## 3 Results  
The observed GNSS displacements were filtered by means of the procedure previously explained in Section 2.2. As already stated, we independently processed the displacement time series of each coordinate (x, y, and z) and point (PT1, 2, 3 and 4), leading to twelve different analyses. In the following, the way in which the filtering algorithm has been applied is described, also explaining the choice and calibration of the different algorithm parameters. As for the polynomial interpolation (Equation (2)), a maximum degree \(\overline{N}=3\) was considered. In other words, only linear, quadratic, and cubic functions were considered as possible trends. The iterative approach for determining the optimal degree \(N\) of the interpolating functions was performed by considering a \(5\%\) significance level for the \(t\)-test. Table 1 shows the resulting degree for each coordinate of each point. Regarding the periodical deterministic component, the first step was to choose the threshold frequencies \(\overline{f}_{L}\) and \(\overline{f}_{H}\), in order to define the low and high frequency ranges. These two ranges were defined by considering periods longer than 25 days (for the low-frequency range) and smaller than 3 days (for the high-frequency range). To identify the periodical trend, a threshold \(\overline{r}=2\) was chosen (see Equation (5)) and therefore this trend was reconstructed from the Fourier coefficients having a power that is at least double the one of the background signal. An example of this analysis is reported in Figure 3, considering the point PT1, showing the empirical and moving-median ASDs and flagging with stars the identified periodical frequencies that were modeled and removed from the signal by means of Equations (6) and (7). \begin{table}
\begin{tabular}{c c c c c} \hline
**Coordinate** & **PT1** & **PT2** & **PT3** & **PT4** \\ \hline x & cubic & cubic & cubic & cubic \\ y & cubic & cubic & cubic & quadratic \\ z & cubic & cubic & cubic & cubic \\ \hline \end{tabular}
\end{table} 
Table 1: Degree of the polynomial trends used to reduce the observed displacements. Note that, to apply the discrete Fourier transform (DFT), it was required to work with regularly sampled data. In general, this requirement is satisfied thanks to GNSS daily solutions. However, there are few gaps in the data, e.g., related to temporary anomalies of the receivers or errors in the data transfer. In general, missing data were less than 2% of the whole period, apart from time series of the point PT4, in which there was about 10% missing data due to a receiver fault in March 2020. To allow the usage of DFT, these data gaps were filled with values computed by the polynomial deterministic trends estimated at the previous step, without a significant impact on the periodical component analysis. Once the observed displacements were reduced for polynomial and periodical deterministic trends, we applied the stochastic analysis based on the collocation approach. To this aim, the first step was to compute the empirical covariance function and to fit it with a model. The candidate models proposed in this analysis were: exponential, cosine-exponential, Gaussian, cosine-Gaussian and power-Gaussian, shown in Equations (6)-(20), respectively. \[C(\tau)=b_{1}\exp(-b_{2}\tau) \tag{16}\]  
\[C(\tau)=b_{1}\exp(-b_{2}\tau)\cos(2\pi b_{3}\tau) \tag{17}\]  
\[C(\tau)=b_{1}\exp\left(-b_{2}\tau^{2}\right) \tag{18}\]  
\[C(\tau)=b_{1}\exp\left(-b_{2}\tau^{2}\right)\cos(2\pi b_{3}\tau) \tag{19}\]  
\[C(\tau)=b_{1}\exp\left(-b_{2}\tau^{2}\right)\left(1-b_{3}\tau^{2}\right) \tag{20}\]  
Due to the non-linear shape of the covariance models of Equations (6)-(20), the optimal values of the \(b_{i}\) positive parameters were retrieved by a discrete exhaustive optimization after defining admissible bounded ranges of possible values. The best covariance model among the five possibilities was chosen as the one minimizing the square residuals between the empirical and model covariances, namely the one better fitting the empirical covariance function.