## 7 GDOP and Angle of Transmission from the Antenna's Boresight  
Figure 43 shows the GDOP computed when using all the available signals (each identified by a PRN) and only the ones transmitted at angles between 0\({}^{\circ}\) and 30\({}^{\circ}\), 0\({}^{\circ}\) and 37\({}^{\circ}\), 0\({}^{\circ}\) and 52\({}^{\circ}\), 0\({}^{\circ}\) and 62\({}^{\circ}\), 0\({}^{\circ}\) and 75\({}^{\circ}\), and 0\({}^{\circ}\) and 90\({}^{\circ}\), in order to evaluate what is the relation between the angle at which the signals are transmitted from the GPS satellites and their relative geometry with respect to the receiver. Figure 44 is a zoomed-in view of Figure 43. We can see that if we exclude the signals coming from the side lobes of the transmitters' antenna (between 30\({}^{\circ}\) and 90\({}^{\circ}\)), the GDOP worsens drastically: e.g., at 38,000 km from a value of 7 to 83, at 200,000 km from a value of 194 to 1682, at Moon altitude from 1500 to above 30,000. Above 40,000 km, most of the time less than four signals are available (see the orange curve in Figures 43 and 44, which exists only when more than four signals are available) and, when available, the GDOP reaches values dramatically high (above 105). Roughly, we can say that, in order to have at least four signals available at the receiver's position for the full MTO, the receiver should be able to process signals transmitted at an angle up to 52\({}^{\circ}\) from the boresight (the green curve has no discontinuity, while the violet one does, as shown in Figures 43 and 44). Figure 42: Visible satellites (in the line of sight (LOS)) and available satellites at the receiver’s position, for which the corresponding signal is transmitted at an angle between 75\({}^{\circ}\) and 90\({}^{\circ}\) from the transmitter’s boresight. Figure 43: GDOP computed when using all the available signals (each identified by a PRN) and only the ones transmitted at angle between 0\({}^{\circ}\) and 30\({}^{\circ}\), 0\({}^{\circ}\) and 37\({}^{\circ}\), 0\({}^{\circ}\) and 52\({}^{\circ}\), 0\({}^{\circ}\) and 62\({}^{\circ}\), 0\({}^{\circ}\) and 75\({}^{\circ}\), and 0\({}^{\circ}\) and 90\({}^{\circ}\). For a limited number of acquisition/tracking channels (i.e., the SANAG receiver should have at least 12 channels), in the lower part of the MTO it is not possible to process all the available signals; for altitudes lower than 187,000 km more than 12 signals are available, as shown in Figure 11. Figure 45 displays the GDOP in three different cases. In the best case, all the available signals are processed (see the blue curve in Figure 45). In the simplest case (see the green curve in Figure 45), the navigation solution is simply computed using the first 12 signals that the receiver processes (e.g., from 1 to 29, the first available PRNs). However, this is not optimal because, among the available signals, there are 12 that ensure the best GDOP. Theoretically, the receiver could predict the combination of 12 signals (among the ones available) that provides the best GDOP, but the associated computational burden may be too high to be performed in real time on board. A less computationally expensive but still effective method is to find the four signals that provide the lowest GDOP and include them in the 12 processed (the brown curve in Figure 45). Figure 46 is a zoomed-in view of Figure 45 between 80,000 km and 105,000 km. For instance, we can see that including the four available signals that give the best GDOP in the 12 processed, instead of just using the first 12 signals that the receiver finds chronologically, provides a GDOP of 35 instead of 48 at 96,000 km (see Figure 46) and of 129 instead of 155 at 187,500 km (see Figure 47).