# Effective Surface Roughness Impact in Polarimetric GNSS-R Soil Moisture Retrievals
## 2 Theoretical Review  
For a polarimetric GNSS-R receiver, the total received power at \(p\) polarization is the sum of the coherent and incoherent terms:  
\[P_{tot_{p}}=P_{cp}+P_{ip} \tag{1}\]  
\[P_{c_{p}}(\theta)=\Gamma_{p}(\theta)\cdot\frac{\lambda^{2}P_{t}G_{t}G_{r}}{(4 \pi)^{2}(R_{t}+R_{r})^{2}} \tag{2}\]  
\[P_{i_{p}}(\theta,\tau,f_{d})=\frac{\lambda^{2}P_{t}G_{t}}{(4\pi)^{3}}\int\frac {G_{r}}{R_{t}^{2}R_{r}^{2}}\cdot\sigma_{p}(\theta)\cdot\chi^{2}(\tau,f_{d})dA \tag{3}\]  
Being \(\lambda\) the signal wavelength; \(P_{t}\) the transmitted power; \(G_{t}\) the transmitter antenna gain in the direction transmitter-specular point; \(G_{r}\) is the receiver antenna gain; \(R_{t}\) and \(R_{r}\) are the distances from the transmitter and the receiver to the specular point, respectively; \(\Gamma_{p}\) is the reflectivity at \(p\)-polarization, which is proportional to the Fresnel reflection coefficient at the incidence angle \(\theta\); \(\sigma_{p}\) is the incoherent bistatic radar cross section at \(p\)-polarization; and \(\chi^{2}\) is the Woodward Ambiguity Function (WAF). Different models have been broadly explored by the community to model the effects of surface roughness [31; 32; 33; 35; 38; 39; 40]. The PO model was validated at high frequencies (e.g., X-band, [41]), where the surface rms was directly affected by the reflection coefficient of bistatic radar. This approximation has been employed in the GNSS-R field as a proxy to correct the effect of surface roughness [32]. The PO model has been validated to be principally sensitive to the coherent component of the signal [41], Chapter 5 in [42]. The Geometric Optics (GO) model models the incoherent scattering, and hence the incoherent component of the signal, detailed in Chapter 10 in [42]. In recent years, several studies have pointed out that in limited geographical areas, the CYGNSS data is mainly sensitive to the surface slope rms. This indicates that the reflectivity attenuation can be better modeled by means of the GO model [35; 39] instead of the PO model. In this manuscript, we utilize global SMAP soil moisture data and the SMAP-R data to analytically compute and compare the surface rms and the surface slope rms under the PO and the GO models. For the coherent part (Equation (2)) the GNSS signal collected over land can be modeled using the PO approximation of the KA model as [33; 35; 43]:  
\[\Gamma_{p}(\theta)=\big{|}R_{p}(\theta)\big{|}^{2}e^{-4k^{2}\sigma_{p}^{2} \cos{(\theta)}^{2}}e^{-\frac{2\tau}{\cos{(\theta)}}}, \tag{4}\]  
where \(\Gamma_{p}(\theta)\) from Equation (4) is the \"coherent\" reflectivity of the GNSS-R signal at \(p\) polarization; \(\big{|}R_{p}(\theta)\big{|}^{2}\) is the Fresnel reflection coefficient at \(p\) polarization, which depends on the incidence angle \(\theta\), and, over land, on the clay/sand/silt soil content, and the moisture value of the soil [44]; \(k\) is the wavenumber, \(\frac{2\pi}{\lambda}\); \(\sigma_{h}\) is the root mean square (rms) of the surface height; \(\tau\) is the vegetation optical depth (VOD) from SMAP, which is retrieved using Moderate Resolution Imaging Spectroradiometer multi-spectral data [13]. For the incoherent part (Equation (3)), the high-frequency limit for the GO model under the KA is applied. This model assumes the reflected signal is quasi-specular reflections affected by moderate roughness [35]:  
\[\sigma_{p}(\theta)=\frac{(4\pi)^{3}\cdot R_{t}^{2}R_{r}^{2}}{P_{t}\lambda^{2} G_{r}G_{t}}\cdot\sum_{r,f}P_{p}(\theta,\tau,f_{d}). \tag{5}\]  
\[\Gamma_{p}^{inc}(\theta)=\sigma_{p}(\theta)\cdot\frac{(R_{t}+R_{r})^{2}}{4 \pi\cdot R_{t}^{2}R_{r}^{2}} \tag{6}\]  
\[\Gamma_{p}^{inc}(\theta)\approx\frac{\big{|}R_{p}(\theta)\big{|}^{2}}{s^{2}} \cdot e^{-2\tau/\cos{(\theta)}} \tag{7}\]  
Being \(\sigma_{p}\) the average bistatic radar cross section at a box 6 C/A chips and \(\pm 1.8\) kHz from the peak, and \(s\) is proportional to the roughness surface slope standard deviation, or DEMSLPSTD if following SMAP's mission nomenclature for this term [29].