# Broadcast Ephemeris with Centimetric Accuracy: Test Results for GPS, Galileo, Beidou and Glonass
## 2 Mathematical Model and Results for GPS, Galileo, BeiDou
### Results for GPS  
Figure 2 shows the pre-fit and post-fit residuals for G01 (Block IIF). The pre-fit residuals are computed using the Broadcast Ephemeris blocks of appropriate Toe and Toe, and the post-fit residuals are computed with the tuned parameters computed with Equation (1). In Figure 2, the pre-fit differences are represented with colored dots, one color for each two-hour ephemeris block, with the exception of the first block where the broadcast ephemeris with Toe = 2 h has been used for an interval of 4 h, from 00:00 to 04:00, for test purposes. The vertical lines have a two-hour spacing. The diamonds represent the post-fit residuals, after the adjustment of the seven Helmert parameters and of the 11 sets of seven arc parameters. In Section 5, we will discuss in more detail the oscillatory pattern of the post-fit residuals, in connection to an improvement of the model to include the contribution of the third zonal harmonic of the gravity field. Likewise, it will be pointed out that the higher noise in the first 4 h of the post-fit residuals of the clock corrections suggests that the second order clock polynomial is unable to track the high frequency noise of the satellite clock, if the fit interval is as long as 4 h, as it may be expected. This will be further discussed in Section 4 in relation to the comparison with high rate clocks.  
The overall improvement in doing the fit described by Equation (1) is summarized as follows: the mean and rms (root mean square spread) of the pre-fit residuals is \(-\)0.013 \(\pm\) 0.917 m, and 0.000 \(\pm\) 0.051 m for the post-fit residuals. This statistic includes the spatial and temporal coordinates. The adjusted clock polynomials fit the SP3 clock data with 0.007 m rms, whereas for the spatial coordinates the corresponding figure is a factor of seven larger.  
Figure 3 describes the epoch-wise corrections to the 11 sets of seven arc parameters, and the corrections to two of the three estimated clock parameters (\(a2\) is not shown but computed). The correction to the mean anomaly \(M_{0}\) means that at the epoch Toe the satellite is moved back or forth along the orbit, relative to the orbital phase predicted by the broadcast ephemeris. Table 1 finally provides the seven Helmert parameters for this daily set, and their estimated formal error 1 sigma.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline  & **Tx** & **Ty** & **Tz** & **Rx** & **Ry** & **Rz** & **Scale** \\  & **(m)** & **(m)** & **(m)** & **(mas)** & **(mas)** & **(mas)** & **(mas)** & **Scale** \\ \hline Estimated & \(-\)0.09 & 0.20 & \(-\)0.39 & 1.2 & 1.2 & \(-\)11.1 & 0.60 \(\times\) 10\({}^{-08}\) \\
1 sigma formal error & 0.04 & 0.04 & 0.02 & 0.00 & 0.00 & 0.00 & 8.48 \(\times\) 10\({}^{-10}\) \\ \hline \end{tabular}
\end{table}
Table 1: Helmert parameters relating the origin, orientation and scale of the Broadcast reference frame of G01 relative to the IGS14 frame of the SP3 precise ephemeris, 2 January 2020. ’mas’ stands for milliarcsec.  
Figure 2: _Cont._  
Figure 2: Pre-fit (dots: different colours for different ephemeris blocks) and post-fit (diamonds) residuals of (**a**) X, (**b**) Y, (**c**) Z, and (**d**) T for G01. Vertical line spacing is every two hours. The rectangle indicates a 4 h rather than 2 h arc. The residuals will be discussed in Section 4.  
Figure 3: Cont.