# Spaceborne GNSS-R for Sea Ice Classification Using Machine Learning Classifiers
## 3 Theory and Methods
### Spaceborne GNSS-R Features
#### 3.1.2 Features Derived from DDM  
Besides the surface reflectivity, the other seven GNSS-R observables are chosen for sea ice classification. GNSS-R observables are defined as characteristics derived from the DDMs that describe their power and shape. The DDM average (DDMA), which represents the mean signal-to-noise ratio (SNR) values over the point of peak SNR (Figure 4a), is applied in this study to classify sea ice types. As shown in Figure 4a, the DDMA expands 3 delay bins and 3 Doppler shifts centered at the peak SNR point. The distribution of DDMA in 5 days (from 11 to 15 February 2018) is presented in Figure 4b compared with the OSISAF SIT map. As illustrated in [23; 39], the integrated delay waveform (IDW) is defined as the summation of 20 delay waveforms, which are the cross-sections of DDM at 20 different Doppler shifts. The cross-section of zero Doppler shift is defined as the central delay waveform (CDW) of the DDM. The degree of difference between IDW and CDW is described by differential delay waveform (DDW), which is depicted as follows,  
\[\mathrm{DDW}=\mathrm{NIDW}-\mathrm{NCDW}, \tag{7}\]  
where NIDW and NCDW represent the normalized IDW and CDW respectively. The NIDW is related to the power spreading characteristics caused by surface roughness. The  
Figure 4: (**a**) The range of calculating Delay-Doppler Map average (DDMA) is marked by the white box, which expands 3 delay bins and 3 Doppler shifts centered at the peak signal-to-noise ratio (SNR) point. (**b**) The distribution of DDMA compared with the OSISAF SIT map. The open water (OW) is described by the light blue color. The white and gray represent the first-year ice (FYI) and multi-year ice (MYI). Figure 3: The (**a**) TDS-1 surface reflectivity and corresponding (**b**) incidence angle distribution in Arctic regions in 5 days (from 11 to 15 February 2018). The open water (OW) is described as light blue. The white and gray represent the first-year ice (FYI) and multi-year ice (MYI). delay waveforms from delay bins \(-5\) to \(20\) over OW, FYI, and MYI surfaces are shown in Figure 5.  
Features extracted from delay waveforms are described as follows:  
* RESC (Right Edge Slope of CDW). The fitting slope of NCDW with 5 delay bins starting from the zero delay one is defined as RESC, which is depicted by the slope of the blue line in Figure 6a. * RESI (Right Edge Slope of IDW). The fitting slope of NIDW with 5 delay bins starting from the zero delay one is defined as RESI, which is depicted by the slope of the green line in Figure 6a. * RESD (Right Edge Slope of DDW). The fitting slope of DDW with 5 delay bins starting from the zero delay one is defined as RESD, which is depicted by the slope of the magenta line in Figure 6a. * REWC (Right Edge Waveform Summation of CDW). The summation of NCDW values (marked with blue diamond dots in Figure 6b) from the delay bins 0 to 6 is defined as REWC. * REWI (Right Edge Waveform Summation of IDW)]. The summation of NCDW values (marked with green cross dots in Figure 6b) from the delay bins 0 to 6 is defined as REWI. * REWD (Right Edge Waveform Summation of DDW)]. The summation of DDW values (marked with magenta circle dots in Figure 6b) from the delay bins 0 to 6 is defined as REWD. Figure 5: (**a**) Normalized central delay waveform (NCDW), (**b**) normalized integrated delay waveform (NIDW), and (**c**) differential delay waveform (DDW) of samples over open water (OW), first-year ice (FYI), and multi-year ice (MYI) surfaces respectively. Figure 6: (**a**) GNSS-R Features demonstration of RESC (Right Edge Slope of CDW), RESI (Right Edge Slope of IDW), and RESD (Right Edge Slope of DDW). The fitting slope of blue, green, and magenta lines are defined as RESC, RESI, and RESD, respectively. (**b**) GNSS-R Features demonstration of REWC (Right Edge Waveform Summation of CDW), REWI (Right Edge Waveform Summation of IDW), and REWD (Right Edge Waveform Summation of DDW). The waveform summation of the blue, green, and magenta dots are defined as REWC, REWI, and REWD, respectively.