# Impact of the Elevation Angle on CYGNSS GNSS-R
## 3 Coherent and Incoherent Scattering in GNSS-R  
The scattered electromagnetic field over land and ocean surfaces is composed of both a coherent and an incoherent contribution in different proportions depending on the dielectric and geometrical properties of the scattering medium, and the directions of incoming and outgoing electromagnetic waves. In the specular direction, the p-polarized bistatic scattering coefficient \(\sigma_{\rm p}^{0}\)[39] can be assumed to be composed of an incoherent \(\sigma_{\rm p,inch}^{0}\) and a coherent \(\sigma_{\rm p,coh}^{0}\) term as it follows [40]:  
\[\sigma_{\rm p}^{0}=\sigma_{\rm p,inch}^{0}+\sigma_{\rm p,coh}^{0} \tag{1}\]  
The estimation of \(\sigma_{\rm p}^{0}\) based on theoretical models requires several hypotheses, such as those regarding the dielectric permittivity of the lower media (land, ocean,...), and the distribution of the surface elevation slopes \(\rm Zs=\sigma_{RMS}/L\)[41]. The root mean square (RMS) of the surface height variation \(\sigma_{RMS}\) and the surface correlation length \(L\) describe the statistical variation of the random component of the surface height relative to a reference surface, which determines the degree of surface roughness. The incoherent scattering term is dominant for moderate-to-strong surface roughness conditions, while the scattering is completely coherent for a flat surface. The power of the incident and the scattered waves is related through the reflectivity of the scattering medium. The reflectivity is a physical property of the medium. It depends on its dielectric properties and the surface roughness. The reflectivity can be indirectly linked to, e.g., the SMC, and it is used as a fundamental observable in GNSS-R. GNSS-R is a sort of bistatic radar that allows for the collection of scattered GNSS signals along the specular direction. GNSS satellites transmit relatively low-power signals, and the scattering is strong only over an area around the nominal specular point (\(\theta_{\rm e,s}=\theta_{\rm e,i}=\theta_{\rm e}\)). GNSS signals are emitted with right-hand circular polarization (RHCP), although with a certain degree of ellipticity. After scattering over a surface, they become left-HCP (LHCP) for elevation angles \(\theta_{\rm e}\) higher than the Brewster angle, while they are mainly RHCP for lower \(\theta_{\rm e}\). Additionally, the scattering over biomass introduces a significant degree of depolarization. RHCP and LHCP reflected signals should be considered simultaneously in this case. The signal power collected by a GNSS reflectometer for LHCP (e.g., CyGNSS) is expressed by the so-called delay Doppler maps (DDMs), that can be derived under the bistatic radar equation as it follows [42]:  
\[\left\langle\left|Y_{\rm r}(\tau,f)\right|^{2}\right\rangle=\frac{\rm P_{T} \lambda^{2}}{\left(4\pi\right)^{3}}\iint\frac{G_{T}G_{R}|\chi(\tau,f)|^{2} \sigma_{LHCP}^{0}}{\rm d}^{2}\overline{\rho} \tag{2}\]where \(\mathrm{P_{T}}\) is the transmitted power, \(\mathrm{G_{T}}\) and \(\mathrm{G_{R}}\) are the transmitter and receiver antenna gains, \(\chi\) is the Woodward ambiguity function (WAF), \(\mathrm{R_{T}}\) and \(\mathrm{R_{R}}\) are, respectively, the ranges from the transmitter and the receiver to the specular point, \(\mathrm{\sigma_{L}^{0}}\) is the bistatic scattering coefficient at LHCP, and \(\overline{\mathrm{\rho}}\) is the positioning vector of the scattering point. GNSS-R LHCP-reflectivity \(\mathrm{\Gamma_{LHCP}}\) includes both coherent \(\mathrm{\Gamma_{LHCP,coh}}\) and incoherent \(\mathrm{\Gamma_{LHCP,incoh}}\) contributions in different proportions depending on satellites' elevation angle, and the biomass, topography, and surface roughness.