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Zhang et al. Satell Navig (2021) 2:11 Page 4 of 10

where Results
Te present section starts with a description of the
˜
d r,j (i) = d r,j (1) + d r,j (i) (9) experimental data and processing strategies, then pro-
ceeds with the results and analysis, and ends with some
˜
with d r,j (i) = d r,j (i) − d r,j (1). concluding remarks.
˜
Note that if the d r,j (i) is ignored, the (rank-defcient)
design matrix of Eq. (8) turns out to be the same as that Experiment setup
˜
of Eq. (2). In the following d r,j (i) is assumed as a time- Te data for this analysis were collected at four stations
varying parameter. But there will be a rank defciency with dual-frequency observations and a 30-s sampling
between dt r (i) , d r,j (i) and a parameters. Te receiver interval. Te detailed information is given in Table 1.
s
r,j
code biases at the frst epoch are chosen as datum to Note that stations ALIC and MTDN have thermometers
˜
eliminate the rank defciency, thus the estimated d r,j (i) to gauge the air temperatures, and the receivers at sta-
are the variations of receiver code bias with respect to tions NOT1 and MEDI were both connected to a high
the frst epoch, see Eq. (9). As for other parameters, the performance external frequency standard (H-maser).
procedures described in the preceding subsection are Our data processing strategies are as follows. Firstly,
closely followed to overcome the rank defciency prob- the original as well as the modifed PPP were conducted,
lem, thereby yielding
solving for the parameters by means of Kalman flter.
˜
s s s Te estimated parameters included the ZTDs (a random
˜
�p (i) = m τ r (i) + dt r (i) + µ j ˜ι (i) + d r,j (i)
r,j r r walk process with a variance rate of 1 × 10 m /s), the
−7
2
s s s s (10)
˜
�φ (i) = m τ r (i) + dt r (i) − µ j ˜ι (i) + ˜ a biased receiver clocks (time-varying), the biased slant
r,j r r r,j
ionosphere delay (time-varying), the biased ambigui-
with ties (time-constant), and the biased receiver code biases
(time-varying for the modifed PPP). Te elevation cut-
˜
dt r (i) = dt r (i) + d r,IF (1) of angle was set as 10°, and the P1-C1 satellite DCB
s
s
˜ ι (i) = ι (i) + d r,GF (1) − d s
r r GF corrections provided by the IGS were applied to the C1
s

˜ a s r,j = a s r,j + d IF − d r,IF (1) − µ j d s GF − d r,GF (1) observations. Te IGS fnal orbit and clock products were
(11) employed to correct for the satellite orbital and clock
where a tilde marks the biased, but estimable param- errors. Te other critical corrections to raw observa-
eters. Note the diference to the estimable parameters in tions were also considered, including the solid Earth tide,
Eqs. (5) and (6). the phase wind-up efects, and the satellite and receiver
Equation (10) is a full-rank system, representing the phase center ofsets and variations.
full-rank functional model constructed for the modifed Second, Te variations of the code observations were
˜
PPP. Note that, the estimable receiver code biases d r,j (i) estimated on each frequency epoch by epoch. Te results
only appear in the observation equations at the second would allow us to verify the ability of the modifed PPP to
epoch and beyond, i.e., i ≥ 2 , for those at the frst epoch directly measure the temporal variability of the receiver
get lumped with the parameters given in Eq. (11). Te code biases for each observable type.
estimability of d r,j (i) implies that any temporal varia- Tird, the time-varying receiver code biases are not
˜
tions in receiver code biases shall fully enter into d r,j (i) , considered in Global Ionosphere Maps (GIMs) provided
˜
and thus have no negative impacts on the estimation of by the IGS, and the GF carrier phase observables are
remaining parameters. almost unafected by measurements noise and multipath.
Te Diference of Slant Total Electron Content (DSTEC)
obtained from the GF carrier phase observations is thus
chosen as the reference to evaluate the performance of
Table 1 The information on the collected GPS data used in this study
Station Receiver Antenna type Temperature data Frequency Period Latitude, longitude
standard
ALIC LEICA GR25 LEIAR25.R3 ✓ Internal DOY 002, 2017 − 23.67°, 133.89°
MTDN TRIMBLE NETR9 TRM59800.00 ✓ Internal DOY 005, 2018 − 22.13°, 131.49°
NOT1 LEICA GR30 LEIAR20 H-maser DOY 001, 2018 36.88°, 14.99°
MEDI LEICA GR10 LEIAR20 H-maser DOY 009, 2018 44.52°, 11.65°