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GPS Solutions

, Volume 21, Issue 2, pp 439–450 | Cite as

GPS, Galileo, QZSS and IRNSS differential ISBs: estimation and application

  • Dennis Odijk
  • Nandakumaran Nadarajah
  • Safoora Zaminpardaz
  • Peter J. G. Teunissen
Original Article

Abstract

Knowledge of inter-system biases (ISBs) is essential to combine observations of multiple global and regional navigation satellite systems (GNSS/RNSS) in an optimal way. Earlier studies based on GPS, Galileo, BDS and QZSS have demonstrated that the performance of multi-GNSS real-time kinematic positioning is improved when the differential ISBs (DISBs) corresponding to signals of different constellations but transmitted at identical frequencies can be calibrated, such that only one common pivot satellite is sufficient for inter-system ambiguity resolution at that particular frequency. Recently, many new GNSS satellites have been launched. At the beginning of 2016, there were 12 Galileo IOV/FOC satellites and 12 GPS Block IIF satellites in orbit, while the Indian Regional Navigation Satellite System (IRNSS) had five satellites launched of which four are operational. More launches are scheduled for the coming years. As a continuation of the earlier studies, we analyze the magnitude and stability of the DISBs corresponding to these new satellites. For IRNSS this article presents for the first time DISBs with respect to the L5/E5a signals of GPS, Galileo and QZSS for a mixed-receiver baseline. It is furthermore demonstrated that single-frequency (L5/E5a) ambiguity resolution is tremendously improved when the multi-GNSS observations are all differenced with respect to a common pivot satellite, compared to classical differencing for which a pivot satellite is selected for each constellation.

Keywords

GPS Galileo QZSS IRNSS Multi-GNSS Differential inter-system bias (DISB) RTK 

Introduction

With the multitude of global and regional navigation satellite systems (GNSS/RNSS) being modernized or developed, many more satellites and frequencies are becoming available that benefit high-precision relative applications, such as real-time kinematic (RTK) positioning. Having a subset of frequencies that are shared between the constellations allows differencing of observations between these constellations (also referred to as tightly or inter-system differencing), resulting in additional double-differenced (DD) observations compared to differencing the observations of each constellation independently (Julien et al. 2003). Hence, this results in a stronger positioning model and better performance of integer ambiguity resolution, which is the key to high-precision positioning.

However, when differencing observations between constellations one has to account for inter-system biases (ISBs), due to a difference in receiver hardware delays between the signals of different constellations, as well as a difference in system time between constellations (Hegarty et al. 2004). As this article is restricted to relative positioning, we do not have to bother about this system time offset since it gets eliminated. More details on the determination of absolute ISBs can be found in Torre and Caporali (2015) and Jiang et al. (2016). Concerning the relative or differential ISBs (DISBs), Odijk and Teunissen (2013a, b) demonstrated that for combinations of GPS, Galileo, BDS and QZSS observations these cancel for baselines with receiver pairs of the same manufacturer. These DISBs do in principle not cancel when the baseline is formed by receivers of different manufacturers, i.e., mixed receivers. Fortunately, Odijk and Teunissen (2013a, b), Melgard et al. (2013) and Paziewski and Wielgosz (2015) demonstrated that these mixed-receiver DISBs are very stable in time and can be calibrated. As a consequence, also mixed-receiver baselines can be processed in the same way as baselines consisting of identical receivers. Results of improved RTK ambiguity resolution with calibrated DISBs are given in Odijk and Teunissen (2013a), Odolinski et al. (2015) and Nadarajah et al. (2015).

This research is motivated by the recent launches of new GNSS satellites. By the end of 2015, 8 Galileo full-operational capability (FOC) satellites were in orbit next to the four in-orbit validation (IOV) satellites. In addition to these new Galileo satellites, during the recent years also GPS Block IIF satellites were launched, bringing the number of GPS satellites transmitting the L5 signal to 12 at the beginning of 2016. Moreover, five satellites of the Indian Regional Navigational Satellite System (IRNSS) have been put in orbit since 2013, of which four satellites are operational. So far these new satellites were not yet included in the DISB analyses, and for IRNSS it is even the first time that DISBs are estimated relative to the other constellations.

As the DISB analyses are restricted to CDMA signals between constellations that have identical frequencies, Table 1 gives an overview of those frequencies of GPS, GLONASS, Galileo, BDS, QZSS and IRNSS. This research is restricted to DISBs for the L5/E5a frequency that is shared between GPS, Galileo, QZSS and IRNSS; see the right column of Table 1.
Table 1

Overview of GNSS frequencies (CDMA signals)

Frequency (MHz)

2492.028

1575.42

1561.098

1278.75

1268.52

1227.60

1207.14

1202.025

1191.795

1176.45

GPS (G)

 

L1

   

L2

   

L5

GLONASS (R)

       

L3

  

Galileo (E)

 

E1

 

E6

  

E5b

 

E5

E5a

BDS (C)

  

B1

 

B3

 

B2

   

QZSS (J)

 

L1

 

LEX

 

L2

   

L5

IRNSS (I)

S

        

L5

The next section reviews the mixed-constellation GNSS model that underlies the estimation and application of the DISBs. In the subsequent section, the outcomes of the DISB estimation based on real GPS, Galileo, QZSS and IRNSS data are presented. That section also gives examples of the performance of integer ambiguity resolution and relative positioning for a short baseline for which a priori knowledge on the DISBs is incorporated.

Mixed-constellation GNSS model: estimation and application of DISBs

This section first reviews the estimation of DISB parameters and after that their application. We distinguish between estimation and application for short baselines, ignoring the differential ionospheric delays, and long baselines, for which these delays cannot be ignored.

Estimation of DISBs

Assume two GNSS constellations, denoted as A and B, of which signals are tracked on identical frequencies. For relative GNSS positioning applications, it is a common procedure to eliminate receiver and satellite specific biases by double differencing the code and phase observations with respect to a pivot satellite and a pivot receiver. In a multi-GNSS environment, this then relates to selecting a pivot satellite for each constellation: In case of two constellations A and B, these are denoted as 1 A and 1 B , as shown in Fig. 1 (left). This double differencing separately per constellation is referred to as the classical differencing approach.
Fig. 1

Multi-constellation differencing: (left) a pivot satellite per constellation, i.e., satellite 1 A for constellation A and satellite 1 B for constellation B; (right) one pivot satellite 1 A for both constellations. Satellite u belongs to constellation A and satellite v to constellation B. The pivot receiver is denoted as 1 and the rover receiver as r

Alternatively, instead of this classical differencing, one could difference the code and phase observations between constellations such that the model is based on inter-system differencing. Assuming the pivot for both constellations A and B to be satellite 1 A , the model of mixed-constellation DD observation equations can be given as follows, for a short baseline, assuming differential atmospheric delays to be absent (Odijk and Teunissen 2013a):
$$\begin{aligned} E\left( {\phi_{j}^{{1_{A} u}} } \right) &= \rho^{{1_{A} u}} + \lambda_{j} a_{j}^{{1_{A} u}} \hfill \\ E\left( {\phi_{j}^{{1_{A} v}} } \right) &= \rho^{{1_{A} v}} + \lambda_{j} a_{j}^{{1_{B} v}} + \lambda_{j} \bar{\delta }_{j}^{AB} \hfill \\ E\left( {p_{j}^{{1_{A} u}} } \right) &= \rho^{{1_{A} u}} \hfill \\ E\left( {p_{j}^{{1_{A} v}} } \right) &= \rho^{{1_{A} v}} + d_{j}^{AB} \hfill \\ \end{aligned}$$
(1)
Here E(.) denotes the mathematical expectation, \(\phi_{j}^{{1_{A} u}}\) and \(p_{j}^{{1_{A} u}}\) the vectors of DD phase and code observables at frequency j for constellation A, with satellite index \(u = 2_{A} , \ldots ,m_{A}\). \(\phi_{j}^{{1_{A} v}}\) and \(p_{j}^{{1_{A} v}}\) are their counterparts for constellation B, with satellite index \(v = 1_{B} , \ldots ,m_{B}\). \(\rho_{{}}^{{1_{A} u}}\) and \(\rho_{{}}^{{1_{A} v}}\) denote the lumped combination of DD ranges and DD tropospheric delays for both constellations, \(\lambda_{j}\) the wavelength corresponding to frequency j, \(a_{j}^{{1_{A} u}}\) and \(a_{j}^{{1_{B} v}}\) the DD ambiguities for both constellations, and finally \(\bar{\delta }_{j}^{AB}\) and \(d_{j}^{AB}\) the phase and code DISB parameters. Note that all above DD observables and parameters are receiver-dependent, but for notational convenience the receiver indices have been omitted. The estimable DD ambiguities, i.e., \(a_{j}^{{1_{A} u}}\) and \(a_{j}^{{1_{B} v}}\), have the property to be integer, and resolving these integers is the key to high-precision RTK positioning solutions.
The estimable DISB parameters in (1) are defined as:
$$\begin{aligned} \bar{\delta }_{j}^{AB} & = \bar{\delta }_{j}^{B} - \bar{\delta }_{j}^{A} ;\quad {\text{with}}\quad \bar{\delta }_{j}^{S} = \delta_{j}^{S} + a_{j}^{{1_{S} }} ,\quad j \ge 1 \\ d_{j}^{AB} & = d_{j}^{B} - d_{j}^{A} ,\quad j \ge 1 \\ \end{aligned}$$
(2)
Here \(\delta_{j}^{S}\) and \(d_{j}^{S}\), with \(S \in \left\{ {A,B} \right\}\), denote the frequency- and constellation-dependent receiver phase and code hardware biases. The phase DISB parameter is denoted using a bar, i.e., \(\bar{\delta }_{j}^{AB}\), as it can only be estimated lumped to the DD ambiguity \(a_{j}^{{1_{A} 1_{B} }}\). This is due to a rank deficiency in the model based on the observation Eq. (1). As a consequence, satellite 1 B does not have an estimable ambiguity parameter, as it is lumped to the phase DISB. The estimable DISBs can thus be interpreted as double differences, but then between receivers and constellations. Compared to the classical DD model, the inter-system model has two more observables for each frequency, corresponding to the pivot satellite of B, but it also has two more unknown parameters per frequency, i.e., the phase and code DISBs. Also note that the DD observables between the two constellations in the inter-system model become correlated, as they all share the same pivot satellite. Furthermore, note that the model based on (1) cannot be directly used to solve the receiver position, as the observation equations need to be linearized.
The observation model based on (1) is only valid for sufficiently short baselines, typically of a few kilometers at maximum. For longer baselines, the ionospheric delays need to be parametrized as well, resulting in the following inter-system DD observation equations (Teunissen et al. 2016):
$$\begin{aligned} E\left( {\phi_{j}^{{1_{A} u}} } \right) & = \rho^{{1_{A} u}} + \lambda_{j} a_{j}^{{1_{A} u}} - \mu_{j} I^{{1_{A} u}} \\ E\left( {\phi_{j}^{{1_{A} v}} } \right) & = \rho^{{1_{A} v}} + \lambda_{j} a_{j}^{{1_{B} v}} + \lambda_{j} \bar{\bar{\delta }}_{j}^{AB} - \mu_{j} \bar{\bar{I}}^{{1_{A} v}} \\ E\left( {p_{j}^{{1_{A} u}} } \right) & = \rho^{{1_{A} u}} + \mu_{j} I^{{1_{A} u}} \\ E\left( {p_{j}^{{1_{A} v}} } \right) & = \rho^{{1_{A} v}} + {d}_{IF}^{AB} + \bar{\bar{d}}_{j}^{AB} + \mu_{j} \bar{\bar{I}}^{{1_{A} v}} \\ \end{aligned}$$
(3)
Because of the ionospheric parameters, this long-baseline model requires at least two shared frequencies, whereas the model based on (1) is already solvable using a single frequency. The estimable DD ionospheric parameters are denoted as \(I_{{}}^{{1_{A} u}}\) for constellation A. It is assumed that these ionospheric parameters apply to the first frequency (i.e., j = 1). For the other frequencies, these ionospheric parameters need to be multiplied by a frequency-dependent coefficient, denoted as \(\mu_{j} = \lambda_{j}^{2} /\lambda_{1}^{2}\). The presence of the ionospheric parameters adds a rank deficiency to the model, which has as consequence that the interpretation and the estimability of the phase and code ISB parameters, as well as of the ionospheric parameters of constellation B, are changed:
$$\begin{aligned} \bar{\bar{\delta }}_{j}^{AB} &= \bar{\bar{\delta }}_{j}^{B} - \bar{\bar{\delta }}_{j}^{A} ;\quad {\text{with }}\quad \bar{\bar{\delta }}_{j}^{S} = \delta_{j}^{S} + \frac{{\mu_{j} }}{{\lambda_{j} }}d_{GF}^{S} + a_{j}^{{1_{S} }} ,\quad j \ge 1 \\ d_{IF}^{AB} &= d_{IF}^{B} - d_{IF}^{A} \\ \bar{\bar{d}}_{j}^{AB} &= \bar{\bar{d}}_{j}^{B} - \bar{\bar{d}}_{j}^{A} ;\quad {\text{with}}\quad \bar{\bar{d}}_{j}^{S} = d_{j}^{S} - \left( {\mu_{j} d_{GF}^{S} + d_{IF}^{S} } \right), \quad j \ge 3 \\ \bar{\bar{I}}^{{1_{A} v}} &= I^{{1_{A} v}} + d_{GF}^{B} - d_{GF}^{A} \\ \end{aligned}$$
(4)
To distinguish the estimable phase and code DISB parameters from their counterparts in the short-baseline model (1), in the long-baseline model (3) they are denoted using a double bar. Also the estimable ionospheric parameter for constellation B gets a double bar. Furthermore, in (4) the geometry-free and ionospheric-free differential receiver code biases for one constellation \(S \in \left\{ {A,B} \right\}\) are defined as:
$$\begin{array}{lll} {d_{GF}^{S} }&=& - \frac{1}{{\mu_{2} - \mu_{1} }}\left( {d_{1}^{S} - d_{2}^{S} } \right)\quad {\text{and}}\\ d_{IF}^{S}&=&\frac{{\mu_{2} }}{{\mu_{2} - \mu_{1} }}d_{1}^{S} - \frac{{\mu_{1} }}{{\mu_{2} - \mu_{1} }}d_{2}^{S}\end{array}$$
(5)
Note that the difference \(d_{1}^{S} - d_{2}^{S}\) is also known as the between-receiver differential code bias (DCB) between the first two frequencies (Zhang and Teunissen 2015).
The ionospheric-free code DISB, i.e., \(d_{IF}^{AB}\), is one of the estimable parameters in the long-baseline model, as shown in (3), but this is not the case for the geometry-free code DISB, i.e., \(d_{GF}^{AB}\). This geometry-free code DISB is inestimable, as it is constrained to overcome the rank deficiency in the long-baseline model (as S-basis; see Odijk et al. 2016). Consequently, its scaled version is lumped to the parameters that are involved in the rank deficiency, i.e., the phase and code DISBs and the ionospheric parameters of constellation B, as shown in (4). It follows from (5) that the estimable ionospheric-free code DISB is also a function of this geometry-free code DISB:
$$d_{IF}^{AB} = d_{j}^{AB} - \mu_{j} d_{GF}^{AB} , \quad j = 1,2$$
(6)
Because of this relation, there are different types of code DISB parameters in the long-baseline model (3). The first type is \(d_{IF}^{AB}\) and is always estimable with at least two frequencies. With more than two frequencies available, a second type of code DISB parameter becomes estimable, denoted as \(\bar{\bar{d}}_{j}^{AB} , j \ge 3\). It is noted that the first type corresponds to the ionospheric-free code DISB parameter given in Montenbruck et al. (2011), with the difference that here we assume that both constellations track identical frequencies. In Montenbruck et al. (2011), this is not necessarily the case, as their DISB corresponds to the L1 and L2 frequencies of GPS and the E1 and E5a frequencies of Galileo. Finally, we note that, instead of the multi-GNSS long-baseline model (3), other ionospheric-float parametrizations exist as well, such as those given by Odolinski et al. (2014) and Yuan and Zhang (2014).

In the case one has multi-GNSS data of more than two constellations that share identical frequencies, the models (1) and (3) can be easily extended, parameterizing DISBs per combination of two constellations (e.g., GPS + Galileo; GPS + QZSS; GPS + IRNSS).

Application of DISBs

From the previous section, it follows that the DISB parameters have a different estimability and interpretation depending on whether the observation equations for a short baseline (1) or for a long baseline (3) are applied for their determination. This difference has influence on their applicability, and this can be shown as follows. First, consider the following relations between the estimable short-baseline DISB parameters, i.e., \(\bar{\delta }_{j \ge 1}^{AB}\) and \(d_{j \ge 1}^{AB}\) at the left side of the equal sign, and the long-baseline DISB parameters, i.e., \(\bar{\bar{\delta }}_{j \ge 1}^{AB}\), \(d_{IF}^{AB}\) and \(\bar{\bar{d}}_{j \ge 3}^{AB}\) at the right side of the equal sign:
$$\begin{aligned} \bar{\delta }_{j \ge 1}^{AB} & = \bar{\bar{\delta }}_{j \ge 1}^{AB} - \frac{{\mu_{j} }}{{\lambda_{j} }}d_{GF}^{AB} \\ d_{j \ge 1}^{AB} & = d_{IF}^{AB} + \bar{\bar{d}}_{j \ge 3}^{AB} + \mu_{j} d_{GF}^{AB} \\ \end{aligned}$$
(7)
From the above, it follows that the short-baseline DISB for phase is equal to its long-baseline counterparts, minus the (scaled) geometry-free DISB, i.e., \(d_{GF}^{AB}\). The short-baseline DISB for code is equal to ionospheric-free DISB from the long-baseline model, plus, in case of more than two frequencies, a code DISB, and the (scaled) geometry-free DISB as well. Thus, to convert the long-baseline DISBs to their short-baseline counterparts the geometry-free DISB is required for both phase and code. Unfortunately, this term cannot be determined from the long-baseline model, as it is fixed in order to remove the rank deficiency of the model. This also explains why the long-baseline model has one estimable DISB parameter less than the short-baseline model. It is therefore not possible to calibrate short baselines with DISBs that are determined from a long-baseline model.

The other way around, i.e., calibrating long baselines by using short-baseline DISBs, is a different story. In that case one does not need to know the geometry-free DISB term. The reason for this is that this geometry-free DISB parameter is multiplied by the ionospheric coefficient vector \({-}\mu_{j}\) for phase and \(\mu_{j}\) for code, as shown in (7), which implies that it will be lumped to the estimable ionospheric parameters in the long-baseline model. Alternatively, it will be eliminated when the ionospheric-free combination is taken. Thus, the DISBs that are determined from short baselines can be applied to long-baseline models.

Let us now assume that DISBs that are estimated from the short-baseline model can be used to calibrate another baseline, such that we have the following DISB corrections available per frequency: \(\bar{\delta }_{j}^{AB}\) and \(d_{j}^{AB}\). Recall from the previous subsection that the phase DISB correction is biased by an ambiguity, which is integer but unknown. However, this is not a problem, as we may subtract an arbitrary integer from the correction and apply a fractional correction for the phase DISB. Thus, let the fractional phase DISB correction be denoted as \(\Delta \delta_{j}^{AB} = \bar{\delta }_{j}^{AB} - \bar{a}_{j}^{{1_{A} 1_{B} }}\), with \(\bar{a}_{j}^{{1_{A} 1_{B} }}\) the arbitrary integer, which is usually the closest integer, the mixed-constellation DD observation equations can be given as follows, applying the short-baseline DISB corrections to the observations of B:
$$\begin{aligned} E\left( {\phi _{j}^{{1_{A} u}} } \right) & = \rho ^{{1_{A} u}} + \lambda _{j} a_{j}^{{1_{A} u}} \left[ { - \mu _{j} I^{{1_{A} u}} } \right] \\ E\left( {\phi _{j}^{{1_{A} v}} - \lambda _{j} \Delta \delta _{j}^{{AB}} } \right) & = \rho ^{{1_{A} v}} + \lambda _{j} \bar{a}_{j}^{{1_{A} v}} \left[ { - \mu _{j} I^{{1_{A} v}} } \right] \\ E\left( {p_{j}^{{1_{A} u}} } \right) & = \rho ^{{1_{A} u}} \left[ { + \mu _{j} I^{{1_{A} u}} } \right] \\ E\left( {p_{j}^{{1_{A} v}} - d_{j}^{{AB}} } \right) & = \rho ^{{1_{A} v}} \left[ { + \mu _{j} I^{{1_{A} v}} } \right] \\ \end{aligned}$$
(8)
The above DISB-corrected model now applies to both short and long baselines, where in case of short baselines the DD ionospheric parameters are absent, while these are present for long baselines. To distinguish this, in (8) these parameters are denoted using square brackets. The absence or presence of these ionosphere parameters does, however, not affect the interpretation of the other parameters. The estimable ambiguities of constellation have the following interpretation: \(\bar{a}_{j}^{{1_{A} v}} = a_{j}^{{1_{B} v}} + \bar{a}_{j}^{{1_{A} 1_{B} }}\); i.e., they are a combination of the DD ambiguity having the pivot satellite of B and the arbitrarily chosen integer. If this chosen integer actually equals the true integer, the estimable ambiguities of B are the true DD ambiguities of this constellation with respect to A; i.e., \(\bar{a}_{j}^{{1_{A} v}} = a_{j}^{{1_{A} v}}\). However, irrespective of the actual choice for \(\bar{a}_{j}^{{1_{A} 1_{B} }}\), as long as it is an integer, \(\bar{a}_{j}^{{1_{A} v}}\) is an integer parameter as well.

Note that the inter-system DD observation equations in (8) are all relative to pivot satellite 1 A of constellation A, and thus, the observations of both constellations at the identical frequencies can be processed as if one constellation, as shown in Fig. 1 (right). In fact, this means that one additional satellite is available compared to models (1) and (3). This strengthens the model and benefits ambiguity resolution and the position estimation. As a consequence of knowing the short-baseline DISBs, the differential ionospheric delays corresponding to constellation B are freed from the geometry-free code DISB. That is why the estimable DD ionospheric parameters corresponding to constellation B, denoted using a double bar in (3), have been replaced by DD ionospheric parameters without bar in the observation Eq. (8).

Estimation and application of DISBs between GPS, Galileo, QZSS and IRNSS

In this section, the estimation of code and phase DISBs is applied to the shared L5/E5a frequencies between GPS, Galileo, QZSS and IRNSS. Similar to earlier DISB studies, zero and very short baselines are measured, with the differential receiver position either absent or known, such that a strong model remains for the estimation of DISBs, which can be carried out in epoch-by-epoch mode.

IRNSS data are tracked by some of the multi-GNSS receivers stationed at the campus of Curtin University in Perth (Australia), such as listed in Table 2 and shown in Fig. 2. Of the IRNSS-enabled receivers CUT3, CUAA and CUCC are Javad TRE-G3TH receivers, whereas CUCS is a Septentrio PolaRx5, the latest generation of Septentrio multi-GNSS receivers at the time of this writing. Note that CUCC and CUCS are connected to the same antenna and thus form a zero baseline. Receivers CUT3 and CUAA are separated by about 8 m. Note from Table 2 that Javad uses combined data and pilot tracking for GPS, Galileo and QZSS, while Septentrio employs pilot-only signals. For IRNSS, however, both Javad and Septentrio track data-only signals.
Table 2

Used multi-GNSS receivers and the constellations they observe (G = GPS, E = Galileo, J = QZSS, I = IRNSS), together with their observables including tracking mode (corresponding to their RINEX 3.03 notation; IGS and RTCM-SC104 2015)

Receiver ID

Receiver type

G

E

J

I

CUT3

Javad TRE-G3TH

L5X

L5X

L5X

L5I*

CUAA

Javad TRE-G3TH

L5X

L5X

L5X

L5I*

CUCS

Septentrio PolaRx5

L5Q

L5Q

L5Q

L5A

CUCC

Javad TRE-G3TH

L5X

L5X

L5X

L5I*

Explanation of the tracking modes: L5X = data + pilot signal for GPS, Galileo and QZSS; L5Q = pilot signal for GPS, Galileo and QZSS; L5I/L5A = data signal for IRNSS

* L5I as tracked by the Javad receivers is not conform RINEX 3.03. According to Javad, this is equivalent to L5A, i.e., the data signal of IRNSS (Javad 2016)

Fig. 2

Some of the multi-GNSS receivers stationed at the roof of building 402 at the campus of Curtin University in Perth

Estimation of the DISBs

First, for the 8 m baseline CUT3-CUAA it is verified whether the DISBs are indeed zero, as is expected for a pair of identical receiver types. Next, DISBs for the mixed zero-baseline CUCS-CUCC are estimated.

Figure 3 depicts sky plots, i.e., azimuth versus elevation, at CUT3 for the four constellations during the day of November 21, 2015. In particular, the 8-shaped Highly inclined Elliptical Orbit (HEO) of the QZSS satellite is visible, as well as the three satellites of IRNSS that have Inclined Geosynchronous Satellite Orbits (IGSO). The fourth IRNSS satellite, i.e., I03 in Fig. 3, is a geostationary (GEO) satellite. The sky plots only show those satellites that are actually used for the DISB estimation. Thus, for GPS only the Block IIF satellites that transmit L5 are shown. As Galileo’s fourth IOV satellite, i.e., E20, is not transmitting on the E5a frequency, but only on E1 due to a loss of power that occurred in 2014 (Cameron and Reynolds 2014), it is not shown here. Galileo FOC satellite E14, which is shown in the sky plot, is one of the two satellites that were originally injected in an anomalous orbit (Hellemans 2014). Because of this, there are no broadcast ephemerides available for this satellite, and therefore, we used the precise orbit product from CODE (Center for Orbit Determination in Europe) for the purpose of DISB estimation.
Fig. 3

Sky plots at CUT3 of used GPS Block IIF (top left), Galileo (top right), QZSS (bottom left) and IRNSS (bottom right) satellites, during November 21, 2015, above a cutoff elevation of 5°

Figure 4 (top left) shows the number of GPS, Galileo, QZSS and IRNSS satellites tracked at CUT3 above 5° cutoff elevation. From the figure, it can be seen that the number of GPS satellites varies between 3 and 6, whereas the number of Galileo satellites varies between 0 during some part of the day and 3. The QZSS satellite is tracked the entire day, although there are no data tracked at one instance close to epoch 500, at which the QZSS satellite is close to the horizon. The number of tracked IRNSS satellites is 3–4. The other graphs in Fig. 4 show the estimated DISBs for GPS–IRNSS, Galileo–IRNSS and QZSS–IRNSS, all based on a data sampling interval of 30 s. Although the mean of these DISBs over the day is close to zero indeed, it can be seen that—especially for the combinations Galileo–IRNSS and QZSS–IRNSS—at certain times the estimated DISBs show quite a noisy behavior, for example around 3:00 hours UTC, as well as around 6:00 hours UTC for the code DISBs of QZSS–IRNSS. The cause for this is the low elevation of the satellites during these times. Between 3:00 and 6:00 hours UTC the elevation of the QZSS satellite is only between 5° and 10° (see Fig. 4; bottom right) causing high noise on the QZSS observations, possibly in combination with low-elevation multi-path biases. A similar behavior applies to Galileo satellite E26 around 3:00 hours UTC, which is tracked just above the cutoff elevation of 5°, in local southeast, as shown in Fig. 3.
Fig. 4

Number of GPS Block IIF, Galileo, QZSS and IRNSS satellites tracked on November 21, 2015, above 5° cutoff elevation (top left) versus estimated GPS–IRNSS DISBs (top right), Galileo–IRNSS DISBs (bottom left) and QZSS–IRNSS DISBs (bottom right) for 8 m baseline CUT3–CUAA based on identical Javad TRE-G3TH receivers. For each GNSS combination, the phase DISB is shown in the upper graph, the code DISB in the middle graph and the elevations in the bottom graph

The estimated DISBs for the mixed-receiver zero-baseline CUCS-CUCC during January 17, 2016, are shown in Fig. 5, again based on a sampling interval of 30 s. This more recent day was chosen since these receivers were only installed at the beginning of 2016. From the top left graph, it follows that toward the end of this day there were no Galileo satellites tracked above the 5° cutoff elevation, which explains the gap in the Galileo–IRNSS DISB time series. As this is a zero baseline for which differential atmospheric errors are completely absent and multi-path errors are very minor, the noise in these ISB time series is less than in those of the nonzero-baseline CUT3-CUAA. As summarized in Table 3, for this Septentrio–Javad combination the estimated-phase DISBs are 0.50 cycle for both GPS–IRNSS and QZSS–IRNSS, while it is estimated as zero for the Galileo–IRNSS combination. The code DISBs of the three constellations with respect to IRNSS are all estimated within the level of 1.3–1.4 m. It is noted that the phase DISBs in Table 3 are rounded to two decimals (in cycle), as 0.01 cyc equals about 2 mm, which corresponds to the standard deviation of an undifferenced phase observation (in zenith). For the code DISBs one decimal is sufficient, as it corresponds to 1 dm and thus falls within the precision level of a few decimeters for an undifferenced code observation.
Fig. 5

Number of GPS Block IIF, Galileo, QZSS and IRNSS satellites tracked above 5° cutoff elevation (top left) versus estimated GPS–IRNSS DISBs (top right), Galileo–IRNSS DISBs (bottom left) and QZSS–IRNSS DISBs (bottom right) for zero-baseline CUCS–CUCC based on Septentrio PolaRx5 and Javad TRE-G3TH receivers. For each GNSS combination, the phase DISB is shown in the upper graph, the code DISB in the middle graph and the elevations in the bottom graph

Table 3

Estimated L5/E5a DISBs for phase (in cycles) and code (in meters) between GPS, Galileo, QZSS and IRNSS, based on the Septentrio–Javad receiver combination

 

GPS

Galileo

QZSS

Galileo

0.50 cyc

0.0 m

  

QZSS

0.00 cyc

0.0 m

0.50 cyc

0.0 m

 

IRNSS

0.50 cyc

−1.3 m

0.00 cyc

−1.3 m

0.50 cyc

−1.4 m

An empty cell means a zero DISB for combinations of same constellations, or opposite of sign for combinations of mixed constellations

Application of the DISBs

As mentioned earlier, the whole purpose of DISB calibration is to improve the performance of multi-GNSS ambiguity resolution which is essential to high-precision mixed-constellation RTK positioning. This is tested for a short, i.e., 352 m, mixed baseline measured at the campus of Curtin University, measured during the full day of January 17, 2016, between stations CUCS (Septentrio PolaRx5) and SPA7 (Javad TRE-G3TH). Both receivers are tracking GPS, Galileo, QZSS and IRNSS data, and for the purpose of demonstration, we only used the L5/E5a observations of each constellation. Broadcast ephemerides were used for all constellations.

In order to evaluate the performance of multi-GNSS, single-frequency ambiguity resolution, we first computed a reference set of integer ambiguities during the full day by solving the short-baseline model with both receiver positions fixed, as precise coordinates for them are available. These reference ambiguities are then used to compare the outcomes of epoch-by-epoch ambiguity resolution, first based on a model in which each constellation defines its own pivot satellite, i.e., classical differencing, as shown in (1), and next based on a model in which all constellations are differenced relative to the pivot satellite of GPS, i.e., inter-system differencing, as shown in (8). The phase and code DISBs are calibrated to the values in Table 3. Table 4 presents the results of ambiguity resolution in terms of empirical success rates, starting with GPS only and then adding the other constellations one by one. The success rate is defined as the number of epochs with correctly resolved integer ambiguities divided by the total number of epochs. Cutoff elevation is set to 5°, and the data sampling interval is 30 s.
Table 4

Empirical integer ambiguity success rates for short-baseline CUCS-SPA7 based on L5/E5a data only

 

Classical differencing (%)

Inter-system differencing (%)

GPS only

72/2880 = 2.5

GPS + Galileo

77/2880 = 2.7

368/2880 = 12.8

GPS + Galileo + QZSS

77/2880 = 2.7

1100/2880 = 38.2

GPS + Galileo + QZSS + IRNSS

831/2880 = 28.9

2755/2880 = 95.7

For GPS only, for 2.5 % of the epochs during the day the correct ambiguities could be resolved. This poor performance is particularly due to having 4 or less GPS Block IIF satellites available for most of the time during the day (see also Fig. 5; top left). With Galileo added the empirical ambiguity success rate increases slightly to 2.7 % when based on classical differencing. This marginal improvement is due to the 1–2 Galileo satellites that are at most in view during most parts of the day. A higher success rate of 12.8 % is achieved, however, when the Galileo data are differenced with respect to the GPS pivot satellite. Adding QZSS data to GPS + Galileo with classical differencing results in a success rate that is identical to GPS + Galileo, i.e., 2.7 %. The reason is that in this case the QZSS data do not contribute, as there is only one satellite and thus no differences can be formed. The QZSS data clearly contribute when they are differenced relative to the GPS pivot satellite, tremendously increasing the success rate to 38.2 %. The best results are obtained when the 3–4 IRNSS satellites are added to GPS + Galileo + QZSS. With classical differencing the success rate increases to 28.9 %, but when the observations are inter-system differenced, the success rate is a large 95.7 %, which means for almost all epochs during the day the ambiguities are correctly resolved.

Figure 6 shows the horizontal position scatter and the vertical position time series of SPA7 based on inter-system differencing for the four cases investigated, i.e., GPS, GPS + Galileo, GPS + Galileo + QZSS and GPS + Galileo + QZSS + IRNSS. The graphs depict both float and fixed position solutions, with the fixed positions also showing the solutions based on the wrong integer ambiguities, next to the correct solutions. From the graphs it can be seen that the availability of the positioning solutions clearly increases when going from one to multiple constellations. The improvement in precision of the float solution is also visible: In case of GPS only, the root mean square (RMS) of the horizontal components is at the level of 1 m, whereas for GPS + Galileo + QZSS + IRNSS it is improved to a level of 30–40 cm. After correct ambiguity fixing, in all cases the horizontal RMS is about 5 mm (in East and North component) and the vertical RMS is at the level of 1 cm.
Fig. 6

Horizontal (E = East vs. N = North) position scatter and vertical (U = Up) time series for station SPA7, based on GPS only (top left), GPS + Galileo (top right), GPS + Galileo + QZSS (bottom left) and GPS + Galileo + QZSS + IRNSS (bottom right), with the multi-GNSS position solutions based on inter-system differencing. The float solutions are depicted in gray, whereas the fixed solutions are in either red (wrong integer solutions) or green (correct integer solutions)

Conclusions

We extended the determination of differential ISBs (DISBs) between multi-GNSS phase and code observations to new GNSS satellites, following the recent launches of GPS Block IIF satellites, Galileo FOC satellites and regional IRNSS satellites. For IRNSS we presented for the first time estimated DISBs at its L5 frequency and relative to GPS L5, Galileo E5a and QZSS L5. For a baseline with identical receiver pairs, it could be confirmed that the DISBs of IRNSS with respect to the other constellations are indeed absent. The DISBs based on a mixed-receiver combination are estimated as nonzero but seem to be stable, which is in line with the DISBs for mixed receivers based on combinations of GPS, BDS, Galileo and QZSS. In addition to the estimation, we focused on the applicability of DISBs. It was first theoretically demonstrated that DISBs which are determined from zero or short baselines can also be applied to calibrate long baselines for which the differential ionospheric delays cannot be ignored. The other way around, the calibration of short baselines using DISBs determined from long baselines, is however not possible, which is due to a lack of information, as the geometry-free combination of the code DISBs on the first two frequencies is needed for that. Application of nonzero DISBs estimated from a zero baseline to a 1-km baseline consisting of mixed receivers demonstrated that the success rate of single-frequency (L5/E5a) GPS + Galileo + QZSS + IRNSS ambiguity resolution based on inter-system differencing, i.e., relative to the pivot satellite of GPS, is tremendously, i.e., 67 %, higher than the ambiguity success rate corresponding to classically differenced observations, where a pivot satellite is selected for each of the four constellations. Also the availability of positioning improves with inter-system differencing, which is benefitting RTK in environments where GNSS signals are obstructed.

Notes

Acknowledgments

Part of this work has been carried out in the context of the Positioning Program Project 1.19 “Multi-GNSS PPP-RTK Network Processing” of the Cooperative Research Centre for Spatial Information (CRC-SI) in Australia. P.J.G. Teunissen is the recipient of an Australian Research Council (ARC) Federation Fellowship (Project No. FF0883188). This support is gratefully acknowledged.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Dennis Odijk
    • 1
  • Nandakumaran Nadarajah
    • 1
  • Safoora Zaminpardaz
    • 1
  • Peter J. G. Teunissen
    • 1
    • 2
  1. 1.GNSS Research CentreCurtin UniversityPerthAustralia
  2. 2.Department of Geoscience and Remote SensingDelft University of TechnologyDelftThe Netherlands

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