# Multilayer Model in Soil Moisture Content Retrieval Using GNSS Interferometric Reflectometry
## 2 Theory
Moreover, \(\mu_{1}\) and \(\epsilon_{1}\) are the permeability and permittivity of the air above the soil, respectively. \(\mu_{2}\) and \(\epsilon_{2}\) are the permeability and permittivity of the soil, respectively. \(\theta_{d}\) and \(\theta_{\tau}\) represent the incidence angle and the refraction angle, respectively. Furthermore, the Fresnel reflection and refraction coefficients of horizontal polarization can be defined as:  
\[\Gamma_{\parallel}=\frac{\eta_{1}\cos\theta_{d}-\eta_{2}\cos\theta_{\tau}}{ \eta_{1}\cos\theta_{d}+\eta_{2}\cos\theta_{\tau}} \tag{7}\]  
and  
Figure 1.0 (  
The phenomenon of the entire transmission may occur when electromagnetic waves are irradiated into the interfaces of two distinct media types, as can be observed from the literature cited in [3]. The critical angles of total transmission for the vertical and horizontal polarizations differ, and the incidence angle when the two polarizations are transmitted may be computed as follows:  
Figure 1.1  
and  
Figure 1.2  
where \(\alpha\) represents the critical incident angle and is complementary to the satellite elevation angle. They mean the permeability and permittivity of the upper and lower soil when we discuss the signal propagating inside the soil. We substitute the air and soil permittivity and permeability when the soil humidity is from 5% to 40%, \(\alpha_{v}\), generally between 65\({}^{\circ}\) and 85\({}^{\circ}\), and \(\alpha_{h}\) close to 0\({}^{\circ}\). The receiver can only now receive direct signals from line-of-sight propagation due to the entire transmission phenomena of vertical polarization at high incidence angles (i.e., low satellite elevation angles). In order to determine the soil moisture layers, the GNSS signals reflected by distinct soil layers can be independently collected using the vertical linear polarization total transmission characteristic. The interference signal's SNR oscillation waveform, which was captured using a vertically linearly polarized GNSS antenna, is shown in Figure 1.3 The simulation is set to sink mode, and a 20 cm layer of soil moisture is lowered from 30% to 20%. In Figure 2, there is a \"notch\" where the satellite elevation angle is 15\({}^{\circ}\), with its amplitude significantly decreasing in comparison to other height angles. The larger portion of Figure 2 demonstrates the presence of three distinct frequency components relating to the satellite's elevation angle at the problem location. The expanded portion of the picture demonstrates that there are three distinct frequency components related to the satellite's elevation angle at the defect site. We divide the upper and lower sides of Equation (12) by \(\varepsilon_{1}\), and it can be rewritten as:  
Figure 1.4  
Figure 1.5 Vertically polarized interference signal–SNR oscillation waveform. where \(k=\varepsilon_{2}/\varepsilon_{1}\). As the soil moisture drops, the permittivity for the sink mode falls, and the ratio \(k\) rises. The angle at which full transmission takes place will decrease with the drop in \(\sin\ a_{\varphi}\), according to the sine function's monotonicity between \(0^{\circ}\) and \(90^{\circ}\). The interference signal SNR created by the reflected signal and the direct signals at the bottom, middle, and surface soil, respectively, are shown in Figure 2's magnified Sections 1-3. The outcomes of the comparison between the single-layer and multi-layer models are shown in Figure 1.6 In contrast to the multi-layer model, the single-layer model only has one \"notch\" in the SNR of the interference signal. With the aid of the vertical polarization's total transmission characteristic, it is possible to measure the soil moisture with a depth gradient.