# Effective Surface Roughness Impact in Polarimetric GNSS-R Soil Moisture Retrievals
## 4 Results
### Dual-Polarization Differential Roughness Estimates  
Solving the roughness effect for a single polarization gives us the possibility for single-polarization soil moisture retrievals using any polarization combination. However, the algorithm is highly dependent on the accuracy of the roughness map. To overcome this issue, combined polarimetric retrievals can be used to mitigate the effect of roughness. This approach has been proposed by several pieces of research [28; 42; 60; 61] to allow a higher quality soil moisture estimation without the need to accurately model the surface roughness, reducing the error. As in the single-polarization case, we define an error term by comparing the measured reflectivity ratio with the Fresnel reflection coefficient ratio:  
\[Q_{pq}[dB]=10\text{log}_{10}\frac{\Gamma_{p}}{\Gamma_{q}}-10\text{log}_{10} \frac{\left|R_{p}(\theta)\right|^{2}}{\left|R_{q}(\theta)\right|^{2}}, \tag{30}\]  
Being \(p\) and \(q\) two orthogonal polarizations, and \(Q\) polarization coupling factor [28]. Note that, VOD and roughness parameters are neglected here, and both are simplified by the \(Q\) parameter.  
Results for \(Q_{pq}\) for \(p=\) [H,R] and \(q=\) [V,L] for both \(\Gamma^{coh}\) and \(\Gamma^{inc}\) are shown in Figure 6. As can be seen, results are very similar for the coherent and incoherent cases. For the HV case, the correlation between the coherent and the incoherent is 0.88, with an RMSD of  
Figure 5: (**a**) Soil moisture and bare soil reflectivity fitting function simulated from a soil composition of Sand = 40%, Clay = 20%, and Silt = 60% [30] at an incidence angle of 40\({}^{\circ}\), (**b**) error in the estimation of the bare soil reflectivity as a function of the error in the rms height.  
0.61 dB. The average \(Q_{HV}\) value is \(-2.2\) dB, and the standard deviation is 1.3 dB. First, these results indicate that the SMAP-R captured H/V ratio presents a \(\sim\)2.29 dB bias with respect to the theoretical one. The analysis of this bias was discussed in [52] to be produced by the GPS L2 antenna axial ratio for blocks IIR or newer [62,63].  
For \(Q_{RL}\) the average value is 2.72 dB with a standard deviation of 3.86 dB, and the correlation between the coherent and incoherent cases is 0.87 and 1.33 dB, respectively.  
In this case, the \(Q_{HV}\) seems uncorrelated with any surface parameter (e.g., DEM), but it does correlate with some parameters of the reflection, as the incoherent reflectivity from Equations (5)-(7), once calibrated by VOD:  
\[Q_{HV}^{inc}[dB]=0.1364\cdot 10\text{log}_{10}\left(\frac{\overset{inc}{H}}{e^{ \frac{-2\tau}{\text{cos}(\theta)}}}\right)+0.35, \tag{31}\]  
With an RMSD of 0.97 dB and a correlation coefficient of 0.64. If VOD is not compensated, the correlation coefficient is lowered down to 0.55 and the RMSD increases up to 1.09 dB. Results for the coherent part show a lower correlation coefficient (0.4) and larger RMSD (1.38), which could indicate that this parameter is more sensitive to incoherent scattering rather than specular scattering.  
On the contrary, \(Q_{RL}\) is correlated with vegetation, notably in areas with large vegetation, e.g., Congo or Amazonian rainforests. VOD and \(Q_{RL}\) are related via the linear relationship:  
\[Q_{RL}^{inc}[dB]=7.7612\cdot\tau+0.48. \tag{32}\]  
In this case, the correlation is 0.65, and the RMSD is 2.93 dB. The scatter-density plots of both fits are shown in Figure 7.  
Figure 6: Polarization decoupling factor (Q) in dB units, for coherent and incoherent HV and RL ratios.