The GNSS integer ambiguities: estimation and validation

oundary of the aperture pull-in region Ω0,E could be chosen as:
y = ϱ · 1
arg min
2 z∈Zn \{0} z2Qâ with ϱ =
1
2
µ
min
z∈Zn \{0} zQâ
(5.59)
since then y 2 Qâ = µ2 . Again the aperture parameter for OIA estimation is approximated
with equation (5.58), so that y is also an element of the boundary of Ω0,OIA. It
is important to note that this approach will not work in the case the ellipsoidal pull-in
regions overlap, since then y /∈ S0. In that case ϱ in equation (5.59) should be set equal
to one, so that y is on the boundary of S0.
Figure 5.18 also shows the OIA aperture parameter as function of the fail rate as approximated
using EIA. Instead of choosing y as in equation (5.59), one could also choose
y = ϱ· 1
2u with u the adjacent integer with the largest distance to the true integer a = 0.
With the first option, the approximated aperture parameter will be too large, with the
second option the approximated aperture parameter will be too small. For this example,
approximation with IAB estimation works better, but still not good.
5.7.2 Determination of the aperture parameter using simulations
As explained at the beginning of this section, computation of the aperture parameter µ
for a fixed fail rate is not possible for most IA estimators. An alternative is to determine
µ numerically. Therefore, simulated data are required. The procedure will be described
here for the OIA estimator, but a similar approach can be followed for other IA estimators.
The procedure is as follows:
1. Generate N samples of normally distributed float ambiguities:
âi ∼ N(0, Qâ), i = 1, . . . , N
2. Determine the ILS solution:
ǎi, ˇɛi = âi − ǎi, ri = fˇɛ(ˇɛi)
fâ(ˇɛi)
3. Choose the fail rate: Pf = β ≤ Pf,LS
The fail rate as function of µ is given by: Pf (µ) = Nf
N
with Nf = N
1 if ri ≤ µ ∧ ǎi = 0
ω(ri, ǎi) and ω(ri, ǎi,) =
0 otherwise
4. Choose:
i=1
µ1 = (min(ri) − 10 −16 ) since this results in: Nf = 0 ⇒ Pf (µ1) = 0
µ2 = max(ri) since this results in: Pf (µ2) = Pf,LS
5. Use a root finding method to find µ ∈ [µ1, µ2] so that Pf (µ) − β = 0.
This will give the solution since Pf (µ1) − β < 0 and Pf (µ2) − β > 0, and the fail
rate is monotonically increasing for increasing µ, as will be shown below.
118 Integer Aperture estimation