Model Modification of Verhulst Inverse-Function Forecasting Model and Probabilistic Forecast for Landslide Failure Time
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摘要: 滑坡时间预测是滑坡灾害防治的重要组成部分，但由于滑坡演化存在不确定性，准确预测滑坡的发生极为困难.Verhulst反函数模型是一种常用的滑坡时间预测模型，但其存在计算起始时刻选择不当会造成监测数据拟合质量差和预测精度低的问题.针对这一不足，提出了一种改进的Verhulst反函数模型（MVIF模型），并进行近实时概率预测分析.结果表明：（1）MVIF模型改善了原模型对计算起始时刻选择苛刻的问题；（2）MVIF模型预测精度较高，在滑坡进入中等加速变形阶段之后可进行可靠的预测；（3）预测滑坡时间与破坏概率结合提供了一种新的滑坡预报准则.该研究可为蠕滑型滑坡的预警预报提供有价值的参考.Abstract: Landslide time-of-failure forecast is an essential part of landslide disaster prevention and control. However, due to the uncertainty of the landslide evolution process, it is challenging to forecast the occurrence time of landslide events accurately. The Verhulst inverse-function model is a common landslide time-of-failure forecasting model, but the model suffers from the problems of poor fitting quality and low forecasting accuracy of displacement monitoring data caused by the improper selection of the calculation starting points. To address this deficiency, an improved Verhulst inverse-function model (MVIF model) is proposed and analyzed for near real-time probabilistic forecast.The results show that (1) the MVIF model addresses the problem of harsh selection of the calculation starting points in the original model; (2) the MVIF model has high forecasting accuracy and can make reliable forecasts after the landslide enters the medium accelerating deformation phase; (3) the combination of predicted landslide time and failure probability provides a new forecasting criterion. This study can provide valuable reference for early warning and forecast of creeping landslides.
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图 1 VIF模型（a）及MVIF模型（b）的预测曲线示意
Fig. 1. Schematic diagram of the forecasting curves of the VIF model (a) and the MVIF model (b)
图 3 加速变形阶段不同时期MVIF模型的模型不确定性{μm, σm}
Fig. 3. Model uncertainty {μm, σm} of the MVIF model for different periods of accelerating deformation phase
图 4 周至滑坡位移监测数据及模型预测曲线
Fig. 4. Displacement monitoring data recorded from the Zhouzhi landslide and model forecasting curves
图 6 周至滑坡预测滑坡时间的概率分布
Fig. 6. Probabilistic distribution of the forecasting landslide time of the Zhouzhi landslide
图 7 茂县滑坡位移监测数据及模型预测曲线
Fig. 7. Displacement monitoring data recorded from the Maoxian landslide and model forecasting curves
图 9 茂县滑坡预测滑坡时间的概率分布
Fig. 9. Probabilistic distribution of the forecasting landslide time of the Maoxian landslide
图 10 案例Failure#1和Failure#3的预测结果
Fig. 10. Forecasting results for the cases of Failure#1 and Failure#3
图 12 Mesnil-Val崩塌和黑方台滑坡的预测结果
Fig. 12. Forecasting results of the Mesnil-Val collapse and the Heifangtai landslide
图 13 基于MVIF模型的分阶段预测决策
Fig. 13. Phased forecasting decision principle based on the MVIF model
图 14 周至滑坡和茂县滑坡的滑坡时间预报分析
Fig. 14. Analysis of landslide time-of-failure forecasts for the Zhouzhi landslide and the Maoxian landslide
表 1 滑坡案例信息与最后一次观测的预测结果
Table 1. Information of landslide cases and forecasting results of final observation
序号 滑坡名称 Tf/d μf/d σf/d 预测滑坡时间95%CI/d 参考文献 仅考虑观测不确定性 考虑模型不确定性 1 Vajont 725 725.11 6.60E-03 725.10 725.12 723.54 727.26 Seguí et al.（2020） 2 Preonzo 673 673.10 8.30E-03 673.08 673.12 671.53 675.25 Loew et al.（2017） 3 Mt. Beni 249 246.09 3.80E-01 245.35 246.83 244.37 248.39 Gigli et al.（2011） 4 Nchanga Open Pit 220.3 220.71 2.00E-01 220.32 221.10 219.10 222.90 Naismith and Wessels（2005） 5 白什乡 196 196.20 3.40E-02 196.13 196.27 194.63 198.35 许强和曾裕平（2009） 6 党川6号滑坡 154.2 154.00 8.80E-02 153.83 154.17 152.42 156.16 王智伟等（2020） 7 美国明尼苏达州2号国道 113 113.26 1.80E-02 113.22 113.30 111.69 115.41 陈贺等（2019） 8 平庄西煤矿4·17 106 106.05 8.80E-03 106.03 106.07 104.48 108.20 王东等（2020） 9 金龙沟 104.7 103.14 2.90E-01 102.57 103.71 101.48 105.38 冷超勤和张扬（2016） 10 Ohto 89 87.97 6.70E-05 87.97 87.97 86.40 90.12 Suwa et al.（2010） 11 New Tredegar 71 73.08 7.60E-02 72.93 73.23 71.50 75.24 Carey et al.（2007） 12 平庄西煤矿1·18 70 69.86 1.20E-02 69.84 69.88 68.29 72.01 何满潮等（2012） 13 Takabayama 64.1 64.15 3.60E-02 64.08 64.22 62.58 66.30 Saito（1979） 14 Kagemori quarry 49 47.07 1.60E-03 47.07 47.07 45.50 49.22 Yamaguchi and Shimotani（1986） 15 Agoyama 26.1 24.79 2.20E-05 24.79 24.79 23.22 26.94 Saito（1979） 16 Carlà2018 22.7 22.72 8.00E-03 22.70 22.74 21.15 24.87 Carlà et al.（2018） 17 龙井村 21.5 21.51 2.70E-08 21.51 21.51 19.94 23.66 亓星等（2020） 18 Ingelsberg 14.7 14.72 8.60E-08 14.72 14.72 13.15 16.87 Kieffer et al.（2016） 19 Mount Owen Mine 12 11.66 9.40E-03 11.64 11.68 10.09 13.81 Harries et al.（2006） 20 Dosan Line 3.4 3.38 1.20E-04 3.38 3.38 1.81 5.53 Saito（1965） 
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表 2 周至滑坡MVIF模型和VIF模型的模型参数
Table 2. Model parameters of the MVIF and VIF models in the Zhouzhi landslide
模型参数 A或a B或b C tf（d） Tf（d） 拟合优度R2 MVIF-1 325.58 -0.11 -4.83 68.05 67.19 0.999 8 MVIF-2 325.59 -0.11 -5.00 68.05 0.999 7 VIF-1 83.63 1.25 - 66.90 0.897 6 VIF-2 194.70 2.90 - 67.14 0.992 8
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表 3 茂县滑坡MVIF模型和VIF模型的模型参数
Table 3. Model parameters of the MVIF and VIF models in the Maoxian landslide
模型参数 A或a B或b C tf（d） Tf（d） 拟合优度R2 MVIF-1 9.89 -3.38 -6.04 427.03 426 0.996 2 MVIF-2 9.04 -11.36 -12.63 425.54 0.995 5 VIF-1 3.43 0.0082 - 418.29 0.855 5 VIF-2 8.33 0.020 - 416.50 0.995 1
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