  The lack of efficiency in urban diffusion is a debated issue, important for
biologists, urban specialists, planners and statisticians, both in developed
and new developing countries. Many approaches have been considered to measure
urban sprawl, i.e. chaotic urban expansion; such idea of chaos is here linked
to the concept of entropy. Entropy, firstly introduced in information theory,
rapidly became a standard tool in ecology, biology and geography to measure the
degree of heterogeneity among observations; in these contexts, entropy measures
should include spatial information. The aim of this paper is to employ a
rigorous spatial entropy based approach to measure urban sprawl associated to
the diffusion of metropolitan cities. In order to assess the performance of the
considered measures, a comparative study is run over alternative urban
scenarios; afterwards, measures are used to quantify the degree of disorder in
the urban expansion of three cities in Europe. Results are easily interpretable
and can be used both as an absolute measure of urban sprawl and for comparison
over space and time.
