Geoscience Reference
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according to (8-75) and (8-103), and on the other hand the geoidal undula-
tions by
∆
g
c
∗
S
(
ψ
)
dσ
+
δW
γ
R
4
πγ
0
N
=
(8-105)
geoid
σ
according to the ordinary Stokes formula applied to ∆
g
c
∗
and (8-5). Since
the height anomalies refer to the elevation
h
, the function
S
(
R
+
h, ψ
) replaces
in (8-104) the original function of Stokes
S
(
ψ
)
≡ S
(
R, ψ
), which occurs in
(8-105) because the geoidal undulation refers to zero elevation. We could
use
γ
0
in (8-104) as well. Summarizing, we have the following steps:
1. Computation of the free-air anomaly at ground level, ∆
g
, according to
(8-23).
2. Computation of the isostatic anomaly at ground level, ∆
g
c
, according
to (8-101).
3. Downward continuation of ∆
g
c
by (8-54), where ∆
g
and ∆
g
harmonic
are
replaced by ∆
g
c
and ∆
g
c
∗
. The resulting isostatic anomalies at sea
level, ∆
g
c
∗
, may now be used for two purposes: either for
4a. the determination of the physical surface of the earth according to
(8-104), or for
4b. the determination of the geoid according to (8-105).
An error in the assumed density of the masses below the earth's surface
affects the geoidal undulations as determined from (8-105) but does not
influence the height anomalies resulting from (8-104). This is clear because
a wrong guess of the density means only that the masses above sea level are
not completely removed, which is no worse than not removing them at all
when using free-air anomalies.
This method is of particular interest for practical computations, as we
will see later. It has become popular by the name
“remove-restore method”
,
invented by K. Colic and others, see Sect. 11.1.
An almost final remark on free-air reduction
The apparently so simple topic of free-air reduction in reality is formidably
complex and complicated. Therefore, it is not possible to treat it in one
block. The problem is rather like a mountain which can only be investigated
by accessing it from various sides. An initial glance has been given as early
as in Chap. 3, and the reader is asked to return to the paragraph “The
many facets of free-air reduction” in Sect. 3.9. Now it is much easier to un-
derstand the remarks made there. What we now understand as harmonic
continuation offers a possibility to
interpret free-air reduction as a mass-
transporting gravity reduction
: the topographic masses are transported into
