The GNSS integer ambiguities: estimation and validation

The single difference approach can be seen as a kind of compromise. The amount of
observations is reduced with respect to the undifferenced approach, but not as much as
with the double difference approach. The same holds for the unknown parameters. An
advantage of the single difference approach is that in the case of a single baseline model
(two receivers) no correlation between the observations is introduced, see section 2.4.
As explained before, for high-precision applications it is important that the carrier phase
ambiguities are resolved as integers. It has been shown that the parameterization of
the unknown ambiguities is such that these parameters are known to be integers in the
single difference and double difference approach.
2.4 GNSS stochastic model
The stochastic model describes the precision of the observations. As a starting point
the following assumptions are made:
E{e s r,j(t)e s r,j(t)} = σ 2 p,j no satellite dependent weighting
E{e s r,i (t)esr,j (t)} = 0 no frequency dependent correlation (2.56)
E{e s r,j (t1)e s r,j (t2)} = 0 no time correlation
Often, the code standard deviation is chosen identical for all frequencies, σp,j = σp ∀j.
The same assumptions are made for the undifferenced phase observations, with the phase
standard deviation denoted as σφ.
The vc-matrix of the undifferenced observations of one satellite and one receiver becomes
then:
Cpφ = diag(σ 2 p,1, . . . , σ 2 p,f , σ 2 φ,1, . . . , σ 2 φ,f , σ 2 I ) (2.57)
where σ2 I is the variance of the ionospheric pseudo-observation, which has to be included
if the ionosphere-weighted model is considered.
For the single differenced observations, the vc-matrices of the observations of both
receivers must be added according to the propagation law of variances, see appendix
A.2.1:
C sd
pφ = Cpφ,q + Cpφ,r
(2.58)
If the standard deviations are assumed equal for both receivers, all variances must simply
be multiplied by 2.
The single difference vc-matrix corresponding to the functional models in equations
(2.45) and (2.46) becomes:
Qy = C sd
pφ ⊗ Im
(2.59)
The vc-matrix of the double difference models in equations (2.54) and (2.55) is obtained
again with the propagation law of variances and the properties of the Kronecker product
20 GNSS observation model and quality control