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ʱ
max
derived for the
Pulacayo orebody by means of singularity analysis differ greatly from previous
estimates based on the binomial/
p
model. From application of the method of
moments it would follow that
d
In Sect.
11.6.1
it was pointed out that estimates of
ʱ
min
and
¼
0.121,
ʱ
min
¼
0.835, and
ʱ
max
¼
1.186. The latter
ʱ
min
¼
ʱ
max
¼
two estimates differ not only from
0.835 and
1.719 derived in Sect.
ʱ
max
¼
ʱ
max
¼
1.402 on the right side of
the multifractal spectrum in Fig.
11.7
. Obviously, the estimate
d
1.186 also is less than the estimate
¼
0.121 is much
too small. Using absolute values of differences between successive values, de Wijs
(
1951
) had already derived the larger value
d
¼0.205. Use of the estimates of
ʱ
min
or
ʱ
max
based on the full convergence local singularities derived by the Chen
algorithm yielded
d
0.392, respectively (Sect.
11.6.1
). Clearly,
the binomial/
p
model has limited range of applicability and a more flexible model
with additional parameters should be used. The accelerated dispersion model of
Sect.
12.5.1
offers one possible explanation. Another approach consists of using the
Lovejoy-Schertzer
¼
0.369 and
d
¼
-model. These authors have successfully applied this model to
the 118 Pulacayo zinc values as will be discussed in the next section.
The three-parameter
α
-model
is a generalization of
the conservative
α
two-parameter (0,
C
1
,
) universal canonical multifractal (Schertzer and Lovejoy
1991b
, p. 59) model. So-called extremal L´vy variables play an important role in a
two-parameter approach. In their large-value tail these random variables have
probability distributions of the Pareto form
P
(
Xx
)
α
|
x
|
α
(
2). They can be
used to generate multiplicative cascades starting from uniform random variables as
outlined in Box
12.3
. Frequency distributions of values belonging to multifractal
fields of this type already can have high-value tails that are thicker than logbinomial
and lognormal tails.
/
α
<
Box 12.3: Extremal L´vy Variables
Suppose
W
is a uniform random variable within the interval [0, 1] and
V
W
1/
α
; so that
·
v
1
α
if
v
¼
the probability
P
(
V
¼
v
)
¼
α
1, and
n
P
(
V
¼
v
)
¼
0if
v
<
1(
cf
. Wilson et al.
1991
, APPENDIX B). Then,
i
¼1
V
i
∑
representing the sum of
n
such random variables
V
i
(i
,
n
) has a
L´vy limit distribution with high-value tails. The L´vy index
α
defines
another index
¼
1, 2,
...
α
0
by means of the relation 1/
α
0
¼
+1/
1. Suppose that
ʼ
and
α
c
are two constants and a new randomvariable is defined as
Y
¼
c
·(
ʼ
V
). Then,
with Laplacian characteristic
n
1
=
α
X
n
w
1
=
α
i
1
for large
n
:
Y
¼
i
¼1
c
α
α
1
function
E
(
e
Y
·
q
)
)·(
cq
)
α
}. It follows, after some manipulation,
¼
exp{
·
ʓ
(
α
α
1
=
α
C
1
ʓ
2
α
that
c
¼
resulting in the two-parameter universal
form
ð
Þ
C
1
α
0 C
α
Kq
ðÞ
¼
ð
q
α
q
Þ;
0
2
;
0
<
C
1
<
E
.
α
characterizes the degree of multifractality and the codimension
C
1
characterizes the sparseness of the mean field (Lovejoy and Schertzer
2010
). In
the three-parameter Lovejoy-Schertzer
The L´vy index
α
-model a third parameter
H
is added that
α
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