Fractals - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 370

Fig. 10.2 “Perimeter”
measured for Fig. 1.10a by
method illustrated in bottom
diagram of Fig. 1.9
(Case 1). Results for smaller
area have smaller total
perimeters. The subarea of
Fig. 1.9 gave results shown
as Case 1a but more detailed
contour map of Sparky
Sandstone thickness for
same subarea gave results
shown as Case 2. See text
for measurements of fractal
dimensions (Source:
Agterberg 1980 , Fig. 7)

Brownian motion of a particle in the plane. Any cross-section of the landscape of
Fig. 10.1 , such as the jagged line topping the black stripe at the bottom, is an ordinary
Brownian function with the following property: It represents the distance (measured in
the vertical direction) in a given direction between (1) a particle subject to Brownian
motion in the plane, and (2) an arbitrarily selected starting point as a function of time
(plotted in the horizontal direction). Every profile of the ordinary Brownian landscape
of Fig. 10.1 , and also any contour created by intersecting the landscape with a
horizontal plane, has D 1.5. More complicated generating mechanisms result in

other types of Brownian landscapes that are either smoother or more regular.
Agterberg ( 1980 ) subjected the pattern of Fig. 1.12 to the following measure-
ment procedure. The “perimeter” of the thickness contour was measured by using
the method explained in Fig. 1.11 for seven different unit radii. When the unit
radius was 1.5 miles (2.4 km) of less, the measured perimeter provides a good
approximation of the combined length of the contours in Fig. 1.12 . For radius
greater than 1.5 miles (2.4 km), a shorter “perimeter” was measured. These
measurements are shown in Fig. 10.2 (Case 1) with a logarithmic scale along
both axes. Lengths cannot be measured when the radius becomes too large;
e.g.,

¼

10 miles (16 km), and measurements then should not be considered. The
pattern of Fig. 1.12 suggests a curve that approximately coincides with the straight
line fitted by least squares to the four points shown as crosses. The fractal dimension
is equal to 1 minus the slope of the straight line (see, e.g., Mandelbrot 1983 ). Hence,
D

>

1.34. This result could be corroborated as follows: Burnett and Adams ( 1977 ,
Fig. 5b on p. 347) also published a more detailed map of the 30 ft (9.14 m) contour
thickness contour of the Sparky sandstone for a subarea (not shown here) of study
area shown in Fig. 1.12 . The results labeled 1a and 2 in Fig. 10.2 are based on the
part of Fig. 1.12 for this subarea and the other large-scale map, respectively.
Estimates of the fractal dimension based on these other two sets of measurements
of the perimeter were D 1.34, respectively. The pattern labeled 2 on

Fig. 1.12 illustrates that estimated length of the perimeter for unit radius set at

1.38 and D