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Fig. 11.8 (a) Histogram
method applied to pattern of
n
¼
14; (b) ditto with
d
¼
0.4 and
n
¼
30. In both
diagrams (a and b), the
larger values belong to
limiting form of
f
(
α
) that
provides an upper bound
and would be reached
exactly for
n
¼
1
a
2.5
2
1.5
1
0.5
.
Difference with limiting
form decreases slowly for
increasing
n
. Note that
method of moments
provides estimates of
f
(
0
0
0.5
1
1.5
2
2.5
3
3.5
4
α
b
2.5
α
)
for
n
(Source:
Agterberg
2001
, Fig. 4)
!1
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
α
Fig.
11.8a
. For infinitely large value
n
, a limiting multifractal spectrum
f
(
α
) (for
n
) can be derived in analytical form. It is noted that it is paradoxical that the
corresponding frequency distribution does not exist because it has infinite variance
when the original variance formula of de Wijs is applied. However, values of
the limiting form of
f
(
¼
1
α
) (for
n
¼
1
) are shown in Fig.
11.8a
for comparison with
the
f
(
14. There are systematic discrepancies between the two
spectra of Fig.
11.8a
. However, at the extremes,
α
) values for
n
¼
log
2
{2/(1 + d)}
2
α
min
¼
¼
1.03 and
d)}
2
α
max
¼
0, the two spectra coincide. The other
point of equality occurs at the center where
f
(
log
2
{2/(1
¼
3.47 with
f
(
α
)
¼
α
)
¼
2.
30. The histogram values are closer to
the theoretical limit values in Fig.
11.8b
, but it is obvious that convergence is
exceedingly slow. On the other hand, the 4-step method immediately resulted
in
f
(
Figure
11.8b
shows similar results for
n
¼
) values (Fig.
11.8c
) approximately coinciding with the limit values of
Fig.
11.8a, b
, illustrating that the method of moments is to be preferred for
derivation of the limiting form of the multifractal spectrum
f
(
α
).
Because the realization of the model of de Wijs for a specific value of
n
is
discrete, the multifractal spectrum can be readily interpreted. Each
f
(
α
) (for
n
¼
1
) value
represents the fractal dimension of a subset of cells with the same element
α
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