## 2 Method
### Azimuth Offset  
In order to derive an offset in azimuth direction, the phase history of a specific target is traced during an overpass. Therefore, the detected phase of complex SAR data containing the target response is compared with the predicted phase derived from the geometry using orbit data. This is done by converting the propagation time (\(\tau_{geometric}\)) between satellite and reference target into a phase value using the radar frequency (\(f_{radar}\)) as given in Equation (3):  
\[\phi=-\tau_{geometric}\cdot 2\pi\cdot f_{radar}. \tag{3}\]  
The backscatter from a point target is shifted in frequency by an amount proportional to the relative velocity between satellite and target (\(\vec{\mathbf{v}}_{SC}-\vec{\mathbf{v}}_{target}\)). This Doppler frequency \(f_{d}\) is described by the SAR Doppler Equation (4) and related to the phase by \(f_{d}=-\mathrm{d}\phi\) / dt:  
\[f_{d}=-\frac{2\,f_{radar}}{c}\,\frac{(\vec{\mathbb{R}}_{SC}-\vec{\mathbb{R}}_{ target})\cdot(\vec{\mathbf{v}}_{SC}-\vec{\mathbf{v}}_{target})}{|\vec{\mathbb{R}}_{SC}- \vec{\mathbb{R}}_{target}|}. \tag{4}\]  
The Doppler frequency \(f_{d}\) changes its sign when the spacecraft passes the target. At \(f_{d}=0\), the minimum distance between spacecraft and target is reached. We define this time as reference time \(t=0\). For \(t<0\), the satellite moves in the direction toward the target; for \(t>0\), the satellite moves away from it. It should be noted that the spacecraft attitude and the antenna pointing have no impact on the Doppler frequency in Equation (4) and the geometric phase in Equation (3), which are related to the spacecraft's center of gravity. An antenna mispointing impacts the transmitted and received gain of the antenna w.r.t. the target direction but not the measured phases of pulses containing the target response. The relative motion between the antenna phase center and the spacecraft's center of gravity is very small for Sentinel-1 and negligible for our study. Thus, the proposed method for deriving the azimuth offset focuses on phases not on amplitudes and verifies the azimuth time of the acquired target phase history using the related geometry including orbit state vectors and target position. Figure 3 (top) depicts the phase history of a point target for a single overpass. The observed SAR data phase shown in blue is wrapped with values between \(-180^{\circ}\) and \(180^{\circ}\); the predicted phase (black line) is overlaid presuming the same phase offset at \(t=0\). The maximum of the predicted phase (related to shortest propagation time \(\tau_{geometric}\)) corresponds to the minimum distance between satellite and target, which is expected at \(t=0\). By subtracting the predicted geometric phase from the measured SAR data phase, a low difference less than \(5^{\circ}\) remains in this case as depicted in Figure 3 (bottom). Note that, if the phase difference is computed using complex math an explicit phase, unwrapping is not necessary. The low variation and the absence of a ramp within the phase difference indicate that the azimuth timing measured by the SAR instrument matches well with that derived by the geometry; consequently, no significant azimuth offset is expected after azimuth compression. The azimuth compression is performed with a matched filter focusing on the phase. For this task, the measured phase history is correlated with the one predicted from geometry. For our analysis, it is sufficient to use the target response derived from the maximum amplitude of each range line (as seen in Figure 3) instead of composing a two-dimensional, full focused SAR image. The correlation result contains the azimuth compressed data as a function of azimuth shift, i.e., the difference between measured and predicted phase history. The azimuth shift can be directly converted to an azimuth offset (in meters) by multiplying with the satellite velocity on ground (\(\sim\)6836 m/s for Sentinel-1).