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Du et al. Satell Navig (2021) 2:3 Page 14 of 22

Authors’ contributions simulated on the GPS L1 measurements of one or two
JW, CR and AE had the idea for the article. YD performed the literature search, satellites during epochs 2000–3000, with random magni-
conducted the experiments and corresponding analysis, and wrote the manu‑
script. All authors provided critical feedback. All authors read and approved tudes from 0.05 to 1.05 m (see Table 3). Te Positioning
the fnal manuscript. Errors (PE) without FDE procedures are shown in Fig. 2.

Funding
This work is supported by the Australian Research Council (ARC) Project No. Figure 3 shows the fault detection tests, i.e. checking
DP170103341. if the absolute post-ft residuals are less than empirical
thresholds, for code and carrier-phase measurements
Availability of data and materials
Data used in this manuscript are available upon request. with the PPP-Wizard software. Te positioning errors
(after convergence) with “Simple FDE” and “Advanced
Competing interests FDE” are shown respectively in Fig. 4a and b.
The authors declare that they have no competing interests.
Author details
1 School of Civil and Environmental Engineering, University of New South
2
Wales, Sydney, Australia. School of Earth and Planetary Sciences, Curtin For comparison purposes, an FDE procedure based
University, Perth, Australia. on the Chi-square test was performed for the same
dataset. Te corresponding test statistics and position-
Appendix ing errors (after convergence) are shown in Figs. 5 and
Example 1: Comparison of “Simple FDE”, “Advanced FDE” 6, respectively. Te excluded measurements by difer-
and Chi‑square test based FDE ent FDE methods, compared with simulated faults, are
Te dataset shown in this example is a one-hour static shown in Table 3. As can be seen in Figs. 3, 4, 5 and 6 and
dataset observed by IGS station STFU from 19:00 to Table 3, all outliers except that on G09 for epoch 2870
20:00 GPST (GPS Time) on January 1, 2020, with 1-s are detected and identifed by diferent methods. For the
sample interval. Te data was processed in kinematic “Simple FDE” there are a few undetected faults and many
foat-PPP mode with the uncombined model using the wrong exclusions, introducing large jumps in the position
modifed PPP-Wizard software. Multi-frequency obser- solutions. On the other hand, the “Advanced FDE” and
vations of GPS and Galileo constellation (with 16–17 sat- Chi-square test can identify all large faults (above 0.9 m
ellites available and over 70 observations in total at each in this case) and provide better results. It can also be
epoch) were adopted and GRG (Groupe de Recherche seen from Fig. 5 that the Chi-square test adopts a vary-
de Géodésie Spatiale of CNES) multi-GNSS fnal prod- ing threshold to account for satellite geometry (number
FA
ucts were applied. Ten groups of outliers were randomly of measurements) changes for a given P .