# Effective Surface Roughness Impact in Polarimetric GNSS-R Soil Moisture Retrievals
## 5 Discussion  
### Land Reflection: Coherent vs. Incoherent Approach  
In this study, the coherent and incoherent components of the GNSS-R waveform have been separated and either the rms height for the PO model and surface slope and for the GO model, have been retrieved. This is feasible thanks to the SMAP high antenna gain, which allows the detection of the GNSS reflection even without incoherent integration. However, coherent component retrieval is not trivial for other systems that require longer incoherent integrations, and the signal coherency cannot be preserved due to terrain fluctuations. In such cases, incoherent components should be used to estimate the roughness. As it has been presented in Section 4.1, the coherent component and the total power waveform present a \(\sim\)\(-9\) dB consistent bias and an ubRMSD of 3.4 dB. Computing the roughness with coherent or incoherent components might impact a bias in the retrieved magnitude. Results in Section 4.2 shows that the roughness coefficient retrieved using the coherent at 30 ms integration, and the incoherent component can be easily related using a logarithmic model. This indicates that neither the reflected waveform is coherent nor incoherent, but as modeled in Equation (1), a combination of both, but being the incoherent component dominant.  
The PO model bases the roughness effect on the rms height of the surface, while the GO model is a high-frequency approximation for the PO model when the surface is rough enough with respect to the signal wavelength. In this regard, limits were defined in the Kirchhoff scattering model to define the regimes of the PO and GO models. In some of the literature, the limit for the PO model is defined for \(\frac{2\pi}{\lambda}\sigma_{h}<3\), while for the GO model, the limit is defined for \(\frac{2\pi}{\lambda}\sigma_{h}\geq 3\) ([42], p. 428). Our findings on \(\sigma_{h}\) point out a roughness comprised between 4 and 8 cm. For GPS L2C band, \(\frac{2\pi}{\lambda}\sigma_{h}=[1.07,2.14]\), for \(\sigma_{h}=[4,8]\) cm, and for GPS L1, the values are \(\frac{2\pi}{\lambda}\sigma_{h}=[1.32,2.64]\). However, other studies have validated using experimental data that the limit for PO and GO approximations differ from the originally proposed value [64], being PO limited by \(\frac{2\pi}{\lambda}\sigma_{h}<0.75\) and GO limited by \(\frac{2\pi}{\lambda}\sigma_{h}>1.75\). In this second case, and given the range of \(\sigma_{h}\) found in this study, and supported by other studies [39; 57], it seems that GNSS reflections can be better approximated by a high--frequency approximation of the PO model (i.e., the GO model) rather than by the classical PO model, as also confirmed by the results in Section 4.2. This can be also explained by the higher power received in the total power waveform compared to the coherent component part, and the larger correlation of the GO model with the SMAP DEMSLPSTD product. Hence, both results seem to indicate that, in general, the specular forward-scattering over land at GPS bands could be better modeled by the GO approximation due to large-scale roughness variations, rather than small roughness effects modeled using the PO approximation. Note that advanced models such as the Analytical Kirchhoff Solution (AKS), have shown promising results in modeling the impact of roughness on GNSS-R signals [57]. However, these models lack a closed analytical solution that can isolate the roughness parameter to study its relationship with physical quantities for global retrievals.