# 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
\begin{array}{c}\varepsilon=\varepsilon^{\prime}-j\varepsilon^{ \prime\prime}\\ \varepsilon^{\prime}=1.15*\left[1+\frac{\rho_{b}}{\rho_{s}}(\varepsilon_{s}^ {\prime})+m_{v}^{\beta^{1}}\varepsilon_{fw}^{\prime\prime\prime}-m_{v}\right] ^{1/\alpha}-0.68\\ \varepsilon^{\prime\prime}=\left[m_{v}^{\beta^{\prime\prime}}\varepsilon_{fw}^ {\prime\prime\prime}\right]^{1/\alpha}\end{array} \right. \tag{12}\]  
where \(\rho_{b}\) is the solid soil volume density; \(\rho_{s}\) is the specific gravity value of solid soil (2.66 g/cm\({}^{3}\)); the \(\alpha\) value is 0.65, which is the empirical constant; \(\varepsilon^{\prime}_{fw}\) and \(\varepsilon^{\prime\prime}_{fw}\) are the permittivity of free water for the real part and imaginary part respectively; \(\beta^{\prime}\) and \(\beta^{\prime\prime}\) are empirical constants related to soil type, and the expression is [38]:  
\[\left\{ \begin{array}{c}\beta^{\prime}=1.2748-0.5195-0.152C\\ \beta^{\prime\prime}=1.3380-0.603S-0.166C\end{array} \right. \tag{13}\]  
where \(S\) and \(C\) are sandy and clay contents respectively. Figure 7 shows the depth of penetration of BDS and GPS in different frequency bands when S and C are 50% respectively, considering different soil moisture conditions except for extremely dry soil, of which the penetration depth is between 0 and 20 cm. When the soil moisture increases from 0 to 0.2 cm\({}^{3}\)/cm\({}^{3}\), the depth of penetration rapidly decreased to within 8 cm. When the soil moisture changes from 0.2 to 1.0 cm\({}^{3}\)/cm\({}^{3}\), the penetration distance slowly declined. When the soil moisture is at higher values (VSM \(>\) 0.5 cm\({}^{3}\)/cm\({}^{3}\)), the probe depth was reduced to within 5 cm. Therefore, during the process of collecting soil moisture data, two depth data of 2.5 and 7.5 cm are generally selected for collection. Further analysis shows that for BDS and GPS and different frequency bands, when the soil moisture is the same, the higher the carrier frequency of the signal is, the smaller the penetration distance is. This is related to the fact that the higher the frequency of the electromagnetic wave is, the faster it decays in the soil. The BDS B1 band is equal to the wavelength of the GPS L1 band (-19 cm), so the penetration distance values of the two are relatively consistent. Similarly, the range of variation between GPS L2C, L5, and BDS B2 bands is small (24-25.5 cm). Therefore, the difference in penetration distance is not significant. Similarly, for the example of the L1 band of GPS, Figure 8 shows the changes in the signal penetration distances in four typical soil types (sandy soils, clay loam, silty soils, and silty clay) with soil moisture. This result indicates that for all soil types, the penetration distance decreases as the soil moisture increases. When the soil is dry (VSM \(<\) 0.2 cm\({}^{3}\)/cm\({}^{3}\)), the soil composition has a greater influence on the depth of penetration. With the increase of soil moisture, the soil composition has little effect on penetration distance. When the soil moisture exceeds 0.2 cm\({}^{3}\)/cm\({}^{3}\), the penetration distance will be shorter than 6 cm. In addition, after estimating the soil moisture through GNSS-R, the penetration distance of the signal in the soil can be determined based on the signal frequency and the soil type; it is then multiplied by \(\cos\theta_{1}\) to obtain the detection depth. The GNSS-R soil moisture remote sensing result is an average value of soil moisture within the detection depth range.