The GNSS integer ambiguities: estimation and validation

3.4.2 Baseline probabilities
The quality of the fixed baseline solution could be evaluated by means of the probability
that ˇb lies in a certain convex region Eb ⊂ Rp probability follows from equation (3.58) as:
symmetric with respect to b. This
P ( ˇb ∈ Eb) =
fˆb|â (x|z)dxP (ǎ = z) (3.62)
z∈Zn Eb
The infinite sum in the equation cannot be evaluated exactly and therefore in practice
a lower and upper bound can be used, (Teunissen 1999b):
P ( ˆ b(a) ∈ Eb)P (ǎ = a) ≤ P ( ˇ b ∈ Eb) ≤ P ( ˆ b(a) ∈ Eb) (3.63)
These bounds become tight when the success rate P (ǎ = a) is close to one. In that
case also the following is true:
P ( ˇ b ∈ Eb) ≈ P ( ˆ b(a) ∈ Eb) ≥ P ( ˆ b ∈ Eb) (3.64)
The vc-matrix of the conditional baseline estimator, Qˆ b|â is often used as a measure of
the fixed baseline precision. Therefore, this vc-matrix will be used in order to define the
shape of the confidence region Eb, so that it takes the ellipsoidal form:
Eb =
x ∈ R n | (x − b) T Q −1
ˆ b|â (x − b) ≤ β 2
(3.65)
The size of the region can thus be varied by the choice of β. Then:
P ( ˇb ∈ Eb) = P (ˇb − b 2 Qˆ ≤ β
b|â 2 ) =
P (χ 2 (p, λz) ≤ β 2 )P (ǎ = z) (3.66)
z∈Z n
with χ 2 (p, λz) the non-central χ 2 -distribution with p degrees of freedom and noncentrality
parameter λz:
λz = ∇ˇbz 2 Qˆ , ∇
b|â ˇbz = Qˆbâ Q −1
â (z − a) (3.67)
The interval of (3.63) becomes now:
with
α1 ≤ P ( ˇ b − b 2 Qˆ b|â ≤ β 2 ) ≤ α2 (3.68)
α1 = α2P (ǎ = a) and α2 = P (χ 2 (p, 0) ≤ β 2 )
By choosing the confidence level α1, the success rate can be used to determine α2,
which finally can be used to compute β with the aid of the χ 2 -distribution.
Equation (3.68) is useful in order to decide whether or not one can have enough confidence
in the fixed solution, although it will be difficult to know what value of β is
Quality of the fixed baseline estimator 55