# Navigation in GEO, HEO, and Lunar Trajectory Using Multi-GNSS Sidelobe Signals
## 2 Observation Simulation Strategy
### Observation Simulation  
The abovementioned transmitter and antenna radiation pattern simulations determined whether a satellite could be tracked and acquired by the receiver. The other satellite visible condition was that line-of-sight (LOS) was not blocked by the Earth, meaning that the off-boresight angle was larger than the Earth block angle, which is shown in Figure 6. The Earth block angle varied with GNSS systems because of satellite altitude. Earth block angles of different GNSS orbits are illustrated in Table 6.  
Once a GNSS satellite was visible for spacecraft, the pseudorange observation could be calculated as:  
\[P_{r,j}^{s}=\rho_{r,j}^{s}+dt_{r}-dt^{s}+T_{r}^{s}+I_{r,j}^{s}+\varepsilon_{r,j} ^{s} \tag{3}\]  
where \(P_{r,j}^{s}\) is the pseudorange; indices \(s\), \(r\), and \(j\) refer to the satellite, receiver, and carrier frequency, respectively; \(\rho_{r,j}^{s}\) represents the geometric distance between the satellite  
Figure 5: Receiver antenna gain with off-boresight angles. Figure 6: Earth block angle. and the receiver \(r\); \(dt_{r}\) and \(dt^{s}\) denote clock offset of the receiver and satellite; \(T_{r}^{s}\) is the slant tropospheric delay; \(I_{r,j}^{s}\) is the ionospheric delay at frequency \(j;\varepsilon_{r,j}^{s}\) denotes the code measurement noise. The GNSS-spacecraft geometric distance was calculated from the positions of the mass centers of GNSS satellite and spacecraft receiver. GNSS satellite position was corrected by the satellite antennas' phase center offsets (PCO) and phase center variations (PCV), which were documented in the jigs-08 antenna information [35]. The Center for Orbit Determination in Europe (CODE) 30 s interval multi-GNSS clock offset products were used to simulate multi-GNSS satellite and receiver clock errors. For atmospheric delay, since the altitudes of GEO spacecraft were not lower than GNSS altitudes, signals may pass troposphere and ionosphere twice when the transmitted signals were from the other side of the Earth. This phenomenon occurs when the nadir angle of the spacecraft is small enough and close to the Earth block angle. Observations of GNSS satellites may be affected by some large variations because ray bending is likely to happen due to fraction [36]. For the HEO spacecraft, since the altitude was partly below GNSS, transmitted signals from the same side of the Earth could be received. The perigee attitude of HEO was approximately 496.7 km; these signals could not pass through the troposphere and pass through the ionosphere once. When transmitted signals were from the other side of the Earth, the tropospheric and ionospheric delay may have occurred twice, which was the same situation as with the GEO spacecraft. For both kind of orbits, the duration that the signal passed the atmosphere only accounted for a small proportion. Therefore, when processing single-frequency signals, the signals that passed through the ionosphere were simply removed; for dual- or multi-frequency signals, ionosphere-free (IF) combination was used to eliminate the ionosphere effects. The lunar trajectory probe departs from the Earth, with increasing altitudes that were lower than GNSS from the departing epochs and higher than GNSS at last. The lowest altitude was higher than the troposphere bound. The fraction of the epochs that the probe was below the ionosphere upper bound was 0.2%. Therefore, the ionospheric delay of transmitted signals from the same side of the Earth was negligible. Transmitted signals from the other side of the Earth, which may have passed twice of the ionosphere but which, however, only accounted for a small proportion, were also ignored. Lower received signal power resulted in larger code measurement noise. According to the experiment results of the Chinese No.2 Telecommunication Technology Test Satellite (TJS-2) and the Chinese No.5 Telecommunication Technology Test Satellite (TJS-5), we set the code measurement noise as follows [37; 38]:  
\[\varepsilon=\left\{