Geoscience Reference
In-Depth Information
the study area, counting how many other points occur within these circles, and
averaging the results. The relation between cluster densities then is according to a
power-law that plots as a straight line with slope
D
c
2 on log-log paper if the
fractal model with
D
c
as its fractal dimension is satisfied.
Carlson (
1991
) obtained the following results: for distances less than 15 km,
cluster density determination as applied to 4,775 precious metal deposits gave
D
c
¼
0.83 versus
D
b
¼
0.50 for box-counting. For greater distances between
15 and 1,000 km, the two methods yielded
D
c
¼
1.51. Note that
for short distances,
D
b
is less than
D
c
but for larger distances, it is the other way
around. For both kinds of fractal cluster, the dimension seemed to be bifractal.
Carlson commented that differences between
D
b
and
D
c
are “troubling but common
in measuring fractal cluster dimensions”. It is mentioned here that fractal dimensions
for short distances generally are too low because of “roll-off” effects that affect
D
b
effects, the
D
b
¼
1.17 versus
D
b
¼
1.51 estimate may be most accurate one. Carlson (
1991
) interpreted
his results in terms of a bifractal model assuming that fractal hydrothermal and
fracture systems are effective over scales from about 15-1,000 km. Various examples
of geological bifractals in other types of applications can be found in Korvin (
1992
).
Differences in fractal point pattern dimension can be due to different reasons.
One explanation is that
D
b
and
D
c
are not necessarily the same; e.g., they are
expected to be different for a multifractal cluster. With respect to the differences
between
D
b
and
D
c
: due to limited resolution of maps, the measurements on fractals
normally become biased for small distances (
or
r
) and this can cause measured
fractal dimensions to be biased. Theoretically, a fractal cluster model has the
property of self-similarity in that point patterns for any enlarged subarea would be
similar even if the zooming-in is repeated indefinitely. In practice, graphical
representations become increasingly incomplete with increased enlargement.
This type of bias does not only apply to fractal point patterns but to other types
of fractals as well. This well- known “roll-off” effect has been studied in detail by
structural geologists (e.g., Walsh et al.
1991
; Pickering et al.
1995
;Blenkinsop
1995
). It already was discussed earlier in this chapter for contour maps
by using a continuous curve that asymptotically approaches the fractal straight
line on log-log paper. However, for smaller samples, the observed “roll-off”
effect may create an artificial sequence resembling two or more separate
straight-line segments. Downward bias in
D
b
toward the origin is stronger than
that in
D
c
over short distances.
On the other hand, estimates of
D
c
for large distances may become less precise
and biased downward more strongly than those of
D
b
if edge effects are not taken
into account. Unless Ripley's
K
(
r
) function is used, it is difficult to account for edge
effects in
D
c
because: (1) much useful information is discarded if all largest circles
around points (along with all smaller circles within the largest circles) are required
to be fully contained within the study area, and (2) the boundary of the study area
may not have a simple (rectangular or circular) shape but can be highly irregular
and then would have to be approximated by a polygon representing a curvilinear
E
Search WWH ::
Custom Search