# 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
### Specular Reflection Point of GNSS Signals  
In the geometric relationship between the direct signal and the reflected signal, the point that has the shortest path among the transmitter, surface, and receiver is called the specular reflection point, at which the incidence angle of the GNSS signal is equal to the signal's reflection angle. Thus, the specular reflection point serves as an important reference point for GNSS-R measurement and modeling. In ground-based observations, the antenna is generally 2 to 3 m in height, and the observation surface can be regarded as a plane. Then, the specular reflection point can be calculated using the satellite's elevation angle and azimuth angle based on simple geometric relationships. In air- and satellite-based observations, the observation platform is high above the ground, and the observed surface must be viewed as a curved one. The WGS84 ellipsoid corresponding to the WGS84 coordinate system used by GPS broadcast ephemeris has been used as a simple and effective geometric approximation of the Earth's surface. Figure 1 shows the geometry of the GNSS-R observation system. When the location of the transmitter (T) and receiver (R) are known, the specular reflection point (S) can be determined using an iterative method. In this method, the location where the line connecting the center of the earth (O) and the receiver (R) intersects the ground is used as the initial point (S'). Then, the ground point S' is continuously moved toward the bisector of the angle formed by S'R and S'T. When the mobile measurement of S' is smaller than the set threshold value, S' is determined as the specular reflection point (S), where the bisector of the angle formed by SR and ST is the normal line of Point S that passes through the curved surface. However, the approximation of the Earth's surface using the WGS84 ellipsoid leads to a certain error, and geoid and terrain height must also be taken into account. The geoid is an isotropic gravitational surface that corresponds to the global average sea level and is more in line with the natural shape of the Earth. The heights of an ellipsoid and of a geoid can differ by tens of meters. The height measured relative to the geoid is called altitude, which is a common height measurement used in many topographic maps and digital data sets [28]. Figure 2 shows the influence of geoid and terrain height on the specular reflection point, where the ellipsoidal height (ET) is the sum of the geoid height (EG) and altitude (GT). Considering only the ellipsoid surface while neglecting the geoid and terrain height will cause a certain deflection in determining the specular reflection point. As shown in Figure 2, Se is the specular reflection point when only the ellipsoid is taken into account, and St is the specular reflection point when both the geoid and the terrain elevation are considered. Assuming the height difference between the Earth's true surface and the WGS84 ellipsoid is \(\Delta h\), then the latitude and longitude deflections of the specular reflection point (\(\Delta lat\), \(\Delta lon\)) are as follows [29]:  
\[\Delta lat=\frac{\Delta h\cdot\cot\gamma\cdot\cos\varphi}{R_{e}},\quad\Delta lon =\frac{\Delta h\cdot\cot\gamma\cdot\sin\varphi}{R_{e}\cdot\cos y} \tag{1}\]  
where \(\gamma\) and \(\varphi\) are the elevation angle and azimuth of the GNSS satellite relative to the specular reflection point, respectively, and \(y\) is the latitude at the specular reflection point. Then, the radius (\(Re\)) of the ellipsoid can be calculated from the major semi-axis (\(ae\)) and the minor semi-axis (\(be\)):  
\[R_{e}^{2}=\frac{b_{e}^{2}}{1+(b_{e}^{2}/a_{e}^{2})\cdot\sin y} \tag{2}\]  
Figure 1: Geometry of the GNSS-R observation system. Figure 2: Effect of geoid and terrain height on the specular reflection point. Based on the XYZ coordinates in the Earth-centered Earth-fixed (ECEF) coordinate system of the GNSS satellite and the receiver, the latitude and longitude of the specular reflection point on the WGS84 ellipsoid can be obtained.