# Inverse Estimation of Soil Hydraulic Parameters in a Landslide Deposit Based on a DE-MC Approach

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Unsaturated Soil Hydrology in Hydro-Mechanical Model

#### 2.2. Differential Evolution Markov Chain (DE-MC)

#### 2.3. Study Area

#### 2.4. Modeling Strategies

## 3. Results

#### 3.1. Synthetic Experiment

#### 3.2. Real Case Simulation

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Location of the in situ monitoring site of the landslide in Yindongzi valley, Dujiangyan Country, Sichuan Province, China.

**Figure 2.**NSE of each iteration step in synthetic experiment: (

**a**) DE-MC 20 chains; (

**b**) DE-MC 80 chains; (

**c**) MCMC 1 chain.

**Figure 3.**Comparison of the prior and posterior PDF of the parameters with the true values (black dotted line denotes true value).

**Figure 5.**Ensemble of simulated soil moisture content (red lines) in comparison with true values (black round dots).

Parameter | Synthetic Experiment | Real Case Simulation | |
---|---|---|---|

Prior Parameter Ranges | True | Prior Parameter Ranges | |

${\mathsf{\theta}}_{r}\left({\mathrm{m}}^{3}{\mathrm{m}}^{-3}\right)$ | 0.015~0.025 | 0.020 | 0.015~0.025 |

${\mathsf{\theta}}_{s}\left({\mathrm{m}}^{3}{\mathrm{m}}^{-3}\right)$ | 0.380~0.500 | 0.417 | 0.350~0.50 |

$\alpha \left({\mathrm{m}}^{-1}\right)$ | 13.0~14.5 | 13.8 | 13.0~14.5 |

$n\left(-\right)$ | 1.000~2.00 | 1.592 | 1.000~2.500 |

${K}_{s}\left({\mathrm{m}\mathrm{day}}^{-1}\right)$ | 4.50~5.50 | 5.04 | 4.50~5.50 |

Parameter | Mean Value | Standard Deviation | ||||||
---|---|---|---|---|---|---|---|---|

Prior | Posterior | Prior | Posterior | |||||

DE-MC 20 Chains | DE-MC 80 Chains | MCMC 1 Chain | DE-MC 20 Chains | DE-MC 80 Chains | MCMC 1 Chain | |||

${\mathsf{\theta}}_{r}\left({\mathrm{m}}^{3}{\mathrm{m}}^{-3}\right)$ | 0.020 | 0.020 | 0.022 | 0.022 | 0.003 | 0.003 | 0.003 | - |

${\mathsf{\theta}}_{s}\left({\mathrm{m}}^{3}{\mathrm{m}}^{-3}\right)$ | 0.44 | 0.43 | 0.451 | 0.389 | 0.039 | 0.020 | 0.022 | - |

$\alpha \left({\mathrm{m}}^{3}{\mathrm{m}}^{-3}\right)$ | 13.75 | 13.814 | 14.189 | 14.41 | 0.476 | 0.413 | 0.264 | - |

$n\left(-\right)$ | 1.75 | 1.604 | 1.645 | 1.587 | 0.437 | 0.056 | 0.048 | - |

${K}_{s}\left({\mathrm{m}\mathrm{day}}^{-1}\right)$ | 5.00 | 5.082 | 4.874 | 4.546 | 0.293 | 0.349 | 0.304 | - |

Saturated Degree | True | DE-MC 20 Chains | DE-MC 80 Chains | MCMC | |
---|---|---|---|---|---|

${\mathrm{log}}_{10}\left(-h\right)$ | $\mathrm{\Theta}$= 0.2 | 2.039 | 1.854~2.382 | 1.803~2.461 | 2.039 |

$\mathrm{\Theta}$= 0.5 | 1.324 | 1.251~1.482 | 1.210~1.524 | 1.324 | |

$\mathrm{\Theta}$= 0.8 | 0.808 | 0.783~0.871 | 0.756~0.895 | 0.808 | |

${\mathrm{log}}_{10}\left(K\right)$ | $\mathrm{\Theta}$= 0.2 | −4.268 | −5.110~−3.762 | −5.284~−3.723 | −4.365 |

$\mathrm{\Theta}$= 0.5 | −1.884 | −2.345~−1.593 | −2.445~−1.576 | −1.957 | |

$\mathrm{\Theta}$= 0.8 | −0.531 | −0.792~−0.357 | −0.855~−0.349 | −0.592 |

Parameter | Mean Value | Standard Deviation | ||
---|---|---|---|---|

Prior | Posterior | Prior | Posterior | |

${\mathsf{\theta}}_{r}\left({\mathrm{m}}^{3}{\mathrm{m}}^{-3}\right)$ | 0.021 | 0.023 | 0.003 | 0.002 |

${\mathsf{\theta}}_{s}\left({\mathrm{m}}^{3}{\mathrm{m}}^{-3}\right)$ | 0.437 | 0.391 | 0.035 | 0.016 |

$\alpha \left({\mathrm{m}}^{-1}\right)$ | 13.721 | 13.576 | 0.434 | 0.451 |

$n\left(-\right)$ | 1.913 | 2.209 | 0.461 | 0.101 |

${K}_{s}\left({\mathrm{m}\mathrm{day}}^{-1}\right)$ | 5.045 | 5.296 | 0.288 | 0.169 |

**Table 5.**The ranges of logarithmic transformed pore water pressure head and hydraulic conductivity in the ensemble prediction of SWCC and HCF at specific saturated degrees.

Saturated Degree | Prior | DE-MC 20 Chains | |
---|---|---|---|

${\mathrm{log}}_{10}\left(-h\right)$ | $\mathrm{\Theta}$= 0.2 | 1.296~7.211 | 1.350~1.607 |

$\mathrm{\Theta}$= 0.5 | 0.975~3.593 | 1.011~1.118 | |

$\mathrm{\Theta}$= 0.8 | 0.701~1.696 | 0.711~0.751 | |

${\mathrm{log}}_{10}\left(K\right)$ | $\mathrm{\Theta}$= 0.2 | −8.435~−2.383 | −3.216~−2.509 |

$\mathrm{\Theta}$= 0.5 | −4.061~−0.802 | −1.296~−0.888 | |

$\mathrm{\Theta}$= 0.8 | −1.707~−0.093 | −0.199~0.044 |

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## Share and Cite

**MDPI and ACS Style**

Chen, S.; Yan, H.; Shao, W.; Yu, W.; Wei, L.; Yang, Z.; Su, Y.; Kan, G.; Luo, S.
Inverse Estimation of Soil Hydraulic Parameters in a Landslide Deposit Based on a DE-MC Approach. *Water* **2022**, *14*, 3693.
https://doi.org/10.3390/w14223693

**AMA Style**

Chen S, Yan H, Shao W, Yu W, Wei L, Yang Z, Su Y, Kan G, Luo S.
Inverse Estimation of Soil Hydraulic Parameters in a Landslide Deposit Based on a DE-MC Approach. *Water*. 2022; 14(22):3693.
https://doi.org/10.3390/w14223693

**Chicago/Turabian Style**

Chen, Sijie, Haiwen Yan, Wei Shao, Wenjun Yu, Lingna Wei, Zongji Yang, Ye Su, Guangyuan Kan, and Shaohui Luo.
2022. "Inverse Estimation of Soil Hydraulic Parameters in a Landslide Deposit Based on a DE-MC Approach" *Water* 14, no. 22: 3693.
https://doi.org/10.3390/w14223693