The GNSS integer ambiguities: estimation and validation

The double difference observation vector and parameter vectors are now derived from
the single difference vectors (denoted with superscript sd) of equations (2.29)-(2.34) as
follows:
y = I (2f+1) ⊗ D T y sd
ρ = D T ρ sd
T = D T T sd
I = D T I sd
a = If ⊗ D T a sd
(2.49)
(2.50)
(2.51)
(2.52)
(2.53)
This gives the ionosphere-weighted double difference geometry-free model:
⎛⎡
e2f
E{y} = ⊗ Im−1 ρ + ⎝⎣
0
µ
⎤ ⎞ ⎛⎡
−µ ⎦ ⊗ Im−1⎠
I + ⎝⎣
1
0
⎤ ⎞
Λ⎦
⊗ Im−1⎠
a (2.54)
0
with ρ the lumped tropospheric and range parameters.
Similarly, the geometry-based model is obtained as:
⎛⎡
⎤ ⎞
µ
e2f
e2f
E{y} = ⊗ G ∆rqr + ⊗ Ψ T + ⎝⎣−µ
⎦ ⊗ Im−1⎠
I
0
0
1
⎛⎡
⎤ ⎞
0
+ ⎝⎣Λ⎦
⊗ Im−1⎠
a
0
with G = D T G sd , and T contains the ZTD.
(2.55)
The double difference approach offers the advantage of less (or even no) rank deficiencies
in the mathematical model. However, rank deficiencies can also be resolved in other ways
as explained in the preceding section.
A disadvantage is that users will be interested in the undifferenced parameters, and
therefore one has to be careful with the interpretation of the results.
As double differencing eliminates the satellite and receiver clock errors, it is not possible
to estimate them explicitly. This implies that it is not possible to model their behavior
in time. But these parameters are rarely needed, so in many practical situations this will
not be considered as a problem.
An important disadvantage of the double difference approach is that all receivers must
track the same satellites. In practice this is not always the case, especially with long
baselines. But also when the receivers are located quite close to each other, one of the
receivers may not track some of the satellites tracked by the other receiver(s), e.g. due
to blocking of signals by local obstacles. In such cases using the double difference model
results in loss of information, as observations from a satellite only tracked by one of the
receivers cannot be used.
GNSS functional model 19