The GNSS integer ambiguities: estimation and validation

and difficult to predict. Therefore, the wet delays are commonly mapped to zenith
tropospheric delay (ZTD) parameters and these are then estimated. An overview of
available mapping functions can be found in Kleijer (2004).
For short time spans this ZTD parameter can be considered constant. The parameter
vector T consists then of the ZTD parameter of one receiver. The mapping function is
denoted as ψ s r. This gives for the partial design matrix related to the ZTD parameter:
e2f ⊗ ψ 1 r · · · ψ m r
T = e2f ⊗ Ψ (2.41)
Note that the a priori tropospheric corrections and the ZTD of the reference station q
are added to the approximate observation equations in (2.22).
the ZTD is not estimated if it is assumed that the a priori model can be fully relied on.
This is referred to as the troposphere-fixed approach. If, on the other hand, the ZTD is
estimated, this is referred to as the troposphere-float approach.
The rank deficiency caused by the phase receiver clocks δt s r and the ambiguities is solved
by the following transformations:
dtqr =
cdtqr,1 + λ1M 1 qr,1 · · · cdtqr,f + λf M 1 T qr,f
cδtqr,1 + λ1M 1 qr,1 · · · cδtqr,f + λf M 1 T qr,f
a = N 12
qr,1 · · · N 1m
qr,1
· · ·
12 Nqr,f · · · N 1m
T qr,f
(2.42)
(2.43)
Due to this reparameterization, the single difference ambiguities are transformed to
double difference ambiguities from which it is known that they are of integer nature
since the initial phases in receiver and satellite are cancelled out: M ts ts
qr,j = Nqr,j . Note
that the code receiver clock parameters are not transformed.
Finally, a rank deficiency is present due to the inclusion of the ionospheric parameters.
In Odijk (2002) it is described how to set up the so-called ionosphere-weighted model.
The approach is to include a vector of ionospheric pseudo-observations, consisting of a
priori estimates, to the vector of observations:
⎛ ⎞
P
y = ⎝Φ⎠
(2.44)
I
If the a priori information is considered exact, the a priori estimates can simply be subtracted
from the observations and there are no ionospheric parameters to be estimated.
This is referred to as the ionosphere-fixed model. If the baseline is shorter than 10 km,
it can be assumed that the ionospheric delay on the signal of one satellite to the two
receivers is identical, i.e. I s r = I s q . Also then the ionospheric parameters can be removed
from the single difference model. If, on the other hand, the ionospheric behavior must
be considered completely unknown, e.g. when the baseline is very long, the weight of
the ionospheric pseudo-observations is set equal to zero, which is equivalent to setting
the standard deviation of the pseudo-observations to infinity. This is referred to as the
ionosphere-float model.
GNSS functional model 17