The GNSS integer ambiguities: estimation and validation

Figure 5.12: 2-D example for IAB estimation. Float samples for which ā = ǎ are shown
as black dots. Left: Pf = 0.001; Right: Pf = 0.025. Also shown is the IA least-squares
pull-in region for the same fail rates (in grey).
with
µSz,LS = {x ∈ Rn | 1
µ (x − z) ∈ S0,LS}
S0,LS = {x ∈ Rn | x2 Qâ ≤ x − u2Qâ , ∀u ∈ Zn }
The aperture parameter is µ, 0 ≤ µ ≤ 1.
(5.31)
The first step for computing the IALS estimate is now to compute the ILS solution ǎ
given the float solution â, and then the ambiguity residuals are up-scaled to 1
µ ˇɛ. This
up-scaled residual vector is used for verification of:
1
u = arg min
z∈Zn µ ˇɛ − z2Qâ (5.32)
If u equals the zero vector, the float solution resides in the aperture pull-in region Ω0,LS,
and thus ā = ǎ, otherwise ā = â.
Since the ILS estimator is optimal, the IALS estimator can be expected to be a better
choice than the IAB estimator. The IALS estimate can be computed using the LAMBDA
method, and, as for the IAB estimator, the shape of the aperture pull-in region is
independent of the chosen aperture parameter. Furthermore, in the Gaussian case it is
known that the aperture parameter acts as a scale factor on the vc-matrix, since
Ps = fâ(x + a)dx
µS0,LS
= µ n
=
S0,LS
S0,LS
fâ(µx + a)dx
1
| 1
µ 2 Qâ|(2π) n
exp{−
2
1
2 x21 µ 2 Qâ }
Integer Aperture Bootstrapping and Least-Squares 105