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Wang et al. Satell Navig (2021) 2:9 Page 4 of 11
G,s G,s G G,s G G,s G
�P = u · �x + c · d ˜ t + M · Z + γ · ˜ I + ε
r,f r r r r f r,1 P,f
C,s C,s C C C,s C C,s C (5)
�P = u · �x + c · d ˜ t + c · ISB + M · Z + γ · ˜ I + ε
r,f r r r r r f r,1 P,f
E,s E,s G E E,s E E,s E
�P = u · �x + c · d ˜ t + c · ISB + M · Z + γ · ˜ I + ε
r,f r r r r r f r,1 P,f
G,s G,s G G,s G G,s G G,s G
˜
�L = u · �x + c · d ˜ t + M · Z − γ · ˜ I + · N + ε
r,f r r r r f r,1 r r,f L,f
C,s C,s G C C,s C C,s C C,s C
�L = u · �x + c · d ˜ t + c · ISB + M · Z − γ · ˜ I + · N ˜ + ε (6)
r,f r r r r r f r,1 r r,f L,f
E,s E,s G E E,s E E,s E ˜ E,s E
�L = u · �x + c · d ˜ t + c · ISB + M · Z − γ · ˜ I + · N + ε
r,f r r r r r f r,1 r r,f L,f
where ISB and ISB are the inter-system bias for BDS-2 and (1,−1), which have the low noise and long wave-
E
C
r
r
and Galileo with respect to GPS, respectively. Normally, lengths, are selected to improve the accuracy of the esti-
the ISBs are regarded as constant which refects the sta- mated ambiguities and reduce the impacts of ionospheric
bility of receiver hardware code delays. However, the errors (Li et al. 2018; Xiao et al. 2019). Hence, the ambi-
datum biases of the satellite clocks between diferent guities are combined as:
GNSS systems are also assimilated into the ISBs. Te q,s
˜
stability of ISBs is strongly correlated to specifc satel- N r,(4,−3) = 4 −3 N ˜ s r,1 (9)
q,s
˜
lite clock products. Correspondingly, the variations of N r,(1,−1) 1 −1 N ˜ s r,2
ISBs are regarded as a white noise process. All estimated q,s q,s
˜
˜
parameters in the Multi-GNSS PPP model are expressed Here, the combined ambiguities, N r,(4,−3) and N r,(1,−1) ,
as vector X: are defned as new NL and WL ambiguities. Te corre-
T
(7)
E
C
C
G
G
C
G
E
˜
˜
˜
˜
˜
˜
X = x, c · d ˜ t , c · ISB ,c · ISB ,Z r , I , I , I , N , N , N E r,f
r,1
r,f
r,f
r
r
r
r,1
r,1
˜
where I denotes the vector of the ionospheric delay sponding FCBs are reformed as:
r,1
parameters for all observed satellites and N denotes the
˜
r,f q q
vector of the ambiguity parameters. B (4,−3) = 4 −3 B 1 q (10)
q
Additionally, the stations in a reference network are B (1,−1) 1 −1 B 2
used to extract the foat ambiguities for FCBs estimation. Since the ambiguities have the same structure in Eq. (7),
Te stations’ coordinates are fxed to their references the fractional parts of ambiguities can be formulated as:
which are obtained in IGS SINEX fles or from the pre-
liminary static PPP processing for the stations not listed n = N − N = B r − B s (11)
˜ s
s
in SNX fles. r r
where n denotes the FCB measurement which is the
Fractional cycle bias estimation fractional part of the real-value ambiguity solution N .
˜ s
r
s
Te ambiguity parameter in Eq. (4) can be rewritten as: N presents the integer part of the real-value ambiguity,
r
which contains the original integer ambiguity and the
˜ q,s q,s q q,s
N r,f = N r,f + B r,f − B f integer part of the code and phase delays from satellite to
q q q q q q q s
B = b − (γ + 1)/(γ − 1) · d + 2/(γ − 1) · d receiver; B r and B are the FCBs for receiver r and satellite
r,f r,f 2 2 r,1 2 r,2
s, respectively. All FCB measurements in Eq. (11) from a
q,s q,s q q q,s q q,s
B = b − (γ + 1)/(γ − 1) · d + 2/(γ − 1) · d
f f 2 2 1 2 2 reference network of m stations and n satellites tracked
(8) can be expressed as:
q,s
q
q,s
where N is an integer value, B and B are the respec-
r,f r,f f
tive receiver and satellite FCBs. Due to a strong correla-
tion between ionospheric delays and ambiguities, which
can be seen from Eq. (6), the accuracy of the estimated
ambiguities on each frequency will be degraded by iono-
spheric errors. Te combinations with integers (4,−3)
