The GNSS integer ambiguities: estimation and validation

2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
Figure 3.3: Two-dimensional pull-in regions in the case of rounding.
Examples of integer estimators that belong to the class of admissible integer estimators
are integer rounding (R), integer bootstrapping (B), and integer least-squares (ILS).
Each of these estimators will be described briefly in the following sections.
3.1.1 Integer rounding
The simplest way to obtain an integer vector from the real-valued float solution is to
round each entry of â to its nearest integer. The corresponding integer estimator, ǎR,
reads then:
ǎR =
⎛ ⎞
[â1]
⎜
⎝ .
⎟
. ⎠ (3.8)
[ân]
where [·] denotes rounding to the nearest integer. It can be easily shown that this
estimator is admissible, see Teunissen (1999a).
Since each component of the real-valued ambiguity vector is rounded to the nearest
integer, the absolute value of the maximum difference between the float and fixed ambi-
guities is 1
2 . The pull-in region Sz,R that corresponds to this integer estimator is therefore
given as:
Sz,R =
n
i=1
x ∈ R n | |xi − zi| ≤ 1
, ∀z ∈ Z
2
n
(3.9)
So, the pull-in regions are the n-dimensional unit cubes centered at z ∈ Z n . For the
two-dimensional case this is visualized in figure 3.3.
30 Integer ambiguity resolution