# Effective Surface Roughness Impact in Polarimetric GNSS-R Soil Moisture Retrievals
## 3 Materials and Methods
### Multiple Polarization Roughness Retrieval  
Assuming that the cross-polarization component is negligible in the specular direction over bare soil (see Figure 8 from [51]), the Stokes parameters of the reflection can be used to generate any polarization, e.g., H, V, R, or L. It is important to note that the cross-polarization component is not negligible over vegetated areas, as shown in Figure 12 from [51]. In such cases, this component is, at L-band, \(\sim\)30 dB lower than the HH component in the specular direction, but, roughly 10 dB smaller in a non-specular configuration. For the sake of simplicity, we will disregard this cross-polarization component, allowing us to directly estimate HH and VV from the Stokes parameters without the need for additional modeling. Consequently, the \(p\)-polarization reflectivity using the Stokes parameter is defined as \(\vec{l}_{p}\) for \(p\) = {H,V,R,L}, for both the incoherent and coherent components of the signal:  
\[\vec{\Gamma_{H}} =\frac{\vec{\Gamma_{0}}}{2}\cdot\left(1+\frac{\vec{S_{1}}}{\vec{S_ {0}}}\right) \tag{17}\] \[\vec{\Gamma_{V}} =\frac{\vec{\Gamma_{0}}}{2}\cdot\left(1-\frac{\vec{S_{1}}}{\vec{S _{0}}}\right)\] (18) \[\vec{\Gamma_{R}} =\frac{\vec{\Gamma_{0}}}{2}\cdot\left(1+\frac{\vec{S_{3}}}{\vec{S _{0}}}\right) \tag{19}\] \begin{table}
\begin{tabular}{l l l} \hline \hline
**Dataset** & **Variable/File** & **Dates** \\ \hline SPL3 SMP\_E & Vegetation Optical Depth & 2018–2019 \\ \hline SPL3 SMP\_E & Soil Moisture SCA-V & 2018–2019 \\ \hline SMAP\_11\_13\_ANC\_STATIC & DEM \(*\) std \((\sigma)\) at 9 km DEMSTD\_M09\_003 & Static, 2015 \\ \hline SMAP\_11\_13\_ANC\_STATIC & DEM slope \(\sigma\) at 9 km DEMSLPSTD\_M09\_003 & Static, 2015 \\ \hline GLDAS Soil Fraction 0.25\({}^{\circ}\) & GLDASp4\_soilfraction\_025d \({}^{1}\) & Static, 2000 \\ \hline \hline \end{tabular} \({}^{*}\)DEM: Digital Elevation Model. \({}^{1}\)[[https://ldas.gsfc.nasa.gov/sites/default/files/ldas/gldas/SOILS/GLDASp4](https://ldas.gsfc.nasa.gov/sites/default/files/ldas/gldas/SOILS/GLDASp4)]([https://ldas.gsfc.nasa.gov/sites/default/files/ldas/gldas/SOILS/GLDASp4](https://ldas.gsfc.nasa.gov/sites/default/files/ldas/gldas/SOILS/GLDASp4)) \({}_{\text{soilfraction\_025d.nc4}}\), accessed on 1 October 2022.  
\end{table} 
Table 1: Datasets used to evaluate SMAP-R capabilities. \[\vec{\Gamma_{L}}=\frac{\vec{\Gamma_{0}}}{2}\cdot\left(1-\frac{\vec{S_{3}}}{S_{0}}\right) \tag{20}\]  
Being \(\vec{\Gamma_{0}}\) the total power normalized reflectivity (from \(\vec{S_{0}}\)), which has been computed by calibrating the power received by SMAP-R [47; 52]. In this case, \(\vec{\Gamma_{0}}\) is retrieved for the coherent component by isolating \(\Gamma\) from Equation (4), and using \(\vec{S_{0}}^{coh}\) as the coherent received power. For the incoherent component case, \(\vec{\Gamma_{0}}^{inc}\) is computed from Equations (5) to (7), isolating \(\Gamma\) and using \(\vec{S_{0}}^{inc}\) as the total received incoherent power:  
\[\vec{\Gamma_{0}}^{coh}=\vec{S_{0}}^{coh}\cdot\frac{\left(4\pi\right)^{2}(R_{t }+R_{r})^{2}}{\lambda^{2}P_{1}G_{t}G_{r}}, \tag{21}\]  
\[\sigma_{0}^{inc}=\frac{\vec{S_{0}}^{inc}R_{r}^{2}R_{r}^{2}(4\pi)^{3}}{P_{t} \lambda^{2}G_{t}G_{R}}, \tag{22}\]  
\[\vec{\Gamma_{0}}^{inc}=\sigma_{0}^{inc}\cdot\frac{\left(R_{t}+R_{r}\right)^{ 2}}{4\pi\cdot R_{t}^{2}R_{r}^{2}}. \tag{23}\]  
Note that H, V, R, and L stand for the reflectivity or reflection coefficient of an RHCP-transmitted signal received either in H, V, R, or L polarizations. Further information on the H, V, R, and L reflectivity products, their magnitude, and their correlation to other products can be found in [47; 52; 53].