# 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
Air-based observation platforms generally use manned aircrafts or drones to carry relevant loads, and the height is about 1 km. Because the spatial resolution of the airborne GNSS-R at an elevation angle of 5\({}^{\circ}\) is only 3151 m\({}^{2}\), the effect of the Earth's curvature is negligible in the estimation, and the first Fresnel zone still can be approximated as a plane. Analysis of Table 1 shows that when the platform height is fixed, the spatial resolution and the satellite elevation angle show a decreasing relationship; when the satellite elevation angle is fixed, the spatial resolution increases with the height of the platform. At the same time, the GNSS-R technology is not based on the observation of points in a very small space, and the selection of different elevation angles and platform heights can meet the needs of different spatial resolutions. The above discussion is based on the premise of a smooth ground, where the area of the first Fresnel zone is precisely the size of the GZ. However, surface state cannot be considered homogeneous at medium to large scales especially in case of a spaceborne receiver configuration, where the conventional GNSS-R DDM is not enough to include nonhomogeneous areas. [PERSON] proposed an inversion scheme based on the two-dimensional (2-D) truncated singular-value decomposition (TSVD) to reconstruct the NRCS field from noisy DDMs (both thermal and speckle noise are accounted for) related to a simulated marine scenario, which includes different nonhomogeneous areas and discussed an almost uniform spatial resolution that characterizes the resolution of the reconstructed image [PERSON] _et al._ (2018). For airborne receivers or receivers with lower altitudes, since their altitude is much smaller than the height of the GNSS satellites, the effect of the curvature of the earth can be ignored. The GZ can be regarded as the cut of the earth passing the specular reflection point and mainly concentrated in the first equi-delay ellipse, even the surface is rough. In order to obtain surface roughness information from the waveform, the GZ must contain several equal-pitch annular zones (three or more), and when the receiver height increases, the flash zone can contain multiple equi-delays, which describe the spatial resolution of a large surface. In the case that the ground surface has a certain gradient, the cosine value of the slope needs to be taken into account on top of the area of the orthographic projection. It needs to be pointed out that the frequency of the 1561.098 MHz) of the BDS satellite is very close to that of the L1 band of the GPS, the spatial resolutions of GNSS-R remote sensing under the same conditions are essentially identical. Whether the observation platform is ground based, air based or satellite based, the spatial resolution of soil moisture remote sensing via GNSS-R can be evaluated at between the conventional satellite remote sensing and the resolution of the ground-site measurement, which complements the advantages of various methods in the observational scale. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Observation Platform**} & \multicolumn{6}{c}{**Satellite Elevation Angle (\({}^{\circ}\))**} \\ \cline{2-7}  & **5** & **10** & **20** & **30** & **50** & **70** \\ \hline Ground-based (2 m) & 16.3 & 7.7 & 3.7 & 2.3 & 1.4 & 1.2 \\ Air-based (1 km) & 3151 & 1581 & 803 & 500 & 316 & 265 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Spatial resolutions (m\({}^{2}\)) of GNSS-R in the B1 band of GPS at different observation platform heights and satellite elevation angles.