Bernese GPS Software Version 5.0 - Bernese GNSS Software

8.3.4 QIF (Quasi Ionosphere-Free) Algorithm
8.3.4.1 Theory of QIF Strategy
8.3 Ambiguity Resolution Algorithms
We neglect the troposphere bias in Eqns. (2.37a) and (2.37b) and do not explicitly write
the receiver and satellite indices k,ℓ,i,j. Then the simplified form of the double difference
observation equations reads as
L1 = ̺ − I + λ1 n1 (8.17)
L2 = ̺ − f2 1
f 2 2
· I + λ2 n2
(8.18)
The corresponding equation for the ionosphere-free linear combination may thus be written
as
L3 = ̺ + B3 = ̺ +
c
(f1 n1 − f2 n2) . (8.19)
f2 1 − f2 2
The initial least-squares adjustment using both frequencies L1 and L2 gives real-valued
ambiguity estimates b1 and b2 and we may compute the corresponding ionosphere-free bias
˜B3 as
˜B3 =
c
f 2 1 − f2 2
(f1 b1 − f2 b2) . (8.20)
This bias may be expressed in narrow-lane cycles (one cycle corresponding to a wavelength
of λ3 = c/(f1 + f2) ≈ 11 cm, see Section 2.3):
˜ b3 = ˜ B3
= ˜ B3 · f1 + f2
= f1
f1 − f2
b1 − f2
f1 − f2
λ3 c
= β1 b1 + β2 b2 . (8.21)
Denoting the correct (resolved) integer ambiguity values by n1i and n2j (i and j are not the
satellite indices) and introducing the associated L3-bias
b3ij = β1 n1i + β2 n2j
we may use the difference between the real-valued and integer L3-bias
b2
(8.22)
d3ij = | ˜ b3 − b3ij| (8.23)
as a criterion for the selection of the “best” pair of integers n1i,n2j. However, many pairs
n1i,n2j give differences d3ij of the same (small) order of magnitude. These pairs lie on a
narrow band in the (n1,n2) space. The equation for the center line of this band is
β1 ni1 + β2 n2j = ˜ b3 . (8.24)
The band-width is essentially given by the rms of the bias ˜ b3. A unique solution only results
if it is possible to limit the search range. The principle is shown in Figure 8.8. The solid line
corresponding to Eqn. (8.24) goes through the real valued estimate (b1,b2) (shown as o in
figure) as well as through the point (n1i,n2j) which is accepted as “true” solution. This line
represents an ionosphere–free combination (constant ionosphere-free bias). The second solid
in Figure 8.8 represents the constant wide-lane L5 ambiguity (accepted as “true” value) and
goes through the point (n1i,n2j), too. The dashed rectangle represents a search range in
(n1,n2) space and the dashed trapezoid represents the search range in (n1,n5) space, see
Eqn. (8.29).
Bernese GPS Software Version 5.0 Page 177