## Theory and Methods  
### The Bistatic GNSS-R Soil Moisture Retrieval Method  
The GPS satellite, ground surface, and receiver constitute a bistatic radar system as described in Figure 1. The right-handed circularly polarized antenna receives the direct signal and the left-handed circularly polarized antenna receives the reflected signal. The soil reflectivity is obtained by measuring the power of the reflected GPS signal. In the meanwhile, the surface roughness causes scattering from a glistening zone that contributes to non-coherent power around the specular reflection point. As the roughness increases, the scattering occurs and the incoherent component of the reflected signal increases. For perfect flat surfaces, Fresnel reflection is satisfied and the received power are coherent. As we assumed a smooth surface in this study, the power we received is predominated with LHCP coherent component. Then the soil moisture retrieval method using reflected signal power is based on the inversion of the bistatic radar equation:  
\[P_{lr}^{c}=\frac{P_{t}G_{t}}{4\pi(R_{st}+R_{rs})^{2}}\frac{G_{r}\lambda^{2}}{4 \pi}\Gamma_{lr} \tag{1}\]  
where subscript \(lr\) represents the scattering when the satellite incident signal is right-hand polarized and inverts the polarization to LH after surface reflection, \(P_{t}\) is the power of the transmitted signal, \(G^{t}\) is the transmitter antenna gain, \(G^{r}\) is the gain of the receiver antenna, and \(\lambda\) is the wavelength (19.042 cm for GPS L1 signal). \(R_{rs}\) and \(R_{st}\) is the distance between the receiver, the specular point, and the satellite respectively. \(\Gamma_{lr}\) is the power reflectivity of the reflecting surface. The term \(\Gamma_{lr}\) in Equation (1) (smooth surface) decreases due to increasing roughness, which can be written as [48]:  
\[\Gamma_{lr}(\theta)=\big{|}R_{lr}(\theta)\big{|}^{2}\chi(z) \tag{2}\]  
where \(R_{lr}\) is the Fresnel reflection coefficient, \(\chi(z)\) is the probability density function of the surface height. In the condition of the flat surface \(\chi(z)=1\), the reflectivity \(\Gamma_{lr}\) becomes the amplitude squared of the Fresnel reflection coefficient \(R_{lr}\). Combining (1) and (2), the processed SNR of peak power can be written as:  
\[SNR_{peak}^{refl}=\frac{P_{lr}^{c}G_{p}}{P_{n}}=\frac{P_{r}^{t}G^{t}G^{r} \lambda^{2}G_{p}}{{(4\pi)}^{2}{(R_{st}+R_{rs})}^{2}P_{n}}|R_{lr}|^{2} \tag{3}\]  
where \(P_{n}\) is the noise power and \(G_{p}\) is the processing gain due to the de-spread of the GPS C/A code. The Fresnel reflection coefficient \(R_{lr}\) can be expressed as linear polarization modes [49]:  
\[R_{lr}=R_{rl}{=}\frac{1}{2}(R_{vv}-R_{hh}) \tag{4}\]  
where \(R_{hh}\) and \(R_{vv}\) are the Fresnel coefficients for horizontal and vertical polarization [50]:  
\[R_{hh}(\theta)=\frac{\cos\theta-\sqrt{\varepsilon_{r}-\sin^{2}\theta}}{\cos \theta+\sqrt{\varepsilon_{r}-\sin^{2}\theta}} \tag{5}\]  
\[R_{vv}(\theta)=\frac{\varepsilon_{r}\cos\theta-\sqrt{\varepsilon_{r}-\sin^{2} \theta}}{\varepsilon_{r}\cos\theta+\sqrt{\varepsilon_{r}-\sin^{2}\theta}} \tag{6}\]  
where \(\theta\) is the incident angle, in which \(\varepsilon_{r}\) is the dielectric constant of the surface, \(\varepsilon_{r}=\varepsilon/\varepsilon_{0}-j60\lambda\sigma\). \(\varepsilon_{0}\) is the free-space permittivity, \(\sigma\) is the electric conductivity, and \(\lambda\) is the wavelength. In the case of dry terrain or almost dry, the imaginary part of the permittivity can be neglected [51; 52]. With this hypothesis, the real part of the permittivity can be obtained from Equations (3)-(6), when the reflected signals are received [51]. Here the input variables are \(SNR_{refle}^{peak}\), \(P_{n}\), \(G_{r}\), \(\theta\) respectively as shown in Figure 2.  
Figure 1: Bistatic radar geometry. In Figure 2, the soil reflectivity is obtained by calculating the correlation power of both the reflected signal and the direct signal. In this study, a coherent integration (1 ms) was determined by the length of the GPS PRN (pseudorandom noise) code sequence.