The GNSS integer ambiguities: estimation and validation

Simulation and examples B
Throughout this work several examples are based on simulations for which the vc-matrix
of the float solution is required as input. In this appendix the method of simulation and
the underlying models for the input are described.
B.1 Simulation
Simulation is a useful method for evaluation of the success rate and fail rate. Although
in practice it would be too time-consuming, the method is used in this research in
order to evaluate the results of different ambiguity resolution methods. Simulation
goes as follows. Since it is assumed that the float solution is normally distributed, the
probabilities are independent of the mean, so one can use N(0, Q) and draw samples
from this distribution.
For example, if only float ambiguities need to be generated, the first step is to use a random
generator to generate n independent samples from the univariate standard normal
distribution N(0, 1), and then collect these in a vector s. This vector is transformed by
means of â = Gs, with G equal to the Cholesky factor of Qâ = GG T . The result is a
sample â from N(0, Qâ), and this sample is used as input for the integer estimator that
is considered. If the output of this estimator equals the null vector, then it is correct,
otherwise it is incorrect. This process can be repeated N number of times, and one can
count how many times the null vector is obtained as solution, say Ns times, and how
often the outcome equals a nonzero integer vector, say Nf times. The approximations
of the success rate and fail rate follow than as:
Ps = Ns
N , Pf = Nf
N
In order to get good approximations, the number of samples N must be chosen sufficiently
large, see (Teunissen 1998c).
The random generator used in this research is the Matlab R○ routine mvnrnd from the
Statistics toolbox.
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