求解岩体裂隙产状三维概率分布的数值方法

测线法是一种广泛使用的裂隙几何特征野外观测技术，但它获得的一维产状观测数据不能代表三维空间内的概率分布.在实测裂隙倾向和倾角之间相互独立的假设基础上，借用概率论和微积分建立了一维数据和三维分布的数值解关系式，进而提出一种由一维观测数据求解三维概率分布的方法.该方法的实现步骤是:（1）通过关系式数值求解产状的三维累积概率；（2）使用如Kolmogorov-Smirnov逼近法对累积概率进行分布形式和分布参数的估计.结合两类裂隙（层理面和节理面）的观测数据，比较了本文方法与Fouché方法的求解误差，并调查了样本容量对本文方法求解误差的影响.结果表明，本文方法求解误差更低.样本容量接近150时，可实现最低求解误差；当超过150时，求解误差不会随样本容量的增加而显著降低.同时，应用于互不平行的裂隙个体如节理面时，本文方法效果明显.而应用于近似平行的裂隙个体如层理面时，效果不明显. 

A Numerical Method for Solving Three-Dimensional Probability Distribution of Rockmass Fracture Orientations

The scanline mapping is a widely-used 1D field technique for fracture geometry observation. However, the 1D orientation observations from this technique poorly represent the 3D probability distribution. In this work, a numerical method for solving the 3D probability distribution of orientations is presented. It makes the assumption of observed dip direction-angle independence and adopts a mathematical relationship between the 1D observations and the 3D distribution. This method follows a two-step procedure that first using the relationship to solve the 3D cumulative, and then estimating the distribution type and parameters over the probabilities by employing the Kolmogorov-Smirnov approximation. Two cases of fractures (bedding planes and joints) illustrate that the presented method provides a smaller-error solution in comparison with the Fouché method. The minimum solution error of the presented method can be attained when the sample size is closely 150; if the sample size exceeds this value, the solution error will not decrease significantly as sample size increases. Moreover, the effectiveness of the presented method is investigated. The results show that the presented method performs effectively when applied to non-parallel fracture individuals, e.g. joints, whereas with low effectiveness when applied to sub-parallel fracture individuals, e.g. bedding planes.