Global Positioning System Reference
In-Depth Information
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where we have denoted the base satellite with a superscript 1; the superscript
q
varies
over the secondary satellites. These estimates are rounded to the nearest integer.
Next we compute the correction to the positions
x
p,i
using sequential least-squares
(Table 4.2):
A
s
N
−
p
A
s
−
1
A
s
x
p,i
+
˜
s,i
(7.150)
N
−
p
A
s
P
s
+
∆
x
p,i
=
x
p,i
−
Th
e dimension of
˜
s,i
equals the number of additional satellites. If the set of primary
am
biguities used to generate
x
p,i
is the correct one, then the respective secondary
am
biguities
b
s,i
are correct and, consequently,
x
p,i
falls
ou
tside a tolerance region, whose size is a function of the accuracy of
x
0
, then
b
s,i
and
co
nsequently
b
p,i
are rejected and the search continues with (7.148) using a different
tri
al set
b
p,j
. If the set is acceptable, the residuals for the combined solution of primary
an
d secondary observations are
∆
x
p,i
should be zero. If
∆
[28
v
p
=
A
p
∆
x
P
(7.151)
Lin
—
5.6
——
Lon
PgE
A
s
x
p
x
P
+
˜
v
s
=
+ ∆
(7.152)
s
Th
e quadratic function
v
T
Pv
v
p
P
p
v
p
+
v
s
P
s
v
s
=
(7.153)
ca
n be used to discern several qualifying solutions.
[28
7.
8.2 LAMBDA
Teunissen (1993) introduced the least-squares ambiguity decorrelation adjustment
(LAMBDA) method. The LAMBDA technique, which has been referred to as the
integer least-squares estimator, is the estimator that has the highest probability of
correct integer estimation among all possible admissible integer estimators (Teunis-
sen, 1999). This probabilistic justification of LAMBDA in addition to its speed has
resulted in a high popularity and general acceptance of the technique. This section
merely highlights some features of the LAMBDA algorithm. The reader is refered to
Jonge and Tiberius (1996) for details of implementation. At the core of LAMBDA is
the
Z
transformation
Z
T
b
z
=
(7.154)
Z
T
b
z
=
(7.155)
Z
T
Q
b
Z
Q
z
=
(7.156)
where
Z
is a regular and square matrix. In order for integers to be preserved, i.e., the
integers
b
should be mapped into integers
z
and vice versa, it is necessary that the
