The GNSS integer ambiguities: estimation and validation

success rate and fail rate. From equation (5.4) it follows that:
Pf =
fâ(x)dx
z∈Zn \{a} µSz,B
=
z∈Zn \{0} (2π)
S0,B
1
2 n
=
1
| 1
µ 2 Qâ|
z∈Zn \{0}
F −1 (2π)
(S0,B)
1
2 n
1
| 1
µ 2 D|
exp{− 1 1
x +
2
exp{− 1 1
y +
2
µ z 21 µ 2 Qâ }dx
µ L−1z 2 1
µ 2 D}dy
where for the third equality the transformation F : x = Ly is used, with L the unit
lower triangular matrix of Qâ = LDL T . This gives the transformed pull-in region
F −1 (S0,B) = {y ∈ R n | | c T i y |≤ 1
, i = 1, . . . , n}
2
Note that 1
µ 2 D is a diagonal matrix having the scaled conditional variances 1
µ 2 σ2 i|I as its
entries, and the transformed pull-in region has become an origin-centered n-dimensional
cube with all side lengths equal to 1. The multivariate integral can therefore be written
as a product of one-dimensional integrals:
Pf = n
1
√ exp{−
2π 1
yi +
2
1
µ cTi L−1z 1
µ σ 2 }dy
i|I
z∈Zn \{0} i=1
=
n
z∈Zn \{0} i=1
=
n
z∈Zn \{0} i=1
i=1
|yi|≤ 1
2
1
µ σ i|I
µ+2cT i L−1 z
2σ i|I
− µ−2cT i L−1 z
2σ i|I
Φ
µ − 2c T i L −1 z
2σ i|I
1
√ 2π exp{− 1
2 v2 }dv
+ Φ
µ + 2c T i L −1 z
2σ i|I
− 1
In a similar way it can be shown that the IAB success rate is given as:
n
Ps = 2Φ
µ
− 1
2-D example
2σ i|I
(5.28)
(5.29)
Figure 5.12 shows in black all float samples for which ā = ǎ for two different fail rates.
5.4.2 Integer Aperture Least-Squares
The IA Least-Squares (IALS) estimator can be defined in a similar way as the IAB
estimator, cf. (Teunissen 2004b). The aperture pull-in region is then defined as:
Ωz,LS = µSz,LS
(5.30)
104 Integer Aperture estimation