The GNSS integer ambiguities: estimation and validation

5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
σ=0
σ=0.1
σ=0.2
σ=∞
σ=0.3
0
−0.5 0
x
0.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
σ=0.4
σ=0.5
σ=1
σ=0.3
0.6
−0.5 0
x
0.5
Figure 3.9: PDF of ˇɛ for different values of σ.
the untransformed vc-matrices, but not for the Z-transformed vc-matrices. However,
the shape of the PDF near the boundaries of the pull-in region is clearly different from
the shape of the PDF of â. This shows that in all cases the fixed ambiguity estimator
should not be considered deterministic, since that would result in the assumption that
fˇɛ(x) := fâ(x + ǎ). This is only true if all the probability mass of â is located in the
pull-in region Sa.
Figure 3.11 shows the ellipses that correspond to the vc-matrices of â and ˆz, and to the
vc-matrices of the ambiguity residuals.
Approximation
In section 3.3.1 it was explained that only an approximation of the PDF of ˇɛ is possible by
replacing the infinite sum over all integers in equation (3.48) by a sum over a finite set of
integers. It is investigated here how good the approximations work for different choices
of λ, which determines the integer set, see equation (3.51). For that purpose, 10,000
samples of float ambiguities were generated using simulation for various vc-matrices, see
appendix B. The corresponding ambiguity residuals were determined, so that the PDF
of these parameters could be determined.
The results for the 2-D case are shown in figure 3.12. Besides Qˆz 02 01, also 1
4 Qˆz 02 01
and 4Qˆz 02 01 were used in order to study the effect of higher/lower precision. The
The ambiguity residuals 49