# Spaceborne GNSS-R for Sea Ice Classification Using Machine Learning Classifiers
## 3 Theory and Methods  
### Spaceborne GNSS-R Features  
#### 3.1.1 Surface Reflectivity  
The GNSS-R instrument (e.g., SGR-ReSI) receives signals directly transmitted from GNSS constellations and scattered from the Earth's surface. For the TDS-1 mission, the signals observed are the L1 C/A codes with a center frequency of 1.575 GHz. The L-band signal is useful for remote sensing applications due to its insensitivity to the precipitation and atmosphere. The specular reflections are dominant when the surface is relatively flat and smooth. As demonstrated in [15], the GNSS-R signal received over sea ice is usually \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Orbit & quasi-Sun synchronous \\ Altitude & \(\sim\)635 km \\ Inclination & 98.4\({}^{\circ}\) \\ Delay pixels/resolution & 128/244 ns \\ Doppler pixels/resolution & 20/500 Hz \\ Coherent integration time & 1 ms \\ Incoherent integration time & 1 s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Orbit and instrument specifications of TDS-1 [45]. Figure 2: Typical TechDemoSat-1 (TDS-1) Delay-Doppler Maps (DDMs) collected over (**a**) sea ice type (SIT), which includes open water (OW), first-year ice (FYI), and multi-year ice (MYI)) and (**b**) sea ice concentration (SIC) maps on 26 November 2015. The magenta, blue and green plots represent the ground tracks of OW, FYI, and MYI respectively. The continuous DDMs from index 709 to 714 for the OW-FYI transition area is marked with the cyan rectangle. Similarly, the FYI-MYI transition area with DDMs from index 726 to 731 is presented in the figure. The SIT and SIC maps are obtained from the Ocean and Sea Ice Satellite Application Facility (OSISAF). The OW is depicted by the light blue color. FYI and MYI are demonstrated by white and gray respectively in Figure 2a. coherent due to its smooth surface. Thus, the coherent surface reflectivity \(SR_{coh}\) at the specular point (SP) can be modeled as [46; 47],  
\[SR_{coh}=\frac{\left(4\pi\right)^{2}\left(R_{t}+R_{r}\right)^{2}}{\left(\lambda \right)^{2}G_{r}G_{t}P_{t}}P_{coh}, \tag{1}\]  
where \(P_{coh}\) is the coherently received power reflected by the surface, \(R_{r}\) and \(R_{t}\) are the distances from the receiver and transmitter to the SP respectively, \(G_{r}\) and \(G_{t}\) are the antenna gain of the receiver and transmitter, \(\lambda\) is the GNSS signal wavelength, and \(P_{t}\) is the power transmitted by GNSS satellites. Most of the variables in (1) can be obtained from the TDS-1 L1b data. \(R_{r}\) and \(R_{t}\) can be easily calculated according to the positions of the transmitter, receiver, and SP, which are stored in the metadata. \(G_{r}\) at SP can be directly extracted from the metadata. The noise floor is determined using the average value of the first four rows of DDM [23]. The transmitted signal power can be derived using [48],  
\[P_{t}G_{t}=4\pi R_{d}^{2}G_{d}P_{d}, \tag{2}\]  
where \(P_{t}G_{t}\) is also termed as effective isotropic radiated power (EIRP). \(P_{d}\) is the direct power, \(R_{d}\) is the distance from the transmitter to receiver, and \(G_{d}\) is the zenith antenna gain, which is set as 4 dB in this study according to the Merrbys documentation [45; 49]. As demonstrated in [28], the received power originates from a region surrounding the SP, which is usually the first Fresnel zone. As demonstrated in [50], the Fresnel reflectivity \(SR_{coh}\) can also be modeled as,  
\[SR_{coh}=|\mathcal{R}|^{2}\cdot exp\left[-\left(\frac{4\pi}{\lambda}cos\theta_ {i}\sigma\right)^{2}\right], \tag{3}\]  
where \(\theta_{i}\) stands for the incidence angle and \(\sigma\) is the surface root mean square (RMS) height. The second term in Equation (3) represents the surface roughness.