The GNSS integer ambiguities: estimation and validation

probability
1
0.8
0.6
0.4
0.2
0
200
20
0 0.2 0.4 0.6
ε
0.8 1 1.2
Figure 4.6: Probabilities P (| ˆ b − b| ≤ ε) (dashed), P (| ˇ b − b| ≤ ε) (solid), P (| ˜ b − b| ≤ ε)
(+-signs) for different number of epochs.
The probabilities shown in figure 4.6 are determined by counting the number of solutions
that fall within a certain interval, but it is also interesting to compare the estimators on
a sample by sample basis. In order to do so, one could determine for each sample which
of the three estimators is closest to the true b, and then count for each estimator how
often it was better than the other estimators.
In table 4.1 the probabilities
P1 = P (| ˇ b−b| ≤ | ˆ b−b|), P2 = P (| ˜ b−b| ≤ | ˆ b−b|), and P3 = P (| ˜ b−b| ≤ | ˇ b−b|)
are given for different number of epochs. It follows that the probability that ˜ b is better
than the corresponding ˆ b is larger or equal to the probability that ˇ b is better than ˆ b. That
is because the ambiguity residuals that are used to compute the fixed and BIE baseline
estimator, see equations (3.4) and (4.8), have the same sign, and |â−ã| ≤ |â−ǎ| as was
shown in figure 4.5. If the float baseline solution is already close to the true solution,
it is possible that ˜ b is closer to b, but that ˇ b is pulled ’over’ the true solution such that
ˇ b − b has the opposite sign as ˆ b − b and | ˇ b − b| > | ˆ b − b|.
From the relationships between the BIE and the fixed ambiguity residuals it follows that
if ˇ b is better than ˆ b, then also ˜ b will be better than ˆ b. In the case of very low precision
ã ≈ â and ˜ b ≈ ˆ b, so that P2 is approximately equal to one – in the example this is the
case for k = 1 and k = 2. If the precision is high, ã ≈ ǎ and as a result P3 becomes
very high.
Note that these results do not necessarily hold true for the higher dimensional case
(n > 1).
Comparison of the float, fixed, and BIE estimators 77
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