Keywords

1 Introduction

The Compass/Beidou Navigation Satellite System is a global navigation satellite system (GNSS), which is independently developed, deployed, and operated by China. This system is still in progress, and is designed to provide global coverage around 2020. At present, only the Beidou regional navigation satellite system has been built up, which can provide coverage in the Asia–Pacific region with Positioning, Navigation, Timing (PNT), and short-message communication service capabilities [1, 2].

The PNT performance of GNSS is directly related to the satellite orbit accuracy. The precise orbit determination (POD) for BeiDou satellites can reach a radial accuracy of decimeter level currently, which is limited in tracking hour and geometric strength of regional configuration of ground-tracks observing in China area [3–5]. The POD performance can be improved by adopting some appropriate data processing techniques, such as differenced data processing. Differences of the original observations allow to eliminate or to reduce some biases, such as clock offset of satellites and receivers. The single-difference is simply defined as determining the relative differences between a pair of receivers measuring satellite simultaneously, and the double-difference means differencing between a pair of receivers and a pair of satellites [6].

In this paper a multi-satellite POD approach of processing double difference observations of phase and pseudorange observations for BeiDou regional navigation satellite system is presented, and the orbit accuracy assessment is performed by the overlapping difference and post fit of satellite laser ranging (SLR) residuals. In addition, to investigate the contribution of the configuration of ground-tracks geometry to the POD accuracy, measurements from several IGS Multi-GNSS Experiment (IGS M-GEX) monitoring stations with BeiDou tracking capability are processed. And some initially results are obtained.

2 Observation Modeling

The BeiDou satellites transmit navigation signals in Quadrature Phase-Shift Keying (QPSK) modulation on a total of three frequency bands (B1, B2, B3) [1, 7], which can form many high-quality combinations for navigation data process. Refer to paper [8], the Beidou pseudorange at high elevation can reach an accuracy of 20 ~ 40 cm, and the accuracy of phase at high elevation is about 3 mm.

2.1 Double-Difference Observations Modeling

Beidou navigation system’s original phase observations can be modeled as

$$ \phi_{r}^{s} (t) = f{{\rho (t_{s} ,t_{r} )} \mathord{\left/ {\vphantom {{\rho (t_{s} ,t_{r} )} c}} \right. \kern-0pt} c}\, - \,f\delta t_{s} \, + \,f\delta t_{r} \, + \,\Updelta \phi_{\text{trop}} \, - \,\Updelta \phi_{\text{iono}} \, + \,\Updelta \phi_{\text{rel}} \, + \,\Updelta \phi_{\text{mult}} \, + \,N_{r}^{s} \, + \,\varepsilon (t) $$
(7.1)

where, \( \phi_{r}^{s} \) is the phase observation from satellite \( s \) to receiver \( r,\;t \) is the observation epoch, \( f\,\left( {=\begin{array}{*{20}c} {\text{B1,}} \hfill & {\text{B2,}} \hfill & {\text{B3}} \hfill \\ \end{array} } \right) \) is the signal frequency, \( \rho \left( {t_{s} ,t_{r} } \right) \) is the geometric distance from satellite to receiver, \( c \) is velocity of light, \( \delta t_{s} \) is the satellite clock offset, \( \delta t_{r} \) is the receiver clock offset, \( \Updelta \phi_{\text{trop}} \) is tropospheric delay error, \( \Updelta \phi_{\text{iono}} \) is ionospheric delay error, \( \Updelta \phi_{\text{rel}} \) is relativistic effects error, \( \Updelta \phi_{\text{mult}} \) is multipath effects error, \( N_{r}^{s} \) is integer ambiguity, and \( \varepsilon \left( t \right) \) is measurement noise.

The double-difference phase between stations and satellites is defined as the difference between satellites of the single-difference, and the single-difference can be got by 2 stations \( \left( {m,\;n} \right) \) observing the same satellite in the same epoch. The double-difference phase is given by

$$ \text{DD}(m,\:n,\:j,\:k)\, = \,(\phi _{{mj}} \, - \,\phi _{{nj}} )\, - \,(\phi _{{mk}} \, - \,\phi _{{nk}} ) $$
(7.2)

The best feature of double-difference observation is the clock offset error was completely eliminated. Relativistic effects and multipath effects error also was significantly weakened. If the 2 stations in double-difference observation are close to each other, the tropospheric and ionospheric effects could be weakened. Substituting Eq. (7.1) into (7.2), we get

$$ \begin{aligned} {\text{DD}}\left( {m,n,j,k} \right) &= f{{\left( {\rho _{m}^{j} - \rho _{n}^{j} - \rho _{m}^{k} + \rho _{n}^{k} } \right)} \mathord{\left/ {\vphantom {{\left( {\rho _{m}^{j} - \rho _{n}^{j} - \rho _{m}^{k} + \rho _{n}^{k} } \right)} c}} \right. \kern-\nulldelimiterspace} c} + \left( {\Delta \phi _{{\text{trop}}}^{{mj}} - \Delta \phi _{{\text{trop}}}^{{nj}} - \Delta \phi _{{\text{trop}}}^{{mk}} + \Delta \phi _{{\text{trop}}}^{{nk}} } \right) \\ & \quad -\left( {\Delta \phi _{{\text{iono}}}^{{mj}} - \Delta \phi _{{\text{iono}}}^{{nj}} - \Delta \phi _{{\text{iono}}}^{{mk}} + \Delta \phi _{{\text{iono}}}^{{nk}} } \right) + N_{m}^{j} - N_{n}^{j} - N_{m}^{k} + N_{n}^{k} + \text{dd}(\varepsilon ) \\ \end{aligned}$$

Where \( {\text{dd}}\left( \varepsilon \right) \) is the measurement noise of combined observations. The noise’s variance is magnified after combination, and the noise contains the residual multipath error.

2.2 Ionospheric-Free Combination

Ionospheric delay error is the maximum error in Beidou navigation system observation, if the clock offset error could be eliminated. It depends on the total electron content (TEC) in ionosphere. The first order term of ionospheric delay error is the main part, and can be eliminated by dual-frequency data combination, which is

$$ {\text{IF}}\left( {\phi_{1} ,\phi_{2} } \right)\, = \,\frac{1}{{f_{1}^{2} \, - \,f_{2}^{2} }}\left( {f_{1}^{2} \phi_{1} \, - \,f_{2}^{2} \phi_{2} } \right) $$
(7.3)

The second order term is only about 0.1 % of the first, and can be eliminated by triple-frequency data combination. That is

$$ \begin{aligned} {\text{DIF}}\left( {\phi_{1} ,\;\phi_{2} ,\;\phi_{3} } \right) = &\, {\text{IF}}\left( {\phi_{1} ,\;\phi_{2} } \right)\, - \,{\text{IF}}\left( {\phi_{2} ,\;\phi_{3} } \right) \\ = & \left( {\frac{{f_{1}^{2} }}{{f_{1}^{2} \, - \,f_{2}^{2} }}\, - \,\frac{{f_{1}^{2} }}{{f_{1}^{2} \, - \,f_{3}^{2} }}} \right) \cdot \phi_{1} \, - \,\left( {\frac{{f_{2}^{2} }}{{f_{1}^{2} \, - \,f_{2}^{2} }}} \right) \cdot \phi_{2} \, + \,\left( {\frac{{f_{3}^{2} }}{{f_{1}^{2} \, - \,f_{3}^{2} }}} \right)\phi_{3} \\ \end{aligned}$$
(7.4)

2.3 Tropospheric Delay Modelling

The zenith path delay (ZPD) due to tropospheric refraction is of the order of 2.3 m (or about 8 ns) for a station at sea level and standard atmospheric conditions, and can reach 20 m when elevation angle less than 10° [6]. Tropospheric delay contains two parts, the dry component and the wet component. We can calculate the tropospheric zenith delay by a priori model, and convert to the other directions using the mapping function. The dry component accounts for more than 90 % in tropospheric zenith delay. It could be almost completely corrected by using typical models, such as Hopfield model [9] or Saastamoinen model [10]. However, an uniform effective wet component correction model is difficult to build, due to the regional and seasonal changing characteristics of atmospheric humidity. In precise orbit determination, the wet component of tropospheric zenith delay is usually taken as a solve-for parameter in calculation. The whole tropospheric delay model can be written as

$$ \Updelta \rho_{r}^{s} \left( {t,\;z} \right)\, = \,\Updelta \rho_{{{\text{apr}},\;k}} \left( {z_{r}^{s} } \right)\, + \,\Updelta \rho_{k} \left( t \right)f\left( {z_{r}^{s} } \right) $$
(7.5)

where, \( \Updelta \rho_{r}^{s} \) is the tropospheric path delay from receiver \( r \) to satellite \( s,\,z \) is the elevation of the observation scenario, \( \Updelta \rho_{{{\text{apr,}}\;k}} \left( {z_{r}^{s} } \right) \) is the slant delay according to an a priori model, \( \Updelta \rho_{k} \left( t \right) \) is the solve-for zenith path delay parameter, and \( f\left( {z_{r}^{s} } \right) \) is the mapping function of \( \Updelta \rho_{k} \left( t \right) \).

2.4 Data Weighting

In satellite orbit determination, the numerical process method of code pseudorange and carrier phase measurements is weighting fusion. A typical prior weight is inversely proportional to measurement noise, and the measurement noise can be written as

$$ \frac{{\sigma_{c} }}{{\sigma_{p} }}\, = \,100 $$
(7.6)

where, \( \sigma_{c} \) is the measurement noise of code pseudorange, and \( \sigma_{p} \) is the measurement noise of carrier phase.

Observations at low elevations are generally much more susceptible to tropospheric refraction and multipath effects than those at high elevations. The unmodeled systematic errors decrease the quality of results. Using low-elevation observations, however, may also improve the estimation of tropospheric zenith delays and, consequently, the vertical component of station positions [11]. In order to optimize the use of low-elevation observations, there is an elevation-dependent weighting of observations, which is

$$ w({\text{z}})\, = \,\cos^{2} ({\text{z}}) $$
(7.7)

3 POD Strategies

We apply the dynamic method to determine BeiDou satellites’ orbit. The arc length is defined as 3 days. The ionospheric-free combination observations are used to form the double-difference observation equations. And the baselines are selected according to the criterion: the two stations have the maximum common-view satellites. The Least-Squares method is adopted for parameter estimation. The POD strategies are detailed in Table 7.1.

Table 7.1 System resulting data of standard experiment
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A brief flow of data processing is showed as follows (Fig. 7.1).

Fig. 7.1
figure 1

A brief flow of data processing

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4 POD Test Based on Regional Tracking Net

4.1 Overlapping Difference

A group of data from 07/17/2012 to 07/27/2012, domestic observation network data for 11 days, is selected for analysis. The orbit determination strategy is proposed in Part 3, determining a set of orbits with every 3 days observations (9 sets of orbits in total). So there is orbital overlap for 1 day between the set D the set D + 2. According to statistics orbit determination principle, the two sets of orbit, set D and D + 2, are mutually independence under ideal conditions. Therefore the consistency of the overlapping arcs orbits can be an index of the orbital accuracy in some way. Figure 7.2 shows the orbital overlapping differences of the Beidou operating satellites (C03, C05, C07, C10) in radial direction, and Fig. 7.3 shows the statistics of the three-dimensional orbital overlapping differences.

Fig. 7.2
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Radial overlapping difference of BeiDou satellites

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Fig. 7.3
figure 3

Overlapping difference of BeiDou C08

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The results in Figs. 7.2 and 7.3 demonstrate the orbital differences in radial direction are about 20 cm, and about 1 m of three-dimensional position.

4.2 Independent SLR Residuals

Another method to estimate the precision of orbit determination is utilizing high-precision Satellite Laser Ranging (SLR) data for external checking on the results of the orbit determination. Independent SLR residuals directly reflect the precision of orbit determination. Supposing we get the SLR observation \( \rho_{o} \) at the time \( t, \) it can be expressed as

$$ \Updelta \rho \, = \,\rho_{o} \, - \,(\rho_{c} \, + \,\Updelta \rho_{\text{stides}} \, + \,\Updelta \rho_{\text{oloading}} \, + \,\Updelta \rho_{\text{atm}} \, + \,\Updelta \rho_{\text{rel}} \, + \,\Updelta \rho_{\text{ec}} \, + \,\Updelta \rho_{\text{st}} \, + \,\varepsilon $$
(7.8)

where, \( \rho_{c} \) is the distance from satellite to stations calculated by the POD results, \( \Updelta \rho_{\text{stides}} \) and \( \Updelta \rho_{\text{oloading}} \) are the influence of earth tides and sea tides to stations respectively, \( \Updelta \rho_{\text{atm}} \) is the atmospheric delay error which can be determined by Marini/Murray Model, \( \Updelta \rho_{\text{rel}} \) is the general relativity effects error, \( \Updelta \rho_{\text{ec}} \) and \( \Updelta \rho_{\text{st}} \) are the eccentric corrections and the displacement of observation stations respectively, \( \varepsilon \) is measurement error.

The SLR data we can get is relatively few. In the POD arcs mentioned in Part 4.1, we only have a small amount of data in July 23 (Station 7237) and July 24 (Station 7821). The post fit of SLR residuals were shown in Fig. 7.4. They indicate that the SLR residuals of Station 7821 are all less than 15 cm, while the SLR residuals of Station 7821 vary in 13 ~ 35 cm. One possible reason is the elevation angles of Station 7821 are relatively lower.

Fig. 7.4
figure 4

Independent SLR residuals of BeiDou C08

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5 IGS Multi-GNSS Experiment

In 2011, IGS initiated and organized an experiment utilize more than one navigation systems navigation, named IGS M-GEX. They tracked and monitored several GNSS signals at the same time in daily operations to analysis the performance and signal characteristics of the GNSS constellations emerging in recent years. Dozens of GNSS multimode tracking stations have been established in this experiment. Part of these stations can track Beidou Satellites, and they are shown in Table 7.2.

Table 7.2 M-GEX monitoring stations with BeiDou tracking capability
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An analysis of Beidou satellite tracking data from these stations was presented in Ref. [8], and it preliminarily assessed the performance of Beidou regional navigation system. The results using orbit overlap method indicated that the precision of Beidou satellite orbits in three-dimensional position is 1 ~ 10 m.

The M-GEX experiment provided valuable observations, which can be applied to study the influence of the layout of the station on the Beidou satellites POD precision. First, we analyzed the visibility of the satellite for tracking stations listed in Table 7.2. In the 6 stations listed in Table 7.2, all the stations except Station CUT0 are in the high latitudes. Actually, their tracking range for Beidou satellite is smaller, and there is low elevation observation data in almost all visible arcs. Figures 7.5 and 7.6 depicts the visibility of Beidou satellites to Station CUT0 and Station KIR8 for one day (Cut-off Elevation = 10°), respectively.

Fig. 7.5
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Visibility of BeiDou satellites at station CUT0

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Fig. 7.6
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Visibility of BeiDou satellites at station KIR8

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As illustrated in Fig. 7.5, the station CUT0 can observe two BeiDou GEOstationary (GEO) satellites (C03, C05), as well as all BeiDou Inclining Geostationary Synchronized Orbit (IGSO) satellites. Because it is located in southern hemisphere, this observing site should quietly enhance the geometric strength of the domestic regional tracking net. However, KIR8 has poor BeiDou tracking capability (Fig. 7.6). Almost no GEO satellite can be seen at this station. And most of the IGSO satellites are at low elevation. Obviously, measurements of KIR8 have limited contribution to POD of BeiDou satellites.

When integrating the observations obtained from M-GEX stations and observations from the domestic regional tracking net to carry out the POD test at the same data period of time defined in Sect. 7.4, we can obtain a new result (Figs. 7.7 and 7.8).

Fig. 7.7
figure 7

Radial overlapping difference of BeiDou satellites, supported by IGS M-GEX monitoring stations

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Fig. 7.8
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Overlapping difference of BeiDou C08, supported by IGS M-GEX monitoring stations

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In this two figures (Figs. 7.7 and 7.8), most of the radial overlapping differences of BeiDou satellites have decreased within 20 cm, and 3-D position differences have fall within 1 m, according to Figs. 7.2 and 7.3. As a result, though most of the IGS M-GEX monitoring stations have limited tracking capability for BeiDou satellites, they have improved the orbit accuracy a lot, compared to the orbit derived from domestic tracking data only. The benefit of offshore monitoring stations is exhibited here preliminarily.

6 Summary and Conclusions

The multi-satellite POD for BeiDou regional navigation satellite system based on double-difference observation processing is discussed in this paper. Tests using overlapping differences and SLR residuals suggest the double-difference solutions of BeiDou regional ground-tracks net have achieved 20 cm radial accuracy, which can satisfy the needs of our domestic navigation users in the near future. With several IGS M-GEX monitoring stations observing BeiDou satellites supplied, the higher orbital accuracy can be obtained. Therefore, to improve the PNT service capability, it is of critical strategic significance to build up more BeiDou monitoring stations in the global region.