The GNSS integer ambiguities: estimation and validation

Implementation aspects of IALS estimation D
The procedure outlined in section 5.7.3 can also be applied for the IALS estimator, but
is more complicated as will be explained here. The problem is the determination of the
aperture parameter such that the float ambiguity estimate will be on the boundary of
the aperture pull-in region. From the definition of the ILS pull-in region in equation
(3.17) follows that ˆx will be on the boundary of Sˇx if:
(ˇx2
− ˇx)
T Q −1
ˆx (ˆx − ˇx)
ˇx2 − ˇx 2 Qˆx
= 1
2
with ˇx and ˇx2 the best and second-best integers in the integer least-squares sense.
(D.1)
From the definition of the IALS estimator in section 5.4.2 it follows that ˆx − ˇx is on
the boundary of the aperture pull-in region Ω0 if 1
µ (ˆx − ˇx) is on the boundary of S0.
Furthermore, the following will be used:
⎧
1
⎨arg
min
z∈Zn µ
⎩
(ˆx − ˇx) − z2 = 0 Qˆx
1
arg min µ (ˆx − ˇx) − z2 (D.2)
= u Qˆx
z∈Z n \{0}
These results are used in equation (D.1) to find an expression for µ:
u
T Q −1
ˆx 1
µ (ˆx − ˇx)
1
=
2
=⇒
2u
µ =
T Q −1
ˆx (ˆx − ˇx)
u 2 Qˆx
u 2 Qˆx
(D.3)
The problem is that u can only be determined if µ is known. Therefore as a first guess,
one could use u 0 = ˇx2 − ˇx and then compute the corresponding µ 0 with equation (D.3)
by using u 0 instead of u. If
arg min
z∈Zn 1
\{0} µ 0 (ˆx − ˇx) − z2Qˆx = u (D.4)
and u = ˇx2 − ˇx it means that indeed µ 0 = µ. If, on the other hand, u = ˇx2 − ˇx it means
that although 1
µ 0 (ˆx − ˇx) is on the plane that is bounding S0, this vector is not on the
boundary of S0. Hence, 1
µ 0 (ˆx − ˇx) ∈ Su. This implies that µ 0 < µ and that 1
µ (ˆx − ˇx)
is on the boundary of Su and S0. Now that u is known, µ can finally be computed with
equation (D.3).
The procedure for IALS estimation is outlined below.
159