There exist various ways of computing or approximating the optimal success-
rate, two of which will be given. One way of obtaining the success-rate is by
simulation. Using a random generator, a large number of real-valued ambiquity
vectors are generated from the origin-centered probability distribution p (x) of
the float solution. For each of these generated vectors, the corresponding inteqer
least-squares solution is computed using the LAMBDA method. The percentage
ot integer solutions that coincide with the origin yields the success-rate. The number
ot generated samples must be large enough in order to obtain a close enough
approximation to the success-rate. a
/-B=n
'1' J
Lustrumboek "The 5th Element"
integer estimators will have different pull-in regions. The pull-in region of inteqer
rounding, tor instance, equals a square. The optimal pull-in region, the one of
integer least-squares, is shown in the figure. If we denote the probability density
function ot the float ambiguities as pa (x) and the pull-in region of the correct
integer ambiguity vector as Ra, the ambiguity success-rate can be written in
formula form as
Succes - rate J pa (x)dx
R
A second option of inferring the success-rate is to compute a sharp lower bound
ot the probability of correct integer least-squares estimation. A sharp and easy-
to-compute lower bound (LB) is given in [5] and reads:
i=i
1
1
20
-1
2(Jil
-
-
X 1 —z2
succes - rate with <E>(x) ~j=e 2 dz
-~"v/2 n
u piuuuu 01 n Terms irne number ot ambiguities 4> is the standard
normal cumulative probability distribution and er is the standard deviation of
ambiguity i, conditioned on all previous ambiguities, indicated by The conditional
standard deviations follow directly from the triangular decomposition of the float
ambiguity variance-covariance matrix Q"1 |_DI_T as one over the square root
ot the elements of diaaonal matrix D. This decomposition is already made in the
computations for the LAMBDA method, and hence available at no extra cost.
For this lower bound to be sharp it is essential that the variance-covariance
lu °L LAMBDA-transformed ambiguities is used for the computation of
the conditional standard deviations, as they have an improved precision and
decreased correlation over the original double difference ambiguities.
This approximation to the success-rate can be computed straightforwardly and if
it is sufficiently large, say 0.99 or 0.999, it is guaranteed that the actual success-
rate ot the integer least-squares method is at least equally high and thus very
close to I .U. As it provides a lower bound, one can safely rely on this
approximation. 7
It is clear that the ambiguity success-rate can be evaluated once the GNSS
nVC!'0naAliann stoc^stic models are known. Hence, similar to the usaqe of
Dilution Of Precision (DOP) measures, it can be computed without having the
actual measurements available, thus prior to actual field operation. By means of
the success-rate the user is given a rigorous way of assessing how often he or she
can expect ambiguity resolution to be successful. Only when the success-rate is
close enough to one, is one allowed to proceed as if the estimated inteqer
ambiguities are non-stochastic.
The success-rate depends of course, as any other formal reliability measure, on
he correctness ot the assumptions which underly the model used. Misspecifications
in the model may lead to unrealistic values for the success-rate. For instance,
