The GNSS integer ambiguities: estimation and validation

Figure 5.11: 2-D example for PTIA 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).
5.4.1 Integer Aperture Bootstrapping
Integer bootstrapping is an easy-to-evaluate integer estimator, for which exact evaluation
of the success rate is possible, see section 3.1.2. It will be shown that this is also the
case by defining the IA bootstrapped (IAB) estimator by choosing the aperture pull-in
regions as down-scaled versions of the bootstrapped pull-in region Sz,B as defined in
equation (3.12), cf. (Teunissen 2004a):
with
Ωz,B = µSz,B, ∀z ∈ Z n
µSz,B = {x ∈ R n | 1
µ (x − z) ∈ S0,B}
S0,B = {x ∈ R n | |c T i L−1 x| ≤ 1
2
The aperture parameter is µ, 0 ≤ µ ≤ 1.
, i = 1, . . . , n}
(5.25)
(5.26)
Like for the original bootstrapped estimator, computation of the IA bootstrapped solution
is quite simple. First the bootstrapped solution for the given float solution â is computed.
The aperture pull-in region ΩǎB,B is then identified, and one needs to verify whether or
not the float solution resides in it. This is equivalent to verifying whether or not
1
µ (â − ǎB) = 1
µ ˇɛB ∈ S0,B
(5.27)
This can be done by applying the bootstrapped procedure again, but now applied to
the up-scaled version of the ambiguity residuals, 1
µ ˇɛB. If the outcome is the zero vector,
then ā = ǎB, otherwise ā = â.
It is now possible to derive closed-form expressions for the computation of the IAB
Integer Aperture Bootstrapping and Least-Squares 103