# 3D Monitoring of Buildings Using TerraSAR-X InSAR, DInSAR and PolSAR Capacities
## 4 Measuring the Deformation Trough Differential Interferometry
### Discussion on the Impact of the Atmosphere on the Precision of the Estimated Deformation  
The fact that waves backscattered by the base of the buildings propagate trough different atmospheric conditions than the waves backscattered by the top of the buildings will impact the phase difference. The unwrapped phase differences as computed in this study will thus depend on the height and the deformation of the buildings, but also on the atmospheric delay. In this section, we estimate the maximum deformation error introduced by neglecting the changes in atmospheric condition.  
Two layers of the atmosphere can impact our deformation measurement: the ionosphere and the troposphere. At X-band, the ionospheric delay is very small and the ionosphere variation have a large spatial extend, impacting the same way waves backscattered by the base and the top of the buildings. The error induced by the ionosphere can thus be neglected.  
To assess the tropospheric delay variation between the top and the ground of the studied buildings, we model the tropospheric delay as exponentially decreasing with the height:  
\[\delta_{\text{tropo}}(h,\delta_{\text{ZPD}})=\frac{\delta_{\text{ZPD}}}{\cos( \theta)}e^{-\frac{h-h_{\text{ZPD}}}{H}} \tag{3}\]  
where \(\delta_{\text{ZPD}}\) is the global tropospheric delay, also called Zenith Path Delay (ZPD), \(\theta\) the look angle and \(H\) is the height of the troposphere. As the DLR teams in [28], we are taking \(H\) = 6000 m. The Zenith path delay can be measured from ground stations of the Global Navigation Satellite System. For the EUREF Permanent Network station at Marne la Vallee (\(h_{\text{ZPD}}\) = 160 m), the smallest ZPD is 2340 mm and occurs in winter while the biggest ZPD is 2440 mm and occurs in summer.  
For the Eiffel Tower, the residual delay between the pole and the second floor is:  
\[(\delta_{\text{tropo}}(320,2440)-\delta_{\text{tropo}}(320,2340))-(\delta_{ \text{tropo}}(160,2440)-\delta_{\text{tropo}}(160,2340))=-3.2\text{mm}\]  
For the Mirabeau Tower, located at an altitude of 33 m with a height of 70 m, the residual tropospheric delay is \(-1.6\) mm. The atmospheric delay results in millimeter error for the DInSAR deformation measurement.  
This error on the deformation measurement has the same order of magnitude than the one induced by a one meter error on the height. Along with the residual error on the PS phases, the precision of the estimated deformation is below 5 mm, which is below the deformation variations illustrated in Figure 6.  
This section demonstrates that the X-band is favorable to the measurement of complex structural deformations at different points of a building, given its height. This height can be assumed to be known elsewhere, or calculated using the whole data stack according to the method of the previous section. Errors of 1 m on the building height lead to negligible inaccuracies in the final deformation estimate, as well as the noncompensation of atmospheric effects locally between the top and bottom of the building.  
The deformation measured using SAR images is linked with the scattering mechanisms involved. However, in urban area and particularly at X-band, these mechanisms can be difficult to interpret. The next section investigates the contribution of polarimetry to refine the analysis of the scattering mechanisms in urban area that could lead to a better understanding of the structure of buildings as seen by the SAR sensor.