The GNSS integer ambiguities: estimation and validation

2
1
0
−1
−2
−2 −1 0 1 2
Figure 3.4: Two-dimensional pull-in regions in the case of bootstrapping.
3.1.2 Integer bootstrapping
A generalization of the integer rounding method is the sequential integer rounding
method, also referred to as the integer bootstrapping method, see e.g (Blewitt 1989;
Dong and Bock 1989). In contrast to integer rounding, the integer bootstrapping estimator
takes the correlation between the ambiguities into account. It follows from
a sequential conditional least-squares adjustment with a conditioning on the integer
ambiguity values from the previous steps (Teunissen 1993; Teunissen 1998b). The components
of the bootstrapped estimator are given as:
ǎ1,B = [â1]
ǎ2,B = [â 2|1] = [â2 − σâ2â1 σ−2
â1 (â1 − ǎ1,B)]
. (3.10)
n−1
ǎn,B = [ân|N ] = ân −
i=1
σânâ i|I σ −2
â i|I (â i|I − ǎi,B)
where â i|I stands for the ith ambiguity obtained through a conditioning on the previous
I = {1, . . . , (i − 1)} sequentially rounded ambiguities. Since, the first entry is simply
rounded to the nearest integer, one should start with the most precise float ambiguity.
The real-valued sequential least-squares solution can be obtained by means of the triangular
decomposition of the vc-matrix of the ambiguities: Qâ = LDL T , where L denotes
a unit lower triangular matrix with entries
lj,i = σâjâ i|I σ −2
â i|I
(3.11)
and D a diagonal matrix with the conditional variances σ 2 â i|I as its entries. Since by
definition |â i|I − [â i|I]| ≤ 1
2 , the pull-in region Sz,B that belongs to the bootstrapped
Integer estimation 31