# Broadcast Ephemeris with Centimetric Accuracy: Test Results for GPS, Galileo, Beidou and Glonass
## 2 Mathematical Model and Results for GPS, Galileo, BeiDou
### Results for Beidou C07  
We address here C07, an example of Beidou GNSS, that is an SV in an IGSO orbit: this is a geosynchronous orbit with an inclination comparable to that of GPS or Galileo. The orbit pattern in ECEF coordinates is such that the X and Y coordinates sample a limited range of values. Consequently, one expects a rank deficiency in the normal equations and that the translational Helmert parameters are poorly constrained by the lack of geometry. We have therefore constrained the translational Helmert parameters to zero and solved for Helmert rotations and scale (Table 3), and the seven arc parameters and three clock parameters at intervals of 2 h. For Beidou, the precise SP3 ephemeris computed by CODE \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **Tx (m)** & **Ty (m)** & **Tz (m)** & **Rx (mas)** & **Ry (mas)** & **Rz (mas)** & **Scale** \\ \hline Estimated & \(-\)0.05 & \(-\)0.03 & \(-\)0.04 & 0.3 & \(-\)0.6 & \(-\)0.8 & 0.70 \(\times\) 10\({}^{-10}\) \\
1 sigma formal error & 0.02 & 0.02 & 0.02 & 0.29 & 0.17 & 0.06 & 4.68 \(\times\) 10\({}^{-10}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Helmert parameters relating the origin, orientation and scale of the Broadcast reference frame of E01 relative to the IGS14 frame of the SP3 precise ephemeris, 2 January 2020. Figure 4: Pre-fit (dots, different colors for different ephemeris blocks) and post-fit (diamonds) residuals of (**a**) X, (**b**) Y, (**c**) Z, and (**d**) T for E01. were used, being those of CNES not yet available. This also provides a way of using data from more than one Center. One more aspect to consider is that this SV is modeled with the GPS type of ephemeris which is optimized on an orbital radius of ca. 26,000 km, whereas the geosynchronous orbit has a radius of roughly 42,000 km. It follows that the perturbations due to the Earth's gravity field will be attenuated, and those due to the Sun and the Moon will tend to increase. Figure 5: _Cont._ \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **Tx (m)** & **Ty (m)** & **Tz (m)** & **Rx (mas)** & **Ry (mas)** & **Rz (mas)** & **Scale** \\ \hline Estimated & 0.00 & 0.00 & 0.00 & \(-4.5\) & 0.7 & 5.0 & \(2.29\times 10^{-8}\) \\
1 sigma formal error & 0.00 & 0.00 & 0.00 & 1.70 & 3.20 & 0.06 & \(1.73\times 10^{-9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Helmert parameters relating the origin, orientation and scale of the Broadcast reference frame of C07 relative to the IGS14 frame of the SP3 precise ephemeris for SV C07, 2 January 2020. The scale factor accounts for the Center of Mass–Antenna Phase Center correction in the radial direction. The translational parameters were constrained to zero due to lack of geometry in a IGSO orbit. Figure 5: Best fitting corrections to **(a)**the clock polynomial parameters, **(b)** Mean anomaly at Toe and amplitude of the cosine and sine perturbations along track due to the second zonal harmonic (Cuc, Cus), and amplitude of the cosine and sine perturbations in the **(c)** radial (Crc, Crs) and **(d)** cross track (Cic, Cis) directions. To convert to radians the scale factor 26 \(\times\) 10\({}^{6}\) m should be used. Error bars are 1 sigma formal uncertainties. Satellite E01 for day 2 January 2020. Figures 6 and 7 summarize our results for C07. The broadcast ephemeris shows a discrepancy relative to the precise orbit of the order of the meter, and there is a constant bias in the clock corrections (Figure 6). After the fit we observe that the post-fit residuals for both coordinates and clock agree very closely with the corresponding precise ephemeris, suggesting that the correction to the 12 sets of seven parameters each (mean anomaly at epoch Toe; amplitudes and phases of the second zonal components) have successfully accounted for the modeling imperfections of the broadcast message. Figure 6: Pre-fit (dots, different colours refer to different ephemeris blocks) and post-fit (diamonds) residuals of (**a**) X, (**b**) Y, (**c**) Z for C07.