## 5 Joint Rock Slope Stability Analysis Using the FLAC3D Model
### Spearman and Kendall Correlation Coefficients of Joint Slope  
Figures 16-18 depict Spearman and Kendall correlation coefficients for the joint horizontal, vertical, and total displacements, revealing some compelling insights. Several monitoring points show strong correlations ranges of \(+1.0\) and \(-1.0\). As explained by [PERSON] et al., these values indicate a strong statistical dependence between the stability reduction factor and displacements. The strong positive correlations are consistent with the expectation that decreasing stability corresponds to increasing surface displacements [56]. Conversely, strong negative correlations, such as the horizontal data from point 1, suggest an inverse relationship, where displacements decrease as stability decreases [54]. This deviation from the expected pattern reflects the complexity of slope destabilization. The consistency of very strong positive total displacement correlations implies a predominant overall trend of movement corresponding to deteriorating stability. However, the points still show sufficient variability to avoid overgeneralizing the behavior. The novelty of this analysis lies in quantifying the local differences in tilt response beyond qualitative observations alone. By documenting the unique correlations, monitoring can better distinguish zone-specific instability progression to design appropriate stabilization interventions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Monitoring Point** & **Displacement Type** & **Mean \$ (mm)** & **Standard Deviation \$ (mm)** & **Range \$ (mm)** \\ \hline
1 & Horizontal & \(-6.6\times 10^{-7}\) & 6.1 \(\times\)\(10^{-7}\) & 0.000002 \\
2 & Horizontal & 0.006448 & 0.016617 & 0.044128 \\
3 & Horizontal & \(-3.8\times 10^{-7}\) & 4.15 \(\times\)\(10^{-7}\) & 0.000001 \\
4 & Horizontal & 0.006469 & 0.016675 & 0.044280 \\
5 & Horizontal & \(3.4\times 10^{-8}\) & 1.03 \(\times\)\(10^{-7}\) & 0.000000 \\
1 & Vertical & \(-5.2\times 10^{-7}\) & 4.79 \(\times\)\(10^{-7}\) & 0.000001 \\
2 & Vertical & 0.003248 & 0.008346 & 0.022173 \\
3 & Vertical & \(5.29\times 10^{-7}\) & 5.52 \(\times\)\(10^{-7}\) & 0.000001 \\
4 & Vertical & 0.003228 & 0.008291 & 0.022028 \\
5 & Vertical & \(-7.6\times 10^{-8}\) & 8.36 \(\times\)\(10^{-8}\) & 0.000000 \\
1 & Total & \(8.39\times 10^{-7}\) & 7.75 \(\times\)\(10^{-7}\) & 0.000002 \\
2 & Total & 0.00722 & 0.018596 & 0.049385 \\
3 & Total & \(6.67\times 10^{-7}\) & 6.71 \(\times\)\(10^{-7}\) & 0.000002 \\
4 & Total & 0.00723 & 0.018623 & 0.049456 \\
5 & Total & \(1.16\times 10^{-7}\) & 9.92 \(\times\)\(10^{-8}\) & 0.000000 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Basic statistical properties of joint slope for displacements (horizontal, vertical, and total). The major limitations of the study can be summarized as follows: The numerical slope models make significant simplifying assumptions and do not fully capture real-world complexities in terms of soil/rock heterogeneities, hydrogeological factors, weathering effects, etc. In addition, the limited monitoring point coverage provides limited spatial insights, while the micron-scale displacement resolutions pose detection challenges in the field. Furthermore, the limited set of variables and validation data set limitations warrant a cautious application of the presented correlation concepts and failure criteria for field use cases. To address these limitations, substantial further refinements through integrated physical-numerical modeling and calibrations against instrumented slope case histories are imperative before these simulations can reliably inform early warning methodologies. Specifically, the integration of field measurements, such as slope inclinometers, extensometers, piezometers, rainfall gauges, and topographic surveys, along with laboratory testing of soil and rock samples, would provide a more comprehensive understanding of the slope behavior and failure mechanisms.