# High Spatio-Temporal Resolution CYGNSS Soil Moisture Estimates Using Artificial Neural Networks
## 2 Theoretical Background
### Inversion of the Bistatic Radar Equations  
An ideal GNSS-R based SM retrieval approach would rely on inversion of the bistatic radar equations to acquire the surface reflectivity. The surface reflectivity would be corrected for the vegetation cover and surface roughness effects to obtain a Fresnel reflection coefficient. Fresnel reflection coefficient could then be related to SM with the help of Fresnel reflection equations. For cases where specular reflections are fully dominant, the coherent component of the bistatic received power can be written as follows [11; 29; 30; 31]:  
\[P_{RL}^{coh}=\left(\frac{\lambda}{4\pi}\right)^{2}\frac{P_{t}G_{t}G_{r}}{(r_{ st}+r_{sr})^{2}}\Gamma_{RL}(\theta_{i}), \tag{1}\]  
where \(P_{RL}^{coh}\) denotes the coherently received power. The subscripts \(R\) and \(L\) stand for the right-hand circularly polarized (RHCP) GNSS transmit antenna and the LHCP downward-looking GNSS-R antenna, respectively [11]. \(\lambda\) is the free space wavelength, \(P_{t}\) is the peak power of the transmitted GNSS signals, \(G_{t}\) is the gain of the transmitter antenna, \(G_{r}\) is the gain of the receiver antenna. \(r_{st}\) is the distance between the specular reflection point and the GNSS transmitter, while \(r_{sr}\) is the distance between the specular reflection point and the GNSS-R receiver. \(\Gamma_{RL}(\theta_{i})\) denotes the specular reflectivity at a local incidence angle of \(\theta_{i}\). The incoherent component of the bistatic received power can be written as follows [32; 33]:  
\[P_{RL}^{inc}=\frac{(\lambda)^{2}}{(4\pi)^{3}}\frac{P_{t}G_{t}G_{r}}{r_{st}^{2} r_{sr}}\sigma_{RL} \tag{2}\]  
where \(P_{RL}^{inc}\) denotes the bistatic received power due to the the diffuse scattering over the surface. \(\sigma_{RL}\) is the bistatic radar cross section (BRCS) in m\({}^{2}\). BRCS can be further defined as follows:  
\[\sigma_{RL}=A_{s}\sigma_{RL}^{0}, \tag{3}\]  
where the quantity \(A_{s}\) is the contributing surface area (frequently called the glistening zone [11]), from where the diffuse scattering originates. \(\sigma_{RL}^{0}\) is the normalized BRCS (NBRCS), which includes the spreading loss and the path-dependent phase terms for diffuse mechanisms (such as single scattering or multi-scattering) in the various multi-path directions for the scattering particles (mainly due to the vegetation canopy) and the surfaces (topography and roughness) [31]. The bistatic received signals are assumed to be dominated by the coherent reflections when the surface is relatively flat (no topographic relief) and smooth (weak roughness), having no or non-heavy vegetation cover [11; 12; 22; 28; 34]. The surface reflectivity in this case can be obtained by directly solving (1) for \(\Gamma_{RL}(\theta_{i})\), as shown below:\[\Gamma_{RL}(\theta_{i})=\left(\frac{4\pi}{\lambda}\right)^{2}\frac{P_{RL}^{coh}(r_{ st}+r_{sr})^{2}}{P_{t}G_{t}G_{r}}. \tag{4}\]  
Furthermore, it can also be computed by substituting Equation (1) into Equation (2) with \(P_{RL}^{coh}=P_{RL}^{inc}\) (i.e., equating the right-hand-sides of two equations to each other) and obtaining \(\Gamma_{RL}(\theta_{i})\) as a function of \(\sigma_{RL}\)[35] as follows:  
\[\Gamma_{RL}(\theta_{i})=\frac{\sigma_{RL}(r_{st}+r_{sr})^{2}}{(4\pi)r_{st}^{2 }r_{sr}^{2}}. \tag{5}\]  
Above \(\Gamma_{RL}\) calculation (correction for the path loss and the \(4\pi\) term) functions as a correction to \(\sigma_{RL}\) for coherency assumption, whereas \(\sigma_{RL}\) is originally computed assuming incoherency. After obtaining the surface reflectivity, \(\Gamma_{RL}(\theta_{i})\), by using either (4) or (5), the Fresnel reflection coefficient, \(\Re_{RL}(\theta_{i})\), should be derived from \(\Gamma_{RL}(\theta_{i})\) for SM retrieval. This is because \(\Re_{RL}(\theta_{i})\) is mainly driven by the moisture content of the soil (SM) [4].