## 2 Proposed Method
### Determination of Reflecting Ground Coverage for GNSS-R Off-Nadir Antennae  
The nadir LHCP antennae are used to receive the GNSS signals that are reflected from the Earth's surface. The sizes and the locations of the reflecting ground coverages are closely related to the beamwidth and the face direction of each of the LHCP antennae. The beamwidth angles are limited by some of the antenna characteristics, along with the frequency and power of the received signals, while the face direction is related to the observation modes of the missions. First, we compare the antenna observation modes of the current GNSS-R missions. TDS-1 (TechDemoSat-1), which is the precursor of the CYGNSS mission, has only one nadir antenna that uses a fixed array of four flared spiral elements. However, each micro-satellite in the CYGNSS constellation has two tilted antenna elements. Figure 1 shows the TDS-1 and CYGNSS nadir antenna observation modes. Obviously, the two-antenna mode should be selected because it can receive more GNSS reflected signals than the single antenna mode when the same instruments are used. Additionally, in the processing of GNSS-R data, the transformation between spatial location on the Earth surface and location in the DDM has ambiguities outside the specular region [41]. When two antennae face different directions, to some extent, they can receive and distinguish the signals reflected from locations in different direction on the Earth surface. It will make a positive impact on the solution of the typical ambiguity problem for GNSS-R.  
From Figure 1, the reflecting ground coverage for the two kinds of antenna modes are both finite, and are limited by the antenna beamwidth. Using TDS-1 as an example, the beamwidths are \(34^{\circ}\times 35^{\circ}\) at GPS L1, with a peak gain of 13.8 dBi, and \(49^{\circ}\times 41^{\circ}\) at GPS L2, with a peak gain of 10.8 dBi [23]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Mean Orbit Parameter** & **Symbol** & **Values** \\ \hline nodal day & D & 13 days \\ revolutions & N & 194 cycles \\ Semi-major & \(a\) & 6,887,905.7860 m \\ eccentricity & \(e\) & \(0.621198\times 10^{-3}\) \\ inclination & \(i\) & \(35^{\circ}\) \\ Argument of perigee & \(\omega\) & \(90^{\circ}\) \\ Longitude of the ascending node & \(\Omega\) & 0.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Orbit parameters of the GNSS-R LEO constellation. However, no files have been released that contain the definitive beamwidths of the CYGNSS antennae. Therefore, in this paper, the beamwidth angles used for the GNSS-R LEO nadir antennae are the same as those of TDS-1. Figure 2 shows a schematic view of a CYGNSS microsatellite and its double nadir antennae. In Figures 2 and 3, the +X and +Z axes point in the directions of flight and nadir, respectively. The +Y axis points towards the normal of the orbit plane and the orientations follow the right-hand rule. The antennae are off-pointing from the nadir by 28\({}^{\circ}\). At the GPS L1 frequency, the elevation beamwidth is 34\({}^{\circ}\) and the azimuth beamwidth is 35\({}^{\circ}\). The corresponding values at the GPS L2 frequency are 49\({}^{\circ}\) and 41\({}^{\circ}\), respectively. Figure 1: A sketch for single-antenna observation mode of TDS-1 (**left**) and dual-antenna observation mode of CYGNSS (**right**). Figure 3: Available reflecting ground coverage for one of the two off-nadir LHCP antennae. Only the signals reflected from specular points which located in the green area can be received by the GNSS-R instrument on the LEO observatory. Figure 2: Schematic of spacecraft of one CYGNSS observatory and its two off-nadir LHCP antennae (yellow). Figure 3 shows the available observation area for a single CYGNSS LHCP antenna on the Earth's surface. The specular points that are located within the green area can be contacted, while the points that lie outside the green area cannot be contacted.