GPS satellite clock estimation using global atomic clock network

Abstract

We report the GPS satellite clock estimation using 20 globally distributed receivers with an external hydrogen maser atomic clock. By applying corrections for the Sagnac effect, the relativistic effect due to orbit eccentricity, tropospheric and ionospheric delays, satellite and receiver antenna phase center offsets and variations, solid earth tides, ocean tide loading, phase wind-up effect, and P1-C1 bias, our satellite clock results matches the IGS final clock product within  ± 1.4 ns with comparable frequency stability for an averaging time of less than 1000 sec and a 10–30% worse frequency stability for an averaging time of greater than 1000 sec, on MJD 58244. This small atomic clock network results in a fast computation that becomes increasingly appealing when the real-time satellite orbit and clock estimation is needed and as the GNSS constellations and the GNSS signals expand.

Appendices

Appendix

Appendix A: performance of our method at day boundaries

When comparing ground high-precision atomic clocks using GPS carrier phase, such as hydrogen masers, the day boundary discontinuity at the level of 100 – 200 ps often occurs because the GPS data are processed on a daily basis which results in different estimations of phase ambiguities between two consecutive days (Yao and Levine 2012). This discontinuity problem becomes less serious for the satellite clock estimation since the noise of GPS satellite clocks obscures it.

Thanks to the forward-only processing in our method, we can avoid the discontinuity at day boundaries, as shown by Fig.

Fig. 8

Performance of our method at day boundaries. The time period ranges from MJD 58244.0 to MJD 58254.0. PTBB and OPMT (see Fig. 1) are receivers at NMIs

8. Remember that PTBB and OPMT (see the map in Fig. 1) are receivers referencing to external UTC(k). The bias of ~ 215.5 ns between the two receivers is caused by the time delays from cables connecting to receivers. We can see that there are six obvious day boundary discontinuities (i.e., at time tags of 1.0, 2.0, 3.0, 5.0, 6.0, and 9.0) in the IGS result. In contrast, our result exhibits no day boundary discontinuities. We also want to mention that the blue curve and the red curve have pattern similarities (e.g., both curves have a dent at 2.5 d and a spike at 4.1 d), which indicates that our method is able to observe the behavior of UTC(k). Admittedly, our result is noisier than the IGS result, consistent with what we have found in Fig. 7.

Appendix B: satellite clock estimation using 20 receivers without external atomic clocks

In this appendix, we investigate the satellite clock estimation performance using 20 receivers without external atomic clocks (for the sake of simplicity, we call these receivers “plain receivers”), instead of using 20 receivers with external hydrogen masers discussed in the body text. A plain receiver typically has its internal quartz clock disciplined to the GPS system time with an accuracy of a few nanoseconds (Misra 1996). To make a fair comparison between the disciplined quartz clock network and the atomic clock network, the distribution of the 20 plain receivers is nearly the same as the distribution shown in Fig. 1. To be specific, these plain receivers are VALD, BLYT, WILL, BAKE, SCH2, HNLC, BRAZ, FALK, ZAMB, RABT, VIS0, QAQ1, GRAZ, MOBN, POL2, YSSK, SHAO, PIMO, MRO1, and MOBS. We use the same algorithm described in Section “Method of Computing Satellite Clocks” to process the GPS data. The noise parameters for receiver clocks in the Kalman filter are adjusted accordingly.

An intuitive impression of the disciplined quartz clock network is that this network cannot accurately estimate satellite clocks since these quartz clocks are orders of magnitude noisier than satellite clocks. The quartz clocks are disciplined to the GPS system time formed by satellite clocks, and thus, lack of independence. As a commonsense, it is impossible to evaluate a standard using a device calibrated by this standard. However, for our case, the function of the disciplined quartz clock network is not to evaluate the satellite clocks directly but to establish links between satellite clocks. One satellite can be linked to other satellites via receivers. In other words, satellite clocks are monitoring each other with the help of this disciplined quartz clock network. Therefore, this network, though receiver clocks are noisy, still enables the estimation of satellite clocks. Nevertheless, the performance of this disciplined quart clock network is expected to be inferior to that of the atomic clock network because of the following reasons. First, considering the disciplined quartz clocks have little weights, the number of effective clocks becomes 32 (i.e., the number of satellite clocks). In contrast, the number of effective clocks in the atomic clock network method is 32 + 20 = 52 (note: 20 is the number of receivers). Therefore, the disciplined quartz clock network method is not as robust as the atomic clock network method. More importantly, whenever we need to re-estimate the phase ambiguity, such as at the occurrence of cycle slips, the rise of a satellite, or measurement outliers, the uncertainty of the phase-ambiguity estimation in the atomic clock network method is only a few centimeters, as mentioned in the section on Method of Computing Satellite Clocks. However, this small uncertainty does not apply to the disciplined quartz clock network method, because the $\Delta {t}_{i}$ uncertainty in (5) becomes as large as a few nanoseconds. Therefore, the uncertainty of the phase-ambiguity estimation shall be at the level of 1 meter, much larger than that in the atomic clock network method. Since phase ambiguity correlates with clocks, this large phase-ambiguity uncertainty makes satellite clock estimation susceptible. In addition, the atomic clock network method can easily identify the measurement anomalies and then remove/de-weight the measurements, while the disciplined quartz clock network method can hardly identify those anomalies that are below the noise of the disciplined quartz clock.

The black curves in Fig.

Fig. 9

Comparison between our result using disciplined quartz clocks (black curves) and our result using hydrogen masers (blue curves). The blue and red curves are the same as those in Fig. 7

9 show our result using the disciplined quartz clock network. We can see that the frequency stability of IIR Rb clocks, IIR-M Rb clocks, and IIF Rb clocks is approximately 60% noisier than if the atomic clock network was used, for an averaging time of 1000 sec – 12 hours. This is consistent with the above analysis that we are able to observe the satellite clocks using disciplined quartz clocks, but not as good as using high-precision atomic clocks. As for the frequency stability of IIF Cs clocks (Fig. 9(d)), the black and blue curves are nearly the same. This can be understood by the fact that the IIF Cs clocks are among the noisiest clocks in the GPS constellation. The advantages brought by the atomic clock network, as discussed in the previous paragraph, are obscured by the noise of IIF Cs clocks.