The GNSS integer ambiguities: estimation and validation

estimator is given by:
n
Sz,B = x ∈ R n | |c T i L −1 (x − z)| ≤ 1
, ∀z ∈ Zn
2
i=1
(3.12)
with ci the canonical vector having a one as its ith entry. Note that the bootstrapped
and rounded solution become identical if the ambiguity vc-matrix, Qâ, is diagonal.
The shape of the pull-in region in the two-dimensional case is a parallelogram, see figure
3.4. For more dimensions it will be the multivariate version of a parallelogram.
3.1.3 Integer least-squares
Optimizing on the integer nature of the ambiguity parameters, cf.(Teunissen 1999a),
involves solving a non-standard least-squares problem, referred to as integer least-squares
(ILS) in Teunissen (1993). The solution is obtained by solving the following minimization
problem:
min
z,ζ y − Az − Bζ2 Qy , z ∈ Zn , ζ ∈ R p
The following orthogonal decomposition can be used:
y − Az − Bζ 2 Qy = ê2Qy + â − z
(3.2)
2 Qâ +
(3.3)
ˆb(z) − ζ 2 Qˆb|â
(3.4)
(3.13)
(3.14)
with the residuals of the float solution ê = y − Aâ − Bˆb, and the conditional baseline
estimator ˆb(z) = ˆb − Qˆbâ Q −1
â (â − z).
It follows from equation (3.14) that the solution of the minimization problem in equation
(3.13) is solved using the three step procedure described in section 3.1. The first term
on the right-hand side follows from the float solution (3.2). Taking into account the
integer nature of the ambiguities means that the second term on the right-hand side of
equation (3.14) needs to be minimized, conditioned on z ∈ Z n . This corresponds to
the ambiguity resolution step in (3.3), so that z = ǎ. Finally, solving for the last term
corresponds to fixing the baseline as in equation (3.4). Then the last term in equation
(3.14) becomes equal to zero, so that indeed the minimization problem is solved.
The integer least-squares problem focuses on the second step and can now be defined
as:
ǎLS = arg min
z∈Z nâ − z2 Qâ
(3.15)
where ǎLS ∈ Z n is the fixed ILS ambiguity solution. This solution cannot be ’computed’
as with integer rounding and integer bootstrapping. Instead an integer search is required
to obtain the solution.
The ILS pull-in region Sz,LS is defined as the collection of all x ∈ R n that are closer
to z than to any other integer grid point in R n , where the distance is measured in the
32 Integer ambiguity resolution