The GNSS integer ambiguities: estimation and validation

1
0.5
0
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2 −2 x
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−2 x
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Figure 3.7: The joint and marginal distribution of â, ǎ and ˇɛ: PDF fâ(x) (top left); joint
PDF fâ,ǎ(x, z) (top middle); PMF P (ǎ = z) (top right); PDF fˇɛ,ǎ(x, z) (bottom middle);
PDF fˇɛ(x) (bottom right).
3.2.2 Success rates
The success rate, Ps, equals the probability that the ambiguities are fixed to the correct
integers. It follows from equation (3.28) as:
Ps = P (ǎ = a) = fâ(x)dx (3.33)
Sa
It is a very important measure, since the fixed solution should only be used if there is
enough confidence in this solution. In Teunissen (1999a) it was proven that:
P (ǎLS = a) ≥ P (ǎ = a) (3.34)
for any admissible integer estimator ǎ. Therefore, the ILS estimator is optimal in the
class of admissible integer estimators. Furthermore, it was shown in Teunissen (1998d)
that:
P (ǎR = a) ≤ P (ǎB = a) (3.35)
Unfortunately, it is very complicated to evaluate equation (3.33) for the integer leastsquares
estimator because of the complex integration region. Therefore, approximations
38 Integer ambiguity resolution