## 4 Homogeneous Soil Slope (Simple Slope) Stability Analysis Using the FLAC3D Model
### Spearman and Kendall Correlation Coefficients of Simple Slope  
Figures 8-10 depict Spearman's rho correlation coefficients ranging from 0.31 to 0.76, while [PERSON]'s tau values range from 0.29 to 0.64, indicating variable displacement-stability relationships across the horizontal, vertical, and total displacements. The application of nonparametric Spearman and Kendall correlation coefficients provides a quantitative measure of the monotonic relationship between the reduction factor and displacement values [55]. As explained by [PERSON] et al., these statistical tests assess the strength and directionality of potential correlations between monitoring points [56]. The results show moderately positive but variable correlations in the horizontal, vertical, and total displacements. This is consistent with the work of [PERSON] et al., who found that displacement correlations with stability factors can vary significantly depending on the direction and location of the measurement [49]. The variability suggests localization of slope motion rather than broad instability. As highlighted in Crosta and Agliardi, differential displacements suggest that slope failure may begin in isolated sections before spreading [54]. Targeted monitoring helps to characterize these specific unstable zones. Additionally, the mix of positive and weakly correlated data points confirms the complexity of mass movement initiation [56].  
\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 & 0.006846 & 0.007076 & 0.0185 \\
2 & Horizontal & 17.64841 & 49.84159 & 141 \\
3 & Horizontal & 13.26401 & 37.47101 & 106 \\
4 & Horizontal & 0.012809 & 0.012851 &  
Going beyond visual inspection alone, the quantitative correlation coefficients allow for objective comparisons of where destabilization is occurring.  
The novelty of this approach is that it complements existing stability indicators with detailed statistical dependence tests. This provides precise, mathematical confirmation of associations across monitored slope locations and measurement directions.  
As explained by [PERSON] et al., correlation coefficients (r) range from \(-1\) to \(+1\) and quantify the linear dependence between variables. Values approaching \(+/-1\) represent strong correlations, while those tending toward \(0\) indicate weaker or nonlinear relationships [56]. The direction of the association is also crucial. Positive correlations indicate that as one variable increases, the other tends to increase as well. This is consistent with the expectation that greater slope instability (lower reduction factor) promotes higher displacement rates [54]. Negative correlations, on the other hand, imply an inverse relationship between the variables. In this study, the variability in correlation strength and directionality between monitoring locations provides insight into the progression of slope movement. [PERSON]  
Figure 8: Correlation coefficients for horizontal displacement of soil slope.  
Figure 9: Correlation coefficients for vertical displacement of soil slope.  
et al. found that displacement behavior can vary locally, even on the same slope, under similar conditions. Weak correlations suggest negligible or subtle destabilization effects, while stronger positive correlations are consistent with accelerated zone-specific failures as stability declines [56]. By distinguishing between localized stable and unstable areas through targeted statistical testing, these monitoring data allow for better characterization of the initial progression of slope failures. The displacement correlations can serve as an early warning system, identifying specific regions that warrant greater attention prior to mass failures.