Multifractal models and their formal properties in urban geography

Résumé En

Fractal analysis for exploring the spatial organization of settlement patterns is used since a coupleof years. On the one hand, scaling behavior turned out to be a suitable approach for characterizingsuch patterns, but town sections, issued from different periods of urban history or corresponding toparticular planning concepts show different types of scaling behavior what aided classifying urbanpatterns. However, on the scale of agglomerations these different scaling behaviors are mixed. Thatincites asking whether multifractal approaches could be of interest when considering urban patternsor settlement systems, as local properties like different degrees of concentration of build-up sitescould be better taken into account and several researches focused on using multifractal analysis inthese contexts.However, the use of fractal geometry is not restricted to measuring scaling behavior. Their intrinsicgeometric properties are of interest, too, when considering urbanization. Indeed, the particularmultiscale feature of fractals brings another perspective on spatial organization. Hence, fractal urbanmodels have been explored in the context of urban economics, and, more recently, proposed forplanning purposes.However, we must be aware, that multifractality is a rather polysemic notion. We can use it whileconsidering the two-dimensional distribution of build-up space or network branches, but also whenlooking at the weights, like the heights of buildings or traffic loads, on a two-dimensional fractalor uniform support, which is soil occupation. In the first case we can still refer to self-similarity,what is no longer true in the second case where self-affinity comes into play. We distinguish thesedifferent topics and focus on their formal properties, which are nevertheless linked to the commonbackground of multifractal theory. More explicitly, we consider the properties of Sierpinski carpetscombining different scaling factors, a model which is of interest for describing central place hierarchyas well as for planning concepts. We show, too, that there exist analytically ambiguities while usingthe Minkowski and the Hausdorff approach for measuring fractality for those objects. We consider,too, by referring the so-called Poincarré-maps, how different degrees of concentration of build-upspace across scales can be treated formally in a coherent way.