## 2 Method
### Impulse Response Extraction of a Range Line  
The transmission pulse of Sentinel-1 is a chirp signal defined by a pulse length, a start frequency and a frequency ramp rate. The time range between subsequent transmission pulses is indicated by the pulse repetition interval (PRI). The received signal is acquired within a certain sampling (or receiving) window between two successive transmission pulses. This sequence of samples is stored for each received signal as a range line. The reciprocal of the range sampling rate defines the time interval between consecutive samples of a range line; the related time axis is called range or fast time.  
The round trip time (\(\tau_{SAR}\)) defines the pulse propagation time as detected by the SAR instrument. It can be derived for each sample of a range line using Equation (1). The \(rank\) is the number of PRIs between the pulse transmission and the corresponding reception of its echo. The start time of the receiving window within the PRI is considered by \(dt_{0}\); \(\tau_{sample}\) represents the fast time within the receiving window of the related pulse echo:  
\[\tau_{SAR}\,=\,rank\cdot PRI\,+\,dt_{0}\,+\,\tau_{sample}. \tag{1}\]  
In order to extract the impulse response from L0 data, the received signal is correlated with an ideal chirp (match filter) in the time domain. The output gives the range compressed signal with complex values for the full range line. As the study focuses rather on propagation times and related phases than on absolute backscattering power, no gain normalization steps are required for this approach. Note that, in general, the matched filter procedures for SAR data processing are transferred to the frequency domain to optimize the performance of the calculations [18], but with the same expected results. This frequency domain approach is also used for the operational Sentinel-1 SAR processor.  
Figure 1: SAR observation scheme containing main parameters that have to be considered for geometric retrieval: the orbit state vectors and the target position define the geometry between spacecraft and target; the pulse timeline stored in the raw data headers contains information about transmitted (\(tx\)) and received (\(rx\)) SAR pulses. Additional delays are caused by propagation effects through the troposphere and ionosphere.  
A high radar backscatter due to a transponder or corner reflector is easily detectable within the range compressed signal. Within the complex data stream, the target responses show a higher amplitude compared to the background noise as depicted for the corner reflector response at 131.84 us in Figure 2. Furthermore, the signal propagation time of the detected pulse can be (roughly) estimated by using the annotated timestamp of the first sample, the range sampling frequency and the sample number containing the target response. This information is used to identify the related target response as the (local) maximum amplitude within the receiving window. To achieve a higher resolution for the receiving time \(\tau_{sample}\), the complex data is interpolated within a certain range around the peak.  
The peak amplitude of the interpolated data is determined for each analyzed range line; the corresponding phase and the range time are stored for the analysis of the offsets in range and azimuth.