# Multilayer Model in Soil Moisture Content Retrieval Using GNSS Interferometric Reflectometry
## 3 Simulation Analysis
### Added Noise  
The interference signal SNR recorded in the experiment will really be impacted by the system and environmental noises. We introduced noise into the measured SNR to bring it closer to the actual experiment. A mix of white noise and flicker noise seems to be the best model for the noise characteristics of the GPS, according to the [23; 24]. The black line in Figure 9 depicts the SNR of the interference signal after adding a group of Gaussian white noise with an average value of 1 dB. According to Figure 9, there are several burrs in the time series of the interference signal SNR after the addition of Gaussian white noise, which will impair the measurement of the notch of the vertical signal transmission. In this work, the interference signal's  
Figure 8: The power reflectivity results for the two modes with different layers. Figure 9: SNR time sequence of the interference signal with Gaussian white noise and its envelope by Hilbert transform. envelope is obtained using the Hilbert transform to lessen the impact of burr. Hilbert transforms take the unprocessed signal as:  
\[x(t)=A(t)cos(2\pi ft+\varphi_{r}(t))=A(t)cos(wt+\varphi_{r}(t)) \tag{16}\]  
Hilbert transform:  
\[\hat{x}(t)=H[x(t)]=\frac{1}{\pi}\int\limits_{-\infty}^{\infty}\frac{x(\tau)}{t- \tau}d\tau \tag{17}\]  
and its frequency characteristic is:  
\[\hat{X}(jw)=X(jw)H(jw) \tag{18}\]  
where  
\[H(jw)= \begin{cases}-jw\geq 0\\ +jw\leq 0\end{cases}  \tag{19}\]  
The Hilbert transform, which is identical to an orthogonal filter, shifts the phase by \(-90^{\circ}\) for all positive frequency components and \(+90^{\circ}\) for all negative frequency components, as shown by the equation above, but the amplitude remains constant. As a result, an analytical signal that has the original reflected oscillation signal acting as its real component and the signal's Hilbert transformation as its imaginary component may be created. \[\hat{x}(t)=x(t)+j\hat{x}(t) \tag{20}\]  
substitute Equation (16) into Equation (20):  
\[\hat{x}(t)=\text{A}(t)\cos(wt+\varphi_{r}(t))+jA(t)sin(wt+\varphi_{r}(t))=A(t) e^{j(wt+\varphi_{r}(t))}=A(t)e^{j\varphi_{r}(t)}e^{jwt} \tag{21}\]  
where \(e^{jwt}\) represents the complex carrier signal, and \(e^{j\varphi_{r}(t)}\) represents the complex envelope. Finally, the envelope signal can be obtained by calculating the absolute value of the analytic signal. The SNR envelope of the interference signal, as determined by the Hilbert transform, is shown as a red line in Figure 9. It is clear that the signal SNR following the Hilbert transformation preserves the frequency and phase properties of the original interference signal while lessening the impacts of roughness on the amplitude. The signal through the Hilbert transform has a stronger resemblance to the SNR sequence of the interference signal without noise than in Figure 4.  
The high-frequency components of the interference signal will be present given the presence of noise. In this study, distinct frequency components are extracted and high-frequency components are eliminated using the empirical mode decomposition (EMD) approach. EMD breaks down the signal into a number of intrinsic mode functions based on the time scale of the local aspects of the signal (IMF). The results of the EMD decomposition and its spectral properties are shown in Figure 10. The 4 panels on the right represent the matching spectrum, while the 4 panels on the left represent the IMF1, 3, 6, and 7 components. The spectrum shows that IMF1's high-frequency component includes a large number of spectral peaks. Low-frequency components include IM3, 6, and 7, of which IMF6 and 7 should be direct current (DC) components due to their lower frequencies. IMF3's spectrum peak emerges at 5 m which shows that the soil surface is where the GNSS signal for this component is reflected. Additionally, other IMF components did not exhibit spectral peaks at approximately 5 m, and the SNR of the IMF3 exhibits a \"notch\" when the satellite elevation angle is around \(18^{\circ}\).