Page 179 - 卫星导航2021年第1-2合期

Zhang et al. Satell Navig (2021) 2:11 Page 3 of 10

and dt (i) the receiver and satellite clock ofsets, d r,j and dt r (i) = dt r (i) + d r,IF
s
d the frequency-dependent receiver and satellite code ι (i) = ι (i) + d r,IF − d s (4)
s
s
s
,j
biases, ι (i) the (frst-order) slant ionosphere delay on the r r GF
s
r

s
2
frst frequency ( µ j = 2 ) and a the (non-integer) the code observation equation is of full-rank and reads,
j 1 r,j
ambiguity with j the wavelength of frequency j . Note �p (i) = m τ (i) + dt r (i) + µ j ι (i) (5)
s
s
s
that all quantities are in unit of range, and the time-con- r,j r r r
stant parameters do not have an epoch index i. with dt r (i) the biased receiver clock and ι (i) the biased
s
For positioning purposes the primary parameter of slant ionosphere delay in (4). r
interest is the three-dimensional receiver position vec- Inserting dt r (i) and ι (i) into the phase observation
s
r
tor. To employ the least squares adjustment one needs to equation and lumping a with d , we eliminate the rank
s
s
r,j
linearize the above observation equations. Te linearized defciency and then obtain ,IF
form of Eq. (1) is as follows (Teunissen and Kleusberg
s
s
s
2012): �φ (i) = m τ (i) + dt r (i) − µ j ι (i) + a s r,j (6)
r r
r
r,j
s
s
s
s
�p (i) = m τ r (i) + dt r (i) + µ j ι (i) + d r,j − d + d s with
r,j r r j IF
s
s
s
�φ (i) = m τ r (i) + dt r (i) − µ j ι (i) + a s r,j + d s IF a s = a s + d s − d r,IF − µ j d s (7)
r,j
r
r

(2) r,j r,j IF GF − d r,GF
s
s
where �p (i) and �φ (i) are the code and phase being the biased ambiguity.
r,j
r,j
observables that are corrected for the approximate Equations (5) and (6) represent the full-rank func-
receiver-satellite ranges and the satellite clocks. Note that tional model for the original PPP, in which all estimable
the satellite positions and clocks computed using the IGS parameters are interpreted by Eqs. (4) and (7). Recall
fnal products are not part of the parameters, and the that it is a common practice to use the (recursive) least-
receiver coordinates are fxed to the values in the IGS squares estimator to solve for the parameters. Bearing
Solution Independent Exchange (SINEX) product and do this in mind, a few remarks on the efects of time-varying
not need to be estimated. Also the derived model shall receiver code biases are thus in order. Firstly, the time-
s
not change if one further considers the receiver positions constant of a , an implicit assumption made to exploit
r,j
as unknown parameters. Te ionosphere-free satellite the very precise phase observable, becomes invalid. Tis,
s
code bias, d , appears in both code and phase observa- in turn, can introduce larger errors in comparison with
,IF
tions, because of its presence in the introduced satellite the formal errors of all parameters. Secondly, the use of
clocks (Zumberge et al. 1997b). dt r (i) for timing and time transfer can be susceptible to
In the following two sections, only the dual-frequency batch boundary discontinuities (Collins et al. 2010; Defr-
case is considered (that is, f = 2 ) to keep the presenta- aigne and Bruyninx 2007), because time averaging of d r,IF
tion of the functional model as simple as possible. induces clock datum changes between batches. Te vari-
ability of d r,IF , which does not average out to zero, limits
the use of dt r (i) for frequency transfer, inducing worse
The original version of PPP frequency stability than one would expect theoretically
Equation (2) represents a rank-defcient system, implying (Bruyninx et al. 1999). Tirdly, when using the thin-layer
that some of the parameters are not unbiased estimable, ionosphere model for determining vertical TEC from
s
but only combinations of them. Tis becomes clearer if ι (i) , a temporal variation of d r,GF is partially responsi-
r
we write (Zhang et al. 2012), ble for the occurrence of the model error efects (Brunini
and Azpilicueta 2009).
s
d r,j − d + d s = d r,IF + µ j d r,GF − µ j d s (3)
j IF GF
with d r,IF the ionosphere-free receiver code bias, d r,GF The modifed version of PPP
and d s ,GF the geometry-free receiver and satellite code Let us consider again Eq. (2), but replace d r,j by d r,j (i) ,
biases. It then follows, d r,IF is not separable from dt r (i) , implying that the receiver code bias is allowed to vary
s
s
and the ι (i) , d r,GF and d ,GF are not separable from one freely over time. Tus
r
another. Te idea is therefore to reduce the number
of parameters by lumping some of them together. For �p r,j (i) = m r τ r (i) + dt r (i) + µ j ι r (i) + d r,j (1) + ˜ d r,j (i) − d j + d IF
s
s
s
s
s
instance, with the lumped parameters, s s s s s
�φ r,j (i) = m r τ r (i) + dt r (i) − µ j ι r (i) + a r,j + d IF
(8)