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| A Copula method for modeling the cross-correlated orientations of rock mass discontinuities |
| JIANG Shuihua1，OUYANG Su1，ZHENG Jun2，HUANG Jinsong1，ZHOU Chuangbing1 |
| (1. School of Infrastructure Engineering，Nanchang University，Nanchang，Jiangxi 330031，China；
2. College of Civil Engineering and Architecture，Zhejiang University，Hangzhou，Zhejiang 310058，China) |
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Abstract 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.
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2022, Vol. 41 
