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is particularly appropriate (see also Sect. 6.5).
Practically it is usually adequate to consider only the linear approxima-
tion by using (8-57). In many cases it is even possible to neglect the correc-
tion
( g/∂h ) h , identifying the sea-level free-air anomalies ∆ g harmonic with
the corresponding ground-level anomalies ∆ g . In agreement with Sect. 3.9,
these free-air anomalies ∆ g harmonic =∆ g may also be considered approxima-
tions to condensation anomalies in the sense of Helmert. This approximation
is particularly sucient for the external gravity field, spherical harmonics,
and geoidal undulations or height anomalies. For deflections of the vertical,
it is often necessary to use a more careful approach, such as the consideration
of the indirect effect with mass-transporting gravity reductions (Sect. 8.2)
or the modern methods of Sect. 8.9.
In high and steep mountains, the approach of Molodensky and others
through free-air anomalies encounters practical diculties, such as unrelia-
bility of interpolation, large corrections, and other computational problems.
To avoid this, isostatic reduction in the modern sense shoud be used. Thus
the clash between “conventional” (geoid) and “modern” (Molodensky-type)
ideas gives way to an important synthesis. For another synthesis, see least-
squares collocation in Sects. 10.2 and 11.2.
For further study, especially of the historic aspects, the reader is referred
to the topic by Molodenski et al. (1962) and the M.S. Molodensky Anniver-
sary Volume edited by Moritz and Yurkina (2000).
Part II: Astrogeodetic methods according to
Molodensky
8.12
Some background
The computation of a detailed geoid, or of a detailed gravity potential field,
in limited areas, especially in mountainous regions, has not been very much
in the focus of attention recently. There may be various reasons for this.
For decades now, global geoid determinations, either from satellite data
or from a combination of satellite and gravimetric data have been in the
center of interest (Lerch et al. 1979, Reigber et al. 1983, Rapp 1981). Even
(almost) purely gravimetric global and local geoids have been successfully
computed (March and Chang 1979), between the classic Heiskanen (1957)
and the modern local geoid (Kuhtreiber 2002 b). An excellent recent reference
volume is that by Tsiavos (2002).
Over the oceans, the geoid is now known to an accuracy of perhaps a
few centimeters, due to satellite altimetry. Unfortunately, satellite altimetry
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