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Zhang et al. Satell Navig (2021) 2:11 Page 7 of 10
where the variations of receiver code biases become esti-
mable, the ambiguity estimates converge to constant rea-
sonably and the residuals of code observations follow the
normal distribution (see Fig. 6).
At this stage it is proved that MPPP model can get
correct ambiguity estimates and reasonable code obser-
vation residuals. In the following sections the improve-
ments in STEC retrieval and timing based on the MPPP
model will be investigated.
Retrieval of STEC parameter
As mentioned earlier, also see Ciraolo et al. (2007), if the
short-term variability of receiver DCBs is not properly
Fig. 4 Estimated ambiguity parameters of L1 frequency at station
MTDN using the MPPP model. Diferent colors correspond to diferent managed, it can have signifcant impact on the STEC
satellites results. In order to evaluate the performance of the MPPP
in retrieving STEC, we compared the DSTECs obtained
by the original and modifed PPP models was compared
with those derived by using the GF carrier phase obser-
vations to minimize any impact from multipath or code
biases. Te corresponding results are shown in Fig. 7,
where diferent colors represent diferent satellites.
It can be clearly seen that the STECs retrieved by the
MPPP (right column) are more accurate and stable than
those by the original PPP model (left column). Table 2
summarizes the Root-Mean-Square (RMS) values for the
original PPP-DSTEC as well as MPPP-DSTEC. Note that
the most obvious improvement of 96% occurs at station
MTDN, which also has the signifcant variations of code
biases on P1 and P2 as shown in Fig. 1. Te other three
stations (ALIC, NOTI and MEDI) also show consider-
able improvements. Considering the fact that the MPPP
results are free of the efects due to variability of code
Fig. 5 P1 (magenta dots) and P2 (green dots) residuals of code biases, we can thus interpret the diferences as the errors
observations at station MTDN using the original PPP model
to which the original PPP results are subject.
Timing
Generally, the colored signature of the code noise is an
interfering factor for the medium and long-term (from
some hours to some days) stability of receiver clock solu-
tions (Martínez-Belda et al. 2011; Martinez-belda et al.
2012). Te MPPP (see Eq. 10) can theoretically elimi-
nate the efect caused by the time-varying receiver code
biases. To validate its efectiveness, the data at station
NOT1 on day 001 in 2018 and station MEDI on day 009
in 2018 were used. Figure 8 shows the Allan deviation of
receiver clock ofsets at station NOT1 and MEDI, respec-
tively, which are both connected to the external Hydro-
gen atomic clock.
Te blue lines and red lines represent the Allan devia-
tion of the PPP-derived and MPPP-derived receiver
Fig. 6 P1 (magenta dots) and P2 (green dots) residuals of code clock ofsets, respectively. It can be clearly seen that
observations at station MTDN using the MPPP model
the frequency stability of the MPPP-derived solu-
tions is improved signifcantly relative to the original