Geoscience Reference
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the interior of the earth in such a way that the exterior potential remains
unchanged. This is not unlike the Rudzki reduction, where
the geoid
remains
unchanged. Whereas the Rudzki reduction is, however, “constructive” in the
sense that a way of performing it can be described, our present interpreta-
tion of free-air reduction as harmonic continuation is nonconstructive, it is
an “improperly posed” inverse problem; cf. Anger and Moritz (2003) and
www.inas.tugraz.at/forschung/InverseProblems/AngerMoritz.html, as well
as Fig. 8.10.
S
topographic masses
(a)
geoid
W
W
( = )
W
(
W
¼%
= -4
)
W=
0
S
S
(b)
harmonic
geoid
(c)
cogeoid
W
harmonic
W
c
W
c
(
W
harmonic
= 0)
W
( = )
W
W
c
( = )
W
( = )
c
W
harmonic
=W
0
W=
0
Fig. 8.10. (a) Geoid and topographic masses, (b) mass displacement in gravity
reduction, (c) “ill-defined” mass displacement in free-air reduction as harmonic
continuation
Important remark
The isostatic gravity anomalies and the topographic-isostatically reduced
deflections of the vertical (Sect. 8.14) are fundamental for least-squares col-
location in mountain areas (Sect. 11.2).
Thus, the spatial approach due
to Molodensky is basic even for least-squares collocation!
