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

ambiguity-fxed positioning solutions (Ge et al. 2008; biases on the fractional parts of ambiguities are rigor-
Collins et al. 2008; Laurichesse et al. 2009). Based on ously considered. Te integer recovery clocks proposed
an empirical assumption that the Uncalibrated Phase by Collins et al. (2008) and Laurichesse et al. (2009) are
Delays (UPD) are relatively stable in time, the Single- used to recover the integer characteristics for the NL
Diference (SD) FCBs of the Wide Lane (WL) and Nar- foat ambiguities in PPP for ambiguity resolution. Tese
row Lane (NL) foat ambiguities between satellites are two methods assimilated the NL FCBs into clocks esti-
estimated from a reference network (Ge et al. 2008). Te mates, which increased the computation of clock prod-
WL FCBs are determined by averaging the fractional ucts. In the FCB method, the original International GNSS
parts of all WL ambiguities from the same satellites. Te Service (IGS) precise clock products are used to separate
Melbourne–Wübbena (MW) measurements are used to the FCBs from ambiguities in an independent process-
derive the wide-lane ambiguities (Melbourne 1985; Wüb- ing. Hence, the FCB method is widely adopted in many
bena 1985). Generally, the WL FCBs are stable over a few studies.
days, or even several months (Gabor and Nerem 1999). Most of the FCB estimation results are derived with the
After determining the WL FCBs, the integer part of the ionospheric-free PPP model. Recently, the uncombined
WL ambiguities is used to estimate the foat narrow-lane PPP model with raw pseudorange and phase observa-
ambiguities. Similarly, the NL FCBs are also determined tions has attracted a great attention. Te uncombined
by averaging the fractional parts of the NL ambiguities. PPP model can directly estimate the raw ambiguities on
Due to the short wavelength of the narrow-lane ambi- each frequency and is fexible for processing multi-fre-
guity, for instance, about 10 cm for Global Positioning quency observations. Its performance was demonstrated
System (GPS), the NL FCBs are not as stable as the WL, in the single- or dual-frequency PPP and PPP-RTK
which are proposed to estimate 15 min mean values. (Real-Time Kinematic) (Wübbena et al. 2005; Li et al.
Instead of estimating the SD FCB from the same satel- 2011; Zhang et al. 2011; Chen et al. 2015; Teunissen
lite pairs, Li and Zhang (2012) proposed that the Zero- and Khodabandeh 2015; Lou et al. 2016). Li et al. (2013)
Diference (ZD) FCBs can be estimated together in the presented the FCB estimates with the uncombined PPP
adjustment system using the least-square method with all model using the GPS L1 and L2 raw observations, which
FCB measurements (Li and Zhang 2012). Tis approach improved the performance of generating FCBs, and con-
is adopted for the WL and NL FCBs estimation, which sequently improved the positioning accuracy of the PPP
signifcantly improved the accuracy of estimated FCBs. AR (Li et al. 2013). Gu et al. (2015a) determined the FCBs
For the NL FCBs, the NL foat ambiguities derivated from in the WL and NL combinations using the uncombined
the ionospheric-free foat ambiguities in PPP and the WL PPP model and demonstrated the ionosphere charac-
integer values, are directly used in the FCB estimator. teristics in the PPP AR (Gu et al. 2015a). Tis approach
Te low accuracy of the NL foat ambiguities degrades is also used for the triple-frequency FCBs estimation of
the performance of FCB estimation. Te ambiguity-fxed BeiDou-2 Navigation Satellite System (BDS-2) (Gu et al.
solutions from a GNSS network, where their double-dif- 2015b; Li et al. 2018). Xiao et al. (2019) further presented
ference ambiguities are got, are used for the NL FCB esti- a triple-frequency FCB model for the PPP ambiguity
mation (Geng et al. 2012). Additionally, the Kalman flter resolution with raw observations, which is verifed with
is also adopted for the FCB estimation epoch by epoch, Galileo Navigation Satellite System (Galileo) and BDS-2.
which signifcantly speeds up the computation and is Te initial positioning performance of the PPP AR with
suitable for real-time applications (Xiao et al. 2018). the uncombined model is also demonstrated (Wang et al.
Distinguished from the FCB model proposed by Ge 2020).
et al. (2008) and Laurichesse et al. (2009) presented the For implementing the PPP AR, the Centre National
integer phase clock model, which consists of the ion- d’Etudes Spatiales (CNES) in French and the School of
ospheric-free combination and the MW combination Geodesy and Geomatics at Wuhan University (SGG-
(Laurichesse et al. 2009). Te NL FCBs are not estimated WHU) have been routinely generating the integer recov-
but assimilated into the estimates of satellite clock of- ery clocks and FCBs for public PPP applications (Loyer
sets. Hence, the NL ambiguities in the network solutions et al. 2012; Li et al. 2016). Recently, SGG presented the
are directly fxed to the nearest integers, and the FCB- PPP AR with GPS, BeiDou Navigation Satellite System
contained clock products based on the ambiguity-fxed (BDS), Galileo, and Quasi-Zenith Satellite System (QZSS)
solutions are determined (Laurichesse 2011). Similarly, with multi-GNSS FCBs using the precise satellite orbits
Collins et al. (2008) estimated the integer recovery clocks and clocks from diferent IGS centers (Hu et al. 2020).
for pseudorange and phase measurements, which is Geng et al. (2019b) proposed a modifed phase clock/bias
named as the decoupled clock model (Collins et al 2008). model to improve the PPP AR and provided the phase
In this model, the efects from time-varying parts of code clocks and daily phase biases based on the Center for