# 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
### Spatial Resolution of Remote Sensing Via GNSS-R  
In the GNSS-R system, the ground is set as the X-Y plane, and the direction of the receiver pointing to the zenith is set as the Z axis to establish the right-handed rectangular coordinate system, as shown in Figure 4. The GNSS signal travels the shortest path by passing the specular reflection point (x\({}_{0}\),0,0) in a grazing angle of \(\gamma\) and then being reflected to reach the receiver at the height of \(h\). Assuming that the signal that is reflected at the point (x,y,0) is also received and that the difference between the travel distance of the signal passing through the point (x,y,0) and that of the signal passing through the specular reflection point is \(\delta\), then \(\delta\) satisfies the following formula [30]:  
Figure 4: Schematic diagram of equi-delay ellipses of the reflected GNSS signal. Figure 3: Calculation procedure of the specular reflection point in GNSS-R.  
\[\delta=\sqrt{x^{2}+y^{2}+h^{2}}-x\cos\gamma-h\sin\gamma \tag{3}\]  
The trajectory of the point (x,y) on the X-Y plane is an ellipse, and its major semi-axis and minor semi-axis \(b\) are:  
\[a=\frac{\sqrt{\delta^{2}+2\delta h\sin\gamma}}{\sin^{2}\gamma},\quad b=\frac{ \sqrt{\delta^{2}+2\delta h\sin\gamma}}{\sin\gamma} \tag{4}\]  
The coordinates of the center of the ellipse are:  
\[\left(\frac{\delta+h\sin\gamma}{\sin\gamma\tan\gamma},\quad 0\right)\]  
The GNSS-R configuration can be considered as a bistatic radar system where the size and the shape of the scattering scene, called Glistening Zone (GZ), depend on system configuration, i.e., receiver altitude and observation angle, receiver antenna and surface roughness [31]. The signal at the receiver, mainly coming from the SP, is correlated with a set of local replicas of the GNSS code shifted in time delay and Doppler frequency. Hence, the received signal is mapped in the Delay-Doppler domain to generate the so-called Delay-Doppler Map (DDM). In case of airborne or fixed platform receiver configuration, the Doppler effect can be neglected. For the height of the receiver is 10 m, the grazing angle of the GNSS signal is 30\({}^{\circ}\), and \(\delta\) is set to \(\lambda/2\), \(\lambda\), \(3\lambda/2\), \(2\lambda\), and \(5\lambda/2\) (for the L1 band of GPS, \(\lambda\) = 0.19 m); the size and shape of the equi-delay ellipses are shown in Figure 5. In this figure, the coordinates of the projection of the receiver on the Earth's surface is (0,0), and the specular reflection point location is labeled \"\(\times\)\". The corresponding \(\delta\) to the equi-delay ellipses increases sequentially from the inside outward. If the relationship between the ground surface and the incident wavelength is in accordance with the Fraunhofer criterion, i.e., the surface is approximately smooth to the L-band microwave, then a typical approximation is to replace \(\delta\) with \(\lambda/2\) so that the location and size of the ellipse can be determined. In this case, the ellipse is called the first Fresnel zone. Because it is the \"first reflection zone\" of the bistatic radar, the size of the area can be viewed as the spatial resolution of remote sensing via GNSS-R [32]. \[S=\pi ab=\frac{\pi\big{(}\delta^{2}+2\delta h\sin\gamma\big{)}}{\sin^{3} \gamma}=\frac{\pi\lambda(\lambda+4h\sin\gamma)}{4\sin^{3}\gamma} \tag{5}\]  
Figure 5: The shape of Equi-delay ellipses at a receiver height of 10 m and a grazing angle of 30\({}^{\circ}\). Formula (5) can be regarded as the spatial pixel size of soil moisture remote sensing via GNSS-R. Taking the L1 (1575.42 MHz) band of GPS as an example, the elevation angles of the GNSS satellite are 5, 10, 20, 30, 50 and 70\({}^{\circ}\), and the spatial resolutions calculated using Formula (5) are shown in Table 1. For ground-based observation platforms, the height of the antenna phase center from the ground is generally controlled between 1.5 and 2.5 m to exclude interference from the surrounding environment when the signal is at a low satellite elevation angle.