Fractal Geography

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  • Format: Hardcover
  • Copyright: 2012-01-17
  • Publisher: Iste/Hermes Science Pub

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Our daily universe is rough and infinitely diverse. The fractal approach clarifies and orders these disparities. It helps us to envisage new explanations of geographical phenomena, which are, however, considered as definitely understood. Written for use by geographers and researchers from similar disciplines, such as ecologists, economists, historians and sociologists, this book presents the algorithms best adapted to the phenomena encountered, and proposes case studies illustrating their applications in concrete situations. An appendix is also provided that develops programs written in Mathematica.

Author Biography

Andr Dauphin is Emeritus Professor at University of Nice Sophia-Antipolis, France.

Table of Contents

Introductionp. xi
A Fractal Worldp. 1
Fractals pervade into geographyp. 2
From geosciences to physical geographyp. 3
Urban geography: a big beneficiaryp. 6
Forms of fractal processesp. 10
Some fractal forms that make use of the principle of allometryp. 11
Time series and processes are also fractalp. 12
Rank-size rules are generally fractal structuresp. 14
First reflections on the link between power laws and fractalsp. 14
Brief introduction into power lawsp. 15
Some power laws recognized before the fractal erap. 17
Conclusionp. 19
Auto-similar and Self-affine Fractalsp. 21
The rarity of auto-similar terrestrial formsp. 22
Yet more classes of self-affine fractal forms and processesp. 24
Brownian, fractional Brownian and multi-fractional Brownian motionp. 25
Levy modelsp. 32
Four examples of generalizations for simulating realistic formsp. 35
Conclusionp. 37
From the Fractal Dimension to Multifractal Spectrumsp. 39
Two extensions of the fractal dimension: lacunarity and codimensionp. 40
Some territorial textures differentiated by their lacunarityp. 40
Codimension as a relative fractal dimensionp. 41
Some corrections to the power laws: semifractals, parabolic fractals and log-periodic distributionsp. 43
Semifractals and double or truncated Pareto distributionsp. 43
The parabolic fractal modelp. 45
Log-periodic distributionsp. 46
A routine technique in medical imaging: fractal scanningp. 48
Multifractals used to describe all the irregularities of a set defined by measurementp. 50
Definition and characteristics of a multifractalp. 50
Two functions to interpret: generalized dimension spectrum and singularity spectrump. 52
An approach that is classical in geosciences but exceptional in social sciencesp. 54
Three potential generalizationsp. 56
Conclusionp. 57
Calculation and Interpretation of Fractal Dimensionsp. 59
Test data representing three categories of fractals: black and white maps, grayscale Landsat images and pluviometric chronicle seriesp. 60
A first incontrovertible stage: determination of the fractal class of the geographical phenomenon studiedp. 62
Successive tests using Fourier or .wavelet decompositionsp. 63
p. 73
Some algorithms for the calculation of the fractal dimensions of auto-similar objectsp. 75
Box counting, information and area measurement dimensions for auto-similar objectsp. 75
A geographically inconclusive application from perceptionp. 78
The fractal dimensions of objects and self-affine processesp. 80
A multitude of algorithmsp. 80
High irregularity of decadal rainfall for Barcelona and Beirutp. 84
Conclusionp. 85
The Fractal Dimensions of Rank-size Distributionsp. 87
Three test series: rainfall heights, urban hierarchies and attendance figures for major French museumsp. 88
The equivalence of the Zipf, Pareto and Power lawsp. 89
Three strategies for adjusting the rank-size distribution curvep. 92
A visual approach using graphsp. 92
Adjusting the only linear part of the curvep. 95
Choosing the best adjustment, and therefore the most pertinent lawp. 96
Which rank-size distribution should be used for Italian towns, the main French agglomerations and all French communes?p. 98
Conclusionp. 101
Calculation and Interpretation of Multifractal Spectrumsp. 103
Three data sets for testing multifractality: a chronicle series, a rank-size distribution and satellite imagesp. 104
Distinguishing multifractal and monofractal phenomenap. 104
An initial imperfect visual testp. 105
A second statistical test: generalized correlation dimensionsp. 107
Various algorithms for calculation of the singularity spectrump. 111
Generalized box-counting and variogram methodsp. 111
Methods derived from wavelet treatmentp. 112
Interpretation of singularity spectrumsp. 113
Possible generalizations of the multifractal approachp. 116
Conclusionp. 118
Geographical Explanation of Fractal Forms and Dynamicsp. 121
Turbulence generates fractal perturbations and multifractal pluviometric fieldsp. 122
The fractality of natural hazards and catastrophic impactsp. 126
Other explanations from fields of physical geographyp. 128
A new geography of populationsp. 129
Harmonization of town growth distributionsp. 131
Development and urban hierarchiesp. 132
Understanding the formation of communication and social networksp. 136
Conclusionp. 137
Using Complexity Theory to Explain a Fractal Worldp. 139
A bottomless pit debatep. 140
General mechanisms for explaining power lawsp. 143
Four theories on fractal universalityp. 144
Critical self-organization theoryp. 144
Bejan's constructal theoryp. 151
Nottale's scale relativity theoryp. 153
A general theory of morphogenesisp. 154
Chaos and fractal analysis theoryp. 163
Conclusionp. 164
Land-use Planning and Managing a Fractal Environmentp. 167
Fractals, extreme values and riskp. 168
Under estimated hazards in preliminary risk assessmentsp. 168
Fractal networks, fighting epidemics and Internet breakdownsp. 171
Fractals, segmentation and identification of objects in image processingp. 173
New image processing toolsp. 173
p. 177
Fractals, optimization and land managementp. 177
Fractal beauty and landscapinngp. 179
Conclusionp. 180
Conclusionp. 183
Some tools and methods for quantifying and qualifying multiscale coarseness and irregularityp. 184
A recap on geographical irregularities and disparitiesp. 186
A paradigm that gives rise to new land-use management practicesp. 189
Appendicesp. 191
Preliminary thoughts on fractal analysis softwarep. 191
Instructions for the following programsp. 192
Software programs for the visual approach of a satellite or cartographic series or imagep. 193
Software programs for calculating fractal dimensions for a chronicle or frequency seriesp. 198
Software programs for calculating the fractal dimensions of a satellite image or mapp. 208
Software programs for calculating multifractal spectrums of a series and an imagep. 213
Bibliographyp. 221
Indexp. 239
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