# Impact of the Elevation Angle on CYGNSS GNSS-R
## 3 Coherent and Incoherent Scattering in GNSS-R
The coherently reflected power is related with \(\mathrm{\Gamma_{LHCP,coh}}\) as in [43]:  
\[\left\langle\left|Y_{\mathrm{r,coh}}(\tau,\mathrm{f})\right|^{2}\right\rangle =\frac{\mathrm{P_{T}}\lambda^{2}\mathrm{G_{T}}\mathrm{G_{R}}\mathrm{|\chi( \tau,\mathrm{f})|}^{2}}{\left(4\pi\right)^{2}\left(\mathrm{R_{T}}+\mathrm{R_ {R}}\right)^{2}}\mathrm{\Gamma_{LHCP,coh}} \tag{3}\]  
The surface roughness has a strong impact on the coherent reflectivity term \(\mathrm{\Gamma_{LHCP,coh}}\) that can be theoretically modelled as in [40]:  
\[\mathrm{\Gamma_{LHCP,coh}}=\left|\mathrm{R_{LHCP}}(\theta_{\mathrm{e}})\right| ^{2}\exp(-(2\mathrm{k}\sigma_{\mathrm{RMS}}\mathrm{sin}\theta_{\mathrm{e}})^{ 2}) \tag{4}\]  
where \(\mathrm{R_{LHCP}}\) is the polarized Fresnel reflection coefficient and \(\mathrm{k}\) is the signal wavenumber. The impact of \(\theta_{\mathrm{e}}\) on \(\mathrm{\Gamma_{LHCP,coh}}\) is twofold; on one hand it affects the Fresnel reflection coefficient, and on the other hand, it effects the exponential decaying factor. For a completely flat surface, \(\mathrm{\sigma_{RMS}}\) is zero, and \(\mathrm{\Gamma_{LHCP,coh}}\) is equal to the square of the amplitude of the Fresnel reflection coefficient. \(\mathrm{R_{LHCP}}\) is linked to the dielectric constant of the soil, and thus with SMC. For a given roughness and a fixed geometry, \(\mathrm{\Gamma_{LHCP,coh}}\) should increases with larger SMC. On the other hand, \(\mathrm{R_{LHCP}}\) is nearly constant for \(\theta_{\mathrm{e}}>40^{\circ}\) and it decreases significantly for lower angles. However, the exponential factor increases significantly with lower \(\theta_{\mathrm{e}}\). As such, \(\mathrm{\Gamma_{LHCP,coh}}\) could be expected to be higher for grazing angles, even over areas with moderate roughness levels. Figure 1 shows the theoretical relationship (Equation (4)) between \(\mathrm{\Gamma_{LHCP,coh}}\) and \(\theta_{\mathrm{e}}\) for different levels of SMC. The [PERSON]'s model [44] is used to relate the SMC with the dielectric constant of the soil, assuming a soil composition of 50% clay and 20% sand. It appears \(\mathrm{\Gamma_{LHCP,coh}}\) increases for lower \(\theta_{\mathrm{e}}\) and it decreases for higher \(\mathrm{\sigma_{RMS}}\). On the other hand, the influence of \(\theta_{\mathrm{e}}\) is higher as \(\mathrm{\sigma_{RMS}}\) increases. It could be expected that the coherently reflected power as collected from a spaceborne platform increases at grazing angles because it is roughly independent of the platform's height (\(\mathrm{R_{T}}>>\mathrm{R_{R}}\)). Figure 1: _Cont._  
The expression of \(\Gamma_{\text{LHCP,inoch}}\) is quite complex and not compact. More detailed information on incoherent scattering over rough surfaces can be found in Reference [40]. The effects of biomass should be also considered for modeling the incoherent scattered field over vegetated areas. A comprehensive method can be found in Reference [29]. Finally, the impact of rough topography in \(\text{Y}_{\text{r}}\) can be understood as a sort of impulse response in the delay and Doppler domains to be convolved with the WAF [45]. Rough topography reduces the spreading of the DDMs, while surface roughness increases the spreading with larger roughness because more specular reflection points contribute to the total scattered field. As such, Equation (4) is valid in regions without rough topography.