# 3D Monitoring of Buildings Using TerraSAR-X InSAR, DInSAR and PolSAR Capacities
## 4 Measuring the Deformation Trough Differential Interferometry
### Analysis of the Estimated Deformation  
#### 4.3.1 Comparison with Weather Data  
All parameter values have been collected from the archives of the Meteo France weather data [25; 26]. They have been recorded by the Paris weather station situated in the parc Montsouris, south east from the studied area. Four weather parameters have been used to validate our deformation measurements:  
* The time between the acquisition and sun set \(\Delta_{\text{sunset}}\), which enables to account for the periodicity in the measured deformations.
* The minimal temperature the day of the acquisition \(T_{\text{min}}\), linked to the thermal expansion in the measured deformation.
* The number of sunny hours during the acquisition day \(\Delta_{\text{sun}}\), that can be linked to the bending of the building caused by sunlight.
* The maximal gust during the hour of the acquisition \(v_{\text{wind}}\), that can account for the deformation due to wind.  
The normalized correlation between the weather data and the deformations, listed in Table 4, shows different result for the Mirabeau tower compared to the three other structures.  
For the Mirabeau Tower, the high normalized correlation between the measured deformation and the duration between the acquisition time and the sunset shows well the periodicity of the measured deformations. The sign minus is explained by the fact that the deformation are minimal in winter, when the sunset occurs before the acquisition and thus \(\Delta_{\text{sunset}}>0\). These deformations seem mainly due to the thermal expansion and the sunlight, given the high correlation with \(T_{\text{min}}\) and the medium correlation with \(\Delta_{\text{sun}}\). There is no correlation with \(v_{\text{wind}}\). Indeed, structures respond very quickly to wind forcing and no records of the wind information precisely at the acquisition time are available.  
For the other buildings, the correlation with the time between the acquisition and the sunset is low, highlighting the absence of periodicity in the measured deformations. Thermal expansion seems to be the major origin of the measured deformation. However, the normalized correlation between the measured deformation and minimal temperature \(T_{\text{min}}\) is around 0.5, meaning that the temperature does not explain alone the deformations. The correlation with the sunlight values is smaller than expected especially for the Eiffel Tower for which it is the principal deformation cause according to experts. The difference between the behavior of the Mirabeau Tower and the other towers can be explained by internal structure informations to which we do not have access, or by the different impact of other buildings adjacent to the site and shadow effects.  
The measured correlations underline the exogenous nature of the phase anomaly leading the unrealistic InSAR heights. However, the validation process does not allow to separate atmospheric changes impacting the phase and recorded through weather parameters from buildings deformations due to weather modifications. The estimated deformations have thus been compared to in situ measurements in Section 4.3.2 and the maximum error potentially induced by the atmosphere is assessed in Section 4.4.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\Delta_{\text{sunset}}\) & \(T_{\text{min}}\) & \(\Delta_{\text{sun}}\) & \(v_{\text{wind}}\) \\ \hline Eiffel Tower & \(-0.30\) & \(0.52\) & \(-0.09\) & \(-0.06\) \\ Cristal Tower & \(-0.35\) & \(0.53\) & \(0.01\) & \(-0.00\) \\ Keller Tower & \(-0.27\) & \(0.48\) & \(0.02\) & \(-0.15\) \\ Mirabeau Tower & \(-0.73\) & \(0.81\) & \(0.53\) & \(0.08\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Normalized correlation with the deformation