Exploring the deviations from scale-invariance of spatial distributions of buildings using a Geographically Weighted Fractal Analysis. An application to twenty French middle-size metropolitan areas

In the early twentieth century a handful of French geographers and historians famously suggested that mainland France comprised two agrarian systems: enclosed field systems with scattered settlements in the central and western France, and openfield systems with grouped settlements in eastern France. This division between grouped and scattered settlements can still be found on the outskirts of urban areas. The objective of this paper is to determine whether the shape of urban areas varies with the type of built patterns in their periphery. To this end, we identify and characterise the local and global deviations from scale-invariance of built patterns in metropolitan France. We propose a new method –Geographically Weighted Fractal Analysis – that can characterize built patterns at a fine spatial resolution without making any a priori distinction between urban patterns and suburban or rural patterns. By applying GWFA to the spatial distribution of buildings throughout mainland France we identify six geographically consistent types of built patterns that are distinctive in the way buildings are either concentrated or dispersed across scales. The relationship between the local built textures and the global shape of twenty metropolitan areas is then analysed statistically. It is found that the proportion of dispersed (or concentrated) outer suburban built patterns in metropolitan areas is closely related to the distance threshold that marks the morphological limit of their urban areas