# Analysis of Key Issues on GNSS-R Soil Moisture Retrieval Based on Different Antenna Patterns
## 2 Theoretical Analysis of Key Issues on Soil Moisture Retrieval
### GNSS-R Soil Detection Depth of GNSS Signals  
For electromagnetic waves with a given frequency, the soil dielectric constant is generally a function of soil moisture, bulk density, soil composition, temperature and salinity, with soil moisture exerting the greatest effect [34]. The propagation distance of an electromagnetic wave in the medium until the electromagnetic wave energy is attenuated to \(1/e\) is referred to as the penetration distance (\(\delta\,p\) in Figure 6). The attenuation of electromagnetic wave energy is affected by soil moisture, which changes the signal's penetration distance. If the soil is not very dry, the penetration distance of an electromagnetic wave is in the range of 0.1\(\lambda\) to \(\lambda\), where \(\lambda\) corresponds to the volumetric soil moisture mv \(\approx\) 0.04 cm\({}^{3}\)/cm\({}^{3}\) and while 0.1\(\lambda\) corresponds to very moist soil. In Figure 6, the detection depth is \(\delta\,p\)-cos \(\theta_{1}\), where \(\theta_{1}\) is the refraction angle of the signal with an incident angle of \(\theta_{0}\) in the soil. For soils with a uniform dielectric constant, 63% of the energy of the signal radiating in the direction with an angle of \(\theta_{0}\) to the normal line after interacting with the soil are from the detected depth. Thus, the estimation based on the reflected signal is primarily the soil moisture information in the range of the detection depth. According to electromagnetic wave theory, the formula of plane wave propagation in a homogeneous lossy medium is as follows [35]:  
\[\stackrel{{\rightarrow}}{{E}}(z)=\stackrel{{ \rightarrow}}{{E_{0}}}(z)\cdot exp(-j\omega\sqrt{\mu\epsilon_{c}}\cdot z) \tag{6}\]  
where \(\stackrel{{\rightarrow}}{{E}}\) is the electric field vector; \(z\) indicates the propagation direction; and \(\omega\), \(\mu\), \(\epsilon_{c}\), and \(j\) are the angular frequency, permeability, complex permittivity, and imaginary unit, respectively. Then, the propagation constant (\(j\omega\sqrt{\mu\epsilon_{c}}\)) can be expressed as:  
\[j\omega\sqrt{\mu\epsilon_{c}}=j\omega\sqrt{\mu\epsilon_{0}\cdot\epsilon_{r}}=j \cdot\frac{2\pi}{\lambda}\sqrt{\epsilon_{r}}=\alpha+j\cdot\beta \tag{7}\]  
where \(\epsilon_{r}\) is the relative complex permittivity and can be expressed as \(\epsilon_{r}=\epsilon_{r}^{\prime}-j\cdot\epsilon_{r}^{\prime\prime}\cdot \epsilon_{0}\), \(\alpha\), and \(\beta\) are the complex permittivity of air, the attenuation constant, and the phase constant, respectively. Then, the attenuation constant \(\alpha\) can be expressed as:  
\[\alpha=\frac{2\pi}{\lambda}|Im(\sqrt{\epsilon_{r}})|=\frac{\pi\epsilon_{r}^{ \prime\prime}}{\lambda\sqrt{\epsilon_{r}^{\prime}}} \tag{8}\]  
Figure 6: Propagation of an electromagnetic wave and its penetration distance in soil. For the microwave band, the imaginary part of the complex permittivity of soil is much smaller than the real part. Therefore, \(\sqrt{e_{r}^{\prime}-j\cdot e_{r}^{\prime\prime}}\) can be performed with Taylor series expansion, where \(Im(\cdot)\) represents the solution of the imaginary part of the complex number in the brackets. Assuming that the power of the electromagnetic wave is \(p(0)\) at the soil-air interface, the power of the electromagnetic wave after traveling in the soil a distance of z satisfies Formula (9) [36]:  
\[p(z)=p(0)\cdot\mathrm{e}^{-2\alpha z} \tag{9}\]  
When \(p(z)/p(0)=1/\mathrm{e}\), z = \(\delta\)\(p\):  
\[\delta_{p}=\frac{1}{2\alpha}=\frac{\lambda\sqrt{e_{r}^{\prime}}}{2\pi e_{r}^{ \prime\prime}} \tag{10}\]  
The relationship between the incident angle \(\theta_{0}\) and the angle of refraction \(\theta_{1}\) can be determined through the refractive index of the microwave in the soil (\(n_{1}\)):  
\[\frac{\sin\theta_{0}}{\sin\theta_{1}}=n_{1}=\sqrt{e_{r}^{\prime}} \tag{11}\]  
When the frequency of the electromagnetic wave is 1.4 GHz, [PERSON] et al. presented the calculation formulas for the real and imaginary parts of the complex permittivity of soil [37]:  
\[\left\{