The GNSS integer ambiguities: estimation and validation

2
1
0
−1
−2
−2 −1 0 1 2
Figure 3.5: Two-dimensional pull-in regions in the case of integer least-squares.
metric of the vc-matrix Qâ. The pull-in region that belongs to the integer z follows as:
Sz,LS = x ∈ R n | x − z 2 Qâ ≤ x − u2Qâ , ∀u ∈ Zn
(3.16)
A similar representation as for the bootstrapped pull-in region can be obtained by using:
x − z 2 Qâ ≤ x − u2Qâ ⇐⇒ (u − z)T Q −1 1
â (x − z) ≤
2 u − z2Qâ , ∀u ∈ Zn
With this result, it follows that
Sz,LS =
x ∈ R n | |c T Q −1 1
â (x − z)| ≤
2 c2
Qâ ∀z ∈ Z n
c∈Z n
(3.17)
The ILS pull-in regions are thus constructed as intersecting half-spaces, which are
bounded by the planes orthogonal to (u − z), u ∈ Zn and passing through 1
2 (u + z).
It can be shown that at most 2n − 1 pairs of such half-spaces are needed for the construction.
The integer vectors u must then be the 2n − 1 adjacent integers. So, for the
2-dimensional case this means three pairs are needed, and the ILS pull-in regions are
hexagons, see figure 3.5.
3.1.4 The LAMBDA method
The ILS procedure is mechanized in the LAMBDA (Least-Squares AMBiguity Decorrelation
Adjustment) method, see (Teunissen 1993; Teunissen 1995; De Jonge and Tiberius
1996).
As mentioned in section 3.1.3, the integer ambiguity solution is obtained by an integer
search. Therefore, a search space is defined as:
Ωa = a ∈ Z n | (â − a) T Q −1
â (â − a) ≤ χ2
(3.18)
Integer estimation 33