# Five-frequency Galileo long-baseline ambiguity resolution with multipath mitigation

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## Abstract

For long-baseline over several hundreds of kilometers, the ionospheric delays that cannot be fully removed by differencing observations between receivers hampers rapid ambiguity resolution. Compared with forming ionospheric-free linear combination using dual- or triple-frequency observations, estimating ionospheric delays using uncombined observations keeps all the information of the observations and allows extension of the strategy to any number of frequencies. As the number of frequencies has increased for the various GNSSs, it is possible to study long-baseline ambiguity resolution performance using up to five frequencies with uncombined observations. We make use of real Galileo observations on five frequencies with a sampling interval of 1 s. Two long baselines continuously receiving signals from six Galileo satellites during corresponding test time intervals were processed to study the formal and empirical ambiguity success rates in case of full ambiguity resolution (FAR). The multipath effects are mitigated using the measurements of another day when the constellation repeats. Compared to the results using multipath-uncorrected Galileo observations, it is found that the multipath mitigation plays an important role in improving the empirical ambiguity success rates. A high number of frequencies are also found to be helpful to achieve high ambiguity success rate within a short time. Using multipath-uncorrected observations on two, three, four and five frequencies, the mean empirical success rates are found to be about 73, 88, 91, and 95% at 10 s, respectively, while the values are increased to higher than 86, 95, 98, and 99% after mitigating the multipath effects.

## Keywords

Ambiguity resolution Long-baseline Galileo Five frequencies## Introduction

In relative positioning using the Global Navigation Satellite System (GNSS), precise positioning results with accuracy of millimeter to centimeter can be achieved after correctly resolving the phase ambiguities. Depending on the ionospheric activities, for baselines with lengths up to 10–20 km, the atmospheric delays, i.e., the tropospheric delays and the ionospheric delays, can be significantly reduced by forming single differencing observations between receivers. The phase ambiguities can thus be resolved within short convergence time, or even in one epoch only (Tiberius et al. 2002). For longer baselines, however, the remaining differential atmospheric delays may strongly hinder successful integer ambiguity resolution (IAR).

Forming the ionospheric-free linear combination removes the first-order term of the ionospheric delays, which accounts for about 99% of the total ionospheric delays (Elmas et al. 2011). However, the ionospheric-free linear combination has the drawback that it has less flexibility to further strengthen the model, e.g., to constrain the temporal or spatial ionospheric behaviors (Mervart et al. 2013; Teunissen and Khodabandeh 2015). Furthermore, as stated in Teunissen and Odijk (2003), not all the ionospheric-free linear combinations could preserve the integer property of the ambiguities. In the GPS triple-frequency case, e.g., only a particular subset of the ionospheric-free linear combinations allow the parametrization of integer ambiguities. One can therefore alternatively estimate the ionospheric delays using uncombined GNSS observations. Processing based on uncombined observations keeps all the information in the observation equations and is easy to be extended to any number of frequencies (Odijk et al. 2016). In the dual-frequency case, however, this model is too weak for fast ambiguity resolution (Odijk et al. 2014a). Applying ionospheric constraint based on a regional ionospheric model during ionospheric quiet days, according to Zhang et al. (2017), can accelerate the ambiguity resolution from about 18 min to about 5 min on average using 30-s dual-frequency GPS observations for baselines with an average distance of around 95 km.

With the fast development of various GNSS during the last ten years, it is possible to exploit the benefits of the increasing number of frequencies on long-baseline ambiguity resolution. Diverse studies have been performed based on simulated multi-frequency GNSS signals. Jonkman et al. (2000) investigated the impact of a third GPS frequency on long-baseline ambiguity resolution based on the ambiguity resolution success rates using the geometry-free model. It was found that the third frequency helps to achieve substantial reduction in time for reliably fixing the ambiguities. Using simulated 1 Hz triple-frequency GPS and Galileo data, according to Zhang et al. (2003), it takes about 70 and 35 s on average to correctly fix the ambiguities for baseline of 50 km in GPS-only and Galileo-only triple-frequency cases, respectively, with the mean Time-To-Fix-Ambiguities (TTFA) rapidly increasing with the baseline length. Based on a hardware simulation, Sauer et al. (2004) have shown a mean TTFA of around 85, 60, and 50 s using two, three, and four frequencies of simulated Galileo 1 Hz data for a 85 km baseline, respectively, with the TTFA increasing with the baseline length. Using triple-frequency GPS real/semi-generated observations for baselines over 200 km, with an ionosphere-weighted model applied, the IAR time was found to be around 20–25 min using 30 s data (Ning et al. 2016). Apart from these investigations, studies were also performed for long-baseline ambiguity resolution using uncombined GNSS observations. Using uncombined observations when solving for ambiguities with cascade ambiguity resolution (CAS), over 39 s is required to resolve the ambiguities on average for baselines longer than 250 km using simulated Galileo 1 Hz data on four frequencies (Ji et al. 2013). Odijk et al. (2014a) have performed a study to predict the success rates of long-baseline GPS and Galileo ambiguity resolution using uncombined observations with simulated receiver and satellite geometries on three frequencies. Using triple-frequency Galileo-only signals on E1, E5a, and E5b, the formal TTFA is found to be around 20 and 10 min using 30- and 10-s data, respectively. Making use of ionospheric modeling such as single layer model (Schaer 1999) and simulated Galileo signals, the Galileo-only triple-frequency test case with 5-s data requires about 5 min for 500 km ground-based single-baseline to resolve ambiguities (Nardo et al. 2016). Despite different underlying models and sampling rates of the data, the phase ambiguities of medium and long baselines are mostly reported to be resolved in less than or about 60 epochs on average using simulated Galileo signals on three or four frequencies.

The European Galileo system is providing data on five frequencies for 15 operational In-Orbit Validation (IOV) and Full Operational Capability (FOC) satellites by the end of September 2017 (Galileo Constellation 2017). Instead of using simulated signals, it is now possible to study and compare the long-baseline ambiguity resolution performance using real Galileo data on two, three, four, and all five frequencies. In this contribution, undifferenced and uncombined observations are used with the slant ionospheric delays estimated as spatially and temporally unlinked parameters, so that the processing does not set conditions for the ionospheric activity. The observation model is easy to be extended to any number of frequencies. The real Galileo observations used in this study are influenced by code and phase multipath effects. Making use of Galileo observations of another day, the procedure to compute the multipath corrections and to mitigate the multipath effects is presented. The ambiguity success rates (SRs) of two long baselines consisting of continuously operating reference stations (CORS) without and with the multipath mitigation are analyzed and compared using different number of frequencies and at different processing times. We remark that the approach proposed in this contribution is applicable to CORS network ambiguity resolution with known coordinates, where the rover positioning is not performed.

In the subsequent section, we first introduce the strategy of the full integer ambiguity resolution using the Least-Squares Ambiguity Decorrelation Adjustment (LAMBDA) method (Teunissen 1993, 1995). After that, the data selection and the process for multipath mitigation are explained. The formal and empirical success rates are then analyzed and compared using two, three, four, and five Galileo frequencies. The results are discussed and concluded.

## Processing strategy

During the processing, the precise Galileo satellite orbits of the Multi-GNSS Experiment (MGEX) (MGEX 2017; Montenbruck et al. 2014, 2017) calculated by CNES/CLS/GRGS and German Research Centre for Geosciences are also used as known parameters. As to the receiver phase center offsets (PCOs) and phase center variations (PCVs), the ones in igs08.atx (Montenbruck et al. 2015) for GPS L1 are used for Galileo E1 and those for GPS L2 are used for the other Galileo frequencies. The a priori ZTDs are computed based on the Saastamoinen model (Saastamoinen 1972), and the Ifadis mapping function (Ifadis 1986) is used as the troposphere mapping function. Since this study concentrates on the ambiguity resolution performance of CORS using signals on different number of frequencies, the coordinates of the stations provided by Geoscience Australia (GA, Geoscience Australia 2017) are assumed to be known. With the strong model with known baselines, we show the ambiguity success rates that can be achieved for Galileo long-baseline processing without and with multipath mitigation, as well as the rate changes with increasing number of frequencies.

*S*-basis parameters are constrained (Teunissen et al. 2010), so that estimable parameters are formed and a full-rank design matrix is generated. After reformulation, the O–C terms of the phase and the code observations are represented by

Parameter | Interpretation |
---|---|

\({\text{d}}{\tilde {t}_{r \ne 1}}({t_i})\) | \({\text{d}}{t_{1r}}\left( {{t_i}} \right)+{d_{1r,{\text{IF}}}}\left( {{t_i}} \right)\) |

\({\text{d}}{\tilde {t}^s}({t_i})\) | \({\text{d}}{t^s}\left( {{t_i}} \right)+d_{{,{\text{IF}}}}^{s}\left( {{t_i}} \right) - ({\text{d}}{t_1}\left( {{t_i}} \right)+{d_{1,{\text{IF}}}}\left( {{t_i}} \right))\) |

\(\tilde {\iota }_{r}^{s}({t_i})\) | \(\iota _{r}^{s}\left( {{t_i}} \right)+{d_{r,{\text{GF}}}}\left( {{t_i}} \right) - d_{{,{\text{GF}}}}^{s}\left( {{t_i}} \right)\) |

\({\tilde {\delta }_{r \ne 1,j}}({t_i})\) | \({\delta _{1r,j}}\left( {{t_i}} \right)+{\mu _j}{d_{1r,{\text{GF}}}}\left( {{t_i}} \right) - {d_{1r,{\text{IF}}}}\left( {{t_i}} \right)+{\lambda _j}a_{{1r,j}}^{1}+m_{{{\phi _{1r,j}}}}^{1}\left( {{t_i}} \right)\) |

\(\tilde {\delta }_{{,j}}^{s}({t_i})\) | \(\delta _{{,j}}^{s}\left( {{t_i}} \right)+{\mu _j}(d_{{,{\text{GF}}}}^{s}\left( {{t_i}} \right) - {d_{1,{\text{GF}}}}\left( {{t_i}} \right)) - (d_{{,{\text{IF}}}}^{s}\left( {{t_i}} \right) - {d_{1,{\text{IF}}}}\left( {{t_i}} \right)) - {\delta _{1,j}}\left( {{t_i}} \right) - {\lambda _j}a_{{1,j}}^{s}- m_{{{\phi _{1,j}}}}^{s}\left( {{t_i}} \right)\) |

\({\tilde {d}_{r \ne 1,j>2}}({t_i})\) | \({d_{1r,j}}\left( {{t_i}} \right) - ({d_{1r,{\text{IF}}}}\left( {{t_i}} \right)+{\mu _j}{d_{1r,{\text{GF}}}}\left( {{t_i}} \right))+m_{{{p_{1r,j}}}}^{1}\left( {{t_i}} \right)\) |

\(\tilde {d}_{{,j>2}}^{s}({t_i})\) | \(d_{{,j}}^{s}\left( {{t_i}} \right) - \left( {d_{{,{\text{IF}}}}^{s}\left( {{t_i}} \right)+{\mu _j}d_{{,{\text{GF}}}}^{s}\left( {{t_i}} \right)} \right) - \left( {{d_{1,j}}\left( {{t_i}} \right) - \left( {{d_{1,{\text{IF}}}}\left( {{t_i}} \right)+{\mu _j}{d_{1,{\text{GF}}}}\left( {{t_i}} \right)} \right)} \right) - m_{{{p_{1,j}}}}^{s}\left( {{t_i}} \right)\) |

\(\tilde {a}_{{r \ne 1,j}}^{{s \ne 1}}\) | \(a_{{1r,j}}^{s} - a_{{1r,j}}^{1}\) |

| \({\text{d}}{t_1}\left( {{t_i}} \right),~{\delta _{1,j}}\left( {{t_i}} \right),~{d_{1,j}}\left( {{t_i}} \right),~{\bar {d}_{r \ne 1,~j=1,2}}\left( {{t_i}} \right),~\bar {d}_{{,j=1,2}}^{s}\left( {{t_i}} \right),~a_{{1,j}}^{s},~a_{{r,j}}^{1}\) |

## Data selection

Two long baselines in West-Australia are used for the processing as shown in Fig. 1. Galileo-only observations of 1 Hz are processed on two, three, four, and five frequencies. The elevation mask is set to be 10°. The length of the baselines, the time intervals used for the processing, and the number of the continuously tracked Galileo satellites are listed in Table 2. Septentrio receivers are used for both baselines. The Galileo frequency combinations used for the processing are listed in Table 3 with the frequency values given in RINEX (2015). The skyplots of the Galileo satellites for the stations FROY and PTHL are shown in Fig. 2.

Details of the baselines used for the Galileo processing

Baseline | Baseline length (km) | Date | Time (GPST) | No. of satellites |
---|---|---|---|---|

FROY–KUNU | 416 | April 19, 2017 | 18:12:20–19:29:36 | 6 |

May 29, 2017 | 15:29:57–16:46:45 | 6 | ||

PTHL–TOMP | 288 | April 24, 2017 | 1:26:23–3:04:28 | 6 |

June 2, 2017 | 22:44:42–23:59:59 | 6 |

Frequency combinations used for the processing

Number of frequencies | Signals |
---|---|

2 | E1, E5a |

3 | E1, E5a, E5b |

4 | E1, E5a, E5, E5b |

5 | E1, E5a, E5, E5b, E6 |

## Multipath mitigation

Figure 3 (top) shows the multipath combination \(\text{MP}_{{1r,j}}^{{1s}}\) on E5 for the baseline FROY–KUNU and the satellite pair E12 and E11 from 18:12:20 to 19:29:36 on April 19, 2017, plotted with the blue line, and at the shifted time points \(T_{i}^{s}\) on May 9, 2017, plotted with the red line. To obtain the terms \(\Delta \phi _{{1r,i}}^{{1s}}\) and \(\check{a}_{{1r,i}}^{{1s}}\) in (12), the phase O–C terms and fixed ambiguities on E1 are also used for computing the multipath combination. We see that the multipath combination \(\text{MP}_{{1r,j}}^{{1s}}\) on April 19 has repeated behaviors on May 9, 2017. After forming differences between the blue and the red lines in Fig. 3 (top), as illustrated in the bottom panel of the figure, the multipath effects are reduced. The remaining systematic patterns are supposed to be caused by two reasons. First, by computing the double-differenced multipath combination on May 9, 2017, the shifted time points \(T_{i}^{{s \ne 1}}\) are used for both the reference satellite and the satellite \(s\). This means that the multipath changes between \(T_{i}^{1}\) and \(T_{i}^{s}\) are not considered for the reference satellite. Second, the same satellite does not fly over exactly the same location after the searched time shift. The corresponding distances range from kilometers to tens of kilometers.

The multipath effects that are clearly observed in Fig. 3 may have significant influences on the ambiguity resolution performance. In the following two subsections, the procedure to mitigate multipath effects in the phase and code observations is explained in detail. In order to mitigate the multipath effects for the test time intervals, the Galileo measurements of the same station pairs, May 9 for the baseline FROY–KUNU and May 14, 2017 for the baseline PTHL–TOMP, are used with a time shift of about 20 days minus 80–82 min. As an example, Fig. 4 shows the deviations of the time shifts \(\Delta T_{i}^{s}\) from 20 days for all the six Galileo satellites and baseline FROY–KUNU in the tested time interval for April 19, 2017 (Table 2). We see that \(\Delta T_{i}^{s}\) does not only differ for different satellites, but also changes during the tested time interval. Allowing a short initialization phase of the filter, the processing on the day used for multipath mitigation begins at 60 s earlier than the smallest \(T_{i}^{{s \ne 1}}\) and ends at the largest \(T_{i}^{{s \ne 1}}\). The entire processing interval on the day used for multipath mitigation is denoted as \(T\). Since the multipath corrections are to be computed on double-difference level at the shifted time point \(T_{i}^{{s \ne 1}}\) for satellite \(s \ne 1\), the shifted time points of the reference satellite \(T_{i}^{1}\) are thus not considered when defining \(T\).

### Residuals on the day used to compute multipath corrections

For long baselines, the change of the between-receiver atmospheric delays in time hampers the mitigation of the multipath effects. Since the estimable tropospheric and ionospheric delays are estimated in the observation (3) and (4), we may utilize the double-differenced atmospheric estimates in \(T\) to reduce the multipath effects in the test time intervals \(t\) (Table 2).

### Multipath corrections applied to current data

Parameter | Interpretation |
---|---|

\({\tilde {\tilde {\tau }}_r}({t_i})\) | \({\tau _r}\left( {{t_i}} \right) - E\left( {\Delta {\check{\tau} _r}\left( {T_{i}^{2}} \right)} \right)\) |

\({\text{d}}{\tilde {\tilde {t}}_{r \ne 1}}({t_i})\) | \({\text{d}}{t_{1r}}\left( {{t_i}} \right)+{d_{1r,IF}}\left( {{t_i}} \right)+g_{r}^{1}\left( {T_{i}^{1}} \right)E\left( {\Delta {\check{\tau} _r}\left( {T_{i}^{2}} \right)} \right) - g_{1}^{1}\left( {T_{i}^{1}} \right)E\left( {\Delta {\check{\tau} _1}\left( {T_{i}^{2}} \right)} \right)\) |

\({\text{d}}{\tilde {\tilde {t}}^s}({t_i})\) | \({\text{d}}{t^s}\left( {{t_i}} \right)+d_{{,{\text{IF}}}}^{s}\left( {{t_i}} \right) - \left( {{\text{d}}{t_1}\left( {{t_i}} \right)+{d_{1,{\text{IF}}}}\left( {{t_i}} \right)} \right) - g_{1}^{s}\left( {T_{i}^{s}} \right)E\left( {\Delta {\check{\tau} _1}\left( {T_{i}^{2}} \right)} \right)\) |

\(\tilde {\tilde {\iota }}_{r}^{s}({t_i})\) | \(\iota _{r}^{s}\left( {{t_i}} \right)+{d_{r,{\text{GF}}}}\left( {{t_i}} \right) - d_{{,{\text{GF}}}}^{s}\left( {{t_i}} \right) - E\left( {\Delta \check{\tilde {\iota }}_{{1r}}^{{1s}}\left( {T_{i}^{s}} \right)} \right)\) |

Due to the influences of the multipath effects during \(T\), the expectations of \(\Delta {\check{\tau} _r}\) and \(\Delta \check{\tilde {\iota }}_{{1r}}^{{1s}}\) in (21) and (22) do not equal to zero. Compared to the case without multipath mitigation, the new estimable parameters in Table 4 during the current processing time interval \(t\) are thus still influenced by the multipath effects during \(T\), which are generally within millimeters at the zenith direction for Galileo phase signals (Cai et al. 2016; Zaminpardaz and Teunissen 2017). However, since the double-differenced multipath during \(t\) is significantly mitigated by subtracting the residuals during \(T\), the ambiguity resolution during \(t\) is accelerated. The results will be discussed in the next section.

Instead of using the observation equations in (3) and (4), the corrected observation (26) and (27) are used for the processing in Kalman filter. The variance matrix of the corrected observations is approximated to a diagonal matrix with the diagonal elements computed using the exponential elevation weighting function given in (15). The undifferenced zenith-referenced a priori standard deviations of the multipath-corrected Galileo observations reported in Zaminpardaz and Teunissen (2017) for Septentrio receivers are used in this study. After subtraction of the residuals at \(T_{i}^{s}\), the undifferenced a priori standard deviations are multiplied with a factor of \(\sqrt 2\).

## Analysis of the results

Formal and empirical analysis are performed using the data from both baselines and all tested time intervals listed in Table 2. A time window of 15 min is defined for each round of processing and is shifted by 30 s before restarting the next round of processing. The empirical results using multipath-mitigated and -uncorrected observations are compared with the formal results in the end of the section.

### Formal analysis

Figure 5 shows the mean formal BSRs \({\bar {P}_{\text{F}}}\) for different frequency combinations using all data sets. The horizontal black-dashed lines mark the \({\bar {P}_{\text{F}}}\) of 90 and 99%. From Fig. 5, we see that the \({\bar {P}_{\text{F}}}\) increase with the increasing number of frequencies. The mean formal BSR \({\bar {P}_{\text{F}}}\) at \({t_i}\) of 5, 10, and 20 s are listed in Table 5 for different frequency combinations. Using three or more frequencies, the \({\bar {P}_{\text{F}}}\) are larger than 99.7% after 5 s. After 20 s, all frequency combinations deliver \({\bar {P}_{\text{F}}}\) larger than 99.9%. We remark that the formal BSRs are computed based on the assumption that the multipath effects are completely removed from the observations.

Mean formal bootstrapping success rates (BSRs) \({\bar {P}_{\text{F}}}\) in percent given in (28) at 5, 10, and 20 s of the processing

Frequencies | 5 s | 10 s | 20 s |
---|---|---|---|

E1 + E5a | 97.20 | 99.64 | 99.97 |

E1 + E5a + E5b | 99.76 | 99.99 | > 99.99 |

E1 + E5a + E5 + E5b | > 99.99 | > 99.99 | > 99.99 |

E1 + E5a + E5 + E5b + E6 | > 99.99 | > 99.99 | > 99.99 |

### Empirical analysis

Figure 6 shows the mean empirical success rates (SRs) \({\bar {P}_{\text{E}}}\) using both baselines and all time windows in the tested time intervals. In the top panel, the multipath effects are mitigated as described in the last section. In contrast to the multipath-mitigated case, we also compute the \({\bar {P}_{\text{E}}}\) without correcting the multipath effects using the zenith-referenced a priori standard deviations of the multipath-uncorrected observations given in Zaminpardaz and Teunissen (2017) for Septentrio receivers.

From Fig. 6, we see that the multipath mitigation has increased the mean empirical SRs compared to the case using multipath-uncorrected observations. In both cases, a higher number of frequencies is helpful to increase the SRs of ambiguity fixing. The mean empirical SRs \({\bar {P}_{\text{E}}}\) at 5, 10, 20 s, 1, and 2 min are listed in Table 6 using multipath-mitigated and -uncorrected observations, respectively. After the multipath mitigation, the \({\bar {P}_{\text{E}}}\) is higher than 95, 98, and 99% at 10 s using three, four, and five frequencies, respectively, while the values remain at around 88, 91, and 95% using multipath-uncorrected observations, respectively.

Mean empirical success rates (SRs) \({\bar {P}_{\text{E}}}\) in percent given in (28) at 5, 10, 20 s, 1, and 2 min of the processing using multipath-mitigated and multipath-uncorrected observations

Frequencies | 5 s | 10 s | 20 s | 1 min | 2 min |
---|---|---|---|---|---|

Multipath-mitigated | |||||

E1 + E5a | 84.34 | 86.26 | 88.38 | 91.84 | 95.51 |

E1 + E5a + E5b | 92.19 | 95.32 | 96.00 | 97.28 | 98.09 |

E1 + E5a + E5 + E5b | 97.31 | 98.31 | 98.52 | 99.03 | 99.45 |

E1 + E5a + E5 + E5b + E6 | 98.43 | 99.23 | 99.37 | 99.66 | 99.82 |

Multipath-uncorrected | |||||

E1 + E5a | 72.07 | 72.51 | 74.90 | 79.37 | 83.78 |

E1 + E5a + E5b | 87.09 | 87.66 | 88.16 | 89.63 | 90.79 |

E1 + E5a + E5 + E5b | 90.47 | 91.11 | 91.55 | 92.50 | 93.32 |

E1 + E5a + E5 + E5b + E6 | 94.56 | 94.99 | 95.05 | 95.24 | 95.58 |

Comparing the mean formal BSRs in the last subsection and the mean empirical SRs using multipath-mitigated and -uncorrected observations, we observe that the differences after the multipath mitigation are reduced. Table 7 lists the differences between the mean formal BSRs and the mean empirical SRs with and without multipath mitigation. At \({t_i}\) of 10 s, the differences in mean SRs amount to around 12, 9, and 5% using multipath-uncorrected observations on three, four, and five frequencies, respectively, while they are reduced to below 5, 2, and 1% using multipath-mitigated observations.

Differences between the mean formal and empirical success rates (SRs) in percent using multipath-mitigated and multipath-uncorrected observations

Frequencies | 5 s | 10 s | 20 s | 1 min | 2 min |
---|---|---|---|---|---|

Multipath-mitigated | |||||

E1 + E5a | 12.86 | 13.39 | 11.59 | 8.16 | 4.49 |

E1 + E5a + E5b | 7.57 | 4.67 | 4.00 | 2.72 | 1.91 |

E1 + E5a + E5 + E5b | 2.69 | 1.69 | 1.48 | 0.97 | 0.55 |

E1 + E5a + E5 + E5b + E6 | 1.57 | 0.77 | 0.63 | 0.34 | 0.18 |

Multipath-uncorrected | |||||

E1 + E5a | 25.14 | 27.13 | 25.07 | 20.63 | 16.22 |

E1 + E5a + E5b | 12.67 | 12.33 | 11.84 | 10.37 | 9.21 |

E1 + E5a + E5 + E5b | 9.52 | 8.89 | 8.45 | 7.50 | 6.68 |

E1 + E5a + E5 + E5b + E6 | 5.44 | 5.01 | 4.95 | 4.76 | 4.42 |

## Conclusions

For long-baseline ambiguity resolution, the ionospheric delays that cannot be removed by generating between-station differences hampers rapid ambiguity resolution. The first-order terms of the ionospheric delays can either be eliminated by forming linear combinations using multi-frequency data, or estimated as float values using uncombined phase and code observations. The latter strategy has the advantage that it keeps all information in the observation equations and is easy to be expanded to any number of frequencies. The dual-frequency case is, however, found to be too weak to use such a model. With the increasing number of the frequencies in current GNSS, it is now possible to exploit the benefits of real multi-frequency data in long-baseline ambiguity resolution using uncombined phase and code observations.

The current Galileo system is providing signals on five frequencies on 15 operational IOV and FOC satellites by the end of September 2017. In this study, real Galileo data on all five frequencies for two baselines of several hundreds of kilometers in Australia are used for the processing. Using both multipath-mitigated and -uncorrected phase and the code observations, we studied the formal and the empirical ambiguity success rates for long baselines with different frequency combinations. We observed that the multipath mitigation procedure is important for improving the empirical ambiguity success rates compared to the cases using multipath-uncorrected observations. A higher number of frequencies are also found to be essential for achieving high success rates within short time. Using 1 Hz data after multipath mitigation, the mean empirical success rates are found to be above 95, 98, and 99% after 10 s using three, four, and five frequencies, respectively.

## Notes

### Acknowledgements

We would like to thank the International GNSS Service (IGS) and Geoscience Australia for providing the MGEX orbit products, the observation data and the station coordinates. The MGEX orbit products were obtained through the online archives of the Crustal Dynamics Data Information System (CDDIS), NASA Goddard Space Flight Center, Greenbelt, MD, USA. ftp://cddis.gsfc.nasa.gov/gnss/products/mgex/. We are also grateful for the contributions of our colleagues in the GNSS Research Centre, Curtin University, on the development of the Curtin PPP-RTK Software. PJG Teunissen is recipient of an Australian Research Council (ARC) Federation Fellowship (project number FF0883188).

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