## 2 Materials and Methods
### GNSS-IR Technology for Inversion of Soil Moisture  
SNR observations are the core computational data of GNSS-IR technology, an index describing the signal quality of GNSS antennas. It is mainly influenced by the combination of elements, such as receiver antenna gain, satellite altitude angle, and multipath effect [42].  
The GNSS receiver receives both direct and reflected signals from GNSS satellites. The continuous movement of GNSS satellites makes the GNSS direct reflection signal constantly change, which makes the characteristic parameters of the interference waveform constantly change over time, and ordinary geodetic receivers will record these changes in the form of SNR [43]. Therefore, the study of SNR can estimate soil moisture through the change of the characteristic parameters of the interference effect. The ground multipath error model is shown in Figure 1.  
From Figure 1, the phase difference between the direct and reflected signals of the GNSS satellite can be deduced as:  
\[\alpha=\frac{4\pi h}{\lambda}\sin\beta \tag{1}\]  
where \(h\) denotes the vertical height of the GNSS receiving antenna from the ground and \(\beta\) denotes the angle \(\lambda\) between the GNSS signal and the ground surface is the L carrier wavelength. Further study reveals that the SNR can be expressed in terms of direct and reflected signals as follows.  
\[SNR^{2}=Q_{d}^{2}+Q_{m}^{2}+2Q_{d}Q_{m}\cos\alpha \tag{2}\]  
In Equation (2), \(\alpha\) denotes the phase difference between the direct and reflected signals of GNSS satellites, and \(Q_{d}\) and \(Q_{m}\) denote the amplitudes of the direct and reflected signals of GNSS satellites, respectively. [PERSON] et al. [44] found that after removing the direct signal amplitude \(Q_{d}\), only the reflected signal amplitude \(Q_{m}\) is retained in the SNR observations. There is a certain sine or cosine relationship between \(Q_{m}\) and \(\sin\beta\), which can be expressed as:  
\[SNR_{D}=Q_{m}\bigg{(}\frac{4\pi h}{\lambda}\sin\beta+\varphi_{m}\bigg{)} \tag{3}\]  
In Equation (3), \(\varphi_{m}\) denotes the relative delayed phase. In the case where \(SNR_{D}\) is known, we can use the Lomb-Scargle spectral analysis transform to find the frequency \(\frac{4\pi h}{\lambda}\) and then solve for the magnitude \(Q_{m}\) and the relative phase \(\varphi_{m}\) using a least squares fit.  
Figure 1: Ground multipath error model.  
A strong correlation between \(\varphi_{m}\) and soil moisture values was found by [PERSON] et al. [45], which is the inverse the best parameter for inversion of surface soil moisture.  
Based on this, [PERSON] et al. [46] smoothed phase \(\varphi_{m}\) using a moving average filter to remove the expected phase variation due to vegetation and add the residual water content in the soil (\(SMC_{r}\)) to produce a phase \(\varphi_{sm}\) that reflects only the variation in soil moisture. The phase \(\varphi_{sm}\) was then related to soil moisture \(SM\) is expressed as follows:  
\[SM=S_{\varphi_{sm}}+SMC_{r} \tag{4}\]  
In Equation (4), \(S\) is the expected slope (between soil moisture and phase), \(\varphi_{sm}\) indicates the phase change due to soil moisture, and \(SMC_{r}\) is available through public data [47].