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

## 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.

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

*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.

*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):

*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.

_{ 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.

*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:

*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:

*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:

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

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.

*B*:

*B*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.

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* |

### 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.

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 |

### 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.

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.

## 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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