## 4 Model-Based Calibration
### Interferometric Baselines  
The calibration procedure described in the preceding section yields refined estimates of the absolute antenna element phase centers. These estimates are sufficient to provide consistently high geometric accuracy in the focused SAR imagery. They are still less accurate, however, than the limit set by residual errors in the assumed sensor and reference target positions. This residual inaccuracy is on the order of a few centimetres for position information derived from differential Global Positioning System (GPS) measurements. As such, it is still far too large for interferometric applications. In interferometry, phase differences, rather than absolute positions, are of central importance. This section introduces a calibration procedure that precisely determines interferometric baselines, i.e., the spatial separation of antenna element phase centers, to ensure phase consistency among all the channels of a multi-channel SAR sensor. Baseline calibration, as illustrated in Figure 7, is based on the residual interferometric phase between two simultaneously acquired data channels \(pq\) and \(uv\). It is obtained as  
\[\delta\phi_{pq,uv}^{s}(az)=\arg\left(e^{j\delta\phi_{pq}^{s}(az)}e^{j\delta \phi_{uv}^{s}\big{(}t_{uv}^{-1}\big{(}t_{pq}(az)\big{)}\big{)}}\right), \tag{35}\]  
where \(\delta\phi_{pq}^{s}(az)\) denotes the residual phase of (15) and \(t_{uv}^{-1}\big{(}t_{pq}(az)\big{)}\) is used to achive the co-registration of the channels \(pq\) and \(uv\) in time, as in (32). For a given channel combination \((pq,uv)\), this results in a set of phase vectors  
\[\left\{\delta\phi_{pq,uv}^{1}(az)\,,\delta\phi_{pq,uv}^{2}(az)\,,\dots\right\} \tag{36}\]  
corresponding to all available reference targets \(s=1,2,\dots\). To be of use in 3D baseline refinement, these phase differences must be made consistent by unwrapping them. To this end, each phase measurement \(\delta\phi_{pq,uv}^{s}(az)\) is associated with a 2D coordinate  
\[\left(\alpha_{p}^{s}\Big{(}az,\Delta\vec{A}_{p}\Big{)}\,,\beta_{p}^{s}\Big{(} az,\Delta\vec{A}_{p}\Big{)}\right)\mapsto\delta\phi_{pq,uv}^{s}(az) \tag{37}\]  
that parameterizes the azimuth variant angular propagation direction towards the target as in Equation (6). Once the interferometric phase measurements comprising a set such as that of (36) have been thus embedded in the 2D angular domain, a phase unwrapping algorithm for sparse grids (In view of phase offset calibration, described in Section 4.5, the unwrapping process should only add or subtract multiples of \(2\pi\) and not introduce arbitrary phase offsets. ), for example [27] or [28], is applied to obtain unwrapped phase vectors  
\[\left\{\delta\phi_{pq,uv}^{1,UW}(az)\,,\delta\phi_{pq,uv}^{1,UW}(az)\,,\dots \right\}. \tag{38}\]The error model functional \(E_{pq,uv}^{s,BL}(az)\) used for baseline refinement aims to minimize residual interferometric phase errors \(\delta\phi_{pq,uv}^{s,IIW}(az)\) by introducing antenna element phase center position adjustments \(\Delta\vec{L}_{e}^{BL}\), which are incremental refinements of the corrections \(\Delta\vec{L}_{e}\) introduced in the previous section. \[E_{pq,uv}^{s,BL}(az)=w_{pq,uv}^{s}(az)\left(\delta\phi_{pq,uv}^{s,IIW}(az)+\frac{ 2\pi f_{0}}{(1+\Delta\epsilon)\,c}\Delta los_{pq,uv}^{s}(az)-\Delta\phi_{pq,uv} \right), \tag{39}\]  
where  
\[\Delta los_{pq,uv}^{s}(az)=\sum_{e\in\{p,A\}}\vec{los}_{e}^{s}(az)\cdot\Delta \vec{L}_{e}^{BL}-\sum_{e\in\{u,v\}}\vec{los}_{e}^{s}\Big{(}t_{uv}^{-1}\big{(}t _{pq}(az)\big{)}\Big{)}\cdot\Delta\vec{L}_{e}^{BL} \tag{40}\]  
denotes the azimuth variant, cumulative change in line of sight distance to the reference target \(s\) due to lever arm corrections \(\Delta\vec{L}_{e}^{BL}\) with \(e\in\{p,q,u,v\}\). The variable \(\Delta\phi_{pq,uv}\) absorbs arbitrary constant phase offsets specific to the channel combination under consideration, while the differential coherence weight \(w_{pq,uv}^{s}(az)\) is defined in (34).