# Interpretation of Bridge Health Monitoring Data from Satellite InSAR Technology
## 3 Method
### Calculation of Vertical and Horizontal Displacement  
Suppose the displacements of an area of the Earth's surface along the LoS of satellites within a specific time interval are known for ascending and descending geometries. In that case, it is possible to derive the East-West and vertical displacements of the PSs in the same area using trigonometric calculations that depend solely on the incidence angles of the two geometries. More information on the assumptions and the procedure to perform this calculation can be found in [29]. In our study, we only analysed SAR images acquired in the descending geometry because the dataset of SAR images acquired in the ascending geometry is highly incomplete. Therefore, we cannot utilise the procedure mentioned above to extract the vertical and horizontal displacements of the bridge. However, we can study the bridge behaviour under the two extreme conditions: (i) the deck experiences exclusively vertical displacements, and (ii) the deck experiences exclusively horizontal displacements while being aware that the actual behaviour of the bridge is a combination of these two. In the case of these two extreme conditions, it is possible to project the measured PS displacements along the LoS (\(d_{\text{LoS}}\)) onto the vertical (\(d_{\text{V}}\)) and horizontal (\(d_{\text{H}}\)) components of the LoS, given the incidence angle of the LoS, which is \(\theta=33.9449^{\circ}\) for our dataset. Furthermore, in the case of exclusively horizontal displacements along the longitudinal direction of the bridge, it is possible to project horizontal displacements \(d_{\text{H}}\) onto the longitudinal direction of the bridge (\(d_{\text{L}}\)), knowing the angle between the satellite's orbit direction and the North-South direction (heading angle, \(\beta=10.6095^{\circ}\)) and the angle between the bridge's direction and the North-South direction (bridge orientation, \(\zeta=-33.5^{\circ}\)). Figure 9 shows these projections for displacements derived from a right-looking satellite in descending geometry and for the actual orientation of the Po River Bridge. Figure 9: (**a**) Projection of displacements along the LoS (\(d_{\text{LoS}}\)) onto the vertical direction (\(d_{\text{V}}\)) assuming exclusively vertical displacements of the bridge; (**b**) projection of \(d_{\text{LoS}}\) onto the horizontal component of the LoS (\(d_{\text{H}}\)) assuming exclusively horizontal displacements of the bridge; (**c**) projection of \(d_{\text{H}}\) onto the longitudinal direction of the bridge (\(d_{\text{L}}\)) assuming exclusively horizontal longitudinal displacements of the bridge. Although strong, the hypothesis of exclusively longitudinal horizontal displacements is plausible for the Po River Bridge portion in the floodplain, as no loads induce transversal displacements to the bridge's direction. Indeed, displacements induced by wind have a frequency and variability too high to be observed with the 16-day acquisition frequency determined by the satellite's revisit time. The only displacements induced by environmental loads observable on this bridge portion at this acquisition frequency are those resulting from seasonal temperature variations. A different scenario applies to the displacements of the bridge portion in the riverbed, which can also be influenced by the hydraulic forces of the Po River on the piers, transverse to the bridge's direction, varying with changes in the water level measured by the hydrometer located upstream of the bridge. Regarding our dataset, the CSK satellite that acquired the SAR images moved from North to South with a right-looking angle.