Geoscience Reference
In-Depth Information
component with standard deviation of 0.25 was retrieved after about
k
¼
120
iterations. It corresponds to the line for
k
120 in Fig.
12.29
. Other values of
k
reproduced standard deviations for the white noise component that are either too
large or too small.
The solid line in Fig.
12.29
can be regarded as a smoothed version of the original
concentration values and
c
k
patterns for
k
¼
¼
1, 5 and 10 shown in the same diagram.
The experiment indicates that
k
¼
120 represents an approximate optimum upper
limit for the number of iterations to be used in the Chen algorithm. Further increases
of
k
yield patterns of
c
k
that are smoother, do not seem to provide a good fit with
variances that are smaller than 0.25. Ultimately, with
k
approaching infinity, the
pattern of
c
k
would become a straight line as in the previous application to the
Pulacayo orebody (Agterberg
2012a
, Fig. 19). The computer experiment described
in this section indicates that full convergence is not desirable in all applications.
12.7 Universal Multifractals
Already in the 1980s, Schertzer and Lovejoy (
1985
) pointed out that the binomial/
p
model (model of de Wijs) can be regarded as a “micro-canonical” version of their
α
-model in which the strict condition of local preservation of mass is replaced by
the more general condition of preservation of mass within larger neighborhoods
(preservation of ensemble averages). Cascades of this type can result in lognormals
with Pareto tails. In the three-parameter Lovejoy-Schertzer
α
-model,
α
(bold alpha)
but represents the L
´
vy index that, together with
the “codimension”
C
1
and the “deviation from conservation”
H
, characterizes a
universal multifractal field. Examples of applicability of universal multifractals to
geological processes that took place in the Earth's crust have been given by
Lovejoy and Schertzer (
2007
).
The model originally developed by Schertzer and Lovejoy (
1991a
,
b
) is based on
the concept of a multifractal field
does not represent the singularity
ʱ
is the
so-called scale ratio with
L
representing the largest scale that can be set equal to
1 without loss of generality (Cheng and Agterberg
1996
). The field
ˁ
ʻ
with codimension
C
1
(
ʳ
) where
ʻ
¼
L
/
E
ˁ
ʻ
can be a flux
or density in physics. In geochemical applications it is the element concen-
tration value of a chemical element. It is characterized by its probability distribution
P
q
ʻ
ˁ
ʻ
ʻ
ʳ
Þ/ʻ
C
1
ðÞ
with statistical moments
E
(
K
(
q
)
. The relations between
ð
ˁ
)
/ ʻ
K
(
q
),
C
1
(
ʳ
) and the field order
ʳ
are:
Kq
ðÞ
¼
q
ʳ
C
1
ðÞ
g
;
C
1
ðÞ
¼
max
q
q
ʳ
Kq
ðÞ
max
ʳ
f
f
g
Cheng and Agterberg (
1996
) have made a systematic comparison of the bino-
universal multifractal model of Schertzer and Lovejoy (
1991a
,
b
).
Search WWH ::
Custom Search