Lucht
Ambiguity success-rate
Figure 2: Using single frequency pseudo-range and carrier phase datathe phase ambiguity
of the geometry-free GPS model is estimated in 1800 sinale epoch experiments at a 1
second interval. The histogram at left shows the residuals of the (double difference) pseudo-
range measurementsthe noise is at the decimeter level. The histogram in the middle
concerns the float ambiguity. It is primarily the noise in the pseudo-range which is reflected
in the noise of the float ambiguity, and as the LI-wavelength is about 2 decimeter, the
corresponding uncertainty in the float ambiguity is at the cycle level. In both the graph at
left and in the middle, the formal Gaussian probability distribution is also shown. Finally
the integer ambiguity was computed for each experiment, and yields the histogram at
right. In this case the integer ambiguity is estimated correctly (value 4) in only 43% of the
experiments.
If one wants to treat the computed integer ambiguities as deterministic variates,
as is done in practice, one will have to ensure that their uncertainty is sufficiently
small to be indeed neglected. This is the case when the frequency with which
estimated integer ambiguities coincide with the correct but unknown values, is
sufficiently large. This concept is formalized in a probabilistic measure, referred
to as the ambiguity success-rote. The success-rate is a number between 0 and 1
or 0% and 100%, and it expresses the chance, or probability, that the integer
ambiguities are correctly estimated.
The ambiguity success-rate depends on three contributing factors: the observation
equations (functional model), the precision of the observables (the stochastic
model), and the chosen method ot integer estimation. Changes in any one of
these will affect the success-rate. The first two contributing Factors reflect the
strength of the data model and they are given once the measurement set-up is
known. As to the method of integer estimation, one has a variety of options
available. However, since different methods of integer estimation will generally
result in different success-rates, one might wish to use the method that maximizes
the success-rate. It has recently been proven, see [3], that the integer least-squares
estimator has the largest success-rate of all admissible integer estimators. The
success-rate of the LAMBDA method is therefore larger than, or at least as large
as any other integer ambiguity estimator.
Figure 3 shows a two-dimensional example of how the success-rate of the integer
least-squares ambiguities is to be obtained. The figure shows the probability
density function of the float ambiguities at left and the corresponding discrete
distribution of the integer least-squares ambiguities at right. The probability density
function can be computed once the GNSS functional- and stochastic models are
known. In case the GNSS data are assumed to be Gaussian distributed, the
shape of the distribution is completely specified by the variance-covariance matrix
of the float ambiguities. In the example the standard deviations of the two
ambiguities are about 0.3 cycle. The corresponding success-rate follows then as
the integral of the probability density function over the area of the convex polygon.
This area is referred to as the ambiguity pull-in region. It contains all locations of
the float ambiguities that get pulled to the correct integer solution. Different
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