# Comparison of Conventional Deterministic and Entropy-Based Methods for Predicting Sediment Concentration in Debris Flow

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## Abstract

**:**

## 1. Introduction

## 2. Conventional Deterministic Methods for Determining the Distribution of Sediment Concentration in Debris Flow

## 3. Entropy-Based Method for Determining the Distribution of Sediment Concentration in Debris Flow

#### 3.1. General Index Entropy-Based Sediment Concentration

#### 3.2. Renyi Entropy-Based Sediment Concentration

#### 3.3. Tsallis Entropy-Based Sediment Concentration

#### 3.4. Shannon Entropy-Based Sediment Concentration

## 4. Results of Comparisons and Discussion

#### 4.1. Collected Experimental Data Sets

^{3}, and the angle of internal friction, $\varphi $, was 38°. The sediment concentration in the experiment was measured over the entire depth of the flow using a camera and an imaging system via the transparent sidewall of the flume. In this experiment, the bed sediment concentration, ${c}_{b}$, was 0.59. More experimental details can be found in the work of Tsubaki et al. [18]. For the mean sediment concentration in debris flow, the observed data from the laboratory experiment by Takahashi [8] were adopted in this study to test the validity of three conventional deterministic methods (Equations (5)–(7)) and four entropy-based expressions (Equations (23), (36), (42), and (48)) due to the limited experimental data from the literature. In the experiment by Takahashi [8], where $\mathrm{tan}\varphi $ = 0.75, the density of particles, ${\rho}_{s}$, was 2600 kg/m

^{3}, and the bed sediment concentration, ${c}_{b}$, was 0.756. By tilting the board, different values of the inclination angle of the channel bed from the horizontal $\theta $ were generated for observing changes in ${c}_{D}$. More information regarding the experiment can be found in the work of Takahashi [8].

#### 4.2. Comparison Results

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Comparison of the method of Tsubaki et al. [18] (Equation (1)), the general index entropy-based expression (Equation (22)), the Renyi entropy-based expression (Equation (35)), the Tsallis entropy-based expression (Equation (41)), and the Shannon entropy-based expression (Equation (47)) with observed data from Tsubaki et al. [18] at angles of (

**a**) 5° and (

**b**) 9°.

**Figure 3.**Comparison of the formulae of Takahashi [8] (Equation (5)), Ou and Mizuyama [19] (Equation (6)), and Lien and Tsai [12] (Equation (7)) and the Tsallis entropy-based expression (Equation (42)) with observed data from Takahashi [8]. Here, the general index entropy-based expression (Equation (23)), the Renyi entropy-based expression (Equation (36)), and the Shannon entropy-based expression (Equation (48)) are not shown because they did not fit the data points completely.

Name | Inclination Angle of 5° | Inclination Angle of 9° | ||
---|---|---|---|---|

Fitting Coefficient Values | Fitting Result | Fitting Coefficient Values | Fitting Result | |

$\mathit{R}$ | $\mathit{R}$ | |||

Theoretical formula of Tsubaki et al. [18] | No | 10.41 | No | - |

General index entropy-based expression | $\alpha $ = 1.5, $M$ = 3 | 5.95 | $\alpha $ = 1.4, $M$ = 0.25 | 7.70 |

Renyi entropy-based expression | ${\alpha}_{R}$ = 0.45, ${M}_{R}$ = 4 | 6.55 | ${\alpha}_{R}$ = 0.75, ${M}_{R}$ = 2 | 10.86 |

Tsallis entropy-based expression | $m$ = 3, ${M}_{T}$ = 0.5 | 7.09 | $m$ = 1.6, ${M}_{T}$ = 1 | 7.09 |

Shannon entropy-based expression | ${M}_{S}$ = 0.5 | 6.51 | ${M}_{S}$ = 3 | 9.46 |

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**MDPI and ACS Style**

Zhu, Z.; Wang, H.; Pang, B.; Dou, J.; Peng, D.
Comparison of Conventional Deterministic and Entropy-Based Methods for Predicting Sediment Concentration in Debris Flow. *Water* **2019**, *11*, 439.
https://doi.org/10.3390/w11030439

**AMA Style**

Zhu Z, Wang H, Pang B, Dou J, Peng D.
Comparison of Conventional Deterministic and Entropy-Based Methods for Predicting Sediment Concentration in Debris Flow. *Water*. 2019; 11(3):439.
https://doi.org/10.3390/w11030439

**Chicago/Turabian Style**

Zhu, Zhongfan, Hongrui Wang, Bo Pang, Jie Dou, and Dingzhi Peng.
2019. "Comparison of Conventional Deterministic and Entropy-Based Methods for Predicting Sediment Concentration in Debris Flow" *Water* 11, no. 3: 439.
https://doi.org/10.3390/w11030439