A discussion on the calculation method of instability probability of landslide due to rainfall
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摘要:
为了解决滑坡风险评价中的滑坡失稳概率计算问题，利用前人在降雨阈值的研究成果，结合气象学中降雨概率分布理论，以云南省盐津县庙坝滑坡为例进行计算，建立降雨型滑坡失稳概率计算模型。结果表明，盐津县降雨型滑坡的降雨阈值类型为累积降雨量-历时关系阈值，即为单日降雨阈值，降雨阈值为29.7 mm；盐津县在当日降雨量达到或超过阈值水平时可能诱发滑坡，对滑坡影响的滞后天数最大为5天；庙坝滑坡在8月20—25日6天内单日降雨达到或超过29.7 mm的降雨概率为46.49%；庙坝滑坡在8月25日因前5天或当天单日降雨量超过29.7 mm而失稳的概率为0.2853%。
Abstract:In order to solve the landslide risk evaluation of landslide failure instability probability calculation problem, this paper summarizes the research achievements of previous researchers in rainfall threshold and, in combination with the previous research results in a threshold rainfall, uses the theory of statistical to perform coupling analysis of the historical record of the rainfall landslide and rainfall data, so as to create a rainfall landslide failure instability probability calculation model, with Miaoba landslide in Yanjin County of Yunnan Province as an example for demonstration.The results show that the rainfall threshold type of Yanjin County rainfall landslide is cumulative rainfall duration threshold, which means the one-day rainfall threshold, with the rainfall threshold being 29.7 mm; most of rainfall landslide in Yanjin County is caused by the daily rainfall which reaches rainfall threshold within 5 days; the possibility that the rain reaches or exceeds 29.7 mm from August 20th to 25th is 46.49%;the possibility of the failure of Miaoba landslide in August 25th is 0.2853%.
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表 1 三种常用降雨概率分布函数及统计参数
Table 1. Three commonly used rainfall probability distribution function and statistical parameter
概率曲线及统计参数 公式 参数 P-Ⅲ型 $P(X \ge x) = \frac{{{\beta ^\alpha }}}{{\Gamma (\alpha )}}\int\limits_x^{ + \infty } {{{(x - {a_0})}^{\alpha - 1}}{e^{ - \beta (x - {a_0})}}} dx$ ${a_0} = E\left( x \right)\left( {1 - \frac{{2{C_V}}}{{{C_S}}}} \right)$ $\alpha = \frac{4}{{C_S^2}}$ $\beta = \frac{2}{{E(x){C_V}{C_S}}}$ 指数分布 $P\left( {X \ge x} \right) = \int\limits_x^{ + \infty } {f(x)dx = } \int\limits_x^{ + \infty } {\alpha {e^{ - \alpha (x - \beta )}}dx} = {e^{ - \alpha (x - \beta )}}$ $\alpha = \frac{1}{b} = \frac{1}{{E(x)}}, {\rm{ \mathit{ β} = (}}{{\rm{X}}_{\rm{i}}}{{\rm{)}}_{\min }}$ 耿贝尔分布 $P(X \ge x) = 1 - {e^{[ - {e^{ - a(x - b)}}]}}$ $a = \frac{{1.2825}}{\sigma }, b = E(x) - 0.45005\sigma $ 统计参数 $E(x) = x = \frac{1}{n}, \sum\limits_{i = 1}^n {{x_i}, {K_i} = \frac{{{x_i}}}{x}} $ ${C_v} = \sqrt {\frac{1}{{n - 1}}\sum\limits_{i = 1}^n {{{({K_i} - 1)}^2}} } , {C_S} = \frac{{\sum\limits_{i = 1}^n {{{({K_i} - 1)}^3}} }}{{\left( {n - 3} \right)C_v^3}}, \sigma = \sqrt {\frac{{\sum\limits_{i = 1}^n {{{({x_i} - x)}^2}} }}{{n - 1}}} $ 
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表 2 8月20—25日降雨概率分布曲线拟合参数结果
Table 2. Fitting parameter results of rainfall probability distribution curve from August 20th to 25th
分布类型及参数 8月20日 8月21日 8月22日 8月23日 8月24日 8月25日 皮尔逊Ⅲ型分布 E(x) 6.3 3.6 9.4 8.6 7.8 7.6 Cv 0.6 0.6 0.6 0.6 0.6 0.6 Cs 1.2 1.2 1.2 1.2 1.2 1.2 s 12.6 8.9 15.8 17.9 19.4 10.8 指数分布 a 0 0 0 0 0 0 b 6.4 3.6 9.5 8.5 7.9 7.7 s 10.8 7.8 7.8 15.2 9.9 8.3 耿贝尔分布 a 0.08366 0.12396 0.06349 0.05908 0.07936 0.0885 b -0.4682 -1.04356 0.44242 -1.00348 0.65429 1.1432 s 9.9 7.1 11 13.3 8.4 6.9 统计参数 E(Si) 6.3 3.6 9.6 8.6 7.8 7.6 D(Si) 235.0 107.0 408.0 471.2 261.2 210.2 Max(Si) 97.2 60.7 103.8 103.9 84.0 52.4 Min(Si) 0.0 0.0 0.0 0.0 0.0 0.0
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表 3 8月20—25日降雨概率分布曲线及超过阈值概率
Table 3. The probability distribution curve of rainfall and the threshold exceeding the probability from August 20th to 25th
日期 超过概率分布公式 P(X≥x=29.7) 8月20日 $P(X \ge x) = 1 - {e^{[ - {e^{ - 0.08366(x + 0.46820)}}]}}$ 0.07702 8月21日 $P(X \ge x) = 1 - {e^{[ - {e^{ - 0.12396(x + 1.04356)}}]}}$ 0.02188 8月22日 $P(X \ge x) = 1 - {e^{ - 0.10526x}}$ 0.04388 8月23日 $P(X \ge x) = 1 - {e^{[ - {e^{ - 0.05908(x + 1.00348)}}]}}$ 0.15041 8月24日 $P(X \ge x) = 1 - {e^{[ - {e^{ - 0.07936(x + 0.65429)}}]}}$ 0.09494 8月25日 $P(X \ge x) = 1 - {e^{[ - {e^{ - 0.08850(x + 1014320)}}]}}$ 0.07677
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