Multifractal characterization of urban form and growth: the case of Beijing

Abstract

Urban form takes on properties similar to random growing fractals and can be described in terms of fractal geometry. However, a model of simple fractals is not effectual enough to characterize both the global and local features of urban patterns. In this paper multifractal measurements are employed to model urban form and analyze urban growth. The capacity dimension Da, information dimension D\, and correlation dimension Di of a city's pattern can be estimated utilizing tbe box-counting method. If Da> D\> D2 significantly, the city can be treated as a system of multifractals, and two sets of fractal parameters, including global and local parameters, can be used to spatially analyze urban growth. In tbis case study, multifractal geometry was applied to Beijing city, China. The results based on the remote-sensing images taken in 1988, 1992, 1999, 2006, and 2009 show that the urban landscape of Beijing bears multiscaling fractal attributes. The dimension spectrum curves show several abnormal aspects, especially the upper limit of the global dimension breaks through the Euclidean dimension of embedding space and the local dimension fails to converge in a proper way. The geographical features of Beijing's spatiotemporal evolution are discussed, and the conclusions may be instructive for spatial optimization and city planning in the future.

Understanding the Fractal Dimensions of Urban Forms through Spatial Entropy

“…Two types of box counting can be applied to fractal dimension estimation of cities [31,42]: one is to fix the largest box [46], and the other is to unfix the largest box [44]. In the former way, the size and shape of the first level box do no change for different years; in the latter way, the size and shape of the largest box change with urban growth in different years.…”

“…The original data are extracted from the remote sensing images, including three Landsat TM images and two Landsat ETM+ images. The ground resolution of these images is 30 m [42]. The functional box-counting method can be used to measure the spatial entropy and fractal dimension (Figure 3).…”

“…This method was originally adopted [43] to analyze radar rain data, and Chen [26] improved this method in urban studies by replacing the largest box (the first level box) with arbitrary area with the largest box with a measure area of an urban system. In fact, the functional box-counting method can be termed Rectangle Space Subdivision (RSS) method [42,44]. The geometrical basis of RSS is the recursive subdivision of space and the cascade structure of hierarchies [30,45].…”

A Comparison Between Two Ols-Based Approaches to Estimating Urban Multifractal Parameters

“…In order to characterize a multifractal pattern, two sets of parameters are required: One is the global parameters, and the other is the local parameters (Feder, 1988;Chen and Wang, 2013;Mandelbrot, 1999;Meakin, 1998;Stanley and Meakin, 1988). New alternative measure has been put forward (Grech, 2016).…”

“…The global parameters include generalized correlation dimension and mass exponent, while the local parameters comprise singularity exponent and fractal dimension of fractal subsets in a complex fractal set (Chen, 2014;Feder, 1988). The basic global parameters are capacity dimension, information dimension, and correlation dimension (Chen and Wang, 2013;Murcio et al, 2015;Williams, 1997). In a given set of fractal dimensions, such as the set of generalized correlation dimensions, a clear numerical relationship exists between different dimension values.…”