## 5 Results and Discussion
### Geometric Accuracy  
Sections 4.3 and 4.4 introduce corrections to the assumed multi-aperture antenna array geometry. While the baseline corrections of Section 4.4 establish inter-channel phase consistency and are generally small, the phase center positions refinements \(\Delta\vec{\mathbf{L}}_{e}\) and tropospheric propagation correction \(\Delta c\) of Section 4.3 are potentially large and can have a measurable impact on the geometric accuracy of the focused SAR imagery. Figure 12 illustrates the impact of antenna phase center calibration on a set of five independent polarimetric L-band acquisitions. The polar plots in the first two rows visualize absolute residual phase errors \(\delta\chi^{s}_{pq}(az)\) within the range histories of trihedral reflectors in the two co-polar channels. As implied by (18) and illustrated in Figure 4, these phase errors are directly related to residual range delay or, equivalently, range position mismatch. The top-left plot suggests that the measured range of reference targets in the HH channel is systematically biased: residual errors clearly show a trend across the sensor swath. This trend is confirmed by the corresponding target range position mismatch measured in focused SAR imagery, as illustrated by the turquoise lines in the bottom-left plot. A comparison of these results with those on the right side of the Figure shows that the estimated phase center position corrections \(\Delta\vec{\mathbf{L}}_{e}\) effectively compensate for this trend. Not only does the correction make the measured target range positions more accurate, but the geometry of the two co-polarized channels has become highly consistent. This is the intended effect of the differential error term (32). The residual errors measured in focused SAR imagery after calibration, as shown in the bottom-right plot, are largely consistent with the uncertainties to be expected in the sensor navigation data and in the reference target phase center positions obtained via Differential GPS (DGPS). While Figure 4 primarily shows the target range mismatch as a function of the off-nadir angle, range error variations do also occur as a function of squint. The top row of plots in Figure 13 emphasize  
Figure 12: The impact of antenna element phase center position calibration on phase and position errors. The measurements shown are derived from the co-polar channels of five independent F-SAR calibration data acquisitions at L-band. The first and second row visualize the absolute residual phase \(\delta\chi^{s}_{pq}(az)\) derived from range compressed raw data in different polarizations, while the last row shows the reference target range position error as measured in focused SAR imagery. In the latter plots, each target is represented by a point and lines connect targets in the same channel (see the legend in Figure 13). such variations by showing the residual \(\delta\chi^{s}_{pq}(az)\) after the mean value has been subtracted from each individual range history. The bottom row of plots, meanwhile, show the target azimuth position mismatch. The information in these plots is closely related: a linear phase error along the range history shifts the response in the focused image. As in the range dimension, phase center position calibration improves the overall geometric accuracy in azimuth. Simultaneously, it ensures consistent results across all polarization channels. The results presented in this section indicate that the proposed phase center calibration model is able to improve the geometric fidelity of focused SAR imagery. It is not, however, possible to quantitatively evaluate the accuracy of the actual position estimates derived as the true phase center positions are simply not known. Qualitatively, however, the results appear plausible. The refined phase centers lie within one or two wavelengths distance from the geometric center of the physical antenna apertures. This holds true for all sixteen phase centers of the F-SAR antenna mount.