Fractals - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 383

shape instead of by a rectangle, thus creating additional complications as discussed
in Sect. 1.5.3 . These two edge effect problems can be avoided by using Ripley's
K ( r ) function. Although estimates of D b based on the largest boxes are relatively
imprecise, because there are relatively few of them, they can have the advantage of
remaining unbiased provided that exactly the same study area is covered by all
box-counting grids.
The results obtained by Carlson ( 1991 ) have been criticized by various authors but
Raines ( 2008 ) in a comprehensive review concluded (a) in several other applications,
differences between D b and D c are less than those found by Carlson, and (b) in
general, the bifractal cluster model would be satisfied because dimensions for shorter
distances are indeed less than those for larger distances. Raines ( 2008 ) was able to
reduce bias in fractal cluster estimation to some extent by using GIS -based methods.
However, it is possible to improve fractal estimation procedures more significantly by
adopting the methods developed by mathematical statisticians as will be shown in the
next section on the basis of a practical example.

10.2.1 Cluster Density Determination of Gold Deposits
in the Kirkland Lake Area on the Canadian Shield

Figure 10.11 (from Agterberg et al. 1993 ) shows the locations of 295 gold deposits
(mines and occurrences) in a rectangular area of 4,185 km 2 in the vicinity of
Timmins and Kirkland Lake in the Abitibi area on the Precambrian Canadian
Shield. It is part of a much larger study area (

89,600 km 2 ) containing 1,306 ¼

gold deposits with D c ¼ 1.514 to be discussed in more detail in the next section.

Cluster density estimation applied to Fig. 10.11 gives D c ¼ 1.493, which is close to

the estimate based on the larger area. Figure 10.11 shows large subareas without
any gold deposits (lacunarity). This is either because the bedrock containing gold
deposits in these subareas is covered by relatively thick Quaternary deposits or it
consists of barren rock types such as Archean gneiss belts without gold deposits.
The pattern of Fig. 10.11 is anisotropic because of strong east-west structural
trends. By means of additional experiments to be summarized later it can be
shown that neither lacunarity nor anisotropy significantly alters the estimate of
D c ( 1.5).

Most problems of lack of precision and accuracy outlined in the previous section
can be avoided by adopting methods of statistical point-process modeling (Diggle
1983 ; Ripley 1988 ; Agterberg 1994 ). In this type of modeling, the first-order and
second-order properties of a spatial point process for events are described by its
intensity function ʻ 2 ( x , y ) where x and

y represent the locations of two event points in the plane. For an isotropic, stationary
point process,

ʻ 1 ( x ) and second-order intensity function

¼ ʻ 2 ( r ) where r is distance between x and y .A
powerful way to characterize the second-order properties of a point process is
provided by Ripley's K ( r ) function, which is proportional to expected number of

ʻ 1 ( x ) ¼ ʻ and ʻ 2 ( x , y )