## 4 Engineering Application
### Verification of Discontinuity Sets' Clustering Effect  
The measured data of the structural planes at a depth of 765 m in this gold mine are used to validate the effectiveness and reliability of the automatic identification method for the structural planes proposed in this study. The FCM-ABC method is employed to cluster the joint parameters of each group together. The measured group size is 81, and the relevant parameters are set as follows: \(m=2\), \(c=2\)-\(7\), \(limit=100\), and \(e=200\). The two-dimensional distribution of joints on the surface of the original rock mass is plotted \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Trace Length (m)**} & \multicolumn{3}{c}{**Dip Direction (\({}^{\circ}\))**} & \multicolumn{3}{c}{**Dip Angle (\({}^{\circ}\))**} & \multicolumn{3}{c}{**Spacing (m)**} \\ \cline{2-13}
**Group** & **Distribution** & **Mean** & **Standard** & **Distribution** & **Mean** & **Standard** & **Distribution** & **Mean** & **Standard** & **Distribution** & **Mean** & **Standard** \\  & **Model** & **Value** & **Deviation** & **Model** & **Value** & **Deviation** & **Model** & **Value** & **Deviation** & **Model** & **Value** & **Deviation** \\ \hline No. 1 & Normal & 0.44 & 0.103 & Normal & 251.88 & 39.4 & Normal & 56.65 & 16.06 & Negative & 0.34 & 0.35 \\ \hline No. 2 & Normal & 0.25 & 0.06 & Normal & 49.21 & 19.75 & Normal & 42.70 & 16.92 & Negative & 0.11 & 0.07 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometric parameter values of structural planes and their distribution models. Figure 6: Illustration of information collection for structural planes of rock masses: (**a**) left and right views during photography; (**b**) 3D reconstruction map; (**c**) identification of structural plane; (**d**) stereographic projection; and (**e**) trace distribution of structural planes. in Figure 7, showing two significant groups of dominant structural planes, which are used for comparison. The clustering results of the field discontinuity data automatically computed by the self-designed MATLAB code using different \(c\) values are presented in Figure 8, and the obtained values of the three cluster validity test indicators under different \(c\) values are provided in Table 2. Apparently, for the tested sample data, from \(c=2\) to \(c=7\), when \(c=2\), the classification entropy coefficient \(H_{m}\) is the closest to 0, the fuzzy classification coefficient \(F_{m}\) is the closest to 1, and the fuzzy hypervolume \(F_{h}\) is the smallest, indicating that the clustering effect is the best in this case and the grouping result is very satisfactory. According to the calculation results of the cluster center when \(c=2\), the average occurrence of the two sets of predominant discontinuity sets is determined, i.e., the dip direction of the first set of discontinuities is \(248.16^{\circ}\), and the dip angle is \(58.17^{\circ}\); the second set of discontinuities has a dip direction of \(52.33^{\circ}\) and a dip angle of \(44.86^{\circ}\). The clustering calculation results are quite comparable to the actual measured results (Table 1 and Figure 7), which implies that the FCM-ABC method has a high solving accuracy and also verifies the correctness of the new algorithm. Moreover, the correlation between the values of the cluster validity test indicators and \(c\) values is intuitively presented in Figure 9. Evidently, with the increase in the \(c\) value, \(H_{m}\) and \(F_{h}\) show an overall increasing trend, while \(F_{m}\) shows a decreasing trend. This indicates that the clustering effect is best when \(c=2\), which is identical to the real situation of the two dominant structural planes. The selection of the \(c\) value has a significant influence on the clustering results. Hence, choosing an appropriate \(c\) value is crucial.