岩体不连续面互相关产状模拟的Copula函数方法

A Copula method for modeling the cross-correlated orientations of rock mass discontinuities

为准确描述岩体不连续面产状(倾角和倾向)的统计特性,并考虑倾角和倾向之间的互相关性,提出不连续面互相关产状模拟的Copula函数方法,通过确定拟合不连续面产状实测数据最优的边缘概率分布和Copula函数(相关结构),建立不连续面倾角和倾向的二维联合概率分布函数。同时与传统的Fisher分布和二维经验分布方法的计算结果进行可视化对比分析,采用极射赤平投影方法在赤平投影图上比较实测产状和不同方法模拟的产状之间的差别。最后,通过4个案例验证了提出方法的有效性。结果表明:传统的Fisher分布和二维经验分布方法不能准确描述不连续面产状之间的互相关性,而提出的Copula函数方法能够基于少量的实测数据更加灵活地构造出具有任意边缘分布类型和相关结构的产状联合概率分布函数,更加真实地表征不连续面产状之间的互相关性,克服了以往构建不连续面产状概率分布模型时将倾角和倾向视作2个独立变量的局限性。

For accurate characterization of the statistical properties of orientations(i.e.,dip angle and dip direction)of a rock mass discontinuity,a Copula method is proposed to model the discontinuity orientations accounting for the cross-correlation between the dip angle and the dip direction. Based on the optimal fittings of the marginal probability distributions and Copula functions(i.e.,correlation structures) of the discontinuity orientations from the measurement data,a two-dimensional joint probability density function of the dip angle and the dip direction can be constructed. In the meantime,a visual comparison between the results obtained from the proposed method with those obtained from the traditional Fisher distribution and bivariate empirical distribution methods is conducted. The measured and simulated discontinuity orientations are compared on stereographic projection maps by using the stereographic projection method. Finally,four examples are investigated to illustrate the effectiveness of the proposed method. The results indicate that the traditional Fisher distribution and the bivariate empirical distribution methods cannot effectively characterize the cross-correlation between the discontinuity orientations,while the proposed method can be in a more flexible way to construct the joint probability density function of the discontinuity orientations that follow arbitrary marginal distributions and correlation structures based on a small amount of measurement data. In short,the proposed approach can better depict the cross-correlation between the dip angle and the dip direction,and circumvent the limitations of treating the dip angle and the dip direction as two independent variables in constructing the probability distribution models of the discontinuity orientations.