Characterization of Doppler collision and its impact on carrier phase ambiguity resolution using geostationary satellites

Abstract

Doppler collision is a unique phenomenon in GNSS where tracking errors are introduced in the measurements due to cross-correlation between two or more satellites. It occurs when the relative Doppler frequencies of the satellites are less than the code loop bandwidth. Doppler collisions affect geostationary satellites for longer durations, due to their small line-of-sight velocities. This is a major concern for regional constellations such as IRNSS where geostationary satellites form a major part of the space segment. Doppler collision error resembles code multipath and if not mitigated could affect the ability to use pseudoranges of geostationary satellites in RTK positioning. We describe likely conditions for Doppler collision, derive a Doppler collision error envelope for L1 C/A code geostationary pseudorange measurements, and then demonstrate the effect using simulated and live signals. Results indicate that the error due to Doppler collision is not purely biased and varies with a mean value close to zero. The novelty includes analysis of Doppler collision effects on an RTK solution using geostationary satellites with emphasis on ambiguity convergence time. De-weighting of geostationary observations is proposed to reduce the impact of Doppler collision.

Appendix

Doppler collision error is determined by the cross-correlation function and the relative code phase. The error follows a bound similarly to a multipath error envelope (Spilker et al. 1996; Cox et al. 1999). In order to obtain Doppler collision error, code phases of the primary peak and the cross-correlation peak are obtained using true ranges. The relative code phase will determine the overlapping geographical area and the magnitude of Doppler collision.

If the user position is not known, the true ranges cannot be determined. In such cases, pseudorange code phases can be used to generate the error envelope. However, the pseudorange code phases are affected by the presence of multiple GEO satellites. Consider a case of only two WAAS satellites in view. The observed code phases of the two satellites are given by

$$\tau_{eff1} = \tau_{1} \pm \Delta \tau_{DC}$$

(13)

$$\tau_{eff2} = \tau_{2} \mp \Delta \tau_{DC} ,$$

(14)

where $\tau_{1}$ and $\tau_{2}$ are the pseudorange code phases for the first and the second satellite, respectively, with no Doppler collision conditions, and $\Delta \tau_{DC}$ is code phase error due to Doppler collision. The effect of Doppler collision on desired satellite is exactly equal but opposite in magnitude on the second satellite. The Doppler collision error follows a multipath envelope, and thus the code measurement errors (in chips) are given by (Ray 2000)

$$\tau_{\text{error}} = \pm \frac{{\alpha \tau_{d} }}{1 + \alpha },\,{\text{in}}\,{\text{the}}\,{\text{range}}\,{\text{of}}\,0 \le \tau_{d} < (1 + \alpha )T_{d}$$

(15)

$$\tau_{\text{error}} = \pm \alpha T_{d} ,\,{\text{in}}\,{\text{the}}\,{\text{range}}\,{\text{of}}\,(1 \pm \alpha )T_{d} \le \tau_{d} < Tc - (1 \mp \alpha )T_{d}$$

(16)

$$\tau_{\text{error}} = \pm \frac{{\alpha (T_{c} + T_{d} - \tau_{d} )}}{(2 \mp \alpha )},\,{\text{in}}\,{\text{the}}\,{\text{range}}\,{\text{of}}\,T_{c} - (1 \mp \alpha )T_{d} \le \tau_{d} < T_{c} + T_{d} ,$$

(17)

where α is the ratio of the cross-correlation peak to primary peak, $\tau_{d}$ is the code delay of cross-correlation peak, $T_{d}$ is the chip spacing, and $T_{c}$ is the code length.

If the observed code phase is used, the relative delay will be affected by two times $\Delta \tau_{DC}$ and the code delay of the cross-correlation peak during Doppler collision is

$$\tau_{d}^{DC} = \tau_{d} \pm 2\Delta \tau_{DC} .$$

(18)

Thus the offset in code measurement error is given as

$$\Delta \tau_{\text{error}} = \mp \frac{\alpha }{1 + \alpha }2\Delta \tau_{DC} ,\,{\text{in}}\,{\text{the}}\,{\text{range}}\,{\text{of}}\,0 \le \tau_{d} < (1 + \alpha )T_{d}$$

(19)

$$\Delta \tau_{\text{error}} = 0,\,{\text{in}}\,{\text{the}}\,{\text{range}}\,{\text{of}}\,(1 \pm \alpha )T_{d} \le \tau_{d} < Tc - (1 \mp \alpha )T_{d}$$

(20)

$$\Delta \tau_{\text{error}} = \pm \frac{{\alpha (T_{c} + T_{d} \mp 2\Delta \tau_{DC} )}}{(2 \mp \alpha )},\,{\text{in}}\,{\text{the}}\,{\text{range}}\,{\text{of}}\,T_{c} - (1 \mp \alpha )T_{d} \le \tau_{d} < T_{c} + T_{d} .$$

(21)

If the relative signal strength of the WAAS satellites is zero, α = 0.0616. For a standard correlator $T_{d} = 1$ and the maximum error due to Doppler collision is $\Delta \tau_{DC} =$ 0.0304 chip (~9 m). Thus, the maximum offset in Doppler collision error caused by approximating the true range code phases with pseudorange code phases is given by

$$\Delta \tau_{\text{error}} = 0. 0 0 3 5\,{\text{chip}} \cong 1.03\,{\text{m}},\,{\text{in}}\,{\text{the}}\,{\text{range}}\,{\text{of}}\,0 \le \tau_{d} < (1 + \alpha )T_{d}$$

(22)

$$\Delta \tau_{\text{error}} = 0. 0 {\text{ chip,}}\,{\text{in}}\,{\text{the}}\,{\text{range}}\,{\text{of}}\,(1 \pm \alpha )T_{d} \le \tau_{d} < Tc - (1 \mp \alpha )T_{d}$$

(23)

$$\Delta \tau_{\text{error}} = 0. 0 0 1 8\,{\text{chip}} \cong 0.53\,{\text{m}},\,{\text{in}}\,{\text{the}}\,{\text{range}}\,{\text{of}}\,T_{c} - (1 \mp \alpha )T_{d} \le \tau_{d} < T_{c} + T_{d} .$$

(24)

Considering that the error envelope varies up to 9 m, an offset less than 1 m is acceptable. The error envelope is computed using pseudorange code phases for Rx1; the reference receiver used in Doppler collision simulation, Fig. 4, is shown in Fig. 22. The pseudorange-based error envelope is compared against the error envelope obtained using the true geometric ranges. The difference between the envelopes never exceeds 1.1 m and has a standard deviation of 0.24 m. The variation is largest during rising and fall of the error envelope.

Fig. 22

Doppler collision error envelope using code phase from pseudorange and geometric range