## 5 Discussion
### Convergence Observations--Number of Simulations
Figure 20: Effective weighted simulation count \(N^{weighted}\) versus monitoring data distribution standard deviation \(\sigma_{mnn}\). envelopes indicate visually where the convergence is assured. When the shading remains narrow and entirely above the 200-simulation reference line, the effective sample size \(N^{weighted}\) is essentially stable and the weighted analysis is converged. Where the shading widens appreciably (below 200-simulation line), the large dispersion in \(N^{weighted}\) denotes practical non-convergence, indicating that a larger nominal simulation count \(N\) or a less restrictive \(\sigma_{mon}\) is necessary. Influence of the Monitoring Data Span on Axial Strain \(\varepsilon_{1}\) and Factor of Safety FS  
In the weighted analysis, the measured axial strain (\(\varepsilon_{1}\) = 2.45%) is represented by a normal distribution. Its standard deviation (\(\sigma\)) controls how strictly the weighting procedure steers the Monte Carlo analysis toward the monitored value. To quantify this effect, four different spans of axial strain distribution are examined, and a probabilistic analysis is rerun with N = 10\({}^{6}\) simulations to ensure convergence even for very small (quasi-deterministic analysis) standard deviations (Table 3). The probability-density curves in Figure 21 illustrate how the choice of monitoring-data span (\(\sigma\)) governs the influence of the weighting procedure:  
* Quasi-deterministic span: The PDF collapses almost into a spike at \(\varepsilon_{1}\) = 2.45%. Only a handful of simulations retain appreciable weight, confirming that an extremely narrow span drives the analysis toward a virtually deterministic match with the observation (extremely high KL = 5.4 nats confirms this), but at the cost of a very large number of simulations necessary to secure convergence. * Tight span: The distribution remains centered reasonably close to the measured strain (KL = 2.9 nats still indicates a very strong pull), yet preserves a finite (small, but not negligible) spread. This span still filters out more than 98% of the baseline tail, providing a strong monitoring pull while needing only around 10\({}^{4}\) simulations for reliable statistics. * Wide span: The peak shifts only modestly toward the monitoring (KL = 0.3 nats); a residual right-hand tail survives. The monitoring data influence the results, but the baseline variability is far from eliminated. * Very-wide span: The weighted curve is virtually indistinguishable from the baseline histogram (KL \(\approx\) 0 indicates negligible information gain), demonstrating that an observation span comparable to the full baseline scatter leaves the analysis effectively unweighted. Figure 22 mirrors the span trend for axial strain but now in terms of the factor of safety distributions. As the monitoring data standard deviation contracts from \"very-wide\" to \"quasi-deterministic\", the entire probability-density function of the factor of safety (FS) shifts leftward and tightens: \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Monitoring Data** & \(\sigma\)/\(\varepsilon_{1,range}\) & \(\sigma\) **(Absolute)** &
\begin{tabular}{c} **Qualitative** \\ **Description** \\ \end{tabular} \\ \hline Quasi-det & 0.1\% & 0.0002 & quasi-deterministic \\ Tight & 1\% & 0.002 & narrow \\ Wide & 10\% & 0.02 & moderate \\ Very wide & 100\% & 0.2 & full baseline range \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Monitoring data span definitions. The chosen \(\varepsilon_{1,range}\) is \(\sim\)0.20, according to baseline MC results (Figure 15). * Quasi-deterministic span: the green PDF peaks at \(\mu\approx 1.45\) with the smallest spread, representing the strongest monitoring pull (KL = 1.99 nats) but achievable only with around a million-simulation analysis. * Tight span: the red curve moves decisively (KL = 1.56 nats) into the lower FS regime (\(\mu\approx 1.48\)) and its variance drops by roughly one quarter; the results now reflect the lower effective friction angle mobilized at high confinement.