Gravity and magnetic methods - Geophysics for the Mineral Exploration Geoscientist - page 127

3.7.4.3 Tilt derivatives
Shallow sources produce large amplitudes in the vertical
and horizontal gradients. The large amplitude range pre-
sents a problem for display. Ratios of the derivatives of
each class of source have similar amplitudes, so the reso-
lution of both classes can be balanced by dividing the
vertical derivative by the amplitude of the total horizontal
derivative. Furthermore, the ratio can be treated as an
angle and the inverse tangent function applied to attenuate
high amplitudes (see Amplitude scaling in Section 2.7.4.4 ) .
This is known as the tilt derivative (TDR) (see Miller and
Singh, 1994 ) , and at a location (x, y) it is given by:

geophysics, rock density is a bulk property. It depends on
the entire mineral assemblage and, in relevant rock types,
their total porosity.
We present here an overview of the density of various
rock types and geological environments, at the same time
considering geological processes that produce changes in
the density of minerals and rocks.

3.8.1 Densities of low-porosity rocks

The bulk density of a rock ( ρ bulk ), i.e. the entire rock

including both matrix and pore contents, can be calculated
from the densities and proportions of its constituents as
follows:

2
4

3
5 ð

s

,
∂

2
+

2

f
dz

∂ f f

∂

tan 1 TDR ð x y Þ¼ 3 : 24 Þ

,

∂ x y

∂

ρ i

X

n

V i
V

ρ bulk ¼ ð : Þ

3 25

where f is either the gravity or the magnetic field.

For gravity anomalies and vertical magnetised bodies the
tilt derivative is positive over the source and negative
outside of it. Its form mimics the 2D shape of the anomaly
source with the zero-value contour line delineating the
upper boundaries of the source ( Figs. 3.25g and 3.26h ).
Its shape is more complex for non-polar magnetisation.
The amplitude scaling properties of the enhancement
are well demonstrated in the Broken Hill magnetic data
in areas of weak TMI, e.g. (5) and (6) in Fig. 3.28g ; com-
pare these with the responses of

i ¼ 1

where the rock comprises n components and V i = V is the

volume fraction of the ith component with density ρ i .

Where a rock has minimal porosity, it is the density and
relative amounts of the constituent minerals which control
its density.
The density of a particular mineral species depends on
the masses of its constituent atoms and how closely they
are packed together within its crystal lattice. The majority
of rock-forming minerals contain the elements Al, Fe, Mg,
Ca, K, Na, C, O and Si which have comparatively low mass
numbers. The largest is that of Fe with a mass number of
about 56 whilst the others have mass numbers between 12
and about 40. Metal sulphides and oxides contain heavier
elements, such as Zn, Ni, Cu and Pb, whose mass numbers
range from about 59 to 207 and are therefore expected to
be denser than the rock-forming minerals, as is confirmed
by Fig. 3.29 .
Minerals that have a range of compositions, in particu-
lar in their cations, for example the feldspars, olivines
etc., will vary in density according to their composition.
Figure 3.31a shows density variations in olivine. The Fe-
rich varieties have signi cantly higher densities, re ecting
the high mass number of Fe. A further cause of density
variations are impurities in the crystal lattice, causing
even those minerals that have a speci c composition
to vary in density. Clearly some variation in the density
of a particular mineral species is almost always to be
expected.
The influence of crystal structure on density is demon-
strated by the carbon allotropes. The tightly packed

the other derivative

enhancements.

3.8 Density in the geological environment

An understanding of the density of geological materials
and geological causes for variations in density is crucial
for making geologically realistic interpretations of gravity
data.
Rock density affects the Earth field, but in the

wider geophysical context density is also an important
control on radiometric and seismic responses (see Sections
4.2.3 and 6.6 ). Figures 3.29 and 3.30 show the variations in
density for materials commonly encountered in the geo-
logical environment. By convention, the average density of
the continental crust is taken to be 2.67 g/cm 3 (see Hinze,
2003 ) , which is consistent with the data presented here.
Densities fall within a very limited range when compared
with the other physical properties relevant to geophysics,
notably magnetic and electrical properties (see Sections 3.9
and 5.3 ) which vary by many orders of magnitude. Also,
unlike most other physical properties

' s gravity

important in