## 3 Methodology
As mentioned above, in this study we employ and compare CYGNSS SMC retrieval using both ICESat-2 and SMAP BSR estimates. Regarding the VOD parameter \(\tau\), the main factors influencing this variable are the signal polarization and the incidence angle (\(\theta_{i}\)) [46]. Note that, since for low vegetated areas the VOD parameter \(\tau\) holds a linear relation with the VWC, it can be related to the Normalized Difference Vegetation Index (NDVI), or other spectral indices [30; 54; 55]. Despite many efforts that have been done for evaluating VWC by means of passive microwave data [11; 65; 66], the VOD parameter \(\tau\) remains difficult to estimate. This is due to its linear relation to VWC, by means of a scaling parameter, which is function of the type and structure of the canopy, and the polarization and wavelength of the signal, which is estimated experimentally [65; 67]. Unfortunately, the problem of determining VOD is to find a proper relation with VWC. The authors in [65] revealed that, independently of the kind of vegetation, only a small variation of a certain scale factor produces an insignificant error for VOD in the L band range (18-21 cm). In any case, VOD is correlated to VWC under experimental circumstances, which is the case in SMAP/Sentinel 1 L2 data, and the use of this variable is preferred. Further studies on this hypothesis can be found in [35; 46], showing that leaves of vegetation are almost transparent for the L band, and attenuation is mainly due to other vegetation constituents, such as branches or trunks. Then, the reflectivity corrected from BSR and VOD (equivalent to VWC), i.e., \(R_{tr}(\theta)\), is employed to estimate soil permittivity (\(\varepsilon_{r}\)) by means of the Fresnel reflection coefficients, that are related to the reflectivity linear polarization modes [31; 45; 47; 51]:  
\[R_{lr}=R_{tl}=\frac{1}{2}(R_{vv}-R_{th}) \tag{6a}\] \[R_{rr}=R_{ll}=\frac{1}{2}(R_{vv}+R_{hh}) \tag{6b}\]  
In these equations, \(R_{vv}\), and \(R_{hh}\) are the Fresnel coefficients for vertical and horizontal polarization, respectively. The subscripts, lr and \(rl\) hold for circular cross-polarized reflections, while \(rr\) and \(ll\) for co-polarized reflections. For soil surfaces, the Fresnel reflectivity (\(R_{vv}\) and \(R_{hh}\)) is function of soil permittivity or dielectric constant \(\varepsilon_{r}\) and the incidence angle (\(\theta\)), from which SMC is retrieved, i.e., \(SMC=f\) (\(\varepsilon_{r}\)). The conversion from the Fresnel reflectivity to the real part of permittivity (\(\varepsilon_{r}\)) has no direct solution. However, [31] suggests two solutions that allow solving either for \(R_{vv}\) or \(R_{hh}\), based on the models given in [45] or [67], respectively. These models are suitable for low incidence or high elevation angles, which is the case of satellite viewing properties and, particularly, for observations of GYGNSS SPs. Once the dielectric constant is retrieved, the volumetric SMC can be derived from the Topp model [68; 69]. The advantage of this model is that clay, sand, and silt proportions, i.e., textural properties of the soil, are unnecessary. In summary, the SMC retrieval methodology suggested in this study is presented in Figure 6 as follows:  
1. Values for Equation (3) can be obtained from GYGNSS products. 2. BSR values are obtained from ICESTA2 or SMAP products. 3. VWC or VOD are obtained from the SMAP products. 4. Surface reflectivity \(R_{tr}(\theta)\) is corrected by means of the previous values using Equation (5). 5. \(R_{vv}\) and \(R_{hh}\) Fresnel coefficients (Equation (6)) are solved for low incidence angles (\(\theta_{\text{i}}<35^{\circ}\)) where \(|R_{vv}|=|R_{hh}|\). 6. The real part of permittivity (\(\varepsilon_{r}\)) is solved by either the method of [45] for \(R_{vv}\) or by the method of [67] for \(R_{hh}\). 7. SMC is finally derived applying the Toppmodel [68]. Figure 6: Flowchart of GNSS-R SMC retrieval methodology.