Global Positioning System Reference
In-Depth Information
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
0
0
−
cos
λ
0
M
ξ
=
0
0
−
sin
λ
0
(8.18)
cos
λ
0
sin
λ
0
0
0
−
cos
ϕ
0
−
sin
ϕ
0
sin
λ
0
M
η
=
cos
ϕ
0
0
sin
ϕ
0
cos
λ
0
(8.19)
sin
ϕ
0
sin
λ
0
−
sin
ϕ
0
cos
λ
0
0
If,
again, second-order terms in scale and rotations and their products are neglected,
th
e model (8.14) becomes
t
+
u
+
s (
u
−
u
0
)
+
(
1
+
s)(
M
−
I
)(
u
−
u
0
)
−
x
=
o
(8.20)
[30
Models 2 and 3 differ in that the rotations in model 3 are around the local geodetic
co
ordinate axes at
u
0
. The rotations
(
η
ξ
α
ψ
ω
,
,
)
are
(ε,
,
)
as related as follows:
Lin
—
-1.
——
Nor
PgE
η
ξ
α
ε
ψ
ω
=
R
2
(
90
−
ϕ
0
)
R
3
(
λ
0
)
(8.21)
Models 1 and 2 use the same rotation angles. The translations for models 1 and 2 are
related as
[30
t
2
=
t
1
−
u
0
+
(
1
+
s)
Ru
0
(8.22)
according to (8.8) and (8.12). Only
t
1
, i.e., the translation vector of the origin as
estimated from model 1, corresponds to the geometric vector between the origins
of the coordinate systems
(x)
and
(u)
. The translational component of model 2,
t
2
, is a function of
u
0
, as shown in (8.22). Because models 2 and 3 use the same
u
0
, both yield identical translational components. It is not necessary that all seven
parameters always be estimated. In small areas it might be sufficient to estimate only
the translation components.
8.3 COMBINATION THROUGH ROTATION AND SCALING
Assume a situation in which a network of terrestrial observations, such as horizontal
angles, slant distances, zenith angles, leveled height differences, and geoid undula-
tions, are available. Assume further that the relative positions of some of these net-
work stations have been determined with GPS. As a first step one could carry out
separate minimal or inner constraint solutions for the terrestrial observations and the
GPS vectors, as a matter of quality control. When combining both sets of observations
in one adjustment, the definition of the coordinate systems might become important.
