Geoscience Reference
In-Depth Information
There are three different ways in which a multifractal spectrum can be
computed: histogram method, method of moments and direct determination. The
first two methods are described in Evertsz and Mandelbrot (
1992
). The histogram
method is intuitionally more appealing because it is easy to understand. However,
in practice it is better to use the method of moments with Legendre transform. This
method produces better results faster than the histogram method although the latter
can be useful in the study of frequency distributions of multifractals. The third
method (direct determination) developed by Chhabra and Jensen (
1989
) is useful
but will not be applied in this chapter.
Box 11.2: Derivation of the Multifractal Spectrum
Evertsz and Mandelbrot (
1992
) make a clear distinction between (1) H¨lder
lim
E
!
0
log
e
ʼ
f
B
x
ðÞ
g
exponent or singularity
α
ðÞ
¼
x
at a point
x
,and
log
e
E
log
e
ʼ
f
B
x
ðÞ
g
(2) “coarse” H¨lder exponent
α
¼
measured for a volume
log
e
E
B
x
(
) around the point
x
. The mass-partition function is:
ˇ
q
(
)
¼
E
E
¼
R
N
E
(
q
i
E
α
)
q
dx
with
N
E
(
¼
E
f
(
α
)
where
f
(
∑
N(
E
)
ʼ
α
)(
α
)
α
)representsfractal
Z
qαf
ðÞ
d
dimension. It follows that lim
E
!
0
ˇ
q
ðÞ
¼
α
keeping in mind that
α
E
¼
α
q
) also is a function of
q
. At the extremum:
∂
q
α
f
f
ðÞ
g
¼ 0and
ʴ
f
ðÞ
(
ʴq
¼
q
∂
α
), it follows that
ʴ˄
ðÞ
ʴα
q
for any moment
q
. Writing
˄
(
q
)
¼
q
α
f
(
α
¼
q
.The
multifractal spectrum satisfies:
f
(
α
)
¼
q
α ˄
(
q
). If the multifractal spectrum
f
(
α
) exists, the so-called codimension satisfies
C
1
¼
f
(
α
)-
E
where
E
is the
Euclidian dimension (
cf
. Evertsz and Mandelbrot
1992
).
11.2.1 Method of Moments
In practice, a feature such as element concentration in rock samples is measured in
blocks of different sizes. The mass-partition function
,
q
) then is the sum of all
measurements raised to the power
q
for blocks with cell edge
ˇ
(
E
. Terms such as
“mass-partition function” were borrowed from physical chemistry (see, e.g.,
Evertsz and Mandelbrot
1992
). The slopes of the lines on a log-log plot of partition
function against size measure are estimates of the mass exponent
E
˄
(
q
). The singu-
larity
α
(
q
) is the first derivative of
˄
(
q
) with respect to
q
, and the fractal dimension
satisfies
f
(
(
q
). The model of de Wijs results in a log-binomial
frequency distribution that converges to a lognormal distribution. The model of
de Wijs can be generalized in various ways. It is useful for simulating spatial
multifractal patterns. The tails of the negative binomial are thinner than those of
the lognormal. The high-value tail of the lognormal, in turn, is thinner than the tail
α
)
¼
q
α
(
q
)
˄
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