The GNSS integer ambiguities: estimation and validation

Unfortunately, this relatively simple approach cannot be applied when integer parameters
are involved in the estimation process, since the integer estimators do not have a Gaussian
distribution, even if the model is linear and the data are normally distributed. Instead of
the vc-matrices, the parameter distribution itself has to be used in order to obtain the
appropriate measures that can be used to validate the integer parameter solution.
In the past, the problem of non-Gaussian parameter distributions was circumvented by
simply ignoring the randomness of the fixed ambiguity estimator. Several testing procedures
for the validation of the fixed solution have been proposed using this approach.
This section will give an overview and evaluation of these procedures, taken from (Verhagen
2004).
3.5.1 Integer validation
The first step in validating the float and fixed solution is to define three classes of
hypotheses, (Teunissen 1998b). In practice the following hypotheses are used:
H1 : y = Aa + Bb + e, a ∈ R n , b ∈ R p , e ∈ R m
H2 : y ∈ R m , (3.71)
H3 : y = Aǎ + Bb + e, b ∈ R p , e ∈ R m
with Qy = σ 2 Gy for all hypotheses.
In hypothesis H1 it is assumed that all unknown parameters are real-valued. In other
words, the constraint that the ambiguities should be integer-valued is ignored. The float
solution is thus the result of least-squares estimation under H1.
In the third hypothesis the integer constraint is considered by assuming that the correct
integer values of the ambiguities are known. The values are chosen equal to the fixed
ambiguities obtained with integer least-squares. Hence, the least-squares solution under
H3 corresponds to the fixed solution. Clearly, the first hypothesis is more relaxed than
the third hypothesis, i.e. H3 ⊂ H1.
The second hypothesis does not put any restrictions on y and is thus the most relaxed
hypothesis.
It will be assumed that the m-vector of observations y is normally distributed with a zeromean
residual vector e. The matrix Gy is the cofactor matrix of the variance-covariance
matrix Qy, and σ 2 is the variance factor of unit weight. The unbiased estimates of σ 2
under H1 and H3 respectively are given by:
H1 : ˆσ 2 = êT G −1
y ê
m − n − p =
H3 : ˇσ 2 = ěT G −1
y ě
m − p = ˇ Ω
m − p
ˆΩ
m − n − p
(3.72)
The denominators in (3.72) are equal to the redundancies under the corresponding
hypotheses.
58 Integer ambiguity resolution