Geoscience Reference
In-Depth Information
8 Modern views on the
determination of the figure of
the earth
8.1
Introduction
In the preceding chapters we have usually followed what might be called the
conservative approach to the problems of physical geodesy using classical
observations. The geodetic measurements - astronomical coordinates and
azimuths, horizontal angles, gravity observations, etc. - are reduced to the
geoid, and the “geodetic boundary-value problem” is solved for the geoid by
means of Stokes' integral and similar formulas. The geoid then serves as a
basis for establishing the position of points of the earth's surface.
The advantage of this approach is that the geoid is a level surface, capable
of a simple definition in terms of the physically meaningful and geodetically
important potential
W
. The geoid represents the most obvious mathematical
formulation of a horizontal surface at sea level. This is why the use of the
geoid simplifies geodetic problems and makes them accessible to geometrical
intuition.
The disadvantage is that the potential
W
inside the earth, and hence
the geoid
W
=
constant, depends on the density
because of Poisson's
Eq. (2-9),
4
πG
+2
ω
2
.
∆
W
=
−
(8-1)
Therefore, in order to determine or to use the geoid, the density of the
masses at every point between the geoid and the ground must be known, at
least theoretically. This is clearly impossible, and therefore some assumptions
concerning the density must be made, which is unsatisfactory theoretically,
even though the practical influence of these assumptions is usually rather
small.
For this reason it is of basic importance that
M.S. Molodensky
in 1945 was
able to show that the physical surface of the earth can be determined from
geodetic measurements alone, without using the density of the earth's crust.
This requires that the concept of the geoid be abandoned. The mathematical
formulation becomes more abstract and more dicult. Both the gravimetric
method and the astrogeodetic method can be modified for this purpose. The
gravity anomalies and the deflections of the vertical now refer to the ground,
