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

land-vehicle navigation (Rabbou and El-Rabbany 2015; (Liu et al. 2017, 2018) and time transfer (Ge et al. 2019;
Wielgosz et al. 2005). In recent years, the efcient and Tu et al. 2019). Tis latter assumption causes adverse
fexible PPP technique has also played a crucial role in efects in these applications will be shown. In this regard,
several non-positioning applications, especially in atmos- the Modifed PPP (MPPP) functional model in iono-
pheric (ionosphere and troposphere) sounding (Rovira- spheric STEC retrieval and timing will be the focus of the
Garcia et al. 2015; Shi et al. 2014; Yuan et al. 2014) as well research. Te precise satellite orbit and clock products
as time and frequency transfer (Defraigne et al. 2007; Ge from an external provider such as the IGS are applied as
et al. 2019; Orgiazzi et al. 2005; Tu et al. 2019). deterministic corrections when conducting PPP, hence
In the implementation of PPP, one needs to formulate the MPPP proposed in this work is to be considered as a
the functional model (i.e., observation equations), relat- geometry-based method. Te model can simultaneously
ing the GPS observations to the parameters to be esti- obtain the variations of receiver code biases at two difer-
mated. One assumption underlying this formulation is ent frequencies using raw observations. In contrast, the
that the receiver code biases do not change signifcantly MCCL method that is actually a kind of geometry-free
over time (Banville and Langley 2011b; Håkansson et al. method (Zhang et al. 2019) can only detect the between-
2017). A vast amount of work has cast considerable doubt epoch fuctuations of Receiver Diferential Code Biases
on the validity of this assumption (Bruyninx et al. 1999; (RDCBs) and cannot be applicable to time or frequency
Coster et al. 2013; Wanninger et al. 2017) and found that transfer because the receiver clock parameter is elimi-
the phenomenon of receiver code bias variation, which nated in the geometry-free combinations of code and
is closely related to the ambient temperature, is wide- phase observations.
spread. For example, the Geometry-Free (GF) receiver Te organization of this work proceeds as follows. In
code biases, known as receiver Diferential Code Biases “Methods” section, the full-rank functional model for the
(DCBs) (Montenbruck et al. 2014; Sardon et al. 1994), has original as well as the modifed PPP are constructed, fol-
been found to exhibit an apparent intraday variability of lowed by the interpretation of the estimable parameters.
4 ns to 9 ns (Ciraolo et al. 2007), far exceeding the code “Results” secrtion provides numerical insights into the
noise level (depending on the receiver type, but gener- ability of the MPPP to exclude the efects that the short-
ally smaller than 1 ns). If the assumption of time-constant term temporal variability of the receiver code biases has
receiver code biases is taken, PPP solutions will be biased on ambiguity estimation, ionospheric STEC retrieval and
and hence the performances in PPP applications will be timing. “Conclusions” section concludes the study with a
degraded. For the ionospheric Slant Total Electron Con- short discussion.
tent (STEC) retrieval, the Modifed Carrier-to-Code Lev-
eling (MCCL) method (Zhang et al. 2019), as well as the Methods
integer-levelling procedure (Banville and Langley 2011a; In this section, the rank-defcient system of GPS obser-
Banville et al. 2012) can both efectively eliminate the vation equations is taken as a starting point, and then
efect of receiver code bias variations. As far as the GPS- the two PPP models are presented (i.e., original and
based timing application is concerned, there are still a modifed), focusing mainly on developing the functional
limited amount of research focusing on how to cope with model and interpreting the estimable parameters.
the time-varying receiver code biases.
In this study, a modifed version of PPP is proposed, GPS observation equations
whose functional model contains time-varying receiver Let us consider a scenario where m satellites, transmit-
code biases, and is formulated based on raw observa- ting signals on f frequencies, are tracked by a single
tions instead of the Ionosphere-Free (IF) combinations. receiver over t epochs. Under this scenario, the system of
Te rank defciencies are removed, which are caused by observation equations has the following form (Leick et al.
the functional model and lead to the non-estimability of 2015),
all the parameters, by fxing a minimum set of parame- s s s s s s
ters to their prior values, resulting in a full-rank system p (i) = l (i) + m τ r (i) + dt r (i) − dt (i) + µ j ι (i) + d r,j − d j
r,j
r
r
r
s
s
s
s
s
of observation equations (Odijk et al. 2016; Teunissen φ (i) = l (i) + m τ r (i) + dt r (i) − dt (i) − µ j ι (i) + a s r,j
r
r
r
r,j
1985). In this system of observation equations, the tem- (1)
poral variations of the receiver code biases on two fre- with r , s = 1, . . . , m , j = 1, . . . , f and i = 1, . . . , t being
quencies become estimable. It will be shown that the the receiver, satellite, frequency and epoch indices,
ambiguity parameters, ionospheric delay, and receiver respectively, and where p (i) and φ (i) denote, respec-
s
s
r,j
r,j
clock ofset directly absorb the combination of time- tively, the code and phase observables, l (i) the receiver-
s
r
varying receiver code biases, which are normally con- satellite range, τ (i) the zenith troposphere delay and m
s
r
r
sidered as constant in subsequent ionospheric modeling the corresponding troposphere mapping function, dt r (i)