# Broadcast Ephemeris with Centimetric Accuracy: Test Results for GPS, Galileo, Beidou and Glonass

^{*}

## Abstract

**:**

## 1. Introduction

_{2}zonal. Because the numerical integration is carried out in a rotating frame, the force includes also the centrifugal and Coriolis terms. Third body perturbations, typically caused by the gravity of the Moon and the Sun, are modeled by accelerations kept constant during the validity time of the message, which, for Glonass, is 30 min. Thus, there are in each Ephemeris Message three ECEF positions and three velocities at epoch Toe, and three constant Lunisolar accelerations, for a total of nine parameters. The clock offset is again modeled by a quadratic polynomial with a time Toc as reference. An important feature of the Glonass approach is that the Reference Time scale tracks the Leap Second, so that the computation epoch must be offset by the number of Leap Seconds corresponding to the epoch [7].

^{14}m

^{3}s

^{−2}, whereas GPS and QZSS use μ = 3.9860050 × 10

^{14}m

^{3}s

^{−2}.

^{−8}) [23].

## 2. Mathematical Model and Results for GPS, Galileo, BeiDou

^{T}H (

^{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.

#### 2.1. Results for GPS

#### 2.2. Results for Galileo

_{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.

#### 2.3. Results for Beidou C07

^{−7}s (Figure 7a). The rms of the post-fit residuals is dominated by the spatial component: the clock polynomials fit with a typical rms of 0.002 m, about a factor of eight smaller than the rms of the spatial components.

#### 2.4. Results for Beidou C12

## 3. Mathematical Model and Results for Glonass

_{2}plus the centrifugal and Coriolis terms, since the equations of motion are integrated in a rotating frame. Glonass broadcasts three additional terms, the Lunisolar accelerations, which are constant accelerations during the validity time of the broadcast message, normally 30 min.

## 4. Polynomial Clock Model vs. IGS/MGEX High Rate Clocks

^{−11}and 10

^{−12}for the Cesium or Rubidium clocks. For Galileo’s Passive Hydrogen Maser, the frequency stability is somewhat smaller than 10

^{−12}[18,19,24,25,26]. This means that two consecutive, non-overlapping time segments of 100 s nominal length will have a 1 sigma difference of 1 ns to 0.1 ns for a two-sample Allan stability of 10

^{−11}and 10

^{−12}respectively. It follows that our clock polynomials need to be tested against high rate clock estimates. The IGS/MGEX makes available such files with sampling rate of 30 s. In the rest of this section, we will therefore compute differences in the clock corrections, between our polynomial values based upon 15 min sampling and high rate IGS clocks based on 30 s sampling.

## 5. Improving the Broadcast Model for GPS-like and Glonass-like Messages

_{2}) components.

_{2}), then we can write:

^{l}, ${e}^{q}$ and $si{n}^{p}\left(I\right)$, $co{s}^{p}\left(I\right)$, and abs(q) < 5 typically. ${\psi}_{0}$, $\dot{{\psi}_{0}}$ are the phase and frequency of the harmonic perturbation, and $M,\dot{\omega},n,\dot{\Omega},\dot{\theta}$ are respectively the mean anomaly, the rate of the perigee, the mean motion, the rate of the ascending node and the Earth rotation rate. Secular terms arise when m = 0 and $\left(l-2p\right)=q=0$ that is for even zonal harmonics. Because the rate of perigee and node are of the order of ${10}^{-3}n$ and the period is $n\cong 2\dot{\theta}$, the period P of a perturbation is primarily determined by:

_{3}zonal (l = 3, m = 0), for p = q = 0 we have perturbations with period close to 4 h. The ${J}_{3}$ term, accounts for the pear-shaped figure of equilibrium of the Earth, that is, the lack of symmetry between north and south hemispheres. The conventional values from the JGM-3 gravity model [28] are:

## 6. Results for Volume Calculations

## 7. Discussion

_{3}) of the Earth’s gravity field. Including this perturbation in the navigation message would help in increasing the validity time and have a more random spread of the post-fit residuals.

## 8. Conclusions

_{3}component of the Earth’s gravity field.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**Helmert parameters relating the GPS broadcast reference frame to the ITRF2014 Reference frame of the precise ephemeris of CNES. Average values for January 2020.

TX (m) | Std_TX (m) | TY (m) | Std_TY (m) | TZ (m) | Std_TZ (m) | RX (mas) | Std_RX (mas) | RY (mas) | Std_RY (mas) | RZ (mas) | Std_RZ (mas) | Sc (ppm) | Std_Sc (ppm) | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

G01 | 0.030 | 0.150 | −0.154 | 0.196 | −0.015 | 0.056 | −1.531 | 1.569 | 1.101 | 2.508 | 10.197 | 3.345 | 0.018 | 0.000 |

G02 | −0.008 | 0.167 | −0.020 | 0.187 | −0.057 | 0.112 | −0.001 | 1.318 | 0.105 | 2.334 | 0.215 | 7.045 | 0.033 | 0.001 |

G03 | 0.161 | 0.216 | 0.040 | 0.259 | −0.248 | 0.184 | −0.696 | 2.532 | 1.599 | 3.782 | −7.023 | 7.854 | 0.015 | 0.001 |

G04 | −0.070 | 0.126 | 0.315 | 0.203 | 0.005 | 0.110 | 1.027 | 1.430 | 3.241 | 2.680 | 5.033 | 2.941 | −0.039 | 0.000 |

G05 | −0.010 | 0.096 | −0.008 | 0.084 | 0.147 | 0.077 | −0.059 | 1.380 | −0.034 | 1.098 | −0.707 | 3.871 | 0.031 | 0.000 |

G06 | −0.059 | 0.075 | 0.083 | 0.125 | −0.097 | 0.078 | −1.028 | 1.324 | −0.469 | 1.713 | −2.255 | 4.417 | 0.014 | 0.000 |

G07 | −0.019 | 0.098 | 0.069 | 0.174 | −0.167 | 0.094 | 0.001 | 1.142 | 0.702 | 2.166 | 2.844 | 4.397 | 0.031 | 0.000 |

G08 | −0.071 | 0.116 | 0.066 | 0.175 | 0.064 | 0.176 | −0.472 | 0.934 | −0.781 | 2.385 | 6.746 | 8.891 | 0.017 | 0.001 |

G09 | 0.070 | 0.165 | −0.083 | 0.225 | −0.237 | 0.093 | −0.998 | 2.112 | −0.606 | 2.164 | 3.085 | 7.022 | 0.012 | 0.001 |

G10 | −0.007 | 0.174 | −0.050 | 0.124 | −0.104 | 0.089 | 0.335 | 2.486 | 0.061 | 1.337 | 0.730 | 3.393 | 0.015 | 0.001 |

G11 | −0.103 | 0.135 | −0.027 | 0.112 | 0.168 | 0.066 | −1.161 | 1.320 | −0.688 | 1.557 | 7.600 | 3.838 | −0.017 | 0.001 |

G12 | 0.033 | 0.149 | −0.020 | 0.080 | 0.020 | 0.103 | −0.447 | 1.858 | 0.084 | 1.292 | −4.620 | 3.282 | 0.033 | 0.001 |

G13 | 0.034 | 0.156 | 0.031 | 0.084 | 0.007 | 0.088 | 0.387 | 1.643 | 0.196 | 1.451 | −0.749 | 3.911 | −0.010 | 0.001 |

G14 | 0.050 | 0.225 | 0.003 | 0.073 | −0.117 | 0.066 | −0.191 | 0.858 | −0.650 | 3.300 | 0.078 | 4.042 | −0.022 | 0.000 |

G15 | −0.059 | 0.092 | −0.006 | 0.158 | −0.054 | 0.107 | −0.748 | 1.312 | −0.352 | 2.224 | 1.194 | 4.491 | 0.024 | 0.001 |

G16 | 0.015 | 0.202 | 0.072 | 0.099 | −0.006 | 0.090 | −0.101 | 2.528 | −0.859 | 1.336 | −4.817 | 5.834 | −0.007 | 0.001 |

G17 | 0.148 | 0.199 | 0.139 | 0.277 | −0.036 | 0.062 | −0.697 | 2.086 | −2.526 | 3.561 | −9.596 | 6.593 | 0.030 | 0.001 |

G19 | 0.028 | 0.173 | 0.013 | 0.080 | 0.027 | 0.053 | −0.090 | 1.000 | −0.386 | 2.211 | −1.517 | 3.482 | 0.031 | 0.001 |

G19 | 0.028 | 0.173 | 0.013 | 0.080 | 0.027 | 0.053 | −0.090 | 1.000 | −0.386 | 2.211 | −1.517 | 3.482 | 0.031 | 0.001 |

G20 | −0.053 | 0.155 | 0.015 | 0.155 | 0.066 | 0.092 | −0.075 | 2.165 | 0.720 | 1.443 | −5.388 | 2.972 | −0.012 | 0.000 |

G21 | −0.042 | 0.101 | −0.062 | 0.187 | −0.021 | 0.077 | 0.615 | 1.368 | −0.542 | 2.141 | 4.565 | 3.996 | −0.006 | 0.001 |

G22 | 0.021 | 0.154 | −0.028 | 0.087 | 0.061 | 0.080 | −0.325 | 1.688 | 0.047 | 1.138 | −0.465 | 4.762 | 0.031 | 0.000 |

G23 | 0.042 | 0.070 | 0.155 | 0.148 | −0.149 | 0.066 | −0.004 | 0.798 | 1.510 | 1.865 | 6.234 | 4.165 | 0.050 | 0.000 |

G24 | −0.021 | 0.471 | −0.044 | 0.271 | −0.144 | 0.214 | −0.529 | 1.672 | −0.305 | 6.123 | −0.451 | 8.189 | 0.012 | 0.002 |

G25 | 0.029 | 0.054 | 0.134 | 0.134 | −0.087 | 0.097 | 1.275 | 1.557 | 0.411 | 0.911 | −3.036 | 4.794 | 0.015 | 0.000 |

G26 | 0.112 | 0.127 | 0.096 | 0.093 | 0.031 | 0.128 | 1.320 | 1.493 | −0.649 | 1.055 | −2.084 | 5.556 | 0.014 | 0.001 |

G27 | −0.032 | 0.143 | −0.028 | 0.080 | 0.178 | 0.091 | −0.560 | 1.797 | −0.252 | 1.278 | 0.115 | 3.791 | 0.016 | 0.001 |

G28 | 0.077 | 0.304 | −0.014 | 0.194 | −0.042 | 0.145 | −0.472 | 3.445 | −1.127 | 1.850 | −2.172 | 4.414 | −0.022 | 0.001 |

G29 | 0.097 | 0.203 | −0.048 | 0.129 | 0.061 | 0.080 | −1.340 | 2.466 | −0.135 | 1.404 | −5.017 | 9.345 | 0.030 | 0.001 |

G30 | 0.024 | 0.092 | 0.072 | 0.093 | −0.086 | 0.087 | −0.187 | 1.315 | 0.769 | 1.232 | 3.065 | 4.180 | 0.017 | 0.000 |

G31 | −0.072 | 0.164 | 0.025 | 0.092 | −0.172 | 0.075 | −1.005 | 1.904 | 0.195 | 1.537 | −4.561 | 3.012 | 0.035 | 0.001 |

G32 | 0.030 | 0.257 | 0.051 | 0.254 | −0.181 | 0.052 | −0.598 | 1.934 | −0.712 | 3.800 | 4.074 | 4.608 | 0.017 | 0.000 |

Average | 0.013 | 0.025 | −0.036 | −0.264 | −0.023 | −0.006 | 0.014 | |||||||

Std | 0.064 | 0.086 | 0.109 | 0.689 | 0.995 | 4.502 | 0.021 |

**Table A2.**Helmert parameters relating the Glonass reference frame to the ITRF2014 reference frame of the precise ephemeris of CNES. Average values for January 2020.

TX (m) | Std_TX (m) | TY (m) | Std_TY (m) | TZ (m) | Std_TZ (m) | RX (mas) | Std_RX (mas) | RY (mas) | Std_RY (mas) | RZ (mas) | Std_RZ (mas) | Sc (ppm) | Std_Sc (ppm) | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

R01 | 0.030 | 0.134 | −0.013 | 0.140 | 0.231 | 0.186 | 0.225 | 2.535 | −0.115 | 1.508 | 2.958 | 6.688 | −0.081 | 0.001 |

R02 | −0.008 | 0.218 | −0.048 | 0.190 | 0.045 | 0.272 | 0.183 | 3.020 | −0.156 | 3.289 | −11.250 | 8.906 | −0.080 | 0.002 |

R03 | −0.050 | 0.155 | −0.019 | 0.145 | 0.028 | 0.159 | −0.386 | 2.257 | −0.768 | 2.295 | 0.885 | 5.835 | −0.080 | 0.001 |

R04 | 0.055 | 0.130 | −0.017 | 0.111 | 0.407 | 0.210 | 0.210 | 1.709 | −0.039 | 1.687 | −9.333 | 7.088 | −0.085 | 0.002 |

R05 | −0.003 | 0.103 | −0.015 | 0.117 | 0.136 | 0.180 | 0.176 | 1.755 | −0.404 | 1.969 | −6.234 | 7.836 | −0.082 | 0.001 |

R07 | −0.022 | 0.107 | −0.059 | 0.135 | 0.035 | 0.258 | 0.061 | 2.456 | −0.865 | 2.348 | 1.454 | 12.849 | −0.080 | 0.002 |

R08 | −0.088 | 0.188 | −0.070 | 0.238 | 0.158 | 0.238 | −0.079 | 3.245 | −0.785 | 2.524 | 0.275 | 9.565 | −0.080 | 0.001 |

R09 | 0.015 | 0.157 | 0.057 | 0.125 | −0.008 | 0.272 | −0.243 | 2.630 | 0.720 | 1.653 | 16.964 | 5.741 | −0.057 | 0.002 |

R11 | 0.021 | 0.274 | 0.083 | 0.187 | −0.050 | 0.247 | 0.318 | 3.861 | −0.392 | 3.380 | 8.649 | 10.854 | −0.082 | 0.002 |

R11 | 0.021 | 0.274 | 0.083 | 0.187 | −0.050 | 0.247 | 0.318 | 3.861 | −0.392 | 3.380 | 8.649 | 10.854 | −0.082 | 0.002 |

R12 | −0.008 | 0.127 | 0.023 | 0.123 | 0.076 | 0.214 | −0.424 | 2.142 | −0.585 | 1.808 | 7.451 | 10.115 | −0.079 | 0.002 |

R13 | 0.020 | 0.273 | −0.009 | 0.279 | 0.161 | 0.437 | 0.028 | 4.544 | −0.090 | 4.261 | 10.536 | 17.612 | −0.081 | 0.003 |

R14 | 0.029 | 0.137 | 0.021 | 0.168 | 0.031 | 0.241 | 0.494 | 2.411 | −0.998 | 1.869 | −3.003 | 9.143 | −0.083 | 0.001 |

R15 | −0.002 | 0.124 | 0.020 | 0.131 | 0.093 | 0.211 | 0.068 | 1.951 | −0.713 | 1.509 | 4.912 | 8.398 | −0.081 | 0.002 |

R16 | 0.040 | 0.138 | 0.027 | 0.141 | 0.145 | 0.240 | 0.459 | 2.449 | −0.687 | 1.717 | −4.259 | 9.195 | −0.083 | 0.001 |

R17 | −0.028 | 0.161 | −0.041 | 0.161 | −0.522 | 0.145 | 0.041 | 1.854 | −0.919 | 3.291 | 7.842 | 8.461 | −0.079 | 0.002 |

R18 | −0.023 | 0.183 | −0.056 | 0.159 | −0.330 | 0.123 | −0.107 | 1.529 | −0.978 | 3.257 | 2.069 | 8.527 | −0.081 | 0.001 |

R19 | 0.031 | 0.184 | −0.052 | 0.145 | −0.411 | 0.184 | 0.088 | 1.791 | −0.441 | 3.590 | −2.504 | 11.598 | −0.079 | 0.003 |

R20 | 0.029 | 0.203 | −0.043 | 0.187 | −0.301 | 0.150 | 0.496 | 1.863 | −0.525 | 3.990 | 2.336 | 9.321 | −0.095 | 0.001 |

R21 | −0.014 | 0.167 | −0.027 | 0.143 | −0.306 | 0.118 | 0.089 | 1.646 | −0.915 | 2.784 | −5.814 | 6.011 | −0.080 | 0.001 |

R22 | 0.184 | 0.273 | −0.040 | 0.230 | 0.219 | 0.177 | 0.782 | 2.639 | 1.377 | 5.168 | −4.685 | 12.145 | −0.080 | 0.002 |

R23 | −0.029 | 0.240 | −0.023 | 0.210 | −0.313 | 0.207 | −0.289 | 1.875 | −0.343 | 4.151 | 8.951 | 11.344 | −0.080 | 0.002 |

R24 | ||||||||||||||

Average | 0.009 | −0.010 | −0.024 | 0.114 | −0.410 | 1.675 | −0.080 | |||||||

std | 0.051 | 0.044 | 0.239 | 0.301 | 0.567 | 7.142 | 0.006 |

**Table A3.**Helmert parameters relating the Galileo reference frame to the ITRF2014 reference frame of the precise ephemeris of CNES. Average values for January 2020.

TX (m) | Std_TX (m) | TY (m) | Std_TY (m) | TZ (m) | Std_TZ (m) | RX (mas) | Std_RX (mas) | RY (mas) | Std_RY (mas) | RZ (mas) | Std_RZ (mas) | Sc (ppm) | Std_Sc (ppm) | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

E01 | −0.021 | 0.073 | −0.024 | 0.054 | 0.038 | 0.063 | −0.161 | 0.537 | 0.153 | 0.625 | 0.684 | 0.708 | −0.003 | 0.000 |

E02 | −0.004 | 0.051 | −0.019 | 0.057 | 0.009 | 0.052 | −0.267 | 0.430 | 0.287 | 0.777 | 0.250 | 0.695 | −0.004 | 0.000 |

E03 | −0.002 | 0.043 | 0.012 | 0.070 | 0.020 | 0.066 | −0.291 | 0.484 | −0.092 | 0.611 | 0.668 | 0.556 | −0.005 | 0.000 |

E04 | −0.013 | 0.065 | −0.021 | 0.072 | 0.054 | 0.103 | −0.188 | 0.624 | −0.289 | 0.596 | 0.531 | 1.188 | −0.005 | 0.000 |

E05 | 0.002 | 0.045 | −0.003 | 0.061 | 0.093 | 0.061 | −0.134 | 0.621 | 0.054 | 0.644 | 0.528 | 0.665 | −0.007 | 0.000 |

E07 | −0.023 | 0.063 | −0.019 | 0.052 | 0.051 | 0.066 | 0.062 | 0.546 | −0.160 | 0.432 | 0.800 | 0.711 | −0.002 | 0.000 |

E08 | −0.006 | 0.047 | −0.016 | 0.069 | 0.020 | 0.076 | −0.108 | 0.514 | 0.132 | 0.488 | 0.441 | 0.553 | −0.005 | 0.000 |

E09 | −0.023 | 0.051 | −0.018 | 0.054 | 0.018 | 0.066 | −0.088 | 0.624 | 0.008 | 0.530 | 0.216 | 0.634 | −0.004 | 0.000 |

E11 | −0.001 | 0.071 | −0.018 | 0.110 | 0.035 | 0.118 | −0.337 | 1.068 | 0.030 | 1.009 | 0.660 | 1.419 | −0.001 | 0.000 |

E12 | −0.025 | 0.070 | 0.017 | 0.061 | 0.056 | 0.061 | −0.235 | 0.613 | 0.144 | 0.772 | 1.593 | 0.994 | −0.002 | 0.000 |

E13 | −0.003 | 0.057 | 0.010 | 0.064 | −0.003 | 0.050 | −0.393 | 0.537 | −0.080 | 0.620 | 0.963 | 1.066 | −0.008 | 0.000 |

E14 | ||||||||||||||

E15 | 0.008 | 0.047 | 0.009 | 0.070 | −0.029 | 0.051 | −0.230 | 0.579 | 0.224 | 0.607 | 0.854 | 0.927 | −0.005 | 0.000 |

E18 | ||||||||||||||

E19 | −0.027 | 0.055 | 0.022 | 0.066 | 0.274 | 0.073 | −0.316 | 0.574 | 0.097 | 0.896 | 0.085 | 1.018 | −0.001 | 0.000 |

E21 | −0.051 | 0.072 | −0.036 | 0.062 | 0.006 | 0.074 | −0.136 | 0.496 | −0.099 | 0.842 | 0.370 | 0.580 | −0.009 | 0.001 |

E24 | −0.016 | 0.076 | −0.002 | 0.083 | 0.019 | 0.107 | −0.137 | 0.710 | −0.050 | 0.685 | 0.181 | 0.769 | −0.003 | 0.001 |

E25 | −0.020 | 0.066 | 0.008 | 0.058 | 0.002 | 0.067 | −0.213 | 0.483 | −0.109 | 0.596 | 0.042 | 0.435 | −0.006 | 0.000 |

E26 | 0.008 | 0.068 | −0.033 | 0.062 | 0.026 | 0.057 | −0.431 | 0.643 | 0.036 | 0.590 | 1.011 | 0.841 | −0.006 | 0.000 |

E27 | −0.030 | 0.063 | 0.011 | 0.081 | −0.007 | 0.075 | −0.277 | 0.612 | −0.037 | 0.520 | 0.441 | 0.941 | −0.005 | 0.000 |

E30 | −0.036 | 0.054 | −0.013 | 0.057 | 0.005 | 0.065 | −0.149 | 0.511 | 0.138 | 0.718 | 0.271 | 0.869 | −0.003 | 0.000 |

E31 | −0.050 | 0.070 | −0.025 | 0.063 | 0.053 | 0.073 | −0.235 | 0.540 | −0.066 | 0.550 | 0.574 | 0.724 | −0.005 | 0.000 |

E33 | −0.008 | 0.069 | −0.005 | 0.073 | 0.023 | 0.066 | −0.434 | 0.848 | 0.042 | 0.633 | 1.062 | 0.680 | −0.005 | 0.000 |

E36 | −0.001 | 0.058 | 0.006 | 0.065 | 0.014 | 0.089 | −0.329 | 0.671 | 0.101 | 0.480 | 0.474 | 0.496 | −0.005 | 0.000 |

Average | −0.015 | −0.007 | 0.035 | −0.229 | 0.021 | 0.577 | −0.005 | |||||||

std | 0.017 | 0.017 | 0.059 | 0.120 | 0.135 | 0.370 | 0.002 |

**Table A4.**Helmert parameters relating the Beidou reference frame to the ITRF2014 reference frame of the precise ephemeris of CODE. Average values for January 2020.

Orbit Type | TX (m) | Std_TX (m) | TY (m) | Std_TY (m) | TZ (m) | Std_TZ (m) | RX (mas) | Std_RX (mas) | RY (mas) | Std_RY (mas) | RZ (mas) | Std_RZ (mas) | Sc (ppm) | Std_Sc (ppm) | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

IGSO | C06 | −0.010 | 0.591 | −0.303 | 0.338 | 0.362 | 0.209 | 2.230 | 1.760 | 3.522 | 1.523 | −5.739 | 5.415 | −0.027 | 0.000 |

IGSO | C07 | −0.636 | 0.573 | −1.247 | 0.454 | 0.115 | 0.335 | 2.232 | 3.277 | 2.154 | 3.446 | 0.176 | 8.256 | −0.002 | 0.002 |

IGSO | C08 | −0.998 | 0.927 | −1.903 | 1.667 | 0.078 | 0.478 | −2.057 | 3.929 | 4.229 | 3.180 | −5.988 | 7.477 | −0.003 | 0.001 |

IGSO | C09 | 0.339 | 0.730 | −0.066 | 0.273 | 0.135 | 0.289 | 1.982 | 1.276 | 0.476 | 2.170 | −5.809 | 5.322 | −0.030 | 0.000 |

IGSO | C10 | −1.417 | 0.821 | −0.835 | 0.213 | 0.152 | 0.309 | 2.311 | 1.303 | −2.750 | 3.054 | 6.918 | 5.200 | −0.019 | 0.001 |

MEO | C11 | −0.126 | 0.563 | 0.124 | 0.503 | −0.402 | 0.393 | 1.675 | 4.108 | −1.418 | 8.841 | −1.229 | 9.970 | −0.028 | 0.002 |

MEO | C12 | −0.145 | 0.650 | 0.226 | 0.507 | −0.467 | 0.465 | 3.745 | 4.368 | −1.500 | 8.715 | −0.165 | 9.271 | −0.020 | 0.002 |

IGSO | C13 | −1.244 | 0.525 | −0.751 | 0.257 | 0.114 | 0.203 | −0.420 | 1.298 | 2.481 | 1.600 | −1.932 | 3.658 | −0.036 | 0.000 |

MEO | C14 | −0.055 | 0.499 | 0.286 | 0.560 | −0.709 | 0.487 | 2.284 | 4.577 | 0.454 | 6.516 | 3.984 | 10.634 | −0.028 | 0.005 |

IGSO | C16 | 0.122 | 0.598 | 0.049 | 0.346 | 0.254 | 0.180 | 2.229 | 1.698 | 3.084 | 1.443 | −5.497 | 5.389 | −0.021 | 0.000 |

Average | −0.417 | −0.442 | −0.037 | 1.621 | 1.073 | −1.528 | −0.021 | ||||||||

std | 0.613 | 0.725 | 0.356 | 1.645 | 2.388 | 4.453 | 0.011 |

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**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 Toc for the clock.

**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.**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 × 10

^{6}m should be used. Error bars are 1 sigma formal uncertainties. Satellite G01 for day 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.

**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 × 10

^{6}m should be used. Error bars are 1 sigma formal uncertainties. Satellite E01 for day 2 January 2020.

**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. For T (plot (

**d**)) pre-fit (dots) refer to the left y-axis and post-fit (diamonds) to the right y-axis.

**Figure 7.**(

**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 42 × 10

^{6}m should be used. Error bars are 1 sigma formal uncertainties. Satellite C07 for day 2 January 2020.

**Figure 8.**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 C12. Post-fits of the time offset T (diamonds) are plotted on the right y-axis.

**Figure 9.**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 × 10

^{6}m should be used. Error bars are 1 sigma formal uncertainties. Satellite C12 for day 2 January 2020.

**Figure 10.**Structure of the partial derivative matrix H of the pre-fit discrepancies relative to the arc parameters. The arc parameters are indexed with the time Toc (Toe is equivalent) and the nominal values are dependent on the individual arcs being 2 h or 1 h long.

**Figure 11.**Pre-fit (dots) and post-fit (diamonds) residuals of (

**a**) X, (

**b**) Y, (

**c**) Z, and (

**d**) T for R01. Length of arc for fit is 1 h. Different colours denote different ephemeris blocks.

**Figure 12.**Corrections to the broadcast (

**a**) clock, (

**b**) positions, (

**c**) velocities, and (

**d**) lunisolar accelerations for R01 using CNES precise orbits as reference. The vector of state (3 clock parameters + 9 orbit parameters) is estimated at intervals of 1 h. Rate of clock drift (a

_{2}) was computed but is not shown.

**Figure 13.**Plots of the differences between IGS high rate clocks and the prediction of our clock polynomials based on 15 min sampling. Update rate of the clock polynomials are 2 h for (

**a**) G01, (

**b**) E01, (

**c**) 1 h for R01, (

**d**) C07, and (

**e**) C12. The mean and rms of the residuals are provided for each satellite.

**Figure 14.**Zoom of the first 4 h of the post-fit residuals of G01 shown in Figure 2 (red diamonds) showing an oscillatory pattern ((

**a**) X, (

**b**) Y, and (

**c**) Z). The blue dots represent an attempt to model the oscillation with a signal driven by the third zonal harmonics of the Earth’s gravity field J

_{3}.

**Figure 15.**Spectrum of the pre-fit (top) and post-fit (bottom) residuals of the coordinates of the GPS satellites relative to the CNES precise ephemeris, projected on the radial, tangential and across track basis. Average values for January 2020.

**Figure 16.**Spectrum of the pre-fit (top) and post-fit (bottom) residuals of the coordinates of the Glonass satellites relative to the CNES precise ephemeris, projected on the radial, tangential and across track basis. Average values for January 2020.

**Figure 17.**Spectrum of the pre-fit (top) and post-fit (bottom) residuals of the coordinates of the Galileo satellites relative to the CNES precise ephemeris, projected on the radial, tangential and across track basis. Average values for January 2020.

**Figure 18.**Spectrum of the pre-fit (top) and post-fit (bottom) residuals of the coordinates of the Beidou satellites (IGSO and MEO) relative to the CODE precise ephemeris, projected on the radial, tangential and across track basis. Average values for January 2020.

**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.

Tx (m) | Ty (m) | Tz (m) | Rx (mas) | Ry (mas) | Rz (mas) | Scale | |
---|---|---|---|---|---|---|---|

Estimated | −0.09 | 0.20 | −0.39 | 1.2 | 1.2 | −11.1 | 0.60 × 10^{−08} |

1 sigma formal error | 0.04 | 0.04 | 0.02 | 0.00 | 0.00 | 0.00 | 8.48 × 10^{−10} |

**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.

Tx (m) | Ty (m) | Tz (m) | Rx (mas) | Ry (mas) | Rz (mas) | Scale | |
---|---|---|---|---|---|---|---|

Estimated | −0.05 | −0.03 | −0.04 | 0.3 | −0.6 | −0.8 | 0.70 × 10^{−10} |

1 sigma formal error | 0.02 | 0.02 | 0.02 | 0.29 | 0.17 | 0.06 | 4.68 × 10^{−10} |

**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.

Tx (m) | Ty (m) | Tz (m) | Rx (mas) | Ry (mas) | Rz (mas) | Scale | |
---|---|---|---|---|---|---|---|

Estimated | 0.00 | 0.00 | 0.00 | −4.5 | 0.7 | 5.0 | 2.29 × 10^{−8} |

1 sigma formal error | 0.00 | 0.00 | 0.00 | 1.70 | 3.20 | 0.06 | 1.73 × 10^{−9} |

**Table 4.**Helmert parameters relating the origin, orientation and scale of the Broadcast reference frame of C12 relative to the IGS14 frame of the SP3 precise ephemeris for SV C12, 2 January 2020. The scale factor accounts for the Center of Mass–Antenna Phase Center correction in the radial direction.

Tx (m) | Ty (m) | Tz (m) | Rx (mas) | Ry (mas) | Rz (mas) | Scale | |
---|---|---|---|---|---|---|---|

Estimated | 0.26 | 0.02 | −0.70 | −0.6 | 1.9 | −2.8 | 1.25 × 10^{−8} |

1 sigma formal error | 0.06 | 0.05 | 0.02 | 0.53 | 0.69 | 0.12 | 1.17 × 10^{−9} |

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**MDPI and ACS Style**

Caporali, A.; Zurutuza, J.
Broadcast Ephemeris with Centimetric Accuracy: Test Results for GPS, Galileo, Beidou and Glonass. *Remote Sens.* **2021**, *13*, 4185.
https://doi.org/10.3390/rs13204185

**AMA Style**

Caporali A, Zurutuza J.
Broadcast Ephemeris with Centimetric Accuracy: Test Results for GPS, Galileo, Beidou and Glonass. *Remote Sensing*. 2021; 13(20):4185.
https://doi.org/10.3390/rs13204185

**Chicago/Turabian Style**

Caporali, Alessandro, and Joaquin Zurutuza.
2021. "Broadcast Ephemeris with Centimetric Accuracy: Test Results for GPS, Galileo, Beidou and Glonass" *Remote Sensing* 13, no. 20: 4185.
https://doi.org/10.3390/rs13204185