Geology Reference
In-Depth Information
to employ techniques which perform the iteration
automatically.
The most flexible of such methods is
non-linear opti-
mization
(Al-Chalabi 1972). All variables (body points,
density contrasts, regional field) may be allowed to vary
within defined limits. The method then attempts to
minimize some function
F
which defines the goodness
of fit, for example
0
Observed
Calculated
-100
-200
-300
-400
-500
n
10 km
Â
)
2
(
A South
North A'
F
=
g
-
g
D
D
obs
calc
i
i
i
=
1
-0.10
12 km
-0.16 Mg m
-3
-0.13
where
D
g
obs
and
D
g
calc
are series of
n
observed and calcu-
lated values.
The minimization proceeds by altering the values
of the variables within their stated limits to produce a
successively smaller value for
F
for each iteration. The
technique is elegant and successful but expensive in
computer time.
Other such automatic techniques involve the simu-
lation of the observed profile by a thin layer of variable
density. This
equivalent layer
is then progressively ex-
panded so that the whole body is of a uniform, specified
density contrast. The body then has the form of a series
of vertical prisms in either two or three dimensions
which extend either above, below or symmetrically
around the original equivalent layer. Such methods are
less flexible than the non-linear optimization technique
in that usually only a single density contrast may be
specified and the model produced must either have a
specified base or top or be symmetrical about a central
horizontal plane.
Fig. 6.23
A two-dimensional interpretation of the gravity
anomaly of the Bodmin Moor granite, southwest England. See
Fig. 6.27 for location. (After Bott & Scott 1964.)
northerly increase in the density of the granite; a possible
alternative, however, would be a northerly thinning of a
granite body of constant density contrast.
Two-dimensional methods can sometimes be extend-
ed to three-dimensional bodies by applying end-correc-
tion factors to account for the restricted extent of the
causative body in the strike direction (Cady 1980). The
end-correction factors are, however, only approxima-
tions and full three-dimensional modelling is preferable.
The gravity anomaly of a three-dimensional body
may be calculated by dividing the body into a series of
horizontal slices and approximating each slice by a poly-
gon (Talwani & Ewing 1960). Alternatively the body
may be constructed out of a suite of right rectangular
prisms.
However a model calculation is performed, indirect
interpretation involves four steps:
1.
Construction of a reasonable model.
2.
Computation of its gravity anomaly.
3.
Comparison of computed with observed anomaly.
4.
Alteration of model to improve correspondence of
observed and calculated anomalies and return to step 2.
The process is thus iterative and the goodness of fit be-
tween observed and calculated anomalies is gradually
improved. Step 4 can be performed manually for bodies
of relatively simple geometry so that an interpretation
is readily accomplished using interactive routines on a
personal computer (Götze & Lahmeyer 1988). Bodies of
complex geometry in two- or three-dimensions are not
so simply dealt with and in such cases it is advantageous
6.11 Elementary potential theory and
potential field manipulation
Gravitational and magnetic fields are both potential
fields. In general the potential at any point is defined as
the work necessary to move a unit mass or pole from an
infinite distance to that point through the ambient field.
Potential fields obey Laplace's equation which states that
the sum of the rates of change of the field gradient in
three orthogonal directions is zero. In a normal Carte-
sian coordinate system with horizontal axes
x
,
y
and a
vertical axis
z
, Laplace's equation is stated
2
2
2
∂
∂
A
x
∂
∂
A
y
∂
∂
A
z
+
+
=
0
(6.23)
2
2
2
