## 2 Description of the GNSS-R Experiment
The Polarimetric Ratio (PR) is computed to properly evaluate the fluctuations of the dielectric properties of scatterers, since theoretically it can cancel out the effects of soil surface roughness without biomass [18], although it does not do so perfectly for arbitrary scattering media. The PR at linear polarization is high when soil moisture content is weak and reversely. On the other hand, at circular polarization the ratio increases with the moisture [18]. It is estimated as the ratio of the peak of the reflected signal power waveform at H-Pol over the peak of the reflected signal power waveform at V-Pol, after proper compensation of the noise power floor and the antenna radiation pattern:  
\[\mathrm{PR}=\frac{\langle\left|\mathrm{Y^{ref}}\right|^{2}\rangle}{\langle \left|\mathrm{Y^{ref}}\right|^{2}\rangle}. \tag{3}\]  
The theoretical definition of the PR is provided in [32] as:  
\[\mathrm{PR}=\sigma_{\mathrm{H}}/\sigma_{\mathrm{V}}, \tag{4}\]  
where \(\sigma_{\mathrm{H}}\) and \(\sigma_{\mathrm{V}}\) are the bistatic radar cross-sections of H and V respectively. Over land surfaces, there are two contributions to the waveforms: coherent and incoherent scattered fields in different proportions. In this scenario, the Kirchhoff Approximation in the Physical  
Figure 2: Signal processing chain to convert SMAP Level 1A “high rate” data to Delay Doppler Maps (DDMs). The SMAP orbit, attitude, and antenna pointing are used to predict times when specular reflections will be visible. These “high rate” data is filtered by these times, and only relevant portions converted to I/Q for GPS L2C DDM generation. Optics Approximation (KPO) provides a reasonable theoretical framework to analyze the results in regions without biomass. The incoherent bistatic scattering coefficient as provided by the KPO is [33]:  
\[\sigma_{\mathrm{qp}}^{0,\mathrm{inc}}=(|\mathrm{a}_{0}|\mathrm{k}l/2)^{2}\mathrm{ e}^{-\mathrm{a}_{2}^{2}\sigma^{2}}\sum_{n=1}^{\infty}\frac{(\mathrm{q}_{2}^{2} \sigma^{2})}{n!n}\mathrm{e}^{-\frac{(\mathrm{q}_{2}^{2}+\mathrm{q}_{2}^{2}) \mathrm{i}^{2}}{4n}}, \tag{5}\]  
where \(\mathrm{k}=\frac{2\pi}{\mathcal{N}}\), \(l\) is the surface correlation length, \(\sigma\) is the surface height standard deviation, and \(\mathrm{q}\) is the bistatic scattering coefficient. In the specular direction, the coefficient \(\mathrm{a}_{0}\) can be computed for the co-polar component H:  
\[\mathrm{a}_{0,\mathrm{H}}=-2\mathrm{R}_{\mathrm{H}}(\mathrm{\theta}_{\mathrm{ inc}})\mathrm{cos}\,\mathrm{\theta}_{\mathrm{inc}}, \tag{6}\]  
and for the co-polar component V:  
\[\mathrm{a}_{0,\mathrm{V}}=2\mathrm{R}_{\mathrm{V}}(\mathrm{\theta}_{\mathrm{ inc}})\mathrm{cos}\,\mathrm{\theta}_{\mathrm{inc}}, \tag{7}\]  
where \(\mathrm{\theta}_{\mathrm{inc}}\) is the incidence angle, and \(\mathrm{R}_{\mathrm{H}}\) and \(\mathrm{R}_{\mathrm{V}}\) are the Fresnel reflection coefficients at H-Pol and V-Pol respectively:  
\[\mathrm{R}_{\mathrm{H}}=\frac{\sqrt{1-\sin^{2}\left(\mathrm{\theta}_{\mathrm{ inc}}\right)}-\sqrt{\mathrm{\varepsilon}_{\mathrm{r}}-\sin^{2}\left(\mathrm{ \theta}_{\mathrm{inc}}\right)}}{\sqrt{1-\sin^{2}\left(\mathrm{\theta}_{\mathrm{ inc}}\right)}+\sqrt{\mathrm{\varepsilon}_{\mathrm{r}}-\sin^{2}\left(\mathrm{ \theta}_{\mathrm{inc}}\right)}}, \tag{8}\]  
and  
Figure 1:0  
where \(\mathrm{\varepsilon}_{\mathrm{r}}\) is the dielectric constant. For the cross-polar components HV and VH, \(\mathrm{a}_{0}=0\)[33]. Therefore, upon substitution of (5), (6) and (7) in (4), the Polarimetric Ratio is found to be independent of the surface roughness:  
Figure 1:1  
In the case of the coherent component of the scattered field [33] (p. 1008), the PR is obtained as the ratio of the H reflectivity over the V reflectivity as:  
Figure 1:2  
Equation (11) shows that in the case of the coherent scattering, the PR is also theoretically independent of the soil surface roughness because the surface height standard deviation is independent of the signal polarization.