Fractals - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 381

exploration “plays” in Alberta with study area boundaries that were approximated
by multi-edge polygons instead of being rectangular in shape. Not only do these
“plays” have irregular boundaries, they may contain islands of terrain judged to be
unfavorable for drilling but situated within the study area. These edge correction
methods can equally well be applied in fractal point process modeling. This section
advocates the use of such methods, which account for edge effects, allow irregu-
larly shaped study areas and, generally, are much more precise than various
box-counting methods more commonly used for the statistical treatment of fractal
point patterns if the domains of study are not rectangular in shape.
A closely related topic not covered in this chapter is that fractal point patterns
can be multifractal. The spatial distribution of gold mineral occurrences in the Iskut
River map sheet, northwestern British Columbia, is multifractal instead of
monofractal (Cheng 1994 ; Cheng and Agterberg 1995 ). Although theory of
multifractal point processes was developed in these latter two publications, there
have been relatively few other applications along these lines, although generaliza-
tions to multifractal point pattern modeling can have advantages similar to those
documented in the relatively many existing multifractal applications to measures or
chemical element concentration methods (Mandelbrot 1999 ; Cheng 2012 ). The
topic of multifractal point patterns will be discussed in the next chapter. The points
in a point pattern can have different values. This has led to the modeling of
“marked” point processes (Cressie 2001 ). This topic also could be of importance
in resource potential modeling where the points represent mineral deposits or oil
pools with positively skewed size-frequency distributions and possess other fea-
tures that differ greatly from point to point.
Carlson ( 1991 ) applied a fractal cluster model to hydrothermal precious-metal
mines in the Basin and Range Province, western United States. The underlying
theoretical model is clearly explained in Feder ( 1988 , Chap. 3 ). For the spatial
distribution of points in the plane this model requires that the number of points
within distance r from an arbitrary point is proportional to r Dc where D c represents
the so-called cluster fractal dimension which is less than the Euclidian dimension of
the embedding space (¼2 for two-dimensional space). It implies that the cluster
density decreases with increasing r . As mentioned before, a basic difference
between this fractal model and commonly used statistical point-process models
such as the Poisson and Neyman-Scott models is that the latter are not fractal
because their dimension remains equal

to the Euclidian dimension ( ¼ 2) for

increasing r .
Two methods (box-counting, and cluster density determination) are commonly
used for the fractal modeling of point clusters. The first method consists of
superimposing square grids with different spacings ( ) on the study region. The

number of boxes ( N E ) containing one or more points is counted for every square
grid. The number N Є decreases with increasing value of

E

, mainly because there are
fewer larger boxes. If the box-counting fractal model is satisfied, this decrease
satisfies a power-law relation and shows as a straight line with slope - D b on a
log-log plot where D b represents the fractal dimension. Cluster density determina-
tion can be performed by centering circles with different radii ( r ) on all points in

E