## 2 Materials and Methods
### Displacement Time Series Analysis  
The time series of the daily estimated displacements of each point along the three local XYZ axes needed to be filtered to properly manage the observation noise [40; 41], fill possible gaps, and allow for a straightforward modeling and interpretation of the displacements. The filtering was performed by applying the collocation approach [42; 43],  
Figure 1: Map of the monitored points on the dam, showing the orientation of the axes of the local reference system. Figure 2: Steel pillars supporting the antennas for points PT3 (**left**) and PT4 (**right**). The other points are materialized with the same kinds of structures. thus modeling the signal in time as a stationary random process. A requirement for such an approach is that the signal to be treated is zero-mean, which is not guaranteed in the case of time series regarding point displacements. To fulfill this requirement, a remove-compute-restore scheme was adopted, where the first step consists of removing a deterministic trend from the observed data by modeling it as the sum of a polynomial and a periodical component. After this stage, the observed residual displacements were filtered by collocation, and finally, the deterministic trend was added back to the filtered time series to estimate the displacement. Given the proposed scheme, we wrote the observation equation as:  
\[d_{o}^{(jk)}(t)=p^{(j,k)}(t)+q^{(jk)}(t)+y^{(j,k)}(t)+v(t) \tag{1}\]  
where, considering a point \(j\) and a coordinate \(k\), \(p^{(j,k)}(t)\) and \(q^{(j,k)}(t)\) are the polynomial and periodic deterministic trends, respectively, \(y^{(j,k)}(t)\) the residual displacement to be stochastically modeled, and \(v(t)\) the observation noise. Note that each \((j,\ k)\) combination was independently processed. Therefore, for the sake of simplicity, the \((j,\ k)\) indexes are omitted in the notation of the following equations. The estimation of the polynomial trend was performed by least squares adjustment [44], expressing the polynomial as:  
\[p(t)=\sum_{i=0}^{N}a_{i}t^{i} \tag{2}\]  
where \(a_{i}\) are the coefficients to be estimated, and \(N\) is the degree of the polynomial. The least squares adjustment was solved disregarding possible periodical components and residual displacements. To avoid an over parametrization of the interpolating function, the optimal polynomial degree \(N\) needed to be calibrated. To this aim, a first estimate of the set of \(a_{i}\) coefficients was performed with \(N=\overline{N}\), where \(\overline{N}\) is the maximum allowable degree. Then, a statistical test (\(t\)-test) to check whether the estimated value of the \(N\)-degree coefficient \(a_{N}\) was significantly different from zero was applied. If the test hypothesis was accepted, the current \(N\) degree of the polynomial was chosen as optimal; otherwise, the least squares adjustment was performed again after reducing the current degree \(N\) by 1. This procedure was iterated until the null hypothesis on the \(N\)-degree coefficient was rejected. As for the periodical component of the signal \(q(t)\), it was determined by means of a Fourier analysis of the polynomial-reduced displacements \(u(t)\), namely:  
\[U(f)=\mathcal{F}(u(t))=\mathcal{F}(d_{o}(t)-\dot{p}(t)) \tag{3}\]  
where \(\mathcal{F}(\cdot)\) is the Fourier transform operator, \(f\) is the frequency, and \(\dot{p}(t)\) is the previously estimated polynomial deterministic trend. Note that possible data gaps were filled by using the previously estimated polynomial trend to make the discrete Fourier transform applicable. These interpolated values were only used for the purpose of Fourier analysis. Given the transformed residuals \(U(f)\), the empirical Amplitude Spectral Density (ASD) of \(u(t)\) was computed as:  
\[A_{u}(f)=|U(f)| \tag{4}\]  
From Equation (4), a smoothed model \(\bar{A}_{u}\) of the ASD was determined by applying a moving median filter.