Geoscience Reference
In-Depth Information
2
distribution with three degrees of freedom).This suggests that the fitted curve
has frequencies that may be too low in the 20-50 % range. The shape of the
histogram obviously depends on the boundaries of the area selected for statistical
analysis. It depends even more strongly on cell size as illustrated in Fig.
12.48
in
which the relative frequency of zeros is reduced from 67 to 25 %. Later in this
section the probnormal frequency distribution will be fitted to these data.
A simplified fractal/multifractal analysis was performed on the larger Abitibi
data set using the method of moments. Multifractal analysis is appropriate if the
pattern being studied is self-similar or scale-independent. A (mono-) fractal can
arise as a special case of a self-similar multifractal. Suppose that a grid of cells with
length of cell side
ˇ
2
(3)
ˇ
0
:
05
(3)
ˇ
test gave
¼
14.7, to be compared with
¼
7.8 (95 % fractile of
is superimposed on a pattern, and that amount of pattern in a
cell is called the cell's “measure”
ʼ
є
. A self-similar multifractal can be character-
ized by its multifractal spectrum in which the fractal dimension
f
(
є
ʱ
) is plotted
є
ʱ
against
the singularity (or H¨ lder exponent)
ʱ
. The measure satisfies
ʼ
є
~
f(
ʱ
)
where ~ denotes proportionality, and the fractal dimension
f
(
ʱ
) satisfies
N
є
~
є
where
N
є
represents number of cells with singularity approximately equal to
.
The method of moments to determine the multifractal spectrum consists of three
consecutive steps. Initially, power moment sums are calculated for different cell
sizes. In our application, square cells with sizes of 10
ʱ
10 km, 20
20 km and
40
40 km were used. As unit for the measure (amount of acidic volcanics per
cell), decimal fraction per (10
10 km) cell was used. The power moment sums are
plotted against length of cell side using log-log paper. Figure
12.50
shows results
for power moments
q
between
1 and 5. Logarithms (base 10) were used to plot the
power moment sums, and the three cell sizes are labelled 1, 2, and 3 in Fig.
12.51
.
The underlying pattern would be multifractal if the power moment sums exhibit
straight line patterns on log-log paper. This condition seems to be satisfied in
Fig.
12.50
. The second step in the method of moments consists of assuming that
the straight line slopes represent multifractal mass exponents (
˄
or “tau”) of the
pattern for the power moments
q
. The first derivative of
˄
(
q
) with respect to
q
then
(
q
) representing the singularity.
Figure
12.51
is the plot of
ʱ
yields an estimate of
(
q
) versus
q
using the slope estimates of Fig.
12.50
.
The pattern of Fig.
12.51
is closely approximated by the straight line
˄
˄
(
q
)
¼
0.4149
0.41 representing a (mono-) fractal
with constant singularity of 0.41 rather than a multifractal with different values of
ʱ
and
f
(
q
0.4136. This would imply
ʱ
(
q
)
¼
). The final step in the method of moments consists of constructing the
multifractal spectrum that is a plot of
f
(
ʱ
ʱ
) versus
ʱ
using the relation:
f
(
ʱ
)
¼
q
ʱ
(
q
)
(
q
). In the application to Abitibi felsic metavolcanics, the multifractal
spectrum is reduced to a single spike representing a fractal. The intercept of the
straight line, which also is 0.41, can be regarded as the (constant) fractal dimension
f
(
˄
) of the pattern.
As in the example of Fig.
12.46
, the broken line in Figure
12.52
was derived
previously (Fig.
12.39
) by estimating
ʱ
from the mean and variance. In this
earlier application, the variance was estimated from the sample of 768 cell values.
ʲ
and
ˁ
Search WWH ::
Custom Search