## 2 Materials and Methods
### Displacement Time Series Analysis
The ratio between the empirical and filtered ASDs  
\[r(f)=\frac{A_{u}(f)}{\bar{A}_{u}(f)} \tag{5}\]  
was then used to identify the main frequencies carrying a periodical contribution in the signal. This periodical effect was decomposed in two parts, one related to low frequencies,i.e., in the range \(|f|<\overline{f}_{L}\), and one related to high frequencies, i.e., in the range \(|f|>\overline{f}_{H}\). Note that, according to the proposed model, the condition \(\overline{f}_{H}\geq\overline{f}_{L}\) must always be satisfied. The choice of introducing two thresholds allows for flexibility in the data analysis, making it possible to adapt the analysis to the physical behavior of the considered phenomenon or structure, e.g., by excluding middle frequencies, i.e., \(\overline{f}_{H}>|f|>\overline{f}_{L}\), from the analysis. Given the ratio \(r(f)\) of Equation (5), a threshold \(\overline{r}>1\) can be defined to identify the frequencies of the dominating periodical contributions, i.e., the frequencies satisfying the condition \(r(f)>\overline{r}\). The periodical trends were then computed by an inverse Fourier transform of the coefficients \(U(f)\) corresponding to the identified frequencies that are also included in the chosen frequency ranges (low or high frequencies). This translates into the following estimation equations:  
\[\hat{q}_{L}(t) = \mathcal{F}^{-1}\bigg{(}U(f)\bigg{(}1-\frac{1}{r(f)}\bigg{)} \chi_{L}(f)\bigg{)} \tag{6}\] \[\hat{q}_{H}(t) = \mathcal{F}^{-1}\bigg{(}U(f)\bigg{(}1-\frac{1}{r(f)}\bigg{)}\chi_ {H}(f)\bigg{)} \tag{7}\]  
where the characteristic functions \(\chi_{L}(f)\) and \(\chi_{H}(f)\) are defined as:  
\[\chi_{L}(f)= \begin{cases}1&|f|<\overline{f}_{L}\wedge\ r(f)>\overline{r}\\ 0&\text{otherwise}\end{cases}  \tag{8}\]  
\[\chi_{H}(f)= \begin{cases}1&|f|>\overline{f}_{H}\wedge\ r(f)>\overline{r}\\ 0&\text{otherwise}\end{cases}  \tag{9}\]  
Note that the introduction of the scale factor \(\Big{(}1-\frac{1}{r(f)}\Big{)}\) before the inverse Fourier transform in Equations (6) and (7) was required to empirically distinguish the periodical from the stochastic component under the assumption that the ASD of the stochastic component was equal to the moving-median ASD. In summary, the overall periodical effect was determined as:  
\[\hat{q}(t)=\hat{q}_{L}(t)+\hat{q}_{H}(t) \tag{10}\]  
In the following, the detected low and high-frequency components were treated with a different approach. In particular, the former was assumed to be related to the physical phenomenon, while the latter to the noise. Therefore, they were both removed from the observations, but only the low-frequency periodical component \(\hat{q}_{L}(t)\) was subsequently restored after the collocation step. Once polynomial and periodical deterministic trends were computed, they were used to reduce the signal and isolate the observations of  
covariance function was used to determine an analytical covariance model \(\hat{C}(\tau)\), along with the determination of \(\sigma_{\eta}^{2}\). In particular, \(\hat{C}(\tau)\) was determined by a best fit procedure on \(C^{\epsilon}(\tau)\) for \(|\tau|>0\) since at \(\tau=0\), the empirical covariance combines signal and noise, according to the assumption of a white \(\eta(t)\). Once the model \(\hat{C}(\tau)\) was defined, the variance of the noise was computed as:  
\[\hat{\sigma}_{\eta}^{2}=C^{\epsilon}(0)-\hat{C}(0) \tag{13}\]  
The best covariance model was chosen among several candidate models as the one with the lowest residuals between the empirical and analytical covariances.