## 4 Azimuth \"Bistatic Residual\" Correction  
During the S1A in-orbit commissioning phase, an unexpectedly high azimuth geolocation error was discovered. Subsequent investigation shed light on the issue and led to the implementation of another necessary timing correction, this time related to the S1A operational SAR Instrument Processing Facility (IPF) itself. As illustrated in Figure 5a for SM mode, the satellite moves a certain distance along its orbit during the time between pulse transmission and echo reception. This is referred to as the _bistatic effect_ or _start-stop approximation_, and it has been understood for many years [15]. For S1A, this distance is typically \(\sim\)43 m at mid-scene, given e.g., a spacecraft velocity of 7592 m/s and an echo travel time (\"fast time\") at mid-scene of \(\sim\)5.7 \(\times\) 10\({}^{-3}\) s (beam S4 product). Usually, a SAR system records the time when the radar echo is received, providing this in the product annotations. While this method provides a good first approximation of the azimuth times corresponding to the image lines, they are nonetheless consistently too \"late\". The S1 IPF improves this method by shifting the time stamps to the moment when the side-looking sensor was actually aimed at the imaged line, as indicated in Figure 5a. In this way, the azimuth time stamps correspond to the time when the _mid-range_ samples were imaged, which is a better approximation for the image line as a whole. The correction performed is thus a perfect correction at mid-range but contains an inherent error residual proportional to the slant range offset from this point. The timing correction is indicated by the gray arrow in Figure 5a.  
Critically, the magnitude of the correction applied by the S1 IPF is made under the mid-range assumption, _i.e._, for a spacecraft travel time corresponding to the mid-range echo travel time, as this is the optimal choice for the image as a whole. However, the correction is insufficient when high accuracy is required for a specific target within the scene, as illustrated in Figure 5b. Here, point target PT is imaged at the same zero Doppler time as in Figure 5a, but the increased echo travel time relative to mid-range implies that a larger bistatic timing shift is needed relative to the echo receive moment. The bistatic shift required for PT is shown as a blue arrow in Figure 5b. We call the _difference_ between the arrow lengths in Figure 5b the _bistatic residual error_. It can be as large as \(\sim\)0.7 m (for a target near the SM beam S6 edge). The residual error can be expressed in seconds or meters, as given in Equations (1) and (2):  
\[\text{bistatic residual error [s]}=\frac{\text{fastTime}_{\text{PT}}\;-\; \text{fastTime}_{\text{mid}}}{2} \tag{1}\]  
\[\text{bistatic residual error [m]}=\text{bistatic residual error [s]}\;\times\; \text{platform velocity [m/s]} \tag{2}\]  
In the above equations, the fast time (echo travel time) for a given range sample at the point target PT, is calculated as follows:  
\[\text{fastTime}_{\text{PT}}\text{[s]}=\;\text{sr}_{\text{near}}\text{[s]}+ \left(\frac{\text{rgSampleNr}}{\text{rgSamplingRate [Hz]}}\right) \tag{3}\]  
Here, \(\text{Stnear}\) is the fast time at the near-range edge. It is obtained from the product annotations (auxiliary file XML file, parameter _slantRangeTime_ in the section product:imageAnnotation:imageInformation).  
rgSampleNr is the range sample number, where the first sample is numbered 0. The range sampling rate (rgSampleRate) is derived from the S1A product parameter _rangeSamplingRate_ in the annotation XML file under product:generalAnnotation:productInformation [16].  
The azimuth timing correction in Equation (1) is integrated into the geolocation processor as a small shift in the predicted time for a given CR.