# Multilayer Model in Soil Moisture Content Retrieval Using GNSS Interferometric Reflectometry
## 2 Theory  
Both the direct signal from line-of-sight propagation and the signal reflected off the reflecting surface may be picked up by the geodetic GNSS receiver. The reflecting surface affects these reflected GNSS signal qualities, and effective signal extraction and processing may be utilized to depict the physical characteristics of the reflective surface. When two electromagnetic waves of the same frequency and phase arrive at the same location in space, they will interfere with one another. In ground-based GNSS, there is little change in frequency between direct and reflected signals. An SNR data model for the interference of a single reflected signal and a direct signal is shown in the literature [8]. The direct signal is separately interfered with by the signal reflected by each layer of soil, and the SNR data model of multilayer soil interference signal is generated as Equation (1) shows. \[SNR=A_{d}^{2}+\sum_{i=1}^{n}A_{m_{i}}^{2}+2\sum_{i=1}^{n}A_{d}A_{m_{i}}\cos\varphi _{i} \tag{1}\]  
where \(A_{d}\) and \(A_{m_{i}}\) represent the amplitude of the direct and reflected GNSS signals from the \(i\)th layer. Moreover, \(\varphi_{i}\) represents the phase difference between the direct and reflected signals from the \(i\)th layer. The signal reflected by the multi-layer soil is picked up by the GNSS receiver, which interferes with the directed signal; the oscillations naturally occur as a result. The schematic representation of the GNSS signal's propagation route via the multi-layer soil model is shown in Figure 1. Figure 1 demonstrates that when a GNSS signal interacts with the dielectric surface, both penetration and reflection take place. The signal will continue to transmit after being reflected. In general, there are two types of soil surface moisture distributions: sinking types (such as rainfall), in which the humidity falls with depth; and reverse osmosis types (such as river tidal flats), in which the humidity increases with depth. Satellite electromagnetic waves may be thought of as plane waves for a reflection route. The travel delays of the direct and reflected signals are equal to the differences between  
Figure 1: Schematic diagram of the propagation path of the GNSS signal in the multi-layer soil model. the red and green lines and may be expressed according to [22], based on the geometric connection in Figure 1:  
\[\delta_{i}=\frac{H_{i}}{\sin\theta}-\frac{H_{i}}{\sin\theta}\cos 2\theta_{i} \tag{2}\]  
where \(H\) represents the vertical height of the GNSS receiver antenna from the soil surface, and \(\theta\) represents the elevation angle of the satellite. After simplifying Equation (2), we can have:  
\[\delta_{i}=2H_{i}\sin\theta_{i} \tag{3}\]  
the oscillation frequency of the interference signal SNR with the sine of the elevation angle as the horizontal axis can be calculated by the following formula [22]  
\[f_{m_{i}}=\frac{d(\delta_{i}/\lambda)}{d\sin\theta_{i}}=\frac{2H_{i}}{\lambda} \tag{4}\]  
where \(\lambda\) represents the wavelength of the electromagnetic wave carrying the GNSS signal. Equation (4) shows that the vertical height of the GNSS antenna above the soil surface affects the oscillation frequency of the interference signal. For generally level ground, its roughness has minimal impact on the signal frequency, and the primary interference comes from the nearby flora. Refraction and reflection occur when an electromagnetic wave illuminates the boundary between two media types. The following are the definitions of the vertical polarization's Fresnel reflection and refraction coefficients:  
\[\Gamma_{\perp}=\frac{\eta_{2}\cos\theta_{d}-\eta_{1}\cos\theta_{\tau}}{\eta_{ 2}\cos\theta_{d}+\eta_{1}\cos\theta_{\tau}} \tag{5}\]  
and  
\[T_{\perp}=\frac{2\eta_{2}\cos\theta_{d}}{\eta_{2}\cos\theta_{d}+\eta_{1}\cos \theta_{\tau}} \tag{6}\]  
where \(\eta_{1}=\sqrt{\mu_{1}/\epsilon_{1}}\) and \(\eta_{2}=\sqrt{\mu_{2}/\epsilon_{2}}\) represent the electromagnetic wave impedance coefficient, respectively.