The GNSS integer ambiguities: estimation and validation

ambiguity residual
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float ambiguity
Figure 4.5: Ambiguity residuals for fixed (dashed) and BIE (solid) estimators for different
number of epochs.
with Γ 2 =
z∈Z
exp{− 1
2σ 2 (x − z) 2 }. The inequality follows from:
F (z, z) = 0, ∀z ∈ Z
F (u, z) + F (z, u) ≥ 0, ∀u, z ∈ Z, u = z
2
10
20
30
200
60
(4.23)
The results in equations (4.17), (4.21) and (4.22) imply that ã will always fall in the
grey region in figure 4.4, and thus that indeed |â − ã| ≤ |â − ǎ|. This means that ã will
always lie in-between â and ǎ, and thus that the BIE and the fixed estimator are pulled in
the same direction in the one-dimensional case. This is shown in figure 4.5 for different
precisions (i.e. for different k). The ambiguity residuals are defined as ˇɛ = â − ǎ and
˜ɛ = â − ã.
Figure 4.6 shows the probability that the baseline estimators will be within a certain
interval 2ε that is centered at the true baseline b, again for different precisions. It can be
seen that for high success rates indeed relation P ( ˇ b ∈ Eb) ≥ P ( ˆ b ∈ Eb) is true, with Eb
a convex region centered at b, cf. (Teunissen 2003c), which means that the probability
that the fixed baseline will be closer to the true but unknown baseline is larger than that
of the float baseline. However, for lower success rates some probability mass for ˇ b can be
located far from b because of the multi-modal distribution. Ideally, the probability should
be high for small ε and reach its maximum as soon as possible. For lower success rates,
the float and fixed estimators will only fulfill one of these conditions. The probability
for the BIE estimator always falls more or less in-between those of the float and fixed
estimators.
76 Best Integer Equivariant estimation