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Du et al. Satell Navig (2021) 2:3 Page 6 of 22

of payload silicon material, satellite induced code-car- For single-frequency users or uncombined PPP, iono-
rier divergence, leap second anomaly and so on (Bhatti spheric delay variability is the greatest challenge for
and Ochieng 2007; Martins 2014; Ochieng et al. 2003; PPP using an empirical ionospheric model (Chen and
Tomas et al. 2011). Some of the faults, e.g. satellite Gao 2005; Montenbruck 2003; Øvstedal 2002), or
clock jump and drift, can result in ranging errors of up ionospheric delay estimation based on a local CORS
to several kilometres and sometimes loss of signal track- network (Chen and Gao 2005; Li et al. 2011; Li et al.
ing (Bhatti and Ochieng 2007). A conservative empirical 2012). Due to its irregular spatiotemporal patterns,
assumption on the probability of satellite and signal faults the ionospheric delay is difcult to describe by either
for GPS, which is suggested by GPS SPS Performance deterministic or stochastic models (Shi et al. 2012).
−5
Standard (5th edition), is less than 1 × 10 /h per satellite However, studies show that the uncertainty of iono-
in total (U.S. Department of Defense 2020). Regarding spheric delay estimation can be reduced by increas-
other GNSS systems, the a priori probabilities of satel- ing the density of the local CORS network used in
lite and signal faults are still under investigation, hence its estimation (Murrian et al. 2016). Te characteris-
they are not yet specifed. It should be noted that the IGS tics and risks of ionospheric storms are discussed in
Multi-GNSS Experiment (MGEX) and the International Imparato et al. (2018b).
GNSS Monitoring and Assessment Service (iGMAS),
which was developed by China, play a signifcant role in • Tropospheric errors. In the conventional PPP algo-
monitoring and evaluating satellite and signal anomalies rithm, the hydrostatic (dry) delay is corrected for
for multi-GNSS systems (Fan et al. 2019; Huang et al. using an empirical model, and the (zenith) wet delay
2018; Ouyang et al. 2019; Ye et al. 2017). Tese faults is estimated as a free parameter. Tropospheric delay
have been investigated extensively in the literature, e.g. variations will increase during a storm. Te tropo-
Bhatti and Ochieng (2007), and Imparato et al. (2018b). sphere can be assumed to be horizontally strati-
fed and azimuthally symmetric; however, ignoring
Medium (Atmosphere) horizontal gradients may introduce range errors at
the decimetre-level at low elevation angles and will
• Ionospheric errors. Most of the ionospheric efect can generally map into the horizontal position bias (Col-
be eliminated through dual-frequency Ionosphere- lins and Langley 1998; Kjørsvik et al. 2006). Gradient
Free (IF) combinations or the Group and Phase Iono- parameters can be estimated together with zenith
spheric Calibration (GRAPHIC) approach (Yunck wet delay; however, this will introduce additional
1992). Te main threat to dual-frequency PPP users parameters which can weaken the model strength.
comes from ionospheric scintillation (Datta-Barua Similarly, external tropospheric delay corrections can
et al. 2003; Kintner et al. 2009; SBAS Ionospheric also be estimated with the data from a regional CORS
Working Group 2010). Te occurrence of scintilla- network in real-time to reduce the time for PPP solu-
tion is more frequent at low and high latitudes than tion convergence and ambiguity fxing (Hadas et al.
at mid-latitudes. Scintillation mostly happens after 2013; Li et al. 2011; Shi et al. 2014).
sunset and may last for a few hours during solar max-
imum years and exhibits seasonal variation (Conker
et al. 2003; Guo et al. 2017). Ionospheric scintillation
can cause positioning degradation in three ways: (1) Products (corrections)
loss of lock of tracked satellite signals, (2) abnormal
measurement blunders, and (3) frequent cycle slips • Errors in real-time corrections. Precise orbit and
which are difcult to detect due to the high rate of clock products for real-time users are derived from
total electron content (TEC) variation (Zhang et al. the measurements made by a tracking network and
2013). Typically, scintillations only afect a few satel- are routinely provided by IGS and the analysis cen-
lites at a time; the probability of two satellites simul- tres (AC). Te IGS products are the combined solu-
taneously having a Rate of TEC Index (ROTI) greater tions generated by processing the individual solutions
than 3 Total Electron Content Units (TECU) per of the participating ACs. Te combination results in
minute is about 2% (Imparato et al. 2018b; Jacob- a higher quality and reliability than that of any single
sen and Dähnn 2014). Signal loss due to ionospheric AC’s product (Dow et al. 2009; IGS 2019). Te typi-
scintillation was studied in terms of its temporal cal accuracy (RMSE, i.e. Root Mean Square Error) of
and spatial behaviours in Liu et al. (2017); however, the IGS Real-Time Service (RTS) products can reach
the probability of such a risk has not been analysed 5 cm for orbits and 300 ps for clocks (IGS 2014).
(Imparato et al. 2018b). However, outliers are still present in the IGS com-