## 3 Global Positioning System (GPS) L1 Receiver: 2D Pattern vs. 3D Pattern
### Availability and Geometric Dilution of Precision (GDOP)  
In this study the \(i\)-th GPS observation is considered available if the corresponding \(i\)-th GPS L1 C/A signal is acquired and tracked. According to the capability of the SANAG receiver, the \(i\)-th GPS L1 C/A signal:  
1. Is acquired if the corresponding signal power is equal to or stronger than 20 dB-Hz without any interruption in the time interval \([t_{k}-5\) min, \(t_{k}]\), or if it is weaker than 20 dB-Hz but stronger than 15 dB-Hz without any interruption in the time interval \([t_{k}-10\) min, \(t_{k}]\). 2. Is tracked if the corresponding signal power is equal to or stronger than 15 dB-Hz. Note that once the receiver is orbiting above the ionosphere upper bound, the few GPS L1 C/A signals that cross the ionosphere are discarded, since they may be affected by large ionospheric delay due to the double crossing of the ionosphere. Figure 10 illustrates the availability of satellites as a function of the altitude, during the full considered trajectory, for both considered antenna patterns, assuming 10 dBi receiver antenna gain. From this plot we cannot conclude for which pattern the availability is higher, however we can see that when assuming a 3D pattern it happens much more often to lose a satellite for a short time and track it again shortly after, compared to the case when we are using a 2D pattern. This is expected as, in the case of a 3D pattern, the gain might change a lot for different azimuth, more often causing a sudden loss of sight on a satellite. Another way to distinguish the impact of averaging the antenna pattern on the availability is to plot the total number of observations available, as shown in Figure 11. From this plot we can see that, depending on the portion, the availability might be higher for either a 3D or a 2D  
Figure 10: Available GPS L1 signals, during the assumed MTO, for a 10 dBi receiver gain pattern. Figure 9: Fourth-highest received power level of GPS L1 signals, during the considered MTO, for a 0 dBi receiver gain pattern. pattern. However, it is interesting to notice that in the case of a 2D pattern after 200,000 km altitude, the number of satellites available has a lower standard deviation compared to using a 3D pattern. Hence, in the case of a 3D pattern there are more outages, as well as more peaks in the number of satellites. The precise statistics on the availability are computed for the entire MTO as well as for three representative portions, as defined in Table 2. They are reported in Table 3, where the two assumed patterns are more concretely compared. In Figure 12, the GDOP is shown. From this figure, it is difficult to assess which between the 3D or the 2D pattern leads to the worst GDOP; in some portion the 3D pattern has higher GDOP and in some other portions lower GDOP. However, at the end of the MTO for the 2D pattern the GDOP reaches its maximal value of 2.5 \(\times\) 10\({}^{6}\), while, when we assume a  
In Table 3 the statistics of the availability are reported for the three representative 500 min portions (as defined in Table 2) of the trajectory and during the full trajectory, for both assumed patterns. As we can see in the first portion, the availability is slightly higher when assuming a 2D pattern; however, in the second and third portion assuming a 3D pattern would result in slightly higher availability. Finally, as expected, averaging the pattern results in different availability, although not significant; for instance, considering the full trajectory, the difference is less than 1% or less than 0.2 satellites observed on average per epoch (1 s). Another important piece of information related to the availability is the continuity of the signal as time interval duration when the signal is continuously available. This information is very useful for the acquisition module design.