Geoscience Reference
In-Depth Information
δg
, respectively, we reduce these complicated problems to the corresponding
spherical problems, for which the solution is simple and well known.
The similarity of the Molodensky series for the Molodensky problem, on
the one hand, and for the GPS boundary problem, on the other hand, is very
clear because ∆
g
and
δg
have the same analytical and geometric structure.
At the same time, this similarity is very surprising since the two underly-
ing boundary problems are mathematically quite different, as we have seen
in Sect. 8.3 (compare Eqs. (8-12) and (8-13)). Nonetheless, (8-87) does give
the potential as (8-13) requires: by Bruns' theorem, which is the omnipresent
link between geometry and physics, we have
T
=
γζ.
(8-92)
Then
W
=
U
+
T
(8-93)
is the geopotential required by (8-13), and
C
=
W
0
−
W
(8-94)
is the geopotential number, the physical measure of height above sea level,
conventionally obtained by the cumbersome method of leveling, but now
computed in a direct way from gravity data. This is the physical, more
general, equivalent of the geometric determination of the normal height by
H
∗
=
h
ζ
, according to Eq. (8-31).
It can be shown that, in the linear approximation, the Molodensky cor-
rection for the gravity disturbance has the same form as for the gravity
anomaly and can for each quantity be computed using either ∆
g
or
δg
.
The formulas for the Molodensky corrections and their numerical values
are the same to the linear approximation.
All this shows the power of Molodensky's approach even in problems he
never treated himself.
−
8.9
Gravity reduction in the modern theory
In Sect. 8.2, we have considered gravity reductions from the point of view
of the determination of the geoid. It is quite remarkable that these reduc-
tions, such as the Bouguer or the isostatic reduction, can also be incorpo-
rated into the new method of direct determination of the earth's physical
surface, although with essentially changed meaning (Pellinen 1962; Moritz
1965: Sect. 5.2).
