城市地理研究中的单分形、多分形和自仿射分形

doi:10.18306/dlkxjz.2019.01.004

摘要/Abstract

摘要：

分形几何学在城市地理研究中具有广泛的应用,然而很多基本概念却让初学者感到迷惑。如何区分单分形、自仿射分形与多分形,是一个基本而重要的问题。简单分形容易理解,而真实的地理现象很少是单分形的。城市生长过程具有自仿射特征,而城市空间格局却具有多分形性质。作者发现,各种分形的共性在于三个方面：标度律、分数维和熵守恒。论文基于标度、分维和熵守恒公式,借助隐喻城市生长的规则分形来区分单分形、多分形和自仿射分形,讨论分形系统演化的机理、分形与空间自相关和空间异质性的联系,同时澄清一些在地理分形研究中的常见错误概念。最后以城市位序-规模分布为例,说明并对比单分形和多分形在城市地理研究中的建模与应用思路。

关键词: 地理分形, 单分形, 双分形, 多分形, 自仿射分形, 多分维谱, 分形城市

Abstract:

Fractal geometry provides a powerful tool for scale-free spatial modeling and analyses in geography. However, a number of basic concepts are puzzling. The three common fractals, that is, monofractal (unifractal), multifractals, and self-affine fractal, are often misunderstood by students of geography. This article clarifies some confusing fractal concepts for urban fractal modeling and fractal dimension analysis. Using simple mathematical models based on three growing fractals that bear an analogy to urban growth, we can distinguish the three types of common fractal structure. The similarities and differences between monofractal, multifractals, and self-affine fractal are as follows: 1) A monofractal is a simple self-similar fractal that bears only one scaling factor (scaling ratio), and a multifractal object is a complex fractal system that bears at least two scaling factors for different parts. Each scaling factor dominates all different scales and is independent of directions and levels. 2) A self-affine fractal bears different scaling factors in different directions of growth or at different levels of scales. The basic feature of self-affine growing fractal is anisotropy, which differs from the isotropic self-similar growing fractals. 3) Both self-affine fractal and multifractals may possess two scaling factors, but there are essential differences between self-affine fractals and multifractals. A self-affine fractal often takes on the form of bi-fractals, which can be reflected by two scaling ranges on a log-log plot. However, there is only one scaling range for a multifractal pattern. As an example, two-scaling fractal modeling is applied to the rank-size distributions of cities to illustrate the concept of urban multifractals. By comparison with these multifractal models, we can better understand monofractals and self-affine fractals in geographical research.

Key words: geographical fractals, monofractal (unifractal), bi-fractals, multifractals, self-affine fractal, multifractal dimension spectrum, fractal cities

陈彦光. 城市地理研究中的单分形、多分形和自仿射分形[J]. 地理科学进展, 2019, 38(1): 38-49.

Yanguang CHEN. Monofractal, multifractals, and self-affine fractals in urban studies[J]. PROGRESS IN GEOGRAPHY, 2019, 38(1): 38-49.

图/表 7

图1

图2

图3

表1

表2

图4

图5

参考文献 49

[1]	陈彦光. 2008. 分形城市系统: 标度、对称和空间复杂性 [M]. 北京: 科学出版社.
	[Chen Y G.2008. Fractal urban systems: Scaling, symmetry, and spatial complexity. Beijing, China: Science Press. ]
[2]	陈彦光. 2015. 简单、复杂与地理分布模型的选择[J]. 地理科学进展, 34(3): 321-329.
	[Chen Y G.2015. Simplicty, complexity, and mathematical modeling of geographical distributions. Progress in Geography, 34(3): 321-329. ]
[3]	陈彦光. 2017. 城市形态的分维估算与分形判定[J]. 地理科学进展, 36(5): 529-539.
	[Chen Y G.2017. Approaches to estimating fractal dimension and identifying fractals of urban form. Progress in Geography, 36(5): 529-539. ]
[4]	陈勇, 陈嵘, 艾南山, 等. 1993. 城市规模分布的分形研究[J]. 经济地理, 13(3): 48-53.
	[Chen Y, Chen R, Ai N S, et al.1993. On the fractal property of city-size distributions. Economical Geography, 13(3): 48-53. ]
[5]	王放, 李后强. 1995. 非线性人口学导论 [M]. 成都: 四川大学出版社.
	[Wang F, Li Q.1995. Introduction to nonlinear demography . Chengdu, China: Sichuan University Press. ]
[6]	周一星. 1998. 主要经济联系方向论[J]. 城市规划, 22(2): 22-25.
	[Zhou Y X.1998. Major directions of economic linkages: Some theoretical considerations. City Planning Review, 22(2): 22-25. ]
[7]	朱晓华. 2007. 地理空间信息的分形与分维 [M]. 北京: 测绘出版社.
	[Zhu X H.2007. Fractals and fractal dimensions of spatial geo-information . Beijing, China: Surveying and Mapping Press. ]
[8]	Anselin L.1996. The moran scatterplot as an ESDA tool to assess local instability in spatial association[M]// Fischer M, Scholten H J, Unwin D. Spatial analytical perspectives on GIS. London, UK: Taylor & Francis: 111-125.
[9]	Appleby S.1996. Multifractal characterization of the distribution pattern of the human population[J]. Geographical Analysis, 28(2): 147-160.
[10]	Ariza-Villaverde A B, Jimenez-Hornero F J, De Rave E G.2013. Multifractal analysis of axial maps applied to the study of urban morphology[J]. Computers, Environment and Urban Systems, 38: 1-10. doi: 10.1016/j.compenvurbsys.2012.11.001
[11]	Batty M, Longley P A.1994. Fractal cities: A geometry of form and function [M]. London, UK: Academic Press.
[12]	Batty M.2010. Space, scale, and scaling in entropy maximizing[J]. Geographical Analysis, 42(4): 395-421. doi: 10.1111/j.1538-4632.2010.00800.x
[13]	Benguigui L, Blumenfeld-Lieberthal E, Czamanski D.2006. The dynamics of the Tel Aviv morphology[J]. Environment and Planning B: Planning and Design, 33(2): 269-284. doi: 10.1068/b31118
[14]	Benguigui L, Daoud M.1991. Is the suburban railway system a fractal?[J]. Geographical Analysis, 23(4): 362-368.
[15]	Chen Y G.2011. Derivation of the functional relations between fractal dimension and shape indices of urban form[J]. Computers, Environment and Urban Systems, 35(6): 442-451. doi: 10.1016/j.compenvurbsys.2011.05.008
[16]	Chen Y G.2012a. The rank-size scaling law and entropy-maximizing principle[J]. Physica A, 391(3): 767-778. doi: 10.1016/j.physa.2011.07.010
[17]	Chen Y G.2012b. Zipf's law, 1/f noise, and fractal hierarchy[J]. Chaos, Solitons & Fractals, 45(1): 63-73.
[18]	Chen Y G.2015. The distance-decay function of geographical gravity model: Power law or exponential law?[J]. Chaos, Solitons & Fractals, 77: 174-189.
[19]	Chen Y G.2016. The evolution of Zipf's law indicative of city development[J]. Physica A, 443: 555-567. doi: 10.1016/j.physa.2015.09.083
[20]	Chen Y G, Lin J Y.2009. Modeling the self-affine structure and optimization conditions of city systems using the idea from fractals[J]. Chaos, Solitons & Fractals, 41(2): 615-629.
[21]	Chen Y G, Wang J J.2013. Multifractal characterization of urban form and growth: The case of Beijing[J]. Environment and Planning B: Planning and Design, 40(5): 884-904. doi: 10.1068/b36155
[22]	Chen Y G, Zhou Y X.2003. The rank-size rule and fractal hierarchies of cities: Mathematical models and empirical analyses[J]. Environment and Planning B: Planning and Design, 30(6): 799-818. doi: 10.1068/b2948
[23]	Chen Y G, Zhou Y X.2004. Multi-fractal measures of city-size distributions based on the three-parameter Zipf model[J]. Chaos, Solitons & Fractals, 22(4): 793-805.
[24]	Cheng Q.1995. The perimeter-area fractal model and its application in geology[J]. Mathematical Geology, 27(1): 69-82. doi: 10.1007/BF02083568
[25]	Couclelis H.1997. From cellular automata to urban models: New principles for model development and implementation[J]. Environment and Planning B: Planning and Design, 24(2): 165-174. doi: 10.1068/b240165
[26]	Feder J.1988. Fractals [M]. New York: Plenum Press, 1988.
[27]	Frankhauser P.1994. The fractal aspects of urban structures[M]. Paris, French: Economica.
[28]	Frankhauser P.1998. The fractal approach: A new tool for the spatial analysis of urban agglomerations[J]. Population: An English Selection, 10(1): 205-240.
[29]	Goodchild M F, Mark D M.1987. The fractal nature of geographical phenomena[J]. Annals of Association of American Geographers, 77(2): 265-278. doi: 10.1111/j.1467-8306.1987.tb00158.x
[30]	Goodchild M F.1980. Fractals and the accuracy of geographical measures[J]. Mathematical Geology, 12(2): 85-98. doi: 10.1007/BF01035241
[31]	Goodchild M F.2004. GIScience, geography, form, and process[J]. Annals of the Association of American Geographers, 94(4): 709-714.
[32]	Grassberger P.1983. Generalized dimensions of strange attractors[J]. Physics Letters A, 97(6): 227-230. doi: 10.1016/0375-9601(83)90753-3
[33]	Halsey T C, Jensen M H, Kadanoff L P, et al.1986. Fractal measure and their singularities: The characterization of strange sets[J]. Physical Review A, 33: 1141-1151. doi: 10.1103/PhysRevA.33.1141
[34]	Hentschel H E, Procaccia I.1983. The infinite number of generalized dimensions of fractals and strange attractors[J]. Physica D: Nonlinear Phenomena, 8: 435-444. doi: 10.1016/0167-2789(83)90235-X
[35]	Hurst H E, Black R P, Simaika Y M.1965. Long-term storage: An experimental study[M]. London, UK: Constable.
[36]	Imre A R, Bogaert J.2004. The fractal dimension as a measure of the quality of habitat[J]. Acta Biotheoretica, 52(1): 41-56. doi: 10.1023/B:ACBI.0000015911.56850.0f
[37]	Jullien R, Botet R.1987. Aggregation and fractal aggregates [M]. Singapore: World Scientific Publishing.
[38]	Lam N S-N, De Cola L.1993. Fractals in geography[M]. Englewood Cliffs, NJ: PTR Prentice Hall.
[39]	Longley P A, Batty M, Shepherd J.1991. The size, shape and dimension of urban settlements[J]. Transactions of the Institute of British Geographers, 16(1): 75-94. doi: 10.2307/622907
[40]	Mandelbrot B B.1982. The fractal geometry of nature[M]. New York, USA: W. H. Freeman and Company.
[41]	Murcio R, Masucci A P, Arcaute Eet al.2015. Multifractal to monofractal evolution of the London street network[J]. Physical Review E, 92: 062130. doi: 10.1103/PhysRevE. 92.062130.
[42]	Stanley H E, Meakin P.1988. Multifractal phenomena in physics and chemistry[J]. Nature, 335: 405-409. doi: 10.1038/335405a0
[43]	Tobler W.2004. On the first law of geography: A reply[J]. Annals of the Association of American Geographers, 94(2): 304-310. doi: 10.1111/j.1467-8306.2004.09402009.x
[44]	Vicsek T.1989. Fractal growth phenomena [M]. Singapore: World Scientific Publishing.
[45]	Wang J F, Liu X H, Peng L, et al.2012. Cities evolution tree and applications to predicting urban growth[J]. Population & Environment, 33(2-3): 186-201.
[46]	White R, Engelen G.1993. Cellular automata and fractal urban form: A cellular modeling approach to the evolution of urban land-use patterns[J]. Environment and Planning A, 25(8): 1175-1199. doi: 10.1068/a251175
[47]	White R, Engelen G.1994. Urban systems dynamics and cellular automata: Fractal structures between order and chaos[J]. Chaos, Solitons & Fractals, 4(4): 563-583.
[48]	Wilson A G.2000. Complex spatial systems: The modelling foundations of urban and regional analysis[M]. Singapore: Pearson Education.
[49]	Wilson A G.2010. Entropy in urban and regional modelling: Retrospect and prospect[J]. Geographical Analysis, 42(4): 364-394. doi: 10.1111/j.1538-4632.2010.00799.x

矩的阶次q	全局参数		局部参数
矩的阶次q	D_q	τ(q)	α(q)	f(α(q))
-∞	1.7604	-∞	1.7604	0
-100	1.7429	-176.0374	1.7604	0
-2	1.6054	-4.8161	1.6153	1.5855
-1	1.6022	-3.2044	1.6081	1.5963
0	1.5995	1.5995	1.6020	1.5995
1	1.5970	0	1.5970	1.5970
2	1.5949	1.5949	1.5930	1.5910
100	1.5798	156.3975	1.5791	1.5129
∞	1.5791	∞	1.5791	0

矩的阶次q	全局参数		局部参数
矩的阶次q	D_q	τ(q)	α(q)	f(α(q))
-∞	1.3219	-∞	1.3219	0
-50	1.2960	-66.0964	1.3219	0
-2	1.0581	-3.1744	1.1419	0.8905
-1	1.0294	-2.0589	1.0879	0.9710
0	1.0000	-1.0000	1.0294	1.0000
1	0.9710	0.9710	0.9710	0.9710
2	0.9434	0.9434	0.9170	0.8905
50	0.7520	36.8483	0.7370	0
∞	0.7370	∞	0.7370	0