Geoscience Reference
In-Depth Information
and no longer to sea level; the “height anomalies” at ground level take the
place of the geoidal undulations.
These developments have considerably broadened our insight into the
principles of physical geodesy and have also introduced powerful new meth-
ods for tackling classical problems. Hence their basic theoretical significance
is by no means lessened by the fact that many scientists prefer to retain the
geoid because of its conceptual and practical advantages.
In this chapter, we first give a concise survey of the conventional determi-
nation of the geoid by means of gravity reductions, in order to understand
better the modern ideas. After an exposition of Molodensky's theory, we
show how the new methods may be applied to classical problems such as
gravity reduction or the determination of the geoid by gravimetric and as-
trogeodetic methods. It should be mentioned that the terms “modern” and
“conventional” merely serve as convenient labels; they do not imply any con-
notation of value or preferability.
Part I: Gravimetric methods
8.2
Gravity reductions and the geoid
The integrals of Stokes and of Vening Meinesz and similar formulas presup-
pose that the disturbing potential
T
is harmonic on the geoid, which implies
that there are no masses outside the geoid. This assumption - no masses
outside the bounding surface - is necessary if we wish to treat any problem
of physical geodesy as a boundary-value problem in the sense of potential
theory. The reason is that the boundary-value problems of potential theory
always involve harmonic functions, that is, solutions of Laplace's equation
∆
T
=0
.
(8-2)
This is equivalent to ∆
V
=0.Proof:
T
=
W
−
U
(
U
is the normal potential),
∆
W
=2
ω
2
outside the earth (density zero, only rotation, ∆
U
=2
ω
2
for
2
ω
2
= 0). Since then
∆
W
=2
ω
2
rather than zero by Eq. (2-9), it is not quite correct to call the
external gravity potential
W
harmonic as well, but we may nevertheless do
so for simplicity. No misunderstanding is possible.
We know, for instance, that the determination of
T
or
N
from gravity
anomalies ∆
g
may be considered as a third boundary-value problem (see
Sect. 1.13).
Since there are masses outside the geoid, they must be moved inside the
geoid or completely removed before we can apply Stokes' integral or related
∆
U
=2
ω
2
the same reason, hence ∆
T
=∆
W
−
−
