Modern views on the determination of the figure of the earth - Physical Geodesy - page 315

6. Finally, the original height anomalies ζ are obtained by considering the
indirect effect according to

ζ = ζ c + δζ . (8-103)

The purpose of this somewhat complicated procedure is to make use of
the well-known advantages of Bouguer and isostatic anomalies. The Bouguer
anomalies, and even more so the isostatic anomalies, are smoother and more
representative than the free-air anomalies and can, therefore, be interpolated
more easily and more accurately.
The isostatic gravity anomalies ∆ g c in the new sense are thus quite anal-
ogous to the conventional isostatic anomalies; accordingly for any other type
of gravity reduction. The difference is that now the isostatic anomalies, etc.,
refer to the physical surface of the earth as well as the free-air anomalies. If
the isostatic anomalies in this new sense are analytically continued from the
earth's surface down to the geoid, then isostatic anomalies in the conven-
tional sense are obtained. Nowadays, in view of the “remove-restore princi-
ple”, one speaks usually of topographic-isostatic reduction while continuing
to speak of isostatic anomalies.
Hence, the isostatic anomalies according to the conventional definition
(at sea level) and those according to the new definition (at ground level) are
related through analytical continuation . This fact leads to two conclusions.
First, the difference between the isostatic anomalies according to these two
definitions will be small, because the distance along which this analytical
continuation is made is only the height above sea level and because the
isostatic reduction achieves a strong smoothing of the anomalous gravity
field. This difference is considerably smaller than the corresponding differ-
ence between free-air anomalies at ground level and at sea level. This fact
clearly provides a computational advantage if isostatic anomalies are used
in a formula such as (8-74).
Second, we obtain a relation between the conventional and the modern
use of gravity reduction if the method of downward continuation, as discussed
in the preceding section, is applied for obtaining the height anomalies. As we
have just seen, the gravity anomalies ∆ g c ∗ at sea level, obtained by downward
continuation of the isostatic ground-level anomalies ∆ g c , are identical with
the isostatic anomalies in the conventional sense. Hence, we obtain on the
one hand the height anomalies by

∆ g c ∗ S ( R + h, ψ ) dσ + δW
γ

R
4 πγ

ζ = (8-104)

ground

σ