The GNSS integer ambiguities: estimation and validation

µ
200
100
10
2
1
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.81.9 2
p
f
µ
2
1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
1 2 10 100 200
p
f
Figure 5.14: µ as function of the penalty pf with ps = 0 and pu = 1. Left: pf = 1, . . . , 2;
Right: pf = 2, . . . , 200.
2-D example
In order to get an idea of the shape of the aperture space, Qˆz 02 01 from appendix B
is chosen as the vc-matrix, the corresponding PDFs of the float ambiguities and the
ambiguity residuals are determined, and then the inequality (5.48) is used to find Ω0 as
function of µ.
Figure 5.15 shows the results. Furthermore, figure 5.14 shows µ as function of the
penalty pf > 1, where ps = 0 and pu = 1. In this case µ → 1 for pf ↓ pu. The
reasoning behind this choice is that it seems realistic that no penalty is assigned in
the case of correct integer ambiguity resolution, and also that the penalty in the case
of incorrect integer ambiguity estimation is larger than the penalty in the case of ’no
decision’ (undecided). This implies that in order to minimize the average penalty E{p},
the fail rate Pf should be as small as possible in all cases, and the probability of no
decision Pu should also be small, especially if pf ↓ pu. Thus the success rate should be
as large as possible in all cases, which is the case if the PDF of â is peaked. However,
for very small values of µ (close to one), the undecided rate will become large since then
the aperture space will be small. This results in lower success rates.
It can be seen that the shape of Ω0 is a mixture of an ellipse and the shape of the pull-in
region. The reason is that the contour lines of fâ(x) (see top panel of figure 5.15) are
ellipse-shaped, but those of fˇɛ(x) (center panels) are not. It is clear that for larger µ
the aperture space becomes larger.
Penalized Integer Aperture estimation 109