# Combined Physical Process and Deep Learning for Daily Water Level Simulations across Multiple Sites in the Three Gorges Reservoir, China

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Sites

#### 2.2. Physical-Based Hydrological Model

^{3}/s), A is the cross-sectional area (m

^{2}), t is time (s), x is distance(m), z is water level (m), g is gravity acceleration (m/s

^{2}), q is the lateral flow per unit distance (m

^{2}/s), B is the width of the water surface (m), n is the roughness coefficient, and R is the wetted perimeter (m). The control equations are discretized using the Preissmann implicit differential scheme, while the coefficient matrix is solved based on the chasing method.

#### 2.3. Deep Learning Model

#### 2.3.1. Back Propagation Neural Network

#### 2.3.2. Long Short-Term Memory

#### 2.4. Comparative Modes

#### 2.4.1. Support Vector Regression

#### 2.4.2. Classification and Regression Tree

#### 2.5. Model Evaluation Index

^{2}) of the model were calculated.

^{2}, which ranges between 0 and 1.

## 3. Results and Discussion

#### 3.1. Building the Connections between Mainstream and Its Tributaries in TGR

^{2}as evaluation criteria.

^{2}value above 0.9. Within the validation set, the smallest R

^{2}was 0.939 and the largest R

^{2}was 0.999. These results indicate that the BP model can reliably predict the water level of mainstream stations by modeling the mainstream-tributaries relationships.

#### 3.2. Water Level Forecasting Based on the Proposed Model at Different Time Tasks

#### 3.3. Model Comparisons with Conventional Machine Learning Approaches

- (1)
- Introduce multi-feature data and construct water level prediction models based on multi-input multi-output or multi-task learning techniques;
- (2)
- Compare different deep learning techniques for water level prediction tasks, and design more suitable deep learning structures for hydrological data characteristics and patterns.
- (3)
- Investigate the interpretability issues of hybrid models in predicting water level fluctuations in order to reveal underlying causal relationships more effectively.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Overview of the study area in the Three Gorges Reservoir (TGR), China. The numbers adjacent to the dots denote the positional indices of the stations. The red circular dots represent observation stations with historically observed flow data (day). The red triangle represents the Three Gorges Dam, which has historically observed water level data (day). The green circular dots represent the junctures of mainstream and tributary. These dots serve as simulated locations and represent the positions that necessitate prediction. The black arrows represent the flow direction.

**Figure 2.**The building process of the proposed model (PHM-BP-LSTM). The arrows points towards the direction of data generation from PHM and subsequent deep learning analysis.

**Figure 3.**(

**a**) Loss plot for BP model for predicting water level (T) of mainstream stations. (

**b**–

**d**) Loss plot for LSTM model for predicting multi-step (T + 1, T + 3, T + 7) water level (WL) of mainstream stations.

**Figure 4.**The fitting effect of the prediction (predicted data) dam-front (No.29 station) water level and the simulated water level (observed data) of the mechanism model. The inner plot represents the relative error after taking the absolute value.

**Figure 5.**Box plots for the absolute value of the relative error between the water level predicted by the LSTM multi-step model and the simulated value of the mechanism model (the station of the mainstream is shown as the green dots in Figure 1).

**Figure 6.**The fitting effect of the multi-step (T + 1, T + 3, T + 7) time series prediction (predicted data) water level of the dam-front (No. 29 station) and the simulated water level (observed data) of the mechanism model. The inner plot represents the relative error after taking the absolute value.

**Table 1.**Performance of BP model in forecasting historical water levels of mainstream stations (the location index of mainstream stations is shown as the green dots in Figure 1).

Location Index | Training Set | Validation Set | ||||
---|---|---|---|---|---|---|

RMSE | MAE | R^{2} | RMSE | MAE | R^{2} | |

3 | 0.883 | 0.414 | 0.918 | 0.661 | 0.363 | 0.939 |

5 | 0.818 | 0.526 | 0.966 | 0.684 | 0.453 | 0.969 |

7 | 0.355 | 0.239 | 0.995 | 0.311 | 0.218 | 0.996 |

9 | 0.361 | 0.271 | 0.996 | 0.321 | 0.246 | 0.997 |

11 | 0.648 | 0.504 | 0.992 | 0.613 | 0.492 | 0.994 |

13 | 0.751 | 0.459 | 0.992 | 0.763 | 0.497 | 0.993 |

15 | 1.025 | 0.686 | 0.985 | 0.883 | 0.591 | 0.991 |

17 | 0.446 | 0.296 | 0.997 | 0.487 | 0.321 | 0.997 |

19 | 1.453 | 0.939 | 0.974 | 1.275 | 0.845 | 0.984 |

21 | 0.773 | 0.48 | 0.993 | 0.664 | 0.409 | 0.995 |

23 | 0.412 | 0.272 | 0.998 | 0.381 | 0.268 | 0.998 |

25 | 0.232 | 0.142 | 0.999 | 0.256 | 0.163 | 0.999 |

27 | 0.639 | 0.392 | 0.995 | 0.561 | 0.347 | 0.997 |

29 | 0.241 | 0.153 | 0.999 | 0.259 | 0.174 | 0.999 |

Step | Time Lag | Hidden Size | Learning Rate | Num Layers | Batch Size | Activation Function |
---|---|---|---|---|---|---|

T + 1 | 4 | 32 | 0.001 | 1 | 32 | relu |

T + 3 | 18 | 32 | 0.001 | 1 | 32 | relu |

T + 7 | 30 | 32 | 0.0005 | 1 | 32 | relu |

**Table 3.**Performance of the models in multi-step time series forecasting water levels of mainstream stations. Δ represents the increase in RMSE and MAE of SVR and CART compared to LSTM, and—represents the performance decline of SVR and CART compared with LSTM.

Step | Model | Training Set | Validation Set | ||||||
---|---|---|---|---|---|---|---|---|---|

RMSE | MAE | RMSE | MAE | ||||||

T + 1 | LSTM | 0.717 | - | 0.412 | 0.793 | 0.489 | |||

SVR | 0.783 | (−Δ9.2%) | 0.524 | (−Δ27.2%) | 0.858 | (−Δ8.1%) | 0.598 | (−Δ22.3%) | |

CART | 0.764 | (−Δ6.6%) | 0.503 | (−Δ22.1%) | 0.89 | (−Δ12.2%) | 0.619 | (−Δ26.6%) | |

T + 3 | LSTM | 1.23 | - | 0.71 | 1.282 | 0.833 | |||

SVR | 1.377 | (−Δ11.9%) | 0.923 | (−Δ30%) | 1.39 | (−Δ8.4%) | 0.981 | (−Δ17.8%) | |

CART | 1.237 | (−Δ0.6%) | 0.781 | (−Δ10%) | 1.379 | (−Δ7.6%) | 0.94 | (−Δ12.8%) | |

T + 7 | LSTM | 1.909 | - | 1.168 | 1.981 | - | 1.321 | - | |

SVR | 2.132 | (−Δ11.7%) | 1.443 | (−Δ23.5%) | 2.136 | (−Δ7.8%) | 1.53 | (−Δ15.8%) | |

CART | 2.008 | (−Δ5.2%) | 1.333 | (−Δ14.1%) | 2.118 | (−Δ6.9%) | 1.437 | (−Δ8.8%) |

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## Share and Cite

**MDPI and ACS Style**

Xie, M.; Shan, K.; Zeng, S.; Wang, L.; Gong, Z.; Wu, X.; Yang, B.; Shang, M.
Combined Physical Process and Deep Learning for Daily Water Level Simulations across Multiple Sites in the Three Gorges Reservoir, China. *Water* **2023**, *15*, 3191.
https://doi.org/10.3390/w15183191

**AMA Style**

Xie M, Shan K, Zeng S, Wang L, Gong Z, Wu X, Yang B, Shang M.
Combined Physical Process and Deep Learning for Daily Water Level Simulations across Multiple Sites in the Three Gorges Reservoir, China. *Water*. 2023; 15(18):3191.
https://doi.org/10.3390/w15183191

**Chicago/Turabian Style**

Xie, Mingjiang, Kun Shan, Sidong Zeng, Lan Wang, Zhigang Gong, Xuke Wu, Bing Yang, and Mingsheng Shang.
2023. "Combined Physical Process and Deep Learning for Daily Water Level Simulations across Multiple Sites in the Three Gorges Reservoir, China" *Water* 15, no. 18: 3191.
https://doi.org/10.3390/w15183191