## 3 Target Response Model and Analysis
### Residual Errors  
After the expected target response has been accounted for, the analysis result comprises residual errors, i.e., deviations from the expectation, for all combinations of reference target \(s\) and data channel \(pq\): residual intensity variations \(\delta\text{RCS}_{pq}^{s}(az)\), residual phase variations \(\delta\phi_{pq}^{s}(az)\) and residual range delays \(\delta\tau_{pq}^{s}(az)\). The residual \(\delta\text{RCS}_{pq}^{s}(az)\) measures the absolute, unit-less offset, in dB, between the expected and the measured RCS of the reference target. This offset is illustrated in the bottom right plot of Figure 3. Equation (8) has accounted for expected systematic intensity variations due to antenna element illumination patterns and the expected target RCS. In addition, the derivation of the absolute RCS offset from the filtered data \(F_{pq}^{s}(az,rg)\) of (10) must also consider the residual clutter intensity of Equation (13) as follows:  
\[\delta\text{RCS}_{pq}^{s}(az)=10\log_{10}\left[\left(\sum_{\overline{g}}\left| F_{pq}^{s}(az,rg)\right|^{2}\right)-\text{Clutter}_{pq}^{s}\right]. \tag{14}\]  
The residual phase \(\delta\phi_{pq}^{s}(az)\), meanwhile, measures phase errors in the target response that can be attributed to target range errors. The sensor position is affected by inaccuracies in the navigation  
Figure 3: The impact on various analysis steps on the target Radar Cross Section (RCS) over azimuth, for the target of Figure 2. From top to bottom and left to right: (i) The backscatter intensity along the hyperbolic target range history in the original raw data. (ii) The target intensity after clutter filtering. (iii) The target intensity after the transmit and receive antenna patterns have been accounted for and, as a dotted line, the expected target RCS variation. (iv) The residual RCS offset between the expected and the measured target response. data and the antenna element phase center positions. Similarly, there is a residual uncertainty in the assumed reference target position on ground. \[\delta\phi_{pq}^{s}(az)=\arg\left(\sum_{\gamma g}F_{pq}^{s}(az,\gamma g)\right) \tag{15}\]  
In addition to the phase error above, the target range error can be measured directly. This residual delay is given by the offset of the target peak response away from the expected range. After the range cell migration correction of Equation (8), the expected peak is in the origin at \(rg=0\), such that he residual delay is estimated as  
\[\delta\tau_{pq}^{s}(az)=\frac{1}{RSF}\arg\max_{rg}\left(\left|F_{pq}^{s}(az,rg )\right|^{2}\right), \tag{16}\]  
where \(RSF\) denotes the range sampling frequency of the raw data. The peak response is localized with sub-sample accuracy using the spectral diversity co-registration method described in [24] using a delta impulse \(\delta(rg)\) as a reference signal. The residual phase of (15) and the residual delay of (16) measure essentially the same residual error and, in a certain sense, are complimentary: the phase measurement extremely sensitive but is relative in the sense that it is arbitrarily offset and wrapped in the interval \([-\pi,\pi]\). The residual delay, on the other hand, is less sensitive but does represent an absolute error. In an attempt to combine the advantages of both measures, the analysis derives the so-called absolute residual phase. It is defined as the offset, unwrapped residual phase:  
\[\delta\chi_{pq}^{s}(az)=\chi_{0}+\sum_{az}\arg\left(e^{j\frac{\partial}{\partial az }\delta\phi_{pq}^{s}(az)}\right) \tag{17}\]  
The absolute phase offset \(\chi_{0}\) is set to ensure consistency with the residual delay by enforcing the equality  
\[\operatorname*{median}\left(\delta\chi_{pq}^{s}(az)+2\pi f_{0}\delta\tau_{pq}^ {s}(az)\right)=0, \tag{18}\]  
where \(f_{0}\) denotes the radar center frequency. Figure 4 illustrates the information content of the residual phase, the residual delay and the absolute residual phase. Finally, the calibration approach also uses the so-called target coherence.