Global Positioning System Reference
In-Depth Information
sin
cos
sin
cos
de
2
+
α
12
,b
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
α
12
,
0
α
21
,
0
α
21
,
0
α
21
,
0
α
=
dn
1
+
de
1
+
dn
2
−
α
12
,
0
−
v
s
12
,
0
s
12
,
0
s
12
,
0
s
12
,
0
(9.9)
sin
dn
1
+
cos
de
1
sin
cos
α
13
,
0
α
12
,
0
α
31
,
0
α
21
,
0
v
δ
=
−
−
s
13
,
0
s
12
,
0
s
13
,
0
s
12
,
0
sin
cos
α
21
,
0
α
21
,
0
−
dn
2
+
de
2
(9.10)
s
12
,
0
s
12
,
0
sin
cos
de
3
+
δ
213
,b
α
31
,
0
α
31
,
0
δ
213
,
0
−
+
dn
3
−
s
13
,
0
s
13
,
0
[32
α
21
,
0
de
2
+
s
12
,
0
−
s
12
,b
(9.11)
cos
sin
cos
sin
v
s
=−
α
12
,
0
dn
1
+
α
21
,
0
de
1
−
α
21
,
0
dn
2
−
Lin
—
1
——
No
PgE
α
0
,
Th
e quantities
(
β
0
, s
0
)
are computed from the inverse solution. The GLM func-
tio
ns are particularly suitable for this purpose because the inverse solution is non-
ite
rative. The results of the adjustment of the ellipsoidal network are the adjusted
ob
servations
(
α
a
,
β
a
, s
a
)
and the adjusted coordinates
]
T
=
x
a
[
···
ϕ
i,a
λ
i,a
···
(9.12)
[32
The partial derivatives (9.9) to (9.11) are a result of series expansion and are
th
erefore approximations and subject to truncation errors. It is of course necessary
th
at the partial derivatives and the GML functions have the same level of accuracy.
Fi
gure 9.6 shows a test computation for the 4°
×
4° test area. We first use the GML
in
verse solution to compute
(s
ci
,
α
ci
)
between center
P
c
(ϕ
c
,
λ
c
)
and
P
i
(ϕ
i
,
λ
i
)
, and
th
en
(s
ci,d
,
α
ci,d
)
between
P
c
(ϕ
c
+
dϕ
c
,
λ
c
+
d
λ
c
)
and
P
i
(ϕ
i
+
dϕ
i
,
λ
i
+
d
λ
i
)
. The
differentials cause a shift on the ellipsoidal surface of
s
ci
−
s
ci,d
2
+
α
ci,d
2
s
ci
s
i,
GML
=
α
ci
−
(9.13)
Using the linear forms (9.9) and (9.11) (see also Table B.3), we compute
ds
ci
+
d
2
ci
s
ci
s
i,lin
=
α
(9.14)
Th
e differences
s
i,
GML
−
s
i,lin
are contoured in Figure 9.6 for the values
dϕ
c
=
d
1 m. The straight lines at the center in north-south and east-
west directions are the zero contour lines. The other contour lines increase in steps
of 0.1 mm starting at
λ
c
=
dϕ
i
=
d
λ
i
=
−
0
.
4 mm in the southeast and southwest corners and ending at