The GNSS integer ambiguities: estimation and validation

Summary
The GNSS integer ambiguities: estimation and validation
Fast and high precision relative positioning with a Global Navigations Satellite System
(GNSS) is only possible by using the very precise carrier phase measurements. However,
these carrier phases are ambiguous by an unknown number of cycles. The knowledge
that the ambiguities are integer-valued has been exploited in the past 15 years for the
development of integer ambiguity resolution algorithms. Once the ambiguities are fixed
to their integer values, the carrier phase measurements start to act as if they were very
precise pseudorange measurements.
The estimation process consists then of three steps. First a standard least-squares
adjustment is applied in order to arrive at the so-called float solution. All unknown
parameters are estimated as real-valued. In the second step, the integer constraint on
the ambiguities is considered. This means that the float ambiguities are mapped to
integer values. Different choices of the map are possible. The float ambiguities can
simply be rounded to the nearest integer values, or conditionally rounded so that the
correlation between the ambiguities is taken into account. The optimal choice is to use
the integer least-squares estimator, which maximizes the probability of correct integer
estimation. Finally, after fixing the ambiguities to their integer values, the remaining
unknown parameters are adjusted by virtue of their correlation with the ambiguities.
Nowadays, the non-trivial problem of integer ambiguity estimation can be considered
solved. However, a parameter estimation theory is not complete without the appropriate
measures to validate the solution. So, fixing the ambiguities should only be applied if
there is enough confidence in their correctness. The probability of correct integer estimation
can be computed a priori – without the need for actual observations – and is
called the success rate. Only if this success rate is very close to one, the estimated fixed
ambiguities may be considered deterministic. In that case it is possible to define test
statistics in order to validate the fixed solution. If the success rate is not close to one,
in practice a user has to choose between two undesirable situations:
• Integer validation is based on wrong assumptions, since the randomness of the
fixed ambiguities is incorrectly ignored;
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