# Toward global instantaneous decimeter-level positioning using tightly coupled multi-constellation and multi-frequency GNSS

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## Abstract

Autonomous driving represents one of the emerging applications that require both high-precision positions and highly critical timeliness to reach stringent safety standards. We develop a method to potentially achieve global instantaneous decimeter-level positioning by virtue of tightly coupled multi-GNSS triple-frequency observations contributing to precise point positioning (PPP). Inter-system phase biases (ISPBs) for two wide-lane observables are first computed for each station to form inter-GNSS resolvable ambiguities, and then correspondingly two wide-lane fractional-cycle biases (FCBs) are computed for each satellite to recover the integer property of single-station ambiguities. With both ISPB and FCB products, we can accomplish tightly coupled multi-GNSS PPP wide-lane ambiguity resolution (PPP-WAR) using only a single epoch of triple-frequency observations on a global scale. To verify this method, we used 1 month of GPS/BeiDou/Galileo/QZSS data from 107 globally distributed stations and 1 h of such multi-GNSS data collected on a vehicle moving in an urban area. We found that both ISPB and FCB products could be estimated every 24 h with high precisions of around or below 0.1 cycles; 83–98% of their day-to-day variations fell within 0.1 cycles, facilitating their precise predictions for real-time applications. Using these corrections, we achieved both instantly and reliably ambiguity-fixed solutions at 91.2% of all epochs at the 107 stations on average; the resultant single-epoch positions reached a mean accuracy of 0.22 m, 0.18 m and 0.63 m for the east, north and up components, respectively, in case of abundant triple-frequency observations from over 15 satellites. Similarly, for the vehicle-borne test, we obtained instantaneous PPP-WAR solutions at 99.31% of all epochs and achieved a positioning accuracy of 0.29, 0.35 and 0.77 m for the east, north and up components, respectively, which improved significantly the identification of road lanes as opposed to other single-epoch solutions. Finally, we expect that the prospect of instantaneous PPP-WAR in aiding driverless vehicles can be more promising if, whenever possible, integrated with inertial sensors and/or smoothed through multi-epoch data.

## Keywords

Global instantaneous decimeter-level positioning Multi-GNSS Multi-frequency Tightly coupled Precise point positioning Ambiguity resolution## 1 Introduction

The races toward autonomous driving are stimulating innovations in high-precision global navigation satellite system (GNSS). European GNSS Agency (2017) projected a rapid growth of requirement for decimeter-level or better positioning accuracy by autonomous driving. They further stressed that, due to stringent safety standards, in all scenarios which implicate “anywhere, anytime and under any condition” does autonomous driving demand “100% position availability at decimeter level or less”. While this goal is only achievable through fusions of varieties of navigation devices and techniques, GNSS among them plays a unique and indispensable role in providing absolute positions referenced to a global coordinate frame. Due to the great and serious safety concern on self-driving vehicles, the strictest rules of navigating them, such as instantaneous decimeter- or even centimeter-level positioning, should be imposed to assure consumers of the vehicle manufacturers’ safety commitments.

Instantaneous, or single-epoch, high-precision GNSS is resistant to satellite signal interruptions which take place frequently in urban and other GNSS-difficult areas and thus is valued in time- and safety-critical applications. Single-epoch decimeter-level or better positioning is premised on ambiguity-fixed carrier-phase data and is thus mostly promised for ultra-short-baseline solutions (e.g., preferably < 10 km) (e.g., Odolinski et al. 2015; Prochniewicz et al. 2016; Teunissen et al. 2014). This is because precise enough ambiguity estimates for both successful integer-cycle resolution and high-precision positioning are hardly achievable in case of single epochs of data. The exception is that, in case of ultra-short baselines, various predominant spatially correlated GNSS errors (e.g., orbit anomalies, atmosphere refractions, etc.) cancel nearly completely, thus minimizing their contamination on ambiguity estimates. The major drawback of short-baseline solutions is that a very dense network of reference stations has to be pre-established, which is costly and unrealistic in remote areas. Precise point positioning (PPP), in contrast, is able to provide subdecimeter- to centimeter-level positions of global coverage without any nearby reference network (Zumberge et al. 1997). However, it is well known that a long initialization time of up to tens of minutes is required before high-precision positions can be achieved, rendering PPP almost useless in time-critical applications (e.g., Geng et al. 2010; Abd Rabbou and El-Rabbany 2016).

The turning point to enabling wide-area single-epoch high-precision positioning is the advent of multi-frequency GNSS signals. For example, Feng and Li (2010) formulated a strategy where two wide-lane combination GNSS observables, whose ambiguities could be resolved near instantaneously, were used to mitigate double-difference ionosphere delays. Single-epoch decimeter-level relative positioning was then implemented through a third ambiguity-fixed wide-lane observable. Tests through simulated triple-frequency GPS data showed that on average about 30 cm positioning accuracy for the horizontal components and 100 cm for the vertical could be attained epoch by epoch. Later, He et al. (2016) used real triple-frequency BeiDou data on medium baselines (e.g., 50–100 km) to realize the idea of Feng and Li (2010), but did not parameterize atmosphere delays in wide-lane positioning with the assumption that they were minimal overall. A success rate of over 98% was achieved for wide-lane ambiguity resolution using a single epoch of BeiDou data; an instantaneous positioning accuracy of better than 20 cm in the horizontal directions and better than 60 cm in the vertical was reached over a 24-h data span. However, if slant ionosphere delays were estimated in addition to positions and ambiguities, Li et al. (2017) showed that the single-epoch position estimates from medium BeiDou baselines could depart from the truth on average by about 50 cm in the horizontal components and 100 cm in the vertical.

In case of PPP, analogously, Geng and Bock (2013) used simulated triple-frequency GPS data to form an ionosphere-free ambiguity-fixed wide-lane observable of decimeter-level noise, through the aid of an extra-wide-lane observable. This strategy resembles that by Feng and Li (2010) in nature, but is formulated using undifferenced observables. The resulting near instantaneous positions reached an accuracy of a few decimeters (i.e., 20–60 cm) for all three components. Gu et al. (2015), alternatively, commenced from raw triple-frequency BeiDou observables, but later still mapped raw ambiguities into two wide-lane counterparts with the goal of speeding up narrow-lane ambiguity resolution. Due to the poor BeiDou satellite geometry, neither wide-lane ambiguity resolution could be accomplished within a few epochs. Rather, both required tens to hundreds of seconds of data.

In this study, we first integrate GPS, BeiDou, Galileo and Quazi-Zenith Satellite System (QZSS) triple-frequency data to enhance satellite geometry and further tightly couple them to achieve instantaneous decimeter-level positioning on a global scale, especially for the horizontal components. Since the multi-frequency satellite constellations of GPS, Galileo and QZSS are incomplete, a tight coupling by sharing a common reference satellite among all GNSS will improve the availability of resolvable ambiguities and also high-precision positions (e.g., Geng and Shi 2017; Geng et al. 2018; Odijk et al. 2017a). This paper is outlined as follows. Section 2 details the methods we developed to realize tightly coupled multi-GNSS instantaneous decimeter-level PPP. Section 3 exhibits the data and processing strategies. Results on ambiguity resolution achievement and single-epoch positioning performance at static stations and on a mobile vehicle are presented in Sect. 4, which will be followed by a discussion in Sect. 5 about high-dimensional ambiguity resolution. Conclusions and outlook are drawn in Sect. 6.

## 2 Methods

*i*and satellite

*k*including their geometric distance, the clock errors and the troposphere delay; \(\gamma _i^k\) is the first-order slant ionosphere delay on L1; \(g_{\mathrm {s},2}=\dfrac{f_{\mathrm {s},1}}{f_{\mathrm {s},2}}\) and \(g_{\mathrm {s},3}=\dfrac{f_{\mathrm {s},1}}{f_{\mathrm {s},3}}\) where \(f_{\mathrm {s},1}\), \(f_{\mathrm {s},2}\) and \(f_{\mathrm {s},3}\) are the frequencies of L1, L2 and L3 signals from GNSS “s”, respectively; \(N_{i,1}^k\), \(N_{i,2}^k\) and \(N_{i,3}^k\) denote the integer ambiguities; \(\lambda _{\mathrm {s},1}=\dfrac{c}{f_{\mathrm {s},1}}\), \(\lambda _{\mathrm {s},2}=\dfrac{c}{f_{\mathrm {s},2}}\) and \(\lambda _{\mathrm {s},3}=\dfrac{c}{f_{\mathrm {s},3}}\) where

*c*is the speed of light in vacuum; \(d_{i,1}^{\mathrm {s}}\), \(d_{i,2}^{\mathrm {s}}\) and \(d_{i,3}^{\mathrm {s}}\) are GNSS-specific station hardware biases on pseudorange, while \(b_{i,1}^{\mathrm {s}}\), \(b_{i,2}^{\mathrm {s}}\) and \(b_{i,3}^{\mathrm {s}}\) are those on carrier phase; \(d_1^k\), \(d_2^k\) and \(d_3^k\) are satellite hardware biases on pseudorange, while \(b_1^k\), \(b_2^k\) and \(b_3^k\) are those on carrier phase; remaining error terms such as multi-path and higher-order ionosphere delays are ignored for brevity. Throughout this study, we use “L1”, “L2” and “L3” to generally represent L1, L2 and L5 signals from GPS and QZSS, B1, B2 and B3 from BeiDou, or E1, E5a and E5b from Galileo, respectively.

Equations 1 and 2 are the raw observation equations we directly employ in the Kalman filter of PPP, where, however, no station or satellite hardware biases (\(d_{i,1}^{\mathrm {s}}\), \(d_1^k\), \(b_{i,1}^{\mathrm {s}}\), \(b_1^k\), etc.) are parameterized due to their linear dependency on the clock and ambiguity parameters. As a result, they have to be combined with other parameters to be estimated (Odijk et al. 2017b; Zhang et al. 2012). Geng and Bock (2016) exemplified that these hardware biases, if time-invariable, could be assimilated entirely, without any residual fractions, into clocks, ionosphere delays and ambiguities in case of dual-frequency CDMA (code-division multiple access) signals. However, this situation deteriorates on the occasion of involving a third frequency signal. Montenbruck et al. (2011) reported that there existed time-varying inter-frequency clock biases (IFCBs) of up to 40 cm between the L1/L2 and L1/L5 GPS clocks, which was also observed for BeiDou, though much smaller (4 cm only) (e.g., Montenbruck et al. 2013). Guo and Geng (2018) later demonstrated in both theory and practice the connections between the satellite hardware biases and the IFCBs. Due largely to these IFCBs, station and satellite hardware biases cannot be absorbed perfectly into other parameters anymore if we insist on common clocks among the three frequencies in undifferenced data processing. Guo and Geng (2018) therefore proposed to estimate a second satellite clock parameter and a constant receiver clock bias for the third frequency signals. Specifically, the legacy L1/L2 carrier phases from a particular satellite share one satellite clock parameter, while the L3 carrier phase monopolizes the second which is intended to absorb the time-varying IFCBs; meanwhile, a station-specific constant unknown is estimated using the L3 pseudorange to address its differing code bias from those within L1/L2 pseudorange. In this case, we can reconcile all hardware biases when assimilating them into the other parameters to be estimated for the three frequencies. In this paper, however, we do not expand Eqs. 1 or 2 to illustrate how the second satellite clock is parameterized, and interested readers can refer to Guo and Geng (2018) for more mathematics.

### 2.1 Inter-system phase bias (ISPB) estimation

ISPBs can be taken as the station-specific biases that prevent the formation of resolvable inter-GNSS ambiguities (Odijk and Teunissen 2013). They will be quite useful when there are few satellites available within each GNSS, since inter-GNSS ambiguities can be composed in addition to the few intra-GNSS counterparts. This means that the number of resolvable ambiguities is increased, and the availability of ambiguity-fixed solutions is improved consequently (Odijk et al. 2017a). We can hence exploit all carrier-phase data for the highest efficiency of ambiguity resolution. This scenario is especially true regarding the minority of triple-frequency GNSS satellites at the moment. Therefore, for this study, we apply the method developed by Geng et al. (2018) to calculate ISPBs across GNSS.

*i*and

*j*and satellites

*k*and

*q*using the estimates in Eq. 3 to eliminate satellite FCBs, that is

*k*belongs to GNSS “\(\mathrm {s_1}\),” while satellite

*q*to “\(\mathrm {s_2}\)”; \({\bar{\beta }}_{i,\mathrm {ew}}^{\mathrm {s_1s_2}}\) and \({\bar{\beta }}_{j,\mathrm {ew}}^{\mathrm {s_1s_2}}\) are the extra-wide-lane ISPBs at stations

*i*and

*j*, respectively, whereas \({\bar{\beta }}_{i,\mathrm {w}}^{\mathrm {s_1s_2}}\) and \({\bar{\beta }}_{j,\mathrm {w}}^{\mathrm {s_1s_2}}\) are the wide-lane ISPBs. Note that these ISPBs will equate zero if “\(\mathrm {s_1}\)” is the same as “\(\mathrm {s_2}\)”, or in other words satellites

*k*and

*q*come from the same GNSS.

*i*and

*j*are still fractional. Then we can apply integer rounding to Eq. 4 to separate differential ISPBs from the nominal integer ambiguities (Geng et al. 2018), that is

### 2.2 Inter-system fractional-cycle bias (FCB) estimation

*i*between a pair of satellites (

*k*and

*q*) belonging to different GNSS, that is

### 2.3 Instantaneous PPP wide-lane ambiguity resolution (PPP-WAR)

*k*belongs to GPS and satellite

*q*to GPS, BeiDou, Galileo or QZSS. We then obtain the (extra-)wide-lane ambiguity estimates again (Eq. 6) and they need to be corrected for ISPBs and FCBs, that is

One important point worthy of attention is that we should leverage the easily resolved extra-wide-lane ambiguities to improve the resolvability of their wide-lane counterparts. To be specific, after fixing extra-wide-lane ambiguities to integers, we should apply these integers as hard constraints to the normal matrix and hence update the wide-lane ambiguity estimates and their variance–covariance matrix (refer to Dong and Bock 1989). The resulting wide-lane ambiguities are then corrected for wide-lane ISPBs and FCBs preceding their integer-cycle resolution. A successful wide-lane ambiguity resolution afterward using a single epoch of data completes PPP-WAR for instantaneous decimeter-level positioning. We note that here extra-wide-lane and wide-lane ambiguity resolution can also be accomplished in one LAMBDA search where the former has to be resolved first in the bootstrapping.

## 3 Data processing

In Fig. 2, we chose 37 globally distributed stations denoted as crosses to test PPP-WAR. Owing to the sparsity of triple-frequency GNSS stations in America, Africa and Asia, most PPP stations were located in Europe and Australia. Most data processing settings at these PPP stations were the same as those at the 70 reference stations. However, we did not estimate residual ZTDs in instantaneous positioning; ionosphere delays were estimated as white noise parameters. Here we should reiterate that only a single epoch of data was used to compute positions in our “instantaneous” PPP-WAR, suggesting that neither atmosphere nor ambiguity parameters were smoothed over multiple epochs. Furthermore, only ambiguities corresponding to elevations of larger than 10\(^\circ \) were allowed to enter the process of integer-cycle resolution; we searched for their integer candidates using the LAMBDA method; the resulting integer (extra-)wide-lane candidates were then validated using the ratio test with a threshold of 2.0 (Euler and Schaffrin 1990). We also enabled partial ambiguity resolution to exclude possibly biased ambiguities. It was mandated that at most four ambiguities could be precluded, while at least five must be reserved for a LAMBDA search. Of particular note, if partial ambiguity resolution still failed ultimately, we chose to fix the ambiguities to the candidate integers corresponding to the largest ratio value we had found, rather than keep a float epoch. In this case, we could always achieve an ambiguity-fixed solution at each epoch, albeit risking identifying incorrect integers.

## 4 Results

### 4.1 Inter-system phase biases (ISPBs)

Station-specific ISPBs are the key to tightly coupling multi-GNSS carrier-phase data. Collecting the (extra-)wide-lane ambiguity estimates from all 107 stations in Fig. 2, we formed double-difference ambiguities between GPS and non-GPS satellites for all eligible station pairs on each day. Their fractional parts were then separated from all these double-difference ambiguities and afterward averaged for each day with respect to each station pair to estimate differential ISPBs between GPS and BeiDou, GPS and Galileo, and GPS and QZSS. Here we can sense that ISPB uncertainties are subject to the agreement among all involved fractional parts of double-difference ambiguities. In fact, more than 98% of such extra-wide-lane and over 83% of wide-lane fractional parts agreed well within ± 0.15 cycles on average over all stations. The mean RMS of all extra-wide-lane and wide-lane fractional parts after removal of ISPBs was around 0.04 cycles and 0.11 cycles, respectively. These statistics demonstrate that extra-wide-lane ISPBs have been quite precisely determined, while wide-lane ISPBs may have been less but still satisfactory to ensure highly efficient ambiguity resolution.

### 4.2 Inter-system fractional-cycle biases (FCBs)

Once ISPBs are computed, we can estimate inter-system FCBs specific to inter-GNSS satellite pairs. In this study, we first corrected for ISPBs at the 70 reference stations in Fig. 2, from which we further formed single-difference (extra-)wide-lane ambiguity estimates between GPS and non-GPS satellites. The fractional parts of these single-difference ambiguities were identified afterward for each satellite pair and then averaged over all involved stations to compute inter-system satellite FCBs. We achieved that on average over 99% of all fractional parts specific to a given satellite pair agreed quite well to ± 0.15 cycles, and the mean RMS of these fractional parts after removal of FCBs was less than 0.05 cycles. Therefore, it is demonstrated that the inter-system FCB products have been precisely estimated in this study.

Success rates of ambiguity-fixed epochs in cases of tightly and loosely coupled multi-GNSS instantaneous PPP-WAR at all 107 stations over the 31 days

Strategy | EWL fixing | WL fixing | |
---|---|---|---|

Without EWL constraints | With EWL constraints | ||

Tightly coupled | 99.9% (2.1%) | 76.7% (55.6%) | 91.2% (30.9%) |

Loosely coupled | 99.5% (1.7%) | 72.1% (48.6%) | 88.7% (21.7%) |

As in the case for ISPBs, FCBs have also to be predicted over time to enable a true real-time PPP-WAR. We have found that day-to-day FCB variations are as small as 0.1 cycles, and even smaller variations can be achieved between neighboring subdaily, instead of 24-h, FCB estimates, as reported by Geng and Bock (2016) and Gu et al. (2015). Therefore, in practice, both ISPB and FCB predictions should be made carefully by considering their temporal properties over various timescales before a pragmatic and reliable real-time PPP-WAR could be implemented.

### 4.3 Instantaneous PPP-WAR at static stations

Once obtaining the ISPB and FCB corrections, we used all 107 stations in Fig. 2 to test instantaneous PPP-WAR. For each station, we performed (extra-)wide-lane partial ambiguity resolution at every epoch independently. An attempt to resolve ambiguities at a particular epoch would be considered successful if these ambiguities were fixed to the same integers for the succeeding 20 epochs (i.e., 10 min). Table 1 shows the success rates of ambiguity-fixed epochs in cases of tightly and loosely coupled multi-GNSS instantaneous PPP-WAR. Loose coupling means that no ISPBs are introduced, and only intra-system ambiguities are formed for integer-cycle resolution. As expected, fixing extra-wide-lane ambiguities has the highest success rates which reach almost 100% by virtue of their super long wavelengths; this prominent performance is not subject to the satellite number or position dilution of precision (PDOP) values (Table 2). Partial ambiguity resolution contributes negligibly to this achievement since its resultant fixed epochs account for only 1–2% of all successfully fixed epochs. This implies that a full integer-cycle resolution is usually possible for extra-wide-lane ambiguities. However, this is not the case for wide-lane ambiguities. As illustrated by the quantities bracketed in the third column of Table 1, at up to 50% of fixed epochs, only a subset of wide-lane ambiguities are resolved. This implies that it is much more difficult to achieve wide-lane ambiguity resolution using a single epoch; overall, only about 70% of epochs achieve such instantaneity. This can be understood in terms of the much shorter wavelengths of wide-lane observables than those of their extra-wide-lane counterparts.

Success rates of ambiguity-fixed epochs and instantaneous positioning errors (meter) for the east, north and up components with respect to the satellite number and PDOP values at the 107 stations in Fig. 2

Satellite number/PDOP | EWL fixing (%) | WL fixing (%) | East | North | Up |
---|---|---|---|---|---|

6–8 | 99.8 | 89.6 | 0.45/0.62 | 0.46/0.64 | 1.21/1.51 |

9–11 | 99.8 | 90.9 | 0.33/0.50 | 0.34/0.51 | 0.94/1.20 |

12–14 | 99.9 | 91.5 | 0.28/0.50 | 0.27/0.48 | 0.83/1.18 |

\(\ge \) 15 | 99.9 | 92.8 | 0.22/0.47 | 0.18/0.43 | 0.63/1.08 |

2.5–5.0 | 99.8 | 88.5 | 0.50/0.69 | 0.51/0.72 | 1.44/1.87 |

2.0–2.5 | 99.8 | 90.3 | 0.38/0.54 | 0.39/0.55 | 1.04/1.35 |

1.5–2.0 | 99.9 | 91.1 | 0.31/0.48 | 0.31/0.49 | 0.84/1.15 |

1.0–1.5 | 99.9 | 92.3 | 0.24/0.46 | 0.23/0.44 | 0.66/1.02 |

Other than only two stations inspected in Fig. 6, Table 2 shows the mean positioning errors of instantaneous PPP-WAR against the satellite number and PDOP values over all 107 stations. Outlier positions were identified if either horizontal component had an absolute error of over 3 m. In total, 0.47% of solutions are excluded for the statistics in Table 2. We can clearly see that when the satellite number rises from 6–8 to more than 15, the positioning errors of instantaneous PPP-WAR keep declining steadily from over 0.4 m to around 0.2 m in the horizontal and from about 1.2 to 0.6 m in the vertical components. The positioning errors of single-epoch PPP generally follow this tendency as well, but they fall relatively slowly against the satellite number. To be specific, when the satellite number exceeds 9, the RMS of single-epoch PPP stays around 0.5 m in the horizontal and 1.1 m in the vertical components. Similar tendency can also be noticed with respect to the PDOP values. Contrasting the RMS between instantaneous PPP-WAR and single-epoch PPP (i.e., instantaneous PPP without WAR), we can draw a conclusion that WAR plays a more and more important role in ameliorating instantaneous positioning accuracy, with respect to the increasing satellite number and the decreasing PDOP values.

To enhance the finding drawn from Table 2, Fig. 7 investigates how the reduction of RMS positioning errors owing to WAR imposed on single-epoch PPP relates to station locations. The mean RMS reductions in the horizontal components are computed for each station over the 31 days and then color-coded as shown within the filled circles of Fig. 7. Warmer colors denote greater RMS reductions. Meanwhile, the mean number of triple-frequency GNSS satellites that can be tracked at each corner of the globe is gray-coded where larger numbers correspond to darker gray. As expected, due to the concentration of BeiDou and QZSS satellites over Asia, 12–18 satellites can be observed in southeast Asia and Australia. As a result, it is this area that delivers the most significant RMS reductions contributed by imposing WAR on single-epoch PPP solutions. Specifically, most reduction rates are more than 45%. In contrast, Europe stations can only observe about 10 satellites and the RMS reduction rates decline to 30–40%. The worst performance comes to American stations over which only eight satellites can be tracked on average. This poor satellite visibility results in less than 20% of RMS improvement at some stations. Therefore, Fig. 7 reinforces that the positioning accuracy of instantaneous PPP-WAR improves with the increasing number of visible triple-frequency satellites.

### 4.4 Instantaneous PPP-WAR on a mobile vehicle

## 5 Discussions on high-dimensional ambiguity resolution

Table 2 and Fig. 7 demonstrate that the more satellites contribute to instantaneous PPP-WAR, the better positioning accuracy we can achieve. For example, a 20 cm horizontal position accuracy, preferable to autonomous driving, is achieved in case of over 15 satellites. However, too many satellites per epoch will jeopardize both fast ambiguity search and reliable ambiguity validation (Verhagen et al. 2012). To be specific, the search for the integer candidates will slow down or even get stuck on the occasion of several tens of ambiguities injected simultaneously into LAMBDA (e.g., Jazaeri et al. 2012; Lu et al. 2018); even if smoothly through LAMBDA, the integer candidate validation for high-dimensional ambiguities through the ratio tests often malfunctions due to the hard choice of threshold values. Figure 10 shows the mean ratio values over the 31 days at all 107 stations against the number of resolved ambiguities. We can see that the ratio values for both extra-wide-lane and wide-lane ambiguity resolution are decreased with respect to the increasing number of resolved ambiguities per epoch. Although the ratio values for extra-wide-lane ambiguity resolution remain far larger than 2 in case of 21 ambiguities, those for wide-lane ambiguity resolution are only marginally over 2, the pre-defined threshold for the ratio tests in this study. In fact, we did come across a small portion (3.2%) of epochs with wide-lane ratio values below 2; however, we still “recklessly” fixed the wide-lane ambiguities to integers. As a result, about 37% of these epochs were not resolved correctly. One remedy effort is that the ratio test should be built upon a fixed failure rate rather than a fixed critical threshold (Teunissen and Verhagen 2009). Another solution is that an optimum subset of ambiguities is identified first for the LAMBDA search and ratio test, whereas the remainder are fixed afterward once the subset is resolved successfully (e.g., Wang and Feng 2013).

## 6 Conclusions and outlook

We developed a method aiming at global instantaneous decimeter-level positioning by tightly coupling GPS/BeiDou/Galileo/QZSS triple-frequency observations. Station ISPBs and satellite FCBs were computed at the server end to facilitate inter-GNSS ambiguity resolution at a single station; with these augmentation products, extra-wide-lane and wide-lane ambiguities could be resolved using a single epoch of data by PPP users (i.e., “instantaneous PPP-WAR” in short).

To validate this method, we first used 31 days of data from 107 globally distributed static stations. The station-specific daily extra-wide-lane and wide-lane ISPBs were computed with high precisions of 0.04 and 0.11 cycles, respectively. Extra-wide-lane ISPBs changed minimally from day to day with over 95% of variations falling in ± 0.1 cycles, while over 83% within ± 0.1 cycles for wide-lane ISPBs. Similarly, satellite-specific daily inter-system FCBs achieved a high precision of better than 0.05 cycles; over 98% of day-to-day extra-wide-lane FCB variations fell in ± 0.1 cycles, while over 83% within ± 0.1 cycles for wide-lane FCBs. The high stability of both ISPB and FCB products over a couple of days favored their precise prediction for real-time PPP-WAR. With the ISPB and FCB products, we could instantly achieve extra-wide-lane ambiguity-fixed solutions at almost all epochs, and 91.2% of epochs could have all or a maximum subset of their wide-lane ambiguities resolved reliably. We envision that this percentage can be further increased in case of full multi-frequency GNSS constellations.

On instantaneous PPP-WAR positioning accuracy, 0.22 m, 0.18 m and 0.63 m for the east, north and up components, respectively, could be attained when the satellite number exceeded 15. Even in America when most triple-frequency signals came from a limited number of GPS and Galileo satellites (6–8 usually), we were still able to achieve a positioning accuracy of better than 0.5 m in the horizontal directions. In addition, a vehicle-borne PPP-WAR test lasting about 60 min in an urban area was carried out with on average 12 triple-frequency satellites in view. We achieved eligible PPP-WAR solutions at 99.31% of all epochs, and the RMS of positioning differences from truth benchmarks was 0.29, 0.35 and 0.77 m for the east, north and up components, respectively, which improved the identification of road lanes.

In this study, we always carried out single-epoch solutions to highlight the instantaneity of decimeter-level positions for the horizontal component on a global scale. However, we should keep in mind that continuous carrier-phase observations, which are obtainable in most open and semi-open sky-view conditions, can be exploited to smooth out the decimeter-level position errors to better facilitate autonomous driving and other high-precision time-critical applications (e.g., Li et al. 2017). In addition, the integration of PPP-WAR with inertial sensors will also improve the position availability in case of the early recovery from GNSS signal blockages.

## Notes

### Acknowledgements

This study is funded by National Science Foundation of China (41674033), State Key Research and Development Programme (2016YFB0501802) and China Earthquake Instrument Development Project (Y201707). We are grateful to International GNSS Service (IGS), Australian Regional GNSS Network (ARGN) for the GPS/BeiDou/Galileo/QZSS data and the high-quality orbit, clock and Earth rotation parameter products. We thank the high-performance computing facility at Wuhan University where all computational works of this study were accomplished. Thanks also go to the three anonymous reviewers for their valuable comments.

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