Lucht
An example: dual versus triple frequency GPS
even with a high enough success-rate, fixing to the wrong integer ambiguities is
still possible when one or more observations are erroneous. A success-rate close
enough to one does therefore not release one from the obligation of performing
statistical tests for model validation. It does however make it much more easier to
perform such tests. The higher the success-rate, the sooner one is allowed to
apply the classical theory of statistical hypothesis testing.
Figure 3: By taking the integral of the probability density function (at left) over the pull-in
region for each integer vector, the probability is obtained that this vector will result as the
integer least-squares solution. The probabilities are given at right for the integer vectors
between -1 and 1. The integral over the area for the correct integer vector, in this case
(0,0), gives the success-rate. It is about 0.85 in this example.
The success-rate can also serve as a tool in analyzing the benefits of three frequency
GPS or as a design-tool in choosing the three frequencies forthe planned European
'Galileo', see [2]. As such an example, we will compare the performance of the
current dual-frequency GPS with the future triple-frequency GPS, also referred to
as 'modernized GPS'. This comparison will be based on the so-called geometry-
free model. This model is the simplest possible GNSS model that still allows the
estimation of integer carrier phase ambiguities. In its most basic form the model
consists of the double-differenced pseudo range and carrier phase observations
of two receivers to two satellites, parameterized in terms of an unknown double-
differenced satellite-receiver range, unknown ambiguities and an unknown
ionospheric delay. The ionospheric delay is included so as to make the model
applicable for long baselines.
We will first study the dual-frequency success rate in its dependence on a varying
second frequency, whilst the first frequency is kept fixed to the GPS LI frequency.
This is shown in figure 4, at left. First note that the success rate fails to exceed the
very small value of 0.025 within the frequency range shown. This stipulates the
poor performance of instantaneous dual-frequency ambiguity resolution for long
baselines. Hence, for these cases one can not expect dual-frequency ambiguity
resolution to be successful. The figure also shows that the success rate reaches its
minimum when the two frequencies coincide and that the success rate gets larger
when the frequency separation gets larger. This contradicts the popular belief
that ambiguity resolution would benefit from choosing the frequencies close
together. It is of course still true that frequencies with little separation would allow
one to construct a wide-lane with a corresponding very large wavelength. However,
as the figure shows this turns out to be counterproductive as far as the overall
success rate is concerned. In fact, as the figure shows, the success rate will be
identical to zero when the two frequencies coincide. This is understandable when
one recognizes that a nonzero frequency separation is needed per se in order to
be able to estimate the ionospheric delays. When the two frequencies coincide,
the ionospheric delay becomes non-estimable and the variance-covariance matrix
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