The GNSS integer ambiguities: estimation and validation

Introduction 1
1.1 Background
In the past decade the Global Positioning System (GPS) has found widespread use
in different fields such as surveying, navigation and geophysics. The requirements on
precision, reliability and availability of the navigation system for these applications have
become higher and higher. Moreover, the positioning information often needs to be
obtained in (near) real-time. Without the development of integer ambiguity resolution
algorithms, it would never have been possible to meet these requirements with GPS.
Fast and high precision relative positioning with a Global Navigations Satellite System
(GNSS) is namely only possible by using the very precise carrier phase measurements.
However, these carrier phases are ambiguous by an unknown number of cycles. The
ambiguities are known to be integer-valued, and this knowledge has been exploited for
the development of integer ambiguity resolution algorithms. Once the ambiguities are
fixed on 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 fixed ambiguity
estimator 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,
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