# Effective Surface Roughness Impact in Polarimetric GNSS-R Soil Moisture Retrievals
## 2 Theoretical Review
The reflectivity equation in Equation (4) is only valid for the coherent component of the GNSS reflection [41], and Equation (7) is only valid for the incoherent component. Note that for current operational GNSS-R missions (e.g., CYGNSS), the Delay-Doppler Maps (DDM) are generated in real-time on the spacecraft following a total power waveform approach (i.e., performing an incoherent integration), without untangling or separating both coherent and incoherent components. On the contrary, SMAP reflectometer data is a collection of raw IQ data, and all processing is performed on the ground, opening the possibility for coherent component computation following [41; 45]. In this case, the coherent component is calculated by computing the total power waveform and the incoherent component, and then subtracting the values as in [45]. Note that the incident angle dependence has been omitted in the following equations for simplicity:  
\[P_{ip}(\tau,f)=\frac{1}{N}{\sum_{i}}^{N}{|Y_{ip}(\tau,f)-\mu_{ip}|^{2}}, \tag{8}\]  
\[\mu_{p}=\frac{1}{N}{\sum_{i}}^{N}{Y_{ip}(\tau,f)}, \tag{9}\]  
where \(Y_{i}\) is one waveform realization (i.e., coherently integrated at a time interval of 3 ms). \(Y_{ip}\) is the cross-correlation between the signal received and the pseudo-random number identifier of the satellite reflection captured by the SMAP antenna, as detailed in Equation (16) from [46], at a \(p\) polarization. Note that the coherency of the reflection surface decreases as the receiving platform moves across the surface. Hence, the coherent integration time \(N\) should be selected accordingly. Additionally, Equations (8) and (9) can be used to compute the coherent component at a coherent integration time \(M\). This would be beneficial to compare the coherency of the surface at different coherent integration times following:  
\[P_{ip}(\tau,f)=\frac{1}{N}{\sum_{i,i}}^{N}{|Y_{ip}(\tau,f)-\mu_{ip}(\tau,f)|^{ 2}}, \tag{10}\]  
\[P_{c_{p}}(\theta)=\Gamma_{p}(\theta)\cdot\frac{\lambda^{2}P_{t}G_{t}G_{r}}{(4 \pi)^{2}(R_{t}+R_{r})^{2}} \tag{2}\]0  
where \(\mu_{ip}\) is the moving mean of \(Y_{ip}\) of \(M\) samples, and \(M<N\). Note that, in order to implement this technique, the navigation bit sign should be handled [18; 45]. Moreover, for a polarimetric GNSS-R scenario such as SMAP-R, the computation of the Stokes parameters of the signal enables a complete reconstruction and a better understanding of the polarimetric properties of the received signal. The SMAP radar receiver collects data in two polarizations, H and V, and generates four DDMs, with the first two DDMs representing the components at the H and V antennas, respectively. The other two DDMs are the real and imaginary parts of the complex correlation between H and V, respectively. The Stokes parameters are then computed by averaging each of the DDMs over time. The processing methodology for SMAP-R to derive the Stokes parameters is detailed in [18; 47]. Following Equations (8) and (9), we can compute the incoherent part of the four Stokes parameters. Instead of the common incoherent integration used to retrieve the total power waveform, we compute the incoherent component by applying the variance theorem [45]:  
\[P_{c_{p}}(\theta)=\Gamma_{p}(\theta)\cdot\frac{\lambda^{2}P_{t}G_{t}G_{r}}{(4 \pi)^{2}(R_{t}+R_{r})^{2}} \tag{2}\]1  
\[P_{c_{p}}(\theta)=\Gamma_{p}(\theta)\cdot\frac{\lambda^{2}P_{t}G_{t}G_{r}}{(4 \pi)^{2}(R_{t}+R_{r})^{2}} \tag{2}\]2  
\[P_{c_{p}}(\theta)=\Gamma_{p}(\theta)\cdot\frac{\lambda^{2}P_{t}G_{t}G_{r}}{(4 \pi)^{2}(R_{t}+R_{r})^{2}} \tag{2}\]3  
\[P_{c_{p}}(\theta)=\Gamma_{p}(\theta)\cdot\frac{\lambda^{2}P_{t}G_{t}G_{r}}{(4 \pi)^{2}(R_{t}+R_{r})^{2}} \tag{2}\]4  
where the terms \(E_{RH}\) and \(E_{RV}\) are the complex electrical fields received by the H and V polarized SMAP antennas. The incoherent component is computed by means of the variance of the Stokes parameters in time, according to Equations (8) and (9).