## 3 Methodology Concept
### Scope and Applicability  
This section presents the observations and limitations related to the application of the proposed methodology. In Bayesian terms, the baseline Monte Carlo analysis constitutes the prior predictive portrait of dam behavior, summarizing everything believed plausible before new monitoring readings arrive. Monitoring data then enter as the likelihood the statistical measure of how strongly each simulated outcome agrees with the evidence actually captured in the field. The applicability and effectiveness of the proposed methodology is influenced by the relationship between the baseline analysis results (prior) and corresponding monitoring distributions (likelihood). Both the shape and range of these distributions influence the results of the weighted analysis (posterior) which is illustrated in Figures 4 and 5. Moreover, since each simulation's weight is taking a value between 0 and 1, the effective number of weighted simulations can never exceed the original sample size.  
Figure 4 illustrates how the uniformly distributed baseline analysis results are transformed by weighting when the monitoring data follow, respectively, a uniform and a normal distribution. In both cases, the weighted histogram fills the entire overlapping range. Because the monitoring uncertainty covers exactly one-third of the baseline range, precisely one-third of the simulations are retained, and the weighted outcomes match the monitoring data distributions exactly. However, the individual weights are all equal to one for the uniform monitoring distribution, unlike for the normal where far fewer simulations \"survive\" the weighting unscaled.  
Figure 3: Process of weighted Monte Carlo analysis with incorporated monitoring data.  
Figure 5 depicts how weighting reshapes normally distributed baseline analysis results when the monitoring data are uniform and normal. If the monitoring distribution is uniform (Figure 5a), all simulation weights are equal to one, but only the tail of the normal baseline overlaps the monitoring range, so the weighted histogram consists of that portion of baseline analysis outcomes. When both the baseline and monitoring distributions are normal (Figure 5b), the weighted analysis results are confined to the region where the two normal distributions intersect, yielding a very small weighted simulation count.  
Finally, for the weighting to yield meaningful results, the monitoring data distribution must overlap the baseline analysis results' distribution (i.e., have non-zero probability in its domain). If the observed data lie entirely outside the prior range, the weighting produces zero for all simulation weights and the weighted analysis fails to converge.  
Hence, the leverage of the proposed methodology lies in a healthy intersection of the two distributions: the prior must be broad enough to anticipate the range of monitoring outcomes, yet not so diffuse that the corrective influence of monitoring data is drowned in noise. Careful design of the baseline input parameter space and mindful examination of monitoring data and its uncertainty are therefore essential--otherwise, the proposed quasi-Bayesian update degenerates into either an uninformative repetition of the prior or an unstable posterior built on too few retained simulations.  
Figure 4: Weighted analysis results for different relationships between baseline analysis and monitoring data distributions: (**a**) uniform–uniform; (**b**) uniform–normal.  
Figure 5: Weighted analysis results for different relationships between baseline analysis and monitoring data distributions: (**a**) normal–uniform; (**b**) normal–normal.