Geoscience Reference
In-Depth Information
instead of negative values of
q
. It is not possible to raise the smallest possible
observed value for a cell (
0 representing absence of fractures) to a negative
power, and large fluctuations are likely to occur when this limit is approached.
Figure
11.18
shows how the stepwise derivation of the difference method for the
step from Fig.
11.18b, c
was used as follows. Suppose that
¼
˄
(
q
k
1
),
˄
(
q
k
) and
˄
(
q
k
+1
)
˄
(
q
) as shown in Fig.
11.18b
. Then
α
k
(
q
)asitis
are three successive estimates of
q
kþ
1
ðÞ
¼
˄
˄
ð
q
k
1
Þ
shown in Fig.
11.18c
satisfies
α
k
q
. The multifractal spec-
q
kþ
1
q
k
1
trum (Fig.
11.18d
) deviates significantly from a marrow spike showing that the 2-D
distribution of the fractures is multifractal instead of fractal. In Agterberg
et al. (
1996a
), the multifractal semivariogram corresponding to Fig.
11.18
is used
for further statistical analysis of fractures at the surface of the Lac du Bonnet
Batholith. The frequency of fractures per block of granite decreases rapidly in the
downward direction. In Agterberg et al. (
1996b
) and Agterberg (
1997
) multifractal
analysis was applied to fractures observed along boreholes drilled in a larger area
including the area shown in Fig.
11.15
. The results of this 3-D analysis were used to
estimate probabilities that blocks of granite below the surface are entirely or
relatively free of fractures. The practical significance of this study is to help decide
on possible underground sites for storage of nuclear waste.
11.4.2
Iskut River Map Gold Occurrences
The area used for this example is the Iskut River map sheet (British Columbia
Minfile Map 104B). In this area there are 183 Au mineral deposits and occurrences
(Fig.
11.19
). It was previously studied using box-counting and number-size
methods (Cheng et al.
1994c
), fractal pattern integration in Au potential mapping
(Cheng et al.
1994b
) and geochemical anomaly separation (Cheng et al.
1996
;
Different cell sizes (
) ranging from 3 to 10 km were used. Number of Au
occurrences per cell was counted on eight grid maps. The multifractal results are
shown in Fig.
11.20
where
q
ranges from 0 to 4. This relatively narrow range was
used because the mass-partition function does not exhibit clearly defined power-law
relations unless 0
E
q
4. Results for
˄
(
q
) include
˄
(0)
¼
1.335
0.077,
˄
(2)
¼
1.219
0.266. A fractal model would have resulted in
straight lines with slopes interrelated according to
0.037 and
˄
(4)
¼
3.070
˄
(
q
)/(
q
¼
˄
(
p
)/(
p
1)
1). For
˄
¼
˄
˄
˄
¼
example,
for
the fractal model:
(4)/3
(0). However,
(4)/3 +
(0)
0.312
0.098 and this is significantly greater than zero, indicating that the under-
lying model is multifractal instead of monofractal. The slope of the straight line for
q
¼1 in Fig.
11.20a
satisfies
˄
(1) ¼0 representing constant first-order intensity. The
maximum value of:
f
(
α
)
in Fig.
11.20c
occurs at
˄
(0)
¼
1.335
0.077
representing the box-counting dimension of the fractal support.
Search WWH ::
Custom Search