# Application of Rainfall-Runoff Simulation Based on the NARX Dynamic Neural Network Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}, accounting for 3.7% of the total area of the Huaihe basin. The region mainly consists of plains, hills and mountains, which range from 57 m to 1125 m in elevation. The basin belongs to a typical continent monsoon climate which is cold and windy in winter, hot and rainy in summer, and mild in spring and autumn. The average annual temperature ranges from 11.8 ℃ to 13.3 ℃, and the mean annual precipitation/evaporation is approximately 830 mm and 839 mm. Precipitation also has a distinct inter-annual and seasonal distribution, which accounts for 50–80% of the annual total during May–September. Flood disasters are common in the research areas because of the large terrain slopes and rapid water flow collection, which exceeds the flood discharge capacity of the downstream rivers. The spatial distribution map of the station and elevation in the interest area is shown in Figure 1.

#### 2.2. Methodology

#### 2.2.1. Cluster Analysis

#### 2.2.2. NARX Model

^{−10}and 10

^{−7}. If one of the above-mentioned parameters reaches the goals, the NARX process is stopped. More details can be found in [43,47]. The output variable is calculated by the following formula:

#### 2.2.3. Hydrological Model Introduction

_{s}and R

_{sb}are surface runoff and subsurface runoff, respectively; F

_{sat}is the percentage of the saturated area; ${Z}_{\nabla}$ is the soil moisture deficit depth; Q

_{wat}is the net rainfall; R

_{sb,max}is the maximum subsurface runoff when ${Z}_{\nabla}$ equals zero; f is the attenuation coefficient of soil; F

_{sat}is determined by the two-parameter power function, which is established by the relationship between TI and soil moisture.

#### 2.2.4. Model Performance Evaluation

_{obs}(i) and R

_{sim}(i) are the observed and simulated discharge at the ith day; $\overline{R}$

_{obs}and $\overline{R}$

_{sim}are the mean values of the observed and simulated discharge for the time series.

## 3. Results

#### 3.1. Model Performance Comparison in Runoff Simulation

#### 3.2. Rainfall-Runoff Simulation by TOPX Model

#### 3.3. The Comparison of Discharge Simulation by NARX and TOPX Model

## 4. Discussion

## 5. Conclusions

- (1)
- The NARX model is capable of simulating the rainfall-runoff process satisfactorily, capturing the flood peak flow and peak occurrence time properly and reflecting complex nonlinear hydrological dynamin rules based on the four criteria of NSE, CC, RMSE and bias in this study area. Although there is a little underestimation of the peak flow, the discharge simulation is generally consistent with the observation.
- (2)
- By the runoff classification based on cluster analysis, the accuracy of the runoff simulation is further improved due to overcoming the great difference in runoff generation mechanisms caused by different rainfalls. The performance of C-NARX is better than the NARX model, especially during the arid season. In addition, the underestimation of peak flow has also been improved by runoff classification in this study.
- (3)
- Compared with the hydrological model TOPX, the performance of daily runoff simulations driven by the NARX and C-NARX models is better and more efficient in the study watershed. It is feasible to take it as a promising method, which also can be seen as a good reference and replacement for the current rainfall-runoff simulation.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Spatial map of the Linyi watershed, and meteorological stations and hydrological control station.

**Figure 3.**Comparison of simulation and observation values between NARX and C-NARX models in training period with the dry season (

**a**), semi-humid season (

**b**), humid season (

**c**) and all years (

**d**).

**Figure 4.**Discharge results based on the NARX and C-NARX models in the (

**a**) dry season, (

**b**) semi-humid season and (

**c**) humid season.

**Figure 6.**The comparison of runoff simulation results driven by the NARX, C-NARX and TOPX models in the training or calibration period.

**Figure 7.**The comparison of runoff simulation results driven by the three models in the test period.

Month | Average Runoff | Maximum Runoff | Minimum Runoff | Standard Deviation of Runoff |
---|---|---|---|---|

Jan | 14.170 | 18.407 | 9.794 | 2.736 |

Feb | 17.294 | 41.535 | 4.668 | 9.808 |

Mar | 14.784 | 28.122 | 4.550 | 5.794 |

Apr | 12.219 | 49.853 | 4.042 | 10.985 |

May | 17.058 | 81.333 | 2.052 | 16.635 |

Jun | 38.371 | 330.534 | 2.314 | 68.268 |

Jul | 170.795 | 871.948 | 23.526 | 190.304 |

Aug | 213.443 | 909.606 | 47.573 | 211.449 |

Sep | 167.802 | 833.135 | 31.195 | 185.765 |

Oct | 69.677 | 218.905 | 23.805 | 45.769 |

Nov | 25.121 | 40.311 | 14.866 | 7.088 |

Dec | 19.490 | 27.637 | 11.974 | 4.218 |

Classification | Code | Month | Season |
---|---|---|---|

Cluster1 | C-NARX1 | Nov, Dec, Jan, Feb, Mar | Dry/Arid season |

Cluster2 | C-NARX2 | Apr, May, Jun, Oct | Semi-humid season |

Cluster3 | C-NARX3 | Jul, Aug, Sep | Flood/Humid season |

Samples | Classification (Season) | NARX | C-NARX | ||||||
---|---|---|---|---|---|---|---|---|---|

NSE | CC | RMSE | Bias | NSE | CC | RMSE | Bias | ||

Training | Dry | 0.435 | 0.682 | 0.832 | 0.540 | 0.947 | 0.973 | 0.256 | 0.147 |

Semi-humid | 0.720 | 0.852 | 1.308 | 0.637 | 0.923 | 0.962 | 0.682 | 0.349 | |

Humid | 0.886 | 0.942 | 0.541 | 0.308 | 0.942 | 0.971 | 0.385 | 0.221 | |

Validation | Dry | 0.347 | 0.713 | 0.468 | 0.363 | 0.877 | 0.939 | 0.203 | 0.142 |

Semi-humid | 0.066 | 0.484 | 0.742 | 0.498 | 0.284 | 0.719 | 0.649 | 0.437 | |

Humid | 0.500 | 0.790 | 1.071 | 0.464 | 0.562 | 0.833 | 1.002 | 0.513 | |

Test | Dry | −2.267 | 0.237 | 0.885 | 0.495 | 0.455 | 0.762 | 0.361 | 0.230 |

Semi-humid | 0.347 | 0.744 | 0.826 | 0.565 | 0.484 | 0.856 | 0.734 | 0.496 | |

Humid | 0.863 | 0.896 | 0.325 | 0.323 | 0.877 | 0.920 | 0.307 | 0.310 |

Parameter Name | Physical Meaning | Parameter Type | Calibration Value |
---|---|---|---|

f | Decaying parameter | Runoff generation parameters | 169 |

G_{max} | Maximum underground runoff | 55 | |

KSS | Outflow coefficient of free water storage to interflow relationship | 0.010 | |

B | The power of the soil water storage curve | 0.45 | |

K | Ratio of potential evapotranspiration to pan evaporation | Soil moisture calculation | 0.95 |

WM | Averaged soil moisture storage capacity | 160 | |

WUM | Averaged soil moisture storage capacity of the upper layer | 30.0 | |

WLM | Averaged soil moisture storage capacity of the lower layer | 90.0 | |

C | Recession constants of the ground water storage | 0.12 | |

KKG | Recession constants of the ground water storage | Runoff concentration parameters | 0.988 |

UH | Initial value of unit hydrograph | /0,12,26,20, 11,5,3/ |

Samples | Model | NSE | CC | RMSE | Bias |
---|---|---|---|---|---|

Training | TOPX | 0.713 | 0.846 | 1.385 | 0.512 |

NARX | 0.885 | 0.941 | 0.876 | 0.397 | |

C-NARX | 0.949 | 0.974 | 0.580 | 0.237 | |

Validation | TOPX | 0.622 | 0.792 | 1.181 | 0.502 |

NARX | 0.564 | 0.817 | 1.248 | 0.446 | |

C-NARX | 0.625 | 0.857 | 1.156 | 0.409 | |

Test | TOPX | 0.816 | 0.897 | 1.018 | 0.418 |

NARX | 0.895 | 0.923 | 0.767 | 0.347 | |

C-NARX | 0.910 | 0.943 | 0.707 | 0.329 |

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**MDPI and ACS Style**

Shao, Y.; Zhao, J.; Xu, J.; Fu, A.; Li, M.
Application of Rainfall-Runoff Simulation Based on the NARX Dynamic Neural Network Model. *Water* **2022**, *14*, 2082.
https://doi.org/10.3390/w14132082

**AMA Style**

Shao Y, Zhao J, Xu J, Fu A, Li M.
Application of Rainfall-Runoff Simulation Based on the NARX Dynamic Neural Network Model. *Water*. 2022; 14(13):2082.
https://doi.org/10.3390/w14132082

**Chicago/Turabian Style**

Shao, Yuehong, Jun Zhao, Jinchao Xu, Aolin Fu, and Min Li.
2022. "Application of Rainfall-Runoff Simulation Based on the NARX Dynamic Neural Network Model" *Water* 14, no. 13: 2082.
https://doi.org/10.3390/w14132082