Fractal Geography / Edition 1

by André Dauphiné

Overview

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.

Contents

1. A Fractal World.
2. Auto-similar and Self-affine Fractals.
3. From the Fractal Dimension to Multifractal Spectrums.
4. Calculation and Interpretation of Fractal Dimensions.
5. The Fractal Dimensions of Rank-size Distributions.
6. Calculation and Interpretation of Multifractal Spectrums.
7. Geographical Explanation of Fractal Forms and Dynamics.
8. Using Complexity Theory to Explain a Fractal World.
9. Land-use Planning and Managing a Fractal Environment.

Product Details

ISBN-13:	9781848213289
Publisher:	Wiley
Publication date:	01/17/2012
Series:	ISTE Series , #594
Pages:	241
Product dimensions:	6.00(w) x 9.30(h) x 1.10(d)

About the Author

André Dauphiné is Emeritus Professor at the University of Nice Sophia-Antipolis, France.

Table of Contents

Introduction xi

Chapter 1 A Fractal World 1

1.1 Fractals pervade into geography 2

1.1.1 From geosciences to physical geography 3

1.1.2 Urban geography: a big beneficiary 6

1.2 Forms of fractal processes 10

1.2.1 Some fractal forms that make use of the principle of allometry 11

1.2.2 Time series and processes are also fractal 12

1.2.3 Rank-size rules are generally fractal structures 14

1.3 First reflections on the link between power laws and fractals 14

1.3.1 Brief introduction into power laws 15

1.3.2 Some power laws recognized before the fractal era 17

1.4 Conclusion 19

Chapter 2 Auto-similar and Self-affine Fractals 21

2.1 The rarity of auto-similar terrestrial forms 22

2.2 Yet more classes of self-affine fractal forms and processes 24

2.2.1 Brownian, fractional Brownian and multi-fractional Brownian motion 25

2.2.2 Levy models 32

2.2.3 Four examples of generalizations for simulating realistic forms 35

2.3 Conclusion 37

Chapter 3 From the Fractal Dimension to Multifractal Spectrums 39

3.1 Two extensions of the fractal dimension: lacunarity and codimension 40

3.1.1 Some territorial textures differentiated by their lacunarity 40

3.1.2 Codimension as a relative fractal dimension 41

3.2 Some corrections to the power laws: semifractals, parabolic fractals and log-periodic distributions 43

3.2.1 Semifractals and double or truncated Pareto distributions 43

3.2.2 The parabolic fractal model 45

3.2.3 Log-periodic distributions 46

3.3 A routine technique in medical imaging: fractal scanning 48

3.4 Multifractals used to describe all the irregularities of a set defined by measurement 50

3.4.1 Definition and characteristics of a multifractal 50

3.4.2 Two functions to interpret: generalized dimension spectrum and singularity spectrum 52

3.4.3 An approach that is classical in geosciences but exceptional in social sciences 54

3.4.4 Three potential generalizations 56

3.5 Conclusion 57

Chapter 4 Calculation and Interpretation of Fractal Dimensions 59

4.1 Test data representing three categories of fractals: black and white maps, grayscale Landsat images and pluviometric chronicle series 60

4.2 A first incontrovertible stage: determination of the fractal class of the geographical phenomenon studied 62

4.2.1 Successive tests using Fourier or .wavelet decompositions 63

4.2.2Decadal rainfall in Barcelona and Beirut are fractional Gaussian noise 73

4.3 Some algorithms for the calculation of the fractal dimensions of auto-similar objects 75

4.3.1 Box counting, information and area measurement dimensions for auto-similar objects 75

4.3.2 A geographically inconclusive application from perception 78

4.4 The fractal dimensions of objects and self-affine processes 80

4.4.1 A multitude of algorithms 80

4.4.2 High irregularity of decadal rainfall for Barcelona and Beirut 84

4.5 Conclusion 85

Chapter 5 The Fractal Dimensions of Rank-size Distributions 87

5.1 Three test series: rainfall heights, urban hierarchies and attendance figures for major French museums 88

5.2 The equivalence of the Zipf, Pareto and Power laws 89

5.3 Three strategies for adjusting the rank-size distribution curve 92

5.3.1 A visual approach using graphs 92

5.3.2 Adjusting the only linear part of the curve 95

5.3.3 Choosing the best adjustment, and therefore the most pertinent law 96

5.3.4 Which rank-size distribution should be used for Italian towns, the main French agglomerations and all French communes? 98

5.4 Conclusion 101

Chapter 6 Calculation and Interpretation of Multifractal Spectrums 103

6.1 Three data sets for testing multifractality: a chronicle series, a rank-size distribution and satellite images 104

6.2 Distinguishing multifractal and monofractal phenomena 104

6.2.1 An initial imperfect visual test 105

6.2.2 A second statistical test: generalized correlation dimensions 107

6.3 Various algorithms for calculation of the singularity spectrum 111

6.3.1 Generalized box-counting and variogram methods 111

6.3.2 Methods derived from wavelet treatment 112

6.3.3 Interpretation of singularity spectrums 113

6.4 Possible generalizations of the multifractal approach 116

6.5 Conclusion 118

Chapter 7 Geographical Explanation of Fractal Forms and Dynamics 121

7.1 Turbulence generates fractal perturbations and multifractal pluviometric fields 122

7.2 The fractality of natural hazards and catastrophic impacts 126

7.3 Other explanations from fields of physical geography 128

7.4 A new geography of populations 129

7.5 Harmonization of town growth distributions 131

7.6 Development and urban hierarchies 132

7.7 Understanding the formation of communication and social networks 136

7.8 Conclusion 137

Chapter 8 Using Complexity Theory to Explain a Fractal World 139

8.1 A bottomless pit debate 140

8.2 General mechanisms for explaining power laws 143

8.3 Four theories on fractal universality 144

8.3.1 Critical self-organization theory 144

8.3.2 Bejan's constructal theory 151

8.3.3 Nottale's scale relativity theory 153

8.3.4 A general theory of morphogenesis 154

8.3.5 Chaos and fractal analysis theory 163

8.4 Conclusion 164

Chapter 9 Land-use Planning and Managing a Fractal Environment 167

9.1 Fractals, extreme values and risk 168

9.1.1 Under estimated hazards in preliminary risk assessments 168

9.1.2 Fractal networks, fighting epidemics and Internet breakdowns 171

9.2 Fractals, segmentation and identification of objects in image processing 173

9.2.1 New image processing tools 173

9.2.2Some little-used fractal approaches using a GIS 177

9.3 Fractals, optimization and land management 177

9.4 Fractal beauty and landscapinng 179

9.5 Conclusion 180

Conclusion 183

C.1 Some tools and methods for quantifying and qualifying multiscale coarseness and irregularity 184

C.2 A recap on geographical irregularities and disparities 186

C.3 A paradigm that gives rise to new land-use management practices 189

Appendices 191

A.l Preliminary thoughts on fractal analysis software 191

A.2 Instructions for the following programs 192

A.3 Software programs for the visual approach of a satellite or cartographic series or image 193

A.4 Software programs for calculating fractal dimensions for a chronicle or frequency series 198

A.5 Software programs for calculating the fractal dimensions of a satellite image or map 208

A.6 Software programs for calculating multifractal spectrums of a series and an image 213

Bibliography 221

Index 239