Geoscience Reference
In-Depth Information
earth's surface
telluroid
P
W=
P
actual level surface
³
Q
U=
P
normal level surface
"
-
±
normal plumb line
(curved)
actual plumb line
ellipsoidal normal
(straight)
P
0
P''
geoid
W=
0
P
0
0
"
0
N
U=
0
ellipsoid
Q
0
'
Q
0
''
Q
0
Fig. 8.13. The basic geometry
As a warm-up, let us return to basics and remember some main principles
of Molodensky's geometry. Figure 8.13 illustrates the basic quantities. In
the classical theory, the geoid is defined by its deviation
N
from a reference
ellipsoid;
N
is the geoidal height. The geoid is a level surface
W
=
W
0
=
constant of the gravity potential
W
; the ellipsoid is defined to be the level
surface
U
=
U
0
= constant of a normal gravity potential
U
; the constants
W
0
and
U
0
are usually assumed to be equal (Sect. 2.12).
For the modern theory according to Molodensky (Sect. 8.4), to each point
P
of the earth's surface we associate a point
Q
in such a way that
Q
lies on
the straight ellipsoidal normal through
P
and that
U
(
Q
)=
W
(
P
)
.
(8-121)
That is,
Q
is defined such that its normal potential
U
equals the actual
potential
W
of
P
.
This corresponds to the classical relation
U
0
=
U
(
Q
0
)=
W
(
P
0
)=
W
0
(8-122)
mentioned above, by which
U
0
is taken to be equal to
W
0
(Fig.8.13).Bythe
same correspondence, the height anomaly according to Molodensky,
ζ
=
QP ,
(8-123)
