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Wang et al. Satell Navig (2021) 2:9 Page 6 of 11

accuracy of FCB estimation is evaluated by the poste-
180° 120°W 60°W 0° 60°E 120°E 180°
90°N 90°N rior residuals of FCB measurements. After analyzing the
performance of estimated FCBs, the uncombined PPP
60°N 60°N
results are presented with the ambiguity-foat and fxed
30°N 30°N
solutions.
0° 0°
30°S 30°S FCB residuals distributions
60°S 60°S In the proposed method of FCB estimation, the FCBs
for receiver and satellites are estimated separately using
90°S 90°S
180° 120°W 60°W 0° 60°E 120°E 180° the WL and NL ambiguities. Te FCB estimator directly
Fig. 2 Station distribution in the reference network for estimating provides the WL and NL combined FCBs. Hence, the
FCBs. The blue points denote stations tracking GPS satellites. The red accuracy assessment of the FCBs is carried out in the
stars and yellow points are for BDS-2, Galileo, respectively WL and NL combinations. Te distribution of the pos-
terior residuals is an intuitional indicator for the consist-
ency of the FCBs results among diferent measurements.
Data and processing strategies Generally, the residuals are close to zero indicating a high
To assess the performance of the FCB estimations with
the satellite orbit and clock products at Chang’an Univer-
sity based on the iGMAS, 316 stations from iGMAS and
Multi-GNSS EXperiment (MGEX) are selected, shown in (4,−3) (1,−1)
Fig. 2. 0.5
GPS satellites are tracked at all stations, and Galileo RMS = 0.086 RMS = 0.069
satellites are tracked at 190 stations, among which 138 0.4
stations tracked BDS-2 satellites. Te uncombined PPP
is implemented with GPS, Galileo, and BDS-2 observa- 0.3
tions for extracting the foat ambiguities as the measure- Probability
ments in the FCB estimation. After generating FCBs, the 0.2
uncombined PPP AR is achieved for GPS/BDS-2/Galileo
combined solutions. Te elevation-dependent stochastic
model σ =σ /sin (el) is adopted for the GNSS observa- 0.1
2
2
2
0
tions, where σ 0 is 0.003 m and 0.3 m for carrier phase and
code measurements, respectively, and el is the satellite 0.0 0.2 0.4 −0.4 −0.2 0.0 0.2 0.4
−0.4 −0.2 0.0
elevation angle in radian. Among GPS, Galileo and BDS- GPS FCB residuals in cycles
2, the ratio of their observation noise is set as 1:1:3. Te Fig. 3 Histogram of the GPS FCB residuals in the NL and WL linear
Phase Center Ofset (PCO) and the Phase Center Vari- combinations
ations (PCV) published by the European Space Agency
(ESA) are used to correct the observations for BDS-2
satellites, while for the others the corrections are accord- (4,3) (1,1)
ing to the IGS14.atx fle. In PPP processing, the cutof 0.5
angle of satellite elevation is 7° in the foat solutions and RMS = 0.087 RMS = 0.046
15° in the AR, while it is 30° in the FCB estimation for 0.4
quality assurance. For the parameters estimated in PPP,
the receiver clock and Inter-System Bias (ISB) are treated 0.3
as white noise processes, the zenith tropospheric delay Probability
as a random walk process, and the ionospheric delays as 0.2
white noise processes.
0.1
Results and analysis
Te FCB products on Day of Year (DOY) 244, 2019 are 0.0
generated to assess the performance of FCB estima- −0.4 −0.2 0.0 0.2 0.4 −0.4 −0.2 0.0 0.2 0.4
tion. Firstly, the stability of the FCB series over one day Galileo FCB residuals in cycles
is analyzed with the averaged standard deviations for the Fig. 4 Histogram of the Galileo FCB residuals in the NL and WL linear
combined FCBs and raw FCBs on each frequency. Te combinations