GRAVITY METHOD - Field Geophysics - page 62

Example 2.3
Interpreting the anomaly in Figure 2.12 as due to a roughly cylindrical
air-filled cavity in rock of density 2.5 Mg m − 3 and working in metres:
ρ =− 2.5 and the anomaly amplitude = g = 0.05 mGal.
Anomaly half-width = 2 m. Therefore, the depth to the cylinder centre =
h = 2m.
r 2 = ( g × h ) / (0 . 04 × ρ ) = (0 . 05 × 2) / (0 . 04 × 2 . 5) = 1 .

i.e. r = 1m.

2.5.1 The Bouguer plate
The Bouguer plate provides the simplest possible interpretational model.
An easily memorised rule-of-thumb is that the gravity effect of a slab of
material 1 km thick and 1.0 Mg m − 3 denser than its surroundings is about
40 mGal. This is true even when the upper surface of the slab is some distance
below the reading point (as in the case of the second layer in Figure 2.11),
provided that the distance of the station from the nearest edge of the slab is
large compared with the distance to its lower surface. The effect varies in
direct proportion to both thickness and density contrast.

2.5.2 Spheres and cylinders
Less extensive bodies can be modelled by homogeneous spheres or by
homogeneous cylinders with circular cross-sections and horizontal axes. If,
in Figure 2.12, the anomaly is due to a sphere, radius r measured at a point
immediately above its centre, the maximum anomalous field is:

g = 4 π.ρ. Gr 3 / 3 h 2

The factor 4 π G / 3 is about 28, if lengths are measured in kilometres, or
0.028 if lengths are measured in metres. The depth, h ,ofthecentreofthe
sphere is roughly equal to four-thirds of the half-width of the anomaly.
If, however, the source in Figure 2.12 can be modelled as an infinite hor-
izontal cylinder of circular cross-section (an example of a two-dimensional
source), the maximum field is:
g = 2 π.ρ. Gr 2 / h

The factor 2 π G is about 40, if lengths are measured in kilometres, or 0.04
if lengths are measured in metres. The depth, h , of the axis of the cylinder
is equal to the half-width of the anomaly.