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Wang et al. Satell Navig (2021) 2:9 Page 3 of 11

Orbit Determination in Europe (CODE) precise satellite Methodology
orbit products to users who process the PPP with soft- Ambiguity‑foat PPP
ware “PRIDE PPP-AR” (Geng et al. 2019a, b). Hence, the In the uncombined PPP model, the ionospheric delays
generation of FCBs for the PPP AR must be an essential can be either estimated or corrected. Te linearized
work for supporting PPP services. observation equations for pseudorange and phase obser-
Te international GNSS Monitoring and Assessment vations from satellite s to receiver r are described as:

q,s q,s q q,s q,s q q,s q q,s q
�P r,f = u r · �x + c · (dt r − dt ) + M r · Z + γ · I r,1 +d r,f − d f + ε P,f (1)
r
f
q,s q,s q q,s q,s q q,s q q,s q q,s q
�L = u r · �x + c · (dt r − dt ) + M r · Z − γ · I + · N +b − b + ε
r,f r f r,1 f r,f r,f f L,f
System (iGMAS), which is a scientifc project initiated q,s q,s
and led by China under the United Nations framework, where P and L are the respective pseudorange
r,f
r,f
provides the daily satellite precise products for PPP users and phase measurements on the frequency f (f = 1,2),
q,s
to get ambiguity-foat solutions (Jiao et al. 2012). To from which the computed values are removed; u r is the
meet the requirement of stable and reliable positioning, receiver-to-satellite unit vector; x is the vector of the
the ambiguity-fxed solutions should be implemented receiver position corrections to its preliminary position;
q
q,s
with the corresponding FCB products. In our work, dt r and dt are the receiver and satellite clock errors,
q,s
the uncombined PPP model is adopted to generate the respectively; c is the light speed in vacuum;M r is the ele-
FCBs with the fnal satellite orbit and clock corrections vation-dependent mapping function for the tropospheric
q,s
with GPS, Galileo, and BDS-2 observations at the Bei- wet delay from the corresponding zenith one Z ; I is
r,1
r
Dou analysis and service center of Chang’an University, the ionospheric delay along the line-of-sight from a
which is a member of the iGMAS. Te uncombined PPP receiver to a satellite at the frst frequency and
q
q
q
q 2
model is fexible for PPP processing with multi-frequency γ = ( / ) ; is the wavelength for the frequency f of
1
f
f
f
q
q
q,s
and multi-GNSS observations. Te raw ambiguities are a GNSS q; N is the phase ambiguity; d and b are the
r,f
r,f
r,f
directly estimated in the PPP model and then combined receiver hardware delays of code and phase observations,
q,s
q,s
to estimate the phase bias in the FCB estimator. Te ini- respectively; d and b are the satellite hardware delays
f
f
tial results of the FCB generation and the PPP AR with of code and phase observations, respectively; ε P,f and ε L,f
GPS/BDS-2/Galileo data are analyzed to present the are the code and phase measurement noises, respectively,
ambiguity-fxed positioning performance based on the which include the multipath efects (Shi and Gao 2014).
iGMAS. Considering that the ionospheric-free combined
Tis contribution presented the great advantages of observations which contain the satellite code biases are
the multi-GNSS PPP in terms of convergence speed adopted for satellite clock error (parameter) estimates,
and positioning accuracy. Te PPP AR based on iGMAS the satellite clock parameters can be denoted as:
products showed the potential for global users in single
q
q,s
q
q
station positioning services. In this paper, the method for d ˜ t q,s = dt q,s + γ /(γ − 1) · d 1 q,s + d /(1 − γ )
2
2
2
2
FCB estimation and the PPP AR are introduced in detail. (2)
Te integer coefcients (4, − 3) and (1, − 1) are adopted Hence, the receiver clock parameter is denoted as:
to decorrelate the involving parameters, reducing the
q
q
q
q
q
q
q
efects of ionospheric delays. Ten, the processing strat- d ˜ t = dt + γ /(γ − 1) · d r,1 + d /(1 − γ ) (3)
2
r,2
2
r
2
r
egies and experiment data are introduced. To evaluate
the PPP AR performances, the precision of the estimated Te ionospheric delays and ambiguities are reparameter-
ized as:
q,s q,s q q
q,s
˜ I = I + (DCB r − DCB )/(γ − 1)
r,1 r,1 2 (4)
q
q
q,s
q
q,s
N ˜ q,s = N q,s + b q − b q,s − (γ + 1)/(γ − 1) · (d q − d ) + 2/(γ − 1) · (d q − d )
r,f r,f r,f f 2 2 r,1 1 2 r,2 2
s
s
s
FCBs for the three GNSS systems is demonstrated. Te where DCB r = d r,1 − d r,2 and DCB = d − d are the
1
2
daily and hourly positioning accuracy is analyzed in static Diferential Code Biases (DCBs) for respective receiver
and kinematic modes. Finally, conclusions and outworks and satellite. After applying the satellite clock correc-
are discussed. tions, the observations for the uncombined PPP model
with GPS, BDS-2, and Galileo data can be written as: