## 4 Model-Based Calibration
### Antenna Mispointing  
The aim of the element mispointing estimation is to derive optimal values for the roll/pitch/yaw corrections introduced in Equation (7). These are the Euler angles underlying the rotation matrix \(\mathbf{R}_{I\to e}\) that transforms instrument to antenna element coordinates in the antenna model of Equation (6).  
The proposed pointing calibration yields attitude angle corrections that minimize variations in the RCS errors \(\delta\text{RCS}_{pq}^{s}(az)\) of Equation (14). The underlying model aims to explain the residual RCS mismatch as a mispointing of individual antenna elements plus a constant gain offset. The corresponding error function takes the form  
\[E_{pq}^{sAA}(az)=w_{pq}^{s}(az)\,\left(\delta\text{RCS}_{pq}^{s}(az)-\Delta \text{RCS}_{pq}-\sum_{e\in\{p,q\}}\Delta Gain_{e}^{s}(az)\right) \tag{27}\]  
where \(\Delta\text{RCS}_{pq}\) is an unknown constant gain offset characteristic of the channel \(pq\), and \(\Delta Gain_{e}^{s}(az)\) denotes the overall, azimuth variant change in the antenna gain, in dB, due to the pointing correction \(\Delta\vec{A}_{e}\). The gain change is derived from the two-way complex antenna gain \(A_{pq}^{s}(az,f_{r})\) of Equation (6) as follows:  
\[\Delta Gain_{e}^{s}(az)=10\log_{10}\left(\frac{\sum_{f_{r}}\left|A_{pq}^{s}( az,f_{r})\right|^{2}}{\sum_{f_{r}}\left|A_{pq}^{s}(az,f_{r}\right|\Delta\vec{A}_{e} =\overline{\partial})\right|^{2}}\right), \tag{28}\]  
where the summations over \(f_{r}\) are over the radar signal bandwidth and \(A_{pq}^{s}\left(az,f_{r}\right|\Delta\vec{A}_{e}=\overline{\partial}\right)\) denotes the nominal complex antenna gain obtained when all pointing correction angles are left at zero.  
The minimization of errors of the form (27) is complicated by the non-linearity of Equation (28) with respect to the unknown attitude corrections \(\Delta\vec{A}_{e}\). The non-linearity is circumvented by linearizing the error function (27) to obtain  
\[\hat{E}_{pq}^{s,\Delta A}(az)=w_{pq}^{s}(az)\,\left(\delta\text{RCS}_{pq}^{s} (az)-\Delta\text{RCS}_{pq}-\sum_{e\in\{p,q\}}\Delta Gain_{e}^{s}(az)+\vec{ \mathcal{G}}_{e}^{s}(az)\cdot\vec{\mathcal{G}}_{e}^{r,p,q}\right), \tag{29}\]  
in which \(\vec{\mathcal{G}}_{e}^{r,p,q}=\left(\delta_{e}^{roll},\delta_{e}^{pitch}, \delta_{e}^{uw}\right)\) denote unknown incremental refinements to the current mispointing estimate \(\Delta\vec{A}_{e}\), and \(\vec{\mathcal{G}}_{e}^{s}(az)\) contains the corresponding, azimuth variant derivatives of \(\Delta Gain_{e}^{s}(az)\) with respect to the three angular axes. These derivatives are obtained numerically from the change in \(\Delta Gain_{e}^{s}(az)\) when the current estimate \(\vec{\mathcal{G}}_{e}^{s}(az)\) is perturbed in each dimension.  
The system of Equation (22) used for minimization comprises three equations for each antenna element \(e\), with one equation for each angular axis. In addition, it includes one equation corresponding to the constant unknown gain offset in each channel \(pq\). After each step of the fixed point iteration, the corrections \(\vec{\mathcal{G}}_{e}^{r,p,q}\) are applied additively to the current \(\Delta\vec{A}_{e}\) estimate and the values of \(\Delta Gain_{e}^{s}(az)\) and the numerical gradients \(\vec{\mathcal{G}}_{e}^{s}(az)\) are updated accordingly. As indicated in Figure 1, target analysis is repeated to take into account the estimated pointing corrections in (6). Figure 6 illustrates this optimization process. It shows how a pointing estimate minimizes the mismatch between the expected and the measured reference target RCS.