Geoscience Reference
In-Depth Information
Integrating this relation yields the difference of the height anomaly
B
B
∆
g
γ
ζ
B
−
ζ
A
=
−
εds
−
dh
;
(8-150)
A
A
the gravity anomaly ∆
g
refers to the earth's surface according to (8-128).
The first term on the right-hand side represents the Helmert integral (8-143)
of the surface deflection
ε
, and the second term is Molodensky's correction to
the Helmert integral, necessary to obtain height anomalies. This correction
depends on the gravity
g
at the earth's surface.
8.14
Topographic-isostatic reduction of vertical
deflections
For the reasons mentioned at the end of the preceding section, it is nat-
ural to try and find a way which makes use of the clear advantages of the
topographic-isostatic reduction but avoids the problems inherent in a free-air
reduction from the surface point
P
to the geoidal point
P
0
.
In Sect. 8.9, we have treated the reduction of gravity from the modern
point of view. The second formula of (8-95) is
g
c
=
g − δg .
(8-151)
Everything is referred to the ground point
P
,and
δg
=
δg
TI
is the effect
of gravity reduction on
g
,alsoat
P
.Inthe
topographic-isostatic reduction
which we use here exclusively
, it is the gravitational attraction of the to-
pography minus the gravitational attraction of the compensating isostatic
masses, topography minus isostasy.
To get the topographic-isostatic gravity anomaly, we subtract normal
gravity
γ
, also referred to ground level, more precisely, to the corresponding
telluroid point
Q
.Thus,
∆
g
c
=∆
g
−
δg
TI
.
(8-152)
The explanation is trivial: you are standing at point
P
and watch how the
topography is removed to fill the isostatic mass deficits, but by a miracle
you are still hovering at
P
, now in “free air”.
Application to deflections of the vertical
The gravity anomaly is only one component of the anomalous gravity vector,
the other two being the vertical components
ξ
and
η
, both, of course, mul-
tiplied by
γ
to get the dimensions right.
Thus, ξ and η can be isostatically
reduced in exactly the same way.
