## 5 Results and Discussion
### Limits of Applicability  
This section briefly discusses the limits of the estimation algorithms presented in preceding sections. These limits are defined in terms of the magnitude of errors that can be corrected. They are summarized in Table 3.  
The maximum antenna pointing correction \(\Delta\vec{A}_{e}\) is roughly equal to the antenna beam width. It therefore depends on the frequency band and the antenna design. The results presented in Section 5.1, and Figure 10 in particular, demonstrate that the approach can correct for mispointing of a magnitude comparable to the azimuth beam width. This result, however, is close to the limit of what is possible for several reasons. Firstly, the linear approximation of Equation (29) eventually fails: assuming, for simplicity, that the antenna gain in the main beam is roughly quadratic, the 1 st order Taylor \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Positions [cm]**} & \multicolumn{3}{c}{**Baselines [cm]**} & \multicolumn{3}{c}{**Antenna Pointing [deg]**} \\ \cline{2-10}  & \(\delta x\) & \(\delta y\) & \(\delta z\) & \(\delta x\) & \(\delta y\) & \(\delta z\) & \(\delta roll\) & \(\deltapitch\) & \(\delta yaw\) \\ \hline X1h & 0.471 & \(-\)0.108 & 0.952 & 0.000 & 0.061 & 0.065 & & & \\ X1v & 0.481 & \(-\)0.214 & 0.840 & 0.010 & \(-\)0.040 & \(-\)0.047 & & & \\ \hline X2h & 0.473 & \(-\)0.173 & 0.884 & 0.001 & \(-\)0.003 & -0.003 & & & \\ X2v & 0.472 & \(-\)0.164 & 0.894 & 0.001 & 0.005 & 0.006 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The quantitative impact of severe azimuth under-sampling (see Figure 16) on the external calibration of the four phase centers comprising the polarimetric X-band across track interferometer of F-SAR. The columns of the table are as follows: _Positions_ denote the change in phase center coordinates, _Baselines_ are the same changes after the overall mean offset is subtracted, while _Antenna Pointing_ denotes the changes in the antenna pointing angles. Figure 17: A comparison of the residual errors measured in two independent F-SAR X-band acquisitions. The plots juxtapose results after calibration with (red) and without (blue) sub-sampling the input data. The comparison is based on the raw data used for calibration and illustrates the impact of the baseline and pointing changes given in Table 2. Left: interferometric phase errors obtained in the six co-polar channel combinations of Figure 14. Each data point is obtained as an average over the illumination time of a reference target. Lines connect targets in the same interferometric channel combination. Right: the residual RCS mismatch in the four co-polar channels of the interferometer. As before, data points are obtained as an average over the target illumination time. expansion implied in the approximation of (27) by (29) is only valid within the main beam (where the difference of two approximately quadratic functions is approximately linear). When this linear approximation fails, the fixed-point iteration for the pointing estimate will converge to meaningless results or may even diverge entirely. Secondly, very large pointing errors would make it difficult to even localize the reference target response in the raw data. Additional pre-processing steps might be required to bootstrap the calibration process. In the case of the maximum phase center position correction \(\Delta\vec{\textbf{I}}_{e}\), simulations with artificially displaced phase centers suggest that errors of at least 10 m can be corrected: for the X-band data of Section 5.5, the corrected positions agree to within margins similar to those reported in Table 2. Even larger errors were not considered, as they appear of little practical relevance. Eventually, however, the calibration process would fail as the target response lies outside the range interval considered during analysis. Furthermore, these results are also the basis for the limit given for the tropospheric propagation correction \(\Delta c\).