The GNSS integer ambiguities: estimation and validation

on bounding the integration region brings the problem of determining the n closest
independent integers to the zero vector.
The approximation based on ADOP performs quite well for the examples shown here.
Except for example 10 03, this approximation is smaller than the empirical success rate,
and better than the lower bound. The vc-matrix of example 10 03 is obtained for a
relatively long baseline and its dimension is equal to 10. This results in conditional
standard deviations of the transformed ambiguities that may differ considerably from
the approximate average given by ADOP. This explains why the approximation is not so
good in this case.
3.2.3 Bias-affected success rates
In section 2.5.1 the minimal detectable effect (MDE) was introduced as a measure of the
impact of a certain model error on the final solution. Using equation (2.77) the impact
on the float solution can be computed, but for high-precision GNSS applications it would
be more interesting to know the impact on the fixed solution. For that purpose, one
could compute the effect of a bias in the float ambiguity solution, ∇â, on the success
rate. This so-called bias-affected success rate can be computed once ∇â is known, see
(Teunissen 2001a). The bootstrapped success rate in the presence of biases, and after
decorrelation so that the bias is ∇ˆz = Z T ∇â, becomes:
P∇(ˇzB = z) =
=
S0,B
1
|Qâ|(2π) 1 exp{−1
2 n
F −1 (S0,B)
1
1 exp{−1
|D|(2π) 2 n
2 x − ∇ˆz2 Qˆz }dx
2 y − L−1 ∇ˆz 2 D}dy, (3.45)
where the transformation F : y = Lx was used, so that the bootstrapped pull-in region
is transformed as:
F −1 (S0,B) = {y ∈ R n | |c T i y| ≤ 1
, i = 1, . . . , n}
2
with ci the vector with a one as ith entry and zeros otherwise. The transformed pull-in
region equals an origin-centered cube with all sides equal to 1. Since D is a diagonal
matrix, the multivariate integral can be written as:
n
1
P∇(ˇzB = z) =
√ exp{−
σi|I 2π 1
yi − c
2
T i L−1 2
∇ˆz
}dyi
σi|I =
=
i=1
|yi|≤ 1
2
n
i=1
n
i=1
1−2c T i L−1 ∇ˆz
2σ i|I
− 1+2cT i L−1 ∇ˆz
2σ i|I
Φ
1
√ 2π exp{− 1
2 v2 }dv
1 − 2c T i L −1 ∇ˆz
2σ i|I
+ Φ
1 + 2c T i L −1 ∇ˆz
2σ i|I
− 1
(3.46)
Quality of the integer ambiguity solution 45