# Research on the Longitudinal Section of River Restoration Using Probabilistic Theory

^{*}

## Abstract

**:**

^{2}and Root Mean Square Error, all R

^{2}values were over 0.80, while RMSE values were analyzed to be between 0.54 and 2.79. Valid results can be obtained by calculating river characteristic factors.

## 1. Introduction

## 2. Methodology

## 3. Theoretical Background

#### 3.1. Nonlinear Regression

#### 3.2. Entropy Theory

_{1}− 1 is ${b}_{1}$, and λ is ${b}_{2}$.

_{1}is obtained by substituting the probability density function p(i) expressed in Equation (12) into the constraint Equation (6). The result is Equation (13), and the Lagrange multiplier b

_{1}is Equation (14):

_{1}of the probability density function Equation (12) is eliminated by substituting b

_{1}into Equation (14), and the probability density function p(i) follows as Equation (15):

#### 3.3. River Mean Slope Formula Development

_{max}is substituted for $\mathrm{b}\xb7{i}_{\mathrm{max}}$., and M

_{0}for $\mathrm{b}\xb7{i}_{0}$. The result is the same as Equation (17):

_{1}, b

_{2}are eliminated by substituting b

_{1}of Equation (14) into Equation (22), and the result follows as Equation (23):

_{max}as the product of b and i

_{max}, and M

_{0}as the product of b and i

_{0}. Thus, Equation (25) could be written as Equations (26) and (27):

#### 3.4. RMSE

## 4. Application to Real River

#### 4.1. Determination of Parameter by Measured Values

#### 4.2. River Longitudinal Elevation

^{2}. Figure 2 shows the comparison for each method. Table 4 shows the accuracy analysis comparison. Considering that the values of R

^{2}were all above 0.80, the formula suggested in this paper was meaningful in calculating the river elevation and slope.

## 5. Conclusions

- (1)
- The verification of the accuracy of the equation in this paper is based on the nonlinear regression analysis with SPSS 26 and SYSTAT 6.0. The value is 0.8150 to 0.9950. The values show that the equation is valid, and the application to actual rivers is considered significant.
- (2)
- Gacheon river shows the highest accuracy of prediction (close to 0.99) out of all methods. Yoodong river has parts that do not increase monotonically, and all three methods predict that singularity similarly. The suggested formulae are able to predict the section where the slope changes are large.
- (3)
- Since M
_{max}and M_{0}are parameters of the river, once the parameter is calculated the longitudinal section of the river can be obtained before it is destroyed. The reliability of this method can be further enhanced by using the data measured over 40 years or over 100 years. - (4)
- When making calculations using the equation presented in this study, it is easy to calculate the slope and elevation at a random point in the river basin.
- (5)
- It is expected that one use the river longitudinal section obtained through the equation to restore damaged rivers to their longitudinal elevations and ramps before development.
- (6)
- Using the method proposed in this paper, the river elevation can be obtained more accurately, which can help more precisely in the production of a digital elevation model or modify the data in places where it is hard to measure.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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River | Length (km) | ${\mathit{i}}_{0}$ | ${\mathit{i}}_{\mathit{m}\mathit{a}\mathit{x}}$ |
---|---|---|---|

Masan | 18.57 (entire) | $1.1\times {10}^{-1}$ | $0.9\times {10}^{-2}$ |

Samcheonpo | 7 (entire) | $6.3\times {10}^{-2}$ | $1.1\times {10}^{-1}$ |

Gacheon | 21.19 (entire) | $1.3\times {10}^{-2}$ | $3.18\times {10}^{-2}$ |

Gamcheon | 13.13 (entire) | $1.27\times {10}^{-2}$ | $1.1\times {10}^{-2}$ |

Geum | 3.25 (entire) | $3.0\times {10}^{-2}$ | $3.81\times {10}^{-2}$ |

Yudong | 11.31 (entire) | $4.29\times {10}^{-4}$ | $5.25\times {10}^{-3}$ |

Nakdong | 21.42 (selected) | $3.2\times {10}^{-2}$ | $2.5\times {10}^{-2}$ |

River | ${\mathit{M}}_{\mathit{m}\mathit{a}\mathit{x}}+{\mathit{M}}_{0}$ | ${\mathit{M}}_{\mathit{m}\mathit{a}\mathit{x}}-{\mathit{M}}_{0}$ | b | ||
---|---|---|---|---|---|

${\mathit{M}}_{0}$ | ${\mathit{M}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{M}}_{0}$ | ${\mathit{M}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ||

Masan | −44.778 | −8.481 | −53.336 | −8.481 | −104.898 |

Samcheonpo | −1311.532 | −708.388 | −2.056 | 3.428 | −5.911 |

Gacheon | −48.3386 | −21.8701 | −21.2442 | −47.2009 | −602.821 |

Gamcheon | 1 | 0.9944 | −24.804 | 5.113 | −88,000 |

Geum | −2.6198 | −101.655 | −2.6195 | −13.8264 | −456.546 |

Yudong | −15.179 | −9.616 | −2.517 | −8.917 | 39.952 |

Nakdong | −155.335 | −20.034 | −11.929 | −91.464 | −2458.048 |

River | Measured | Predicted | ||
---|---|---|---|---|

${\mathit{M}}_{\mathit{m}\mathit{a}\mathit{x}}+{\mathit{M}}_{0}$ | ${\mathit{M}}_{\mathit{m}\mathit{a}\mathit{x}}-{\mathit{M}}_{0}$ | b | ||

Masan | 0.01818 | 0.022464 | 0.022648 | 0.019296 |

Samcheonpo | 0.054015 | 0.05124 | 0.194054 | 0.054411 |

Gacheon | 0.015634 | 0.015098 | 0.01483 | 0.014972 |

Gamcheon | 0.001508 | 0.001588 | 0.001247 | 0.001284 |

Geum | 0.001591 | 0.001052 | 0.000876 | 0.001145 |

Yudong | 0.003771 | 0.002601 | 0.002624 | 0.002942 |

Nakdong | 0.028358 | 0.003568 | 0.00355 | 0.003488 |

River | ${\mathit{M}}_{\mathit{m}\mathit{a}\mathit{x}}+{\mathit{M}}_{0}$ | ${\mathit{M}}_{\mathit{m}\mathit{a}\mathit{x}}-{\mathit{M}}_{0}$ | b | |||
---|---|---|---|---|---|---|

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

Masan | 0.987 | 2.153 | 0.987 | 2.153 | 0.938 | 5.883 |

Samcheonpo | 0.94 | 0.828 | 0.991 | 2.72 | 0.979 | 4.175 |

Gacheon | 0.995 | 2.794 | 0.987 | 4.908 | 0.987 | 5.142 |

Gamcheon | 0.998 | 0.549 | 0.992 | 1.321 | 0.966 | 2.862 |

Geum | 0.998 | 10.821 | 0.955 | 1.605 | 0.815 | 15.775 |

Yoodong | 0.995 | 2.687 | 0.949 | 0.822 | 0.94 | 3.831 |

Nakdong | 0.993 | 1.369 | 0.995 | 0.834 | 0.994 | 0.801 |

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**MDPI and ACS Style**

Choo, Y.-M.; Kim, J.-M.; An, I.-T. Research on the Longitudinal Section of River Restoration Using Probabilistic Theory. *Entropy* **2021**, *23*, 965.
https://doi.org/10.3390/e23080965

**AMA Style**

Choo Y-M, Kim J-M, An I-T. Research on the Longitudinal Section of River Restoration Using Probabilistic Theory. *Entropy*. 2021; 23(8):965.
https://doi.org/10.3390/e23080965

**Chicago/Turabian Style**

Choo, Yeon-Moon, Ji-Min Kim, and Ik-Tae An. 2021. "Research on the Longitudinal Section of River Restoration Using Probabilistic Theory" *Entropy* 23, no. 8: 965.
https://doi.org/10.3390/e23080965