Geoscience Reference
In-Depth Information
use of local singularities for spatial interpolation (Cheng
2006
). Most of these
techniques were incorporated in the software package GeoDAS (Cheng
2003
).
10.3.1 Concentration-Area (C-A) Method
Suppose that a map pattern in 2-D is fractal with dimension
D
). The reason
for introducing the so-called singularity
ʱ
at this point is that in multifractal
modeling (Chap.
11
) a fractal can be considered as a special case of a multifractal
in which different element concentration values have different values of
¼
f
(
ʱ
.A
multifractal is characterized by its multifractal spectrum which is a plot of
f
(
ʱ
ʱ
)
versus
. If areal distribution of element concentration values in a study region is
fractal, its multifractal spectrum degenerates into a single spike.
For small cell size
ʱ
, the estimated area
A
(
) and concentration value
ˁ
(
)
∈
∈
E
/
E
f
(
ʱ
)+2
;
ʱ
2
. Elimination of
can be expressed as
A
(
)
ˁ
(
)
/
E
gives:
E
E
∈
[2
f
(
ʱ
)]/(
ʱ
2)
where it is assumed that 0
A
(
ˁ
)
/ˁ
f
(
ʱ
)
2and0
ʱ<
2. For
ʱ
¼
2,
ˁ
becomes a constant that is independent of
A
(
ˁ
). The power-law relationship
between
A
(
would plot as a straight line on log-log paper. However, in
Fig.
10.6
, as in many other applications of this type, a good fit only is obtained
when two straight line segments are fitted instead of a single one. In Cheng
et al. (
1994
) it is assumed that, to a first approximation, these two straight line
segments represent separate populations representing “background” and “anomaly”,
respectively. In Sect.
10.1.2
it was discussed that the cumulative frequency distribu-
tion of gold does not provide an indication that there would be two separate
populations. Thus the C-A method can provide a new tool for geochemists to identify
anomalies of element concentration contour maps. For the Mitchell-Sulphurets area,
the thresholds for Au and Cu fall at 400 ppb and 200 ppm, respectively (Sect.
10.1.2
).
In the next section, stream sediment data from the Iskut River area, northwestern
British Columbia, will be analyzed using the C-A method with results that are similar
to those obtained for the bedrock samples from the Mitchell Sulphurets area.
In general, fractal and multifractal modeling can produce useful information on
separating anomalies from geochemical background (fractal C-A method, Cheng
et al.
1994
; and S-A method, Cheng
2001
). At the end of this chapter (Sect.
10.4
), it
will be discussed that patterns such as those shown in Fig.
10.6
probably are not
bifractal but can be explained as a combination of a fractal anomaly pattern that is
superimposed on multifractal background with approximately lognormal element
concentration frequency distribution.
ˁ
)and
ˁ
10.3.2
Iskut River Area Stream Sediments Example
The Iskut River area is located in northwestern British Columbia (Fig.
10.17
). In
this relatively isolated area, 6 Au deposits and 177 Au mineral occurrences of
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