## 2 Baseline Monte Carlo Analysis  
Conventional probabilistic dam safety analyses are typically based on model input parameter distributions established a priori from preliminary investigations and engineering judgement. Meanwhile, the long-term monitoring of dams generates large sets of measurements that capture the structure's actual state and behavior throughout its service life. Incorporating these data into probabilistic models can therefore make safety estimates far more realistic and reliable. The challenge is that monitoring quantities (displacements, pore pressures, etc.) appear in the numerical workflow only as model outputs, not as inputs. Traditionally, they are introduced into a FEM analysis through the calibration of input parameters using the inverse process--back analyses. This is not a simple task, even if the monitoring data are simplified and treated deterministically. Among the reliability-based methods, the Monte Carlo simulation (MCS) is the most transparent because it estimates failure probability by direct random sampling. The procedure consists of repeatedly sampling the random input variables in the safety analysis, solving the FEM model for each sampled set and counting the proportion of realizations in which a predefined failure criterion is met. If a large-enough set of simulations is used, the calculated failure probability is close enough to a theoretical failure probability. For each input variable \(X_{j}\) in the input vector \(X\) a random sample \(\hat{x}_{i}\) is drawn, forming a sample vector \(\hat{x}\) for one simulation. A predefined limit-state function (LSF), usually written \(G(\hat{x})=0\), is then evaluated. If \(G(\hat{x})\leq 0\), the structure is considered to have failed in that realization. This process, sampling a new \(\hat{x}\) and checking the LSF, is repeated \(N\) times. The failure probability \(\mathrm{p}_{f}\) is defined as:  
\[p_{f}=\frac{n(G(\hat{x})\leq 0)}{N} \tag{1}\]  
where \(n(G(\hat{x})\leq 0)\) represents the number of simulations in which failure occurred [32]. Random draws are generated by the inverse-transform method, i.e., mapping a uniform variable \(\mathrm{u}\in(0,1)\) through the cumulative distribution of each input variable. LSF \(G(\hat{x})\) should be defined explicitly for every failure mode considered. Embarkment dams are subject to three broadly recognized failure modes [3]: overtopping, internal erosion (piping) and global slope stability. While overtopping is normally treated as a hydraulic-capacity problem that is mitigated by the spillway design, the LSFs for piping and slope stability must be defined for a structural reliability assessment. They can be defined in the following way, assuming the FEM analysis is performed:  
* For internal erosion (piping), the governing quantity is the maximum hydraulic gradient \(i^{FEM}\) obtained from the seepage analysis model. Failure occurs once this gradient equals or exceeds the material specific critical value \(i^{cr}\); therefore, LSF can be defined as \(G(\hat{x})=i^{cr}-i^{FEM}\). * For global stability, the relevant response in the FEM analysis is the shear-strength-reduction (SSR) factor of safety \(FS^{SSR}\). Failure is triggered when the computed factor of safety drops to unity, hence \(G(\hat{x})=FS^{SSR}-1\). The set of outcomes is post-processed either (i) by assuming an analytic distribution and computing its parameters (for normal distribution that would be mean value \(\upmu\) and standard deviation \(\sigma\)), or, more generally, (ii) by building a histogram. The choice of histogram bin width governs the trade-off between bias and noise in the results--the Freedman-Diaconis rule [33] is a common rule that proposes adequate width. Figure 1 contrasts two extreme binnings of a synthetic data set--one with very few wide bins and one with many narrow bins--alongside the histogram obtained with the Freedman-Diaconis bin width used in this study.