The GNSS integer ambiguities: estimation and validation

estimator
1.5
1
0.5
0
−0.5
−1
−1.5
−1.5 −1 −0.5 0
float ambiguity
0.5 1 1.5
Figure 4.4: Ambiguity estimators as function of the float ambiguity. Solid lines: fixed
estimator; Dashed line: float estimator; the BIE estimator will always fall in the grey area.
one), and smaller than the variance of the fixed estimator. On the other hand, for large
k the variances of the BIE and fixed estimator become equal to each other, and smaller
than that of the float estimator. Only after 30 epochs, when the success rate is larger
than 0.9, the variance of the fixed baseline estimator becomes lower than the variance
of the float estimator.
It can be shown that in the one-dimensional case |â − ã| ≤ |â − ǎ|, ∀â, σ 2 . First note
that:
|â − ã| = 0 if â = 1
z, z ∈ Z (4.21)
2
Furthermore, ã is monotonically increasing with â, since:
dã 1
=
dâ Γ2
z(z − x)σ
z∈Z
−2 exp{− 1
2σ2 (x − z)2 }
exp{−
z∈Z
1
2σ2 (x − z)2 }−
−
zσ
z∈Z
−2 exp{− 1
2σ2 (x − z)2 }
(z − x)σ
z∈Z
−2 exp{− 1
2σ2 (x − z)2
}
= 1
σ2Γ2
u(u − x) exp{−
u∈Zz∈Z
1
2σ2 (x − u)2 } exp{− 1
2σ2 (x − z)2 }
−u(z − x) exp{− 1
2σ2 (x − u)2 } exp{− 1
2σ2 (x − z)2
}
= 1
σ2Γ2
(u 2 − uz) exp{− 1
2σ2 (x − u)2 } exp{− 1
2σ2 (x − z)2 }
u∈Zz∈Z
F (u,z)
≥ 0 (4.22)
Comparison of the float, fixed, and BIE estimators 75