Global Positioning System Reference
In-Depth Information
N
11
L
11
L
11
T
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
N
21
O
O
A
1
PA
1
=
=
=
N
(7.134)
N
21
N
22
L
12
L
22
L
12
L
22
Q
a
Q
ab
N
−
1
Q
x
∗
=
=
(7.135)
Q
ab
Q
b
Q
−
1
b
L
22
L
22
=
(7.136)
The submatrices
L
ij
are part of the Cholesky factor
L
. The relation (7.136) can be
readily verified. In the notation of Section 4.9.4 we state the zero hypothesis
H
0
as
H
0
:
A
2
x
∗
+
2
=
[27
o
(7.137)
Th
ese are
n
conditions, one for each ambiguity. The hypothesis states that a particular
in
teger set is statistically compatible with the estimated ambiguities from the float
so
lution. When constraining the ambiguities the coefficient matrix
A
2
takes on the
si
mple form
A
2
Lin
—
0.4
——
Nor
PgE
=
[
O
]. The identity matrix
I
is of size
n
. The misclosure is
2
b
, where
b
is the set of integer ambiguity values that are to be tested. The
ch
ange in
v
T
Pv
due to the
n
constraints can be written according to (4.279)
=−
b
T
Q
−
b
b
b
b
v
T
Pv
∆
=
−
−
(7.138)
[27
which can be used in the
F
test (4.280)
v
T
Pv
v
T
Pv
∗
∆
df
n
∼
F
n,df
(7.139)
to test acceptance of
H
0
.
v
T
Pv
∗
comes from the float solution and
df
denotes the
degree of freedom of the latter.
In the early days of GPS surveying, a test set
b
of integer values was obtained by
simply rounding the estimated float ambiguities to the nearest integer. This approach
works well for long observation times where many satellites can be observed and
the change in satellite geometry over time significantly improves the float solution.
In such cases, the estimated real-valued ambiguities are already close to integers
and their estimated variances are small. The situation changes drastically when we
attempt to shorten the time of observation, possibly down to the extreme of just one
epoch. It is only the distribution of the satellites in the sky and the availability of
observations at multiple frequencies that adds strength to the geometry in such a
case. The estimated float ambiguities will not necessarily be close to integer, and the
estimates will have large variances and be highly correlated in general. A possible
solution is to find candidate sets
b
i
of integers and compute
v
T
Pv
i
according to
(7.138). Those with the smallest contribution are subjected to the test (7.139). There
are two potential problems with this approach.
∆
