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
[33
Lin
—
-
——
Lon
PgE
Figure 2.9
Equipotential surfaces and the gravity force vector.
The shapes of equipotential surfaces, which are related to the mass distribution
wi
thin the earth through (2.53), have no simple analytic expressions. The plumb
lin
es are normal to the equipotential surfaces and are space curves with finite radii
of
curvature and torsion. The distance along a plumb line from the geoid to a point
is
called the orthometric height,
H
. The orthometric height is often misidentified as
th
e “height above sea level.” Possibly, confusion stems from the specification that the
ge
oid closely approximates the global ocean surface.
Consider a differential line element
d
x
along the plumb line,
[33
dH
.By
no
ting that
H
is reckoned positive upward and
g
points downward, we can rewrite
(2
.56) as
d
x
=
dW
=
g
·
d
x
(2.58)
=
gdH
cos
(
g
,d
x
)
=
gdH
cos
(
180°
)
=−
gdH
This expression relates the change in potential to a change in the orthometric height.
This equation is central in the development of the theory of geometric leveling.
Writing (2.58) as
dW
dH
g
=−
(2.59)
it is obvious that the gravity
g
cannot be constant on the same equipotential surface
because the equipotential surfaces are neither regular nor concentric with respect to
the center of mass of the earth. This is illustrated in Figure 2.10, which shows two
differentially separate equipotential surfaces. It is observed that
dW
dH
1
=
dW
dH
2
g
1
=−
g
2
=−
(2.60)
