The GNSS integer ambiguities: estimation and validation

probability
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
c
b
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
ε
Figure 4.9: Probabilities P (| ˆ b − b| ≤ ε] (dashed), P (| ˇ b − b| ≤ ε] (solid), P (| ˜ b − b| ≤ ε]
(+-signs). a) σp = 1.4m, σφ = 14mm; b) σp = 0.8m, σφ = 8mm; c) σp = 0.6m,
σφ = 6mm.
4.3.3 The geometry-based case
Several geometry-based models were set up and corresponding float samples were generated.
A description of the models is given in appendix B. Only the Z-transformed
vc-matrices, Qˆz nn xx, are used.
First, it is investigated how the integer set Θ λ x must be chosen in order to get a good
approximation for the BIE estimators. For that purpose, 10,000 samples of float ambiguities
were generated and the integer set was determined using (4.10) and (4.14) for
different values of α. Then the corresponding ˜z α were computed. Table 4.3 shows the
difference ˆz − ˜z α 2 Qˆz −ˆz − ˜zαmin 2 Qˆz with αmin = 10 −16 . Also shown are the number of
integer vectors in the integer set. It follows that for all examples the solution converges
to the true BIE estimator for α < 10 −8 .
The BIE estimators are compared based on the squared norms of the ambiguity residuals,
and based on the probability that the baseline estimators are within a certain distance
from the true baseline:
P ( ˙ b − b 2 Q˙ b ≤ ε) (4.24)
where ˙ b is either the float, fixed, or BIE baseline estimator. The vc-matrices Qˇ b and Q˜ b
are determined from the corresponding estimates. Since a user will be mainly interested
in the baseline coordinates, also the following probabilities were analyzed:
P ( ˙ bx − bx ≤ ε) (4.25)
where bx refers only to the three baseline coordinates. So, the distance to the true
Comparison of the float, fixed, and BIE estimators 81
a