# Effective Surface Roughness Impact in Polarimetric GNSS-R Soil Moisture Retrievals
## 4 Results
### Multiple Polarization Roughness Estimates  
The methodology from Section 3.2 is implemented using the coherent and incoherent components from Figure 1, and its corresponding Stokes parameters. The surface rms height (\(\sigma_{h}\)) is estimated from Equation (24), and the slope rms (\(s\)) is estimated from Equation  
Figure 1: **(a)** Total power waveform reflectivity, **(b)** Coherent component reflectivity about \(10\) dB smaller than the total power waveform, and **(c)** difference between **(a,b)**. the average value. Still, this ubRMSD is two times lower than the slope rms standard deviation. This finding confirms that the assumption of negligible cross-polarization does not significantly bias the results, but it should be taken into account for very accurate retrievals. Note that, simulation studies on polarimetric roughness have studied the effects of scattering in HR, VR, LR, and RR, showing that the real and imaginary parts of the Fresnel reflection coefficient at h and v polarization are slightly affected by the surface roughness [58]. Additionally, despite \(\sigma_{h}\) and \(s\) represent different physical phenomena, the correlation between them is noticeable. In this regard, \(\sigma_{h}\) and \(s\) are related by:  
\[\sigma_{h}=a\cdot\ln(s)+b, \tag{26}\]  
where \(a\) and \(b\) are two coefficients that can be estimated via least squares. Table 6 summarizes the coefficients (\(a\) and \(b\)), the correlation coefficient, and the root mean square difference (RMSD) of the fit. Note that, the units of \(\sigma_{h}\) are centimeters. The results are highly consistent for the four different parameters, with a high correlation coefficient, and very similar parameters, \(a\), and \(b\), that link both magnitudes. This similarity between both methods is linked to the fact that the roughness effect, simulated via the coherent or incoherent component, is similar. The scattered wave coherently integrated for 30 ms, and modeled via the PO model produces a coefficient that is proportional to the incoherently averaged waveform modeled using the GO model. This implies that the signal at 30 ms coherent integration is neither coherent nor incoherent, but a mix of both since the GO model has a formulation to relate to the PO model at the selected integration time. It is worth noting that the RMSD between the fitted GO and the PO model is approximately 0.4 cm, equivalent to around 8% of the average rms height. This means that the difference between the PO and the GO models leads to an error of approximately 8% in the retrieved rms height. To further analyze the similarity between the rms height and the rms slope, both retrieved magnitudes are compared to the actual DEMSTD and DEMSLPSTD described in Table 1. Note that, DEMSTD is commonly known as the surface height rms, and DEMSLPSTD is the surface slope rms. Given its similarity with other roughness coefficients, we will only compute this for the L polarization. The scatter plot for both \(\sigma_{h}\) and \(s\) are shown in Figure 4.  
In order to relate the DEM rms height (e.g., DEMSTD) to the PO model, we would require a vertical and horizontal resolution of the DEM better than the signal wavelength ([42], p. 423), which is not feasible. However, several studies dealing with different DEM resolutions conclude that a resolution change basically produced a scale on the rms height product [55; 59] (e.g., if one wants an equivalent resolution of 1 cm from a 9 km product, the resultant rms height shall be scaled by the ratio of resolutions). In our case, the base resolution of the DEM used here is 30 m, and the standard deviation has been computed and upscaled to larger patches of 9 km.