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approximated by the second order difference method. The mass exponent was
determined for pairs of closely spaced values
q
0.001 and successive differ-
ences between two of these values divided by 0.002. A monofractal would show
as a horizontal sequence of points on this diagram.
4. Finally, because it can also be shown that
f
(
α
q
α
(
q
)
˄
(
q
), the multifractal
)
¼
spectrum is obtained with
f
(
α
α
(Fig.
11.5c
). The points plotted
in this diagram coincide with a theoretical limiting curve
f
(
) as a function of
α
) for
n
!1
. The
maximum value of this curve is equal to 2. A monofractal would show as a spike
in the multifractal spectrum. If the object would show a narrow peak, it would
still be multifractal but close to monofractal.
A multifractal case history study in 2-D was provided by Cheng (
1994
) for gold in
the Mitchell-Sulphurets area. The example of Fig.
11.6a
shows how the multifractal
spectrum was obtained for 100 ppb Au cutoff. Different grids with square cells
measuring on a side were superimposed on the study area. The average gold value
was determined for each grid cell containing one or more samples with Au
100 ppb.
Each average value was raised to the power
q
. The sum of the powered values
satisfies a straight-line relationship for any
q
in a multifractal model. This aspect is
verified in Fig.
11.6a
. The slopes of many best-fitting straight lines (including those in
Fig.
11.6a
) are shown as the function
>
˄
(
q
) in Fig.
11.6b
. The first derivative of
˄
(
q
)
with respect to
q
gives
α
(
q
), and the multifractal spectrum
f
(
α
) follows from the
relation
f
(
(
q
) (solid line in Fig.
11.6c
).
The multifractal nature of the gold deposits in the Mitchell-Sulphurets area is
shown in the spectra for different cutoff values in Fig.
11.6c, d
. A fractal model
would have resulted in a spectrum consisting of a single spike characterized by two
constants: the fractal dimension
f
(
α
)
¼
q
α
(
q
)
˄
. The four spectra of
Fig.
11.6c, d
are approximately equal on the left side, which is representative of the
largest concentration values. The point where the multifractal spectrum reaches the
α
α
) and the singularity
α
-axis and the slope of the curve at this point together determine the approximate
area-concentration power-law relation on the right side in Fig.
10.4d
(Agterberg
et al.
1993
; Cheng et al.
1994a
). The maximum value of
f
(
α
) in Fig.
11.6c, d
decreases with increasing cutoff value. In general, if
f
(
2 in 2-D space, it can be
assumed that the multifractal measure is defined on fractal support (
cf
. Feder
1988
).
The preceding four-step method also can be applied to 1-D or 3-D objects.
In Fig.
11.7
the method is applied to the 118 channel samples from the Pulacayo
Mine (original example from Cheng
1994
). A measure
α
)
<
ʼ
i
(
)
¼
E
·x
i
(
) where
x
i
(
),
E
E
E
i
) ranging from 2 to 30 m (total length is 234 m). Some
results of estimating the mass-partition function with
q
ranging from - 34 to
34 are shown in Fig.
11.7a
. The straight lines were fitted by ordinary least squares.
The slopes
¼
1,
...
, 118, for cell sizes (
E
˄
(
q
) of all fitted straight lines with
q
ranging from
35 to 50 are shown
in Fig.
11.7b
. These results include
˄
(0)
¼
0.976
0.011 where the uncertainty is
expressed using the standard deviation (
s
).
˄
(0) represents the box-counting
dimension. It is noted that the slope for
q
¼
1 is approximately equal to 0. This is
as it should be because
0 represents the principle of conservation of total
mass. The second-order mass exponent
˄
(0)
¼
0.019 is important for
geostatistical modeling and will be used extensively in Sect.
11.4
.
˄
(2)
¼
0.979
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