where, € is a constant of proportionality, but not defined where
radius R = 0, and a is the parameter on distance, is able to
accommodate scale independence observed in urban systems
through the notions of fractal geometry (Batty and Kim, 1992).
Given, the limits on the range of (7), the cumulative population
N(R) associated with the density of p(R) can be modeled as
NR) = iR (8)
from which the area A(R) over which density is defined with
respect to distance R from the centre is given as
AB) Bein RE (9)
where a perfect circle of area would have y = x. Both c and y
are constants of proportionality. Applying the principle of self-
similarity evident in fractal geometry, it is possible to show that
the density parameter a is related to the fractal dimension D,
such that D — (2 - «) and that the cumulative population relation
can now be rewritten as
MR) = en” (10)
where D is the fractal dimension measuring both the extent and
the rate at which space is filled by urban development with
increasing distance from the urban centre. From here there are
two sets of techniques which have been developed for
estimating the fractal dimension. The first set emphasis space-
filling (Batty and Kim, 1992), and the second set focuses on
density attenuation (Mesev et al, 1996). Both are based on
squared lattices which are easily derived from classified
remotely-sensed data, but only the second set also generates
density profiles and will be the one examined in this paper.
Dimensions and profiles from linear regression
A well known method which takes into account variance within
the distributions is to first, linearize the power laws in both (7)
and (8), and then perform regression to calculate values for C
and a, c and D respectively. Linearized forms can be expressed
for both discrete densities p; and cumulative populations, N;, in
this way
p; = InG —a ln R, (11)
N:
I
= Inc — Din R; (12)
The slope parameters for both equations measure the rate at
which density attenuates and population increases, each with
respect to distance, respectively. Fractal dimensions are
generated by the intercept parameters, G and c which are, in
turn, affected by the slope parameters, a and 2 - D, in (11) and
(12) respectively. It has duly been noted that slope parameters
may become volatile when confronted with abnormal data sets,
leading to fractal dimensions that could lie beyond the logical
limits associated with generalized space-filling, i.e. 1 « D « 2.
These are considered abnormal, in the sense that data do not
conform to established linear relationships in both cumulative
population and discrete density with respect to distance. An
example of abnormal data would result if physical barriers
restrict development near the central business district (CBD),
and in this case, fractal dimensions of over 2 may be possible.
Similarly, values of less than 1 are possible if “reversals in the
norm" are encountered. This is when discrete density actually
increases with distance from the urban centre. Work on
“constraining” linear regression can be found in Batty and Kim
(1992) and Mesev et al (1995).
Empirical application
The Norwich case study may now be re-started by applying the
linearized fractal modeling equations (11) and (12) to the
thematic residential dwelling types produced through
classification. Hence, the modeling of the measured urban
remotely-sensed data.
Table 2 shows the fractal dimensions and the coefficient of
determination (goodness of fit) for both cumulative and density
profiles. The dimensions are within the hypothesised 1 « D «2
range but much lower than those produced using more
traditional data sources (lists in Batty and Longley, 1994).
These lower dimensions are due to the fact that the higher
spatial resolution of remotely-sensed data have been able to
determine “pockets” of undeveloped residential land within
settlement boundaries. It means that residential areas are not
treated as centinuous areas of development but as more discrete
entities of incremental growth. The residential patterns
produced are therefore more in line with the way land is
incrementally apportioned into residential use by planners and
echoed by idealised fractal growth (Fotheringham et al, 1989).
Between the four dwelling types, terraced has the highest
dimension and mirrors conventional centralized tendency in
British cities. When examining the cumulative and density
profiles (figure 3), it quickly becomes apparent that the density
gradients are less than linearized and this is reflected in the
coefficients of determination in Table 2. Lower r-squared
values for the density profiles are a symptom of the degree of
constraints to urban development. These may be physical
impediments or, as is more likely in the case of Norwich,
planning restrictions which are an important aspect of British
settlements. — Nevertheless, the profiles in figure 3 are good
representations of the ability of remotely-sensed data to
measure the way urban development varies within a settlement.
The finer spatial resolutions of satellite images allow more
detailed intricate variabilities to be highlighted than have yet
been possible. With temporal comparisons it may be possible
to evaluate urban changes with respect to estimated levels of
suburbanisation, decentralisation of economic functions, and
segregation of urban land uses.
CONCLUSIONS
This paper has demonstrated how interaction between image
processing, GIS data, and spatial analysis can be applied to the
measurement and modeling of urban development. What it has
shown is that residential development can be measured using
the standard ML classifier, as long as reliable extraneous data,
preferably handled by a GIS, can be incorporated as
representative a priori probabilities. Once measured, the
structure of residential development and the way residential
560
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996
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