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This generating process has the lognormal distribution as its end product. There
is an obvious similarity between generating processes and the multiplicative cas-
cade processes to be discussed later in this topic in the context of fractals and
multifractals. The Vistelius model of lognormality will be applied in the next
section.
3.2.2 Muskox Layered Intrusion Example
Vistelius (
1960
, p. 11) argued as follows. Suppose that a geological process, at any
time, yields a different population of values. If a given part of the Earth's crust is
sampled, a set of values is obtained which may reect various stages of the same
geological process. In some places, the process may have developed further than in
other places. The sampled population then consists of a mixture of many separate
populations that are, however, interrelated by a single process. Suppose that for
each of these populations, the standard deviation
˃
is proportional to the mean
ʼ
.
This is equivalent to assuming that the coefcient of variation
is constant. In
that situation, the sampled population can have a positively skew frequency distri-
bution because the subpopulations with relatively large means and standard devi-
ations will generate a long tail of large values. The frequency distribution
f
(
x
) then
satises
f
(
x
)
ʳ
¼
˃
/
ʼ
w
i
·
f
i
where each
f
i
(
x
) is the distribution representing the process at
stage
i
;
w
i
is a weighting factor for that stage. It represents the proportion by which
the subpopulation
f
i
(
x
) occurs in the sampled population.
In order to test this theory, Vistelius (
1960
) compiled averages and standard
deviations for phosphorus (as weight percent P
2
O
5
) in granitic rock from various
areas. He found that average and standard deviation are positively correlated with a
correlation coefcient of 0.56. Individual distributions
f
(
x
) are approximately
normal but
¼
ʣ
the joint distribution
f
i
(
x
) with all values lumped together
is
positively skew.
The Vistelius model was tested for the element copper in various layers of the
1,175 my old Muskox layered ultramac gabbroic intrusion, District of Mackenzie,
northern Canada. The layers were formed by crystallization differentiation of
basaltic magma (Smith
1962
; Smith and Kapp
1963
). Individual layers are approx-
imately homogeneous with regard to the major rock-forming minerals. Layers that
are rich in olivine, such as dunite and peridotite, tend to occur near the bottom of the
sequence of layers in the Muskox intrusion, whereas gabbroic layers occur closer to
the top. The sequence is capped by an acidic layer of granophyric composition.
Figure
3.8a
shows the frequency distribution of 116 copper values from rock
samples taken at the surface on a series of gabbros with increasing granophyre
content. These gabbros are situated between a clinopyroxenite layer (at the bottom)
and a mac granophyre layer (at the top). A logarithmic scale or ratio scale was used
in this gure for plotting the trace element copper in ppm (parts per million). A
straight line for logarithms of values indicates approximate lognormality in Fig.
3.8a
.
However, the 116 copper values are for specimens of three different types of gabbros
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