Case studies in geography are strongly dependent on the size of the spatial units used for the analysis. This has been expressed as the Modifiable Areal Unit Problem (MAUP): whatever the phenomenon under consideration, it is impossible to identify a single spatial partition that would be most appropriate to analyze it. In this respect, multifractal analysis may be an interesting tool for geographers. It integrates not just a series of nested spatial resolutions, as fractal analysis does, but also a series of points of view about the quantity of information contained in each spatial unit. In this article, we first expose the mathematical bases of multifractal analysis and we describe how it applies to geographical analyses. We insist on the mathematical notion of dimension, which allows us to describe how multifractal parameters can be used to quantify the MAUP. Then, we use the method to characterize the spatial distribution of population density in France. The main result is a typology map of population density that uses the MAUP as a descriptive tool. This map allows the joint identification of several phenomena: the main cities, the rural settlement patterns, and several types of periurban settlement patterns.

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 mainland France. For this, we propose a new method-Geographically Weighted Fractal Analysis-that can characterise 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.

Even though the study of fractal and multifractal properties has now become an established approach for statistical urban data analysis, the accurate multifractal characterisation of smaller, district-scale spatial units is still a somewhat challenging task. The latter issue is key for understanding complex spatial correlations within urban regions whilst the methodological challenge can be mainly attributed to inhomogeneous data availability over their territories. We demonstrate how the approach proposed here for the multifractal analysis of irregular marked point processes is able to estimate local self-similarity and intermittency exponents in a satisfactory manner via combining methods from classical multifractal and geographical analysis. With the aim of emphasizing general applicability, we first introduce the procedure on synthetic data using a multifractal random field as mark superposed on two distinct spatial distributions. We go on to illustrate the methodology on the example of home prices in the greater Paris region, France. In the context of complex urban systems, our findings proclaim the need for separately tackling processes on the geolocation (support) and any attached value (mark, e.g. home prices) of geospatial data points in an attempt to fully describe the phenomenon under observation. In particular, the results are indicators of the strength of global and local spatial dependency in the housing price structure and how these build distinct layered patterns within and outside of the municipal boundary. The derived properties are of potential urban policy and strategic planning relevance for the timely identification of local vulnerabilities whilst they are also intended to be combinable with existing price indices in the regional economics context.

Even though the past three decades have seen numerous crucial investigations on interurban scaling characteristics, there has been less focus on revealing multiscale properties within municipal or metropolitan structures. We demonstrate how a newly developed methodology, the Geographically Weighted Multiscale Analysis (GWMSA) stemming from the theory of multifractal systems, can be used to analyze small-scale urban environments with respect to their intermittency and roughness simultaneously. To this end, apart from the widely used sand-box method, we introduce wavelet coefficients in the multiscale analysis of urban systems. In more detail, the spatially continuous scanning of the three largest French conurbations—Paris, Marseille, and Lyon—over their territories and at length scales ranging from parcel to neighborhood level will allow to derive and compare globally and locally characteristic scaling exponents. Depending on the feature under analysis, the exponents reveal qualitatively distinct structural properties, whereby the viability of our findings is further verified on four exemplary typologies of multiscale behavior in urban systems. To introduce GWMSA, this paper focuses primarily on morphological characteristics and findings provide a compelling alternative to how we capture and define district-scale spatial organization and interdependencies within urban settlements.