Geoscience Reference
In-Depth Information
cogeoid
,tothepoint
P
c
(hence the notation with upper index c). This
gives the boundary value of gravity at the cogeoid,
g
c
.
5. The shape of the cogeoid is computed from the reduced gravity anoma-
lies
∆
g
c
=
g
c
−
γ
(8-4)
by Stokes' formula, which gives
N
c
=
QP
c
.
6. Finally, the geoid is determined by considering the indirect effect. The
geoidal undulation
N
is thus obtained as
N
=
N
c
+
δN .
(8-5)
Remark
. At first sight it may seem that the masses between the geoid
and the cogeoid should be removed if the cogeoid happens to be below the
geoid, because Stokes' formula is applied to the cogeoid. However, this is
not necessary, and therefore we need not be concerned with a “secondary
indirect effect”. The argument is a little too technical to be presented here;
see Moritz (1965: p. 26).
In principle, every gravity reduction that gives boundary values at the
geoid is equally suited for the determination of the geoid, provided the in-
direct effect is properly taken into account. Thus, the selection of a good
reduction method should be made from other points of view, such as the
geophysical meaning of the reduced gravity anomalies, the simplicity of com-
putation, the feasibility of interpolation between the gravity stations, the
smallness or even absence of the indirect effect, etc. (see Sect. 3.7).
The
Bouguer reduction
corresponds to a complete removal of the ex-
ternal masses. In the
isostatic reduction
, these masses are shifted vertically
downward according to some theory of isostasy. In Helmert's
condensation
reduction
, the external masses are compressed to form a surface layer on
the geoid. The Bouguer reduction and especially the isostatic reduction (in
modern terminology
topographic-isostatic reduction
) are used as auxiliary
quantities for computational purposes, especially to facilitate interpolation.
The
free-air anomaly
is nowadays used in three senses:
1. at ground level (on the physical surface of the earth) it is simply the
gravity anomaly in the sense of Molodensky (Sect. 8.4);
2. at sea level it may be identified with the analytical continuation of
the Molodensky anomaly from ground down to sea level. This will
be considered in detail in Sect. 8.6. A final review will be found in
Sect. 8.15.
