# Interpretation of Bridge Health Monitoring Data from Satellite InSAR Technology
## 4 Results and Discussion
### Structural Interpretation of PS Deformation Periodicity  
Besides studying the deformation velocities of the different bridge sections and the surrounding territory, we are interested in investigating the periodicity of the observed displacements because we expect their correlation with the seasonal air temperature variations in that area. For this purpose, in Figures 12 and 13, we isolated the displacement time series of some PSs of particular interest to observe and discuss them. All the selected PSs have temporal coherence higher than 0.8. Let us begin with the PSs located on the access lane north and south of the viaduct, whose displacement time series are shown in Figure 12a. We can observe that the displacement time series of a PS on the southern access lane (PS 8202) is characterised by a deformation velocity exceeding \(-5\) mm/year along the LoS and limited seasonal periodicity. The limited periodicity aligns with the expectation since this access lane has been constructed with embankment soil, which is not expected to exhibit significant displacements in response to temperature variations. In contrast, the high velocity suggests abnormal displacements of the access lane. The infrastructure operator has recently detected such abnormal displacements and is already addressing this issue. A similar behaviour is observed on the northern access lane, where the periodicity is almost absent, and the deformation velocity is around \(-2\) mm/year, indicating a less pronounced effect than the southern access lane, consistent with the results shown in Figure 10.  
Figure 11: (**a**) Velocity along the LoS and (**b**) temporal coherence of the PSs identified on the A22 Po River Bridge. Next, we examine the displacement time series of some PSs identified along the bridge. Figure 12b illustrates the displacements of the top of Piers 10 and 11, represented by yellow PSs in Figure 10. It is evident that the behaviour is similar for both PSs: a noticeable negative displacement trend along the LoS of approximately \(-2\) mm/year--even though more pronounced in the PSs located at the top of Pier 11--and a more visible periodicity compared to the PSs on the access lanes. The greater periodicity was expected since these are displacements of reinforced concrete piers with a theoretical thermal expansion coefficient of around \(\alpha=15\ 10^{-6}\ ^{\circ}\mathrm{C}^{-1}\). We can estimate the actual thermal expansion coefficient \(\alpha\) based on the observed displacements and the measured temperatures. That would allow us to qualitatively verify whether the displacements obtained through MT-InSAR applied to our dataset are consistent with the expected displacements for a structure of this type. To do this, we perform the following procedure, whose results are reported in Table 1:  
1. Assume that the piers' displacements in response to thermal variations are mainly vertical;
2. Project the observed displacements along the LoS (\(d_{\mathrm{LoS}}\)) onto the vertical direction (\(d_{\mathrm{V}}\)) according to Equation (1), knowing the incidence angle \(\theta=33.9449^{\circ}\): \[d_{\mathrm{V}}=d_{\mathrm{LoS}}/\mathrm{cos}\theta\] (1)
3. Divide the vertical displacements by the height of the piers and the girders (approximately 15 m above the river surface) to obtain the time series of vertical deformation \(\varepsilon\) of Piers 10 and 11;
4. Fit the time series of vertical displacement with Least Square Analysis (LSA) and a linear model that takes as inputs deformations \(\varepsilon\), temperatures \(T\), and the time \(t\) at which the measurements were taken and estimates a purely geometric offset \(\varepsilon_{0}\), the deformation trend \(v\), and the thermal deformation coefficient \(\alpha\). The model is described in Equation (2): \[\varepsilon=\varepsilon_{0}+v\ t+\alpha\ T\] (2)  
Figure 12: Displacement time series of PSs located on (**a**) the access lane to the north (PS 4322, PS 4361) and south (PS 77542, PS 8202) of the bridge; (**b**) the top of Pier 10 (PS 6228, PS 6160) and Pier 11 (PS 6596, PS 6705).