# Utilizing Entropy-Based Method for Rainfall Network Design in Huaihe River Basin, China

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## Abstract

**:**

## 1. Introduction

## 2. Study Area and Dataset

#### 2.1. Study Area

#### 2.2. Data Processing

- (i)
- Firstly, regional rainy days were selected as at least one station having precipitation records ($>0\mathrm{m}\mathrm{m}$). In contrast, regional non-rainy days were defined as all stations in the basin having no precipitation records.
- (ii)
- Secondly, adjacent regional rainy days would be separated by non-rainy days. Thus, effective interval days (${n}_{EI}$) were defined, which was used to accumulate precipitation records from adjacent rainy days.
- (iii)
- Thirdly, autocorrelation tests were implemented to ensure the processed series obeyed the “independent and identically distributed” assumption.

## 3. Methods

#### 3.1. Entropy-Based Indexes for Station Network Optimization

- (1)
- Marginal entropy (ME):

- (2)
- Joint entropy (JE):

- (3)
- Total correlation (TC):

- (1)
- For example, choosing 2 sites from 43 sites would result in 903 (${C}_{43}^{2}=\frac{43!}{41!\times 2!}=903$) possible combinations. In the same way, choosing 6 sites from 43 sites would result in 6,096,454 (${C}_{43}^{6}=\frac{43!}{37!\times 6!}=$ 6,096,454) possible combinations. Therefore, it was not necessary to exhaustively search all possible combinations of the given number of gauges. Instead, we commenced by generating a multitude of potential gauge combinations, specifically 90,000 in this study.
- (2)
- The joint entropy and total correlation for each combination of gauges can be calculated using the processed precipitation data. However, it is important to note that these two information indexes, namely joint entropy and total correlation, need to be computed using an appropriate discretization method, which is investigated in Section 3.2.
- (3)
- Based on these two objective functions, ($\mathrm{max}\left\{H\left({X}_{{S}_{1}},{X}_{{S}_{2}},\dots ,{X}_{{S}_{m}}\right)\right\}$ and $\mathrm{min}\{TC\left({X}_{{S}_{1}},{X}_{{S}_{2}},\dots ,{X}_{{S}_{m}}\right)\}$), a certain number of Pareto solutions (${N}_{Pare}$) can be derived as approximations to the optimal function. This is because it is impossible to satisfy both objective functions simultaneously.
- (4)
- The frequency of station selection is calculated by examining the occurrence of the label (in this case, ${S}_{m}\text{}$represents the label for a specific station) in the Pareto solutions. Since different stations have different frequencies of label occurrence, the selected frequency can be determined using the following calculation:

#### 3.2. Three Kinds of Quantization Methods for Entropy Calculation

## 4. Results

#### 4.1. Analysis of Processed Data in the HRB

#### 4.2. Selection of Data Discretization Method

#### 4.3. The Impact of Time Variability in Precipitation on the Optimization Results of Gauge Networks in the Huaihe River Basin

## 5. Discussion

## 6. Conclusions

- (1)
- Careful selection of discretization technology is the basis for station network optimization. This study compared the network optimization results derived from three kinds of discretization methods, including the floor function-based approach, Scott’s equal bin width histogram (EWH-Sc) approach, and Sturges’s equal bin width histogram (EWH-St) approach (Figure 5). The floor function-based approach with a = 100 was selected as the most suitable discretization method for this study by optimizing the matching degree of the variance and edge entropy sequence of the measured values at each station.
- (2)
- The criterion of maximizing the joint entropy and minimizing the total correlation (maxJE-minTC) was able to generate potential Pareto solution sets for the optimal network. The frequency of selecting sites in the Pareto solution set proposed in this study provides a new approach for characterizing the results of station network optimization.
- (3)
- Due to the trend-caused nonstationarity in almost 75% of all stations in the HRB, taking the impact of temporal variability in the precipitation series on the final rain gauge network optimization results into consideration is of great significance. The analysis results indicated that the degree of nonstationarity in the processed precipitation series is directly proportional to the frequency of station selection.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Study area: Huaihe River Basin with 43 meteorological stations. DEM, digital elevation model.

**Figure 4.**Comparison diagram of information entropy-based index (JE: Joint Entropy and TC: Total Correlation) estimated results.

**Figure 5.**Process of optimization of network design based on the Pareto solution through flood function-based discretization approach with a being 150.

**Figure 6.**Comparison of station network optimization results using different discretization methods.

**Figure 8.**Lollipop chart of Mann–Kendall trend test results for the processed data in all 43 stations. Bold yellow lollipops represent the processed data of stations showed trend-caused nonstationarity at the significance level of 5%. The blue dashed line represents the threshold of Z (1.96) at the significance level of 5%.

**Figure 9.**Time-varying analysis of selection frequency of stations in the Huaihe River Basin under the 30-year sliding window (SW) width condition.

**Figure 10.**Comparison of the processed and original precipitation series for the matching degree between the standard deviation series and marginal entropy series.

**Table 1.**The detailed characteristics of the grouped precipitation data derived from the 16 selected stations.

Station No. | Original Data | Processed Data | ||
---|---|---|---|---|

Maximum (mm) | SD ^{a} (mm) | Maximum | SD | |

S1 | 188.8 | 6.9 | 945.7 | 132.9 |

S2 | 191.3 | 8.4 | 868.6 | 151.2 |

S3 | 288.6 | 8.9 | 1183.7 | 173.3 |

S4 | 189.4 | 7.4 | 818.4 | 131.1 |

S5 | 217.8 | 7.7 | 913.7 | 134.4 |

S6 | 276.2 | 10.9 | 1279.3 | 210.3 |

S7 | 216.7 | 9.7 | 1086.3 | 182.8 |

S8 | 257.7 | 8.6 | 1073.9 | 144.5 |

S9 | 177.2 | 7.9 | 896.9 | 145.4 |

S10 | 206.9 | 10.5 | 1313.7 | 104.4 |

S11 | 232.6 | 9.3 | 1193.9 | 172.4 |

S12 | 285.3 | 9.1 | 1013.3 | 161.3 |

S13 | 226.1 | 9.8 | 1186.8 | 181.1 |

S14 | 225.4 | 8.8 | 1109.7 | 156.2 |

S15 | 363.6 | 8.6 | 1062.9 | 149.6 |

S16 | 263.2 | 9.9 | 1456.0 | 198.3 |

^{a}SD represented standard deviation.

Discretization Approaches | Number |
---|---|

Floor function-based with a = 50 | 27 |

Floor function-based with a = 100 | 29 |

Floor function-based with a = 150 | 48 |

Floor function-based with a = 1000 | 100 |

EWH-Sc | 17 |

EWH-St | 37 |

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**MDPI and ACS Style**

Liu, J.; Li, Y.; Wang, Y.; Xu, P.
Utilizing Entropy-Based Method for Rainfall Network Design in Huaihe River Basin, China. *Water* **2023**, *15*, 3115.
https://doi.org/10.3390/w15173115

**AMA Style**

Liu J, Li Y, Wang Y, Xu P.
Utilizing Entropy-Based Method for Rainfall Network Design in Huaihe River Basin, China. *Water*. 2023; 15(17):3115.
https://doi.org/10.3390/w15173115

**Chicago/Turabian Style**

Liu, Jian, Yanyan Li, Yuankun Wang, and Pengcheng Xu.
2023. "Utilizing Entropy-Based Method for Rainfall Network Design in Huaihe River Basin, China" *Water* 15, no. 17: 3115.
https://doi.org/10.3390/w15173115