The GNSS integer ambiguities: estimation and validation

Integer ambiguity resolution 3
The problem of integer ambiguity resolution has drawn a lot of attention in the past
decades. Many ambiguity resolution algorithms have been published. Not all of these
algorithms will be discussed here; only a brief overview will be given. The first step
of ambiguity resolution is integer estimation. Therefore, section 3.1 starts with the
definition of admissible integer estimators. For the quality description and validation of
the estimators, their distribution functions are required. The distributional properties of
the float and fixed ambiguity will be given in section 3.2, where it is also shown how the
probability of correct integer estimation, the success rate, can be approximated. For the
purpose of validation, the parameter distribution of the ambiguity residuals is required.
This distribution function and its properties will be given in section 3.3. In section 3.4
it is shown how the quality of the fixed baseline estimator can be expressed. Section 3.5
gives an overview of currently available methods for the validation of the fixed ambiguity
solution and their shortcomings. Finally, in section 3.6 a completely different approach
of ambiguity resolution is described, namely the Bayesian approach.
3.1 Integer estimation
Any GNSS observation model can be parameterized in integers and non-integers. This
gives the following system of linear(ized) observation equations:
y = A a + B b + e
m×nn×1 m×pp×1 m×1
m×1
(3.1)
where y is the GPS observation vector of order m, a and b are the unknown parameter
vectors of dimension n and p respectively, and e is the noise vector. The data vector
y usually consists of the observed-minus-computed DD phase and/or code observations
on one, two or three frequencies and accumulated over all observation epochs. The
entries of the parameter vector a will then consist of the unknown integer carrier phase
ambiguities, which are expressed in units of cycles rather than in units of range. It is
known that the entries are integers, so that a ∈ Z n . The remaining unknown parameters
form the entries of the vector b. These parameters may be the unknown baseline
increments and for instance atmospheric (ionospheric, tropospheric) delays, which are
all real-valued, i.e. b ∈ R p . These real-valued parameters are referred to as the baseline
parameters, although the vector b may thus contain other parameters than only the
baseline components.
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