## 3 Target Response Model and Analysis
### Response Analysis  
To provide the inputs required for estimating calibration corrections, the measured target response is compared to the expected target response of (4), as illustrated in Figure 2. Any systematic discrepancy is potentially a sensor calibration issue that should be addressed by estimating corresponding calibration corrections. Such discrepancies will be termed residual errors in the following. Figure 2: An illustration of the processing steps involved in the analysis of the response of a 1.5 m trihedral reflector in SAR data acquired at L-band in VV polarisation. For a given data channel that uses antenna elements \(p\) and \(q\) to transmit and receive echos, respectively, the sensor acquires a 2D echo buffer \(D_{pq}(az,rg)\) defined over range and azimuth sample indices \(rg\) and \(az\), respectively. The normalized response of a target \(s\) is closely related to residual errors in the response. It is given by  
\[D_{pq}^{s}(az,rg)=\mathrm{IFFT}_{f_{r}\to rg}\left[\frac{\mathrm{FFT}_{rg\to f_{r}} \left[D_{pq}(az,rg)\right]}{C_{pq}(az,f_{r})\ M_{pq}^{s}(az,f_{r})}\right], \tag{8}\]  
where FFT and IFFT denote the forward and inverse discrete Fourier transform, respectively. \(M_{pq}^{s}(az,f_{r})\) is the expected target response, defined in terms of the physical range frequency \(f_{r}\) in (4), while \(C_{pq}(az,f_{r})\) is a correction derived from the composition of internal calibration measurements and external calibration corrections:  
\[C_{pq}(az,f_{r})=\mathrm{FFT}_{rg\to f_{r}}\left[C_{pq}(az,rg)\right] \Delta C_{pq}(f_{r})\,. \tag{9}\]  
In the above, \(C_{pq}(az,rg)\) is a 2D buffer of internal calibration pulses recorded alongside the data channel in question. In practice, the SAR sensor might not record an internal calibration signal for each and every pulse. The calibration buffer would then have to be interpolated in azimuth. Alternatively, internal calibration information may not be acquired during the actual data take at all. In this case, the calibration signal takes the form of a 1D chirp replica that is constant over azimuth. \(\Delta C_{pq}(f_{r})\), meanwhile, is a one dimensional, complex valued correction to the range frequency response. This correction is initially unknown and assumed equal to unity until estimates become available during the calibration process. The application of Equation (8) simultaneously achieves all of the processing steps illustrated in Figure 2, apart from the clutter filtering. In view of implementation, it should be noted that the footprint of each individual target is typically relatively small. The calibration process can therefore be speeded up considerably by cropping the data data analysed for each target. The azimuth extent of the response can, for example, be limited by thresholding \(\int\left|M_{pq}^{s}(az,f_{r})\right|^{2}\,\mathrm{d}f_{r}\). Corresponding range limits are then given by the extrema of \(|\overline{r}_{e}^{s}(az)|\) within the valid azimuth extent. To improve the signal to clutter ratio in the target response, analysis also includes an azimuth-adaptive clutter filter. This filter effectively singles out the directions of arrival corresponding to the reference target. The impact of this filtering step is illustrated in the example response of Figure 3. The approach is based on the observation that, since the target phase history is flattened in (8), the Doppler frequency of the response is expected to be constant and close to zero. Significant clutter reduction can therefore be accomplished by simply applying a Gaussian bandpass filter in the Doppler frequency domain:  
\[F_{pq}^{s}(az,rg)=\mathrm{IFFT}_{f_{a}\to az}\left[K(f_{a})\,\mathrm{FFT}_{ az\to f_{a}}\left[D_{pq}^{s}(az,rg)\right]\right]\ \ \mathrm{with}\ \ K(f_{a})=e^{-\frac{(f_{a}-p)^{2}}{2\sigma^{2}}}. \tag{10}\]  
The filter mean \(\mu\) is set to coincide with the peak of the power spectrum of \(\sum_{rg}D_{pq}^{s}(az,rg)\).