The GNSS integer ambiguities: estimation and validation

is close to one. This probability can be computed as follows:
P (
1
|Qâ|(2π) 1 exp{−1
2 n
2 x − a + z2Qâ } > λ)
=fâ(x+z)
= P (x − a + z 2 Qâ < −2 ln(λ |Qâ|(2π) 1
= P (â − a 2 Qâ < χ2 )
2 n
χ
)
2
) (3.51)
This probability can be computed since â − a 2 Qâ ∼ χ2 (n, 0), see appendix A.2.2.
Summarizing, this means that fˇɛ(x) can be approximated by:
with
fˇɛ(x) =
1
|Qâ|(2π) 1
2 n
z∈Θ
exp{− 1
2 x + z2 Qâ }s0(x) (3.52)
Θ = {z ∈ Z n | x + z 2 Qâ < χ2 } (3.53)
where a ∈ Z n is eliminated from equation (3.48), and χ 2 as defined in equation (3.51).
So, the integer set contains all integers z within the ellipsoid centered at −x and its size
governed by χ.
3.3.2 PDF evaluation
Figure 3.9 shows fˇɛ(x) under H0 for different values of σ in the one-dimensional case
(n = 1). Also the extreme cases, σ 2 = 0 and σ 2 → ∞, are shown. In the first case, an
impulse PDF is obtained, in the second case a uniform PDF. Note that the unit of σ is
cycles.
It can be seen that the PDF becomes peaked if the precision is better, i.e. σ ↓ 0. In
that case most of the probability mass of the PDF of â is located in the pull-in region
Sa. This is the case for σ = 0.1. For σ ≥ 0.3 (right panel) the distribution function
becomes flat, and already for σ = 1 the PDF is very close to the uniform distribution.
Figure 3.10 shows the PDFs of â, ˆz and ˇɛ obtained for the three admissible estimators.
The following vc-matrices (original and Z-transformed) were used:
4.9718 3.8733
0.0865 −0.0364
Qâ =
, Qˆz =
(3.54)
3.8733 3.0188
−0.0364 0.0847
These vc-matrices are obtained for the dual-frequency geometry-free GPS model for
one satellite-pair, with undifferenced code and phase standard deviations of 30 cm and
3 mm, respectively. The results were evaluated in Verhagen and Teunissen (2004c).
Especially for the bootstrapped and the ILS estimator, the shape of the PDF of the
residuals ’fits’ the shape of the pull-in region quite well. The PDFs are multi-modal for
48 Integer ambiguity resolution