# Broadcast Ephemeris with Centimetric Accuracy: Test Results for GPS, Galileo, Beidou and Glonass
## 2 Mathematical Model and Results for GPS, Galileo, BeiDou
A partial derivative matrix H is computed numerically by approximating the partial derivatives with finite central differences. The best fitting corrections to the a priori values are computed by solving the linear system of 223 normal equations. The matrix H\({}^{\text{T}}\)H (\({}^{\text{T}}\) stands for transposed), however, is poorly conditioned, as it may be expected. The rotational Helmert parameters tend to be highly correlated with the angles of orientation of the orbit. A well-conditioned matrix is obtained by increasing the weight of the matrix elements corresponding the parameters in red in Equation (1), which are therefore fixed to the broadcast values. In practice we solve for seven global Helmert parameters (i.e., one set for one day) and 12 sets each of seven parameters (the mean anomaly \(M_{0}\) at the reference epoch Toe and three pairs of amplitudes of cosine and sine components of periodic perturbations along track (\(Cuc\), \(Cus\)), radial (\(Crc\), \(Crs\)) and across track (\(Cic\), \(Cis\)) (Figure 1). This is equivalent to constrain the Helmert rotations and solve for the angles defining the spatial orientation of the orbit. In the following, we use this model for data referring to 2 January 2020, as an arbitrary choice. We consider G01 for GPS, E01 for Galileo, C07 for Beidou IGSO and C12 for Beidou MEO. The reference SP3 file is available at the IGS/MGEX servers. We have arbitrarily chosen the CNES SP3 file for GPS and Galileo ([[https://cddis.nasa.gov/archive/gnss/products/mgex/2086/GRG0](https://cddis.nasa.gov/archive/gnss/products/mgex/2086/GRG0) MGXFIN_20200020000_01D15M_ORB.SP3.gz]([https://cddis.nasa.gov/archive/gnss/products/mgex/2086/GRG0](https://cddis.nasa.gov/archive/gnss/products/mgex/2086/GRG0) MGXFIN_20200020000_01D15M_ORB.SP3.gz), accessed on 10 October 2021), and the CODE SP3 file for Beidou ([[https://cddis.nasa.gov/archive/gnss/products/mgex/2086/COD0](https://cddis.nasa.gov/archive/gnss/products/mgex/2086/COD0) MGXFIN_2020002000_01D_05M_ORB.SP3.gz]([https://cddis.nasa.gov/archive/gnss/products/mgex/2086/COD0](https://cddis.nasa.gov/archive/gnss/products/mgex/2086/COD0) MGXFIN_2020002000_01D_05M_ORB.SP3.gz), accessed on 19 September 2021). The Mixed broadcast ephemeris file was downloaded from the BKG  
Figure 1: Schematic structure of the partial derivative matrix for the GPS-like model. The first vertical block represents the partials of \(XYZT\) pre-fit residuals relative to the Helmert parameters, appropriately scaled. The block diagonal part contains 12 blocks each of 4 rows of 18 partial derivatives, using as a priori the broadcast values of the parameters which are indexed with Toe for the coordinates and Toe for the clock. server ([[https://igs.bkg.bund.de/root_ftp/MGEX/BRDC_v3/2020/002/brdm0020.20p.Z](https://igs.bkg.bund.de/root_ftp/MGEX/BRDC_v3/2020/002/brdm0020.20p.Z)]([https://igs.bkg.bund.de/root_ftp/MGEX/BRDC_v3/2020/002/brdm0020.20p.Z](https://igs.bkg.bund.de/root_ftp/MGEX/BRDC_v3/2020/002/brdm0020.20p.Z)), accessed on 10 October 2021). As mentioned in the Introduction, our results will be valid for the specific satellites and epochs only. To extend the validity of the conclusions drawn in this paper beyond the specific satellites and epoch requires volume calculations. In this paper we concentrate on the mathematical approach as a prerequisite for further massive processing. Epoch-dependent departures from the results described here could for example be caused by a different angle between the Sun's geocentric vector and the satellite orbital plane (beta angle), which controls the force caused by the solar radiation pressure.