## 4 Model-Based Calibration
### Phase Center Positions and Troposphere  
Phase center calibration aims to improve the accuracy of the assumed absolute antenna element phase center positions, as parameterized by the vector \(\vec{\mathbf{L}}_{e}\) introduced in the context of (2). In addition, this calibration step provides an estimate for the residual tropospheric propagation delay \(\Delta c\) introduced in Equation (5). The latter allows corrections to phase center geometry to be clearly separated from other effects. In practice, it may be used to discover any systematic error in the assumed value of \(c\) or, when estimated corrections are unreasonably large or highly variable, instabilities in the assumed radar center frequency. It should be noted that, while this approach is sufficient in airborne SAR applications, it would have to be extended to consider ionospheric effects in low frequency/large wavelength space borne scenarios [26]. The underlying model relates phase center corrections \(\Delta\vec{\mathbf{L}}_{e}\) and the tropospheric propagation correction \(\Delta c\) to the observed absolute residual phase of Equation (17). The corresponding error function to be minimized is given by  
\[E^{s,\Delta L}_{pq}(az)=\frac{\left(1+\Delta c\right)\lambda}{2\pi}\delta\chi^{ s}_{pq}(az)-\Delta c\sum_{e\in\{p,q\}}|\vec{\mathbf{r}}^{e}_{e}(az)|+\Delta los^{s}_{ pq}(az)-\Delta r_{pq}, \tag{30}\]  
where \(\lambda=\frac{c}{f_{0}}\), \(\Delta r_{pq}\) is a constant range offset that is related to residual propagation delays within the sensor, and  
\[\Delta los^{s}_{pq}(az)=\sum_{e\in\{p,q\}}\vec{los}^{s}_{e}(az)\cdot\Delta\vec {\mathbf{L}}_{e} \tag{31}\]  
is the range offset corresponding to position corrections \(\Delta\vec{\mathbf{L}}_{e}\) projected onto the line of sight vectors \(\vec{los}^{s}_{e}(az)\) defined in (3). While the error measure (30) aims to introduce corrections that minimize the absolute phase mismatch, it fails to capture an important property of multi-channel SAR: after calibration, any residual errors must be identical in all of the data channels comprising a single acquisition. All channels are equally affected by inaccuracies in the assumed sensor or target positions. To capture this prior knowledge, phase center calibration introduces a second, additional error measure  
\[E^{s,\Delta L}_{pq,uv}(az)=E^{s,\Delta L}_{pq}(az)-E^{s,\Delta L}_{uv}\Big{(}t ^{-1}_{uv}\big{(}t_{pq}(az)\big{)}\Big{)} \tag{32}\]  
that penalizes any difference in the residual mismatch between two simultaneously acquired channels \(pq\) and \(uv\). The composed mapping \(t^{-1}_{uv}(t_{pq}(az))\) serves to co-register the channels \(uv\) and \(pq\) using the relation between azimuth index and continuous time introduced in the context of (2). Figure 6: The impact of antenna pointing calibration on the residual RCS errors along azimuth, illustrated by the response of a single 1.5 m trihedral reflector imaged at C-band in VV polarization. The sub-plots show a superposition of the target backscatter intensity without antenna pattern correction and the antenna gain (left) and the residual RCS error \(\delta\text{RCS}^{s}_{pq}(az)\). To accommodate the additional differential error (32) as well as the coherence weight \(w_{pq}^{s}(az)\) of (19), the fixed point iteration of (22) is modified to include a second summation over simultaneously acquired pairs of channels \((pq,uv)\):  
\[\forall o_{i}\in\mathcal{O}:\ 0=\sum_{az,pq,s}w_{pq}^{s}(az)\ \frac{\partial\Psi\left(E_{pq}^{s \Delta L}(az)^{2}\right)}{\partial o_{i}}+\sum_{az,(pq,uv),s}w_{pq,uv}^{s}(az) \ \frac{\partial\Psi\left(E_{pq,uv}^{s\Delta L}(az)^{2}\right)}{\partial o_{i}}, \tag{33}\]  
where  
\[w_{pq,uv}^{s}(az)=w_{pq}^{s}(az)\ w_{uv}^{s}\left(t_{uv}^{-1}\big{(}t_{pq}(az) \big{)}\right). \tag{34}\]  
Generalizing the pattern of (22), each iteration involves pre-computing two \(\dot{\Psi}\) terms, one for each error measure.