Rényi’s spectra of urban form for different modalities of input data

Morphologies of urban patterns display multifractal scaling. However, what data should be used to represent an urban pattern and its scaling? Here, we calculated Renyi’s generalized dimensions (RGD) spectra using data corresponding to different urban modalities including urban land cover, urban impervious surface, population density, and street intersection points. All data are circa 2010 and we calculated their RGD spectra in six urbanized areas located across the United States. We calculated the RGD spectra by using Hill’s numbers rather than statistical moments which leads to a clear interpretation of generalized dimensions and to spatial visualization of pattern’s multifractality. The results show that patterns of different urban modalities in a given urbanized area are characterized by different RGD spectra and thus have different morphologies. In our six examples, we found that morphologies of patterns of land cover and impervious surface tend to be monofractal, patterns of street intersection points tend to be moderately multifractal, and patterns of population density tend to be strongly multifractal. Spatial visualization supporting this numerical finding is provided. Thus, when studying the multifractality of urban morphology, it is important to choose a modality that is appropriate to the goal of the investigation. Urban areas may have similar morphologies on the basis of one modality but dissimilar on the basis of another. We have found that two out of our six urban areas have similar morphologies on the basis of all four modalities.