Geology Reference
In-Depth Information
the poles exceeds gravity at the equator by some
51 860 gu, with the north-south gravity gradient at
latitude
f
being 8.12 sin 2
f
gu km
-1
.
Clairaut's formula relates gravity to latitude on the ref-
erence spheroid according to an equation of the form
rects for the decrease in gravity with height in free air
resulting from increased distance from the centre of
the Earth, according to Newton's Law. To reduce to
datum an observation taken at height
h
(Fig. 6.12(a)),
FAC
=
3 086
.
h
gu
(
h
in metres
)
)
g
=+
g
(
1
k
sin
2
-
k
sin
2
2
f
f
(6.11)
0
1
2
f
The FAC is positive for an observation point above
datum to correct for the decrease in gravity with
elevation.
The free-air correction accounts solely for variation
in the distance of the observation point from the centre
of the Earth; no account is taken of the gravitational
effect of the rock present between the observation point
and datum. The
Bouguer correction
(BC) removes this
effect by approximating the rock layer beneath the
observation point to an infinite horizontal slab with a
thickness equal to the elevation of the observation above
datum (Fig. 6.12(b)). If
r
is the density of the rock, from
equation (6.8)
where
g
f
is the predicted value of gravity at latitude
f
,
g
0
is the value of gravity at the equator and
k
1
,
k
2
are con-
stants dependent on the shape and speed of rotation of
the Earth. Equation (6.11) is, in fact, an approximation
of an infinite series.The values of
g
0
,
k
1
and
k
2
in current
use define the International Gravity Formula 1967
(
g
0
= 9 780 318 gu,
k
1
= 0.0053024,
k
2
= 0.0000059;
IAG 1971). Prior to 1967 less accurate constants were
employed in the International Gravity Formula (1930).
Results deduced using the earlier formula must be mod-
ified before incorporation into survey data reduced
using the Gravity Formula 1967 by using the relation-
ship
g
f
(1967) -
g
f
(1930) = (136 sin
2
f
- 172) gu.
An alternative, more accurate, representation of the
Gravity Formula 1967 (Mittermayer 1969), in which the
constants are adjusted so as to minimize errors resulting
from the truncation of the series, is
BC
=
2
Gh
=
0 4191
.
gu
in metres
h
p
r
r
(
-
3
)
h
,
in Mg m
r
On land the Bouguer correction must be subtracted,
as the gravitational attraction of the rock between obser-
vation point and datum must be removed from the
observed gravity value. The Bouguer correction of sea
surface observations is positive to account for the lack
of rock between surface and sea bed. The correction
is equivalent to the replacement of the water layer by
material of a specified rock density
r
r
. In this case
(
2
g
f
=
9 780 318 5 1
.
+
0 005278895
.
sin
f
4
)
+
0 000023462
.
sin
gu
f
This form, however, is less suitable if the survey results
are to incorporate pre-1967 data made compatible with
the Gravity Formula 1967 using the above relationship.
The value
g
f
gives the predicted value of gravity at
sea-level at any point on the Earth's surface and is sub-
tracted from the observed gravity to correct for latitude
variation.
BC
=
2
pr r
G
-
)
z
(
r
w
where
z
is the water depth and
r
w
the density of water.
The free-air and Bouguer corrections are often ap-
plied together as the
combined elevation correction
.
The Bouguer correction makes the assumption that
the topography around the gravity station is flat. This is
6.8.3 Elevation corrections
Correction for the differing elevations of gravity stations
is made in three parts.The free-air correction (FAC) cor-
(a)
(b)
(c)
B
A
A
h
h
Datum
Fig. 6.12
(a) The free-air correction for an observation at a height
h
above datum. (b) The Bouguer correction.The shaded region
corresponds to a slab of rock of thickness
h
extending to infinity in both horizontal directions. (c) The terrain correction.
