Geoscience Reference
In-Depth Information
topics with applications to surface fractures. Emphasis is on (a) comparison with
results obtained by the simpler fractal approach, and (b) implications of the
multifractal approach for spatial statistical analysis.
In a paper on fractal measurements, Roach and Fowler (
1993
) presented com-
puter programs to determine the box-counting dimension and other fractal dimen-
sions from patterns. Another method for dealing with this problem when measuring
the so-called mass fractal dimension had been proposed by Lerche (
1993
), with an
application to self-similar fault patterns. An important problem considered in these
papers was to avoid bias related to measurements for small cells at one end, and
measurements for large cells (close to total size of study region) at the other. For a
pattern consisting of line-segments in the plane, the small and large cells yield
biased estimates which are approximately equal to 1 and 2, respectively. The latter
are estimates of Euclidian dimensions instead of fractal dimensions. These arise
from the fact that it is not possible to derive meaningful results from measurements
on cells that are nearly as large as the entire study area. Neither is it possible to
estimate fractal dimensions from measurements on small cells for which the
number of cells with nonzero measurements becomes approximately inversely
proportional to cell area.
A fractal dimension must be obtained as the slope of a straight line on a log-log
plot representing a fractal (for examples, see later); it should not be estimated from
a curve that is gradually changing its slope from 1 to 2. The same type of bias arises
in multifractal modeling and should be avoided when the values of
ˇ
2
(E) are
estimated. A second problem of bias in fractal measurements occurs when the
cells or boxes used for counting or measuring the mass exponents
(
q
) are not
restricted to the study area. Use of cells that contain parts of the boundary of the
study region results in undesirable edge effects. For patterns of fractures, the
importance of this second type of bias was clearly demonstrated by Walsh and
Watterson (
1993
) and Gillespie et al. (
1993
). In the current application, there is the
additional problem that bedrock is not fully exposed (Fig.
11.15
). A procedure
which can be used to avoid simultaneous bias because of lack of exposure and edge
effects is as follows. As illustrated in Fig.
11.16
, fractures can be measured only in
cells or portions of cells underlain by exposed bedrock within the boundaries of the
study region. Suppose that for a square cell with area (
a
) equal to
˄
2
, the following
two measurements are obtained: area of exposed bedrock per cell (
s
i
), and total
length of all fractures per cell (
E
ʼ
i
). The following weighted form then can be used to
correct for bias in the multifractal situation:
X
n ðÞ
q
ˇ
q
ðÞ
¼
w
i
½
ʼ
i
ðÞ=
w
i
where
w
i
¼
s
i
/
a
for
i
¼
1, 2,
...
,
n
(
). If
w
i
¼
1 for all cells, this expression reduces
E
q
i
as was used before. Otherwise, when bedrock is not fully
exposed in the study region, it reduces to the original form only if
q
to:
ˇ
q
(
)
¼
∑
n
(
E
)
ʼ
E
¼
1. The
modified equation implies adherence to the principle of conservation of total mass
within the study region. Note that
s
i
6
¼
a
for a cell represents either area of covered
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