The GNSS integer ambiguities: estimation and validation

7
6
5
4
3
2
1
0
−1
−2
−3
−4
−5
−6
−7
−7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7
1.5
0.5
−0.5
−1.5
−1.5 −0.5 0.5 1.5
Figure 3.6: Two-dimensional example of the search space before (left) and after (right)
the decorrelating Z-transformation. The float ambiguity vector is depicted with a +-sign.
with χ 2 a positive constant that needs to be chosen. The search space is an ndimensional
ellipsoid centered at â. Its shape is governed by the vc-matrix Qâ and
its size by χ 2 . Due to the high correlation between the individual ambiguities, the
search space in the case of GNSS is extremely elongated, so that the search for the
integer solution may take very long. Therefore, the search space is first transformed to a
more spherical shape by means of a decorrelation of the original float ambiguities. This
decorrelation is attained by a transformation:
ˆz = Z T â, Qˆz = Z T QâZ and ǎ = Z −T ˇz (3.19)
This transformation needs to be admissible,i.e. Z and Z −1 must have integer entries,
so that the integer nature of the ambiguities is preserved and the Z-transformation is
volume-preserving with respect to the search space, since the determinant of Z is equal
to ±1. The transformed search space is then given by:
Ωz = z ∈ Z n | (ˆz − z) T Q −1
ˆz (ˆz − z) ≤ χ2
(3.20)
Figure 3.6 shows a two-dimensional example of the search space before and after the
decorrelating Z-transformation. Obviously, the search tree that is required to find the
three integers in the search space will be much smaller after decorrelation.
Using the LDL T -decomposition of Qˆz, the left-hand side of equation (3.20) can be
written as a sum-of-squares:
n (ˆz i|I − zi) 2
≤ χ 2
i=1
σ 2 i|I
(3.21)
with the conditional LS-estimator ˆz i|I, and the conditional variances σ2 i|I that come from
34 Integer ambiguity resolution