# Broadcast Ephemeris with Centimetric Accuracy: Test Results for GPS, Galileo, Beidou and Glonass
## 2 Mathematical Model and Results for GPS, Galileo, BeiDou
### Results for Galileo  
For Galileo we have chosen the E01, an FOC (Full Operational Capability) SV, as an example. The repetition rate of the broadcast ephemeris is for Galileo usually 10 min, but not always at a regular rate. The rate of update of the message is considerably higher than GPS, so that a tailored ephemeris/clock is available at each epoch of observation. We seek broadcast model coefficients which make this high repletion rate unnecessary both for position and clocks. We expect that the validity time of the Galileo navigation message should be in fact comparable with that of GPS, for example. We have chosen those ephemeris blocks with a Toe/Toc closest to the even hours of day 2 January 2020. Figure 4 shows the difference between the raw residuals (original broadcast coordinates and clock minus corresponding SP3 values) described by colored dots (one color for each Toe); the post-fit residuals are shown by diamonds. The clock correction is expressed in meters. By comparison with GPS, it can be noted that the original ephemeris has, relative to the reference precise orbit, a pattern suggesting periodic resets, to keep the broadcast position and time as close as possible to the precise value, in the neighborhood of the respective Toe. In this way the effects of the third zonal harmonic (\(J_{3}\)) are probably less visible than with a longer sampling time, as discussed for G01 in Figure 2. Moving away from this reference epoch, the departure from the reference values increases, so that the ephemeris needs to be replaced by a more recent block near the edge of the validity period. The broadcast ephemeris with best-fitting parameters (post-fit residuals, y axis to the right) tracks instead the precise ephemeris very closely and with no divergence at the edges of the validity interval.  
Figure 3: Best fitting corrections to (**a**) the clock polynomial parameters; (**b**) the Mean anomaly at Toe and amplitude of the cosine and sine perturbations along track (Cuc, Cus) due to the second zonal harmonic; the 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 G01 for day 2 January 2020.  
Figure 4: _Cont._  
The overall improvement for Galileo's E01 in doing the fit described by Equation (1) is summarized as follows: the mean and rms of the pre-fit residuals is \(-\)1.224 \(\pm\) 1.992 m, and 0.000 \(\pm\) 0.014 m for the post-fit residuals. This statistic includes the spatial and temporal coordinates. The adjusted clock polynomials fit the SP3 clock data with 0.002 m rms, whereas for the spatial coordinates the corresponding figure is a factor of seven larger.  
Figure 5 shows the corrections to two clock parameters (a), to the Mean anomaly at epoch Toe and cosine and sine amplitude of the corrections to second harmonic periodic perturbations along track (b). In the bottom left and bottom right plots, we have the cosine and sine amplitudes of the corrections for the radial (c) and across track (d) components, respectively. Table 2 gives the Helmert parameters of the broadcast reference frame relative to the IGS14 frame of the precise ephemeris. Here it is noteworthy that the broadcast frame of E01 has a negligibly small misalignment to IGS14. To account for the CoM-APC offset the offsets in [[https://www.gsc-europa.eu/support-to-developers/galileo-satellite-metadata](https://www.gsc-europa.eu/support-to-developers/galileo-satellite-metadata)]([https://www.gsc-europa.eu/support-to-developers/galileo-satellite-metadata](https://www.gsc-europa.eu/support-to-developers/galileo-satellite-metadata)) (accessed on 10 October 2021) have been used. This calibration implies that the scale factor must be negligibly small, as it results from Table 2.