The GNSS integer ambiguities: estimation and validation

chosen equal to half the number of facets that bound the ILS pull-in region. So, it is
required to know this number, but in practice only the bounds are known:
n ≤ p ≤ 2 n − 1
If p is chosen to be smaller than half the number of bounding facets, it is not guaranteed
that the approximation gives a lower bound. On the other hand, if p is chosen to be
larger than required in order to guarantee that P (â ∈ Ua) = P (â ∈ Sa), the lower bound
is less strict since the lower bound is defined as a product of probabilities which are all
smaller or equal to one. Note that p may become very large when many satellites are
visible. For instance, with 6 visible satellites and two frequencies available, the number
of unknown ambiguities for one epoch is n = 10, and p ≤ 2 n − 1 = 1023.
It is possible to find all adjacent integers, but it is computationally demanding. First,
note that it is not always the case that the 2p adjacent integers are also the 2p closest
integers. Therefore, a large set of integers ci must be selected, in the same way as for
the computation of the upper bound described above, but now with q >> 2(2n − 1).
For each integer in this set it must be checked if it is adjacent, which is the case if 1
2ci lies on the boundary of both the pull-in regions S0 and Sci . This is the case if:
1
2 ci − 0 2 Qâ = 1
2 ci − ci 2 Qâ = min
z∈Zn1 2 ci − z 2 Qâ
(3.43)
Note that if cj is selected as adjacent integer, −cj must not be included in the set
C = (c1 . . . , cp) adjacent integers that is used to compute the lower bound.
Approximation based on ADOP
Until now only lower and upper bounds of the ILS success rate were given. If these
bounds are sharp, it is very useful to know whether an approximation is a lower or upper
bound, since in general a user will choose a minimum required success rate. If the
lower bound is below this minimum, the user will not have enough confidence in the
fixed solution – although there is still a probability that the actual success rate is large
enough. An upper bound could be used in a similar way: if it is below a certain threshold
value, one cannot expect ambiguity resolution to be successful.
Instead of a lower and/or upper bound, it is also possible to use the following approximation:
1
Ps,LS ≈ (2Φ( ) − 1)n
2ADOP
(3.44)
In Teunissen (2003b) it was proven that this is an invariant upper bound of the bootstrapped
success rate, and it is equal to the bootstrapped success rate when all ambiguities
are completely decorrelated. Since the bootstrapped success rate is used as a lower
bound for the ILS success rate, the bootstrapped upper bound is expected to be a good
approximation of the ILS success rate. The reason is that the ADOP approximates the
average precision of the transformed ambiguities, and can therefore be used to replace
the individual conditional standard deviations in equation (3.37).
42 Integer ambiguity resolution