The GNSS integer ambiguities: estimation and validation

2
1
0
−1
−2
−2 −1 0 1 2
0.4
0.2
0
−0.2
−0.4
−0.4 −0.2 0 0.2 0.4
Figure 3.11: Ellipses corresponding to the vc-matrices of the float ambiguities (grey) and
the ambiguity residuals (black). Left: for Qâ; Right: for Qˆz.
horizontal axis gives the value of α:
α = 1 − P (â − a 2 Qâ < χ2 ), (3.55)
see equation (3.51).
The ’correct’ values, f č ɛ (x), were computed for all samples using a very large integer set.
Then the difference with the approximations df = f č ɛ (x) − f α ˇɛ (x) were computed. The
top panels show along the vertical axes:
y = df
,
N
(mean error)
y = arg max(df),
x
(maximum error)
where N is the number of samples.
It follows that for lower precision – top panels – a large integer set is required in order
to get a good approximation. Note that the number of integers in the set Θ depends
on the sample value of ˇɛ, since the ellipsoidal region is centered at −ˇɛ.
The results for the geometry-based examples are summarized in table 3.4. For each
example, four different integer sets were chosen, based on the choice of α, see equation
(3.55). The corresponding value of λ in equation (3.51) is also given. The largest of
these sets was chosen such that the error in the approximation of the PDF was small
enough to be neglected. It follows that the approximation errors have the same order as
α. So, with α = 10 −6 the errors are small, and the number of integers in the set Θ is
also small enough to guarantee reasonable computation times.
3.4 Quality of the fixed baseline estimator
The distributional properties of the ambiguity parameters were discussed in the preceding
sections. A user, however, will of course be mainly interested in the quality of the baseline
solution. In this section, the distributional properties of the fixed baseline estimator will
Quality of the fixed baseline estimator 51