Triple-frequency GPS precise point positioning with rapid ambiguity resolution

Abstract

At present, reliable ambiguity resolution in real-time GPS precise point positioning (PPP) can only be achieved after an initial observation period of a few tens of minutes. In this study, we propose a method where the incoming triple-frequency GPS signals are exploited to enable rapid convergences to ambiguity-fixed solutions in real-time PPP. Specifically, extra-wide-lane ambiguity resolution can be first achieved almost instantaneously with the Melbourne-Wübbena combination observable on L2 and L5. Then the resultant unambiguous extra-wide-lane carrier-phase is combined with the wide-lane carrier-phase on L1 and L2 to form an ionosphere-free observable with a wavelength of about 3.4 m. Although the noise of this observable is around 100 times the raw carrier-phase noise, its wide-lane ambiguity can still be resolved very efficiently, and the resultant ambiguity-fixed observable can assist much better than pseudorange in speeding up succeeding narrow-lane ambiguity resolution. To validate this method, we use an advanced hardware simulator to generate triple-frequency signals and a high-grade receiver to collect 1-Hz data. When the carrier-phase precisions on L1, L2 and L5 are as poor as 1.5, 6.3 and 1.5 mm, respectively, wide-lane ambiguity resolution can still reach a correctness rate of over 99 % within 20 s. As a result, the correctness rate of narrow-lane ambiguity resolution achieves 99 % within 65 s, in contrast to only 64 % within 150 s in dual-frequency PPP. In addition, we also simulate a multipath-contaminated data set and introduce new ambiguities for all satellites every 120 s. We find that when multipath effects are strong, ambiguity-fixed solutions are achieved at 78 % of all epochs in triple-frequency PPP whilst almost no ambiguities are resolved in dual-frequency PPP. Therefore, we demonstrate that triple-frequency PPP has the potential to achieve ambiguity-fixed solutions within a few minutes, or even shorter if raw carrier-phase precisions are around 1 mm. In either case, we conclude that the efficiency of ambiguity resolution in triple-frequency PPP is much higher than that in dual-frequency PPP.

Appendix: Tracking loop measurement errors

The dominant sources for the phase error that originates in a GPS receiver phase-locked loop (PLL) include its thermal noise, oscillator noise and dynamic stress error (Ward et al. 2006). Normally, the PLL thermal noise is taken as the only source of the carrier tracking error, because the oscillator noise can be either transient or negligible, and the dynamic stress error can be ignored for a stationary receiver. In particular, the thermal noise in the unit of length for an arctangent PLL can be approximated as

$$\begin{aligned} \sigma _\mathrm{PLL}=\dfrac{\lambda }{2\pi }\sqrt{\dfrac{W_\mathrm{n}}{C/N_0}\left(1+\dfrac{1}{2T_\mathrm{i}C/N_0}\right)} \end{aligned}$$

(12)

where $\lambda$ is the carrier-phase wavelength; $W_\mathrm{n}$ is the carrier loop noise bandwidth in Hz; $C/N_0$ is the carrier-to-noise ratio expressed as a ratio in Hz, and $10\log C/N_0=(C/N_0)_\mathrm{dB}$ which is expressed in dB-Hz; and $T_\mathrm{i}$ is the predetection integration time in seconds. In this study, Eq. (12) is used to quantitatively assess the nominal precision of the simulated carrier-phase measurements. For the Septentrio PolaRxS Pro receiver in this study, $W_\mathrm{n}$ is 15 Hz and $T_\mathrm{i}$ is 10 ms. $(C/N_0)_\mathrm{dB}$ is derived for each frequency as the mean of the signal strength measurements given by the receiver.

In addition, if there are no multipath or interference effects, the dominant sources of the pseudorange error in a GPS receiver delay-locked loop (DLL) include its thermal noise and dynamic stress error (Braasch and Van Dierendonck 1999). To quantify the precision of the simulated pseudorange measurements, we have to know the early-to-late correlator spacing (ELP) besides $W_\mathrm{n}$, $T_\mathrm{i}$ and $C/N_0$ (Van Dierendonck et al. 1992; Ward et al. 2006). Unfortunately, ELP is kept by Septentrio as proprietary information and we cannot access this quantity. In this case, we tried various possible values for ELP, but found that the calculated nominal precision for the simulated pseudorange measurements is too high, in contrast to the nominal precision of carrier-phase measurements. Finally, we conservatively presume 3 m for the pseudorange precision throughout this study.