# High Spatio-Temporal Resolution CYGNSS Soil Moisture Estimates Using Artificial Neural Networks
## 2 Theoretical Background
### Inversion of the Bistatic Radar Equations
\(\Re_{RL}(\theta_{i})\) can be calculated by correcting \(\Gamma_{RL}(\theta_{i})\) for the vegetation [36] and surface roughness effects assuming the rough surface under the vegetation to be flat and smooth and to follow [PERSON]'s approximation with a Gaussian height distribution [37] as follows:  
\[\Gamma_{RL}(\theta_{i})=\Re_{RL}(\theta_{i})^{2}\gamma^{2}exp(-hcos^{2}(\theta _{i})), \tag{6}\]  
where the exponential term in (6) accounts for the surface roughness effects. The _h-parameter_ is assumed linearly related to the root-mean-square-height surface roughness [38], as follows:  
\[h=2k_{0}s, \tag{7}\]  
where \(k_{0}\) is the angular wavenumber and \(s\) is the surface root-mean-squared (rms) height. The squared transmissivity, \(\gamma^{2}\), in (6) accounts for the wave attenuation as the waves propagate from the top of the vegetation canopy to the ground and then from the ground to the top of the vegetation cover again. The transmissivity depends on the vegetation optical depth, \(\tau\), and the incidence angle as follows:  
\[\gamma=exp(-\tau sec(\theta_{i})). \tag{8}\]  
\(\gamma\) and \(\tau\) (and parameter \(b\) in (9)) are dependent on the electromagnetic signals' polarization, but the polarization notation is waived here for simplicity. The vegetation optical depth has been previously related to vegetation water content (VWC) and a land cover-based proportionality value (\(b\)) that depends on both the vegetation structure and the microwave frequency in the literature [36], and this approach has been successfully applied to the coarse spatial resolution SMOS/SMAP missions [6; 7], as shown below:  
\[\tau=b\times VWC. \tag{9}\]  
VWC was empirically derived from normalized difference vegetation index (NDVI) by the SMAP mission with additional utilization of the minimum and maximum NDVI values of ten-year time-series, and the _stem factor_ parameter that comes from a land cover-based lookup table (LUT) [39] as follows:  
\[VWC=(1.9134\times NDVI^{2}-0.3215\times NDVI)+stemfactor\times\frac{NDVI_{max }-NDVI_{min}}{1-NDVI_{min}}. \tag{10}\]  
Equation (6) could be solved for the Fresnel reflection coefficient, \(\Re_{RL}(\theta_{i})\), substituting equations from (6) to (10). It could then be related to the soil dielectric constant, \(\epsilon_{r}\), with the help of the Fresnel reflection equations as follows:  
\[P_{RL}^{inc}=\frac{(\lambda)^{2}}{(4\pi)^{3}}\frac{P_{t}G_{t}G_{r}}{r_{st}^{2} r_{sr}}\sigma_{RL} \tag{2}\]0where  
\[P_{RL}^{inc}=\frac{(\lambda)^{2}}{(4\pi)^{3}}\frac{P_{t}G_{t}G_{r}}{r_{st}^{2} r_{sr}}\sigma_{RL} \tag{2}\]1  
\[P_{RL}^{inc}=\frac{(\lambda)^{2}}{(4\pi)^{3}}\frac{P_{t}G_{t}G_{r}}{r_{st}^{2} r_{sr}}\sigma_{RL} \tag{2}\]2  
The soil dielectric constant, \(\epsilon_{r}\), can be related to SM with the help of a ground dielectric mixing model by using soil texture information. A number of dielectric mixing models has been developed in the literature such as [PERSON] (2014), [PERSON] (2016), or [PERSON] (2017) models. It should be noted that some of these models might require the use of additional geophysical parameters such as soil temperature ([PERSON], 2016).