The GNSS integer ambiguities: estimation and validation

where y ∈ R m with mean E{y} = Aa + Bb and PDF fy(y), and ˜ θ is the BIE estimator
of θ = l T a a + l T b b.
The BIE estimator can also be written as:
˜θ = l T a ã + l T b ˜ b (4.5)
with
⎧
⎪⎨ ã =
⎪⎩
z∈Zn zwz(y),
z∈Zn wz(y) = 1
˜b = ζwζ(y)dζ,
wζ(y)dζ = 1
R p
R p
(4.6)
where the weighting functions wz(y) and wζ(y) follow from equation (4.4). This shows
that the BIE estimator of the integer parameter vector a is also a weighted sum of all
integer vectors in Zn , just like the admissible integer estimators of (3.6). However, in
this case the weights are not binary (1 or 0); their values are determined by y and its
PDF, so that the BIE weights are real-valued and nonzero for all integer vectors. This
implies that the estimator ã is real-valued, instead of integer-valued. Furthermore, it
implies that there is no need to apply a discrimination test.
In Teunissen (2003f) it was shown that the BIE estimator is unbiased and has minimum
variance, i.e. it has better precision than the best linear unbiased estimator. This result
applied to the baseline estimator gives:
⎧
⎪⎨ D{
⎪⎩
˜b} ≤ D{ ˇb} and E{ ˜b} = E{ ˇb} = b
D{ ˜b} ≤ D{ ˆb} and E{ ˜b} = E{ ˆ (4.7)
b} = b
Hence, the BIE baseline precision is always better than or as good as the precision of its
float and fixed counterparts.
In GNSS applications it is assumed that the data are normally distributed as mentioned
in section 3.1.3. It can be shown that in this case the BIE estimator is given by equation
(4.5) with the following estimators for the ambiguities and the baseline parameters:
⎧ P
⎪⎨ ã =
⎪⎩
z∈Zn z exp{− 1
2 â−z2Q
}
â
P
exp{− 1
2 â−z2 Q }
â
z∈Z n
˜ b = ˆ b − Qˆ bâ Q −1
â
See appendix C.
(â − ã)
= zwz(â)
z∈Z n
(4.8)
Note that the formal expressions in (4.8) are identical to their Bayesian counterparts as
presented in section 3.6, but that the distributional properties of the BIE estimator and
its Bayesian counterpart of course differ. Since the functional form of the non-Bayesian
estimator is identical to the Bayesian solution in the special case of normally distributed
data, the theory of BIE estimation has provided the link with the Bayesian approach of
ambiguity resolution.
The BIE estimator 69