## 2 Materials and Methods
### Extraction of Features  
The scattering components of DDM come from the glistening zone with different delay and Doppler shifts with respect to the specular point. The method proposed in [28] uses the two-dimensional delay waveforms generated from DDMs as basic observables for easier data processing. As introduced in [28], the cross section of 20 different Doppler shifts produces 20 delay waveforms, whose summation refers to the integrated delay waveform (IDW) [25] of the DDM over the Doppler domain. The relationship between the power of scattered signals and time delay is illustrated by DDM, which is described by the model proposed in [45] based on the bistatic radar equation:  
\[DDM=T_{i}^{2}\int\frac{D^{2}\left(\overrightarrow{\rho}\right)}{4\pi R_{T}^{2 }\left(\overrightarrow{\rho}\right)R_{R}^{2}\left(\overrightarrow{\rho} \right)}\left|\Lambda(\tau)\times S(f_{D})\right|^{2}\sigma_{0}\left( \overrightarrow{\rho}\right)d^{2}\rho \tag{1}\]  
where \(T_{i}\) represents the coherent integration time, \(\tau\) represents the time delay, \(D^{2}\) represents the function of power antenna footprint, \(R_{T}\) represents the distance from the scattering point to GNSS transmitters, \(R_{R}\) represents the distance from the receiver to the scattering point, \(\Lambda\) is a triangular function as a function of time delay, S is a sinc function in the frequency domain for GPS C/A codes, \(\sigma_{0}\) represents the normalized bistatic radar cross section, \(f_{D}\) represents the Doppler shift frequency and \(\rho\) represents the vector from the specular reflection point to the scattering point. In the TDS-1 mission, the coherent integration time is 1 ms and the Doppler bandwidth can be described by \(\Delta f_{0}=1/2T_{i}\). If the maximum and minimum Doppler shift of the scattered signal is defined as \(f_{max}\) and \(f_{min}\), respectively, the width of the glistening zone can be described by \(f_{max}-f_{min}\). If the Doppler bandwidth is larger than the width of the glistening zone (i.e., \(\Delta f_{0}>f_{max}-f_{min}\)), the Doppler effects is negligible. The sinc function S is equal to 1 and the cross section with zero Doppler shift (Doppler = 0) is a particularly Central Delay Waveform (CDW) from the DDM. The waveform can be defined as:  
\[CDW=T_{i}^{2}\int\frac{D^{2}\left(\overrightarrow{\rho}\right)R_{T}^{2}\left( \overrightarrow{\rho}\right)}{4\pi R_{T}^{2}\left(\overrightarrow{\rho} \right)R_{R}^{2}\left(\overrightarrow{\rho}\right)}\left|\Lambda(\tau) \right|^{2}\sigma_{0}\left(\overrightarrow{\rho}\right)d^{2}\rho \tag{2}\]  
Another observable termed as differential delay waveform (DDW) was used to describe the degree of difference between CDW and IDW. The DDW between normalized CDW (NCDW) and normalized IDW (NIDW) can be defined as:  
\[DDW=NIDW-NCDW \tag{3}\]  
The IDW is useful to describe the power spreading characteristics due to surface roughness. In order to extract features from DDMs, several data pre-processing schemes presented in the previous study [23; 28] should be applied to subtract the noise floor to obtain normalized DDM (NDDM) with NIDW, NCDW and DDW. Contrary to the previous studies, data with a peak signal-to-noise ratio  
Figure 1: Typical TechDemoSat-1 (TDS-1) Delay-Doppler Maps (DDMs) collected over (**a**) sea water and (**b**) sea ice, respectively. (SNR) above \(-3\) dB are adopted to increase the amount of data. The more relaxed data filtering strategy is also useful to inspect the applicability and generality of the proposed methods. There are no effective signals over the several starting and ending delay bins. Therefore, only a part of delay bins from chips \(-3\) to \(8.75\) (48 delay bins) around the specular point are adopted to extract features. The ground tracks and parts of samples (DDMs and corresponding delay waveforms) of TDS-1 data collected over Baffin Bay on 15 January 2016 are presented in Figure 2. The open water, ice and land are filled with light blue, white and light yellow, respectively.