## 3 Case Study B
### Extraction Results of the Case B  
According to the study by [PERSON] et al. [43], in this section, the number of nearest neighbors is set to \(K=40\) to calculate the point cloud normal vectors using the PCA algorithm. The colors assigned based on the normal vectors in Figure 11 mainly represent three sets of discontinuities in the rock mass (the artificial plane in the upper-right corner). Selecting three training points corresponding to three discontinuity sets shown in Figure 11 as centers, respectively, and generating 200 random normal vectors with angle less than \(maxAGL=5^{\circ}\) from each center forms the training set (Figure 12) for training the improved Naive Bayes classifier established in this study. Figure 13 illustrates the outcomes of discontinuity set extraction utilizing the improved Naive Bayes classifier trained with training sets applied to the entire point cloud, where each color denotes a distinct discontinuity set. It is evident that three discontinuity sets have been successfully extracted, and the majority of the artificial plane (the top-right corner of the point cloud), which does not belong to any discontinuity sets, has been effectively excluded by establishing a minimum logarithmic probability threshold \(minLPR=-114.8513\). The results of extracting individual discontinuities using the HDBSCAN algorithm under the \(minCluster=10\) and \(minDis=200\) are depicted in Figure 14. Herein, individual discontinuities are effectively extracted, with the removal of some excessively small discontinuities and noise points. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Discontinuity** & **[PERSON] et al.** & & & **New** & **[PERSON] et al.** [32] (\({}^{\circ}\)) & & **[PERSON] et al.** [18] (\({}^{\circ}\)) \\
**ID** & [32] (\({}^{\circ}\)) & **[PERSON] et al.** [18] (\({}^{\circ}\)) & **[PERSON] et al.** [18] (\({}^{\circ}\)) & **[PERSON] et al.** [18] (\({}^{\circ}\)) & **[PERSON] et al.** [18] (\({}^{\circ}\)) & **[PERSON] et al.** [18] (\({}^{\circ}\)) \\ \hline
22 & 166.33/76.58 & 166.31/78.73 & 166.64/77.31 & 0.31 & 0.73 & 0.33 & 1.42 \\
23 & 160.2/89.86 & 157.52/86.88 & 159.43/89.14 & 0.77 & 0.72 & 1.91 & 2.26 \\
24 & 173.55/76.85 & 353.07/77.82 & 173.77/77.83 & 0.22 & 0.98 & 0.7 & 0.01 \\
31 & 136.59/82.58 & 314.73/80.04 & 135.33/81.71 & 1.26 & 0.87 & 0.6 & 1.67 \\
32 & 131.23/82.67 & 136.52/89.85 & 137.43/87.91 & 6.2 & 5.24 & 0.91 & 1.94 \\
33 & 143.91/89.7 & 145.62/89.85 & 323.94/89.89 & 0.03 & 0.19 & 1.68 & 0.04 \\
41 & 97.55/63.22 & 285.98/59.84 & 99.33/62.05 & 1.78 & 1.17 & 6.65 & 2.21 \\
42 & 91.07/50.19 & 272.57/47.64 & 272.75/48.46 & 1.68 & 1.73 & 0.18 & 0.82 \\
43 & 96.64/47.97 & 277.31/49.31 & 96.72/47.79 & 0.08 & 0.18 & 0.59 & 1.52 \\
51 & 123.42/76.15 & 305.04/77.62 & 122.67/75.92 & 0.75 & 0.23 & 2.37 & 1.7 \\
52 & 105.75/69.94 & 109.29/76.61 & 106.65/69.93 & 0.9 & 0.01 & 2.64 & 6.68 \\  & \multicolumn{3}{c}{Maximum deviation} & 9.73 & 5.24 & 9.05 & 6.68 \\  & \multicolumn{3}{c}{Average deviation} & 1.31 & 0.93 & 1.92 & 1.60 \\ \hline \hline \end{tabular} Note: \(\Delta\) I DD\(|-\)absolute value of dip direction difference; \(\Delta\) I DA\(|-\)absolute value of dip difference.  
\end{table} 
Table 1: \(Cont.\)  
Figure 10: The point cloud data of case B.  
Figure 11: Training points selection of case B. One color per discontinuity set. Figure 12: Training sets generation of case B.  
Figure 13: Extraction results of the discontinuity set for case B. After obtaining the normal vectors of each discontinuity using the PCA algorithm, and substituting them into Equations (11) and (12), the mean dip direction and dip of each discontinuity set are determined. By comparing these results with those obtained by [PERSON] et al. (2013) (Table 2), it is evident that Set 1 demonstrates a high level of agreement, with differences of 0\({}^{\circ}\) in dip direction and 3\({}^{\circ}\) in dip.