Geoscience Reference
In-Depth Information
3. The free-air anomaly can be theoretically interpreted as an approxi-
mation of the classical condensation anomaly in the sense of Helmert
(Sect. 3.9). This is one of the interpretations of the frequent practice to
simply apply Stokes' formula to the classical free-air anomaly, where
only the standard normal free-air reduction is applied to measured
gravity
g
, see Eq. (8-6) below.
This is pretty rigorously the gravity anomaly in the sense of Molo-
densky (item 1 above), so there is another interpretation of this fre-
quent practice: it is a (conscious or unconscious) use of Molodensky's
method in the zero approximation (i.e., only Stokes' formula without
Molodensky correction
g
1
, see Sect. 8.6). Of course, this works only in
a reasonably flat terrain.
Important remark.
Curiously enough, it helps if the terrain correction
(Sect. 3.4) is applied; this is explained in Moritz (1980 a: Sect. 48) as
some kind of Molodensky correction
g
1
and in Moritz (1990: p. 244)
by isostatic reduction.
Also the
Poincare-Prey reduction
is quite different (Sect. 3.5). It gives
the actual gravity inside the earth. It does not give boundary values but is
used for orthometric heights (Chap. 4).
In all reduction methods it is necessary to know the density of the masses
above the geoid. In practice, this involves some kind of an assumption - for
instance, putting
=2
.
67 g cm
−
3
. A second assumption is usually made in
the free-air reduction, which is part of the reduction of gravity to the geoid:
the actual free-air gravity gradient is assumed to be equal to the normal
gradient
∂γ
∂h
=
0
.
3086 mgal m
−
1
.
−
(8-6)
These two assumptions falsify our results, at least theoretically.
The second assumption can be avoided by using the actual free-air gradi-
ent as computed by the methods of Sect. 2.20. The anomalies ∆
g
to be used
in formula (2-394) must be gravity anomalies reduced to the geoid: gravity
g
after steps l and 2 of the above description, minus normal gravity
γ
on the
ellipsoid. This presupposes that in step 2 a preliminary free-air reduction
using the normal gradient has been applied first.
Deflections of the vertical
The indirect effect affects the deflection of the vertical as well as the geoidal
height. We have found
N
=
N
c
+
δN ,
(8-7)
