## 3 Methodology
### Analysis Method
The Fresnel coefficient connects the observable reflectivity and surface parameter, such as roughness and vegetations, and it is employed to convert the circular polarization properties of GNSS signals into linear polarization waves, which are generally adopted in electromagnetic reflection theory. Equation (3) expresses the relation between the reflectivity and Fresnel coefficient as a function of wave number \(k\), surface roughness \(\sigma\), and elevation angle \(\Theta\). The surface roughness is derived from the Shuttle Radar Topography Mission (SRTM) Digital Elevation Model (DEM) [45], and the wave number and incident angle are obtained from Level 1 CYGNSS data. \[\Gamma=\left|\mathcal{R}\right|^{2}\;\exp((2k\sigma\sin\;\Theta)^{2}) \tag{3}\]  
It is noted that the current electromagnetic scattering theory explains the natural phenomena with linear polarization waves; however, this circumstance exposes the necessity of expanding scattering theory in terms of circular polarization waves to infer accurate geophysical properties while using reflected GNSS signals. The Fresnel coefficient of circularly polarized waves being expressed as a linear combination of horizontal, \(h\), and vertical, \(v\) was discussed by [46]. Fresnel coefficients use the transformation matrix shown in Equation (4):  
\[\left[ \begin{array}{cc}\mathcal{R}_{ll}&\mathcal{R}_{lr}\\ \mathcal{R}_{rl}&\mathcal{R}_{rr}\end{array} \right]=\frac{1}{2}\left[ \begin{array} []{cc}\mathcal{R}_{hh}+\mathcal{R}_{vv}&\mathcal{R}_{hh}-\mathcal{R}_{vv}\\ \mathcal{R}_{hh}-\mathcal{R}_{vv}&\mathcal{R}_{hh}+\mathcal{R}_{vv}\end{array} \right] \tag{4}\]The subscripts on both sides of Equation (4) represent the change in the wave polarization from the latter to the former subscript. For example, \(r\)l represents that the left-hand circular polarization wave changed to right-hand circular polarization. The linear polarized Fresnel coefficient of the horizontal \(\mathcal{R}_{hh}\) and vertical \(\mathcal{R}_{ev}\) direction can be expressed as Equation (5) [47]:  
\[ \begin{array}{ll}\mathcal{R}_{hh}(\theta)&=\frac{\cos\theta-\sqrt{\varepsilon _{r}-\sin^{2}\theta}}{\cos\theta+\sqrt{\varepsilon_{r}-\sin^{2}\theta}}\\ \mathcal{R}_{ev}(\theta)&=\frac{\varepsilon_{r}\cos\theta-\sqrt{\varepsilon _{r}-\sin^{2}\theta}}{\varepsilon_{r}\cos\theta\sqrt{\varepsilon_{r}-\sin^{2} \theta}}\end{array}  \tag{5}\]  
where \(\theta\) is the signal incident angle and \(\varepsilon_{r}\) denotes the dielectric constant, which implies an electrical property of insulating material--a dielectric. The constant is the main parameter that is used to infer soil moisture using GNSS-R. With the inverted Fresnel equation for horizontal polarization, the dielectric constant can be simply expressed as Equation (6) in terms of the surface reflectivity and incident angle, as derived in [48]:  
\[\varepsilon=\sin^{2}\ \theta+\left[\cos\theta\Bigg{(}\frac{-1-\sqrt{\Gamma}}{ \sqrt{\Gamma}-1}\Bigg{)}\right]^{2} \tag{6}\]  
For CYGNSS data, the dielectric constant is computed using Equation (6) and it accumulated on the EASE grid. The detail process will be addressed in the following section.