## 3 Methodology  
In recent years, different approaches have been addressed for GNSS-R SMC retrieval [26; 32; 33; 38; 46]. These approaches usually employ a modification of the bistatic radar model described in [61; 62]; its basic equation can be expressed as [25; 57]:  
\[P_{pq}^{r}=P_{pq}^{c}+P_{pq}^{i} \tag{1}\]  
In this expression, \(P_{pq}^{r}\) is the scattered power received at the reflectometer, and \(P_{pq}^{c}\) and \(P_{pq}^{i}\) are the received coherent and incoherent power, respectively. The subscripts \(p\) and \(q\) stand for the incident and scattered polarizations, respectively. Regarding this, [46; 63] considered that for certain smooth surfaces under specific conditions, the strongest scattering is received from the coherent contribution, relying on the assumption that scattering decreases quickly as it moves away from the nominal SP location. Natural land surfaces are included in this hypothesis and this is the rationale of working with the coherent contribution of GNSS-R.  
Figure 5: SMAP data for the periods (**a**) 1–12 and (**b**) 14–29 of April 2019 plotted at intervals of 500 samples. We employ CYGNSS level 1 data, which was initially designed to study wind and ice over the oceans. However, the above literature has shown its potential to derive land surface reflectivity, which in turn is tightly coupled with SMC (e.g., [19; 34; 42; 44]). In this scheme, for the purpose of this work, we employ GNSS-R models described in [31; 42; 43; 47] for the modeling of CYGNSS DDM reflectivity estimates. Under this assumption, the bistatic radar equation for the coherent component of smooth surfaces from LHCP GPS bistatic microwave signals observed by a receiver can be written as:  
\[P_{lr}^{c}=\frac{P_{t}\lambda^{2}G_{t}G_{r}}{(4\pi)^{2}(R_{t}+R_{t})^{2}}\Gamma _{l}(\theta) \tag{2}\]  
In this equation, \(P_{lr}^{c}\) is the DDM peak value from the analog scattered power (namely, \(P_{DDM}\)). The subscript \(rl\) stands for a scattering mechanism when the incident RHCP signal is scattered by the surface and inverts the polarization to LHCP at the receiver position. \(\Gamma_{lr}\) is the surface reflectivity from which the SMC might be estimated. Solving for reflectivity, Equation (2) can be expressed in the manner specified by [37], which appears more convenient for processing GYGNSS data:  
\[\Gamma_{lr}(\theta)=\frac{(4\pi)^{2}(P_{DDM}-N)(R_{t}+R_{t})^{2}}{\lambda^{2} G_{t}G_{r}P_{t}} \tag{3}\]  
This is the final expression for a CYGNSS SP reflectivity, where measurements of \(\Gamma_{lr}\) need to be calibrated from instrumental bias [37]. For the CYGNSS data employed in this study, we employ a reflectivity constant bias value of 0.15. In Equation (3), we introduced a new variable, \(N\), which refers to the power noise floor of the DDM. For any surface, in general, but in particular for land surfaces, this observed reflectivity is affected by its corresponding BSR parameter. The implementation of the BSR effect in Equation (3) is:  
\[\Gamma_{lr}(\theta)=\big{|}R_{lr}(\theta)\big{|}^{2}\chi(z) \tag{4}\]  
In this equation, \(R_{lr}(\theta)\) is the Fresnel reflection coefficient and \(\chi(z)\) is the probability function for the surface height z and can be modeled by the Gaussian probability density function [47]. Moreover, when natural surfaces are covered with vegetation, the observed reflectivity is attenuated. Therefore, this contribution [38] must be taken into account in Equation (4) as:  
\[\Gamma_{lr}=\big{|}R_{lr}(\theta_{i})\big{|}^{2}e^{(-2\omega\cos\theta_{i})^{2 }}e^{-2\tau/(\cos\theta_{i})^{2}} \tag{5}\]  
In this equation, \(\tau\) stands for the VOD and \(h=4k^{2}\sigma^{2}\), with \(k=2\pi/\lambda\), where \(\lambda\) is the wavelength of the system, and \(\sigma\) is the standard deviation of the surface [64]. This is the expression for surface reflectivity corrected by BSR and VOD and shows that both effects of soil and vegetation are relevant for SMC determination.