Fractal analytical approach of urban form based on spatial correlation function

Abstract

Urban form has been empirically demonstrated to be of scaling invariance and can be described with fractal geometry. However, the rational range of fractal dimension value and the relationships between various fractal indicators of cities are not yet revealed in theory. By mathematical deduction and transform (e.g., Fourier transform), I find that scaling analysis, spectral analysis, and spatial correlation analysis are all associated with fractal concepts and can be integrated into a new approach to fractal analysis of cities. This method can be termed ‘3S analyses’ of urban form. Using the 3S analysis, I derived a set of fractal parameter equations, by which different fractal parameters of cities can be linked up with one another. Each fractal parameter has its own reasonable extent of values. According to the fractal parameter equations, the intersection of the rational ranges of different fractal parameters suggests the proper scale of the fractal dimension of urban patterns, which varies from 1.5 to 2. The fractal dimension equations based on the 3S analysis and the numerical relationships between different fractal parameters are useful for geographers to understand urban evolution and potentially helpful for future city planning.

Introduction

Fractal geometry is a powerful tool in geographical modeling and spatial analysis, and it has been applied to urban studies for a long time [3], [4], [6], [10], [13], [23], [28], [50]. The fractal concept is in essence based on scaling symmetry, and symmetry suggests invariance under some kind of transformation [42]. Thus self-similarity is equivalent to invariance under contraction or dilation, which is termed ‘scaling invariance’. In mathematics, fractal property can be abstracted as scaling relations between linear scales and corresponding measurements. A city can be empirically treated as a fractal system with self-similarity or self-affinity (e.g., [8], [7], [12], [17], [25], [27], [49], [48], [53], [54]). Where urban form is concerned, the relation between the radius (a scale) from a city center and the corresponding urban density (a measurement) may follow the inverse power law indicating a fractal distribution [10], [11], [27]. Smeed’s model on traffic network can be employed to analyze urban growth and estimate fractal dimension of urban patterns [10].

The spatial structure of fractal cities follows power laws or inverse power laws, which implies some kind of scaling relations. Smeed’s model of urban density suggests an inverse power-law distribution [46]. A power-law distribution is a scale-free distribution that cannot be effectively described with the conventional statistical measures such as mean, variance, and covariance. The scale-free distributions can be dealt with by scaling analysis in both theoretical and empirical studies. In theory, scaling analysis is an approach to deriving a particular power-law relation from a certain equation. The derivation process is always based on the property of invariance under scaling transform (contraction/dilation transform). From the power-law relation we can obtain one or more useful parameters, which are termed ‘scaling exponents’ and usually associated with fractal dimension. In practice, scaling analysis indicates an empirical separation process of a system, say, a city, into different aspects by means of scaling exponents [38]. The scaling analysis is very useful and significant in both theoretical and empirical studies of urban patterns and processes.

The inverse power function of urban density mentioned above proved to be a special spatial correlation function [47]. Urban growth and form have been modeled by using the concept of spatial correlation [39], [40]. Spatial correlation is an underlying way of modeling both urban growth and form, and the correlation models can be mathematically analyzed by scaling transform. In fact, based on the density function, a general spatial correlation function of cities can be constructed [18], [21]. If the correlation function is converted into energy spectrum, the urban density will be converted into spectral density through Fourier transform [15]. Thus spatial correlation analysis can be converted into spectral analysis and vice versa [16]. Both spatial correlation analysis and spectral analysis are useful tools in urban studies [14], [21]. The two analytical processes are associated with scaling analysis. A problem is how to combine the scaling analysis, spectral analysis, and spatial correlation analysis with one another to form a new method of spatial analysis of cities.

Especially, in order to apply fractal theory to city planning and urban spatial optimization, the geographical meaning of fractal dimension values must be revealed. In the previous works, the fractal dimension values of urban patterns were discussed by Frankhauser [29], and the statistical relationship between residential satisfaction and the fractal dimension of the built-up residential environments was discussed by Thomas et al. [51]. This paper is devoted to revealing the theoretical relations and proper numerical scales of different fractal parameters of urban form. Based on the inverse power law of urban density, scaling analysis, spectral analysis, and spatial correlation analysis will be integrated to make a new approach to analyzing urban patterns and process (Section 2). As a case study, the methods will be applied to three cities in Yangtze River delta, China (Section 3). The academic contributions of this paper to fractal theory of cities are as follows. First, a new method termed ‘3S analysis’ of fractal city systems is presented, and the analytical procedure is sketched out. Second, a set of fractal parameter equations is derived, and these equations are useful for our understanding urban development. Third, the valid range of fractal indicators of urban form is determined, and the results are potentially helpful for future city planning.