## 4 Discussion  
The filtered displacement time series were compared with respect to available external data, such as the water level in the reservoir and temperature of both the air and water surface. In particular, the time series of the reservoir water level (\(H_{\text{W}}\)) in proximity of the dam crest was available with hourly time sampling and 0.1 cm resolution. Regarding the temperature, air (\(T_{A}\)) and shallow water (\(T_{W}\)) temperatures measured in the dam proximity were both available. The former was observed with hourly time sampling and 0.1 \({}^{\circ}\)C resolution, while the latter with a daily time sampling (observation at 8:00 am) with 1 \({}^{\circ}\)C resolution. To allow for comparison, daily averages of \(H_{\text{W}}\) and \(T_{A}\) were computed. This was not required for \(T_{W}\) due to its sampling rate. As for possible missing data, they were filled by linear interpolation. Figure 5 shows the time series of \(H_{\text{W}}\), \(T_{A}\) and \(T_{W}\). It should be noted that, unsurprisingly, air and shallow water temperatures, namely \(T_{A}\) and \(T_{W}\), matched each other with a linear correlation index \(\rho=+0.93\). Therefore, \(T_{W}\) was not considered for further analyses. A first comparison was performed by computing the linear correlation index of the filtered GNSS displacement time series with the reservoir water level and the air temperature to understand how much these environmental variables are explanatory of the observed dam displacements. As for the water lever, the results are shown in Table 4, where all the  
Figure 5: Time series of the reservoir water level (on the **left**) and time series of air and water temperatures (on the **right**). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Coordinate** & **PT1** & **PT2** & **PT3** & **PT4** \\ \hline x & 0.51 & 0.50 & 0.51 & 0.53 \\ y & 0.34 & 0.32 & 0.34 & 0.36 \\ z & 0.93 & 0.88 & 0.88 & 0.79 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: RMS of the differences between the estimated displacements and corresponding raw GNSS observations. Units are mm. correlation indexes computed between the displacement time series of each coordinate of each point and water level \(H_{W}\) are reported. From Table 4, a significant correlation of about 70% can be noted along the downstream direction (y coordinate) for points PT 1-3, as well as a correlation of about 45% for the altitude direction (z coordinate) for the point PT4 located on the downstream shell. Regarding the correlation between the displacements and air temperature, Table 5 shows the computed linear correlation indexes between the filtered GNSS displacement time series and the air temperature \(T_{A}\). The results of Table 5 highlight a high correlation of the displacement along the crest direction (x coordinate) for all the points (with an index ranging from about 50% to 80%), as well as along the vertical direction (z coordinate) with estimated indexes ranging between about \(-60\%\) and \(-80\%\). The computed correlation indexes suggest that the displacements could be related to a linear combination of the effects of water height and air temperature. Therefore, a combination of two autoregressive models (considering both \(H_{W}\) and \(T_{A}\)) can be exploited. The general form of this model can be expressed as:  
\[\overline{d}^{(j,k)}(t)=\sum_{i=0}^{N_{H}^{(jk)}}b_{i,H}^{(j,k)}H_{w}(t-i)+ \sum_{i=0}^{N_{W}^{(jk)}}b_{i,T}^{(j,k)}T_{A}(t-i)+c^{(jk)} \tag{21}\]  
where the set of parameters \(b_{i}\) and the parameter \(c\) can be estimated by fitting the filtered time series of each \((j,k)\) point-coordinate combination through least squares adjustment. For each time series, the maximum orders \(N_{H}^{(j,k)}\) and \(N_{T}^{(j,k)}\) related to the two environmental components of the model had to be calibrated. This calibration was based on a \(t\)-test on the significance of the estimated parameters. The idea was to choose for each component the maximum order for which all the estimated parameters were significantly different from zero.