Journal of Geo-information Science ›› 2020, Vol. 22 ›› Issue (11): 2140-2151.doi: 10.12082/dqxxkx.2020.190640

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Improvement and Application of Two-dimensional Fractal Model of Urban Growth and Morphology

LI Yatong1,2(), ZHANG Lijun3,*(), YE Shilin1,2, QI Xinhua1,2   

  1. 1. State Key Laboratory for Subtropical Mountain Ecology of the Ministry of Science and Technology and Fujian Province, Fujian Normal University, Fuzhou 350007, China
    2. College of Geographic Sciences, Fujian Normal University, Fuzhou 350007, China
    3. College of Environment and Planning, Henan University, Kaifeng 475000, China
  • Received:2019-11-07 Revised:2020-01-14 Online:2020-11-25 Published:2021-01-25
  • Contact: ZHANG Lijun;
  • Supported by:
    National Natural Science Foundation of China(41901138);National Natural Science Foundation of China(41671536);Humanities and Social Sciences Research Fund of Ministry of Education in China(19YJCZH225)


Rapid urban expansion has brought disorder and low efficiency to the socioeconomic development and the land utilizatopm. However, due to a large number of scale-free phenomena in the urban complex system, it is difficult to measure its morphological characteristics effectively. In essence, fractal is a hierarchical system which is related to the complex network cascade structure. Fractal structure can be used to measure the spatial cycle subdivision of urban geographic system, which plays an important role in the exploration on the law of urban morphology evolution, and provides an effective mathematical tool for the implementation of territorial spatial planning. Based on the urban fractal theory, this paper forms a logical frame for the evolution of urban functional land, through the dual analysis of the radial dimension and the grid dimension representing urban spatial form. By calculating and analyzing the empirical case of Zhengzhou, a typical representative of urban growth and evolution in China, the validity of using the second derivative to automatically identify the scaling range in the radius method is verified. Finally, the paper discusses the structural and functional problems hidden in the urban evolution process, and provides theoretical reference and method enlightenment for the exploration of integrated optimization scheme of the fractal urban system. The results show that: (1) Using the second derivative method to automatically identify the boundary of scaling range in the radius method can significantly improve the feasibility of the fractal model. The fitting accuracy R2 is increased from 0.920 to above 0.996. This method makes a breakthrough attempt on the application bottleneck in the fractal model of radius method, which is difficult to improve the fitting accuracy. This method has obvious advantages in simulating the urban growth, especially the evolution of urban spatial structure, and can judge the expansion speed and mode, so as to evaluate the city's health. (2) The double scale phenomenon exists in the urban spatial structure of Zhengzhou. The radius method model only fits effectively in the circular range of "the edge of urban center-inside the built-up area", which may have the growth characteristics of self-affine or random multi-fractal. (3) From 1982 to 2020, the urban form of Zhengzhou evolves into a central sprawl mode, and the urban center and the periphery present a dualistic trend. The interaction between the systems is insufficient, the system efficiency is lower, and the urban spatial structure can be further refined and upgraded.

Key words: urban growth, urban spatial form, scaling range identification, second derivative method, two-dimensional fractal model, radial dimension, grid dimension, Zhengzhou