Fractals and Chaos in Geology and Geophysics

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Author: Donald L. Turcotte

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mn CM:9209630051 FRRCTRLS I D CHROS I N GEDLDGY lUTHUR : CMBRIDGE 336 EMTH SCIEHCE

The fundamental concepts of fractal geometry and chaotic dynamics, along with the related concepts of multifractals, self-similar time series, wavelets, and self-organized criticality, are introduced in this book, for a broad range of readers interested in complex natural phenomena. Now in a greatly expanded, second edition, this book relates fractals and chaos to a variety of geological and geophysical applications. These include drainage networks and erosion, floods, earthquakes, mineral and petroleum resources, fragmentation, mantle convection, and magnetic field generation. Many advances have been made in the field since the first edition was published. In this new edition coverage of self-organized criticality is expanded and statistics and time series are included to provide a broad background for the reader. All concepts are introduced at the lowest possible level of mathematics consistent with their understanding, so that the reader requires only a background in basic physics and mathematics. Fractals and Chaos in Geology and Geophysics can be used as a text for advanced undergraduate and graduate courses in the physical sciences. Problems are included for the reader to solve.

FRACTALS AND CHAOS IN GEOLOGY AND GEOPHYSICS

FRACTA CHAOS GEOLOGY A GEOPHYSICS Second Edition

DONALD L. TURCOTTE Cornell University

CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S l o Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521561648 O Donald L. Turcotte 1997

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1992 Second edition 1997

A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publica~iondata Turcotte, Donald Lawson. Fractals and chaos in geology and geophysics / Donald L. Turcotte. - 2nd ed. p. cm. Includes bibliographical references (p. 343-70) and index. ISBN 0-521-56164-7 (hc). -ISBN 0-521-56733-5 (pbk.) 1. Geology - Mathematics. 2. Geophysics - Mathematics. 3. Fractals. 4. Chaotic behavior in systems. I. Title. QE33.2.M3T87 1997 96-3 1558 550'. 1'51474 - dc20 CIP ISBN 978-0-521-56164-8 hardback ISBN 978-0-52 1-56733-6 paperback Transferred to digital printing (with amendments) 2007 The color figures within this publication have been removed for this digital reprinting. At the time of going to press the original images were available in color for download from http://www.cambridge.org/9780521567336

CONTENTS

Preface Preface to the second edition 1 Scale invariance

2 Definition of a fractal set 2.1 Deterministic fractals 2.2 Statistical fractals 2.3 Depositional sequences 2.4 Why fractal distributions?

3 Fragmentation 3.1 Background 3.2 Probability and statistics 3.3 Fragmentation data 3.4 Fragmentation models 3.5 Porosity 4 Seismicity and tectonics 4.1 Seismicity 4.2 Faults 4.3 Spatial distribution of earthquakes 4.4 Volcanic eruptions 5 Ore grade and tonnage 5.1 Ore-enrichment models 5.2 Ore-enrichment data 5.3 Petroleum data

6 Fractal clustering 6.1 Clustering 6.2 Pair-correlation techniques 6.3 Lacunarity 6.4 Multifractals

vi

CONTENTS

7 Self-affine fractals 7.1 Definition of a self-affine fractal 7.2 Time series 7.3 Self-affine time series 7.4 Fractional Gaussian noises and fractional Brownian walks 7.5 Fractional log-normal noises and walks 7.6 Rescaled-range (WS)analysis 7.7 Applications of self-affine fractals 8 Geomorphology Drainage networks Fractal trees Growth models Diffusion-limited aggregation (DLA) Models for drainage networks Models for erosion and deposition Floods Wavelets

9 Dynamical systems 9.1 Nonlinear equations 9.2 Bifurcations 10 Logistic map 10.1 Chaos 10.2 Lyapunov exponent 11 Slider-block models

12 Lorenz equations 13 Is mantle convection chaotic?

14 Rikitake dynamo 15 Renormalization group method 15.1 Renormalization 15.2 Percolation clusters 15.3 Applications to fragmentation 15.4 Applications to fault rupture 15.5 Log-periodic behavior 16 Self-organized criticality 16.1 Sand-pile models 16.2 Slider-block models 16.3 Forest-fire models

CONTENTS

17

Where do we stand?

References Appendix A: Glossary of terms Appendix B: Units and symbols Answers to selected problems Index

vii

PREFACE

I was introduced to the world of fractals and renormalization groups by Bob Smalley in 198 1. At that time Bob had transferred from physics to geology at Cornell as a Ph.D. student. He organized a series of seminars and convinced me of the relevance of these techniques to geological and geophysical problems. Although his official Ph.D. research was in observational seismology, Bob completed several renormalization and fractal projects with me. Subsequently, my graduate students Jie Huang and Cheryl Stewart have greatly broadened my views of the world of chaos and dynamical systems. Original research carried out by these students is included throughout this book. The purpose of this book is to introduce the fundamental principles of fractals, chaos, and aspects of dynamical systems in the context of geological and geophysical problems. My goal is to introduce the fundamental concepts at the lowest level of mathematics that is consistent with the understanding and application of the concepts. It is clearly impossible to discuss all aspects of applications. I have tried to make the applications reasonably comprehensible to non-earth scientists but may not have succeeded in all cases. After an introduction, the next seven chapters are devoted to fractals. The fundamental concepts of self-similar fractals are introduced in Chapter 2. Applications of self-similar fractals to fragmentation, seismicity and tectonics, ore grades and tonnage, and clustering are given in the next chapters. Self-affine fractals are introduced in Chapter 7 and are applied to geomorphology in Chapter 8. A brief introduction to dynamical systems is given in Chapter 9. The fundamental concepts of chaos are introduced through the logistic map, slider-block models, the Lorenz equations, mantle convection, and the Rikitake dynamo in the next five chapters. The renormalization group method is introduced in Chapter 15 and self-organized criticality is considered in Chapter 16. Problems are included so that this book can be used as a higher-level undergraduate text or a graduate text, depending upon the background of the

x

PREFACE

students and the material used. Little mathematical background is required for the introduction to self-similar fractals and chaos that includes Chapters 1-6 and 10-1 1. The treatment of self-affine fractals in Chapters 7-8 requires some knowledge of spectral techniques. Chapters 9 and 12-13 require a knowledge of differential equations. I would like to dedicate this book to the memory of Ted Flinn. Until his untimely death in 1989 Ted was Chief of the Geodynamics Branch at NASA Headquarters. In this position, over a period of ten years, he supervised a program that changed routine centimeter-level geodetic position measurements from a dream to a reality. Ted also had the foresight to support research on fractals and chaos applied to crustal dynamics at a time when these subjects were anything but popular. In particular his enthusiasm was instrumental in a conference on earthquakes, fractals, and chaos held at the Asilomar conference facility in January 1989. This conference established a dialogue between physicists, applied mathematicians, and seismologists focused on the applications of dynamical systems to earthquake prediction. I would also like to acknowledge extensive discussions with John Rundle, Charlie Sammis, Chris Barton, and Per Bak. Chapter 8 is largely the result of a collaboration with Bill Newman. And this book could not have been completed without the diligent manuscript preparation of Maria Petricola.

PREFACETOTHE SECOND EDITION

A large number of new results on fractals, chaos, and self-organized criticality applied to problems in geology and geophysics have appeared since the first edition of this book was published in 1992. Evidence for this comes from the large number of new references included in this edition, increasing from 160 to over 500. Because of the rapid advances in knowledge it became evident shortly after the publication of the first edition that a second edition would be required in a few years. A large number of additions and some deletions have been made in preparing this second edition. In Chapter 2 a comprehensive treatment of the completeness of the sedimentary record has been used to introduce the application of fractal techniques to geological problems. To make this textbook more complete, a brief introduction to probability and statistics has been included in Chapter 3. Chapter 4 on seismicity and tectonics has been extensively revised to include the work that has been recently carried out on the spatial distributions of earthquakes and fractures. In Chapter 5 the elegant work by Claude Allkgre and his associates explaining power-law (fractal) distributions of mineral deposits has been added. A major addition to the second edition is the comprehensive treatment of multifractals in Chapter 6. Also added to this chapter on fractal clustering are pair-correlation techniques and lacunarity. One of the most extensive revisions concerns the treatment of self-affine fractals in Chapter 7. An introductory section on time series has been added as well as deterministic examples of self-affine fractals. Additional techniques for generating fractional Gaussian noises and fractional Brownian walks have been added as well as a treatment of rescaled-range (RIS) analyses. Chapter 8 on geomorphology is almost entirely new. An in-depth treatment of fractal trees has been added, including the Tokunaga taxonomy of quantifying side branching. Also added to this chapter are treatments of growth models, models for erosion and deposition, floods, and wavelet filtering techniques. Chapters 9-14 remain essentially unchanged as there have

xii

PREFACE TO THE SECOND EDITION

been relatively few new advances concerning the fundamental aspects of chaos. A major addition to Chapter 15 on the renormalization group method has been the introduction of log-periodic behavior. If the fractal exponent is complex, log-periodic behavior is found. Applications of log-periodic concepts may be one of the most promising avenues for future development. Chapter 15 on self-organized criticality has been extensively revised. Although the entire concept of self-organized criticality is controversial, the models developed appear to cross the gap between the chaotic behavior of low-order systems and the complex, often fractal behavior of high-order systems. The development of forest-fire models is of particular interest and a section on these has been added. Systems that exhibit self-organized critical behavior can satisfy Maxwell-Boltzmann statistics and thus are closely linked to classical statistical mechanics. Extensive discussions and collaborative work with John Rundle have been particularly valuable in developing the sections in Chapter 16 on this subject. Again my graduate students Bruce Malamud, Jon Pelletier, Gleb Morein, Algis Kucinskas, Lesley Greene, Lygia Gomes Da Silva, and Kirk Haselton have made major contributions to the preparation of this second edition. Important contributions were also made by Galena Narkounskaia and Andrei Gabrielov, visiting scientists from the Institute for Earthquake Prediction and Theoretical Geophysics in Moscow. The director of the Institute, Volodya Keilis-Borok, has been an inspiration in providing connections between dynamical systems and earthquake prediction. His Institute brings a level of mathematics to geophysical problems that is not available in the United States. Once again I would like to acknowledge extensive discussions and collaborative research with Bill Newman, John Rundle, Chris Barton, Charlie Sammis, and Claude Allkgre. This second edition could not have been completed without the diligent manuscript preparation of Marilyn Grant and Sue Peterson and the figure preparation of Teresa Howley.

Chapter One

SCALE INVARIANCE

A stone, when it is examined, will be found a mountain in miniature. The fineness of Nature's work is so great, that, into a single block, a foot or two in diameter, she can compress as many changes of form and structure, on a small scale, as she needs for her mountains on a large one; and, taking moss for forests, and grains of crystal for crags, the sur$ace of a stone, in by far the plurality of instances, is more interesting than the surface of an ordinary hill; more fantastic in form, and incomparably richer in colour - the last quality being most noble in stones of good birth (that is to say, fallen from the crystalline mountain ranges). J. Ruskin, Modern Painters, Vol. 5, Chapter 18 (1860)

The scale invariance of geological phenomena is one of the first concepts taught to a student of geology. It is pointed out that an object that defines the scale, i.e., a coin, a rock hammer, a person, must be included whenever a photograph of a geological feature is taken. Without the scale it is often impossible to determine whether the photograph covers 10 cm or 10 km. For example, self-similar folds occur over this range of scales. Another example would be an aerial photograph of a rocky coastline. Without an object with a characteristic dimension, such as a tree or house, the elevation of the photograph cannot be determined. It was in this context that Mandelbrot (1967) introduced the concept of fractals. The length of a rocky coastline is obtained using a measuring rod with a specified length. Because of scale invariance, the length of the coastline increases as the length of the measuring rod decreases according to a power law; the power determines the fractal dimension of the coastline. It is not possible to obtain a specific value for the length of a coastline, owing to all the small indentations down to a scale of millimeters or less. Many geological phenomena are scale invariant. Examples include frequency-size distributions of rock fragments, faults, earthquakes, volcanic eruptions, mineral deposits, and oil fields. A fractal distribution requires that

2

SCALE INVARIANCE

the number of objects larger than a specified size has a power-law dependence on the size. The empirical applicability of power-law statistics to geological phenomena was recognized long before the concept of fractals was conceived. A striking example is the Gutenberg-Richter relation for the frequency-magnitude statistics of earthquakes (Gutenberg and Richter, 1954). The proportionality factor in the relationship between the number of earthquakes and earthquake magnitude is known as the b-value. It has been recognized for nearly 50 years that, almost universally, b = 0.9. It is now accepted that the Gutenberg-Richter relationship is equivalent to a fractal relationship between the number of earthquakes and the characteristic size of the rupture; the value of the fractal dimension D is simply twice the b-value; typically D = 1.8 for distributed seismicity. Power-law distributions are certainly not the only statistical distributions that have been applied to geological phenomena. Other examples include the normal (Gaussian) distribution and the log-normal distribution. However, the power-law distribution is the only distribution that does not include a characteristic length scale. Thus the power-law distribution must be applicable to scale-invariant phenomena. If a specified number of events are statistically independent, the central-limit theorem provides a basis for the applicability of the Gaussian distribution. Scale invariance provides a rational basis for the applicability of the power-law, fractal distribution. Fractal concepts can also be applied to continuous distributions; an example is topography. Mandelbrot (1982) has used fractal concepts to generate synthetic landscapes that look remarkably similar to actual landscapes. The fractal dimension is a measure of the roughness of the features. The earth's topography is a composite of many competing influences. Topography is created by tectonic processes including faulting, folding, and flexure. It is modified and destroyed by erosion and sedimentation. There is considerable empirical evidence that erosion is scale invariant and fractal; a river network is a classic example of a fractal tree. Topography often appears to be complex and chaotic, yet there is order in the complexity. A standard approach to the analysis of a continuous function such as topography along a linear track is to determine the coefficients An in a Fourier series as a function of the wavelength An. If the amplitudes A,, have a power-law dependence on wavelength An,a fractal distribution may result. For topography and bathymetry it is found that, to a good approximation, the Fourier amplitudes are proportional to the wavelengths. This is also true for a Brownian walk, which can be generated by the random walk process as follows. Take a step forward and flip a coin; if tails occurs take a step to the right and if heads occurs take a step to the left; repeat the process. The divergence of the walk or signal increases in proportion to the square root of the number of steps. A spectral analysis of the random walk shows that the Fourier coefficients A,, are proportional to wavelength A,,.

SCALE INVARIANCE

Many geophysical data sets have power-law spectra. These include surface gravity and magnetics as well as topography. Since power-law spectra are defined by two quantities, the amplitude and the slope, these quantities can be used to carry out textural analyses of data sets. The fractal structure can also be used as the basis for interpolation between tracks where data have been obtained. A specific example is the determination of the threedimensional distribution of porosity in an oil reservoir from a series of well logs from oil wells. The philosophy of fractals has been beautifully set forth by their inventor Benoit Mandelbrot (Mandelbrot, 1982). A comprehensive treatment of fractals from the point of view of applications has been given by Feder (1988). Vicsek (1992) has also given an extensive treatment of fractals emphasizing growth phenomena. Kaye (1989, 1993) covers a broad range of fractal problems emphasizing those involving particulate matter. Korvin (1992) has considered many fractal applications in the earth sciences. Although fractal distributions would be useful simply as a means of quantifying scale-invariant distributions, it is now becoming evident that their applicability to geological problems has a more fundamental basis. Lorenz (1963) derived a set of nonlinear differential equations that approximate thermal convection in a fluid. This set of equations was the first to be shown to exhibit chaotic behavior. Infinitesimal variations in initial conditions led to first-order differences in the solutions obtained. This is the definition of chaos. The equations are completely deterministic; however, because of the exponential sensitivity to initial conditions, the evolution of a chaotic solution is not predictable. The evolution of the solution must be treated statistically and the applicable statistics are often fractal. A comprehensive study of problems in chaos has been given by Schuster (1995). The most universal example of chaotic behavior is fluid turbulence. It has long been recognized that turbulent flows must be treated statistically and that the appropriate spectral statistics are fractal. Since the flows in the earth's core that generate the magnetic field are expected to be turbulent, it is not surprising that they are also chaotic. The random reversals of the earth's magnetic field are a characteristic of chaotic behavior. In fact, solutions of a parameterized set of dynamo equations proposed by Rikitake (1958) exhibited spontaneous reversals and were subsequently shown to be examples of deterministic chaos (Cook and Roberts, 1970). Recursion relations can also exhibit chaotic behavior. The classic example is the logistic map studied by May (1976). This simple quadratic relation has an amazing wealth of behavior. As the single parameter in the equation is varied, the period of the recursive solution doubles until the solution becomes fully chaotic. The Lyapunov exponent is the quantitative test of chaotic behavior; it is a measure of whether adjacent solutions converge or diverge. If the Lyapunov exponent is positive, the adjacent solutions diverge

3

4

SCALE INVARIANCE

and chaotic behavior results. The logistic map and similar recursion relations are applicable to population dynamics and other ecological problems. The logistic map also produces fractal sets. Slider-block models have long been recognized as a simple analog for the behavior of a fault. The block is dragged along a surface with a spring and the friction between the surface and the block can result in the stick-slip behavior that is characteristic of faults. Huang and Turcotte (1990a) have shown that a pair of slider blocks exhibits chaotic behavior in a manner that is totally analogous to the chaotic behavior of the logistic map. The two slider blocks are attached to each other by a spring and each is attached to a constant-velocity driver plate by another spring. As long as there is any asymmetry in the problem, for example, nonequal block masses, chaotic behavior can result. This is evidence that the deformation of the crust associated with displacements on faults is chaotic and, thus, is a statistical process. This is entirely consistent with the observation that earthquakes obey fractal statistics. Nonlinearity is a necessary condition for chaotic behavior. It is also a necessary condition for scale invariance and fractal statistics. Historically continuum mechanics has been dominated by the applications of three linear partial differential equations. They have also provided the foundations of geophysics. Outside the regions in which they are created, gravitational fields, electric fields, and magnetic fields all satisfy the Laplace equation. The wave equation provides the basis for understanding the propagation of seismic waves. And the heat equation provides the basis for understanding how heat is transferred within the earth. All of these equations are linear and none generates solutions that are chaotic. Also, the solutions are not scale invariant unless scale-invariant boundary conditions are applied. Two stochastic models that exhibit fractal statistics in a variety of ways are percolation clusters (Stauffer and Aharony, 1992) and diffusion-limited aggregation (DLA) (Vicsek, 1992). In defining a percolation cluster a twodimensional grid of square boxes can be considered. The probability that a site is permeable p is specified, and there is a sudden onset of flow through the grid at a critical value of this probability, pc = 0.59275. This is a critical point and there are a variety of fractal scaling laws valid at and near the critical point. There is observational evidence that distributed seismicity has a strong similarity to percolation clusters. In generating a diffusion-limited aggregation a two-dimensional grid of square boxes can again be considered. A seed cell is placed in one of the boxes. Additional cells are added randomly and follow a random-walk path from box to box until they accrete to the growing cluster of cells by entering a box adjacent to the growing cluster. A sparse dendritic structure results because the random walkers are more likely to accrete near the tips of a cluster rather than in the deep interior. The resulting cluster satisfies fractal statistics

SCALE INVARlANCE

in a variety of ways. Diffusion-limited aggregation has been applied to the dendritic growth of minerals and to the growth of drainage networks. The renormalization group method can be applied to a wide variety of problems that exhibit scale invariance. A relatively simple system is modeled at the smallest scale; the problem is then renormalized (rescaled to utilize the same simple model at the next larger scale). The process is repeated at larger and larger scales. The method can be applied to the analysis of percolation clusters and to model fragmentation, fracture, the concentration of economic ore deposits, and other problems that satisfy fractal statistics. The concept of self-organized criticality was introduced by Bak et al. (1988) in terms of a cellular automata model for avalanches on a sand pile. A natural system is said to be in a state of self-organized criticality if, when perturbed from this state, it evolves naturally back to the state of marginal stability. In the critical state there is no natural length scale so that fractal statistics are applicable. In the sand-pile model the frequency-magnitude statistics of the sand avalanches are fractal. Regional seismicity is taken to be a classic example of a self-organized critical phenomena. Stress is added continuously by the movement of the surface plates of plate tectonics. The stress is dissipated in earthquakes with fractal frequency-magnitude statistics. Scholz (1991) argues that the earth's crust is in a state of self-organized criticality; he cites as evidence the observation that induced seismicity occurs whenever the reservoir behind a large dam is filled. This is evidence that the earth's crust is on the brink of failure. As discussed above, a pair of interacting slider blocks can exhibit chaotic behavior. Large numbers of driven slider blocks are a classic example of self-organized critica!ity. A two-dimensional array of slider blocks is considered. Each block is attached to its four neighbors and to a constant velocity driver plate by springs. Slip events occur chaotically and the frequency-size statistics of the events are generally fractal. By increasing the number of blocks considered, the low-order chaotic system is transformed into a high-order system that exhibits self-organized criticality.

5

ChapterTwo

DEFINITION OF A FRACTAL SET

2.1 Deterministic fractals

Since its original introduction by Mandelbrot (1967), the concept of fractals has found wide applicability. It has brought together under one umbrella a broad range of preexisting concepts from pure mathematics to the most empirical aspects of engineering. It is not clear that a single mathematical definition can encompass all these applications, but we will begin our quantitative discussion by defining a fractal set according to

where Ni is the number of objects (i.e., fragments) with a characteristic linear dimension ri, C is a constant of proportionality, and D is the fractal dimension. The fractal dimension can be an integer, in which case it is equivalent to a Euclidean dimension. The Euclidean dimension of a point is zero, of a line segment is one, of a square is two, and of a cube is three. In general, the fractal dimension is not an integer but a fractional dimension; this is the origin of the term fractal. We now illustrate why it is appropriate to refer to D as a fractal or fractional dimension by using a line segment of unit length. Several examples of fractals are illustrated in Figure 2.1. In Figure 2.1(a) the line segment of unit length at zero order is divided into two parts at first order so that r, = $; one part is retained so that N, = 1. The remaining segment is then divided 1 into two parts at second order so that r, = 4; again one part is retained so that N, = 1. To determine D, (2.1) can be written as


where In is a logarithm to the base e and log is a logarithm to the base 10. In almost all applications we will require the ratio of logarithms; in this case the result is the same if the logarithm to the base e (In) is used or if the logarithm to the base 10 (log) is used. For the example considered in Figure 2.1 (a), In (N2/N,) = In 1 = 0, ln(r,lr,) = In 2, and D = 0, the Euclidean dimension of a point. This construction can be extended to higher and higher orders, but at each order i, i = 1,2, . . . , n, we have In (Ni+,INi)= In 1 = 0. As the order approaches infinity, n + =, the remaining line length approaches zero, rn + 0, becoming a point. Thus the Euclidean dimension of a point, zero, is appropriate. The construction illustrated in Figure 2.l(b) is similar except that the line segment of unit length at zero order is divided into three parts at first order so that r, = f ;one part is retained so that N, = 1. At second order r, = 31 and again N2 = 1. Thus as the order is increased and n + the construction again tends to a point and D = 0. In Figure 2.1 (c) the zero-order line segment of unit length is divided into two parts but both are retained at first order so that r, = $ and N, = 2. The process is repeated at second order so that r, = and N, = 4. From (2.2) we see that D = In 2/ln 2 = 1. Similarly for Figure 2.l(d) we have D = 1; in both cases the fractal dimension is the Euclidean dimension of a line segment. This is appropriate since the remaining segment will be a line segment of unit length as the construction is repeated. However, not all constructions will give integer fractal dimensions; two examples are given in Figures 2.1 (e) and 2.1(f). In Figure 2.1 (e) the zero-order line segment of unit length is divided into three parts at first order so that r, = 31 ; the two end segments are retained so that N, = 2. The process is repeated at second order so that

-

7

Figure 2.1. Illustration of six one-dimensional fractal constructions. At zero order a line segment of unit length is considered. At first order the line segment is divided into an integer number of equal-sized smaller segments and a fraction of these segments is retained. The first-order fractal acts as a generator for higher-order fractals. Each of the retained line segments at first order is further divided into smaller segments using the generator to create a second-order fractal. The first two orders are illustrated but the construction can be carried to any order desired. (a) A line segment is divided into two parts and one is retained; D = In lfln 2 = 0 (fractal dimension of a point). (b) A line segment is divided into three parts and one is retained; D = In lfln 3 = 0 (fractal dimension of a point). (c) A line segment is divided into two parts and both are retained; D = In 21111 2 = 1 (fractal dimension of a line). (d) A line segment is divided into three parts and all three are retained; D = In 3nn 3 = 1 (fractal dimension of a line). (e) A line segment is divided into three parts and two are retained; D = In 2fln 3 = 0.6309 (noninteger fractal dimension; this construction is also known as a Cantor set). (f) A line segment is divided into five parts and three are retained; D = In 3/ln 5 = 0.6826 (noninteger fractal dimension).


r2 = 91 and N2 = 4. From (2.2) we find that D = In 2An 3 = 0.6309. This is known as a Cantor set and has long been regarded by mathematicians as a pathological construction. In Figure 2.l(f) the zero-order line segment is divided into five parts at first order so that r, = the two end segments and the center segment are retained, giving N, = 3. The process is repeated at second order so that r2 = and N2 = 9. From (2.2) we find that D = In 31ln 5 = 0.6826. These two examples have fractal dimensions between the limiting cases of zero and one; thus they have fractional dimensions. Constructions can be devised to give any fractional dimension between zero and one using the method illustrated in Figure 2.1. The iterative process illustrated in Figure 2.1 can be carried out as often as desired, making the remaining line lengths shorter and shorter. The constructions given in Figure 2.1 are scale invariant. Scale invariance is a necessary condition for the applicability of (2.1) since no natural length scale enters this power-law (fractal) relation. As a particular example consider the Cantor set illustrated in Figure 2.l(e). Iterations of the Cantor set up to fifth order, i = 5, are illustrated in Figure 2.2. The first-order Cantor set is used as a "generator" for higher-order sets. Each of the two remaining line segments at first order is replaced by a scaled-down version of the generator to obtain the second-order set, and so forth at higher orders. If n iterations are carried out, then the line length at the nth iteration, rn, is related to the length at the first iteration, r,, by rJr, = (r,lr,)". Thus, as n + =, rn + 0; in this limit the Cantor set illustrated in Figure 2.2 is known as a Cantor "dust," an infinite set of clustered points. The repetitive iteration leading to a dust is known as "curdling."

4;

9

0 order

1 st order

2 nd order

Figure 2.2. Illustration of the Cantor set carried to fifth order. The first-order Cantor set acts as a generator; the straight-line segments at order i are replaced by the generator to obtain the set at order i + 1.

3 rd order

4 th order

5 th order

9


The fractal concepts applied above to a line segment can also be applied to a square. A series of examples is given in Figure 2.3. In each case the zeroorder square is divided into nine squares at first order each with r , = f . At second order the remaining squares are divided into nine squares each with r2 = and so forth. In Figure 2.3(a) only one square is retained, so that N , = N2 = . . . = Nn = 1. From (2.2) D = 0,which is the Euclidean dimension of a point; this is appropriate since as n + w the remaining square will become a point. In Figure 2.3(b) two squares are retained at first order so that r , = N , = 2 and at second order r2 = N2 = 4 . Thus from (2.2), D = In 2/ln 3 = 0.6309, the same result that was obtained from Figure 2.l(e), as expected. Similarly, in Figure 2.3(c) three squares are retained at first order so that r , = f , N , = 3, and at second order r2 = $, N2 = 9; thus D = In 3An 3 = 1. In the limit n -+ = the remaining squares will become a line as in Figure 2.l(d). The Euclidean dimension of a line is found. In Figure 2.3(d), only the center square is removed; thus at first order r , = , N , = 8, and at second order r2 = N2 = 64. From (2.2) we have D = In 81ln 3 = 1.8928. This construction is known as a Sierpinski carpet. In Figure 2.3(e) all nine squares are retained; 1 thus at first order r , = 51 , N , = 9, and at second order r, = 9, N2 = 81. From (2.2) we have D = In 91ln 3 = 2. This is the Euclidean dimension of a square and is appropriate because when we retain all the blocks we continue to re-

6,

5,

4,

4,

3

Figure 2.3. Illustration of five two-dimensiona1 constructions. At zero order a square of unit area is considered. At first order the unit square is divided into nine equal-sized smaller squares with r , = 51 and a fraction of these squares is retained. The first-order fractal acts as a generator for higher-order fractals. Each of the retained squares at first order is divided into smaller sauares using the generator to create a second-order fractal. The 1 first two orders with r, = 5 1 and r, = g illustrated but the construction can be carried to any order desired. (a)N,=l,N,=l,D=lnlAn 3 = 0. (b) N, = 2, N, = 4, D = In 2/ln 3 = 0.6309. (c) N, = 3, N 2 = 9 , D = I n 3 A n 3 = l.(d) N, = 8, N2 = 64, D = In 81111 3 = 1.8928 (known as a Sierpinski carpet). (e) N, = 9, N2=81,D=ln9/ln3=2.

-


10

tain the unit square at all orders. Iterative constructions can be devised to yield any fractal dimensions between 0 and 2; again each construction is scale invariant. The examples for one and two dimensions given in Figures 2.1 and 2.3 can be extended to three dimensions. Two examples are given in Figure 2.4. The Menger sponge is illustrated in Figure 2.4(a). A zero-order solid cube of unit dimensions has square passages with dimensions r , = f cut through the centers of the six sides. At first order six cubes in the center of each side are removed as well as the central cube. Twenty cubes with dimensions r , = remain so that N, = 20. At second order the remaining 20 cubes have square passages with dimensions r2 = ) cut through the centers of their six sides. In each case the six cubes in the centers of each side are removed as well as the center cube. Four hundred cubes with r2 = $ remain so that N2 = 400. From (2.2) we find that D = In 201ln 3 = 2.7268. The Menger sponge can be used as a model for flow in a porous media with a fractal distribution of porosity. Another example of a three-dimensional fractal construction is given in Figure 2.4(b). Again the unit cube is considered at zero order. At first order it is divided into eight equal-sized cubes with r , = $, and two diagonally opposite comer cubes are removed so that N , = 6. At second order each of the remain1 ing six cubes are divided into eight equal-sized smaller cubes with r, = 2. In each case two diagonally opposite corner cubes are removed so that N2 = 36. From (2.2) we find that D = In 61ln 2 = 2.585. We will use this configuration for a variety of applications in later chapters. Iterative constructions can be devised to yield any fractal dimension between 0 and 3; again each construction is scale invariant. The examples given above illustrate how geometrical constructions can give noninteger, non-Euclidean dimensions. However, in each case the

3

Figure 2.4. Illustration of two three-dimensional fractal constructions. (a) At first order the unit cube is divided into 27 equal-sized smaller cubes with r, = 20 cubes are retained so that N, = 20. At second order r, = and 400 out of 729 cubes are retained so that N, = 400; D = In 20lln 3 = 2.727. This construction is known as the Menger sponge. (b) At first order the unit cube is divided into eight equal-sized smaller cubes with r, = Two diagonally opposite cubes are removed so that six cubes are retained and N, = 6. At second order r, = and 36 out of 64 cubes are retained so that N, = 36; D = In 6fln 2 = 2.585.

i,

i.

a


11

structure is not continuous. An example of a continuous fractal construction is the triadic Koch island illustrated in Figure 2.5. At zero order this construction starts with an equilateral triangle with three sides of unit length, No = 3, r, = 1. At first order equilateral triangles with sides of length r , = 51 are placed in the center of each side; there are now 12 sides so that N, = 12. This construction is continued to second order by placing equilateral triangles of length r2 = $ in the center of each side; there are 48 sides so that N, = 48. From (2.2) we have D = In 4Iln 3 = 1.26186. The fractal dimension is between one (the Euclidean dimension of a line) and two (the Euclidean dimension of a surface). This construction can be continued to infinite order; the sides are scale invariant, and a photograph of a side is identical at all scales. To quantify this we consider the length of the perimeter. The length of the perimeter Pi of a fractal island is given by

where ri is the side length at order i and N is the number of sides. Substitution of (2. I) gives

For the triadic Koch island illustrated in Figure 2.5 we have Po = 3, P I = 4, and P2 = = 5.333. Taking the logarithm of (2.4) and substituting these values we find that log4 D = 1 + log(P;+ ,/pi) = 1 + log(413) = 1 + log4 - log3 - l o d r j r ;+ I ) log3 log3 log3

(2.5)

This is the same result that was obtained above using (2.2), as expected. The perimeter of the triadic Koch island increases as i increases. As i approaches infinity, the length of the perimeter also approaches infinity, as indicated by (2.4), since D >I (D is greater than unity). The perimeter of the triadic Koch island in the limit i -+ is continuous but is not differentiable.

Figure 2.5. The triadic Koch island. (a) An equilateral triangle with three sides of unit length. @)Three F les with sides of length r, = J are placed in the center of each side The is now made up of 12 sides and N, = 12. (c) Twelve triangles with sides of length r, = are placed in the center of each side. The perimeter is now made up of 48 sides and N2= 48; D = In 4fln 3 = 1.26186. The length of the perimeter in (a) is Po = 3, in (b) is PI = 16 4, and in (c) is P2 = y = 5.333.

4

12


2.2 Statistical fractals The triadic Koch island can be considered to be a model for measuring the length of a rocky coastline. However, there are several fundamental differences. The primary difference is that the perimeter of the Koch island is deterministic and the perimeter of a coastline is statistical. The perimeter of the Koch island is identically scale invariant at all scales. The perimeter of a rocky coastline will be statistically different at different scales but the differences do not allow the scale to be determined. Thus a rocky coastline is a statistical fractal. A second difference between the triadic Koch island and a rocky coastline is the range of scales over which scale invariance (fractal behavior) extends. Although a Koch island has the maximum scale of the origin triangle, the construction can be extended over an infinite range of scales. A rocky coastline has both a maximum scale and a minimum scale. The maximum scale would typically be 103 to 104 km, the size of the continent or island considered. The minimum scale would be the scale of the grain size of the rocks, typically 1 mm. Thus the scale invariance of a rocky coastline could extend over nine orders of magnitude. The existence of both upper and lower bounds is a characteristic of all naturally occurring fractal systems. In addition, the scale invariance of a coastline will be only approximately scale invariant (fractal), and there will be statistical fluctuations in any measure of fractality. On the other hand, the triadic Koch island is exactly scale invariant (fractal). Mandelbrot (1967) introduced the concept of fractals by using (2.4) to determine the fractal dimension of the west coast of Great Britain. The length of the coastline Pi was determined for a range of measuring rod lengths ri. Mandelbrot (1967) used measurements of the length of the coastline obtained previously by Richardson (1961). Taking a map of a coastline, the length is obtained by using dividers of different lengths ri. Using the scale of the map, the length of the coastline is plotted against the divider length on log-log paper. If the data points define a straight line, the result is a statistical fractal. The result for the west coast of Great Britain is given in Figure 2.6. As shown, the data correlate well with (2.4), taking D = 1.25. This is evidence that the coastline is a fractal and is statistically scale invariant over this range of scales. The technique for obtaining the fractal dimension of a coastline is easily extended to any topography. Contour lines on a topographic map are entirely equivalent to coastlines; the lengths along specified contours Piare obtained using dividers of different lengths ri.The fractal relation (2.4) is generally a good approximation and fractal dimensions can be obtained. As illustrated in Figure 2.7, the fractal dimensions of topography using the ruler (divider) method are generally in the range D = 1.20 0.05 independent of the tec-

+


13

tonic setting and age. Topography is primarily a result of erosional processes; however, in young terrains topography is being created by active tectonic processes. It is not surprising that many of these processes are scale invariant and generate fractal topography. An interesting question, however, is whether erosional processes and tectonic processes each generate topographies with about the same fractal dimension. Bruno et al. (1992, 1994) and Gaonach et al. (1992) showed that the perimeters of basaltic lava flows are also fractal with D = 1.12- 1.42. Details on the use of the ruler (divider) method have been given by Andrle (1992). It should be emphasized that not all topography is fractal (Goodchild, 1980). Young volcanic edifices are one example. Until modified by erosion, both shield and strata volcanoes are generally conical in shape and do not yield well-defined fractal dimensions. Alluvial fans are another example of a nonfractal geomorphic feature. The morphology of alluvial fans can be modeled using the heat equation (Culling, 1960). Because the heat equation is linear, it contains a characteristic length (or time) and cannot give solutions that are scale invariant (fractal). The heat equation can also be used to model the elevation of mid-ocean ridges. The morphology of ocean trenches can be modeled by considering the bending of the elastic lithosphere. Again, the equation governing flexure is linear, introducing a characteristic length, and solutions are not scale invariant (fractal). However, despite these exceptions, most of the earth's topography and bathymetry are best modeled using fractal statistics and are therefore scale invariant.

Figure 2.6. Length P of the west coast of Great Britain as a function of the length r of the measuring rod; data from Mandelbrot (1967). The data are correlated with (2.4) using D = 1.25.

14


Although the ruler (divider) method was the first used to obtain fractal dimensions, it is not the most generally applicable method. The box-counting method has a much wider range of applicability than the ruler method (Pfeiffer and Obert, 1989). For example, it can be applied to a distribution of points as easily as it can be applied to a continuous curve. We now use the

Figure 2.7. The lengths P of specified topographic contours in several mountain belts are given as functions of the length r of the measuring rod. (a) 3000 ft contour of the Cobblestone Mountain quadrangle, Transverse Ranges, California (D = 1.21); (b) 5400 ft contour of the Tatooh Buttes quadrangle, Cascade Mountains, Washington (D = 1.21); (c) 10,000 ft contour of the Byers Peak quadrangle, Rocky Mountains, Colorado (D = 1.15); (d) 1000 ft contour of the Silver Bay quadrangle, Adirondack Mountains, New York (D = 1.19). The straight-line correlations in these log-log plots are with (2.4).


box-counting method to determine the fractal dimension of a rocky coastline. As a specific example we consider the coastline in the Deer Island, Maine, quadrangle illustrated in Figure 2.8(a). The coastline is overlaid with a grid of square boxes; grids of different-size boxes are used. The number of boxes Ni of size ri required to cover the coastline is plotted on log-log paper as a function of ri. If a straight-line correlation is obtained, then (2.2) is used to obtain the applicable fractal dimension. The box-counting method for the coastline given in Figure 2.8(a) is illustrated in Figures 2.8(b) and 2.8(c). The shaded areas are the boxes required to cover the coastline. In Figure 2.8(b) we require 98 boxes with r = 1 km to cover the coastline; in Figure 2.8(c) we require 270 boxes with r = 0.5 km to cover the coastline. The results for a range of box sizes are given in Figure 2.9. The correlation with (2.2) yields D = 1.4. This is somewhat higher than the values given above for other examples. But this is due to the extreme roughness of the coastline used in this example. When the ruler method is applied to this coastline, the same fractal dimension is found. The statistical number-size distribution for a large number of objects can also be fractal. A specific example is rock fragments. For the distribution to be fractal, the number of objects N with a characteristic linear dimension greater than r should satisfy the relation

where D is again the fractal dimension. It is appropriate to use this cumulative relation rather than the set relation (2.1) when the distribution takes on a continuous rather than a discrete set of values. Another example where (2.6) is applicable is the frequency-magnitude distribution of earthquakes. As a statistical representation of a natural phenomenon, (2.6) will be only approximately applicable, with both upper and lower bounds to the range of applicability. A specific example of the applicability of (2.6) is the Korcak (1 940) empirical relation for the number of islands on the earth with an area greater than a specified value. Taking the characteristic length to be the square root of the area of the island, Mandelbrot (1975) showed that (2.6) is a good approximation with D = 1.30. The worldwide frequency-size distribution of lakes is given in Figure 2.10 (Meybeck, 1995). The cumulative number of lakes N with an area A greater than a specified value is given as a function of both area A and the square root of the area r. An excellent correlation is obtained with (2.6) taking D = 1.go. There is a considerable regional variation in this result; Kent and Wong (1982) applied the same approach for the number of lakes in Canada and found a good correlation with (2.6) taking D = 1.55.

15

16


As we discussed, the term fractal dimension stands for fractional dimension. The meaning of this is clear in Figures 2.1-2.3; however, the meaning may be less clear in statistical power-law distributions. Some power-law distributions fall within the limits associated with fractional dimensions, i.e., 0 < D < 3, but others do not. The question that must be addressed is whether

Figure 2.8. (a) Illustration of a rocky coastline: the Deer Island, Maine, quadrangle. (b) The shaded area contains the square boxes with r = 1 km required to cover the coastline; N = 98. (c) The shaded area contains the square boxes with r = 0.5 km required to cover the coastline; N = 270.


17

all power-law distributions that satisfy (2. I ) or (2.6) are fractal. In this book

we define them to be fractal. Such distributions are clearly scale invariant, even if not directly associated with a fractal dimension. This choice eliminates an ambiguity that can lead to considerable confusion when addressing measured data sets. We will continually address this question as we consider specific applications.

Figure 2.9. The number N of square boxes required to cover the coastline in Figure 2.8(a) as a function of the box size r. The correlation with (2.1) yields D = 1.4.

Figure 2.10. Worldwide frequency-size distribution of lakes. The cumulative number of lakes N with an area A greater than a specified value is given as a function of both area and the square root of area r. The straight-line correlation is with the fractal relation (2.6) taking D = 1.90.

18


2.3 Depositional sequences

The relationship between deterministic and statistical fractals will be further illustrated using a classic problem in geology, the deposition of sediments. Many mechanisms are associated with the deposition of sediments, and it certainly can be considered a complex geological process. Gaps in the sedimentary record are recognized on a global basis, and these gaps form the boundaries of various geological epochs. But these gaps appear at all time scales and can be attributed to periods dominated by erosion or lack of deposition. We will introduce a simple model, based on fractal concepts, for sediment deposition. The basis of this model is the devil's staircase. An example of a devil's staircase based on the third-order Cantor set from Figure 2.2, given in Figure 2.11, has the same fractal dimension as the Cantor set. Instead of removing the middle third of each line segment at each order, it is retained as a horizontal segment. The vertical segments are equal upward steps moving from left to right. Taking the total horizontal length to be unity, there is one horizontal step N, = 1 with a length r , = i, there are two horizontal steps N2 = 2 with a length r2 = and there are four horizontal steps N, = 4 with a length r, = &. Thus from (2.2) we have D = In 2/ln 3 = 0.6309. Note that there are 24 steps with r, = ii but 16 of these would be further subdivided if the construction was continued to higher order. The devil's staircase based on the Cantor set can also be obtained as the integral of the Cantor set from 0 to x. The devil's staircase has a strong similarity to the age distribution in a pile of sediments, periods of rapid deposition interspersed with gaps in the sedimentary record (unconformities). For our simplified model of sediment deposition, we assume that the rate of subsidence of the earth's crust is a constant R. Without sediment deposition the water depth yw would increase linearly with time and would be given by yw = Rt. We further assume that the

;,

Figure 2.11. A fractal devil's staircase based on the thirdorder Cantor set illustrated in Figure 2.2. The horizontal step sizes are given by the Cantor set; the vertical step sizes are equal.

f


sediment supply rate is sufficient to keep the surface of the sediments at sea level. With this assumption and a constant rate of subsidence R , the rate of deposition of sediments is also R and the thickness of sediments is ys = Rt. With this simple model the rate of deposition is constant, and there are no gaps in the sedimentary record. However, it is well known that sedimentary sequences are characterized by unconformities (bedding planes), which represent gaps in the sedimentary record. An unconformity represents a period of time during which erosion was occurring and/or a period of time during which no sediment was deposited. One mechanism for generating sedimentary unconformities is to hypothesize variations in sea level. We will first illustrate how harmonic variations in sea level with time can generate gaps (unconformities) in the sedimentary record. Our simple model is illustrated in Figure 2.12. The dashed straight line in Figure 2.11(a) gives the thickness of sediments ys = Rt with R = 1 m d y r and no variations in sea level. After two million years, t = 2 Myr, the thickness of sediments is ys = 2 km. Now assume that the variation in sea level is given by

and we take ysL0 = 400 m and T, = 2 Myr. During the first 500,000 yr sea level is rising, during the next 1,000,000 yr sea level is falling, and during the final 500,000 yr sea level is again rising. If no sedimentation was occurring, the depth of water during a cycle 7, would be given by yw = Rt

+ y,,

sin (2 T tIrJ

and this is the solid line in Figure 2.12(a). We again assume that the rate of sedimentation is sufficiently high that the actual water depth is zero. At t = 0 the rate of subsidence is R = 1 mrnlyr; the rate of sea level rise is 1.26 m d y r so that the rate of sediment deposition is 2.26 m d y r . The thickness of sediments deposited follows the solid curve in Figure 2.12(a). However, at t = 792,000 yr (point a) the rate of sea level fall becomes equal to the rate of subsidence. For the period 792,000 < t < 1,208,000 yr (point b) sea level is falling faster than the subsidence rate. Without erosion the previously deposited sediments would rise above sea level. We assume, however, that erosion is sufficiently rapid that the rising landscape is maintained at sea level. At t = 1,208,000 yr, 70 m of previously accumulated sediments have been eroded. The result is an unconformity and a gap in the sedimentary record. The sediments immediately below the uncomformity were deposited at t = 577,000 yr (point c) and the sediments immediately above the unconformity were deposited at t = 1,208,000 yr (point b), a gap

19

20

Figure 2.12. Illustration of the development of an unconformity during the deposition of sediments. (a) The thickness of sediments ys in a sedimentary basin is given as a function of time t. The dashed line is the thickness of sediments with no sea level change and a constant rate of subsidence R = 1 mmlyr. If sea level varies according to (2.7), the thickness of sediments is given by the solid line. From t = 0 to t = 792,000 yr (point a) deposition occurs and the thickness of sediments is given by (2.8). From t = 792,000 yr to t = 1,208,000 yr (point b) sea level is falling faster than the rate of subsidence and erosion is occumng. Sediments deposited between t = 577,000 yr (point c) and t = 792,000 yr (point a) are eroded, as shown in the cross-hatched region. This erosion creates an unconfonnity; a gap in the sedimentary record lasts from t = 577,000 yr (point c) to t = 1,208,000 yr (point b). (b) For the model given in (a) the age of the sediments T is given as a function of depth y. The unconfonnity corresponds to the gap illustrated in (a).


in the sedimentary record results. From t = 1,208,000 yr to t = 2,000,000 yr sea level is either falling more slowly than the subsidence rate or is rising so that sedimentation occurs. The entire sequence of deposition is illustrated in Figure 2.12a. At the end of deposition, i.e., at t = 2 Myr, the age of the sediments 7 is given as a function of depth y in Figure 2.12(b). The age of the sediments at the base of the sedimentary pile, y = 2 km, is t = 2 Myr. The gap illustrated in Figure 2.12(a) results in the unconformity illustrated in Figure 2.12(b). The age of the sediments above the unconformity is 7 = 792,000 yr (point b) and the age of the sediments below the unconformity is T = 1,423,000 yr (point c).

Age 5 Myr

1


21

A simple harmonic variation of sea level will lead to a periodic sequence of unconformities of equal length. However, there is observational evidence that variations in sea level obey fractal statistics (Hsui et al., 1993) so that it should not be surprising that sea level variations could generate a distribution of unconformities that also obeys fractal statistics. We next develop a fractal model for sediment deposition based on the devil's staircase given in Figure 2.1 1 (Plotnick, 1986; Korvin, 1992, pp. 95-113). We consider the devil's staircase associated with a second-order Cantor set. The age of sediments in this model is given as a function of depth in Figure 2.13(a). Eight kilometers of sediments have been deposited in this model sedimentary basin in a period of 9 Myr so that the mean rate (velocity) of deposition is = 8 kml9 Myr = 0.89 mrnlyr over this period. However, there is a major unconformity at a depth of 4 km. The sediments immediately above this unconformity have an age T = 3 Myr and the sediments immediately below it have an age T = 6 Myr. There are no sediments in the sedimentary pile with ages between T = 3 and 6 Myr. In terms of the Cantor set this is illustrated in Figure 2.13(b). The line of unit length is divided into three parts, and the middle third, representing the period without deposition, is removed. The two remaining parts are placed on top of each other as shown. During the first three million years of deposition (the lower half of the sedimentary section) and the last three million years of deposition (the upper half of the sedimentary section) the mean rates of deposition are = 4 km/3 Myr = 1.33 mdyr. Thus the rate of deposition increases as the period considered decreases. This is shown in Figure 2.13(c).

Age T, M yr

Depth Y km

Figure 2.13. Illustration of a model for sediment deposition based on a devil's staircase associated with a second-order Cantor set. (a) Age of sediments T as a function of depth y. (b) Illustration of how the Cantor set is used to construct the sedimentary pile. (c) Average rate of deposition as a function of the period T considered.

22


There is also an unconformity at a depth of 2 km. The sediments immediately above this unconformity have an age T = 1 Myr and immediately below have an age T = 2 Myr. Similarly there is an unconformity at a depth of 6 krn;the sediments above this unconformity have an age T = 7 Myr and sediments below an age T = 8 Myr. There are no sediments in the pile with ages between T = 8 and 7 Myr, between T = 6 and 3 Myr, and between T = 2 and 1 Myr. This is illustrated in Figure 2.13(a). In terms of the Cantor set, Figure 2.13(b), the two remaining line segments of length are each divided into three parts and the middle thirds are removed. The four remaining segments of length are placed on top of each other as shown. During the periods T = 9 to 8, 7 to 6, 3 to 2, and 1 to 0 Myr the rates of deposition are = 2 km/l Myr = 2 m d y r . This rate is also included in Figure 2.13(c). The rate of deposition clearly has a power-law dependence on the length of the time interval considered. The results illustrated in Figure 2.13 are based on a second-order Cantor set, but the construction can be extended to any order desired and the power-law results given in Figure 2.13(c) would be extended to shorter and shorter time intervals. We now generalize the determination of the rate of deposition as a function of record length and relate it to the fractal dimension of a set. The rate of deposition Ri for a set of order i is given by

5

6

where L, is the thickness of sediments deposited in the period T;. The period in our model is equivalent to the line segment length riin the fractal sets illustrated in Figure 2. l . For the example given in Figure 2.13 we have T,, T, = ~ ~ 1and 3 , T, = ~ d 9The . thickness of sediments Li is given by the number of segments retained at a specified order N so that T;

For the example given in Figure 2.13 we have N, = 2 and L, = Ld2 and N, = 4 and L, = Ld4. Noting the equivalence of our T~and ri in the fractal relation (2. l), we can write the fractal relation

Combining (2.9), (2.10), and (2.1 1) we relate Ri to T~ with the result


which also can be written as

The rate of deposition Ri has a power-law dependence on the time interval over which the deposition occurs. For the case considered above and illustrated in Figure 2.13, we have D = In 2/ln 3 = 0.6309; this is expected since it is the fractal dimension of a Cantor set. Depending on which set is used to construct the devil's staircase model for sedimentation, any fractal dimension between 0 and 1 can be obtained. The model for sediment deposition given above is deterministic, whereas actual deposition is clearly stochastic. Nevertheless, the deterministic model illustrates how fractal variations in sea level or other fractal depositional mechanisms can give a fractal sedimentary sequence. It is also clear that not all gaps in the sedimentary record can be attributed to variations in sea level. Despite these obvious limitations it is of interest to consider the observational data on rates of sediment deposition. In Figure 2.14 the rates of deposition of fluvial sediments R are given as a function of time span T over which deposition occurs (Sadler and Strauss, 1990). These data were based on 5600 deposition rates determined in modern sedimentary basins and ancient stratigraphic sections. The straight-line correlation is with (2.13) taking D = 0.336. A reasonably good correlation is obtained over 8 orders of magnitude in rate and 12 orders of magnitude in time. Clearly such a correlation is only approximate since important complications have not been considered. For example, different depositional mechanisms would be expected to dominate on different time scales. However, the results in Figure 2.14 clearly indicate the strong episodicity of sedimentation and a good correlation with power-law (fractal) statistics. Gardner et al. (1987) have correlated elevation changes with measured intervals for rates of tectonic uplift and erosion. In both cases they find good power-law (fractal) correlations. Snow (1992) has applied the devil's staircase model described above to explain their results. For uplift Gardner et al. (1987) found that the rate of uplift RUis related to the interval T,, by

And from (2.14) we have D = 0.746. For erosion these authors found that the rate of erosion Re is related to the interval T~ by

23

24


And from (2.14) we have D = 0.815. The devil's staircase model for sedimentary completeness could have been carried to higher and higher orders and the rates of deposition would have increased accordingly. Also, the thickness of the layers would have gotten smaller and smaller. In the limit of infinite order, the rate of deposition would be infinity and the layer thickness would be zero, clearly unreasonable from a physical standpoint. Thus when the mathematical construction of a Cantor set is applied to a real physical problem, the deposition of sediments, it is necessary to truncate the model at finite order. Also, it must be emphasized that the layer thicknesses in an actual sedimentary pile will have a statistical distribution (a problem we will consider in a later chapter), rather than having equal thicknesses as in this model. This is a common problem when a deterministic fractal model (a Cantor set) is applied to a statistical fractal problem (a layered stack of sediments in a sedimentary pile).

2.4 Why fractal distributions? In this chapter a number of deterministic fractal sets have been introduced including the Cantor set, the Sierpinski carpet, the Menger sponge, and the triadic Koch island. Two geological applications have been considered, the length of a coastline and the completeness of the sedimentary record. The Koch triadic island illustrates the fractal statistics of the length of the coastline. The devil's staircase based on the Cantor set illustrates the fractal statis-

Figure 2.14. Dependence of rates of deposition of sediments R on the time span T over which deposition occurs. The circles are mean rates of deposition of fluvial sediments from modem sedimentary basins and ancient stratigraphic sections (Sadler and Strauss, 1990). The straight line is the fractal correlation from (2.14) taking D = 0.336.

D E F I N I T I O N OF A FRACTAL S E T

tics of sediment deposition. The erosional processes responsible for the formation of coastlines and the depositional processes responsible for the structure of a sedimentary pile are both extremely complex. But despite the complexity, both examples exhibit fractal behavior to a good approximation. A simple explanation is that a distribution will be fractal if there is no characteristic length in the problem. The fractal distribution is the only statistical distribution that is scale invariant. However, a broad class of nonlinear physical problems involving chaotic behavior and/or self-organized critical behavior invariably yield fractal behavior. One objective in succeeding chapters is to describe physically realistic models that generate fractal behavior.

Problems Problem 2.1. Consider the construction illustrated in Figure 2.l(e). (a) Illustrate the construction at third order. (b) Determine N,, N,, r,, and r,. Problem 2.2. Consider the construction illustrated in Figure 2.l(f). (a) Illustrate the construction at third order. (b) Determine N,, N,, r,, and r,. Problem 2.3. A unit line segment is divided into five equal parts and two are retained. The construction is repeated. (a) Illustrate the construction to third order, i.e., consider i = 1, 2, 3. (b) Determine N,, N,, N,, r,, r,, r,. (c) Determine the fractal dimension. Problem 2.4. A unit line segment is divided into seven equal parts and three are retained. The construction is repeated. (a) Illustrate this construction to second order, i.e., consider i = 1,2. (b) Determine N,, N,, N3, r,, r2, r3. (c) Determine the fractal dimension. Problem 2.5. A unit line segment is divided into seven equal parts and four are retained. The construction is repeated. (a) Illustrate this construction to second order, i.e., consider i = 1,2. (b) Determine N,, N,, N,, r,, r2, r,. (c) Determine the fractal dimension. Problem 2.6. Consider the construction of the Sierpinski carpet illustrated in Figure 2.3(d) at third order. (a) Illustrate the construction at third order. (b) Determine N,, N,, r,, and r,. Problem 2.7. A unit square is divided into four smaller squares of equal size. Two diagonally opposite squares are retained and the construction is repeated. (a) Illustrate the construction to third order, i.e., consider i = 1, 2, 3. (b) Determine N,, N,, N,, r,, r,, r,. (c) Determine the fractal dimension. Problem 2.8. A unit square is divided into nine smaller squares of equal size. The center square and four corner squares are retained and the construction is repeated. This is known as a Koch snowflake. (a) Illustrate the construction to second order, i.e., consider i = 1,2. (b) Determine N,, N,, N,, r,, r,, r,. (c) Determine the fractal dimension.

25

26


Problem 2.9. A unit square is divided into nine smaller squares of equal size and the four corner squares are discarded. The construction is repeated. (a) Illustrate the construction to second order. (b) Determine N,, N,, N,, r,, r,, r,. (c) Determine the fractal dimension. Problem 2.10. A unit square is divided into 16 smaller squares of equal size. The four central squares are removed and the construction is repeated. (a) Illustrate this construction to second order, i.e., consider i = 1, 2. (b) Determine N,, N,, N,, r,, r,, r,. (c) Determine the fractal dimension. Problem 2.1 1. A unit square is divided into 25 smaller squares of equal size. All squares are retained except the central one and the construction is repeated. (a) Illustrate this construction to second order, i.e., consider i = 1, 2. (b) Determine N,, N,, N,, r,, r,, r,. (c) Determine the fractal dimension. Problem 2.12. A unit square is divided into 25 smaller squares of equal size. All the squares on the boundary and the central square are retained and the construction is repeated. (a) Illustrate this construction to second order, i.e., consider i = 1, 2. (b) Determine N,, N,, N,, r,, r,, r,. (c) Determine the fractal dimension. Problem 2.13. A unit cube is divided into 27 smaller cubes of equal volume. All the cubes are retained except for the central one. What is the fractal dimension? Problem 2.14. Consider a variation on the Koch island illustrated in Figure 2.5. At zero order again consider an equilateral triangle with three sides of unit length. At first order this triangle is enlarged so that it is an equilateral triangle with sides of length three. Equilateral triangles with sides of unit length are placed in the center of each side. (a) Illustrate this construction at second order. (b) Determine the areas to second order, i.e., obtain A,, A,, A,. (c) Do the areas given in (b) satisfy the fractal condition (2. I)? If the answer is yes, what is the fractal dimension? Problem 2.1 5. Consider the fractal construction illustrated in Figure 2.15. A

Figure 2.15. Illustration of a fractal construction. (a) The zero-order unit square. (b) The first-order fractal construction.


unit square is considered at zero order and the first-order fractal construction is also illustrated. (a) Illustrate the construction at second order. (b) Determine No, N,, N,, r, r,, r,, Po, P I ,P,. (c) Determine the fractal dimension. Problem 2.16. Assume that the open squares in the Sierpinski carpet illustrated in Figure 2.3(d) represent lakes. (a) Determine the numbers of lakes to the third order, i.e., obtain N,, N,, N, corresponding to r,, r,, r,. (b) Do the numbers of lakes given in (a) satisfy the fractal condition (2. l)? If the answer is yes, what is the fractal dimension? Problem 2.17. Zipf's law (Zipf, 1949) has been applied in a wide variety of problems including the size distribution of cities. This law states that the 2nd largest is $ the size of the largest, the 3rd largest is $ the size of the largest, the 4th largest is the size of the largest, and so forth. Does this distribution satisfy a cumulative fractal distribution and, if so, what is the fractal dimension? Problem 2.18. Construct a second-order devil's staircase based on the fractal construction given in Figure 2.l(f). Problem 2.19. Consider the simple deposition model illustrated in Figure 2.11. Assume that no erosion occurs. What are the ages of the sediments immediately above and below the resulting unconformity? Problem 2.20. Use the second-order Cantor set based on the fractal construction given in Figure 2,l(f) as a model for sedimentation. Assume in this model that 9 km of sediments have been deposited in 25 Myr. (a) At what depths do the two first-order unconformities occur, and what are the ages of the sediments just above and just below the unconformities? (b) At what depths do the six second-order unconformities occur, and what are the ages of the sediments just above and just below the unconformities? (c) Plot the age of the sediments as a function of depth. (d) What are the rates of deposition associated with the periods 25 Myr, 5 Myr, 1 Myr?

27

ChapterThree

FRAGMENTATION

3.1 Background

To illustrate how fractal distributions are applicable to real data sets, we consider fragmentation. Fragmentation plays an important role in a variety of geological phenomena. The earth's crust is fragmented by tectonic processes involving faults, fractures, and joint sets. Rocks are further fragmented by weathering processes. Rocks are also fragmented by explosive processes, both natural and man made. Volcanic eruptions are an example of a natural explosive process. Impacts produce fragmented ejecta. Although fragmentation is of considerable economic importance and many experimental, numerical, and theoretical studies have been camed out on fragmentation, relatively little progress has been made in developing comprehensive theories of fragmentation. A primary reason is that fragmentation involves the initiation and propagation of fractures. Fracture propagation is a highly nonlinear process requiring complex models for even the simplest configuration. Fragmentation involves the interaction between fractures over a wide range of scales. Fragmentation phenomena have been discussed by Grady and Kipp (1987) and Clark (1987). If fragments are produced with a wide range of sizes and if natural scales are not associated with either the fragmented material or the fragmentation process, fractal distributions of number versus size would seem to be expected. Some fractal aspects of fragmentation have been considered by Turcotte (1986a).

3.2 Probability and statistics Clearly the distribution of fragment sizes is a statistical problem. This will also be the case in other applications we will consider. Thus at this point it is appropriate to introduce some of the fundamental concepts of probability and statistics. Data that we will be considering can be divided into two types,

FRAGMENTATION

discrete data and continuous data. Discrete data are generally characterized by a set of n data points {x,,x,, . . .xi, . . . ,xn).Examples include the masses of n fragments and the magnitudes of n earthquakes. It is standard practice to describe the statistical properties of a discrete data set by defining the mean and moments of deviations from the mean. The mean value of the xi, i , is given by

The average squared deviation from the mean is a measure of the spread of the data; this is the variance V and for a discrete set of n data points it is given by

The variance is the second-order moment of the distribution. The standard deviation of the distribution a is simply the square root of the variance

The asymmetry of the data is quantified by the coefficient of skew y, which is the third-order moment

The factor a 3 makes y a nondimensional number. Higher-order moments can be defined but they are generally of little use. It is standard practice in geostatics to fit an empirical statistical distribution to a discrete set of data. This is often done by equating the mean i ,variance V, and skew y of the distribution to that of the data. It should be noted that a variety of definitions for the variance and skew appear in the literature; for example, the n in (3.2) is sometimes replaced by the factor n - 1 (Barlow, 1989, pp. 10-12). As a simple example assume that x takes the values x, = 2, x, = 5, x3 = 9, and x, = 16. From (3.1) the mean is i = 8, from (3.2) the variance is V = 27.5, from (3.3) the standard deviation is a = 5.244, and from (3.4) the skew is y = 1.311.

29

30

FRAGMENTATION

In many cases, each of the values x , , x,, . . . ,xi, . . . ,xn will have a probability of occurring f , , f,, . . . ,&, . . . ,fn. By definition we have

Introducing these probabilities, the mean, variance, standard deviation, and skew of a distribution are given by

+

As an example consider flipping coins. Assign 1 to a head and - 1 to a tail. For a single coin there are two values of x, x = + 1 (a head) and x2 = - 1 (a tail). Since the probabilities of having a head or a tail are equal, we have f , = f2 = 0.5. Next consider flipping two coins. We now have three values for x, x , = + 2 (two heads), x, = 0 (one head and one tail), and x, = -2 (two tails). However, there are two ways to obtain x, = 0, the first coin is a head and the second a tail or the first is a tail and the second a head, whereas there is only one way to obtain x, = + 2 (two heads) and x3 = -2 (two tails). Thus we have f,=0.25,f2=0.5,andf3=0.25.From(3.6) t o ( 3 . 9 ) w e f i n d i = y = 0 , V = 2 , and a = fi.Finally consider flipping three coins. In this case we have x, = 3 (3 heads) with f,= 0.125, x, = 1 (2 heads, 1 tail) with f, = 0.375, x, = - 1 (1 head, 2 tails) with f3 = 0.375, and x, = -3 (3 tails) with f, = 0.125. From (3.6) to (3.9) we find 2 = y = 0, V = 3, and a = fi. We will next consider continuous data. A particular variable can take any value over a specified range, say - = < x < =. An example would be the x-component of the velocity of the gas molecules in a room, vx. It is appropriate to consider the range of velocities -= < v < -, and there will be a X. statistical probability that a particular molecule will have a velocity greater than a specified value vx. In terms of our general distribution the cumulative distribution function F (x,) is the probability Pr that x has a value greater than x,

FRAGMENTATION

--

It should be noted that the usual definition of a cumulative distribution function in probability and statistics is from to x rather than x to m, i.e.,

For most applications in geology and geophysics we are concerned with the "number" larger than a specified value and thus the definition of F(x) given in (3.10) is preferred. The cumulative distribution function is related to the probability distribution functionflx) by

where fix) 6 x is the probability that x lies in the range (x (x + 3 6 ~ )We . also have

-

6x) < x 5

and the probability Pr that x lies between x, and x2 is given by Pr(x, < x 5 x,) =

f(u)du = F(x,)

-

F(x2)

(3.14)

Several widely used statistical distributions will now be discussed. One of the most widely used statistical distributions is the normal distribution; it is also known as the Gaussian distribution or simply as the bell curve (from its shape). Its probability distribution function takes the form

From statistical mechanics (Morse, 1969) the x-component of molecular velocities in a room satisfy a Gaussian distribution of the form

where m is the molecular mass, k the Boltzmann constant, and T the temperature.

31

32

FRAGMENTATION

The mean of the normal distribution is obtained using the relation

Introducing

Y=-

x-x

diu

(3.17) becomes

but

so that i is the mean of the normal distribution. The variance of the normal distribution is obtained from

:v : 1 =

-

x

f(x)dx =

1 o (2n)lD

I:m

[

( x - i)' exp - " 2i 'y ]dx

(3.20)

Introducing (3.18) gives

so that a is the standard deviation of the normal distribution. We define the cumulative distribution function for the normal distribution to be given by (3.13)

Making a substitution of the form of (3.18) into (3.22) gives

FRAGMENTATION

It is convenient to introduce the error function

and the complementary error function

And noting that

the cumulative distribution for the normal distribution becomes

and F (- w) = 1 as expected. The probability that x is in the range x, < x Ix, is given by

The standard form of the normal distribution is obtained by taking i = 0 and a = 1. The probability distribution functionflx) and the cumulative distribution function F(x) for the standard form of normal distribution are given in Figure 3.1. Relevant values are tabulated in Table 3.1. Note that for this symmetric distribution F(0) = 0.5 and F(-=) = 1. It is often necessary to generate a set of random numbers with a specified distribution of values, say a normal (Gaussian) distribution. One way to do this is to use the cumulative distribution function F(x). A set of random numbers with a uniform distribution between 0 and 1 is obtained. Each random number is assumed to be Fi(xi) and the corresponding value of xi is found. If the dependence of F(x) on x given in Figure 3.1 is used, the values of xi will satisfy a normal (Gaussian) distribution.

33

34

FRAGMENTATION

Table 3.1. Values relevant to the normal distribution

For an application in which a finite number of observations have been carried out, it is difficult to apply the probability distribution function directly. A selection of "bins" is chosen and each data point is put in a bin. The number of points in the bins then constitute the statistical distribution. For the normal distribution, (3.28) can be used to establish the probability of a data point being in a bin. If the cumulative distribution is used it is not necessary to "bin" the data. The number of data points with values greater than x is plotted against x. This is the preferred way to handle most geological and geophysical data. The wide applicability of the normal distribution is based on the central limit theorem. This theorem states that if a distribution is the sum of a large

Figure 3.1. The probability distribution functionflx) and the cumulative distribution function F(x) for the standard form of the normal distribution, from (3.15) and (3.22) with 2 = 0 and a = 1.

FRAGMENTATION

number of independent random distributions, the distribution will approach a normal distribution as the number approaches infinity. For example, consider the coin flipping experiment discussed above with x, = + 1 for a head and x2 = - 1 for a tail. As the number of coins approaches infinity the distribution of values approaches a normal distribution. A normal distribution is symmetrical about its mean so that its coefficient of skew is zero, y = 0; also, the independent variable x takes on all values from -M to +-. In many applications a distribution of only positive values is required. One of the most widely used distributions of this type is the log-normal distribution. The log-normal distribution can be obtained directly from the normal distribution simply by taking the logarithm of the normally distributed values [replacing the x in the definition of the normal probability distribution function given in (3.15) with y]; the corresponding log-normal probability distribution function is given by

where the substitution

has been made noting that dy = dxlx andfix) dx =f(y) dy. The values of y are normally distributed with a mean j and a standard deviation uy.Using the definitions of the mean and standard deviation, (3.6) and (3.8), we can relate i and a to j and uywith the result

Since both the standard deviation and the mean are positive for the lognormal distribution, the ratio of the two quantities is a measure of the spread of the distribution

and is known as the coefficient of variation. The coefficient of skew for the log-normal distribution is

35

36

FRAGMENTATION

The values of ayand y are related to 2 and c, by a,,= [ln(l

2 112 + c,)]

These relations with (3.29) specify the log-normal probability distribution function when i and c, have been given. The cumulative distribution function for the log-normal distribution is obtained from (3.13)

And making the substitution

we obtain

The cumulative distribution functions for the log-normal and normal distributions have the same forms when x is replaced by In x, i.e. (3.30). The standard form of the normal distribution was obtained by taking i = 0 and V = 1. All normal distributions have this universal form and can be obtained simply by rescaling. This is not the case for the log-normal distribution. The probability distribution functionsflx) for the log-normal distribution are given in Figure 3.2 for 2 = 1 and c, = 0.25,0.50, 1.00. It is seen that the shape of the log-normal distribution changes systematically with changes in c,. As the value of c, becomes smaller the distribution narrows and the maximum value approaches x = 1. In the limit c, + 0 the distribution is a 6 function centered at x = 1.As c, becomes larger the distribution spreads out and the maximum value occurs at smaller x. Whereas the normal distribution is symmetric with a zero coefficient of skew, the asymmetry and coefficient of skew for the log-normal distribution increase with increasing c, Log-normal distributions are basically a one-parameter family of distributions depending on the appropriate value of c,. This has important impli-

FRAGMENTATION

37

cations in terms of applications. It is often appropriate to approximate the distribution of annual rainfalls at a station by a log-normal distribution. A maritime station, say Seattle, would have little year-to-year variation in rainfall and a small value for c , On the other hand, an arid station, say Phoenix, would have large year-to-year variations in rainfall and a larger value for c,. This dependence on c , for rainfall and river flows is referred to as the Noah effect (Mandelbrot and Wallis, 1968). We will next consider the Pareto distribution, which is closely associated with fractal distributions. The Pareto probability distribution function is given by

The corresponding cumulative distribution function is given by

The standard form of the Pareto distribution is obtained by taking k = 1 so that (3.40) becomes

Figure 3.2. The probability distribution function f(x) for the log-normal distribution with unit mean, % = 1, and several values of the coefficient of variation c,.

38

FRAGMENTATION

and (3.41) becomes

Fb) = (1

;

y)'

y20

The mean of the standard form is given by

=

[

Y a dy (1

+ y)'+l

1

-

a

-

for a > 1

1

This integral does not converge and a mean does not exist for a I1.The variance of the standard form of the Pareto distribution is given by

-

a

(a - l)(a - 2)

for a > 2

(3.45)

This integral does not converge for a 1 2 and the variance does not exist. The Pareto distribution is widely used in economics and is often a good approximation for the distribution of incomes (Ijiri and Simon, 1977). The probability distribution functions for the standard form of the Pareto distribution are given in Figure 3.3 for a = 1, 2, and 3. The power-law tail of the Pareto distribution dies off much more slowly than the tails of the normal or log-normal distributions; this is the characteristic of fractal distributions. For y >> 1we can write (3.43) as

This is clearly quite similar to the statistical fractal relation (2.6)

The power a in the Pareto distribution is equivalent to the fractal dimension

D. This has led some statisticians and others to conclude that fractals are a trivial extension of the Pareto distribution. While there is clearly a close association between statistical fractals and the Pareto distribution, there are

FRAGMENTATION

39

many other aspects of fractal concepts. The wide applicability of scale invariance provides a rational basis for fractal statistics just as the central limit theorem provides a rational basis for Gaussian statistics. An important distinction between the cumulative Pareto distribution (3.41) and the fractal distribution (3.47) is that the former is finite as x + 0 whereas the latter diverges to w as r + 0. Scale invariance implicitly requires this divergence. Many geological and geophysical data sets also have this divergence. As a specific example, consider earthquakes. Data on large earthquakes are often complete, but data on small earthquakes generally do not exist. Even the best seismic networks cannot resolve the very smallest earthquakes that are known to occur. Thus it is impossible to define complete probability or cumulative distribution functions for earthquakes. However, it is possible to determine the number of earthquakes N that have rupture dimensions greater than r, and we will show in Chapter 4 that the frequency-magnitude statistics for earthquakes are fractal and do satisfy (3.47). The final distribution we will consider is an exponential distribution; its probability distribution function is given by

'

vxv-

f(x)

=

I[-"):(

7 ex* xo

x 20

Figure 3.3. The probability distribution function for the standard form of the Pareto distribution with a = 1, 2,3; from (3.42).

m)

40

FRAGMENTATION

where the power v is generally taken to be an integer. The mean of the exponential distribution is given by

Making the substitution

we obtain

where r (v') is the tabulated gamma function (Dwight, 1961, Table 1005). If v = 2 we have r = 0.886 so that 2 = 0.88%. The variance of the exponential distribution is given by

(i)

Again substituting (3.50) we obtain

If v = 2 we have r (2) = 1 so that V = 0.2146~;. The cumulative distribution function for the exponential distribution is obtained from (3.13) with the result

This is known as the Rosin and Rammler (1933) distribution and it is used extensively in geostatistical applications. We can also write

FRAGMENTATION

[

1 - F(x) = 1 - exp -

41

(3'1 -

This is known as the Weibull distribution. Thus the Weibull distribution is entirely equivalent to the Rosin and Rammler distribution. The Rosin and Rammler distribution is the Pr (x' > x) and the Weibull distribution is the Pr (0 < x' < x) when the probability distribution function is given by (3.48). The Rosin and Rammler, F b ) , and the Weibull, 1 - F(y), distributions are given in Figure 3.4 for v = 2 and 4. If (xIx,)" is small, then the exponential in (3.55) can be expanded in a Taylor series to give

where higher powers of (xlx,). have been neglected. Substitution of (3.56) into the Weibull distribution (3.55) gives

Thus for small x the Weibull distribution reduces to a power-law (fractal) distribution. This power-law approximation is also illustrated in Figure 3.4.

Figure 3.4. The Rosin and Rammler distribution F ( y ) from (3.54) with y = xlx, and the Weibull distribution 1 -F(y) from (3.55) are given for v = 2 and 4. Also included is the power-law (fractal) approximation to the Weibull distribution from (3.57).

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FRAGMENTATION

3.3 Fragmentation data Many of the statistical distributions discussed above have been used to represent the frequency-size (mass) distributions of fragments; these include log-normal, Pareto, Rosin and Rammler, Weibull, and power law. In terms of the concepts developed in Chapter 2, it is clear that we would like to relate the number of fragments N to their linear dimension r. Since fragments can occur in a variety of shapes, it is appropriate to define a linear dimension r as the cube root of volume, r = W 3 . Assuming constant density it follows that m r3, where m is the mass of a fragment. However, it is standard practice to give the total mass of fragments with a linear dimension r less than a specified value M (r). The reason for this is that these masses are obtained directly from a sieve or screen analysis; the mass of fragments passing through a sieve with a specified aperture r is M (r). Of course we have

-

the total mass of fragments. In many cases the power-law approximation to the Weibull distribution (3.57) can be used to approximate sieve analyses in the form

This power-law mass relation can be related to the fractal number relation

by taking incremental values (Redner, 1990). Taking the derivative of (3.58) gives

Taking the derivative of (3.59) gives

However, the incremental number is related to the incremental mass by

FRAGMENTATION

43

Substitution of (3.60) and (3.61) into (3.62) gives

When data are obtained by sieve analyses, (3.64) is used to convert mass distributions to number distributions to specify a fractal dimension. Many experimental studies of the frequency-size distributions of fragments have been carried out. Several examples of power-law fragmentation are given in Figure 3.5. A classic study of the frequency-size distribution for broken coal was carried out by Bennett (1936). The frequency-size distribution for the chimney rubble above the PILEDRIVER nuclear explosion in Nevada has been given by Schoutens (1979). This was a 61 kt event at a depth of 457 m in granite. The frequency-size distribution for fragments resulting from the high-velocity impact of a projectile on basalt has been given by Fujiwara et al. (1977). In each of the three examples a good correlation with the fractal relation (2.6) is obtained over two to four orders of magnitude. In each example the fractal dimension for the distribution is near D = 2.5. Further examples of power-law distributions for fragments are given in Table 3.2. It will be seen that a great variety of fragmentation processes can be interpreted in terms of a fractal dimension. Examples include impact shatFigure 3.5. Since fragments have a variety of shapes, the cube root of volume is an objective measure of size. The number N of fragments with cube root of volume greater than r is given as a function of r for broken coal (Bennett, 1936), broken granite from a 61 kt underground nuclear detonation (Schoutens, 1979), and impact ejecta due to a 2.6 km s-1 polycarbonate projectile impacting on basalt (Fujiwara et al., 1977). The best-fit fractal distribution from (3.59) is shown for each data set.

44

FRAGMENTATION

tering, explosive disruption, crushed materials, weathered materials, and volcanic ejecta. The fact that fault gouge has a fractal frequency-size distribution is a particularly striking example of how a natural geological process can result in fractal fragmentation (Sammis et al., 1986; An and Sammis, 1994; Sammis and Steacy, 1995). The relative displacement across a fault zone results in the fragmentation of the wall rock to form a zone of fragmented rock known as fault gouge. This is referred to as comminution since it strongly resembles the fragmentation that takes place in a grinding mill. Sammis et al. (1987) and Sammis and Biegel (1989) have shown that the fault gouge obtained from the Lopez fault zone, San Gabriel Mountains, California, has a fractal dimension D = 2.60 + 0.11 on scales from 0.5 pm to 10 mm. Synthetic fault gauge has also been shown to have a fractal dimension D = 2.60 (Biegel et al., 1989). Sammis et al. (1987) have also suggested that the comminution of the earth's crust has resulted in fractal tectonic fragmentation on scales from millimeters to hundreds of kilometers. We will return to this concept in the next chapter. Studies of the frequency-size distribution of asteroids show that they fit a power-law (fractal) relation to a good approximation taking D €I3 2.5 (Klacka, 1992). Since asteroids are responsible for the impact craters on the surface of the moon, it is not surprising that the frequency-size distribution of lunar craters is also fractal with D e1.4 (Greeley and Gault, 1970). Table 3.2. Fractal dimensions for a variety of fragmented objects

Object

Reference

Fractal dimension D

Artificially crushed quartz Disaggregated gneiss Disaggregated granite FLAT TOP I (chemical explosion, 0.2 kt) PILEDRIVER (nuclear explosion, 62 kt) Broken coal Asteroids Projectile fragmentation of quartzite Projectile fragmentation of basalt Fault gouge Sandy clays Soils Terrace sands and gravels Glacial till Ash and pumice

Hartmann (1969) Hartmann (1969) Hartmann (1969)

1.89 2.13 2.22

Schoutens (1979)

2.42

Schoutens (1979) Bennett (1936) Klacka (1992)

2.50 2.50 2.50

Curran et al. (1977)

2.55

Fujiwara et al. (1977) Sammis and Biegel(1989) Hartmann (1969) Wu et al. (1993) Hartmann (1969) Hartmann (1969) Hartmann (1969)

2.56 2.60 2.61 2.80 2.82 2.88 3.54

FRAGMENTATION

It is seen that the values of the fractal dimension vary considerably, but most lie in the range 2 < D < 3. This range of fractal dimensions can be related to the total volume of fragments and to their surface area. The total volume (mass) of fragments is given by

since r has been defined to be the cube root of the volume. In all cases it is expected that there will be upper and lower limits to the validity of the fractal (power-law) relation for fragmentation. The upper limit rmaxis generally controlled by the size of the object or region that is being fragmented. The lower limit rminis likely to be controlled by the scale of the heterogeneities responsible for fragmentation, for example the grain size. For a power-law (fractal) distribution of sizes, substitution of (3.61) into (3.65) and integration gives

If 0 < D < 3 it is necessary to specify rmax but not rminto obtain a finite volume (mass) of fragments. The volume (mass) of fragments is predominantly in the largest fragments. This is the case for most observed distributions of fragments (see Table 3.2). If D > 3 it is necessary to specify rminbut not rmax. The volume (mass) of the small fragments dominates. The total surface area A of the fragments is given by

where C is a geometrical factor depending on the average shape of the fragments. For a power-law distribution, substitution of (3.61) into (3.67) and integration gives

If 0 < D < 2 it is necessary to specify rmaxbut not rminto obtain a finite total surface area for the fragments. But if D > 2 it is necessary to specify rminto constrain the total surface area to a finite value. Thus for most observed distributions of fragments (see Table 3.2) the surface area of the smallest fragments dominates.

45

46

FRAGMENTATION

3.4 Fragmentation models A simple model illustrates how fragmentation can result in a fractal distribution. This model is illustrated in Figure 3.6; it is based on the concept of renormalization, which will be considered in greater detail in Chapter 15. A cube with a linear dimension h is referred to as a zero-order cell; there are No of these cells. Each zero-order cell may be divided into eight equal-sized, zero-order cubic elements with dimensions h/2. The volume V, of each of these elements is given by

where Vo is the volume of the zero-order cells. The probability that a zeroorder cell will fragment to produce eight zero-order elements is taken to be f. The number of zero-order elements produced by fragmentation is

After fragmentation the number of zero-order cells that have not been fragmented, No,, is given by

Each of the zero-order elements is now taken to be a first-order cell. Each first-order cell may be fragmented into eight equal-sized, first-order cubic elements with dimensions h/4. The fragmentation process is repeated Figure 3.6. Idealized model for fractal fragmentation. A zero-order cubic cell with dimensions h is divided into eight zero-order cubic elements each with dimensions hl2. The probability that a zero-order cell will be fragmented into eight zero-order elements is$ The fragments with dimension hl2 become firstorder cells; each of these has a probability f of being fragmented into first-order elements with dimensions h/4. The process is repeated to higher orders. The basic structure is fractal.

FRAGMENTATION

for these smaller cubes. The problem is renormalized and the cubes with dimension h/2 are treated in exactly the same way that the cubes with linear dimension h were treated above. Each of the fragmented cubic elements with linear dimension h/2 is taken to be a first-order cell; each of these cells is divided into eight first-order cubic elements with linear dimensions h/4 as illustrated in Figure 3.6. The volume of each first-order element is

The probability that a first-order cell will fragment is again taken to be f to preserve scale invariance. The number of fragmented first-order elements is

After fragmentation the number of first-order cells that have not been fragmented is

This process is repeated at successively higher orders. The volume of the nth-order cell Vnis given by

After fragmentation the number of nth-order cells is

Taking the natural logarithm of both sides we can write (3.75) and (3.76) as

Elimination of n from (3.77) and (3.78) gives

47

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FRAGMENTATION

Comparison with (2.1) shows that this is a fractal distribution with

Although this model is very idealized and non-unique, it illustrates the basic principles of how scale-invariant fragmentation leads to a fractal distribution. It also illustrates the principle of renormalization. The division into eight fragments is an arbitrary choice, however; other choices such as the division into two or 16 fragments will give the same result. This model is deterministic rather than statistical. Actual distributions of fragments are continuous rather than discrete but the deterministic model can be related to a "bin" analysis of a statistical distribution. Also, this model relates the probability of fragmentation f to the fractal dimension D but does not place constraints on the value of the fractal dimension. It is of interest to discuss this model in terms of the allowed range of D. The allowed range off is < f < 1 and the equivalent range of D is 0 < D < 3. Thus the concept of fractional dimension introduced in Figure 2.4 appears to be appropriate for fragmentation. However, the data for ash and pumice given in Table 3.2 fall outside this allowed range. Since such distributions are not precluded physically, we will consider this a fractal (scaleinvariant) distribution even though it lies outside the geometrically allowed range. We accept the physical view rather than the mathematical view. Fragmentation is a process with a wide range of applications. Thus many studies have been carried out to prescribe size distributions in terms of basic physics; however, fragmentation is a very complex problem. The results given above indicate that in many cases fragmentation is a scale-invariant process that leads to a fractal distribution. We now turn to a discrete model of fragmentation that does yield a specific fractal dimension. We will consider the fractal cube illustrated in Figure 2.4(b) and use it as a basis for a fragmentation model. This model is illustrated in Figure 3.7. Although the geometry and fractal dimension are the same, the concepts of the two models are quite different. The models given in Figure 2.4 are essentially for a porous (Swiss cheese) configuration. At each scale blocks are removed to create void space. In this chapter we consider fragmentation such that some blocks are retained at each scale but others are fragmented. In the model given in Figure 3.7 two diagonally opposed blocks are retained at each scale. No two blocks of equal size are in direct contact with each other. This is the comminution model for fragmentation proposed by Sammis et al. (1987). It is based on the hypothesis that direct contact between two fragments of near equal size during the fragmentation process will result in the breakup of one of the blocks. It is unlikely that small fragments will break large fragments or that large fragments will break small fragments.

6

FRAGMENTATION

49

For the configuration illustrated in Figure 3.7 we have N, = 2 for r , = h/2,N2 = 12 for r, = h/4, and N, = 72 for r, = h/8. From (2.2) we find that D = In 6/ln 2 = 2.5850. This is the fractal distribution of a discrete set but we wish to compare it with statistical fractals obtained from the actual fragmentation observations. It is therefore of interest to consider also the cumulative statistics for the comminution model. The cumulative number of blocks larger than a specified size for the three highest orders are N I c = 2 for r , = h/2, N2= = 14 for r, = hl4, and N,c = 86 for r, = h/8; Nnc is the cumulative number of the fragments equal to or larger than r,,. The cumulative statistics for the model illustrated in Figure 3.7 are given in Figure 3.8; excellent

Figure 3.7. Illustration of a fractal model for fragmentation. Two diagonally opposite cubes are retained at each scale. With r, = h/2, N , = 2 and r, = h/4, N , = 12 we have D = In 6 t h 2 = 2.5850.

Figure 3.8. Cumulative statistics for the fragmentation model illustrated in Figure 3.7. Correlation with (2.6) gives D = 2.60.

50

FRAGMENTATION

agreement with the fractal relation (2.6) is obtained taking D = 2.60. Thus the fractal dimensions for the discrete set and the cumulative statistics are nearly equal. This comminution model was originally developed for fault gouge. The derived fractal dimension for the model D = 2.60 is in excellent agreement with the measured values for fault gouge described in the last section. It is seen from Figure 3.5 and Table 3.2 that many observed distributions of fragments have fractal distributions near this value. This is evidence that the comminution model may be widely applicable to rock fragmentation. This model may also be applicable to tectonic zones in the earth's crust. The implication is that there is a fractal distribution of tectonic blocks over a wide range of scales. A number of other models have been proposed to explain fractal fragmentation. Steacy and Sammis (1991) developed an automaton that modeled nearest neighbor fragmentation. Palmer and Sanderson (1991) developed a model for crushing ice that accounts for the relative size of contacting fragments. In their model, D = 2.5 has the special significance that fragments of all sizes make equal contributions to the crushing force. Englman et al. (1987, 1988) have obtained a power-law distribution utilizing a maximumentropy model.

3.5 Porosity Most rock has a natural porosity. This porosity often provides the necessary permeability for fluid flow. There are generally two types of porosity, intergranular porosity and fracture porosity. Based on the discussion given above it would not be surprising if both types of porosity exhibited fractal behavior. Fractures are directly related to fragmentation, and detridal rocks are composed of rock grains with a variety of scales. Based on laboratory studies a number of authors have suggested that sandstones have a fractal distribution of porosity (Katz and Thompson, 1985; Krohn and Thompson, 1986; Daccord and Lenormand, 1987; Krohn, 1988a, b; Thompson et al., 1987). Hansen and Skjeltorp (1988) carried out two-dimensional box counting of the pore space in a sandstone and found D = 1.73. Brakensiek et al. (1992) carried out similar studies of the two-dimensional porosity of soils and found D = 1.8. Soils can be considered both in terms of fractal distributions of particle sizes and in terms of fractal distributions of void spaces (Rieu and Sposito, 1991a, b; 5 l e r and Wheatcraft, 1992). Fractal distributions of voids have also been suggested to be applicable to caves (Curl, 1986), karst regions (Laverty, 1987), and sinkholes (Reams, 1992). We previously introduced models with scale-invariant porosity in Figure 2.4. The Menger sponge, Figure 2.4(a), can be taken as a simple model for a

FRAGMENTATION

51

porous medium. However, this model is conceptually somewhat different from that considered in Chapter 2. The model is constructed from solid cubes of density p, and size r,. We construct a first-order Menger sponge from these cubes; the size of the first-order cube is r, = 3r0. The first-order sponge is made up of 20 solid zero-order cubes so that the first-order poros. the ity is 4, = 7/27 and the first-order density is p , = 2 0 ~ 4 2 7 Continuing construction to second order, the size of the cube is r, = 9r, and there are 400 solid cubes of size r, with density p,. Thus the porosity of the second-order Menger sponge is 4, = 3291729 and its density is p, = 4 0 0 ~ 4 7 2 9The . porosity of the nth-order Menger sponge is

which is not a power-law (fractal) relation. The density of the nth-order Menger sponge is

This is a fractal relation and is illustrated in Figure 3.9. For the Menger sponge the fractal dimension is D = In 201111 3 = 2.727. Generalizing (3.81), the porosity 4 for a fractal medium can be related to its fractal dimension by

Figure 3.9. Density dependence p/p, of a Menger sponge as a function of the size of the sponge rlr,. A fractal decrease in the density is found with D = 2.727 from (3.82).

52

FRAGMENTATION

where r is the linear dimension of the sample considered. Similarly, the density of the fractal medium scales with its size according to

The density of a fractal solid systematically decreases with the increasing size of the sample considered. A number of studies of the densities of soil aggregates as a function of size have been carried out. These studies show a systematic decrease in density as the size of the aggregate increases. A sieve analysis is carried out on a soil, and the mean density of each aggregate is found. The results for a sandy loam obtained by Chepil (1950) are given in Figure 3.10. Although there is scatter, the results agree reasonably well with the fractal soil porosity from (3.84) using as the fractal dimension D = 2.869.

Figure 3.10. Density of soil aggregates as a function of their size (Chepil, 1950). The solid line is from (3.84) with D = 2.869.

FRAGMENTATION

Problems

Problem 3.1. Assume that x takes the values xl = 1, x2 = 4, x, = 7, and x, = 8. Determine i , V, u, and y. Problem 3.2. Assume that x takes the values x, = 3, x, = 6, x, = 14, and x, = 17. Determine 2, V, a,and y. Problem 3.3. Flip four coins assigning + 1 to a head and - 1 to a tail. What are the probabilities of obtaining +4, +2,0, -2, -4? What are the variance and standard deviation of this distribution? Problem 3.4. Flip five coins assigning + 1 to a head and - 1 to a tail. What are the probabilities of obtaining +5, +3, +1, -1, -3, -5? What are the variance and standard deviation of this distribution? Problem 3.5. Assume that x takes the following valuesxi with probabilitiesf;:: x, = 2 ,fl =0.25,x2 = 4 , f2=0.5,x,= 8, f3=0.25. Determinei,V,u, and y. Problem 3.6. Assume that x takes the following values xi with probabilities fi: x1 = - 1' f1 = $, x2 = 1,f2 = i. Determine 2, V, u, and y. Problem 3.7. For the standard form of the normal distribution, determine the probability that a value lies in the "bins" -0.141 to 0.141, 0.141 to 0.424 to 0.707, and 0.707 to 0.990. Problem 3.8. For the standard form of the normal distribution, what is the probability that a value is greater than 0.707; greater than -0.707? Problem 3.9. For the standard form of the normal distribution, what are the mean, standard deviation, and coefficient of variation of the corresponding log-normal distribution? Problem 3.10. For a log-normal distribution with i = 1 and cV= 0.5, for what value of x is F(x) = 0.5? Problem 3.11. For a log-normal distribution with 2 = 1 and cV= 1, for what value of x is F(x) = 0.5? Problem 3.12. Derive an expression for the coefficient of variation cVfor the standard form of the Pareto distribution. What is its value for a = 3? Problem 3.13. Derive an expression for the coefficient of variation cVfor the exponential distribution. What is its value for v = 2? Problem 3.14. The distribution function for the power-law mass distribution given by (3.58) is

Assume that the maximum fragment size is r, and that v > 1. Determine the mean fragment radius i and the variance V about this mean. Problem 3.15. Consider a bar of unit length that has a probability f2 of being fragmented into two bars of equal length i. The two smaller bars have

53

54

FRAGMENTATION

the same probability of being fragmented into bars of length this process leads to a fractal distribution with

a. Show that

Show that this result is equivalent to (3.80). Problem 3.16. Consider a cube with a linear dimension h that is divided into 64 cubic elements with a dimension of h/4. The probability of fragmentation is f,,. The smaller cubes have the same probability of being fragmented into cubes with dimensions of hI16. Show that this process leads to a fractal distribution with

Show that this result is equivalent to (3.80). Problem 3.17. Consider a model for fragmentation based on the Menger sponge illustrated in Figure 2.4(a). Seven cubic elements are retained at each scale. Determine N , for r , = h/3, N2 for r, = h/9, and N3 for r, = h/27. What is the fractal dimension of the fragments? Problem 3.18. Consider the fragmentation model illustrated in Figure 3.11. Determine N, for r , = h/2,N, for r2 = h/4, and N3 for r, = h/8. What is the fractal dimension of the fragments? Problem 3.19. A model for fragmentation is constructed from a solid cube that is divided into 27 equal-sized cubes at each scale and six of these cubes are retained; i.e., the cubes in the center of each face are retained.

Figure 3.11. Illustration of a fractal model of fragmentation. Four cubic elements are retained at each scale.

FRAGMENTATION

Determine N , for r , = h/3, N , for r, = h/9, and N, for r, = h/27. What is the fractal dimension of the fragments? Problem 3.20. A model for a porous medium is constructed from solid cubes of density p, and size r,. At first order 26 of these cubes are used to construct a cube with r , = 3r,; the central cube is missing. Determine the and the density p , . Continue the construction and determine porosity and p,. What is the fractal dimension? Problem 3.21. A model for a porous medium is constructed from solid cubes of density p, and size r,. At first order 21 of these cubes are used to construct a cube with r , = 3r0; the cubes in the center of each face are missing (i.e., a Menger sponge with a central cube). Determine porosity and the density p , . Continue the construction and determine 4, and p,. What is the fractal dimension?

+,

+,

+,

55

Chapter Four

SEISMICITY AND TECTONICS

4.1. Seismicity

A variety of tectonic processes are responsible for the creation of topography. These include discontinuous processes such as displacements on faults and continuous processes such as folding. Tectonic processes are extremely complex but they satisfy fractal statistics. Earthquakes are of particular concern because of the serious hazard they present; earthquakes also satisfy fractal statistics in a variety of ways. Seismicity is a classic example of a complex phenomenon that can be quantified using fractal concepts (Turcotte, 1993, 1994a, 1995). According to plate tectonic theory, crustal deformation takes place at the boundaries between the major surface plates. In the idealized plate tectonic model plate boundaries are spreading centers (ocean ridges), subduction zones (ocean trenches), and transform faults (such as the San Andreas fault in California). Relative displacements at subduction zones and transform faults would occur on well-defined faults. Displacements across these faults would be associated with great earthquakes such as the 1906 San Francisco earthquake. However, crustal deformation is more complex and is usually associated with relatively broad zones of deformation. Take the western United States as an example: Although the San Andreas fault is the primary boundary between the Pacific and the North American plates, significant deformation takes place as far east as the Wasatch Front in Utah and the Rio Grande Graben in New Mexico. Active tectonics is occumng throughout the western United States. Distributed seismicity is associated with this mountain building. Even the displacements associated with the San Andreas fault system are distributed over many faults. South of San Francisco and north of Los Angeles the San Andreas fault has significant bends. Deformation associated with these bends is responsible for considerable mountain building and the 1956 Kern County earthquake, the 1971 San Fernando earthquake, the 1989 Loma Prieta earthquake, the 1992 Landers earthquake, and the 1994 Northridge earthquake.


Although the crustal deformation in the western United States may appear to be complex, it does obey fractal statistics in a variety of ways. This is true of all zones of tectonic deformation. We will first consider the frequency-magnitude statistics of earthquakes. Several quantities can be used to specify the size of an earthquake; these include the strain associated with the earthquake and the radiated seismic energy. However, for historical reasons the most commonly used measure of earthquake size is its magnitude. Unfortunately, a variety of different magnitude scales have been proposed; but to a first approximation, the magnitude is the logarithm of the energy radiated and dissipated in an earthquake. Typically great earthquakes have a magnitude m = 8 or larger. The 1992 Landers earthquake had a magnitude m = 7.6 and was the largest earthquake in California since the great 1906 San Francisco earthquake. The 1989 Loma Prieta earthquake had m = 7.1 and the 1994 Northridge earthquake had m = 6.6. Many regions of the world have dense seismic networks that can monitor earthquakes as small as magnitude two or less. The global seismic network is capable of monitoring earthquakes that occur anywhere in the world with a magnitude greater than about four. Various statistical correlations have been used to relate the frequency of occurrence of earthquakes to their magnitude, but the most generally accepted is the log-linear relation (Gutenberg and Richter, 1954)

where b and a are constants, the logarithm is to the base 10, and N is the number of earthquakes per unit time with a magnitude greater than m occurring in a specified area. The Gutenberg-Richter law (4.1) is often written in terms of N, the number of earthquakes in a specified time interval (say 30 years), and the corresponding constant a. The magnitude scale was originally defined in terms of the amplitude of ground motions at a specified distance from an earthquake. Typically the surface wave magnitude was based on the motions generated by surface waves (Love and Rayleigh waves) with a 20-s period, and the body wave magnitude was based on the motions generated by body waves (P and S waves) having periods of 6.8 seconds. The magnitude scale became a popular measure of the strength of earthquakes because of the logarithmic basis, which allows essentially all earthquakes to be classified on a scale of 0-10. Alternative magnitude definitions include the local magnitude and the magnitude determined from the earthquake moment. The frequency-magnitude relation (4.1) is found to be applicable over a wide range of earthquake sizes both globally and locally. The constant b or "b-value" varies from region to region but is generally in the range 0.8 < b < 1.2 (Frohlich and Davis, 1993). The constant a is a measure of the regional level of seismicity.

57

58


The magnitude is an empirical measure of the size of an earthquake. It can be related to the total energy in the seismic waves generated by the earthquake, Es, using the relation log Es = 1.44m + 5.24

(4.2)

where Es is in Joules. The strain released during an earthquake is directly related to the moment M of the earthquake by the definition

where is the shear modulus of the rock in which the fault is embedded, A is the area of the fault break, and 6e is the mean displacement across the fault during the earthquake. The moment of an earthquake can be related to its magnitude by

where c and d are constants. Kanamori and Anderson (1975) have established a theoretical basis for taking c = 1.5. Kanamori (1978) and Hanks and Kanamori (1979) have argued that (4.4) can be taken as a definition of magnitude with c = 1.5 and d = 9.1 (M in joules). This definition is consistent with the definitions of local magnitude and surface wave magnitude but not with the definition of body wave magnitude. It is standard practice today to use long-period (50-200 s) body and/or surface waves to directly determine the scalar moment M and (4.4) is used to obtain a moment magnitude. Kanamori and Anderson (1975) have also shown that it is a good approximation to relate the moment of an earthquake to the area A of the rupture by

where a is a constant. Combining (4. l), (4.4), and (4.5) gives

with

bd log@=C

+ loga-

b -1oga C


59

and (4.6) can be written as

In a specified region the number of earthquakes N per unit time with rupture areas greater than A has a power-law dependence on the area. A comparison with the definition of a fractal given in (2.6) with A r2 shows that the fractal dimension of distributed seismicity is

-

Taking the theoretical relation c = 1.5 we have

Thus the fractal dimension of regional or worldwide seismic activity is simply twice the b-value. The empirical frequency-magnitude relation given in (4.1) is entirely equivalent to a fractal distribution (Aki, 1981). The Gutenberg-Richter frequency-magnitude relation (4.1) has been found to be applicable under a great variety of circumstances. We will first consider its validity on a worldwide basis. The worldwide number of earthquakes per year with magnitudes greater than m is given in Figure 4.1 as a

Figure 4.1. Worldwide number of earthquakes per year, N,with magnitudes greater than m as a function of m. The square root of the rupture area A is also given. The solid line is the cumulative distribution of moment magnitudes from the Harvard Centroid Moment Tensor Catalog for the period January 1977 to June 1989 (Frohlich and Davis, 1993). The dashed line represents (4.1)with b = 1.11 ( D = 2.22) and a = 6 X 108 yr-1.

60


function of m. This data consists of 8719 earthquakes that occurred between January 1977 and June 1989. The data are from the Harvard Centroid Moment Tensor Catalog (Dziewonski et al., 1989) and seismic moments have been converted to moment magnitudes using (4.4). Comparisons of frequency-magnitude statistics from various catalogs have been given by Frohlich and Davis (1993). The worldwide data correlate with (4.1) taking b = 1.11 (D = 2.22) and a = 6 X 108 yr-1. Also given in Figure 4.1 is the equivalent characteristic length All2 obtained from (4.5) taking a = 3.27 X 106 Pa. The data given in Figure 4.1 can be used to estimate the frequency of occurrence of earthquakes of various magnitudes on a worldwide basis. For example, about 10 magnitude-seven earthquakes are expected each year and a single magnitude-eight earthquake can be expected in a year. The deviation of the data from the Gutenberg-Richter law (4.1) at magnitudes less than m = 5.2 can be attributed to the resolution limits of the global seismic network. Regional studies indicate that good correlations are obtained down to at least m = 2. The deviation of the data from the Gutenberg-Richter law at magnitudes greater than m = 7.5 is more controversial. Clearly there must be an upper limit to the size of an earthquake; but the deviations in Figure 4.1 can be attributed either to a real deviation from the correlation line or to the small number of very large earthquakes in the relatively short time span considered. Frequency-magnitude statistics for older earthquakes have been given by Abe (1981) for the period 1904 to 1980 and by Purcaru and Berckhemer (1982) for the period 1920 to 1979. The results given by Abe (198 1) support a systematic reduction of large earthquakes relative to the correlation curve in Figure 4.1, whereas the results given by Purcam and Berckhemer (1982) support the direct extrapolation of the correlation curve to larger earthquakes. Pacheco et al. (1992) have considered the extrapolation problem in detail and favor a systematic reduction of the large earthquakes. There is a physical basis for a change in scaling for large earthquakes. Smaller earthquakes are expected to be nearly equidimensional so that r Al12. However, the depth of large earthquakes is confined by the thickness of the seismogenic zone, say 20 lun,whereas the length can increase virtually without limit. Thus for large earthquake r A. The transition would be expected to occur for All2 = 25 km or m = 7. It is of interest to consider the distribution of seismicity associated with the relative velocity v across a plate boundary. We consider a specified length of the fault zone, which also has a specified depth of rupture. Thus the two plates are assumed to interact over an area Ap. The relative plate velocity v and interaction area A are related to the rupture area A and mean slip displacement 6, in an indiddual earthquake by

-

-

-


where the integral is camed out over the entire distribution of seismicity and &is the number of earthquakes per unit time with magnitudes between rn and rn + dm. The earthquake moment has been introduced from (4.3). We hypothesize that a fractal distribution of seismicity accommodates this relative velocity. From (4.1) and (4.4) we have

and

Taking the derivative of (4.12) gives


Since c > b the integral diverges for large rn so that the maximum-magnitude earthquake mmaxmust be specified. This is the well-known observation that a large fraction of the total moment and energy associated with seismicity occurs in the largest events. Integration of (4.15) gives

A large value of regional strain vA implies either a high level of regional P seismicity (large a) or a large magnitude for the maximum-magnitude earthquake (large rnm,,). This type of relation has been derived by several authors (Smith, 1976; Molnar, 1979; Anderson and Luco, 1983) and has been used to estimate regional strain (Anderson, 1986; Young and Coppersmith, 1985) and to compare levels of seismicity with known strain rates (Hyndman and Weichert, 1983; Singh et al., 1983). As a specific application, we consider the regional seismicity in southe m California. The frequency-magnitude distribution of seismicity in southern California is given in Figure 4.2. The data from the southern California earthquake network are for the period 1932 to 1994, the number of earthquakes per year N with magnitudes greater than rn are given as a function of rn. Over the entire range of 4 < rn < 7.5 the data are in excellent agreement

61

62


with (4. l), taking b = 0.923 (D = 1.846) and a = 1.4 X 105 yr-1. In terms of the linear dimension of the fault rupture, this magnitude range corresponds to a linear size range 0.7 < All2 < 40 km. Also included in Figure 4.2 is the value of N associated with great earthquakes on the southern section of the San Andreas fault. Dates for 10 large earthquakes on this section of the fault have been obtained from radiocarbon dating of faults, folds, and liquifaction features within the marsh and stream deposits on Pallett Creek where it crosses the San Andreas fault 55 km northeast of Los Angeles (Sieh et al., 1989). In addition to historical great earthquakes on January 9, 1857, and December 8, 1812, additional great earthquakes were estimated to have occurred in 1480 15, 1346 2 17, 1100 + 65, 1048 33,997 t 16,797 22,734 2 13, and 67 1 2 13. The mean repeat time is 132 years, giving N = 0.0076 yr-1. The most recent in the sequence of earthquakes occurred in 1857, and the observed offset across the fault associated with this earthquake was 12 m (Sieh and Jahns, 1984). Sieh (1978) estimates that the magnitude of the 1857 earthquake was m = 8.25. Taking the values given above, the recurrence statistics for these large earthquakes are shown by the solid circle in Figure 4.2. An extrapolation of the fractal relation for regional seismicity appears to make a reasonable predic-

+

Figure 4.2. Number of earthquakes per year N occurring in southern California with magnitudes greater than m as a function of m. The solid line is the data from the southern California earthquake network for the period 1932-1994. The straight dashed line is the correlation with (4.1) taking b = 0.923 (D = 1.846) and a = 1.4 X 105. The solid circle is the observed rate of occurrence of great earthquakes in southern California (Sieh et al., 1989).

+

+


tion of great earthquakes on this section of the San Andreas fault. Since this extrapolation is based on the 40 years of data between 1932 and 1972, a relatively large fraction of the main interval of 132 years, it suggests that the value of a for this region may not have a strong dependence on time during the earthquake cycle. This conclusion has a number of important implications. If a great earthquake substantially relieved the regional stress, then it would be expected that the regional seismicity would systematically increase as the stress increased before the next great earthquake. An alternative hypothesis is that an active tectonic zone is continuously in a critical state and that the fractal frequency-magnitude statistics are evidence for this critical behavior. In the critical state the background seismicity, small earthquakes not associated with aftershocks, have little time dependence. This hypothesis will be discussed in Chapter 16. Acceptance of this hypothesis allows the regional background seismicity to be used in assessing seismic hazards (Turcotte, 1989b). The regional frequency-magnitude statistics can be extrapolated to estimate recurrence times for larger magnitude earthquakes. Unfortunately, no information is provided on the largest earthquake to be expected. An important question in seismology is whether the occurrence of large plate-boundary earthquakes can be estimated by extrapolating the regional seismicity as was done above for southern California. This is a subject of considerable controversy. Some authors argue that the large earthquakes occur more often than would be predicted by an extrapolation. To further consider the time dependence of regional seismicity (the time dependence of a), we consider the frequency-magnitude statistics of the regional seismicity in southern California on a yearly basis. Again the number of earthquakes N in each year between 1980 and 1994 with magnitudes greater than m are given in Figure 4.3 as a function of m. In general there is good agreement with (4. I), taking b = 1.05 and a = 2.06 X 105 yr-1. The exceptions can be attributed to the aftershock sequences of the Whittier (1987), Landers (1992), and Northridge (1994) earthquakes. Comparing the correlation lines in Figures 4.2 and 4.3 shows that the correlation line in Figure 4.2 lies somewhat above those in Figure 4.3. This is because the data given in Figure 4.2 include aftershocks. With aftershocks removed, the near uniformity of the background seismicity in southern California illustrated in Figure 4.3 is clearly striking. This is strongly suggestive of a thermodynamic behavior. We will return to this point in Chapter 16. We now relate the seismicity in southern California to the relative velocity across the plate boundary. The data given in Figure 4.2 can be used to predict the regional strain using (4.16). Substituting p. = 3 X 1010 Pa, b = 0.89, c = 1.5,d=9.1, v = 4 8 mmyr-',mmax=8.05, a n d a = 1.4 X lO5yr-1 we find from (4.16) that Ap = 1.5 X lo4 km2. Taking the depth of the seismogenic zone to be 15 km, the length of the seismogenic zone corresponding to

63

64


this area is 730 km. This is about a factor of two larger than the actual length of the San Andreas fault in southern California. This is reasonably good agreement considering the uncertainties in the parameters. However, there are two other factors that can contribute to this discrepancy.

Figure 4.3. The cumulative number of earthquakes N with magnitudes greater than rn for each year between 1980 and 1994 is given as a function of m; the region considered is southern California. (a) 19801984; (b) 1985-1989; (c) 1990-1994. The straightline correlation is with the Gutenberg-Richter relation (4.1) with b = 1.05 and a = 2.06 X 105 yr-1. The relatively large numbers of earthquakes in 1987,1992, and 1994 can be attributed to the aftershocks of the Whittier, Landers, and Northridge earthquakes, respectively. If aftershocks are excluded, the background seismicity in southern California is nearly uniform in time.


1.

2.

65

Southern California is an area of active compressional tectonics. The Transverse Ranges are in this region and are associated with the displacements on the San Andreas fault. The strains associated with the formation of the Transverse Range should be added to the strains associated with strike-slip displacements on the San Andreas fault system. South of San Bernardino, displacements on the San Andreas fault are associated with small and moderate earthquakes; no great earthquakes are believed to occur on this section. With a maximum magnitude of about seven, the expected level of seismicity would be about a factor of five greater than with a maximum magnitude eight earthquake. Thus a higher level of seismicity on this section of the San Andreas could contribute to the high observed level.

Since the eastern United States is a plate interior, the concept of rigid plates would preclude seismicity in the region. However, the plates act as stress guides. The forces that drive plate tectonics are applied at plate boundaries. The negative buoyancy force acting on the descending plate at a subduction zone acts as a "trench pull." Gravitational sliding off an ocean ridge acts as a "ridge push." Because the plates are essentially rigid, these forces are transmitted through their interiors. However, the plates have zones of weakness that will deform under these forces and earthquakes result. Thus earthquakes occur within the interior of the surface plates of plate tectonics, although the frequencies of occurrence are much lower than at plate boundaries. An ex-

M~

(c)

Figure 4.3. (con?.)

66


ample was the three great earthquakes that occurred in the MemphisSt. Louis (New Madrid, Missouri) seismic zone during the winter of 18111812. Nuttli (1983), based on historical records, has estimated that the surface wave magnitudes of these earthquakes were 8.5, 8.4, and 8.8, respectively. This area remains the most active seismic zone in the United States east of the Rocky Mountains. Based on both instrumental and historical records Johnston and Nava (1985) have given the frequency-magnitude statistics for earthquakes in this area for the period 1816-1983. Their results are given in Figure 4.4. The data correlate well with (4.1), taking b = 0.90 (D = 1.80) and a = 2.24 X 103 yr-1. Comparing the data in Figure 4.4 with the data in Figure 4.2 indicates that the probability of having a moderate-sized earthquake in the Memphis-St. Louis seismic zone is about 1/50 of the probability in southern California. Assuming that it is valid to extrapolate the data in Figure 4.4 to larger earthquakes, a magnitude m = 8 would have a recurrence time of about 7000 yr. Although there is certainly a significant range of errors, the results given above indicate that the measured frequency-magnitude statistics associated with the Gutenberg-Richter frequency-magnitude relation (4.1) can be used to assess seismic hazards. The regional b(D) and a values can be used to estimate recurrence times for earthquakes of various magnitudes.

Figure 4.4. The cumulative number of earthquakes per year N occurring in the Memphis-St. Louis (New Madrid, Missouri) seismic zone with magnitudes greater than m as a function of m (Johnston and Nava, 1985). The data are for the period 1816-1983. The open circles represent instrumental data and the solid circles historical data. The dashed line represents (4.1) with b = 0.90 (D = 1.80) and a = 2.24 X 103

N yr"


4.2 Faults

There are two end-member models that give fractal distributions of earthquakes. The first is that there is a fractal distribution of faults and each fault has its own characteristic earthquake. The second is that each fault has a fractal distribution of earthquakes. Observations strongly favor the first hypothesis. On the northern and southern locked sections of the San Andreas fault, there is no evidence for a fractal distribution of earthquakes. Great earthquakes and their associated aftershock sequences occur, but between great earthquakes seismicity is essentially confined to secondary faults. A similar statement can be made about the Parkfield section of the San Andreas fault, where moderate-sized earthquakes occurred in 1881, 1901, 1924, 1934, and 1966. There is no evidence for a fractal distribution of events on this section of the San Andreas fault. We therefore conclude that a reasonable working hypothesis is that each fault has a characteristic earthquake and a fractal distribution of earthquakes implies a fractal distribution of faults. Although we can conclude that the frequency-size distribution of faults is fractal, the fractal dimension is not necessarily the same as that for earthquakes. Equal fractal dimensions would imply that the interval of time between earthquakes is independent of scale. This need not be the case. Tectonic models for a fractal distribution of faults have been proposed by King (1983, 1986), Turcotte (1986b), King et al. (1988), and Hirata (1989a). Fractal distributions of faults that give well-defined b-values have been proposed by Huang and Turcotte (1988) and Hirata (1989b). Before discussing the observational data on spatial distributions of faults, we will discuss the definitions of faults, joints, and fractures. Fractures are generally any crack in a rock. If there is a lateral offset across the fracture, it is a fault; if there is no lateral offset, it is a joint. Because of the grinding (comminution) effect of creating offsets on faults during earthquakes, a zone of brecciated rock (fault gouge) generally develops on the fault. The larger the total offset on the fault, the wider the disrupted zone. It is generally difficult to quantify the frequency-size distributions of faults. This is because the surface exposure is generally limited. Many faults are not recognized until earthquakes occur on them. Coal mining areas provide access to faults and fractures at depth. The cumulative distributions of the number of faults N with lengths greater than r are given in Figure 4.5 for two coal mining areas (Villemin et al., 1995). Correlations with the fractal relation (2.6) are given with D = 1.6. Other compilations of the numberlength statistics of faults and comparisons with power-law correlations have been given by Gudmundson (1987), Hirata (1989a), and Main et al. (1990). Hirata et al. (1987) and Velde et al. (1993) found a fractal distribution of microfractures in laboratory experiments that stressed unfractured granite.

67

68


Systematic studies of the statistics of exposed joint and fault-trace patterns over many orders of magnitude in length scale have been given by Barton (1995). Bedrock exposures were created on Yucca Mountain, Nevada, by the removal of soil and debris creating exposures known as pavements. Barton (1995) mapped these exposures to obtain the two-dimensional distribution of joints and faults. His map of pavement 1000 is reproduced in Figure 4.6. This was located in the densely welded orange brick unit of the Topopah Spring Member of the Miocene Paintbrush Tuff. He used the box-counting method illustrated in Figure 2.8 to determine the fractal dimension of the fracture traces. His result for the pavement illustrated in Figure 4.6 is given in Figure 4.7; the mean fractal dimension is D = 1.7. Barton (1995) analyzed 17 fracture maps over scales ranging from 0.5 mm (microfractures) to 5000 km (transform faults) and found good correlations with fractal statistics with values of D ranging from 1.4 to 1.7. Davy et al. (1990) and Sornette et al. (1993) carried out laboratory simulations of brittle crustal deformation and found D = 1.7 -+ 0.1 for fracture trace networks. The study of the distribution of faults and joints can also be carried out by one-dimensional sampling. Drilling cores provide an excellent data base for this type of study. The intersections of fractures with the core can be rep-

Figure 4.5. Cumulative number of faults N,with lengths r greater than r. The boxes are measurements in the Lorraine coal basin and the circles from the Vernejoul coal field (Villemin et al., 1995). Correlations with the fractal relation (2.6) are given with D = 1.6.


69

resented a s a series of points o n a line, the drill core. A one-dimensional b o x -

counting method is applied in direct analogy to the two-dimensional box counting illustrated in Figure 2.8. The line is divided up into 2, 4, 8, 16, . . . 1 1 1 segments so that r = 1, 2,1 a, 8, G, . . . . In each case the number of line seg-

Figure 4.6. Map of the faults and joints exposed on pavement 1000, Yucca Mountain, Nevada (Barton, 1995). This exposure was located in the densely welded orange brick unit of the Topopah Spring Member of the Miocene Paintbrush Tuff.

70


ments that include points (fractures) N(r) is determined and log N(r) is plotted against log r(or log llr). If a linear or near-linear dependence is found, the slope gives the fractal dimension using (2.2). Barton (1995) analyzed the distribution of gold-bearing, quartz-filled fractures (veins) intersecting exploratory drill holes from tunnels in the Perseverance Mine, Juneau, Alaska. His results for core hole 7- 18, using the onedimensional box-counting technique, are given in Figure 4.8. A good correlation with the fractal relation (2.2) is obtained taking D = 0.59. For the 23 drill holes studied by Barton (1995), good correlations with fractal statistics were obtained, with D ranging from 0.41 to 0.62. Similar studies have been carried out by La Pointe (1988). Velde et al. (1990, 1991), Ledesert et al. (1993), Manning (1994), Boadu and Long (1994a, b), and Magde et al. (1995). It should be emphasized that a wide variety of mechanisms are responsible for the formation of joints and faults and not all would be expected to yield fractal distributions. Limitations of the fractal approach have been discussed by Harris et al. (1991) and Gillespie et al. (1993). The fractal model for fragmentation illustrated in Figure 3.7 can also be applied to tectonic fragmentation (Sammis et al., 1987). As the surface plates of plate tectonics evolve in time, geometrical incompatibilities develop (Dewey, 1975). Simple plate boundaries consisting of ocean ridges, subduction zones, and transform faults cannot evolve in time without overlaps or holes de-

Figure 4.7. Statistics using the box-counting algorithm on the exposed fracture network at Yucca Mountain, Nevada (Barton, 1995), given in Figure 4.6. A correlation with (2.1) is used to obtain the fractal dimension

D = 1.7.


71

veloping. The result is that plate interiors must deform to accommodate the geometrical incompatibilities. Because of the weaker silicic rocks of the continental crust, and the many ancient faults pervading the continental lithosphere, continental parts of surface plates deform much more readily than oceanic parts. This can be easily seen at the boundary between the Pacific and North American plates in the western United States. The adjacent continental North American plate consisting of the western states deforms extensively whereas there is little internal deformation in the adjacent oceanic Pacific plate. Just as the comminution model can be applied to fragmentation, it can also be applied to the deformation of the continental crust. The tectonic forces break the continental crust into a fractal distribution of interacting crustal blocks over a wide range of scales. The crustal blocks are bounded by faults so that a fractal distribution of block sizes can be related to a fractal distribution of faults. To illustrate this we consider the deterministic comminution model for fragmentation given in Figure 3.7. To fragment the single zero-order block of size h requires three orthogonal faults (No = 3) of size r, = h; the result is eight blocks of size hl2. Six of these eight blocks are further fragmented; this requires N, = 3 X 6 = 18 faults of size r, = hl2. The result is 48 blocks of size hl4; 36 of these 48 blocks are further fragmented to

1

llr (rn-')

Figure 4.8. Distribution of quartz-filled fractures (veins) in core hole 7- 18, Perseverance Mine, Juneau, Alaska (Barton, 1995). The number of line segments N of length r that intersect fractures along the core is given as a function of r. This is the one-dimensional box-counting method. The correlation is with the fractal relation (2.2) taking D =

0.59.

72


give blocks of size h/8.This further fragmentation requires N2 = 3 X 36 = 108 faults of size r2 = h/4.From (2.2) we have D = In 6/ln 2 = 2.5850 for the frequency-size distribution of faults in the volume of the block, which is equal to the value for the frequency-size distribution of blocks obtained in Chapter 3. However, for a statistical model we require cumulative statistics. The cumulative frequency-size distribution of faults is essentially identical to the cumulative statistics for fragment volumes given in Figure 3.8. Thus we have D,= 2.60 for this model. This deterministic fragmentation model can also be used to relate onedimensional and two-dimensional measurements to the three-dimensional size-frequency distribution of faults. A surface projection of the comminution model given in Figure 3.7 is illustrated in Figure 4.9.Also illustrated in this figure is the box-counting method applied to the deterministic fault exposure. At zero-order there is one box with ro = h and it covers faults, No = 1. At first order there are four boxes with r , = h/2 and three of them cover faults, N , = 3. At second order there are 16 boxes with r, = h/4 and nine of them cover faults, N2 = 9. From (2.2) we find that D = In 3/ln 2 = 1.5850. Noting that

Figure 4.9. (a) Illustration of the surface exposure of the deterministic fractal fragmentation model given in Figure 3.7. The lines represent a fractal distribution of linear faults that separate a fractal distribution of blocks. method applied to this exposure. (b) At zero order the one box with r = h covers faults, No = 1. (c) At first order three of the four boxes with r = hl2 cover faults, N, = 3. (d) At second order nine of the 16 boxes with r = h/4 cover faults, N, = 9. And from (2.2) we have D = In 3lln 2 = 1S8SO.


we see that the fractal dimension of the cross section, D2 = In 3lln 2, is related to the fractal dimension of the original construction from Figure 3.3., D,= In 6 t h 2, by

Thus we expect, for a statistically self-similar distribution of fault lengths, that the two-dimensional fault dimension of a surface D, will be related to the three-dimensional fractal dimension D,by (4.18). We next consider the fractal dimension of one-dimension transects (drill holes) through the three-dimensional deterministic fracture model given in Figure 3.7. In this case different transects intersect different fault distributions. The four different distributions are given in Figure 4.10. For each of these we carry out a one-dimensional "boxw-countingmethod using line segments of length h, h12, and hl4. For the four distributions we find No = 1, 1, 1, 1 so on average No = 1, we find N , = 1, 1 , 2 , 2 so on average N , = 1.5, and we find N2 = 1 , 2 , 2 , 4 so on average N2 = 2.25. Using the average values we find from (2.2) that D = In 1Slln 2 = 0.5850. Noting that

we see that the average fractal dimension of the linear transects, D l= In 1.5Iln 2 is related to the fractal dimensions of the cross sections D,= In 31ln 2 and the original construction D,= In 6 t h 2 by

Thus we expect for a statistically self-similar distribution of faults that the one-dimensional fractal dimension from a bore hole D l will be related to the three-dimensional fractal dimension D, by (4.20). These results are quite consistent with the values given in Figures 4.7 and Figure 4.8 where fractal dimensions D, = 1.7 and D,= 0.59 were obtained. Barton (1995) found that

Figure 4.10. Illustration of four different onedimensional transects (drill holes) across the deterministic fractal fragmentation model given in Figure 3.7. In each case the one-dimensional boxcounting method is applied. For transect (a) we have No = 1 (rO= h), N , = 1 (ro = h/2), and N2 = 1 (ro = h14); for transect (b) we have No = 1, N , = 1 , N2 = 2 ; for transect (c) we have No = 1, N , = 2, N, = 2, and for transect (d) we have No = 1, N , = 2, N, = 4. On average No = 1 , N , = 1.5, N, = 2.25 so that from (2.2) we have D = In 1.51 In 2 = 0.5850.

74


a three-dimensional vein network at the Perseverance mine sampled in one dimension with drill cores and in two dimensions by surface mapping yielded fractal dimensions of 0.5 and 1.48, respectively. Watanabe and Takahashi (1995) have proposed a more general statistical approach to the determination of the three-dimension fractal distribution of faults from a onedimensional data base. In many oil fields the permeability of the rock is dominated by faults and joints. To determine the feasibility of producing from a field, accurate estimates of the permeability are required. The application of fractal statistics as discussed above can provide the basis for making such estimates. Data on fault and joint spacings intersecting a single well can be used to estimate the full three-dimensional permeability of a field. Although the discussion given above has been based on the applicability of a comminution-based fragmentation model, it should be noted that Sornette et al. (1990) and Sornette and Davy (1991) have offered an alternative model for fractal fracture networks. These authors have suggested that the networks are random growth networks similar to diffusion-limited aggregation (see Chapter 8). Termonia and Meakin (1986) have given another approach to fractal fracturing. Taking the fractal dimension for the distribution of faults in a volume of rock to be D,= 2.6, the number of faults with a characteristic linear dimension greater than r, in a given area, scales with r according to

Similarly we assume that for earthquakes De= 2 so that the number of earthquakes per unit time, in a given area, with a characteristic rupture size greater than r scales with r according to

The average interval between earthquakes T~ on a fault with a characteristic dimension r is given by

Thus the interval between earthquakes on a specified fault is longer for smaller faults. This is generally consistent with observations. If we further assume that faults remain active for a time 7,then the total displacement 6 on a fault of scale r is given by


75

where ae is the displacement in a single event. However

and substitution of (4.23) and (4.25) into (4.24) gives

Walsh and Wattersen (1988) and Marrett and Allmendinger (1991) have compiled measurements of the dependence of total displacement on a fault 6 as a function of fault length r and concluded that there is a power-law (fractal scaling). The results obtained by Marrett and Allmendinger (1991) are given in Figure 4.11. Data from a wide variety of tectonic environments are included. Although there is considerable scatter, a reasonably good correlation with (4.26) is found. It should be emphasized that this correlation must be to some extent fortuitous since T, is unlikely to be a constant in different tectonic settings. Also, Scholz and Cowie (1990), Cowie and Scholz (1992a, b), Scholz et al. (1993), and Dawers et al. (1993) conclude that 6 - r with an

Figure 4.11. Dependence of total fault displacement on fault length; data from Marrett and Allmendinger (1991). The straight line correlation is with (4.26), 8 rl.6.

-

76


additional parameter, the critical shear stress for fault propagation. These authors correlate fault displacement and length in individual tectonic environments and find for each environment a reasonably good correlation with 6 r. They argue that it is misleading to include data from a variety of tectonic environments. In addition, Gillespie et al. (1992) find a universal power-law correlation between fault width and total displacement. Jackson and Sanderson (1992) and Pickering et al. (1994) have concluded that in several examples the number of faults with displacements greater than a specified value satisfy fractal statistics with D = 0.7-1.4. Sedimentary basins are often formed by horizontal extension on suites of normal faults. The horizontal extension thins the continental crust, resulting in the subsidence of the surface and the deposition of a sedimentary pile on the subsiding "basement." A common observation is that the amount of extension associated with "visible" normal faults (for example, on seismic reflection profiles) is significantly less than the amount of extension associated with the observed crustal thinning. Typically only 40-70% of the required extension can be associated with the larger faults on which displacements can be determined. Using a fractal distribution for the number of faults as a function of size and the displacements of these faults as described above, the displacements on small unobserved faults can be determined from the displacements on the larger faults. Walsh et al. (1991) and Marrett and Allmendinger (1992) have argued that this approach can explain the discrepancy. The total strain E in a volume V, is related to the number of faults N,, fault area A,, and total fault displacement 6 by

-

Taking A,

-

r2

along with (4.21) and (4.26) we have

If these statistics are valid in a region, the larger faults dominate in terms of regional strain, but the smaller faults do make a significant contribution.

4.3 Spatial distribution of earthquakes The box-counting method in three dimensions has been applied to the spatial distribution of earthquake aftershocks by Robertson et al. (1995). Aftershocks are particularly well located because extensive arrays of seismometers have been deployed following the main shock. These authors considered the aftershock sequence of the m = 6.1 Joshua Tree earthquake of April 23, 1992 (2600 events in a 20 X 20 X 19 km volume in 160 days), and the after-


77

sequence of the m = 6.2 Big Bear earthquake of June 28, 1992 (818 events in a 20 X 20 X 17 km volume in 375 days). The spatial distributions of these aftershocks are given in Figure 4.12(a, b). The numbers of cubes occupied by one or more earthquakes are given as a function of the cube size in Figure 4.12(c) for the two aftershock sequences; cubes with linear dimensions between 500 m and 20 km were used. The data are in quite good agreement with (2.2) taking D = 2. A fractal dimension of two would be expected if the earthquakes lie on a plane; however, there is considerable three-dimensional structure to the aftershock seshock

Figure 4.12. Box-counting method of cluster analysis applied to the threedimensional distributions of the aftershocks of the Joshua Tree earthquake (JTS) and the Big Bear earthquake (BBS). The spatial distributions of these earthquakes are given in (a) and (b). The number of cubes N in which one or more earthquakes occur is given as a function of the linear dimension of the cube r in (c). The straight line is the fractal correlation (2.2) with D = 2.

78


quences. This led Robertson et al. (1995) to suggest that the earthquakes form the "backbone" of a percolation cluster. The "backbone" of a threedimensional percolation cluster has a fractal dimension near two. A detailed discussion of percolation clusters and the meaning of the "backbone" will be given in Chapter 15. The box-counting technique has been applied to both the temporal and two-dimensional spatial distribution of earthquakes in Japan by Bodri (1993).

4.4Volcanic eruptions We next turn to volcanic eruptions. It is considerably more difficult to quantify a volcanic eruption than it is to quantify an earthquake. There are a variety of types of eruption, and the various types are quantified in dif-

Figure 4.13. Number of volcanic eruptions per year N=with a tephra volume greater than V as a function of V for the period 1975-1 985 (squares) and for the last 200 years (circles) (McClelland et al., 1989). The line represents the correlation with (2.6) taking D = 2.14.


ferent ways. Some eruptions produce primarily magma (liquid rock) while others produce primarily tephra (ash). Utilizing the volume of tephra as a measure of size McClelland et al. (1989) have published frequency-volume statistics for volcanic eruptions. Their results for eruptions during the period 1975-1985 and for historic (last 200 years) eruptions are given in Figure 4.13. The number of eruptions with a volume of tephra greater than a specified value is given as a function of the volume. A reasonably good correlation is obtained with the fractal relation (2.6) by taking D = 2.14. It appears that volcanic eruptions are scale invariant over a significant range of sizes. A single volcano can produce eruptions with a wide spectrum of sizes. Also, volcanoes have a wide spectrum of sizes. The circumstances that determine the volume of tephra in an eruption are poorly understood. Thus models that would provide an explanation of the observed value of D are not available.

Problems Problem 4.1. Determine Es, M, A , and se for an m = 7 earthquake (take c = 1.5, d = 9.1, a = 3.27 X 106Pa, p = 3 X 1OlOPa). Problem 4.2. Determine Es, M, A , and se for an m = 6 earthquake (take c = 1.5, d = 9.1, a = 3.27 X 106Pa, p = 3 X IOIOPa). Problem 4.3. On a worldwide basis how many magnitude-six earthquakes are expected in a year? Problem 4.4. In a region, the recurrence interval -re for a magnitude-six earthquake is 18 months; if b = 0.9 what is the recurrence interval T~ for a magnitude-seven earthquake? Problem 4.5. In a region, the recurrence interval T~ for a magnitude-five earthquake is 10 years; if b = 1 what is the recurrence interval r efor a magnitude-seven earthquake? Problem 4.6. In a subduction zone the length of the seismogenic zone is 1000 km and its depth is 30 km. The convergence velocity is 100 mm yr-1. (a) Determine a if b = 1, c = 1.5, d = 9.1, p= 3 X IO'OPa, and mmax= 8.5. (b) Determine the recurrence time for the magnitude-8.5 earthquake. Problem 4.7. The length of a seismogenic zone on a strike-slip fault is 100 km and its depth is 15 km. (a) Determine a if b = 1, c = 1.5, d = 9.1, p = 3 X 10"JPa, v = 50 mm yr-1, and mmax= 6.2. (b) Determine the recurrence time for the magnitude-6.2 earthquake. Problem 4.8. The characteristic earthquake of magnitude-seven on a fault has a recurrence interval of 200 years; using (4.23), what is the recurrence time for a characteristic earthquake of magnitude six? Take c = 1.5, d = 9.1.

79

80


Problem 4.9. The characteristic earthquake of magnitude six on a fault has a recurrence interval of 400 years; using (4.23), what is the recurrence time for a characteristic earthquake of magnitude four? Take c = 1.5 and d = 9.1. Problem 4.10. Assume that the distribution of rupture zone sizes on a fault is modeled by the Sierpinski carpet, Figure 2.3(d). (a) Determine N, for r, = h/3, N2 for r2 = h/9, and N3 for r3 = h/27. Further assume that the earthquake displacement associated with each rupture zone is proportional to its linear dimension. (b) If the slip velocities on all the zones are equal, determine the b-value for the earthquakes. Problem 4.11. Assume that the distribution of rupture zone sizes on a fault is modeled by the fractal distribution given in Figure 4.9(a). Further assume that the earthquake displacement associated with each rupture zone is proportional to its linear dimension. If the slip velocities on all the zones are equal, determine the b-value for the earthquakes. Problem 4.12. Consider the fragmentation model given in Figure 3.11 as a model for the distribution of fracture and joints in the earth's crust. What are the fractal dimensions for a two-dimensional surface exposure and for a one-dimensional transect? Problem 4.13. Ash eruptions with a volume greater than 1000 km3 are expected to have a profound influence on the global climate. What is the expected recurrence interval for such eruptions? Problem 4.14. The Pinatuba, Philippines, eruption had a tephra volume of 18 km3. What is the expected recurrence interval for this eruption?

Chapter Five

ORE GRADE ANDTONNAGE

5.1 Ore-enrichment models Statistical treatments of ore grade and tonnage for economic ore deposits have provided a basis for estimating ore reserves. The objective is to determine the tonnage of ore with grades above a specified value. The grade is defined as the ratio of the mass of the mineral extracted to the mass of the ore. Evaluations can be made on either a global or a regional basis. Much of the original work on this problem was carried out by Lasky (1950). He argued that ore grade and tonnage obey log-normal distributions. Other authors, however, have suggested that a linear relation is obtained if the logarithm of the tonnage of ore with grades above a specified value is plotted against the logarithm of the grade. The latter is a fractal relation. A fractal relation would be expected if the concentration mechanism is scale invariant. Many different mechanisms are responsible for the concentrations of minerals that lead to economic ore deposits. Probably the most widely applicable mechanisms are associated with hydrothermal circulations. We first consider two simple models that illustrate the log-normal and power-law distributions for tonnage versus grade. De Wijs (1951, 1953) proposed the model for mineral concentration that is illustrated in Figure 5.l(a). In this model an original mass of rock Mo is divided into two equal parts each with a mass M , = M d 2 . The original mass of the rock has a mean mineral concentration Co, which is the ratio of the mass of mineral to mass of rock. As in Chapter 3 we refer to this mass as a zero-order cell. It is hypothesized that the mineral is concentrated into one of the two zero-order elements such that one element is enriched and the other element is depleted. The zero-order elements then become first-order cells, each of which is divided into two first-order elements with mass M , = M d 4 . The mean mineral concentration in the enriched zero-order element C,, is given by

82

ORE GRADE AND TONNAGE

where 4, is the enrichment factor. The first subscript on C refers to the order of cell being considered. The second subscript refers to the amount of enrichment: the lower the number the more the enrichment and the higher the concentration. The subscript on the enrichment factor refers to the fact that each cell is divided into two equal elements; the enrichment factor 4, is greater than unity since C,, must be greater than C,,.A simple mass balance shows that the concentration in the depleted zero-order element is

Figure 5.1. Illustration of two models for the concentration of economic ore deposits. In both models a mass of rock is first divided into two equal parts, then four equal parts, etc. (a) De Wijs (1951, 1953) proposed a successive concentration of minerals into smaller and smaller volumes. Four orders of concentration are illustrated. At each order one-half of each cell is enriched by the ratio 42and the other half is depleted by the ratio 2 4,. This model gives a binominal distribution for tonnage versus grade and in the limit of very small volumes gives a log-normal relation. (b) Turcotte ( 1 9 8 6 ~ ) proposed a similar model, but with the further concentration limited to the highest-grade ores. This leads to a power-law (fractal) distribution of tonnage versus grade.


The enrichment factor must be in the range 1 < 4, < 2. This model is illustrated in Figure 5.l(a). The process of concentration is then repeated at the next order as illustrated in Figure 5.l(a). The zero-order elements become first-order cells and each cell is again divided into two elements of equal mass M, = Md4. The mineral is again concentrated by the same ratio into each first-order element. The enriched first-order element in the enriched first-order cell has a concentration

The depleted first-order element of the enriched first-order cell and the enriched first-order element of the depleted first-order cell both have the same concentrations:

The depleted first-order element of the depleted first-order cell has a concentration

This result is also illustrated in Figure 5.l(a) along with two higher-order cells. This model gives a binomial distribution of ore grades, and in the limit of infinite order reduces to the log-normal distribution given in (3.29). The resulting distribution is not scale invariant; the reason is that the results are dependent on the size of the initial mass of ore chosen and this mass enters into the tonnage-grade relation. We will show in Chapter 6 that the resulting distribution is a multifractal. Cargill et al. (1980, 1981) and Cargill (1981) disagreed with the logarithmic dependence and suggested that a linear relationship is obtained if the logarithm of the tonnage is plotted against the logarithm of the mean grade. A simple model that gives this result was proposed by Turcotte ( 1 9 8 6 ~ and ) is illustrated in Figure 5.l(b). This model follows very closely the model discussed above. Again, an original mass of rock M, is divided into two parts each with a mass M, = M,/2, and it is hypothesized that the mineral is concentrated into one of the two zero-order elements so that (5.1) and (5.2) are applicable. However, at the next step only the enriched element is further fractionated. The problem is renormalized so that the enriched element is treated in exactly the same way at every scale (order). This results in a fractal (scale-invariant) distribution. The concentration of ore into one or the two elements in the enriched first-order cell results in the concentrations given by (5.3) and (5.4). However, the depleted first-order cell continues to

83

84


have the concentration given by (5.2). This result is illustrated in Figure 5.l(b) along with two higher-order cells. The results given in Figure 5.l(b) can be generalized to the nth order with the result

where C,, is the mean ore grade associated with the mass

Taking the natural logarithms of (5.6) and (5.7) gives

and

The elimination of n between (5.8) and (5.9) gives

-

With the density assumed to be constant, M r3, where r is the linear dimension of the ore deposit considered, and we have

Comparison with (2.6) shows that this is a power-law or fractal distribution with

Since the allowed range for 4, is 1 < 4, < 2, the allowed range for the fractal dimension is 0 < D < 3. To be fractal the distribution must be scale invariant. The scale invariance is clearly illustrated in Figure 5.l(b). The con-


centration of ore could be started at any order and the same result would be obtained. The left half at order two looks like order one, the left half at order three looks like order two, etc. This is not true for the distribution illustrated in Figure 5.1 (a). We now generalize this model so that the original mass of rock is divided into two parts, but the masses of the two parts are not equal. The mass of the enriched element M , , is related to the original mass Mo by

and the mass of the depleted element M I , is given by

The mass ratio a can take on the range of values 1 < a < m. The concentration ratio is defined as before and is the ratio of the concentration in the enriched element C , , to the reference concentration C,.

A mass balance shows that the concentration in the depleted element C , , is given by

The enriched zero-order element becomes a first-order cell; this cell is divided into two parts with the enriched part having a mass

7- M0 a2 Mll

M21=

And the concentration in this enriched mass is

The above results are generalized to the nth order to give

85

86


and

Taking the natural logarithms of (5.19) and (5.20) yields

and

Elimination of n between (5.21) and (5.22) gives

Comparison with (2.6) shows that this is a power-law or fractal distribution with

+,

It is clear from (5.25) that depends upon a.It is easy to show that this is reasonable. The case a = 2 was considered above. For a = 8 we have from (5.25)

We now show that (5.26) is entirely equivalent to (5.12). The first-order concentration into one-eighth of the original mass, $, must be equivalent to three orders of the concentration into one-half the original mass, 4,. Thus we can write


It follows that

Thus (5.26) is equivalent to (5.12) and can be derived independently of the mass ratio chosen. Two classic models for the generation of ore deposits lead directly to the fractal distribution derived above. The first is the chromatographic model and the second is the Rayleigh distillation model (Allbgre et al., 1975; All&gre and Lewin, 1995). The chromatographic model can be directly applied to the dissolution-reprecipitation process that occurs during fluid percolation through cooling porous intrusions. Such a mechanism has been applied to explain hydrothermal, epithermal, and skarn mineral deposits. Consider a segment of the crust with an average trace element concentration Co. Fluid circulation occurs enriching a mass fraction a-I with an enrichment factor +a so that the new concentration is given by (5.15). Assume that this enrichment process again affects the already enriched region with the same enrichment factor. The doubly enriched mass fraction is a - 2 and the concentration is given by (5.18). The process is repeated to produce successive enrichments of smaller and smaller segments of the crust. This is the chromatographic model and is identical to the fractal model described above. We next consider Rayleigh distillation. This is the classic model used in geochemistry to explain the extreme enrichment of trace elements observed in some crystalline rocks (Allkgre and Minster, 1978). The basic model considers the solidification of a magma to form the crystalline rock; trace elements are partitioned between the remaining magma and the crystallizing solid. If a trace element is incompatible with the crystalline solid, the residual magma will become progressively more enriched. If an incremental mass of magma 6 M crystallizes, the incremental mass of mineral 6Mmtransferred from the magma to the solid is given by

where M is the mass of magma, Mm is the mass of the mineral in the magma, and KR is the solid-liquid partition coefficient. If KR < 1 the remaining magma is systematically enriched. The allowed range of values for the

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solid-liquid partition coefficient is 0 I KR I 1 . If KR = 1 there is no enrichment of the remaining magma and the concentration of the mineral is constant. If K , = 0 the concentration of the mineral in the solid is zero until the melt contains only the mineral. We can write (5.29) as a differential equation in the form

Integrating with the initial condition that Mm = Mmowhen M = Mo gives

The concentration of the mineral in the enriched residual magma Cmand the concentration of the mineral in the original magma Cmoare given by

and

Substitution of (5.32) and (5.33) into (5.31) yields

This is the classic result for Rayleigh distillation. If KR = 1 then Cm/Cmo= 1 and there is no enrichment as expected. If KR = 0 then Cm= 1 when M/Mo = Cmo;the melt contains only the mineral. Note that Cmis constrained to the range CmoI CmI 1. We now show that this classic result for Rayleigh distillation is entirely equivalent to the fractal relation (5.24). In terms of the fractal model considered above, the enriched element has a mass M - 6M with concentration C , , and the depleted element has a mass 6M with concentration C,,. Thus we have Co = MmIM and C = 6Mm/6M so that the definition of the solid1.2 liquid partition function given in (5.29) becomes


Substitution from (5.16) gives

c$a=a+(l -a)KR Using this result the power in (5.24) is given by

We assume that at each renormalization the fraction of solid is very small so that

with E < < 1. In this limit we can write

In [ a + (1 - a)KR] = In [I

+~

( -1 K,)] = ~ ( -1 K,)

(5.41)


And the substitution of (5.42) into (5.24) gives (5.34). In the limit a + 1 the fractal model is identical to Rayleigh distillation. Furthermore, the substitution of (5.42) into (5.25) gives

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The fractal dimension of the ore deposit is simply related to the solid-liquid partition coefficient of the Rayleigh distillation process. For the allowed range of values for KR,0 I KR I 1, the allowed range for D is 0 ID I 3. In the limit KR + 1 there is little enrichment and D + 0; in the limit KR + 0 there is very strong enrichment and D + 3.

5.2 Ore-enrichment data In each of the enrichment steps in our fractal model the concentration C,, is the mean concentration in the mass of ore M,,. For applications to actual ore deposits we generalize the fractal relation between ore grade and tonnage to

where M is the mass of the highest grade ores, which have a mean concentration The reference mass Mo is the mass of rock from which the ore was derived, which has a mean concentration Co. As in the previous examples of naturally occurring fractal distributions, there are limits to the applicability of (5.44). The lower limit on the ore grade is clearly the regional background grade C , that has been concentrated. However, there is also an upper limit: the grade C cannot exceed unity, which corresponds to pure mineral. The entire subject of tonnage-grade relations has been reviewed by Harris (1984). There is clearly a controversy in the literature between Lasky's law, which gives a log-normal dependence of tonnage on grade, and the power-law or fractal dependence. Lasky (1950) and Musgrove (1965) have argued in favor of the log-normal relation. On the other hand, Cargill et al. (1980, 1981) and Cargill (1981) have argued in favor of the power-law dependence. These authors based their analyses on records of annual production and mean grade. Their results for mercury production in the United States are given in Figure 5.2. The cumulative tonnage of mercury mined prior to a specified date is divided by the cumulative tonnage of ore from which the mercury was obtained to give the cumulative average grade. The data points in Figure 5.2 represent the five-year cumulative average grade (in weight ratio) versus the cumulative tonnage of ore. Using Bureau of Mines records Cargill et al. (1981) found that the total amount of mercury mined between 1890 and 1895 was Mm, and the tonnage of ore from which = this mercury was obtained was M , ; the mean grade for this period was Mm,IM,.The cumulative amount of mercury mined between 1890 and 1900 was Mm, and the cumulative tonnage of ore from which the mercury was mined was M,; the mean cumulative grade for this period was c, = Mm21M2.

c.

c,


91

These computations represent the two data points farthest to the left in Figure 5.2. The other data points represent the inclusion of additional five-year periods in the computations. Cargill et al. (1980, 1981) and Cargill (1981) further hypothesized that the highest-grade ores are usually mined first so that the cumulative ratio of mineral tonnage to ore tonnage at a given time is a good approximation to the mean ore grade of the highest-grade ores. Thus it is appropriate to compare their data directly with the fractal relation (5.44). Excellent agreement is obtained taking D = 2.01. This is strong evidence that the enrichment processes leading up to the formation of mercury deposits are scale invariant. It is also of interest to introduce a reference concentration of mercury into the fractal relation. An appropriate choice is the mean measured concentration in the continental crust. The mean crustal concentration of mercury as = 8 X 10-8 (0.08 ppm). Using this value in given by Taylor (1964) is (5.44) we find that the correlation line in Figure 5.2 is given by

c,

with M in kilograms. According to the fractal model the mercury ore in the United States has been concentrated from continental crust with a mass M, = 4.05 X 10'7 kg. Assuming a mean crustal density of 2.7 X 103 kg m-3, the mercury resources of the United States were concentrated from an original crustal volume of 1.5 X 105 km3. Since the total crustal volume of the United States is approximately 2.7 X 108 km3, the source volume for the mercury deposits is about 0.05 percent of the total. It is concluded that the

Figure 5.2. Dependence of cumulative ore tonnage M on mean grade C for mercury production in the United States (Cargill et al., 1981). Correlation with (5.44) gives a fractal dimension D = 2.01.

92


processes responsible for the enrichment of mercury ore deposits are restricted to a relatively small fraction of the crustal volume. It is seen from Figure 5.2 that the cumulative production of 1.2 X 108 kg of mercury has been obtained from 2 X 1010kg of ore of volume 7.4 X 106 m3. Since the source region has a volume of 1.5 X 105 km3, the fraction of the source region that has been mined is only 5 X 10-8. The results given in Figure 5.2 can also be used to determine how much mercury ore must be mined in the future to produce a specified amount of mercury. To produce the next 1.2 X lo8 kg of mercury will require the processing of about 1.6 X 1011 kg of ore. Using production records of lode gold, Cargill (1981) gave cumulative tonnage-grade data for lode gold production in the United States. The data points in Figure 5.3 represent the five-year cumulative average grade versus the cumulative tonnage of ore for the period 1906-1976. A good correlation with the fractal relation (5.44) is obtained taking D = 1.55. This fractal dimension is somewhat less than the value obtained for mercury, indicating a smaller enrichment factor. Again, the mean crustal concentration is introduced as a reference concentration. Taking = 3 X 10-9 (3 ppb) (Taylor and McLennan, 1985) for gold, we find the correlation line in Figure 5.3 is given by

co

Figure 5.3. Dependence of cumulative ore tonnage M on mean grade for lode gold production in the United States (Cargill, 1981). Correlation with (5.44) gives a fractal dimension D = 1.55.


93

with M in kilograms. According to the fractal model the lode gold in the

United States has been concentrated from a continental crustal mass of 3 X 1018 kg. Assuming a mean crustal density of 2.7 X 103 kg m-3, the gold was concentrated from a crustal volume of 106 km3 or about 0.4 percent of the total crustal volume. Using copper production records in the same way, Cargill et al. (1981) have also given cumulative grade-tonnage data for copper production in the United States. Their results are given in Figure 5.4. The cumulative grade is again given as a function of cumulative ore tonnage at five-year intervals. The data obtained prior to 1920 fall systematically low compared to the later data. Cargill et al. (1981) attributed this systemic deviation from a fractal correlation to the adoption of an improved metallurgical technology for the extraction of copper in the 1920s. A smaller fraction of the available copper was extracted prior to this time so that the data points are low. It is again appropriate to compare these data with the fractal relation (5.44). Assuming the early data to be systematically low, excellent agreement is obtained taking D = 1.16. This fractal dimension is almost a factor of two less than the fractal dimension obtained for mercury ore. This indicates that the applicable enrichment processes concentrate copper less strongly than they do mercury. We again relate the fractal relation for the enrichment to the mean crustal concentration. The mean concentration of copper in the upper crust as given by Taylor and McLennan (1981) is C, = 2.5 X 10-5 (25 ppm). Using this value in (5.44), we find that the correlation line in Figure 5.4 is given by

Figure 5.4. Dependence of cumulative ore tonnage M on mean grade for copper production in the United States (Cargill et al., 1981). Correlation with (5.44) gives a fractal dimension D = 1.16.

94


with M in kilograms. According to the fractal model, the copper ore in the United States has been concentrated from continental crust with a mass M,, = 3.22 X 1019 kg. Assuming a mean upper crustal density of 2.7 X 103 kg m-3, the copper resources of the United States were concentrated from an original crustal volume of 1.19 X 107 km3. This represents about 4 percent of the total crustal volume of the United States. The crustal volume from which copper is enriched is nearly 100 times larger than the volume from which mercury is enriched. It is concluded that the processes responsible for the enrichment of copper are much more widely applicable than those for mercury. As our final example we consider data on the relationship between cumulative tonnage and grade for uranium in the United States. Data for the preproduction inventory as given by the US Department of Energy have been tabulated by Harris (1984, p. 228) in terms of cumulative tonnage and the average grade of this tonnage; these data are tabulated in Figure 5.5. The high-grade data are based on production records and the lower-grade data are based on estimates of reserves. The higher-grade data are in excellent agreement with the fractal relation (5.44) taking D = 1.48. Thus the enrichment of uranium is intermediate between the enrichment of copper and mercury. The predicted cumulative tonnage at lower grades falls below the extrapolation of the fractal relation; this can be attributed to an underestimation of the preproduction inventory at low grades. It is again instructive to relate the fractal relation for the enrichment of uranium to the mean crustal concentration. The mean concentration of uranium in the upper crust as given by Taylor and McLennan (1981) is C, =

Figure 5.5. Dependence of cumulative ore tonnage M on mean grade for uranium production in the United States (Harris, 1984, p. 228). Correlation with (5.44) gives a fractal dimension D = 1.48.


1.25 X (1.25 ppm). Using this value in (5.44), we find that the correlation line in Figure 5.5 is given by

with M in kilograms. According to the fractal model the uranium ore in the United States has been concentrated from continental crust with a mass M, = 6.4 X 10'7 kg. Assuming a mean crustal density of 2.7 X 103 kg m-3, the uranium resources of the United States were concentrated from an original crustal volume of 2.4 X lo5 km3. This represents about 0.09 percent of the crustal volume of the United States. The crustal volume from which uranium is enriched is about a factor of two larger than the crustal volume for mercury but is a factor of 50 less than the crustal volume for copper. In several examples the statistics on ore tonnage versus ore grade have been shown to be fractal to a good approximation. This is not surprising since two of the classic models for the generation of ore deposits, chromatographic and Rayleigh distillation, both lead directly to fractal distributions. The examples considered here yield a considerable range of fractal dimensions: 2.01 for mercury, 1.55 for gold, 1.48 for uranium, and 1.16 for copper. If Rayleigh distillation were applicable then from (5.43), the applicable liquid-solid partition functions would be 0.33 for mercury, 0.48 for gold, 0.49 for uranium, and 0.61 for copper. It should be emphasized, however, that the chromatographic model is a more likely explanation for the concentration of these minerals. Not all mineral deposits and related statistical data satisfy power-law (fractal) distributions. A specific example is the frequency-size distribution of diamonds (Deakin and Boxer, 1986).

5.3 Petroleum data There is also evidence that the frequency-size distribution of oil fields obeys fractal statistics (Barton and Scholz, 1995). Drew et al. (1982) used the rela= 1.67N to estimate the number of fields of order i, N j , in the westtion Ni-, ern Gulf of Mexico. Since the volume of oil in a field of order i is a factor of two greater than the volume of oil in a field of order i - 1, their relation is equivalent to a fractal distribution with D = 2.22. Barton and Scholz (1995) find D = 2.49 for the Frio Strandplain play, onshore Texas. The number-size statistics for oil fields worldwide as compiled by Carmalt and St. John (1984) are given in Figure 5.6. A reasonably good correlation with the fractal relation (2.6) is obtained taking D = 3.3. The large differences between these values for the fractal dimension may be attributed to differences in the

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96


regional geology, but it may also be due to difficulties in the data. It is often difficult to determine whether adjacent fields are truly separate, and data on reserves are often poorly constrained. Nevertheless, the applicability of fractal statistics to petroleum reserves can have important implications. Reserve estimates for petroleum have been obtained by using power-law (fractal) statistics and log-normal statistics. Accepting power-law statistics leads to considerably higher estimates for available reserves (Barton and Scholz, 1995; La Pointe, 1995; Crovelli and Barton, 1995). The model for the concentration of economic ore deposits given above leads to a range of geometrically acceptable fractal dimensions. However, the observed distribution for oil fields falls outside this range. This again illustrates the difficulties associated with restrictions on power-law .(fractal) distributions. As stated previously, we define a power-law statistical distribution as a fractal distribution. It should not be surprising that the frequency-size statistics of oil pools and oil fields are fractal; it was shown in Chapter 2 that topography is generally fractal. One consequence is that the frequency-size statistics of lakes has been found to be fractal (Maybeck, 1995). Because traps for oil involve topography on impermeable sedimentary layers, it is expected that this topography will also be fractal. Thus it is reasonable that the frequency-size distribution of oil pools is fractal. V lo6 bbl oil

Figure 5.6. The number N of oil fields worldwide with a volume of oil greater than V as a function of V. The equivalent number of barrels is also given. The circles represent the data given by Carmalt and St. John (1984) and the line represents the correlation with (2.6) taking D = 3.3.

V km3

ORE GRADE A N D TONNAGE

Barton and Scholz (1995) have examined the spatial distribution of hydrocarbon accumulations and have concluded that they obey fractal statistics. Their results for production from the J sandstone of the Denver basin are given in Figure 5.7. Production from this basin is primarily in the northeast comer of Colorado and the southwest comer of Nebraska. A 40 X 40mile section of the basin is considered and this section is divided into 80 X 80 square cells of size 0.5 miles. The cells with one or more wells are illustrated with black dots in Figure 5.7 as drilled cells. The cells with one or more wells that are either producing or had a show of hydrocarbons but at quantities too small to produce are illustrated with black dots in Figure 5.7 as producing or showing cells.

Figure 5.7. A 40 X 40-mile section of the Denver basin is considered. This section is divided into 80 X 80 cells of dimension 0.5 mile. Wells that penetrated the J sandstone reservoir were considered (Barton and Scholz, 1995) and were listed as dry holes, producing oil or gas, or showing oil or gas. Cells with one or more drilled wells are illustrated as drilled cells by black dots; cells with one or more producing or showing wells are illustrated as producing and showing cells by black dots. The box-counting technique was applied and the number of occupied boxes is given as a function of box size for the two distributions. Good correlations are obtained with the fractal relation (2.2) taking D = 1.80 for the drilled cells and D = 1.43 for the producing and showing cells.

98


The box-counting technique was applied to both the drilling data and the producing and showing data. The number of occupied boxes as a function of the reciprocal of the box size is given in Figure 5.7 for both data sets. In both cases good correlations were obtained with the fractal relation (2.2). For the drilled cells the fractal dimension was D = 1.80; if every cell had been drilled the fractal dimension would have been D = 2.0. For the producing and showing cells the derived fractal dimension was D = 1.43. This result indicates that the complex processes responsible for the generation of petroleum traps leads to a fractal spatial distribution of oil pools. Barton and Scholz (1995) also examined the spatial distribution of hydrocarbon accumulations in the Powder River basin, Wyoming, and found a good correlation with fractal statistics taking D = 1.49. Carlson (1991) examined the spatial distribution of 4775 hydrothermal precious-metal deposits in the western United States and found that the probability-density distribution for these deposits is fractal. Blenkinsop (1994) found similar results for gold deposits in the Zimbabwe Archean craton.

Problems Problem 5.1. Determine the concentration factor $* for an ore deposit with D = 2. Problem 5.2. Determine the concentration factor $, for an ore deposit with D = 1. Problem 5.3. Determine the solid-liquid partition coefficient K , corresponding to an ore deposit with D = 2. Problem 5.4. Determine the solid-liquid partition coefficient K , corresponding to an ore deposit with D = 1. Problem 5.5. Consider the cubic model for mineral concentration illustrated in Figure 3.6. (a) In terms of the enrichment factor $, defined by (5.26) and C,, what is the concentration in the seven depleted zero-order elements? (b) What is the concentration in the seven depleted first-order elements? (c) What is the allowed range for $,? (d) What is the corresponding allowed range for D? Problem 5.6. From the correlation for mercury production given in (5.45), how much pure mercury ( c = 1) would be expected? Problem 5.7. From the correlation for mercury production given in (5.45), determine the total production of mercury when the mean grade of ore that has been mined reaches C = 0.001. Problem 5.8. From the correlation for lode gold production given in (5.46), = 1 ) would be expected in the United States? how much pure gold

(c


Problem 5.9. From the correlation for lode gold production given in (5.46), determine the total amount of lode gold mined to date. Assume that the mean grade of ore mined prior to the present is C = 9 ppm. Problem 5.10. From the correlation for copper production given in (5.47) determine the total production of copper to date. Assume that the mean grade of ore mined prior to the present time is = 0.008. Problem 5.1 1. From the correlation for copper production given in (5.47), how much pure copper (C = 1) would be expected? Problem 5.12. The fractal dimension for the distribution of areas of lakes has been found to be D = 1.55 (Kent and Wong, 1982). Assuming that the mean depth of a lake is proportional to the square root of its area, what is the fractal dimension for the distribution of water volumes in lakes? Problem 5.13. Consider the data for the 40 X 40-mile section of the Denver basin given in Figure 5.7. What fraction of 1 X 1-mile sections would be expected to contain oil?

99

Chapter Six

FRACTAL CLUSTERING

6.1 Clustering

We next relate fractal distributions to probability. This can be done using the sequence of line segments illustrated in Figure 2.1. The objective is to determine the probability that a step of length r will include a line segment. First consider the construction illustrated in Figure 2.1 (a). At zero order the probability that a step of len th r, = 1 will encounter a line segment, p, = 1; at 1 first order we have r , = 2 and p , = 2, and at second order r, = 31 and p, = 41 . Next consider the construction illustrated in Figure 2.l(c). At zero order the probability that a step of len th r, = 1 will encounter a line segment is p, = 1; at first order we have r , = 2 and p , = 1, and at second order r, = 41 and p, = 1. Finally we consider the Cantor set illustrated in Figure 2.l(e). At zero order the probability that a set of length r, = 1 will encounter a line segment is 2 p, = 1 ; at first order we have r , = 31 and p , = 3, and at second order r, = 31 and 4 PZ = 9 . The probability that a step of length ri will include a line segment can be generalized to

f

f

where N is the number of line segments of length ri. Taking C = 1 in (2.1) the number N is related to ri so that we obtain

For the Cantor set the probability that a step of length ri = (f)' encounters a line segment is pi= ($1' so that D = In 2Iln 3 as was obtained previously. The Cantor set is both scale invariant and deterministic. Its deterministic aspect can be eliminated quite easily. A scale-invariant random set is generated by randomly removing one-third of each line rather than always removing the

FRACTAL CLUSTERING

101

middle third. This process is illustrated in Figure 6.1. The fractal dimension is unchanged and the probability relations derived above are still applicable. We will use the examples given above as the basis for studying fractal clustering. We consider a series of point events that occur at specified times. To consider N point events that have occurred in the time interval i0we introduce the natural period T , = i d N . We then introduce a sequence of intervals defined by

Our measure of clustering will be the probability pn that an event occurs in an interval of length i n . As a specific example, consider a uniform (equally spaced in time) se. first event occurs at t, = ries of N events that occur in an interval i OThe i,,/2N, the second event at t, = 3 i d 2 N , the third event at t, = 5rO/2N,and so forth. The probability p,, that an event will occur in an interval is given by

IIIIIIIIIIIIIIIIIIIlllIllllm

11I

Figure 6.1. Illustration of the first six orders of a random Cantor set. At each step, a random third of each solid line is removed.

102

FRACTAL CLUSTERING

If the number of events N is greater than the number of intervals n , we have N > n or T,,> rO/N.In this case an event occurs in every interval so that pn = 1. If the number of events is less than the number of intervals, we have N < n or T,, < T,/N. In this case only N of the n intervals have events so that pn= N/n. Because there is no clustering no interval T~ contains more than one event. A more realistic model for a series of events in time is that their occurrence is completely uncorrelated. The time at which each individual event occurs is random. An example would be telephone calls placed in a city during a given hour. If N point events occur randomly in a time interval T,, it is a Poisson distribution. In the limit of a very large number of events (N + =J), the distribution of intervals between events is given by

where f ( ~ ) dis~the probability that an event will occur after an interval of time between T and T + d r in length. This distribution is clearly not scale invariant since the natural time scale T ~ enters N (6.5). We will next determine the probability p that an interval of length T~ will include an event if N events occur randomly in an interval T,. This is the classic problem of the random distribution of N balls into n boxes. We introduce Pm= m/n where m is the number of intervals that include events and n = T ~ T is , the total number of intervals. We assume that both n and m are integers. The probability that Pmhas a specified value is given by

n

where

fm(pm)= 1; the binomial coefficient is defined by m=O

It is often appropriate to take Nand n to be large numbers. In this limit, fn(Pm) from (6.6) has a strong maximum at a specific value of P,, p. To illustrate the probabilistic approach to clustering we consider a ninthorder random Cantor set. First we rescale so that unit length is the length of a ninth-order element. Thus the length of the zero-order element is 3 9 and there are 29 ninth-order elements. To determine the fractal dimension of the

FRACTAL CLUSTERING

103

clustering by the "box method," we take intervals of length r,, = 2" and determine the fraction p that include at least one ninth-order element as a function of rn. An example is given by the open circles in Figure 6.2. The best-fit straight line has a slope of 0.368 so that p T-0.368 and D = 0.632. The deviation from the exact value D = 0.6309 for the deterministic Cantor set is due to the reduced rate of curdling in the probabilistic set. If the same number of ninth-order elements is uniformly distributed (no clustering), the probability of finding an element with an interval from (6.4) is given by the solid circles in Figure 6.2. In this case, the slope is unity for r < (;)9 and zero for r > (;)9. Thus, D = 0 for r < (;)9, that is, a set of isolated points, and D = 1 for r > ($)9, that is, a line. Fractz! clustering has been applied to seismicity by Sadovskiy et al. (1985) and by Smalley et al. (1987). The latter authors considered the temporal variation of seismicity in several regions near Efate Island in the New Hebrides island arc for the period 1978-1984. One of their examples is given in Figure 6.3. During the period under consideration 49 earthquakes that exceeded the minimum magnitude required for detection occurred in the region. Time intervals T such that 8 min 1 T I524,288 min were considered. The fraction of intervals with earthquakes p as a function of interval length T is given in Figure 6.3(a) as the open circles. The solid line shows the correlation with the fractal relation (6.2) with D = 0.255. The dashed line is the result for uniformly spaced events. The results of a simulation for a random distribution of 49 events in the time interval studied is given in Figure 6.3(b). The random simulation (Poisson distribution) is significantly different from the earthquake data and is close to the uniform distribution.

-

Figure 6.2. The fraction p of steps of length r that include solid lines for a ninth-order random Cantor set is given by the open circles. The unit length is the length of the shortest line; the original line length is 39. The solid circles correspond to a uniform distribution of the same number of lines as given by (6.4). The line corresponds to (6.2) with D = 0.632.

104

FRACTAL CLUSTERING

Fractal clustering of seismic activity in time for California has been found by Lee and Schwarcz (1995). For the historical record of activity in the San Andreas fault zone of central California and the paleoseismic record of the San Gabriel Fault, they found D = 0.43-0.46. For the San Andreas fault zone in southern California, they found D = 0.67. Velde et al. (1990) and Meceron and Velde (1991) have carried out studies of the one-dimensional clustering of joints and faults and found good fractal correlations. They utilized the intersections of the joints and faults with a well. Manning (1994) has studied the one-dimension clustering of metamorphic veins. He found D = 0.46 for wollastonite-quartz veins in marbles, D = 0.81 for actinolite-chlorite veins in oceanic diabases, and D = 0.25-0.63 for epidote-quartz veins in basalts.

Figure 6.3. Fractal cluster analysis of 49 earthquakes that occurred near Efate Island, New Hebrides, in the period 1978-1984 (Smalley er al., 1987). (a) The circles give the fraction of intervals p of length 7 that include an earthquake as a function of 7 . The solid line represents the correlation with (6.2) taking D = 0.255. The broken line is the result for uniformly spaced events. (b) The results for 49 randomly distributed events (Poisson process).

FRACTAL CLUSTERING

Fractal clustering can also be studied in higher dimensions. The application to two dimensions is illustrated by the sequence of constructions given in Figure 2.3. The objective is to determine the probability that a square box of size r encounters a square that has been retained. First consider the construction given in Figure 2.3(a). At zero order the probability that a box of size ro = 1 will include a square is p, = 1; at first order we have r, = and p , = $, and at second order we have r2 = and p, = &. Next consider the construction illustrated in Figure 2.3(b). At zero order the probability that a box of size r,, = 1 will include a retained square is p, = 1; at first order we have r , = $ and p , = $, and at second order we have r2 = $ and p2 = &. Finally we consider the Sierpinski carpet illustrated in Figure 2.3(d). At zero order the probability that a box of size ri = 1 will include a retained square is po = 1; at first order we have r , = f and p, = 8, and at second order we have r, = and

6

5

64

P2=81e The probability that a square box of size ri will include a retained square can be generalized to

and substitution of (2.1) gives

For the Sierpinski carpet the probability that a square box of size ri = (fY will include a retained square is pi = ($)' so that D = In 8nn 3, as was previously found. The Sierpinski carpet can be applied to clustering in two dimensions in the same way that the Cantor set was applied in one dimension. This is directly analogous to the box-counting algorithm discussed in Chapter 2 and illustrated in Figure 2.8. The two-dimensional spatial clustering of intraplate hot spot volcanism (i.e., Hawaii, etc.) has been studied by Jurdy and Stefanick (1990). They found a fractal correlation with D = 1.2. This approach can be extended to three dimensions using cubes of various sizes. The application to three dimensions is illustrated using the Menger sponge given in Figure 2.4(a). The objective is to determine the probability that a cube with size r encounters retained material. At zero order the probability that a cube of size ro = 1 will include material is p - 1;at first order we have 20 01400 r , = 51 and p , = 2 , and at second order we have r, = 9 and p, = m. The probability that a cube of size ri includes retained material can be generalized to

and substitution of (2.1) gives

105

106

FRACTAL CLUSTERING

For the Menger sponge the probability that a cube of size ri = (f)' encounters retained material is pi= (E)' so that D = In 20nn 3 as was previously found. The generalization of (6.2),(6.9)and (6.11)is

6.2 Pair-correlationtechniques Another approach to the clustering of point events is to use the pair-correlation distribution C(r),which is defined to be the number of pairs of points whose separation is between r - $Ar and r + i h r per unit area (Vicsek, 1992). One point is picked and the distances to all other points are determined. The same thing is done for the second point and for all other points. The number of pairs with separations between r - khr and r + t h r is divided by Ar. This result is the pair-conelation distribution C(r) in one dimension. For a two-dimensional distribution the number in each interval A r is divided by r Ar to obtain C(r);for a three-dimensional distribution the number in each interval Ar is divided by r2 Ar to obtain C(r). Two simple deterministic examples illustrate how pair-correlation distributions are determined. First consider the one-dimensional example of four equally s aced points on a line of unit length. The pair-correlation distribution is C ( J )= 6 , C($)= 4, C ( l )= 2. Next consider the two-dimensional example of four points on the corners of a unit square. The pair-correlation distribution is C(1)= 8, ~ ( f =i 4)1 f i = 2 f i . If the points are randomly distributed in space, the pair-correlation distribution is exponential

P

If the points exhibit scale-invariant clustering, a power-law dependence is obtained

where a is related to the fractal dimension of the distribution by

It is seen that (6.14)and (6.15)are entirely equivalent to (6.12).

FRACTAL CLUSTERING

107

We will consider two examples of scale-invariant (fractal) pair-correlation distributions. Our first example is a sixth-order Cantor set with 64 points. The pair-correlation distribution is given in Figure 6.4. A good correlation with (6.14) is obtained taking a = 0.369. From (6.15) we find with d = 1 that D = 0.631, the fractal dimension for a Cantor set is D = In 2/ln 3 = 0.6309. As our second example we consider the fourth-order Koch snowflake illustrated in Figure 6.5, which has 625 points. The pair-correlation distribution is given in Figure 6.6. A good correlation with (6.14) is obtained taking a = 0.58. From (6.15) we find with d = 2 that D = 1.42, and the fractal dimension of the Koch snowflake is D = In 5/ln 3 = 1.465. Again reasonably good agreement is found. Kagan and Knopoff (1980) have determined the pair-correlation distribution for the two-dimensional spatial distribution of worldwide seismicity and found that a = 1 for shallow seismicity so that D = 1 . This is consistent with the D = 2 found for the spatial distribution of aftershocks in California by Robertson et al. (1995) illustrated in Figure 4.12. The pair-correlation technique is entirely equivalent to the box-counting technique for point events, and both methods give the same fractal dimension for scale-invariant distributions.

,-. -

slope=-0 369 a=O 369

L

u m QQ0 4

w o

*

0

\ L.p-L D=l-a=O 631

. L

o9

12

'

18

5

log r

2 1

74

Figure 6.4. The paircorrelation distribution C(r) is given as a function of r for a sixth-order (n = 64) Cantor set. A good correlation with the fractal relation (6.14) is obtained taking a = 0.369. From (6.15) the corresponding fractal dimension fractal dimension is D = 0.631 for the(the Cantor set is D = In 2nn 3 = 0.6309).

108

FRACTAL CLUSTERING

Figure 6.5. Illustration of a fourth-order Koch snowflake (n = 625). This deterministic fractal has a fractal dimension D = In 5An 3 = 1A65.

Figure 6.6. The paircorrelation distribution C(r) is given as a function of r for the fourth-order Koch snowflake illustration in Figure 6.5. A good correlation with the fractal relation (6.14) is obtained taking a = 0.58. From (6.15) we have D = 1.42 (D = 1.465 for the Koch snowflake).

0.0

20.0

40.0

60.0

80.0

col

0.2

0.4

0.6

1 .O

0.8

log r

1.2

1 4

1.6

FRACTAL CLUSTERING

109

6.3 Lacunarity

It is clear that fractal constructs with identical fractal dimensions can have quite different appearances. One example is the deterministic Cantor set illustrated in Figure 2. l (e) compared with the random Cantor set illustrated in Figure 6.1. Third-order examples of these sets are given in Figure 6.7. The difference between these two sets is the distribution of the size of gaps. Mandelbrot (1982) introduced the concept of lacunarity as a quantitative measure of the distribution of gap sizes. Large lacunarity implies large gaps and a clumping of points; small lacunarity implies a more uniform distribution of gap sizes. Also included in Figure 6.7 are examples of a near uniform distribution (near zero lacunarity) and a totally clumped distribution (high lacunarity). In each case a line segment with a length of 27 is divided into 27 equal parts, each of unit length, and 8 are retained. Allain and Cloitre (1991) have introduced a quantitative measure of lacunarity, which we will use below. Alternative measures have been given by Gefen et al. (1984) and by Lin and Yang (1986). The technique given by Allain and Cloitre (1991) is illustrated in Figure 6.8. We consider the thirdorder Cantor set given in Figure 6.7(b). The total length is r, = 27 and individual segments have unit length ( r = I). We consider a moving window of length r, which is translated in unit increments. The total number of steps considered is given by

For the example considered in Figure 6.8, we take r = 9 and N(9) = 19 as shown. The number of remaining line segments covered by a step is s and for the first steps, = 4, second steps, = 3, and so on, as shown. We denote the number of steps of length r that contain s segments by n(s, r). For the example Figure 6.7. A line segment with a length of 27 is divided into 27 equal-sized segments of unit length and 8 are retained. (a) A near uniform distribution of equally spaced elements. (b) A Cantor set as in Figure 2.l(e). (c) A random Cantor set as in Figure 6.1. (d) A clumped distribution with the 8 segments adjacent to each other. The lacunarity increases from top to bottom (a to d).

110

FRACTAL CLUSTERING

considered we have n(0, 9 ) = 1 , n(1, 9 ) = 4, n(2, 9 ) = 8 , n(3, 9 ) = 4 , and n ( 4 , 9 ) = 2. We next define a sequence of probabilities by

a n d ~ e f i n d p ( 0 , 9 ) = & , p ( l , 9 ) = $ , ~ ( 2 , 9 ) = & , ~ (=3& , 9, a) n d p ( 4 , 9 ) = for the example considered. The first- and second-order moments of this 19. distribution are defined according to

2

Figure 6.8. Illustration of the sliding-window method used to determine the lacunarity of a one-dimensional set. This third-order Cantor set contains eight line segments of unit length and a total length r, = 27. The moving window has a length r = 9. The window moves unit steps to the right and there are N(9) = 19 steps as illustrated. The values of s for each step are given. The number of steps that contain s line segments n(s, r) are n(0,9)=1,n(l,9)=4, n(2,9)=8,n(3,9)=4,and n ( 4 , 9 )= 2 .

FRACTAL CLUSTERING

111

For the example illustrated in Figure 6.7 we have M, (9) = $ and M, (9) = %. The lacunarity L is defined in terms of the moments by

The lacunarity for the example considered is L (9) = 1.235. The lacunarities for the four distributions illustrated in Figure 6.7 are given in Figure 6.9 as a function of the window length r ( r = l , 2 , . . . , 27). For r = 1 all the distributions have L (1) = 3.375 [n (0, 1) = 19, n (1, 1) = 8 , p 19 8 8 8 (0, 1) = 27, p (I, 1) = 3 ,M1(l)= 27, M2 (1) = 3 1.AS the value of r increases, the lacunarity decreases toward unity. The near uniform distribution illustrated in Figure 6.7(a) has the lowest lacunarity, near zero for r > 2, as expected. Also, the clumped distribution illustrated in Figure 6.7(d) has a single large gap and has the highest lacunarity, as expected. The random Cantor set has a significantly higher lacunarity than the deterministic Cantor set even though both have the same fractal dimension. Thus lacunarity can be used as a measure of a distribution in addition to its fractal dimension. The method described above for determining the lacunarity in one dimension is easily extended to higher dimensions. As a specific example we consider the Sierpinski carpet illustrated in Figure 2.3(d). The moving-box

0

0 Uniform x Cantor

0

+

0 0

Random Cantor

o Clumped

Figure 6.9. Lacunarities L as a function of step length r for the four distributions given in Figure 6.7.

112

FRACTAL CLUSTERING

method applied to a second-order Sierpinski carpet is illustrated in Figure 6.10. The carpet is a 9 X 9 square, which has been divided into 81 unit squares of which 64 have been retained. The moving box is a r X r square that moves at unit increments until the entire carpet has been covered. The total number of steps required for a two-dimensional problem is Nr = (r,- r

+ 1)2

(6.21)

For the example considered in Figure 6.10 we have r = 9 and r = 3 so that N (3)= 49;the 49 steps are illustrated. For the first step in the upper, left-hand corner, the 3 X 3 box covers 8 remaining unit squares so that s = 8. The values of s for the 49 steps are given. Again we denote the number of steps with a r X r box that contains s unit squares by n (s, r). For the example considered we have n (0,3)= 1, n (3,3)= 4,n (5,3)= 8,n (6,3)= 8,and n (7,3) = 4,n = (8,3) = 24. The corresponding sequence of probabilities is p (0,3)= &,P(3,3)=~,p(5,3)=$,p(6,3)=$,p(7,3)=~,andp(8,3)=~.~rom (6.18)and (6.19)the moments are M, (3)= % and M, (3)= And from (6.20)the lacunarity is L = 1.080. Lacunarity studies can be used in a variety of applications involving texture analysis. Plotnick et al. (1993)give applications to landscape textures. The concept of extending fractal analyses to higher-order moments leads us naturally to the introduction of multifractal methods.

T.

Figure 6.10. Illustration of the moving-box method used to determine the lacunarity of a two-dimensional distribution. This secondorder Sierpinski carpet contains 81 unit squares in a 9 X 9 square carpet. The moving box is a 3 X 3 square. The box moves unit steps to the right until the sweep is completed with seven steps. The box is then moved down a unit step and a second sweep is carried out. In all there are seven sweeps with a total of N ( 3 ) = 49 steps as illustrated. The number of unit squares included in the box at each step s is given for each step. The number of steps that contain s unit squares n ( s , r) are n ( 0 , 3 ) = 1 , n ( 3 , 3 ) = 4, n (5, 3 ) = 8, n (6, 3 ) = 8, n (7, 3 ) = 4 , and n (8, 3 ) = 24.

FRACTAL CLUSTERING

113

6.4 Multifractals

Our analysis of lacunarity introduced higher-order moments to our considerations of self-similarity and fractals. The utilization of higher-order moments of statistical distributions can be generalized utilizing the concept of multifractals (Halsey et al., 1986; Mandelbrot, 1989). We again begin by considering the Cantor set illustrated in Figure 2.l(e). A third-order Cantor set is illustrated in Figure 6.1 l(a). A line segment of unit length is divided into 27 equal parts and 8 line segments are retained. To define a multifractal set, the original line of unit length is divided into n equal segments, and the segments are denoted by i = 1, 2, . . . , n; and the length of each segment is given by r = n-1. The fractionfi of the remaining line in segment i is given by

where Liis the length of line in segment i and L is the total length of line.

Li= L,we have i=l

xfi = 1. The quantityfi is the probability that the ren

n

Since

i=l

maining line segment is found in "box" i.

Figure 6.11. Illustration of the multifractal fractions for a third-order Cantor set. (a) Third order Cantor set. (b) The single fractionf , is given for r = 1 . (c) The three fractions f,-f, are given for r = (d) The nine fractions f,-f,are given for r =

:.

a.

114

FRACTAL CLUSTERING

For the third-order Cantor set of unit length illustrated in Figure 6.11(a), we have L = $. We now determine the values off, for three cases, n = 1 (r = I), n = 3 ( r = 3). and n = 9 (r = i). Taking n = 1 (r = 1) we have i = 1; in this one segment we have L,= $ and from (6.22) obtain f,= 1. This is illustrated in Figure 6.1 l(b). For n = 3 ( r = i) we have i = 1, 2, 3; from Figure 6.11(c) we obtain L,= A, L2 = 0,L3 = $ and from (6.22) find f,= f2= 0, f3= i. With n = 9 ( r = ,$)we have i = 1, 2, 3, 4, 5, 6, 7, 8, 9; from Figure 2 6.11(d) we obtain L,= &, L2 = 0,L3 = $, L4 = L5 = L6 = 0,L, = 27, L8 = 0, L, = &; and from (6.22) find f, f2= 0,f3=:, f4=f5=f6= 0,f, = $, f, = 0, f ='. 9 4

4,

=a,

It is next necessary to define generalized moments M4(r) of the set of fractionsf,(r). This is done using the relation

where the sum is taken over the set of fractions and q is the order of the moment; (6.23) is valid for both integer and noninteger values of q as long as q # 1. The special case q = 1 will be considered below. For the example given in Figure 6.11, we can obtain the moments of the distribution for any order q (except q = 1) using (6.23). We first take q = 0 and find the zeroorder moments for r = I , $,and ,$ with the result:

Note that any finite number raised to the power 0 is 1, but 0 raised to the power of 0 is 0. We next take q = 2 and find the second-order moments for r = 1, and ,$ with the result:

i,

M, ( I ) = 12 = 1

FRACTAL CLUSTERING

Moments of other orders q can be obtained in a similar manner. For the first-order moment M, (r), q = 1, a modified definition of the moment is required. We first write fq

= ffq-I

=f

exp (ln fq-1)

= f exp

[(q - 1 ) ln f ]

In the limit E < < 1 we can write exp= ~ 1

+E

(6.25)

plus higher-order terms. Thus in the limit q comes fq

+ 1 (q

1 0 the fixed pointy = x = 0 is a node. The behavior for a = 1 is illustrated in Figure 9.3(a) and for a = 2 in Figure 9.3(b). If a > 0 and f < 0, the solutions converge on y = x = 0 and the fixed point is a stable node. If a > 0 and f > 0, the solutions diverge from y = x = 0 and the fixed point is an unstable node.

Figure 9.3. Illustration of singular point behavior; (a) and (b) are nodes and (c) is a saddle point.

DYNAMICAL SYSTEMS

If a < 0 in (9.23) and (9.24), the fixed pointy = x = 0 is a saddle point. Its behavior for a = - 1 is illustrated in Figure 9.3(c). Only the singular solutions y = 0 or x = 0 enter or leave the fixed pointy = x = 0. If x = 0 we have

and the fixed point is stable for a < 0. If y = 0 we have

and the fixed point is stable for f < 0. Since a and f must have opposite signs, one singular solution will be stable and the other singular solution will be unstable. We next substitute b = 1, c = - 1, and a =f = a in (9.22) with the result

dy-- x + ay du ax-y Changing to polar coordinates p and 0 we substitute the variables x = p cos 0

y

= p sin 0

giving

With p = p, at 0 = 0 this is integrated to give p=

This solution is a logarithmic spiral and the fixed point at y = x = 0 in (9.27) is known as a spiral.

9.2 Bifurcations We now turn to the subject of bifurcations. Solutions to a set of nonlinear equations generally experience a series of bifurcations as they approach chaotic behavior. These bifurcations occur when a parameter of the system is varied. We first consider the equation

225

226

DYNAMICAL SYSTEMS

where p is considered to be a parameter. The fixed points of this equation obtained by setting dxldt = 0 are

When p is negative there are no real fixed points, and when p is positive there are two real fixed points. The transition at p = 0 from no solutions to two solutions is known as a turning point bifurcation. We examine the stability of the two real roots by linearization. We substitute

into (9.32); after dropping the quadratic term in x , we have

Thus the fixed point x = pl/2 is stable: solutions as they evolve in time converge to it. The fixed point x = - p"2 is unstable: solutions as they evolve in time diverge from it. The corresponding bifurcation diagram is given in Figure 9.4(a). This figure illustrates the meaning of the word bifurcation, to split into two branches. This figure also shows that for x < -p1I2all solutions diverge to x = - = and for - pl/2 < x < + = all solutions converge to the stable fixed point x = ~ 1 1 2 . We now turn to a modified form of the logistic equation (9.1)

where p is again considered to be a parameter. There are fixed points at x = 0 and x = p for all values of p. For p < 0 there is an unstable fixed point at x = (I, and a stable fixed point at x = 0. As p increases the unstable fixed point approaches the origin and coalesces with it when p = 0. For p > 0 the fixed point at x = p is now stable. This is known as a transcritical bifurcation, and an exchange of stabilities between the two fixed points has taken place at p = 0. The corresponding bifurcation diagram is given in Figure 9.4(b). We next turn to a third class of bifurcations, the pitchfork bifurcations. We first consider the supercritical pitchfork bifurcation, an example of which is given by the equation

DYNAMICAL SYSTEMS

227

Note that this equation is invariant under the transformation x' = - x . Thus solutions are symmetric in x and fixed point must appear or disappear in pairs. The fixed points of (9.37) obtained by setting dxldt = 0 are

When IJ, is negative there is a single real fixed point x = 0, and when p, is . transition at p = 0 positive there are three fixed points x = 0, 2 ~ ' 1 2 The from one to three solutions is known, for obvious reasons, as a pitchfork bifurcation (although the word trifurcation would be more appropriate). A stability analysis shows that for (I < 0 the solution x = 0 is stable. For b > 0 this solution is unstable but the other solutions are stable. The corresponding bifurcation diagram is shown in Figure 9.4(c). For < 0 all solutions con-

no solutions

I

1

... _ ----_ ,

UnSlrble branch

unnUble branch

Figure 9.4. (a) Illustration of a turning point bifurcation occurring at y = 0. The stable and unstable fixed points of (9.32) are given as a function of y. (b) Illustration of a transcritical bifurcation and exchange of stabilities occurring at y = 0. The stable and unstable fixed points of (9.36) are given as a function of y. (c) Illustration of a supercritical pitchfork bifurcation occurring at y = 0. The stable and unstable fixed points of (9.36) are given. The transition is from a single stable branch for y < 0 to three branches, two stable and one unstable, for y > 0. (d) Illustration of a subcritical pitchfork bifurcation occurring at y = 0. The stable and unstable fixed points of (9.39) are given. The transition is from three branches, one stable and two unstable, for y < 0 to a single unstable branch for y > 0.

228

DYNAMICAL SYSTEMS

verge to the stable fixed point x = 0. For p > 0 all solutions for x > 0 converge to the stable fixed point x = pllz and all solutions for x < 0 converge to the stable fixed point x = - p"2. An example of a subcritical pitchfork bifurcation is given by the equation

When p is negative ( p < 0) there are two unstable fixed points at x = 2 (-p)'/z and a stable fixed point at x = 0. For positive p ( p > 0) the only fixed point is at x = 0 and it is unstable. In this region all solutions diverge to + w. Since solutions converge to a finite value of x only for p < 0, the term subcritical is used. The applicable bifurcation diagram is shown in Figure 9.4(d). Finally we consider the pair of equations

As in (9.28) and (9.29) it is again appropriate to introduce polar coordinates p and 0 in the xy phase plane: x = p cos 0

(9.42)

y

(9.43)

= p sin

0

Substitution into (9.40) and (9.41) gives

Figure 9.5. Illustration of a Hopf bifurcation at p = 0. The limiting solutions of (9.40) and (9.41) are given for various values of p. For p < 0 the origin x = y = 0 is a stable branch. For p > 0 there are stable limit cycles, which are circles in the xy-plane with radius p = p ' / Z .

stable periodic orbit P -----

unstoble branch

DYNAMICAL SYSTEMS

These equations have the fixed point solution p = 0 (x = y = 0); it is stable for y < 0 and unstable for y > 0. In addition, for y > 0, solutions of (9.44) and (9.45) converge to a circular limit cycle given by

These solutions are illustrated in Figure 9.5. For y < 0, all solutions spiral into the stable fixed point p = 0. For p > 0, solutions for p > p1/2 spiral into circular limit cycle given by (9.46): solutions for p < y112 spiral outward to this circular limit cycle. The transition from a stable branch for y < 0 to a stable limit cycle for y > 0 is a Hopf bifurcation. The van der Pol equation (9.14) also undergoes a Hopf bifurcation at E = 0.

Problems Problem 9.1. For b = c = 0 in (9.20) and (9.21) solve for y(t) and x(t) directly. Show that these solutions reduce to (9.24). Problem 9.2. Derive (9.30) from (9.27)-(9.29). Problem 9.3. Solve (9.36) in the vicinity of the three fixed points. Problem 9.4. Derive (9.44) and (9.45) from (9.40) and (9.4 1). Problem 9.5. Consider the equation

(a) What are the fixed points? (b) Are they stable or unstable? (c) Show that x =

(1 (1

+ xo)e2' - 1 + xo is a solution if x = x, at t = 0. + x,)e2' + 1 - xo

(d) Sketch solutions for xo = -2, 0, 2 and discuss in terms of the fixed points. Problem 9.6. Consider the nondimensional logistic equation (9.3). Deterdx mine the solution in the E phase plane, where i = -. dt

229

230

DYNAMICAL SYSTEMS

dx Problem 9.7. Consider the equation -= x dt

- x3

(a) What are the fixed points? (b) Are they stable or unstable? (c) Show that x

=

X:

xie2'

e2'

+ 1- xi

I

is a solution if x = x, at t = 0.

(d) Sketch solutions for x, = - 1.5, -0.5,0.5, and 1.5. Problem 9.8. Consider the equation

dx -

dt

= sin x

(a) What are the fixed points? (b) Are they stable or unstable?

(c) Show that x = 2 tan-'

[e'tan (31 -

if x = x, at t = 0.

IT 3IT

Sketch the solution for x, = - -. 2' 2 dx Problem 9.9. Consider the equation - = px + x2. Determine the fixed dt points and sketch the bifurcation diagram. What type of bifurcation is this? dx

= x - px3.Determine the fixed dt points and sketch the bifurcation diagram. What type of bifurcation is this? dx Problem 9.11. Consider the equation - = x + px3.Determine the fixed

Problem 9.10. Consider the equation

-

dt

points and sketch the bifurcation diagram. What type of bifurcation is this?

ChapterTen

LOGISTIC M A P

10.1 Chaos

The concept of deterministic chaos is a major revolution in continuum mechanics (Berg6 et al., 1986). Its implications may turn out to be equivalent to the impact of quantum mechanics on atomic and molecular physics. Solutions to problems in solid and fluid mechanics have generally been thought to be deterministic. If initial and boundary conditions on a region are specified, then the time evolution of the solution is completely determined. This is in fact the case for linear equations such as the Laplace equation, the heat conduction equation, and the wave equation. However, the problem of fluid turbulence has remained one of the major unsolved problems in physics. Turbulent flows govern the behavior of the oceans and atmosphere. The appropriate Navier-Stokes equations can be written down, but solutions yielding fully developed turbulence cannot be obtained. It is necessary to treat turbulent flows statistically and to carry out spectral analyses. The concept of deterministic chaos bridges the gap between stable deterministic solutions to equations and deterministic solutions that are unstable to infinitesimal disturbances. Chaotic solutions must also be treated statistically; they evolve in time with exponential sensitivity to initial conditions. A deterministic solution is defined to be chaotic if two solutions that initially differ by a small amount diverge exponentially as they evolve in time. The evolving solutions are predictable only in a statistical sense. A necessary condition that a solution be chaotic is that the governing equations be nonlinear. As our first example of deterministic chaos we consider the logistic map

This is a recursive relation that determines the sequence of values x,, x,, x,, . . . . An initial value, x,, is chosen; this value is substituted into (10.1) as

232

LOGISTIC M A P

x, and x, is determined as xn+,.This value of x, is then substituted as x, and x, is determined as x,+ 1. The process is continued iteratively. This is referred to as a map because the algebraic relation maps out a sequence of values of x,; xorx,, x2, . . . . The procedure is best illustrated by taking a specific example. Assume that a = 1 and x, = 0.5 and substitute these values into (10.1), giving x, = 0.25. This value is then substituted as x, and we find that x, = 0.1875. The iterations of (10.1) were studied by May (1976) and have a remarkable range of behavior depending upon the value of a that is chosen. This is in striking contrast to the rather dull behavior of the logistic differential equation given in (9.1). The logistic map is a simple representation of the population dynamics of a specie with an annual breeding cycle (May and Oster, 1976). The quantity x, is the population of the specie in year n and the parameter a can be interpreted as the average net reproductive rate of a population. To better understand the behavior of the logistic map, it is appropriate to study the functional relation fix)

= ax(1

- x)

(10.2)

The fixed points x, of this equation are obtained by settingfix,) = x, with the result

This is equivalent to setting xn+,= x,. The two fixed points obtained by solving (10.3) are

An essential question is whether the iterative mapping given by (10.1) will evolve to these fixed points. Depending on the behavior offix) in the vicinity of the fixed point, the fixed point can be either stable or unstable. Solutions will iterate toward stable fixed points and will iterate away from unstable fixed points. We introduce

LOGISTIC M A P

233

This is the slope of the functionfix) evaluated at the fixed point x,. If Irl < 1, where (r( is the absolute value of r, the fixed point is attracting (stable), but if Irl > 1 the fixed point is repelling (unstable). For the logistic map from (10.2) we find that

For positive values of a we find that the fixed point at x, = 0 is stable for 0 < a < 1 and unstable for a > 1. The fixed point x, = 1 - a-1 is unstable for 0 < a < 1, stable for 1 < a < 3, and unstable for a > 3. We next examine a sequence of iterations of the logistic map (10.1). As our first example we consider the iteration for a = 0.8 as illustrated in Figure 10.1. The curve represents the functionflx) given by (10.2) for a = 0.8. Taking x, = 0.5 we draw a vertical line; its intersection with the parabolic curve gives x, = 0.2. A horizontal line drawn from this intersection to the diagonal line of unit slope transfers xn+, to xn.A vertical line is drawn to the parabola giving x, = 0.128. Further iterations give x, = 0.0892928, x, = 0.06505567, etc. The sequence iterates to the stable fixed point xf = 0. All iterations converge to x, = 0 for 0 < x, < 1. As our next example we consider two iterations for

Figure 10.1. Illustration of the iteration of the logistic map (10.1) for a = 0.8. The iteration from x, = 0.5 converges on the stable fixed point x, = 0.

23 4

LOGISTIC M A P

a = 2.5 as illustrated in Figure 10.2. The parbolic curveflx) given by (10.2) now intersects the diagonal at the two fixed points given by (10.4) and (10.5),x, = 0 and x, = 0.6. For x, = 0, A = 2.5 and the singular point is unstable; for x, = 0.6, A = 0.5 and the singular point is stable. For x, = 0.1 we find x , = 0.225, x, = 0.43594, x, = 0.61474 and the iteration converages on the fixed point x, = 0.6. For x, = 0.8 we find x , = 0.4 and x, = 0.6 with no further iteration required. All iterations converge to x, = 0.6 for 0 < x, < 1 . This is consistent with the stability of the fixed points discussed above. At a = 3 a flip bifurcation occurs. Both singular points are unstable and the iteration converges on a limit cycle oscillating between x,, and x,. The period of the oscillation doubles from one iteration, n = 1, for a C 3, to two iterations, n = 2, for a > 3. The values of x,, and x, are obtained from the logistic map (10.1) by writing

The limit cycle oscillates between x,, and x,. As an example of the period n = 2 limit cycle, we consider the iteration for a = 3.1 given in Figure 10.3. The iteration from x, = 0.1 approaches the limit cycle that oscillates between x,, = 0.558 and x, = 0.765. The n = 2 limit cycle occurs in the range 3 < a < 3.449479. At a = 3.449479 another flip bifurcation occurs and the period

Figure 10.2. Illustration of the iteration of the logistic map (10.1) for a = 2.5. The iterations from xo = 0.1 and xo = 0.8 converge on the stable fixed point at x, = 0.6.

LOGISTIC M A P

235

doubles again so that n = 4. As an example of the period n = 4 limit cycle, we consider the iteration for a = 3.47, which is given in Figure 10.4. The iteration from x, = 0.1 approaches the n = 4 limit cycle that oscillates between x,, = 0.403, xo = 0.835, x, = 0.479, and x, = 0.866. The n = 4 limit cycle occurs in

Figure 10.3. Illustration of the iteration of the logistic map(lO.l)fora= 3.1.The iteration from x, = 0.1 converges to the n = 2 limit cycle between x,, = 0.558 and x, = 0.765.

Figure 10.4. Illustration of the iteration of the logistic map (10.1) for a = 3.47. The iteration from x, = 0.1 converges to the n = 4 limit cycle between xf, = 0.403, x, = 0.835, x, = 0.479, and x, = 0.866.

236

LOGISTIC M A P

the range 3.449499 < a < 3.544090.At larger values of a higher-order limit cycles are found. They are summarized as follows:

where n is the period of the limit cycle and k is the number of flip bifurcations that have occurred. Period-doubling flip bifurcations occur at a sequence of values a,, where a , = 3, a, = 3.449499, a3 = 3.544090, a, = 3.564407, a, = 3.568759, a, = 3.569692, a, = 3.569891, a, = 3.569934, etc. In the region 3.569946 < a < 4 windows of chaos and multiple cycles occur. The values of a, approximately satisfy the Feigenbaum relation

Where F = 4.669202 is the Feigenbaum constant. This becomes a better approximation as k becomes larger. This relation indicates a fractal-like, scaleinvariant behavior for the period-doubling sequence of bifurcations. The Feigenbaum relation can also be written in the form

Thus the initial values of the period-doubling sequence can be used to predict the onset of chaotic behavior at a=. Taking a , = 3 and a, = 3.449499, we find that a_ = 3.572005 from (10.12).Taking a, and a, = 3.544090, we find a_ = 3.569870. Taking a, and a, = 3.564407, we find a_ = 3.569944. Taking a, and a, = 3.568759, we find a_ = 3.569945. These are clearly converging on the observed value of am= 3.569946. We now turn to the behavior of the logistic map in the region of chaotic behavior. An example illustrating chaotic behavior is given in Figure 10.5 with a = 3.9; one thousand iterations are shown and no convergence to a limit cycle is observed. The behavior is space filling (chaotic) but the range of values of xn is well defined. The maximum value is obtained taking xn = 0.5 with the result xn+, = 0.975. The minimum value is obtained tak-

LOGISTIC M A P

237

ing x,, = 0.975 with the result x n + , = 0.0950625. Thus we have for a = 3.9 that 0.0950625 < x,, < 0.975. For a = 4 the logistic map (10.1) becomes

This iteration can be expressed analytically by taking

Substitution of (10.14) into (10.13) gives

The nth iteration of this relation is

Provided P is not a rational number, the values of x,, jump around randomly and fully chaotic behavior is obtained. The route to chaos and the windows of chaotic behavior of the logistic map are illustrated in the bifurcation diagram given in Figure 10.6. The asymptotic, large n, behavior of the map is illustrated for 2.9 < a < 4.0. At

Figure 10.5. Illustration of the iteration of the logistic map (10.1) for a = 3.9. The iteration from 0.9 gives chaotic behavior; 1000 iterations are shown.

238

LOGISTIC M A P

a = 2.9 the fixed point x, = 0.655 is shown. At a = 3 the fixed point is x, = 0.66767 and the period-doubling flip bifurcation to the n = 2 limit cycle is shown. In the interval 3 < a < 3.449499 the two values of x, corresponding to the n = 2 limit cycle are given. At a = 3.449499 the period-doubling flip bifurcation to the n = 4 limit cycle is shown. In the interval 3.449499 < a < 3.544090 the four values of x, corresponding to the n = 4 limit cycle are given. In the interval 3.544090 < a < 3.569946 an infinite sequence of period-doubling flip bifurcations occurs as n + m. For the higher values of a the windows of chaotic behavior are illustrated by the cloud of points. Chaotic behavior results in an infinite set of random values of x, with a well-defined range of values; this range is clearly illustrated in Figures 10.5 and 10.6. The maximum value of x, is the maximum value offlx) from (10.2) and this maximum is at x = 0.5; thus, we have

= 0.975, which is in agreement with the examTaking a = 3.9 we have xfmax ple given in Figure 10.5. The minimum value of x, is obtained by substituting (10.17) into (10.1) with the result

Figure 10.6. Bifurcation diagram showing the asymptotic behavior for large n of the logistic map (10.1) as a function of a.

LOGISTIC MAP

Taking a = 3.9 we have xhin = 0.0950625, which is again in agreement with the example given in Figure 10.5.

10.2 Lyapunov exponent Chaotic behavior can be quantified in terms of the Lyapunov exponent A. The definition of the Lyapunov exponent is

where dxn is the incremental difference after the nth iteration if dxo is the incremental difference in the initial value. If the Lyapunov exponent is negative, adjacent solutions converge and deterministic solutions are obtained. If the Lyapunov exponent is positive, adjacent solutions diverge exponentially and chaos ensues. To determine the Lyapunov exponent, we consider the incremental divergence in a single iteration by writing (10.1) in the form

where f(x) is the functional form of the mapping; for the logistic map it is given by (10.2). Since Xn+1 = f ( x n )

by definition, (10.20) can be written

And for the logistic map from (10.2) we find

From ( l o . 19) and (10.22) the definition of Lyapunov exponent is

A = lim rn+

m

1 " log, m n=O

-

1H.1

239

240

LOGISTIC M A P

where log, is the logarithm to the base 2. The Lyapunov exponents A for the logistic map (10.1) are given in Figure 10.7 for a range of values for a. The windows of chaotic behavior for 3.569946 < a < 4 where A is positive are clearly illustrated. The Lyapunov exponent goes to zero at each flip bifurcation as shown. Consider as a particular example the iteration for a = 4 given by (10.16). For this case we find (10.25)

dxn = 2 sin ( ~ " I T Pcos ) (2%p)2nndp and dx, = 2 sin I$ cos TTPIT~P

(10.26)

Combining (10.19), (10.25), and (10.26) gives

I

2hn = [sin (TITP)cos ( T n P ) 2n sin IT^ cos I$

-

Although the coefficient is variable, the growth with n as n + requires that A = 1. Thus the Lyapunov exponent for this special case is unity and the iteration is fully chaotic. The role of the Lyapunov exponent is clearly illustrated by the simple linear map

Figure 10.7. Lyapunov exponents A from (10.24) for the logistic map (10.1) as a function of the parameter a.

LOGISTIC M A P

241

The only singular point is at x, = 0 and it is stable if a < 1 and unstable if a > 1. Illustrations of the iteration of this linear map for a = 0.6, 1.2 and x, = 0.8 are given in Figure 10.8. With a = 0.6 we have x , = 0.48, x, = 0.288, x, = 0.1728, and x, = 0.10368, and the solution iterates to the stable fixed point x, = 0. With a = 1.2 we have x , = 0.96, x, = 1.152, x, = 1.3824, and x, = 1.65888, and the solution iterates to x + oo. If there is an incremental difference in x,, 6x0, the incremental difference in x , , 6 x , , is 6 x , = a 6x0; similarly the incremental difference in x,, sx,, is ax, = a s x , = a2 6xo. This can be generalized to give

Combining (10.19) and (10.29) gives

A = - log a log 2 Thus the Lyapunov exponent A is positive for a > 1 and adjacent solutions diverge; the Lyapunov exponent is negative for 0 < a 0: 0 unstable, - p, stable. transcritical. 0 (unstable), + k-112 for p > 0 (stable) supercritical pitchfork. 0 (unstable), (-p)-112 for p, < 0 (stable) subcritical pitchfork. x, = 0 . 1 2 5 , ~=~0 . 0 5 4 7 , ~=~0 . 0 2 5 8 , ~=~0 . 0 1 2 6 , ~=~0. x, = 0.16875, x, = 0.12625, x, = 0.09928, x, = 0.08048, xf = 0. x, = 0 . 3 2 , ~=0.4352,x3 ~ =0.4916,x4= 0.4999,xf=0.5. x, = 0 . 5 2 5 , ~=~0 . 6 2 3 4 , ~=~0 . 5 8 6 9 , ~=~0 . 6 0 6 1 , ~=~0.6. x,, = 0.513045, x, = 0.799455. x,, = 0 . 4 5 1 9 7 , ~=~0.84216. xfmax = 0.925, xfmin = 0.2567. xfmax= 0.95, xfmin= 0.1805. x, = 0.2298, x, = 0.7081, x, = 0.8268, x, = 0.5728, x4 = 0.9788. x, = 0.1070, x, = 0 . 3 8 2 4 , ~=~0.9447, x, = 0 . 2 0 9 1 , ~=~0.6615. r = a. X,

+

(a) x, = 0 unstable, x, = 1 stable for 0 < a < 2, unstable for a > 2. (b) 2.2408, 0.0542,0.9248, 1.1589,0.7196. (c) 2.46302,0.03057. (a) Y = 1. (b) Y = 1.333. (a) Y = 1. (b) Y = -0.3333.

(a) Y, = 1, Y, = 0.5. (b) Y, = 0, Y, = 0.5. (c) Y, = 0.5, Y2 = 1. = 0.5, Y2 = 0. (d) Y, (a) Y, = 1, Y2 = 0.75.(b) Y, = 0.5, Y2 = 0.75.(c) Y, = 0.75, Y, = 1. (d) Y, = 0.75, Y, = 0.5. -6.16, -1.714. 14.21, 7.20. x = y = t ( R - v ) ~ / * ,=z 1. 0.864,0.6023, 1.0555, -0.3309, -0.8103; - 1.1547 < x < 1.1547. 12, 144. = nD/3.

3i.

ANSWERS TO SELECTED PROBLEMS

383

INDEX

accretionary headward growth, 195 acoustic well log, 165 activation, seismic, 329,330 advection-diffusionmodel, 207 aftershock statistics, 304 aftershocks, 76.77, 107,246,304,323,325,328 Omori's law for, 304 spacial distribution of, 77, 107 age distribution of sediments, 18 age of faults, 203 age of shorelines, 203 aggregation, diffusion-limited,4,74, 195, 197, 199,201,207,371 alluvial fans, 13, 203 analysis, screen, 42 sieve, 42 angle of repose, 3 16 angular velocity, 28 1 annual floods, 208,209 annual peak discharge, 208 antipersistent, 136, 137, 138, 152, 161, 162 area, drainage, 193 rupture, 39.58 Arizonia topography, 177 Armenian earthquake, 328 arrays, slider-block, 325, 330, 336 arteries, 193 ash, volcanic, 44.79 fractal dimension of, 44 asperities, 299 asteroids, 44 atmosphere. 23 1 atmospheric pressure, 160 atmospheric temperature, 136, 146, 160 attraction, basin of, 371 attractor, 265,272,284,371,373 Lorenz, 265,272,284 strange, 265,284,373 autocomelation function, 137, 138, 141, 145 automata, cellular, 5,316,317,320,322,324, 325,326 automaton. 50 autoregressive integrated moving average model, 145

autoregressive model, 141 autocorrelation function for, 141 comelogram for, 142 variance of, 141 autoregressive moving average model, 143 autocorrelation function for, 145 variance of, 145 avalanche model, 5,207,3 l6,3 17 avalanches, 3 16,317,321 backbone, percolation, 78,291,336 Bak, Per, 5,316,341 basement, 76 basin, drainage, 181, 193.207 sedimentary, 76 basin of attraction, 37 1 bathymetry, 2, 163, 165, 167, 168,217 correlation dimension of, 167 Fourier spectral analysis of, 2. 163 wavelet analysis of, 217 bedding planes, 19,321 bell curve, 31 Benioff strain, 305 bifurcation, 185, 187,225,226,227,229, 234,236,237,240,243,251, flip, 234,236,238,240 Hopf, 229,263,265,272,273,372 pitchfork, 226,227,228,251,261,263,265, 272,372 subcritical, 228, 272 supercritical,226 transcritical, 226 turning point, 226 bifurcation diagram, 226,227,237,243,251, 284 bifurcation ratio, 185, 187 Big Bear earthquake. 77 bin analysis, 34, 37 binomial coefficient, 102, 126 binomial distribution, 83 block models, slider, 4, 245,247, 254, 325, 330, 336,342

386

INDEX

blocks, pair of slider, 247, 342 multiple slider, 325, 330, 334, 336, 342 slider, 245,247,325,330,334,342 body-wave magnitude, 57.58 Boltzmann constant, 31, 118 boundary layer, thermal, 265,269 Boussinesq approximation. 257,269 box-counting method, 14, 15,50,68,69,70,72, 76.78.98, 107, 120, 124, 132, 135, 147 moving, 111 one-dimensional, 69, 70 three-dimensional, 76 box-counting method for earthquakes, 76 box-counting method for oil pools, 98 box-counting method for pore spaces, 50 box-counting method for self-affine fractals, 132, 135, 147 box-counting method for time series, 147 box dimension, 115, 124 braided rivers, 207 branch numbers, 188 branches, side, 188 branching ratios, 189 bronchial system, 193 Brownian topography, 163,207,208 Brownian walk, 2, 138, 140, 146, 147, 149, 150, 152, 155, 157, 158, 160, 163, 165, 167,207,208,371 correlation dimension of, 167 fractional, 149, 150, 152, 155, 156, 158, 161. 162, 166 as a self-affine fractal, 140, 146, 150 correlation dimension of, 166 method of successive random additions, 152 rescaled range analysis of, 161, 162 log normal, 157, 158 semivariagrams of, 155 standard deviation of, 140 time series, 140 variance of, 140 buoyancy force, 257,265 b-value, 2, 57.59.324 cable, stranded, 299,303, 307 California earthquakes, 61.63.330.335 southern, 61.63.330, 335 Cantor dust, 8.37 1 Cantor set, 8, 18.21.24, 100, 102, 105, 107, 109, 111, 113, 116, 118,321 pair correlation of, 107 random, 100, 102, 109, 1 11 cardiovascular, 193 carpet, Sierpinski, 9.24, 105, 1 11, 118, 119, 29 1 cascade, multiplicative, 120, 12 1, 124, 126, 128 caves, 50 cell, 46.81.83.292.299 cellular-automata model, 5, 3 16.3 17, 320, 322, 324,325,326 center, 223 central-limit theorem, 2, 34, 39

chaos, 3,219,223,225,231,236,237,238,240, 245,25 1,253,256,265,266,269, 272,289,341,37 1,372 deterministic, 219. 223, 23 1 , 245, 256, 266 route to, 237 windows of, 236,237,238,240,25 1 chaotic mantle convection, 269 characteristic earthquake, 67, 322. 325 chromatographic model, 87.95 circulations, hydrothermal, 81 climate, 166, 341 clustering, 100, 103, 104,328, 371, 372 fractal, 100, 103, 104, 328 lacunarity of, 109,372 clustering of faults, 104 clustering of joints, 104 clustering of metamorphic veins, 104 clustering of seismicity, 103, 328 clusters, 4,78, 100, 103, 104,290, 291, 372 percolation, 4.78.290, 291,372 backbone of, 78,291,336 number-size statistics of, 291 coal, 43.44 coal mines, 67 coastline, 1 , 12, 15, 132 fractal dimension of, 1, 12, 15, 132 length of, 1, 12 rocky, 1, 12, 15, 132 roughness of, 15 coefficient of skew, 29,30,35 coefficient of skew for a log-normal distribution, 35 coefficient of thermal expansion, 258 coefficient of variation, 35, 137, 158 coefficient of variation for a log-normal distribution, 35, 158 comminution, 44,48,49,50,67,71,74, 298 comminution model, 48,49,50,71,74,298 complementary error function, 33 complexity, 342 component, periodic, 136 stochastic, 136, 137 trend, 136 concentration, mineral, 8 1.90, 120, 136 ore, 8 1.90, 120, 136 conditional probability, 301 conduction, electrical, 295 heat, 257 conduction solution. 261 conductivity, electrical, 165 thermal, 257 conservation of energy, 223,289 conservation of mass, 202 contact areas, multifractal analysis of, 129 continental drift. 279 continental margin, 321 continuity equation, 256 continuous data, 29.30 continuous processes, 56 continuous time series, 136, 137 contour, topographic, 12, 132 fractal dimensions of, 12

INDEX

convection. 3, 256, 257, 261, 268, 269 chaotic mantle, 269 heat, 257 mantle, 256,269 thermal, 3, 256,261,268, 269 copper, 93.95 core, 279,280 core dynamo, 279, 280 cores, drilling, 68, 136 correlated noises, 140 correlation, pair, 106, 107 of Cantor set, 107 of Koch snowflake, 107 of seismicity, 107 correlation dimensions, 116, 167, 286 of bathymetry, I67 of Brownian walks. 167 of dynamo models. 286 of geysers, 167 of time series, 167 of volcanic eruptions. 167 correlations, 138. 158 long range, 138 time series, 158 correlogram, 137, 142 autoregressive model, 142 counter scaling, 162 craters, 44 impact, 44 lunar, 44 creep, thermally activated, 269 solid state, 258 critical permeability, 4, 290, 295 critical phenomena, 63, 316, 328 critical point, 4, 289, 294, 3 16, 333 critical porosity, 295 critical probability, 4, 290, 295. 372 critical Rayleigh number, 260,264 critical state, 5, 316 criticality, self-organized, 5, 3 16, 3 17, 325, 326, 328,329,336,341 crushed materials, 44 crustal deformation, 56 crustal seismicity, 322 crustal thinning. 76 Culling model, 202, 207 cumulative distribution function, 30, 3 1, 32, 36, 37.40. 146 exponential, 40. 303 log-normal. 36 normal, 32 Pareto, 37 curdling, 8, 103 Curie temperature. 279 cycle, limit, 223,229,234,236,238.25 I, 253,372 damping, linear, 222 nonlinear, 222 dams, induced seismicity by, 328 Darcy's law, 290 day, length of, 136, 217 deformation, crustal, 56

delta, prograding, 203 river, 193 dendritic gold, 198 dendritic growth, 197 dendritic structure, 4, 37 1 density, 136 power spectral, 148, 149, 150, 166, 168. 170, 203 soil. 52 depleted element, 85, 120 deposition of sedimens, 18, 22, 202 rate of, 22 depositional sequence, 18 deposits. epithermal, 87 hydrothermal, 87 mineral, I, 5, 8 1,87,90, I46 ore, 1, 5,81. 87,90, 146 skarn, 87 turbidite, 321 deterministic, 37 1 deterministic chaos, 2 19,223. 23 1 , 245, 256. 266,372 deterministic fractal, 6 determinisstic self-affine fractal, 133 deviation, standard, 29, 30 devil's staircase, 18, 2 1, 23, 24 diamonds, 95 difference equation, 37 1 diffusion equation, 202, 203, 207 diffusion-limited aggregation, 4, 74, 195, 197, l99,2OI,207,37l diffusivity, thermal. 258 dimension, 115, 167, 371 box, 115, 124 correlation, 116. 167, 286 embedding, I67 entropy, 1 16, 1 18 Euclidian, 6 fractal, I, 2 , 6 , 14, 15, 16,22,59,90,92, 1 15, 132, 135,372 of drainage networks, 185 of earthquakes. 59 of time series, 147 fractional, 6, 16, 372 information, 116, 1 18 rnultifractal, 115, 117, 124 self-affine fractal, 135 discharge, annual, 208 flood, 136,208 river, 37, 136, 158, 160,215 discontinuous process, 56 discontinuous time series, 137 discrete data, 29 discrete Fourier transform. 150, 17 1 inverse, 150 discrete time series, 136 disk dynamo, 280 displacement, earthquake, 75 fault, 56, 74 disruptions, explosive. 44 dissipation, minimum energy, 207 dissolution, 87

387

388

INDEX

distillation, Rayleigh, 87, 88, 89.95 distribution, binomial, 83 exponential, 39.40.303 fractal, 3, 15,21,28,38,39,41,84,87,90, 96, 100,208,211,317,341 frequency-magnitude, 15,57,66 gamma, 208.2 1 1 Gaussian, 2.31.33, 137, 138, 140, 145,214, 332 Gumbel, 208,211 Hazen, 208.2 1 1 log-Gumbel, 208,211 log-normal, 2,35,38,42,81,83,90,96, 137, 145, 158,208,211 log-Pearson, 208,211 Maxwell-Boltzmann, 332,333,336 multifractal, 128, 129 normal,2,31,33,38, 137, 138, 145,341, 371,373 number-size, 1, 15, 79, 291 Pareto, 37.38.42.341 Poisson, 102, 103 power-law, 2, 16,38,41,42,50,81,84,90, 96,208,211,341 probability, 3 1 Rosin-Rammler, 40,42 scale-invariant, 3 Weibull, 4 1,42,299,301,303 distribution function, 118 cumulative, 30,31,32,36,37,40, 146 distribution of aftershocks, 77, 107,304,323 distribution of earthquakes, 1.76, 107, 128,320, 324,325,326,327 distribution of faults, 1, 67, 68, 71, 199, 322 distribution of floods, 208,209 distribution of forest fires, 337,339 distribution of fractures, 1.67.68, 129, 199 distribution of fragments, 1, 15.42 distribution of gaps, 109 distribution of incomes, 38 distribution of islands, 15 distribution of joints, 68, 199 distribution of lakes, 15,96 distribution of landslides, 316,321 distribution of lifetimes, 303 distribution of mineral deposits, 1, 81 distribution of oil fields, 1.95 distribution of ore, 1.81 distribution of pore spaces. 129 distribution of porosity, 3, 166, 167 distribution of sand slides, 321 distribution of sediment ages, 18 distribution of slider block slip events, 325, 326 distribution of topography, 129 distribution of volcanic eruptions, 1.79 divider method, 12, 13, 14 doubling, period, 234,236,238,251,371,372 drainage area, 193 drainage basins, 181, 193, 207 drainage networks, 2.5, 181, 185, 191, 194, 199, 207 fractal dimension of, 185 drift, continental, 279

drilling cores, 68, 136 droughts, 158, 208 dust, Cantor, 8, 37 1 dynamic friction, 246,326 dynamical systems, 219, 371 dynamics, population, 4, 219,232,244 dynamo, 3,279,280,282,285 core, 279,280 disk, 280 Rikitaki, 3,279,280,282,342 self-excited, 279 two-disk, 280,282 dynamo equations, 3,282,285 earthquake, 67,246,299,317,322 aftershocks of, 76,77, 107,246,304,323, 325,328 Omori's law for, 304 spacial distribution of, 77. 107 Armenian, 328 Big Bear, 77 characteristic, 67,322, 325 Haicheng, 330 Joshua Tree, 76 Kern County, 56,330 Landers, 56,57,63.329,330 Loma Prieta, 56,57,305,329 Northridge, 56,57,63,330 San Fernando, 56,330 San Francisco, 56,57,330 Tangshan. 330 Whittier, 63 earthquake displacement, 75 earthquake energy, 57,58,61 earthquake magnitude, 57.58.61 bodywave, 57,58 local, 57.58 moment, 57,58,59,60,61,305 surface wave, 57.58 earthquake moment, 57,58,59,60,61,305 earthquake precursors, 305,328 earthquake prediction, 254,307,328,329,342 earthquake rupture area, 39.58 earthquakes, 1.4, 15,39,56,57,58,59,61,65, 74, 103, 107, 128, 145,317,320,324, 325,326,327,328,330 clustering of, 103,328 distribution of, 1.76, 107, 128,320,324, 325,326,327 foreshocksof, 246,323,324 fractal dimension of, 59.76 frequency-magnitudestatistics of, 1, 15, 39, 57,59,60,66,317,324,327 intervals between, 74 Memphis-St. Louis, 66 multifractal distribution of, 128 New Madrid, 66 pair correlations for, 107 Parkfield, 67.254 rupture dimensions of, 39,58 southern California, 61.63.330, 336 spatial distribution of, 76, 128 earth's mantle, viscosity of. 258

INDEX

economic ore deposits, 5, 8 1,90,120, 146 economics, 38 Eden growth, 207 Edward-Wilkinson equation, 207 ejecta, volcanic, 44 elastic lithosphere, 13 elastic rebound, 245,246 electric fields, 4 induced, 28 1 electrical conduction, 295 electrical conductivity, 165 element, 46,81,85,290,292,299,303,307 depleted, 85, 120 enriched, 85, 120 impermiable, 290 permiable, 290 elevation, 163 elevation changes, 23 embedding dimension, 167 energy, 223,289 conservation of, 223,289 earthquake, 57.58.61 energy dissipation, minimum, 207 energy equation, 256 enriched element, 85, 120 enrichment factor, 82,87, 120 entropy, 118,289 maximum, 50 entropy dimension. 116.11 8 entropy method, 207 episodic sedimentation, 23 epithermal mineral deposits, 87 epochs, geological, 18 ergodic, 137 erosion, 2, 13, 18, 19.23, 194,202,208 characteristic time for, 202 error function, 33,203 complementary, 33 eruptions, volcanic, 1.28.78.79. 167, 243 chaotic behavior of, 243 correlation dimension of, 167 Euclidian dimension, 6 exchange of stabilities, 226 expansion, thermal, 256,258 coefficient of, 258 explosion, nuclear, 43.44 explosive processes. 44 exponential, distribution, 39, 303 cumulative, 40,303 mean. 40 probability, 39 variance, 40 failure, time to, 303, 304, 305, 306 fan, alluvial, 13,203 fault, 4.56.74 San Andreas, 56.62.65.67. 104 San Gabriel, 104 fault displacement, 56.74 fault gouge, 44,50 fault rupture, 299 fault scarps, 203 age of, 203

faults, 1, 28.56.67.70. 104,203,245,247,299,

303 age of, 203 clustering of, 104 distribution of, 1.67,68,71, 199.322 fractal distribution of, 1.67.71.322 interacting, 247 length of, 67 normal, 76 number-length statistics, 67, 322 pre-existing. 245 transform, 56.70 Feigenbaum constant, 236,371 Feigenbaum relation, 236 fields, electric, 4 gravitational,4 induced electric, 28 1 magnetic, 4,217,279,342 filling, space, 236 filtering, 214 Fourier, 152 fingering, viscous, 198 first law of thermodynamics, 289 fixed point, 220,221,223,224,225,226,227, 232,242,265,270,294,297,302,372 stability of, 220, 221 stable, 220,225,232,265,294,297,302 unstable, 232,265,294,297,302 flexure, 13 flip bifurcation, 234,236,238,240 flood discharge, 136,208 flood frequency factor, 209 flood hazard, 208 floods, 158,208,209 annual, 208,209 frequency of, 208 partial duration series of. 209 fluid flow, 50 fluid layer. 256,268,269 fluid turbulence, 3, 127,231,268,341 folds, 1, 56 force balance equations, 256 forces, buoyancy, 257,265 inertial, 257 pressure, 257 viscous, 257 foreshocks. 246.323.324 forest-fire models, 336, 337, 339 forest fires, 291.339 distribution of, 337,339 fossil magnetism, 279 Fourier coefficients, 2, 150 Fourier filtering technique, 152 Fourier series, 2. 148 amplitudes of. 2, 148 phases of, 148 Fourier spectrum, 2, 148, 163 of bathmetry, 2, 163 of topography, 2, 163 Fourier transform, 148, 150, 171.214 discrete, 150, 171 inverse, 148, 150 two-dimensional, 17 1, 172

390

INDEX

fractal, 1.2.6, 12,89,304,341,372 deterministic, 6 self-affine, 133 homogeneous, 116, 118 self-affine, 132, 135, 138, 140, 145, 146, 148, 150, 163, 166,372 deterministic, 133 statistical, 140 self-similar, 132 statistical, 12, 38, 289 fractal clustering, 100, 103, 104, 328 fractal dimension, 1.2.6, 12, 14, 15, 16, 22, 59, 90.92, 115, 132, 135, 372 box, 115, 124 correlation, 116, 167,286 entropy, 116, 118 information, 116, 1 18 methods of determining, 12, 14 box counting, 14, 15,50,68,69,70, 72, 76,78,98, 105. 120, 132, 135, 147 divider, 12, 13, 14 ruler, 12, 13, 14 multifractal, 115, 117, 124 of coastlines, 1, 12, 15, 132 of drainage networks, 185 of earthquakes, 59,76 of fractures, 68 of mineral deposits, 5, 81, 87, 90 of oil fields, 95 of ore deposits, 5, 8 1,87,90 of sedimentary sequences, 22.32 1 of time series, 147, 148 of topographic contours, 12 self-affine, 135 fractal distribution, 3, 15, 2 1, 28, 38, 39,4 1, 84, 87.90.95, 100,208, 211,317, 341 of earthquakes, 59.76 of faults, 1, 67, 68, 71, 322 of floods, 208 of fragments, 28,323 of landslides, 32 1 of mineral deposits, 5, S 1, 87.90 of oil fields, 95 of ore concentrations, 5.81.87.90 of sedimentary layer thicknesses, 321 of volcanic ash, 44 fractal fracture surfaces, 166 fractal fragmentation, 42,292, 323 fractal island, 11. 24 fractal landscapes, 181 fractal set, 6, 371 fractal statistics, 12, 38, 181, 289 fractal topography, 132, 208 fractal tree, 2, 181, 187, 188, 299, 302 binary, 188 branch numbers for, 188 branching ratios of, 189 deterministic, 187 side branching of, 188 Tokunaga, 188, 190, 198 fractional Brownian walk, 149, 150, 152, 155, 156, 158, 161, 162, 166

correlation dimension of, 166 fractal dimension of, 149 rescaled range analysis of, 161. 162 self-affine fractal, 150 semivariograms of, 155 successive random additions, 152 wavelet analysis of, 217 fractional dimension, 6. 16, 372 fractional Gaussian noise, 149, 150, 152, 155, 158, 161, 162 Fourier filtering techniques, 152 rescaled range analysis of, 161, 162 semivariograms of, 155 fractional log-normal noises, 157, 158 fractional log-normal walks, 157, 158 fracture networks, 129 multifractal analysis of, 129 fracture surfaces, 166 fractures, 5,28,50,67,68,70,72, 120, 166, 199 distribution of, 1.67.68, 129, I99 fractal dimension of, 68 permability of, 50, 166 porosity of, 50 fragmentation, 5, 28,42,48, 71, 292, 295, 323 fractal, 42, 292, 323 models for, 46,48,49,50,71, 74,292, 323 multifractal analysis of, 129 renormalization group analysis of, 295 tectonic, 44, 70 fragmentation probability, 46, 292, 298 fragments, rock, 1, 15. 28-42 mass of, 42 number of, 42 size distribution of, 1, 15.42 free surface, 260 frequency-magnitude statistics, 15, 57, 66 for earthquakes, 1, 2, 15, 39,57, 59,60,66, 317,324,335 for floods, 208 Gutenberg-Richter, 2,57,59,60, 66, 3 17, 335 frequency-size distribution, 1, 15, 57, 67. 79, 322 for earthquakes, 1, 15,57, 3 17 for faults, 1.67, 322 for floods, 208 for forest fires. 337, 339 for islands, 15 for lakes, 15.96 for landslides, 3 16, 32 1 for mineral deposits, 1, 8 1 for oil fields, 1,95,96 for rock fragments, 1, 15.42 for slider blocks, 325,326 for volcanic eruptions, I, 79 friction, 245, 246,299, 326, 330 dynamic, 246,326 static, 246, 326, 330 velocity weakening, 299,325 gamma distribution, 208.21 1 gamma function, 40

INDEX

gaps, distribution of, 109 sedimentary record, 18, 19,20, 167 Gaussian distribution, 2, 31, 33. 137, 138, 140, 145,214,332 Gaussian white noise, 140, 149, 150, 160, 207 fractional, 149. 150, 152, 155, 158, 161, 162 Fourier filtering technique, 152 semivariograms of, 155 geoid, 170, 217 power spectral density of, 170 wavelet analysis of, 217 geological epochs, 18 geomagnetic field, 2 17 geometric incompatibility, 70 geomorphology, 13, 181,208 geysers, correlation dimension of. 167 glaciers, 2 16 global load sharing, 304 global seismicity, 59 gold, 92.95.98. 198 dendritic, 198 lode, 92 spatial distribution of, 98 gouge, fault, 44,50 grade, ore, 81, 84,87,90, 136, 146 granular, material, 3 16 gravitational fields. 4 gravitationally unstable, 256 gravity, 3, 170 ground water hydrology, 167,290 group, renormalization, 5,289, 290,292,295, 299,316,317 growth, dendritic, 197 Eden, 207 growth models, 195 diffusion-limited aggregation, 4, 74, 195, 197, 199,201,207 growth networks, 74. 195 accretionary, 195 growth phenomena, 3,207 Gumbel distribution, 208, 21 1 log, 208,2 11 Gutenberg-Richter relation, 2,57,59,60, 66, 317,335 gyration, radius of, 196 Haicheng earthquake. 330 harmonic motion, 223 harmonic oscillator, 222 harmonics, spherical, 168 Hausdorff measure, 132, 135, 145, 146, 148,209, 372 Hawaii, 194,243,269 hazard rate, 303 hazards, flood, 208 seismic, 63, 66, 254, 307, 328, 329, 342 volcanic, 79 Hazen distribution, 208. 21 1 headward growth, 195 heat, specific, 257 heat conduction, 257 heat conduction equation. 23 1

heat convection, 257 heat equation, 4, 13,257 height of topography, 132, 136 hiatuses. sedimentary, 18, 19.20, 167 homogeneous fractal, 1 1 6, 118 Hopf bifurcation, 229, 263, 265, 272, 273, 372 Horton's laws, 185 hot spots, 106,269 Hurst exponent, 160, 161. 162,212,372 hydrology, 167,290 hydrothermal circulations, 8 1 hydrothermal mineral deposits, 87 hypsometric curve, 145 impact craters, 44 impacts. 28,43,44 impermeable element, 290 incomes, distribution of, 38 incompatibilities. geometrical, 70 induced electric field, 281 induced seismicity. 5, 129. 328 inductance, mutual, 281 self, 282 inertia, moment of, 282 inertial force, 256 information dimension, 116, 118 interacting faults, 247 interactive systems. 3 16 intergranular porosity, 50 intervals between earthquakes, 74 invariance, scale, 1, 11,39,84,91,289,304, 316, 372,373 invasion percolation, 207 inverse Fourier transform, 148 discrete, 150 island, fractal, 11, 24 Koch, 11,24, 146 volcanic, 194 islands, frequency-size distribution of, 15 isotropic, 132 iterative map, 232 joints, 28.67.68, 70, 104,299 clustering of, 104 distribution of, 68, 199 Joseph effect, 158.2 13 Joshua Tree earthquake, 76 karst regions, 50 Kern county earthquake, 56,330 kinetic theory of gases, 140 Koch island, 11, 24, 146 Koch snowflakes, 25, 107, 195 pair correlation of, 107 Korcak relation, I5 lacunarity, 109, 11I, 112, 372 moving box method for, 111 moving window method for, 109 lag, 137, 138 lake levels, 160 lakes, number-size statistics of, 15, 96

392

INDEX

Landers earthquake, 56,57,63,329,330 landforms, 181 landscapes, 2, 181 fractal, 181 roughness of, 2, 165, 176 synthetic, 2. 173 textures of. 112 landslides, 3 16,32 1 frequency-size distribution, of 316, 321 Langevin equation, 207,208 Laplace equation, 4,23 1 layer, fluid, 256,268,269 thermal boundary, 265,269 Legendre functions, 168 length of coastline, 1, 12 length of day, 136,2 17 length of faults, 67 length of perimeter, 11 length-order ratio, 185, 187 levels, lake, 160 lifetimes. distribution of. 303 limit cycle, 223,229,234,236,238,251,253, 372 linear damping, 222 linear map, 240 linearization, 220,226,259 linearized stability analysis, 220, 259 lithosphere, elastic, 13 load sharing, global. 304 local earthquake magnitude, 57.58 lode gold, 92 logarithmic spiral, 225 logistic equation, 219,226 logistic map, 4.23 1,232,237,293 log-normal distribution, 2,35,38,42, 81, 83,90, 96, 137,145, 158,208,211 coefficient of skew for, 35 coefficient of variation for, 35, 158 cumulative distribution function for, 36 for ore deposits, 83 mean of, 35 probability distribution function for, 35 standard deviation of, 35 variance of, 35 log-normal noises. 157, 158 log-normal walks, 157, 158 log-Pearson distribution, 208.21 1 log-periodic behavior, 303,305,307,342,372 logs, well, 3, 136, 165, 166, 168 Loma Prieta earthquake, 56,57,305, 329 long memory, 137 long-range correlation, 138 Lorenz, Ed, 3,256,341 Lorenz attractor, 265,272,284 Lorenz equations, 3,256,262,263,264,266, 269,270,284,341,372 Lorenz truncation, 268 lunar craters, 44 Lyapunov exponent, 3,239,240,241,25 1.37 1, 372 magma, 87

magma migration, 193 magnetic field, 4,217,279,342 magnetic field polarity, 279 magnetic field reversals, 3.279 correlation dimension of, 286 magnetic surveys, 167 magnetics, 3 magnetism, fossil, 279 natural remanent, 279 magnitude, body wave, 57.58 earthquake, 57,58,61 local, 57,58 moment, 58 surface wave, 57,58 Mandelbrot, Benoit, 1.6, 15,341 mantle convection, 256,269 chaotic, 269 mantle plume, 269,273 mantle viscosity, 258 map, 232,372 iterative, 232 linear, 240 logistic, 4, 231,232,237, 293 recursive, 243,286 tent, 241 triangular, 24 1 margin, continental, 321 marginal stability, 259,3 16 mass conservation,202 mass distribution of fragments, 42 maximum entropy. 50 Maxwell-Boltzmann distribution, 332, 333, 336 mean-field approximation. 304 mean value, 29.30, 137, 138, 141 exponential, 40 Gaussian, 32 log-normal, 35 normal, 32 Pareto, 38 meanders, 195 measure, Hausdorff, 132, 135, 145, 146, 148, 209,372 measuring rod, 1, 12 mechanics, statistical, 289, 330,342 medium, porous. 290 memory, long, 137 short, 137 Memphis earthquakes, 66 Menger sponge, 10,50, 105,295 mercury, 90.9 I. 95 metamorphic veins, clustering of, 104 Mexican hat wavelet, 214 migration, magma, 193 mineral concentration, 8 1.90, 120, 136 mineral deposits, 1,5,81, 87,90, 146 epithermal, 87 fractal dimension of, 90 frequency-sizedistribution of, 1.81 hydrothermal, 87 skarn, 87 spatial distribution of, 98 mines, 67

INDEX

207 mining induced seismicity, 129 model, advective-diffusion,207 avalanche, 5,207,316,317 cellular-automata. 5,316,317,320,322,324, 325,326 comminution, 48,49,50,71,74,298 Culling, 202,207 diffusion limited aggregation, 4,74, 195, 197, 199,201,207 forest fire, 336,337,339 sandpile, 5,316,317 slider-block, 4,245,247,254,325,330,336, 342 stochastic, 4 molecular velocities, 30, 3 1 moment magnitude, 58,60 moment of inertia, 282 moments, 29, 110, 114,372 earthquake, 57,58,59,60,61,305 generalized, 114 seismic, 60, 305 Monte Carlo, 291 mother wavelet, 214 motion, equation of, 222,246,325 moving-average model, 140 autocorrelation function for, 141 autoregressive, 143 autoregressive integrated, 145 variance of, 141 moving-box method, 111 moving-window method, 109 multifractal, 83, 113,372 perfect, 124, 128 multifractal analysis, 120, 125, 129 multifractal analysis of fragmentation, 129 multifractal analysis of well logs, 129 multifractal dimension, 115, 117, 124 multifractal distribution of contact areas, 129 multifractal distribution of earthquakes, 128 multifractal distribution of fractures, 129 multifractal distribution of mining induced seismicity, 129 multifractal distribution of pore spaces, 129 multifractal distribution of void spaces, 129 multifractal scaling, 128 multifractal spectrum, 115 multifractal time series, 129 multifractal topography, 129 multiplicative cascade, 120, 121, 124, 126, 128 mutual inductance, 281

minimum energy dissipation,

Nankai Trough, 254 natural remanent magnetism, 279 Navier-Stokes equations, 23 1 nearest neighbor model, 317 networks, branch numbers, 188 branch ratio, 189 drainage, 2.5, 181, 185, 191, 194, 199, 207 fractal dimension of, 185 river.2.5, 181, 185, 191, 194, 199,207

side branching, 188

neutron activation well logs, 165 New Madrid earthquakes, 66 Nile River, 158 Noah effect, 37, 158,213 node, 224,372 stable, 224 unstable, 224 noises, 140, 149, 150, 152, 155, 157, 158, 161, 162,373 antipersistent, 152 correlated, 140 fractional Gaussian, 149, 150, 152, 155, 158, 161, 162 fractional log-normal, 157, 158 Gaussian white, 140, 149, 150, 160,207 persistent, 140, 152 white, 138, 139, 140, 146, 148, 149, 150, 156,207.373 nondimensional parameters, 222, 248,258, 282 nondimensional variables.. 219.222.246.248. . . . . 258,282,332 nondimensionalization, 219, 222,246, 248,258, 282,326,332 nonlinear damping, 222 nonlinear equations, 4,219,23 1 normal distribution, 2,31,33, 38, 137, 138, 145, 341,371.373 cumulative, 32 mean of. 32 probability distribution for, 3 1 standard deviation of, 32 variance of, 32 normal fault, 76 no slip boundary condition, 260 Northridge earthquake, 56.57.63.330 nuclear explosion, 43,44 number-size distribution, 1, 15,79, 291,317, 337 earthquake, 1,15,57,59,60,66,317,335 fault. 67.322 flood, 208 forest fire, 337,339 island, 15 lake, 15,96 oil field, 1,95 percolation cluster, 291 rock fragment, 1, 15.42 volcanic eruption, 1.79 numerical solutions, 264 Nusselt number. 264 ocean ridge, 13.56.70.269 ocean surface, 146 ocean trench, 13.56.70.269 oceans, 231 oil, spatial distribution of, 97,98 oil fields, 74,95,96,97 fractal dimension of, 95 frequency-size distribution of, 1.95.96 oil reservoirs, 3,96,97 Old Faithful Geyser, correlation dimension of, 167

394

INDEX

Omori's law, 304 order, 6 of rivers, 181 ore concentrations, 1 , 5 , 8 1, 87,90, 120, 136, I46 ore deposits, 1, 5, 81, 87,90, 120, 136, 146 ore grade, 81.84.87.90. 136, 146 ore reserves, 8 1 ore tonnage, 8 1,90,92 Oregon topography, 163, 165, 175, 177 oscillator, harmonic. 222 oxygen isotope ratios, 167 pair-correlation technique, 106, 107 Cantor set, 107 Koch snowflake, I07 seismicity, 107 paleomagnetism, 279 parameters, nondimensional, 222, 248, 258, 282 Pareto distribution, 37, 38.42.341 cumulative, 37 mean of, 38 probability distribution for, 37 standard deviation of, 38 standard form of, 37.38 variance of, 38 Parkfield earthquakes, 67,254 partial duration series, 209 particulate matter, 3 partition coefficient, 87, 90 pattern recognition, 328 pavements, 68 peak discharge, 136,208 percolation, invasion, 207 percolation backbone, 78,291,336 percolation clusters, 4.78, 290, 291, 372 number-size statistics of, 291 percolation model, 333 percolation threshold, 333 perimeter, length of, 11 period, rotational, 136 period doubling, 234,236,238, 25 1.37 1, 372 periodic behavior, 248,250,253,305 periodic component, I36 periodic oscillation, 222, 372 permeability, 4.50.74, 136, 166, 295,372 critical, 4, 290, 295, 372 fracture, 50, 166 permeable element, 290 persistence, 136, 137, 138, 140, 152, 161, 162 strong, 137 weak, 137 petroleum, spatial distribution of, 97, 98 petroleum reserves, 96 petroleum traps, 98 phase plane, 249 phase portrait, 265 phase space, 222,372,373 phase trajectory, 222 pitchfork bifurcation, 226,227, 228.25 I, 261, 263,265,272,372

subcritical, 228, 272 supercritical, 226 plane, phase, 249 plate tectonics, 5, 56.60, 70, 256, 269, 279 plumes, 269, 273 Poincare section, 372 Poisson distribution, 102, 103 polarity, magnetic field, 279 population dynamics, 4, 219, 232, 244 pore spaces, 50 pore spaces, multifractal distribution of, 129 porosity, 3,50, 136, 166, 167,295 critical value of, 295 fracture, 50 intergranular, 50 porosity distribution, 3, 166, 167 porosity logs, 166, 167 porous media, 290 portrait, phase, 265 power law, 1, 372 power-law distribution, 2, 16,38,41,42, 50.81, 84,90,96,208,2 11, 34 1 power-law spectra, 3, 166, 168 power spectral density, 148, 149, 150, 166, 168, 170,203 geoid, 170 topography, 168, 170 two-dimensional, 168, 172 Prandtl number, 258,268,269,270,273 precursor, seismic, 305,328 prediction of earthquakes, 254, 307,328,329, 342 pre-existing faults, 245 pressure, 160,289 atmospheric, 160 pressure forces, 257 probability, 4, 28.30, 100, 102, 113, 301, 372 conditional, 301 critical, 4, 290, 295 distribution, 3 1 exponential, 39 Gaussian, 3 1 log-normal, 35 normal, 31 Pareto, 37 of fragmentation, 46, 292, 298 prograding delta, 203 pull, trench, 65 pumice, 44 push, ridge, 65 quiescence, seismic. 328 radius of gyration, 196 rainfall, 37, I36 random, 102, 136,373 random additions, successive, 152 random Cantor set, 100, 102, 109, 111 random growth networks, 74, 195 random simulation, 103 random walk, 2, 195, 197, 199,207, 371 Rayleigh distillation model, 87, 88, 89.95

INDEX

Rayleigh number, 258, 260, 264, 268, 269, 271. 272,273 critical, 260, 264 rebound, elastic, 245, 246 recursion relations. 3, 293 recursive map, 243,286 regional seismicity, 5, 63, 324 remanent magnetism, 279 renormalization, 47.83.89.289. 373 renormalization group method, 5,289,290,292, 295, 299,316,317 applied to fault rupture, 299 applied to fragmentation, 295 repose, angle of, 3 16 rescaled range analysis, 138, 158, 161, 162,373 of fractional Gaussian noises, 161, 162 reserves, ore, 8 1 petroleum, 96 reservoir, oil, 3.96.97 reservoir storage, 158 resistance, 28 1 reversals, magnetic field, 3,279 ridge, ocean, 13.56.70.269 ridge push, 65 Rikitaki dynamo, 3,279,280,282, 342 rings, tree, 160 river deltas, 193 river discharge, 37, 136, 158, 160, 215 time series of, 158,215 river meanders, 195 river networks, 2, 5, 18 1, 185, 19 1, 194, 199, 207 river sinuosity, 195 rivers, braided, 207 multifractal analysis of, 129 order of, 18 1 rock fragments, 1, 15, 28.42 size distribution of, 1, 15.42 rock surfaces, 166 rocky coastlines, 1, 12, 15, 132 length of, 1, 12 rod. measuring. 1, 12 Rosin and Rammler distribution, 40.42 rotational period, 136 roughness, 2, 15, 165, 176 of coastline, 15 of topography, 2, 165, 176 roughness-range method, 162 route to chaos, 237 ruler method, 12, 13, 14 rupture, fault, 299 rupture area, earthquake, 39.58 saddle point, 225, 373 St. Louis earthquakes, 66 San Andreas fault, 56,62,65,67, 104 San Fernando earthquake, 56.330 San Francisco earthquake, 56.57.330 San Gabriel fault, 104 sand pile model, 5.3 16.3 17 sand piles, 316, 317, 321 scale invariance, 1, 11, 39, 84, 91, 289, 304, 3 16. 372,373

scale invariant distribution, 3 scaling, counter, 162 multifractal, 128 scarpes, earthquake, 203 shoreline, 203 screen analysis, 42 sea-floor bathymetry, 2, 163, 165, 167, 168, 217 sea level, 19, I67 second law of thermodynamics, 289 sedimentary basement, 76 sedimentary basin, 76 sedimentary bedding planes, 19, 321 sedimentary completeness, 24 sedimentary hiatuses, 18, 19.20, 167 sedimentary pile, 25, 76 sedimentary record, 18, 19 gaps in, 18, 19.20, 167 sedimentary sequence, 22.23, 321 fractal dimension of, 22,321 power-law correlation of, 23 thickness statistics of, 321 sedimentary unconformities, 18, 19,20,2 1 sedimentation, episodic, 23 sediments, age distribution of, 18 deposition of, 18.22.202. 321 erosion of, 18, 19.23.202 subsidence of, 76 seismic activation, 329, 330 seismic hazards, 63,66,254,307,328,329,342 seismic moment, 60, 305 seismic precursors, 305, 328 seismic quiescence, 328 seismicity, 1.4, 15, 39.56, 57.58, 59, 61, 65, 74, 103, 107, 128, 145, 317,320, 324, 325,326,327,328.329,330,341 clustering of, 103, 328 crustal, 322 distributed, 1, 76, 107, 128, 320, 324, 325, 326,327 fractal dimension, of 59, 76 global, 59 induced, 5, 129,328 Memphis-St. Louis. 66 pair correlations of, 107 regional, 5.63.324 southern California, 61.63.330, 336 seismograms, 166,217 self affine. 373 self-affine fractal, 132. 135, 138, 140, 145, 146, 148, 150, 163, 166,372 box counting method for, 132, 135, 147 Brownian walk as a, 140 deterministic, 133 fractional Brownian walk as a, 140, 146, 150 fracture surfaces as a, 166 sea level as a. 167 statistical, 140 topography as a, 145, 178 variance of, 146 self-affine fractal dimension, 135 self-affine tiling, 207 self-affine time series, 145

INDEX

self-excited dynamo, 279 self inductance. 282 self-organizedcriticality, 5, 316, 317, 325, 326, 328,329,336,341 self similar, 1,373 self-similar fractal, 132 semivariance, 138, 155 semivariogram, 138, 146, 155 fractional Brownian walk, 155 fractional Gaussian noise, 155 sequencies, depositional, 18 shear stress, 260 shoreline scarps, 203 short memory, 137 side branches, 188 Sierpinski carpet, 9.24, 105, 111, 118, 119,291 sieve analysis, 42 simulation, random, 103 singular point, 241 sinkholes, 50 sinuosity, 195 skarn mineral deposits, 87 skew, coefficient of, 29.30.35 for log-normal distribution, 35 slider-block array, 325,330,336 slider-block model, 4,245,247, 254, 325,330, 336,342 slider blocks, 245,247,325,330,334,342 pair of, 247, 342 multiple, 325,330,334,336,342 slip events, frequency-size statistics of, 325, 326 slumps, 321 snowflake, Koch, 25, 107, 195 soil density, 52 soils, 44.50.52 fractal dimension of, 44 multifractal fragmentation of, 129 solid-state creep, 258 southern California seismicity, 61, 63, 330, 335 space, phase, 222,373 space filling, 236 spatial distribution of aftershocks, 77, 107 of earthquakes, 76, 128 of gold, 98 of mineral deposits, 98 of oil, 97 specific heat, 257 spectra, power-law, 3, 166, 167 spectral analysis, 2, 163 of bathymetry, 2, 163 of topography, 2, 163 two-dimensional, 168 spectral density, power, 148, 149, 150, 166, 168, 170,203 of geoid, 170 of topography, 168, 170 of topography on Venus, 170 two-dimensional, 168, 172 spectrum, Fourier, 2, 148, 163 multifractal, 115 spherical harmonics, 168

spiral, 225 logarithmic, 225 sponge, Menger, 10,50,105,295 spreading centers, 56 spring-mass oscillator, 222 stabilities, exchange of, 226 stability analysis, 220, 221, 227.259.261 linearized, 220,259 marginal, 259, 3 16 stable fixed point, 220,225,232,265,294,297, 302 stable node, 224 staircase, devil's, 18,21,23,24 standard deviation, 29.30 Brownian walk, 140 exponential distribution, 40 Gaussian distribution, 32 log-normal distribution, 35 normal distribution, 32 Pareto distribution, 38 standard form, 33,37,38 state, critical, 5,316 static friction, 246, 326, 330 stationarity, 138 statistical fractal, 12, 38, 289 statistical mechanics, 289,330,342 statistical self-affine fractal, 140 statistics, 28 stick-slip behavior, 245, 246, 249 stiffness parameter, 248,326 Stirling approximation, 127 stochastic, 136, 137, 373 stochastic component, 136, 137 stochastic models, 4 storage, reservoir, 158 strain, 76,305 Benioff, 305 stranded cable, 299,303,307 strange attractor. 265,284,373 stratigraphichiatuses, 18, 19,20, 167 stream function, 257, 260 stream order. 181 stream flow time series, 215 stress, shear, 260 strong persistence, 137 subcritical pitchfork bifurcation, 228,272 subduction zones, 13.56.70.269 subsidence, 76 successive random additions, 152 sunspot numbers. 160 supercritical pitchfork bifurcation, 226 surface, free, 260 ocean, 146 surface wave magnitude, 57,58 surfaces, rock, 166 symmetry. 248 synthetic landscapes, 2, 173 Tangshan earthquake, 330 tectonic fragmentation, 44, 70 tectonic processes, 13.56, 181 tectonic uplift. 23

INDEX

tectonics, 13,50, 56 plate, 5.56.60, 70,256, 269, 279 temperature, 31, 136, 146, 160,289 atmospheric, 136, 146, 160 Curie, 279 tent map, 24 1 tephra, 79 textural analysis, 3, 112 landscape, 112 thermal boundary layer, 265,269 thermal conductivity, 257 thermal convection, 3,256,261,268,269 thermal diffusivity, 258 thermal expansion, 256,258 coefficient of, 258 thermally activated creep, 269 thermodynamics, 289 first law of, 289 second law of, 289 thickness statistics, 321 thinning, crustal, 76 threshold, percolation, 333 tiling, self-affine, 207 time series, 136, 137, 140, 146, 148, 158, 167, 214,373 atmospheric temperature, 136, 146 box-counting method for, 147 Brownian walk. 140 continuous, 136, 137 correlation dimension of, 167 correlations of, 158 discontinuous, 137 discrete, 136 fractal dimension of, 147, 148 multifractal dimension of, 129 river flow, 158,215 self-affine, 145 stream flow, 158,215 time to failure, 303,304,305,306 Tokunaga fractal tree, 188, 190, 198 tonnage, ore, 8 1 tonnage-grade, 8 1,90,92 copper, 93 gold, 92 mercury, 90 uranium, 94 topographic contour, 12, 132 topography, 2, 12, 13,56, 132, 145, 163, 165, 168, 170, 175, 177,207 Arizona, 177 Brownian walk, 163,207,208 elevation of, 132, 136 Fourier spectral analysis of, 163 fractal, 132, 208 height of, 132, 136 multifractal analysis of, 129 Oregon, 163, 165, 175, 177 power spectral density of, 168, 170 roughness of, 2, 165. 176 self-affine, 145, 178 Venus, 170 torque, 28 1

trace elements, 87 trajectory, phase, 222 transcritical bifurcation, 226 transform, discrete Fourier, 150, 171 Fourier, 148, 150, 171,214 inverse discrete Fourier, 150 inverse Fourier, 148 two-dimensional Fourier, 171, 172 wavelet, 214 transform faults. 56, 70 Transverse Ranges, 65 traps, petroleum, 98 trees, fractal, 2, 181, 187, 188, 299, 302 binary, 188 branch numbers for, 188 branching ratios for, 189 deterministic, 187 length-order ratio of, 185 side branching, 188 Tokunaga, 188, 190,198 tree rings, 160 trench, oceanic, 13, 56, 70, 269 trench pull. 65 trend component, 136 triadic Koch island, 11.24 triangular map, 241 tributaries, 185 truncations, 262,268, 270 tuples, 167 turbidite deposits, 321 turbulence, 3, 127, 23 1,268, 34 1 turning-point bifurcation, 226 two-dimensional Fourier transform, 171, 172 unconformities, 18, 19, 20.2 1, 167 unstable. gravitationally, 256 unstable fixed point, 232,265,294, 294,302 unstable node, 224 uplift, 23 uranium, 94.95 van der Pol equation, 221,223,229 variables, nondimensional, 219,222,246,248, 258,282,332 variance, 29.30, 137, 140, 141, 145 autoregressive model. 141 autoregressive moving average model, 145 Brownian walk, 140 exponential, 40 Gaussian, 32 log-normal. 35 moving average model, 141 normal, 32 Pareto, 38 self-affine fractal, 146 two-dimensional spectra, 168 variation, coefficient of, 35, 137, 158 varves, 160 veins, 70, 104, 193 clustering of, 104 velocities, molecular, 30.3 1 velocity, angular, 281

397

398

INDEX

velocity weakening friction, 299,325 Venus, 163. 170 spectral density of topography, 170 viscosity, 257, 258 mantle, 258 viscous fingering, 198 viscous forces, 257 void spaces, multifractal analysis of, 129 volcanic ash, 44.79 fractal dimension of. 44 volcanic edifices, 13 volcanic ejecta, 44 volcanic eruptions, 1,28,78,79, 167,243 correlation dimension of, 167 frequency-volume statistics of, 1, 79 volcanic hazard, 79 volcanic islands. 194 walk, Brownian, 2, 138, 140, 146, 147, 149, 150, 152, 155, 157, 158, 160, 163, 165, 167,207,208,371 correlation dimension of, 167 fractional, 149, 150, 152, 155, 156, 158, 161, 162, 166 log-normal, 157, 158 random, 2, 195, 197, 199,207,371 self-affine fractal, 140, 146 standard deviation of, 140 time series, 140 variance of, 140

wave equation, 4. 23 1 wavelength, 2 wavelet transform, 214, 373 bathymetric profile. 2 17 fractional Brownian walk, 2 17 geiod, 2 17 length of day, 217 seismogram, 2 17 wavelets, 214, 2 17, 373 Mexican hat. 214 mother, 2 14 weak persistence, 137 weather, 268, 341 weathering processes, 44 Weibull distribution, 41,42,299,301,303 Weirstrass-Mandelbrot functions, 152, 165 well logs, 3, 136, 165, 166, 168 acoustic, 165 electrical conductivity, 165 multifractal analysis of, 129 neutron activation, 165 porosity, 136, 166, 167 white noise, 138, 139, 140, 146, 148, 149, 150, 156,207,373 Gaussian, 140, 149, 150, 160, 207 Whittier earthquake, 63 window method, moving, 109 windows of chaos, 236,237,238,240.25 1 Zipf's law, 26

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Chaos and Fractals: New Frontiers of Science

Fractals and Chaos: An Illustrated Course

Fractals and chaos: an illustrated course

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