## 4 Discussion
The results showed that the maximum order was 1 or 0 for both components for all \((j,k)\) point-coordinate combinations, meaning that, at least according to the proposed model, variations in temperature and water level had effects on the displacement with a maximum delay of 1 day. The estimated autoregressive models are shown in Figure 6 (see dashed lines), and in evaluating their performances in explaining the dam displacement, the linear correlation index was computed between the filtered GNSS time series and the autoregressive models for each coordinate and point. The computed indexes, shown in Table 6, highlight that there is a high correlation between observed and modelled displacements, showing that, \begin{table}
\begin{tabular}{c c c c c} \hline
**Coordinate** & **PT1** & **PT2** & **PT3** & **PT4** \\ \hline x & \(-0.13\) & \(-0.21\) & \(-0.49\) & \(-0.16\) \\ y & \(0.68\) & \(0.76\) & \(0.63\) & \(0.06\) \\ z & \(0.04\) & \(0.05\) & \(-0.07\) & \(-0.46\) \\ \hline \end{tabular}
\end{table} 
Table 4: Linear correlation index computed between the filtered GNSS displacement time series and the reservoir water level. \begin{table}
\begin{tabular}{c c c c c} \hline
**Coordinate** & **PT1** & **PT2** & **PT3** & **PT4** \\ \hline x & \(0.83\) & \(0.78\) & \(0.52\) & \(0.82\) \\ y & \(-0.16\) & \(-0.22\) & \(-0.04\) & \(-0.48\) \\ z & \(-0.66\) & \(-0.59\) & \(-0.66\) & \(-0.74\) \\ \hline \end{tabular}
\end{table} 
Table 5: Linear correlation index computed between the filtered GNSS displacement time series and the air temperature. as expected, the main environmental factors determining the dam displacement are the temperature and the water level. This kind of analysis shows how the relationship between the dam displacement and the external factors can be determined by means of autoregressive models. This is a fundamental step to evaluate the expected displacements at monitored points. In fact, knowing the displacements due to some external quantities that can be observed independently, it becomes easier to interpret the overall GNSS displacement time series and, for instance, to calibrate thresholds of an early warning system based on the residual displacements. Finally, spatial analyses were also carried out to determine correlations between the estimated displacements at different PT stations. Along the crest direction, the displacements of PTs 1, 2 and 4 were extremely coherent, while the displacement of PT3 was slightly less coherent, even if the correlation with the other points was very high (\(\rho\) ranging between 85% and 92%, see Table 7). Less evident was the coherency between PT displacements along the stream direction; in this case, the point PT4 placed on the downstream shell showed to be quite uncorrelated with the other points placed on the dam crest (see Table 8). In the vertical direction, PTs 1, 2 and 3 showed to be highly correlated with each other and less correlated with PT4 (see Table 9). \begin{table}
\begin{tabular}{c c c c c} \hline
**Coordinate** & **PT1** & **PT2** & **PT3** & **PT4** \\ \hline x & 0.95 & 0.93 & 0.88 & 0.95 \\ y & 0.79 & 0.90 & 0.67 & 0.53 \\ z & 0.71 & 0.64 & 0.68 & 0.79 \\ \hline \end{tabular}
\end{table} 
Table 6: Linear correlation index computed between the filtered GNSS displacement time series and the autoregressive model of the displacement based on air temperature and water level. Figure 6: Time series of filtered GNSS displacement (solid line) versus the estimated autoregressive model (dashed lines) for each point and coordinate. With such a kind of analysis, at a first approximation, one could imagine that the dam body was rigidly moving because of the high correlation between the displacements of the points, particularly those on the dam crest. However, by properly magnifying the displacements and jointly illustrating them in the XY plane (see Figure 7), a finer interpretation of the overall deformation of the structure in time can be retrieved.