# A Comparative Analysis of Multiple Machine Learning Methods for Flood Routing in the Yangtze River

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Comparative ML Models

#### 2.1.1. Support Vector Regression

#### 2.1.2. Gaussian Process Regression

#### 2.1.3. Random Forest Regression

#### 2.1.4. Multilayer Perceptron

_{j}) can be calculated by the following equation.

_{i}is the output value of the ith node in the previous layer, w

_{ij}is the weight from the ith node in the previous layer to this node, θ

_{j}is the bias of this node and f denotes the activation function.

#### 2.1.5. Long Short-Term Memory

_{t}) can be computed by the following equations.

_{t}and h

_{t}

_{-1}are, respectively, the outputs of the current and prior LSTM cells. F

_{t}is the forgetting gate output of the current LSTM cell. I

_{t}is the input gate output of the current LSTM cell. O

_{t}is the output gate output of the current LSTM cell. G

_{t}is the candidate state set of the current LSTM cell. C

_{t}is the state of the current LSTM cell. x

_{t}is the input datum of the current LSTM cell. w

_{Fx}, w

_{Ix}, w

_{Gx}, w

_{Ox}, w

_{Fh}, w

_{Ih}, w

_{Gh}and w

_{Oh}are the weights of the current LSTM cell. b

_{Fx}, b

_{Ix}, b

_{Gx}, b

_{Ox}, b

_{Fh}, b

_{Ih}, b

_{Gh}and b

_{Oh}are the biases of the current LSTM cell. σ and tanh denote the sigmoid function and hyperbolic tangent function, respectively. ∗ denotes the Hadamard product.

#### 2.1.6. Gated Recurrent Unit

_{t}) can be computed by the following equations.

_{t}and h

_{t}

_{-1}are, respectively, the outputs of the current and prior GRU cells. R

_{t}is the reset gate output of the current GRU cell. U

_{t}is the update gate output of the current GRU cell. K

_{t}is the candidate set of the current GRU cell. x

_{t}is the input datum of the current GRU cell. w

_{Rx}, w

_{Ux}, w

_{Kx}, w

_{Rh}, w

_{Uh}and w

_{Kh}are the weights of the current GRU cell. b

_{Rx}, b

_{Ux}, b

_{Kx}, b

_{Rh}, b

_{Uh}and b

_{Kh}are the biases of the current GRU cell.

_{t}and U

_{t}are first obtained using Equations (8) and (9), respectively, based on the input data of the current GRU cell and the output of the prior GRU cell. Secondly, K

_{t}is obtained using Equation (10) based on the input data and reset gate output of the current GRU cell and the output of the prior GRU cell. Lastly, h

_{t}is obtained using Equation (11) based on the update gate output and the candidate set of the current GRU cell and the output of the prior GRU cell [48]. It can be seen from Figure 1 that the GRU possesses a simpler structure compared to the LSTM. Consequently, it facilitates a faster training rate than the LSTM. Therefore, the GRU has been widely used in the field of artificial intelligence, especially in processing long time series.

#### 2.2. Experimental Methods

#### 2.2.1. Data Normalization

_{t}is the normalized value of the data at time t; x

_{t}is the value of the original datum from the training dataset or the testing dataset at time t; and x

_{max}and x

_{min}are the maximum and minimum values of the original data from the training dataset, respectively.

#### 2.2.2. Efficiency Criteria

_{t}and S

_{t}are, respectively, the observed and simulated values at time t; $\overline{O}$ and $\overline{S}$ are, respectively, the average values of the observed and simulated values; T is the number of O

_{t}; and SD

_{O}and SD

_{S}are the standard deviations of the observed and simulated values, respectively. SD

_{O}and SD

_{S}can be respectively defined using the following equations.

#### 2.2.3. Taylor Diagram

_{0}is the maximum correlation attainable and is identified as 0.9999 in this paper. For any given SD

_{S}, TSS increases monotonically with increasing R; for any given R, TSS increases as SD

_{S}approaches SD

_{O}. The larger the TSS of the model, the better the model performance is.

## 3. Case Study

^{2}, which makes it the third-largest river in the world and the first in China. Above Yichang are the upper reaches of the Yangtze River with a main stream length of over 4500 km and a drainage area of 1 million km

^{2}. From Yichang to Hukou are the middle reaches of the Yangtze River with a main stream length of over 950 km and a drainage area of 0.68 million km

^{2}. Below Hukou are the lower reaches of the Yangtze River with a main stream length of over 930 km and a drainage area of 0.12 million km

^{2}. In this paper, two reaches with complex water systems are selected as case studies to compare and analyze the applicability of different ML methods for flood routing in the Yangtze River. The first case study is a reach in the valley from Cuntan Station to the Three Georges Reservoir (TGR) with a main stream length of over 650 km, which is located at the end of the Upper Yangtze River and characterized by a long distance, a large river bottom drop, a deep canyon, rapid flow and many tributaries, which increases the difficulty of inflow forecasting of the TGR. A schematic diagram of the Yangtze River and hydrological stations is shown in Figure 2.

^{3}) significantly enhances the flood control capacity of the Yangtze River in the Middle and Lower reaches, and the flood control standard of the downstream Jingjiang reach has risen from once every ten years to once every one hundred years. Cuntan Station and Wulong Station are the main control hydrological stations for the inflow hydrograph prediction of the TGR, and Zhicheng Station, Shashi Station and Chenglingji Station are the main flood control stations of the downstream Jingjiang reach. Therefore, the inflow hydrograph prediction of the TGR and the water level hydrograph prediction of Shashi Station for flood routing in the Yangtze River are scientific problems with important academic significance and engineering application value, which play an important role in guaranteeing the flood control security of the Jingjiang reach of the Yangtze River as well as China’s energy security.

^{2}, and the drainage areas above Cuntan Station and Wulong Station are about 867,000 and 88,000 km

^{2}, respectively. The drainage areas above Yichang Station, Zhicheng Station and Shashi Station are over 1.0055, 1.0241 and 1.0320 million km

^{2}, respectively. According to the location relationship between these stations, the daily average discharge series of Cuntan Station and Wulong Station are used for inflow prediction of the TGR so as to consider the influence of the discharge from the Wu River, and the daily average discharge series of Yichang Station and the daily average water level series of Zhicheng Station are used for water level prediction of Shashi Station so as to consider the influence of the discharge from the Qing River.

## 4. Results and Discussion

#### 4.1. Experimental Conditions

#### 4.2. The Inflow Hydrograph Prediction of the TGR

^{2}characterizes the accuracy of the linear fit equation. The larger the R

^{2}is, the more the data points are concentrated on both sides of the linear fit line. The closer the slope of the linear fit line is to 1 and the closer the intercept is to 0, the better the linear fit line matches the ideal fit line, and the better the simulated inflows and the observed inflows of the TGR are matched. Therefore, the slope of the linear fit line of the GRU model is closest to 1, and the linear fit line of the GRU model is more in line with the ideal fit line than those of the other ML models, which means that the simulated inflows of the TGR of the GRU model are closest to the corresponding observed inflows. In order to analyze the distribution pattern of the simulated inflow errors of the TGR of the different ML models, violin plots of the simulated inflow errors by different ML models during the testing period are shown in Figure 5.

^{3}/s), Q2 (from 10,000 to 20,000 m

^{3}/s), Q3 (from 20,000 to 30,000 m

^{3}/s), Q4 (from 30,000 to 40,000 m

^{3}/s), Q5 (from 40,000 to 50,000 m

^{3}/s), Q6 (from 50,000 to 60,000 m

^{3}/s), Q7 (from 60,000 to 70,000 m

^{3}/s) and Q8 (greater than 70,000 m

^{3}/s), and the ribbon diagram of the MAPEs for inflow hydrograph prediction of the TGR by the models during the testing period is shown in Figure 6. For the inflow of the TGR from 30,000 to 50,000 m

^{3}/s, namely, discharge interval Q4 and Q5, all models had a larger MAPE, around 10%. The GPR and RFR models had relatively poor MAPEs for larger inflow, the SVR model had a poor MAPE for a smaller inflow, and the MLP, LSTM and GRU models had relatively better MAPEs for smaller and larger inflows. Overall, the GRU model had a smaller MAPE.

#### 4.3. The Water Level Hydrograph Prediction of Shashi Station

^{3}/s (the obtained MAPE is around 10%), which may be related to the long distance from Cuntan Station to the TGR and the influence of many small tributaries with no observation data for this river.

#### 4.4. Performance Comparison among Three Deep Learning Models with Different Time Lags for Flood Routing

## 5. Conclusions

- (1)
- The ML models were verified as effective and efficient in obtaining accurate flood hydrographs in river flood routing with fewer data (e.g., only flows and water levels that are daily measured). Therefore, the ML models could be widely used for flood routing in complex natural rivers. However, it is important to note that not all ML models were equally effective in flood routing, as some may overfit during the training phase.
- (2)
- The deep learning models, including the MLP, LSTM and GRU models, were more efficient than the SVR, GPR and RFR models. The GRU model, in particular, outperformed the others in almost all efficiency criteria, including MAPE, RMSE, NSE, TSS and KGE. The reductions in MAPE and RMSE were significant, with at least 7.66% and 3.80% for the first case study and 19.51% and 11.76% for the second case study during the testing period.
- (3)
- The model that had higher accuracy may necessitate a longer training time, but the GRU exhibited a faster training rate than the LSTM. Although the training times of the LSTM and GRU were longer than those of the other models, the GRU’s training times were, respectively, 32.19% and 26.14% shorter than those of the LSTM for the two case studies due to its simpler structure and more effortless convergence.
- (4)
- The time lag in flood routing determined the number of input variables of the models, which in turn may have affected the accuracy of flood routing. As a result, the accuracy of flood routing gradually increased and then slightly decreased as the time lag increased for the MLP, LSTM and GRU models. Interestingly, the GRU model performed better than the MLP and LSTM models for different time lags.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 12.**TSSs of the MLP, LSTM and GRU models for water level hydrograph prediction of Shashi Station during the testing period.

**Figure 3.**Taylor diagram depicting the performance of six ML models for inflow hydrograph prediction of the TGR.

**Figure 4.**The scatter plots of the simulated and observed inflows of the TGR by different ML models during the testing period.

**Figure 5.**Violin plots of the simulated inflow errors of the TGR by different ML models during the testing period.

**Figure 6.**Ribbon diagram of the MAPEs for inflow hydrograph prediction of the TGR by the models during the testing period.

**Figure 8.**Scatter plots of the simulated and observed water levels of Shashi Station by different ML models during the testing period.

**Figure 9.**Violin plots of the simulated water level errors of Shashi Station by different ML models during the testing period.

**Figure 10.**Ribbon diagram of the MAPEs for water level hydrograph prediction of Shashi Station by the models during the testing period.

Dataset | Model | MAPE (%) | RMSE (m ^{3}/s) | NSE | R | TSS | KGE | Time (s) |
---|---|---|---|---|---|---|---|---|

Training | LMM | 6.25 | 1834 | 0.9681 | 0.9858 | 0.9701 | 0.9370 | 7.969 |

SVR | 27.19 | 2348 | 0.9478 | 0.9898 | 0.9757 | 0.8506 | 0.057 | |

GPR | 6.28 | 1698 | 0.9727 | 0.9864 | 0.9721 | 0.9662 | 6.159 | |

RFR | 4.72 | 1171 | 0.9870 | 0.9935 | 0.9869 | 0.9852 | 8.992 | |

MLP | 5.12 | 1438 | 0.9804 | 0.9915 | 0.9811 | 0.9462 | 2.370 | |

LSTM | 5.58 | 1404 | 0.9813 | 0.9911 | 0.9820 | 0.9694 | 13.841 | |

GRU | 5.17 | 1401 | 0.9814 | 0.9909 | 0.9819 | 0.9837 | 9.386 | |

Testing | LMM | 6.48 | 2262 | 0.9429 | 0.9733 | 0.9463 | 0.9361 | \ |

SVR | 19.96 | 2301 | 0.9410 | 0.9858 | 0.9687 | 0.8720 | \ | |

GPR | 5.65 | 2044 | 0.9534 | 0.9766 | 0.9531 | 0.9612 | \ | |

RFR | 5.01 | 1889 | 0.9602 | 0.9800 | 0.9601 | 0.9691 | \ | |

MLP | 4.96 | 1763 | 0.9653 | 0.9842 | 0.9672 | 0.9436 | \ | |

LSTM | 5.21 | 1736 | 0.9664 | 0.9840 | 0.9682 | 0.9643 | \ | |

GRU | 4.58 | 1670 | 0.9689 | 0.9848 | 0.9697 | 0.9793 | \ |

Dataset | Model | MAPE (%) | RMSE (m) | NSE | R | TSS | KGE | Time (s) |
---|---|---|---|---|---|---|---|---|

Training | SVR | 1.65 | 0.67 | 0.9601 | 0.9908 | 0.9659 | 0.8796 | 0.014 |

GPR | 0.57 | 0.28 | 0.9932 | 0.9966 | 0.9932 | 0.9882 | 5.291 | |

RFR | 0.30 | 0.15 | 0.9979 | 0.9989 | 0.9979 | 0.9975 | 3.723 | |

MLP | 0.22 | 0.14 | 0.9983 | 0.9992 | 0.9984 | 0.9968 | 3.845 | |

LSTM | 0.23 | 0.15 | 0.9981 | 0.9991 | 0.9982 | 0.9985 | 11.912 | |

GRU | 0.24 | 0.14 | 0.9983 | 0.9993 | 0.9986 | 0.9943 | 8.798 | |

Testing | SVR | 2.64 | 1.02 | 0.9037 | 0.9918 | 0.9483 | 0.8237 | \ |

GPR | 0.99 | 0.39 | 0.9857 | 0.9966 | 0.9889 | 0.9358 | \ | |

RFR | 0.78 | 0.33 | 0.9899 | 0.9967 | 0.9918 | 0.9589 | \ | |

MLP | 0.41 | 0.17 | 0.9972 | 0.9992 | 0.9984 | 0.9869 | \ | |

LSTM | 0.42 | 0.18 | 0.9971 | 0.9991 | 0.9944 | 0.9937 | \ | |

GRU | 0.33 | 0.15 | 0.9980 | 0.9993 | 0.9985 | 0.9966 | \ |

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## Share and Cite

**MDPI and ACS Style**

Zhou, L.; Kang, L.
A Comparative Analysis of Multiple Machine Learning Methods for Flood Routing in the Yangtze River. *Water* **2023**, *15*, 1556.
https://doi.org/10.3390/w15081556

**AMA Style**

Zhou L, Kang L.
A Comparative Analysis of Multiple Machine Learning Methods for Flood Routing in the Yangtze River. *Water*. 2023; 15(8):1556.
https://doi.org/10.3390/w15081556

**Chicago/Turabian Style**

Zhou, Liwei, and Ling Kang.
2023. "A Comparative Analysis of Multiple Machine Learning Methods for Flood Routing in the Yangtze River" *Water* 15, no. 8: 1556.
https://doi.org/10.3390/w15081556