## 2 Models and Assumptions
### Different Antenna Patterns Assumptions Analyzed  
#### 2.3.1 L1 Antenna Patterns  
The published data in [3] for GPS L1 C/A Blocks IIR and IIR-M were used to model these blocks satellites' antenna patterns. The measured antenna gain data and relative phase data for each transmitter of Block IIR and IIR-M are provided for L1 and L2 signals by Lockheed Martin in Appendix B and C of [3]. In [15], we can see that some of the IIR satellites use the legacy antenna pattern, while the remaining part and all the IIR-M satellites use the improved antenna pattern. For the benefit of terrestrial and space GPS users, the improved pattern includes a new design of the L-band elements mounted on the spacecraft's Earth-facing structure. Meanwhile, for the GPS Block IIF satellites (built by Boeing), similar data have not been released. Logically, the antenna patterns used for GPS IIF Block should be at least as good as the improved antenna pattern used from Lockheed Martin. Thus, we decided to use a single 3D pattern for the satellites of Block IIF, where for each azimuth and each elevation the gain is the mean of the gains at the same azimuth and elevation of all the transmitters' patterns with the improved antenna pattern in [15]. This pattern is illustrated in Figure 4, while in Appendix A of [3] all the gain patterns for each satellite of the IIR and IIR-M Blocks are shown.  
In the same figure is shown the assumed 2D antenna pattern for the satellites of Block IIF, which is the mean of the gains for different azimuth (in the same figure). In the same way the 2D patterns were formed for each satellite of block IIR and IIR-M from their respective patterns, shown in Appendix A of [3].  
\begin{table}
\begin{tabular}{c c} \hline \hline \multirow{2}{*}{**Pseudorange**} & \(\rho_{i}=\sqrt{\left(x_{sat_{i}}-x_{u}\right)^{2}+\left(y_{sat_{i}}-y_{u} \right)^{2}+\left(z_{sat_{i}}-z_{u}\right)^{2}+c\delta t_{u}-c\delta t^{i}+A^{ i}+D^{i}+K^{i}+n^{i}}\) \\ \cline{2-3}  & \(x_{sat_{i}},~{}y_{sat_{i}},~{}z_{sat_{i}}\): \(i\)-th satellite coordinates at time of transmission \\ \cline{2-3}  & \(x_{u},~{}y_{u},~{}z_{u}\): the receiver coordinates at time of reception \\ \cline{2-3}  & \(c\delta t_{u}\): receiver clock offset at signal \\  & reception time \\ \cline{2-3}  & \(c\delta t^{i}\): transmitter clock offset at signal transmission time \\ \cline{2-3}  & \(A^{i}\): atmospheric delay (m) & \(\sigma_{A}=7\) m (when the receiver is orbiting below the ionosphere upper bound) \\ \cline{2-3}  & \(D^{i}\): multipath delay (m) & \(\sigma_{D}=0.2\) m \\ \cline{2-3}  & \(K^{i}\): signal-in-space ranging error \\  & (SISRE) (m) & \(\sigma_{K}=0.5\) m \\ \cline{2-3}  & \(n^{i}\): receiver noise (m) & \(\sigma_{n}=\sqrt{\sigma_{DLL}^{2}+0.1^{2}}\) \\ \hline \multirow{4}{*}{**Pseudorange**} & \(\dot{\rho}_{i}=(v_{sat_{i}}-v_{u})\cdot a_{i}+\dot{b}+\sigma_{f}\) \\ \cline{2-3}  & \(v_{sat_{i}}\): \(i\)-th satellite’s velocity vector \\ \cline{2-3}  & \(v_{u}\): receiver’s velocity vector \\ \cline{2-3}  & \(a_{i}\): the line of sight (LOS) unit vector from the receiver to the \(i\)-th GNSS satellite \\ \cline{2-3}  & \(\dot{b}\): receiver clock drift & initial value: 100 m/s \\ \cline{2-3}  & \(\sigma_{f}\): Doppler tracking jitter (m/s) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Observations and assumptions.