of water accumulation. * is the data points of the finite element simulation, the red line is fitted through the data points.

**Figure 5.**Relationship between bulk density and time of soil water accumulation. * is the data points of the finite element simulation, the red line is fitted through the data points.

**Figure 6.**The relationship between application rate of sprinkler irrigation and the duration of water accumulation. * is the data points of the finite element simulation, the red line is fitted through the data points.

**Figure 15.**Relative error of generalized regression neural networks. The red line represents the mean relative error.

Condition | Sand (%) | Silt (%) | Clay (%) | Soil Texture |
---|---|---|---|---|

1 | 55 | 45 | 0 | Sandy loamy soil |

2 | 60 | 30 | 10 | Sandy loamy soil |

3 | 64 | 30 | 6 | Sandy loamy soil |

4 | 68 | 27 | 5 | Sandy loamy soil |

5 | 70 | 15 | 15 | Sandy loamy soil |

6 | 75 | 13 | 12 | Sandy loamy soil |

7 | 80 | 11 | 9 | Sandy loamy soil |

Initial Moisture Content (%) | 6 | 9 | 11 | 12 | 14 | 15 | 17 | 18 |

Application rate of sprinkler irrigation(mm/h) | 30 | 40 | 50 | 60 | 65 | 70 | 75 | 80 |

Soil Texture | Initial Moisture Content (%) | Bulk Density (g/cm^{3}) | Application Rate of Sprinkler Irrigation (mm/h) | Time of Ponding (min) |
---|---|---|---|---|

Condition1 | 6 | 1.5 | 30 | 84 |

Condition1 | 9 | 1.5 | 30 | 75 |

Condition1 | 11 | 1.5 | 30 | 69 |

Condition1 | 12 | 1.5 | 30 | 66 |

Condition1 | 14 | 1.5 | 30 | 60 |

Condition2 | 6 | 1.3 | 30 | 169 |

Condition2 | 6 | 1.4 | 30 | 134 |

Condition2 | 6 | 1.5 | 30 | 113 |

Condition2 | 9 | 1.3 | 30 | 156 |

Condition2 | 9 | 1.4 | 30 | 122 |

Condition3 | 6 | 1.3 | 40 | 93 |

Condition3 | 6 | 1.4 | 40 | 83 |

Condition3 | 6 | 1.5 | 40 | 74 |

Condition3 | 9 | 1.3 | 40 | 87 |

Condition3 | 9 | 1.4 | 40 | 76 |

Condition3 | 9 | 1.5 | 40 | 67 |

Condition3 | 11 | 1.3 | 40 | 90 |

Condition4 | 11 | 1.3 | 50 | 62 |

Condition4 | 11 | 1.4 | 50 | 55 |

Condition4 | 11 | 1.5 | 50 | 51 |

Condition4 | 12 | 1.3 | 50 | 69 |

Condition4 | 12 | 1.4 | 50 | 53 |

Condition4 | 12 | 1.5 | 50 | 49 |

Condition4 | 14 | 1.3 | 50 | 62 |

Condition4 | 14 | 1.4 | 50 | 49 |

Original data (mm) | 85 | 35 | 76.5 | 24 | 53 | 42 | 60 | 62 | 34.67 |

Post-processing data (mm) | 1 | −0.64 | 0.72 | −1 | −0.05 | −0.41 | 0.18 | 0.25 | −0.65 |

Initial Moisture Content (%) | Bulk Density (g/cm^{3}) | Application Rate of Sprinkler Irrigation (mm/h) |
---|---|---|

15 | 1.3 | 60 |

15 | 1.4 | 60 |

16 | 1.3 | 60 |

18 | 1.3 | 70 |

18 | 1.4 | 60 |

18 | 1.5 | 80 |

18 | 1.4 | 80 |

16 | 1.5 | 80 |

Condition | Time of Ponding (min) | Total Irrigation Volume (mm) |
---|---|---|

1 | 56 | 56 |

2 | 50 | 50 |

3 | 55 | 55 |

4 | 40 | 47 |

5 | 44 | 44 |

6 | 30 | 40 |

7 | 32 | 43 |

8 | 31 | 41 |

Condition | Predicted Application Rate of Sprinkler Irrigation (mm/h) |
---|---|

1 | 55 |

2 | 77 |

3 | 53 |

4 | 73 |

5 | 70 |

6 | 84 |

7 | 90 |

8 | 86 |

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**MDPI and ACS Style**

Liu, X.; Zhu, X.; Liang, Z.; Zou, T.
Optimal Sprinkler Application Rate of Water–Fertilizer Integration Machines Based on Radial Basis Function Neural Network. *Water* **2022**, *14*, 2838.
https://doi.org/10.3390/w14182838

**AMA Style**

Liu X, Zhu X, Liang Z, Zou T.
Optimal Sprinkler Application Rate of Water–Fertilizer Integration Machines Based on Radial Basis Function Neural Network. *Water*. 2022; 14(18):2838.
https://doi.org/10.3390/w14182838

**Chicago/Turabian Style**

Liu, Xiaochu, Xiangjin Zhu, Zhongwei Liang, and Tao Zou.
2022. "Optimal Sprinkler Application Rate of Water–Fertilizer Integration Machines Based on Radial Basis Function Neural Network" *Water* 14, no. 18: 2838.
https://doi.org/10.3390/w14182838