From Fractal Urban Pattern Analysis to Fractal Urban Planning Concepts - Computational Approaches for Urban Environments - page 19

of urban patterns exhibits scaling properties, as they are characteristic for fractals.
To this end, several methods have been developed. We focus here on four methods
referring to different types of information about the fractal behavior of empirical
structures and to different properties of empirical textures like urban patterns.
It is clear in real-world textures such as urban patterns that the spatial organiza-
tion, even if it can well be described by fractal measures, does not obey a fractal
law exactly. This prompts us to speak often of “scaling behavior” which can change
at certain scales, vary from one urban district to another, or be disturbed at certain
scales. Mixing different scaling behaviors is reminiscent of multifractal geometry,
where the fractal behavior varies from one site to another.
There are a number of methods used to estimate the fractal dimension of
empirical structures. Not all of these methods obey the same logic, and if the
empirical structures exhibit multifractal properties, the results obtained will not be
the same. In this case, the methods provide complementary information.
The basic idea is to explore the texture under consideration at different scales. For
this purpose, a distance " is introduced and the number of elements N (") lying within
this range is determined. Then, the value " is changed and the procedure repeated.
For fractals, the following power-law relation between the number of elements N (")
and distance (") holds

N."/ D a " D

(2.3)

which is used to estimate the dimension D and the prefactor a . Hence, we assume
that the analysis provides a sequence N (obs) (" i ) of empirical data, the empirical curve,
for a discrete series of "-values called " i serving to estimate the parameters a and
D by which the “theoretical curve” ( 2.3 ) is obtained. Theoretically, this prefactor
corresponds to the measure M introduced previously. For empirical structures,
however, we should expect deviations at different scales from a pure fractal law.
Thus, for each " i value, we should introduce a local version of the fractal law which
reads

N . obs / ." i / D a i " D a (2.4)

i

In Thomas et al. ( 2012 ), it was shown that the global prefractor a is just the
geometrical mean value of these scale-specific prefactor values and so a mean
measure for the object across scales:

n !

1
n

Y

a D a i (2.5)

i D1

Up to now, it has proved difficult to interpret the observed values directly.
Recent tests have shown that the a -values are neither associated with the correlation
between the empirical curve and theoretical curve, i.e., the quality of adjustment,