## Appendix B Zavorotny-Larson Model for the Surface Reflection Coefficient \(\Gamma\)  
In [7], the authors proposed a model for the complex reflection coefficient \(\Gamma=|\Gamma|\exp\left\{\phi_{\mathrm{T}}\right\}\) of the GPS L1 signal for the case of bare soil with a plane surface. \(\Gamma\) is defined like in Expression (30) as the amplitude ratio between the reflected and the LOS signals. The model treats the soil as a single-layer dielectric medium with losses. It also takes into account the polarization dependent antenna gain effect on the reflected signal. The main geometric optic contribution to the amplitude of the reflected signal comes from the first Fresnel zone around the specular reflection point. According to the model, when the antenna boresight is perpendicular to the ground, \(\Gamma\) can be expressed as a function of the satellite elevation \(\theta_{el}\) as:  
\[\Gamma(\theta_{el}) =\exp\left\{-0.5\,k^{2}\sigma_{s_{h}}^{2}\sin^{2}(\theta_{el})\right\}\] \[\qquad\times\left(V_{RR}F_{R}\left(-\theta_{el}\right)+V_{RL}F_{L} \left(-\theta_{el}\right)\right), \tag{67}\]  
where the effect of the surface roughness is incorporated by the exponential term, the so-called coherence loss factor [61]. \(k=2\pi/\lambda\) is the carrier wavenumber, and \(\sigma_{s_{h}}\) is the standard deviation of the surface height. \(F_{R}\left(\theta_{el}\right)\) and \(F_{L}\left(\theta_{el}\right)\) represent the antenna complex amplitude gains for right-hand and left-hand circular polarizations (RHCP and LHCP) at \(\theta_{el}\), respectively; and  
\[V_{RR} =V_{LL}=\frac{1}{2}\left(V_{v}+V_{h}\right), \tag{68}\] \[V_{RL} =V_{LR}=\frac{1}{2}\left(V_{v}-V_{h}\right),\]are the polarization-dependent reflection coefficients, where the first subindex stands for the polarization state (\(R\) for RHCP and \(L\) for LHCP) of the incident wave, while the second subindex stands for the polarization state of the reflected wave. \(V_{v}\) and \(V_{h}\) are the coefficient for the linear vertical and horizontal polarizations, which can be expressed as:  
\[\begin{split} V_{v}&=\frac{\varepsilon\sin\left( \theta_{el}\right)-\sqrt{\varepsilon-\cos^{2}\left(\theta_{el}\right)}}{ \varepsilon\sin\left(\theta_{el}\right)+\sqrt{\varepsilon-\cos^{2}\left( \theta_{el}\right)}},\\ V_{h}&=\frac{\sin\left(\theta_{el}\right)-\sqrt{ \varepsilon-\cos^{2}\left(\theta_{el}\right)}}{\sin\left(\theta_{el}\right)+ \sqrt{\varepsilon-\cos^{2}\left(\theta_{el}\right)}},\end{split} \tag{10}\]  
with:  
\[\varepsilon\approx\varepsilon_{r}-j60\lambda\sigma \tag{11}\]  
the complex dielectric permittivity of the material [81; 82]. In Equation (11), \(\varepsilon_{r}\) represents the real-part of the permittivity, and \(\sigma\) is the conductivity of the reflection surface (in S/m units). The model for \(\Gamma\) can be extended for the multiple layer ground case, where \(\Gamma\) will also depend on the thickness of each layer and their relative permittivity [81; 83]. For example in [84; 85], the authors use a three-layer (air + two soil layers) model for GNSS-R applications working with the GPS L1 band signal. For more details on all of the assumptions made to obtain (12), we refer the interested reader to [7].