# An Analysis of Rainfall Characteristics and Rainfall Flood Relationships in Cities along the Yangtze River Based on Machine Learning: A Case Study of Luzhou

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## Abstract

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## 1. Introduction

## 2. Data and Methods

#### 2.1. Data

#### 2.2. Methods

#### 2.2.1. Construction of a Dynamic Characteristics Matrix for Space–Time Distribution of Heavy Rainfall

#### 2.2.2. Dimensionality Reduction Analysis Based on the Isomap Algorithm

- (1)
- Build an adjacent graph, $G$, by defining input space X as any two samples of vectors ${x}_{i}$ and ${x}_{j}$. Calculate the Euclidean distance between them as $d\left(i,j\right)$. Then, select the k-nearest points of sample ${x}_{i}$ as its neighbors. If two samples, ${x}_{j}$ and ${x}_{i}$, have neighboring points, they are connected. The connection length is the Euclidean distance between them, $d\left(i,j\right)$. This operation is repeated on all points to obtain graph G.
- (2)
- In graph G, for any two sample vectors ${x}_{i}$ and ${x}_{j}$, set the distance between them as ${d}_{G}\left(i,j\right)$. Initialize ${d}_{G}\left(i,j\right)=d\left(i,j\right)$ if ${x}_{i}$ and ${x}_{j}$ are directly connected, otherwise set ${d}_{G}\left(i,j\right)=\infty $. Then, update ${d}_{G}\left(i,j\right)$. For all samples $k=1,2,\cdots N$ in graph G, calculate the shortest path via ${d}_{G}=min\left\{{d}_{G}\left(i,j\right),{d}_{G}\left(i,k\right)+{d}_{G}\left(i,k+1\right)\right\}$. After each iteration, we obtained the shortest path between the sample vectors as ${D}_{G}\left(i,j\right)=\left\{{d}_{G}\left(i,j\right)\right\}$.
- (3)
- Let $B={D}_{G}^{T}{D}_{G}$ and solve the eigenvalue decomposition of matrix B$$B=\mathsf{\Phi}\mathsf{\Lambda}{\mathsf{\Phi}}^{T}$$$${\Omega}^{\prime}={\left[{y}_{1},{y}_{2},{y}_{3}\dots ,{y}_{N}\right]}_{d\times N}={\left[\begin{array}{c}\sqrt{{\lambda}_{1}}{v}_{1}\\ \sqrt{{\lambda}_{2}}{v}_{2}\\ \sqrt{{\lambda}_{3}}{v}_{3}\\ .\\ .\\ .\\ \sqrt{{\lambda}_{d}}{v}_{d}\end{array}\right]}_{d\times N}$$

#### 2.2.3. Dynamic Clustering Analysis

## 3. Results and Discussion

## 4. Conclusions

- Traditional methods of analyzing rainfall characteristics focus on rainfall data or surface rainfall at a single station. However, rainfall processes are characterized by spatial and temporal variation. Traditional methods cannot objectively express rainfall variations in time and space. Machine learning technology can quantitatively describe the dynamic temporal and spatial distributions of various types of rainfall. This is consistent with the climatic characteristics of the region. By comparing the featured rainfall process with the typical actual rainfall process, we found that the temporal and spatial distribution characteristics of the two were similar, and that the featured rainfall process was sufficiently representative of the actual rainfall process. Machine learning techniques can be effectively applied in the study and extraction of rainfall’s spatiotemporal distribution features.
- Rainfall in the Luzhou area can be divided into three types according to the different spatial and temporal distributions. And the flood characteristics formed by different types of rainfall are also different. When analyzing flood characteristics, it is necessary to study the spatial and temporal distribution characteristics of rainfall in the area in order to obtain objective and reasonable conclusions.
- River flooding had a negligible impact on urban flood control and drainage. When the rainstorm center was located upstream of the Yangtze and Tuojing rivers, the movement direction of the rainstorm center was consistent with the flood evolution direction of the Yangtze and Tuojing rivers. The Tuojing River was significantly affected by the top support of the Yangtze River, with flood level changes up to 7–8 m. Both the Yangtze and Tuojing rivers maintained high water levels with a high risk of flooding. This is the most unfavorable rainfall process and must be a focus for flood prevention.
- Although the rainfall–flood relationship obtained in this study applies only to Luzhou, the proposed method is universal. At present, this study only considers Luzhou as an example; the research scope can be further expanded to include the entire basin to obtain more objective results. The results of this study can provide technical support for urban storm risk management, as well as for the design of urban flood control and drainage systems.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Type of Rainfall | Spatial and Temporal Characteristics of Rainfall | Morphological Characteristics of Flood Processes | Characterization of Maximum and Minimum Flood Levels |
---|---|---|---|

Type I | The center of the storm moved upstream from the Tuogang River toward Luzhou | The flooding process is mostly of the single-peak type, and this type of rainfall mainly affects the flood level of the Tuojiang River. | The water level of the Tuo River fluctuated between 1 and 2 m due to rainfall, and the flood level of the Yangtze River remained stable. |

Type II | The center of the storm moved from the upper reaches of the Yangtze River to Luzhou. | The flooding process was mostly a single peak, and this type of rainfall mainly affects the flood level of the Yangtze River. | The water level of the Tuo River is greatly influenced by the water level of the Yangtze River. The water level of the Tuo River fluctuated greatly, with an average change of about 4–8 m. |

Type III | The center of the storm moved toward Luzhou from the upper Yangtze River and the upper Tuo River, respectively. | The flooding process was mostly a single peak. The rainfall process in the upper reaches of the Tuojian River overlapped with that in the upper reaches of the Yangtze River and moved towards Luzhou. | The direction of movement of the rainstorm center was consistent with the flood evolution direction of the Yangtze and Tuojian rivers, which was the most unfavorable rainfall process. |

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**MDPI and ACS Style**

Liu, Y.; Liu, Y.; Wang, J.; Ren, H.; Liu, S.; Hu, W.
An Analysis of Rainfall Characteristics and Rainfall Flood Relationships in Cities along the Yangtze River Based on Machine Learning: A Case Study of Luzhou. *Water* **2023**, *15*, 3755.
https://doi.org/10.3390/w15213755

**AMA Style**

Liu Y, Liu Y, Wang J, Ren H, Liu S, Hu W.
An Analysis of Rainfall Characteristics and Rainfall Flood Relationships in Cities along the Yangtze River Based on Machine Learning: A Case Study of Luzhou. *Water*. 2023; 15(21):3755.
https://doi.org/10.3390/w15213755

**Chicago/Turabian Style**

Liu, Yuanyuan, Yesen Liu, Jiazhuo Wang, Hancheng Ren, Shu Liu, and Wencai Hu.
2023. "An Analysis of Rainfall Characteristics and Rainfall Flood Relationships in Cities along the Yangtze River Based on Machine Learning: A Case Study of Luzhou" *Water* 15, no. 21: 3755.
https://doi.org/10.3390/w15213755