Fractals and Chaos in Geology and Geophysics

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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

DEFINITION OF A FRACTAL SET

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).

DEFINITION OF A FRACTAL SET

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

DEFINITION OF A FRACTAL SET

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.

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DEFINITION OF A FRACTAL SET

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

DEFINITION OF A FRACTAL SET

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

DEFINITION OF A FRACTAL SET

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-

+

DEFINITION OF A FRACTAL SET

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

DEFINITION OF A FRACTAL SET

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).

DEFINITION OF A FRACTAL SET

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

DEFINITION OF A FRACTAL SET

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.

DEFINITION OF A FRACTAL SET

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

DEFINITION OF A FRACTAL SET

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

DEFINITION OF A FRACTAL SET

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).

DEFINITION OF A FRACTAL SET

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

DEFINITION OF A FRACTAL SET

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

DEFINITION OF A FRACTAL SET

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

DEFINITION OF A FRACTAL SET

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

DEFINITION OF A FRACTAL SET

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

DEFINITION OF A FRACTAL SET

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.

DEFINITION OF A FRACTAL SET

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).

42

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.

SEISMICITY AND TECTONICS

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

SEISMICITY AND TECTONICS

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

SEISMICITY AND TECTONICS

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

SEISMICITY AND TECTONICS

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

-

-

-

SEISMICITY AND TECTONICS

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

Substitution of (4.13) and (4.14) into (4.11) 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

SEISMICITY AND TECTONICS

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).

+

+

SEISMICITY AND TECTONICS

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

SEISMICITY AND TECTONICS

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.

SEISMICITY AND TECTONICS

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

SEISMICITY AND TECTONICS

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"

SEISMICITY AND TECTONICS

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

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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.

SEISMICITY AND TECTONICS

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.

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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.

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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.

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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.

SEISMICITY AND TECTONICS

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.

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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

SEISMICITY AND TECTONICS

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.

-

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SEISMICITY AND TECTONICS

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-

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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.

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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.

SEISMICITY AND TECTONICS

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

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SEISMICITY AND TECTONICS

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.

ORE GRADE AND TONNAGE

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

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ORE GRADE AND TONNAGE

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-

ORE GRADE AND TONNAGE

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

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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

ORE GRADE AND TONNAGE

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

ORE GRADE AND TONNAGE

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)

Substitution of (5.40) and (5.41) into (5.38) gives

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,

ORE GRADE AND TONNAGE

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.

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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.

ORE GRADE AND TONNAGE

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.

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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.

ORE GRADE AND TONNAGE

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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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.

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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

ORE GRADE AND TONNAGE

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.

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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 << 1) (6.24) be-

= f [ l + ( 9 - 1) l n f l

Substitution of this result into the definition of the generalized moment (6.23) gives

xfi n

and since

= 1 we have

i=l

This result is valid in the limit q + 1. The generalized multifractal dimension Dq is defined by

where the values of Dq form the multifractal spectrum. For q = 0, Do is referred to as the "box" dimension; it is entirely equivalent to the fractal dimensions considered in previous chapters. This is because the probabilityp

115

116

FRACTAL CLUSTERING

is unity if the box is occupied and is zero if it is not. For q = 1 , D, is referred to as the "information" or "entropy" dimension and for q = 2, D, is referred to as the "correlation" dimension. These specific cases will be discussed in more detail below. If all values of Dq in (6.29) are equal, then a "homogeneous fractal" is defined. If (6.29) gives well-defined power laws, but the values of D, are different, then a "multifractal" is obtained. The definition (6.29)can be rewritten as

This result can be used directly as long as q # 1 . Substitution of (6.23)gives

for q # 1 . The zero-order box dimension is ob~tainedfrom (6.30)taking q = 0 with the result

For the third-order Cantor set illustrated in Figure 6.11 we have M, ( 1 ) = 1 , M, = 2 and Mo = 4 . Taking either rj = 1 and rk = j1 or r.J = j1 and rk = g1 , we find Do = In 2/ln 3, which is identical to the value obtained in Chapter 2.

(3)

(i)

The correlation dimension is obtained from (6.30) taking q = 2 with the result

For the third-order Cantor set we have M,(1) = 1 , ~

1 , ( f )= i, and M2(g) = 21 .

Taking either rj = 1 and rk= 31 or rj = 51 and rk= y1 we find that D,= In 21ln 3, which is again identical to the value obtained in Chapter 2. To obtain an expression for D, it is necessary to combine (6.28) and (6.30).We first consider the quantity In [Mq (rj)lMq(rk)] in the limit q + 1 ; using (6.28)we have

FRACTAL CLUSTERING

However, in the limit E << 1 we have

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

4

1 (q - 1 << 1) (6.34) be-

Substitution of this limit into the definition of the multifractal dimension (6.30) while taking the limit q + 1 gives

where

For the third-order Cantor set we have

s(f)= -

(i)In (f)- OlnO - (f)In (f)= In2

117

118

FRACTAL CLUSTERING

s

(i) = - (a)

In ($) - 0 In 0 - ($)In

(a) - 0 In 0 - 0 In 0

-

0 In 0

where by definition 0 In 0 = 0. Taking either r.J = 1 and r, = 31 or r.J = 51 and

i

r, = in (6.37), we find that again D, = In 2An 3. We have found that Do= D, = D,= In 2lln 3 for a third-order Cantor set. In fact this value is obtained for any value of q and the third-order Cantor set is a homogeneous fractal. Any other order Cantor set would have given the same result and would also be a homogeneous fractal. To interpret the first-order fractal dimension D,it is appropriate to note that the classical statistical mechanics definition of the entropy S of a thermodynamic system is (Morse, 1969, p. 248)

x

& = 1 where is the distribution function, k is the Boltzmann conwith stant, ind the summation is carried out over the allowed states of the system. The similarity between (6.39) and (6.38) is the basis for calling D,the information or entropy dimension. As a second example of a homogeneous fractal we consider the Sierpinski carpet illustrated in Figure 2.3(d). A second-order Sierpinski carpet is illustrated in Figure 6.12(a). The unit square is divided into 81 equal-sized squares with r, = $ and 64 squares are retained. To define a multifractal in this case we divide the unit area into n equal-sized squares with linear dimensions r = n- '2; the squares are denoted by i = 1,2, . . . , n. The fraction& of the remaining area in each square box i is given by

Figure 6.12. Illustration of the multifractal fractions for a second-order Sierpinski carpet. (a) Second-order Sierpinski carpet. (b) The single fractionf,is given for r = 1. (c) The nine fractionsf,-f,are given for

FRACTAL CLUSTERING

n

where A, is the area in box i and A is the total retained area. Again

Ai = A

n

so that

Z J= 1 i=l

For the second-order Sierpinski carpet illustrated in Figure 6.12(a) we have A = We now determine the values of fn for two cases, n = 1 (r = 1) 1 and n = 9 ( r = j). Taking n = 1 we have i = 1; in this one area we have A, = and from (6.40) f , = 1. This is illustrated in Figure 6.12(b). For n = 9 we have i = 1, 2, 3, 4, 5, 6, 7, 8, 9 as illustrated in Figure 6.12(c); we have A , = 8 8 A, = A3 = A4 = 8 , A5 = 0, A6 = A, = A8 = A9 = 81. From (6.40) we have 1 f , = f , = f 3 = f , = 8 , f 5 = 0 , f6=f,=f8=f9=$.From(6.23) we find

g.

3

119

120

FRACTAL CLUSTERING

And from (6.38) we find

(i) (3 (f) (3 (f) (3 (f)

S -

-

-

1n

-

1n

-

1n

-

(f) 1n

(3

And from (6.32), (6.33), and (6.37) we obtain Do = Dl = D, = In 8An 3, the same value obtained in Chapter 2. The Sierpinski carpet is also a homogeneous fractal. Many examples similar to the above can be found. If a problem consists of two regions, one occupied and one not occupied, with a scale-invariant boundary between them, then the box-counting method introduced in Chapter 2 is generally applicable and there is no need for multifractal concepts. However, there are many geological problems in which there is a continuous variation in a quantity rather than a discontinuous variation. The concentration of a mineral has a continuous variation so that a straightforward boxcounting approach is not applicable to its spatial distribution; every box would be occupied so that the fractal dimension would be equal to the spatial dimension. We will now consider a variation on the De Wijs (1951, 1953) model for the concentration variability in ore deposits, which was illustrated in Figure 5.l(a). At each order the mass was divided into two parts and the enriched half was given a concentration C,, = C, (1 < 4, < 2) and the depleted half was given a concentration C,, = (2 - +,)C0. The third-order distribution is given in Figure 6.13(a). A line of unit length is divided into eight equal parts, each segment having been enriched or depleted as indicated, with 4, the enrichment factor. Since the concentration in each part is determined by and (2 - +,), this process is known as a a multiplication by powers of multiplicative cascade. To carry out a multifractal analysis of the third-order distribution, the line of unit length is divided into n equal segments, which are denoted by i = 1,2, . . . , n and r = n-1. The fraction4 is defined by

+,

+,

FRACTAL CLUSTERING

121

where Ciis the mean concentration in segment i and Cois the overall mean concentration. n=4 We now determine the values off;. for n = l(r = I),n = 2 (r= (r = and n = 8 (r = Taking n = 1 we have i = 1, C, = C,, and f,= 1; for n = 2 we have i = 1, 2, C,= C,,C2= (2 - 4,) C,, f,= (1)+2, f2= 1 - (i)+2; for n = 4 we have i = 1, 2, 3, 4,C,= +,2 Co,

i),

a),

k).

+,

(2 - +J cO, c4= (2- +,)2 cO, f,= ($1$2 , f2= f3= (1) +2 [l - (1)+,I, f4= [l - (1)+,I2; for n = 8 we have i = 1, 2, 3,4,5 , 6,7,8, c,= c,,c2= c3= c4= +; (2 - +,) C,,C5= C6= C,= +,(2 - +,)2Co,Cg = (2- +,I3, fl= (i) f2= f3 = f4 = ( f ) +$ - ($1 $21 f 5 = f6 = f7 = (1)+2 [l - (4) +2]2, fa= [l - (1)+2]3. These results are illustrated in Figure

c2= c3= +2

+;

+;9

We can now determine the generalized moments for the De Wijs multiwe find for q = 0 that plicative cascade. From (6.23)

Figure 6.13. Illustrations of the multifractal fractions for a third-order De Wijs multiplicative cascade. (a) Concentrations in the eight equal masses at third order. (b) The single fraction f,is given for r = 1. (c) The two f, are given for fractions f,, r = i. (d) The four fractions fl-f,are given for r = (e) The eight fractions fl-f,are given for r = i.

a.

122

FRACTAL CLUSTERING

+ [(1

- ;+2)3]0

=

1

+ 3 + 3 + 1= 8

where it is assumed that 0 < +, < 2; in particular +, we have

#

0 and # 2. For q = 2

This result can be easily generalized to a De Wijs multiplicative cascade of arbitrary order n, r,, = ($)", and to a moment of arbitrary order q giving

FRACTAL CLUSTERING

+,

with q # 1 and 0 < < 2. Note that this result is also valid for noninteger values of q. For q = 1 from (6.38) we find

s(1)

= -

llnl =O

And again this result can be generalized to a cascade of arbitrary order n with the result

123

124

FRACTAL CLUSTERING

We can now determine the generalized multifractal dimensions for the De Wijs multiplicative cascade. For q = 0 we find from (6.32) that Do = 1 and this result is valid at all orders n. Each of the linear array of boxes considered is occupied by some ore so that we find the box-counting dimension of a line, Do = 1. For q = 2 we find from (6.33) that

And again this result is independent of the order n considered. The fractal dimension for arbitrary moment order q is given by

which is valid for both integer and noninteger values of q except q = 1. For q = 1 we find from (6.37) that

again independent of the order n. The De Wijs multiplicative cascade is a perfect multifractal in that for all values of q, D, is independent of order n. As a specific example we take +, = 1.5 and find that Do = 1, D, .= 0.81 1, and D, = 0.678. Values of Dq for a range of values of 4, are given in Figure 6.14. As 4, + 1 the multifractal dimension Dq approaches unity for all values of q. In this limit each box has the initial concentration of ore Co and the dimension of a straight line is obtained. As +, + 2 the multifractal dimension Dq approaches zero for all values of q except q = 0. In this limit all the ore becomes concentrated into one box so that the fractal dimension of zero is obtained. However, since minute quantities of ore remain in the other boxes, the box-counting dimension, q = 0, remains unity. We now review the steps necessary to carry out a multifractal analysis. As a specific example consider a spatial distribution of faults and joints such as that illustrated in Figure 4.6. The total length of the fractures is L. The required steps are: 1.

The standard box-counting method is applied. The two-dimensional array of joints and faults is overlaid with a grid of square boxes; grids of different size boxes are used.

FRACTAL CLUSTERING

2.

3.

4. 5.

125

The fractions& for each box of size r,, is determined. The length of faults and joints in box i is L,. The fraction of the faults and joints in box i,f,,is determined using (6.22). The generalized moments Mq (r) are obtained using (6.23) if q ;t 1 and (6.28) if q = 1. If the Mq ( r ) have a power-law dependence on r for specified values of q, then the fractal dimensions Dq are obtained using (6.3) if q # 1 and (6.37)if q = 1. The Dq are given as a function of q over the range 0 5 q < -.

It is common practice in multifractal analysis to introduce a scale-invariant probability distribution analysis. This is known as thefla), a curve. To illustrate this approach we will again use the De Wijs multiplicative cascade. Our analysis will be carried out in the limit of very high order n + -. In terms of the order n of the system, the length of the line segment is given by

Figure 6.14. Generalized multifractal dimension D y as a function of q for the De Wijs multiplicative cascade; enrichment factors = 1.25, 1S O , and 1.75 are considered. The results are valid for any order n of the cascade.

+,

126

FRACTAL CLUSTERING

For the De Wijs multiplicative cascade the range of concentrations is given by

with j = 0 , 1, . . . , n. Note that this nomenclature is different from that used above because we assign each concentration a subscript j rather than assigning each line segment a subscript i. We also use a reference concentration Cor to avoid confusion with C, corresponding to j = 0 . For the De Wijs multiplicative cascade the number of line segments with the concentration C,, Ni. is given by

N, =

(7)

=

n! j!(n - j)!

where the binomial coefficient has been defined in (6.7). To illustrate this nomenclature consider the third-order, n = 3, example given in Figure 6.13. From (6.46) we have r = i.From (6.47) and (6.48) we have f o r j = 0: C, = C,,, No = 1 , f o r j = 1 : C, = +;(2 - +,) C , , N, = 3, for j = 2: C, = ( 2 - +2)2 CoryN2 = 3; and for j = 3: C, = ( 2 - +,)3 C,,, N3 = 1 . The probability has been previously introduced in (6.41) and here takes the form

+:

x4 n

Note, however, that we now have

i=O

x n

N, = 1 rather than

f ; = 1 as in the

i=l

previous analysis. Substituting (6.46)and (6.47)into (6.49)gives

We can consider the4 from (6.50)to be analogous to a concentration and the N, from (6.48) to be analogous to the probability of having that concentration. In order to define the flor,), or, curve, we introduce the following definitions o f f i y ) and or,

FRACTAL CLUSTERING

For the De Wijs multiplicative cascade, substitution of (6.46), (6.48), and (6.50) into (6.51) and (6.52) gives

f@,>

=

log n! - log j! - log ( n n log 2

- j)!

+,.

Values of a.andfla,) can be determined for specified values of n, j, and A contihuous dependence offla) on a can be obtained by assuming both j and n to be large integers. In this limit the Stirling approximation gives log n!

= n log n -

logj!

=j l o g j

-j

log (n

- j)! =

(n

n

(6.55) (6.56)

- j)

(6.57)

log (n - j) - (n - j)

Introducing x = jln along with (6.55)-(6.57), (6.53) and (6.54) become

+, x 1% ( 2 -

a = 1 - (1 - x)----log log 2 Aa)

= &[x

log

-

(T)1 - x

$2)

log 2

- log ( 1

-

x)

1

The dependence of fla) on a is obtained by varying x over the range 0 <

x < 1. The dependence offla) on a from (6.58) and (6.59) is given in Figure = 1.2, 1.4, and 1.6. This result is clearly analogous to the depen6.15 for dence of a probability distributionflx) on x, for example the log-normal distribution in Figure 3.2. Also included in this figure are the measured values for one-dimensional sections through the energy-dissipation field in several fully developed turbulent flows (grid turbulence, wake of a circular cylinder, atmospheric turbulence) as given by Meneveau and Sreenivasan (1987). Good agreement with the case = 1.4 ( p , = 0.7) is found.

+,

+,

127

128

FRACTAL CLUSTERING

It is seen that the De Wijs multiplicative cascade generates a perfect multifractal. However, as was shown in Chapter 4, the resulting distribution of ore grade on tonnage is not fractal. In fact, as the construction is carried to infinite order, the tonnage-grade distribution becomes log normal. A lognormal distribution is a multifractal. Neither the log-normal distribution nor the De Wijs construction is scale invariant. Thus a multifractal is not necessarily scale invariant, and in general it will not be scale invariant. Multifractals represent a particular type of scaling that can be associated with multiplicative cascades. Multifractal scaling can be found in a wide variety of geological and geophysical applications. A number of authors have shown that the spatial distribution of earthquakes may obey multifractal statistics (Hirabayashi et al., 1992; Hooge et al., 1994a, b; Blanter and Shnirman, 1994). Hirata and Imoto (1991) found the spatial distribution of microearthquakes in the Kanto region of Japan to be multifractal, and Eneva

Figure 6.15. Theflcx) vs. a curve for the De Wijs multiplicative cascade model in the limit of infinite order. Curves are given for 4, = 1.2, 1.4, 1.6 ( p , = 0.6,0.7, 0.8). The data points are measured values for the energy-dissipation field in fully developed turbulence (Meneveau and Sreenivasan, 1987).

FRACTAL CLUSTERING

(1994) found the spatial distribution of mining-induced seismicity to be mul-

tifractal. Dongsheng et al. (1994) have suggested that there was a systematic variation in the multifractal spectra of the spatial distribution of regional seismicity prior to the great Tanshan (China) earthquake in July 1976. Belfield (1994) found that the clustering of fracture networks can be multifractal, and Sornette et al. (1993) found similar results for fractures generated in a laboratory experiment. Muller (1994) found the distribution of pore spaces in North Sea chalk formations to be multifractal, and PyrakNolte et al. (1992) found that the distribution of contact areas and void spaces in single fractures in granite were multifractal. Perfect et al. (1993) applied multifractal statistics to soil aggregate fragmentation. Saucier (1992) determined the effective permeability of a multifractal porous media. Lovejoy et al. (1995) applied multifractal characterization to topography and Ijjasz-Vasquez et al. (1992) to river basins. Klement et al. (1993) applied multifractal scaling to time series, and Muller (1992) and Saucier and Muller (1993) have applied the technique to the characterization of geological formations using well logs. The applicability of multifractal statistics to a natural phenomenon may provide important clues to the underlying physical processes. However, multifractal statistics are much less useful than monofractal statistics from a practical point of view. A monofractal distribution is characterized by two constants, for example in (2.2). But a full multifractal spectral in principle requires an infinite number of constants. Monofractal applicability implies scale invariance, but multifractal applicability does not.

Problems Problem 6.1. Consider the construction given in Figure 2.l(b). What is the probability that a step of length r includes a line segment for r = 1,

i,

1 L7 99 2 7 '

Problem 6.2. Consider the construction given in Figure 2.l(d). What is the probability that a step of length r includes a line segment for r = 1, 1 I? 97 27 Problem 6.3. Consider the construction given in Figure 2.l(f). What is the probability that a step length r includes a line segment for r = 1, $, &,? Problem 6.4. A line segment is divided into seven equal parts and four are retained. The construction is repeated. What is the probability that a step of length r includes a line segment for r = i , &, & ? Problem 6.5. A line segment is divided into seven equal parts and three are retained. The construction is repeated. What is the probability that a step of length r includes a line segment for r = &,

5,

3, A?

129

130

FRACTAL CLUSTERING

Problem 6.6. Consider the construction given in Figure 2.3(c). What is the probability that a square box with dimensions r includes a retained square when r = 1 1 ?,1 Problem 6.7. A unit square is divided into four smaller squares of equal size. Two diagonally opposite squares are retained and the construction is repeated. What is the probability that a square box with dimensions r includes a retained square when r = 1, $, i? Problem 6.8. 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. What is the probability that a square box with dimensions r includes a retained square when r = 1, i , &? Problem 6.9. Consider the construction given in Figure 2.4(b). What is the probability that a cube with dimensions r includes solid when r = 1, $,

,,

:,

I I? 49

8

'

Problem 6.10. A unit cube is divided into 27 smaller cubes of equal volume. All the cubes are retained except for the central one and the construction is repeated. What is the probability that a cube with dimensions r includes solid when r = 1, &, ;? Problem 6.11. What is the pair-correlation distribution for three equally spaced particles on a line of unit length? Problem 6.12. What is the pair-correlation distribution for three particles on the comers of an equilateral triangle with sides of unit length? Problem 6.13. What is the pair correlation distribution for the eight particles on the corners of a unit cube? Problem 6.14. Consider the third-order Cantor set illustrated in Figure 6.1 l(a). Determine M3 (I), M3 ($)and M3 (6). Write an expression for D3 in terms of ri and r, and determine its value for the third-order Cantor set. Prob!em 6.15. Consider the third-order Cantor set illustrated in Figure 6.11(a). Determine MI, (I), M,,,($), and M,,,($). Write an expression for Dl,, in terms of ri and rj and determine its value for the third-order Cantor set. Problem 6.16. Consider the second-order set illustrated in Figure 2.l(f) ( L = &). Determine L, and fi for n = 1 and n = 5, determine Mo(l), M&), M2(f),411, ~ ( 3Do, ) ~Dl, and D,. Problem 6.17. A line segment is divided into seven equal parts and four are retained (L = $).Determine Li and& for n = 1 and n = 7. Determine Mo (11, Mo ($),M2(lI9M2(+I, s (I), s ($1,Do, Dl, andD2. Problem 6.18. Consider the second-order Sierpinski carpet illustrated in Figure 6.12a. Determine M3 (1) and M3 ($);determine D,. Problem 6.19. A unit square is divided into four smaller squares of equal size. Two diagonally opposite squares are retained, A = Determine Ai

4.

FRACTAL CLUSTERING

(i),

a n d 4 for n = 1 (r = 1) and n = 4 (r = $), determine M,,(I), M,, M2 (I), 4($1, s (I), s (;I, Do,Dl, D2. Problem 6.20.Consider the third-order De Wijs model for ore concentration M3 illustrated in Figure 6.13.Determine expressions for M 3(I), M , M, (Q), and D,in terms of +,. Obtain values of these quantities for

(i),

(a), +, = 1.5.

Problem 6.21.Consider a unit cube, r,, = 1, which is divided into 8 smaller cubes of equal size, r, = Four of the cubes have been enriched by the and four have been depleted by the factor 2 Show that factor f,=f, = f, = f, = and f, = f, = f, = f, = (2 - +,). Determine M, ( I ) , M, ($1, s (11, s ($1, Do, D l , and D, expressions for Mo(I), M o in terms of

4.

+,

:+,

+,.

+,.

(i),

131

Chapter Seven

SELF-AFFINE FRACTALS

7.1 Definition of a self-affine fractal Up to this point we have considered self-similar fractals; we now turn to self-affine fractals (Mandelbrot, 1985). Topography is an example of both. In the two horizontal directions topography is often self-similar; the ruler method can be applied to a coastline or to a contour on a topographic map to define a fractal dimension. The box-counting method can also be applied to a coastline, and square boxes are used to determine a fractal dimension. These are examples of self-similar fractals. Consider next the elevation of topography. Three examples of elevation h as a function of distance x along linear tracks are given in Figure 7.1. The vertical coordinate is statistically related to the horizontal coordinate but systematically has a smaller magnitude. Vertical cross sections of this type are often examples of self-affine fractals (Dubuc et al., 1989a). A statistically self-similar fractal is by definition isotropic. In two dimensions defined by x and y coordinates the results do not depend on the geometrical orientation of the x- and y-axes. This principle was illustrated in Figure 2.8, where the box-counting method was introduced. The fractal dimension of a rocky coastline is independent of the orientation of the boxes. A formal definition of a self-similar fractal in a two-dimensional xy-space is thatflrx, ry) is statistically similar toflx, y) where r is a scaling factor. This result is quantified by applications of the fractal relation (2.6). The number of boxes with dimensions x,, y, required to cover a rocky coastline is N,; the number of boxes with dimensions x, = rx,, y, = ry, required to cover a rocky coastline is N,. If the rocky coastline is a self-similar fractal, we have N2/N1= r-D. A statistically self-affine fractal is not isotropic. A formal definition of a self-affine fractal in a two-dimensional xy-space is that f(rx, Fay) is statistically similar to f(x, y) where Ha is known as the Hausdorff measure; we shall relate Ha to the fractal dimension in what follows. In the box-counting method square boxes become more and more rectangular as their size is increased.

SELF-AFFINE FRACTALS

133

To illustrate this scaling we consider the deterministic example of a selfaffine fractal in Figure 7.2. A rectangular region with width ro and arbitrary height h, is defined. At zero order we consider a straight line between x = 0, y = 0, and x = r,, y = h,. At all orders our fractal construction will begin at x = 0, y = 0, and end at x = r, y = h,. Our first-order, self-affine fractal is defined in Figure 7.2(b). The horizontal coordinate is divided into four equal parts so that r, = rd4, and the vertical coordinate is divided into two equal parts so that h, = hd2. We connect the points (0, O), (r0/4, hd2), (r0/2, O), and (r,, h,). This is the generator for our fractal construction. In the second-order 54 --

V.E. = 1:270

4

394.5

4 0

2

6

4 X,

m

(4

8

10

Figure 7.1. Elevation h as a function of position x along linear tracks. Three examples are given: (a) Earth's topography along a 7,500km track from 70W.55s to 70W,12N. (b) Venus topography in Ovda Regio. (c) Elevation across a small gully on the earth.

134

SELF-AFFINE FRACTALS

fractal illustrated in Figure 7.2(c) each straight-line segment in the firstorder fractal has been replaced by the generator. At second order the horizontal coordinate has been divided into 16 equal parts so that r, = rd16 and the vertical coordinate is divided into four equal parts so that h, = hd4. The use of the generator in this construction is entirely equivalent to the genera-

Figure 7.2. Illustration of a deterministic, self-affine fractal. (a) At zero order a rectangular region of width r, and height h, is considered. A straight-line segment extends from (0,O) to (r,, h,). (b) The first-order self-affine fractal is given. This construction also serves as the generator for higherorder fractals. (c) Each straight-line segment in (b) is replaced by the scaled generator from (b) to give the second-order fractal construction. (d) Each straight-line segment in (c) is replaced by the scaled generator from (b) to give the third-order fractal. The construction can be carried out as far as desired.

SELF-AFFINE FRACTALS

135

tion of the self-similar fractals in Chapter 2. In terms of the formal definition of a self-affine fractal given above, we havef(xl4, yI2) statistically similar to f(x, y). Thus the Hausdorff measure can be obtained from r = 114, rHa = 112, and (114)Ha = 112 or

Our construction of a self-affine fractal is extended to third order in Figure 7.2(d), where each straight line segment of the second-order fractal has been replaced by the generator. We next wish to determine the fractal-dimension of our self-affine fractal. To do this we will use the box-counting method. For self-similar fractals we used square boxes, but for self-affine fractals it will be necessary to use rectangular boxes. At zero order consider a single box with width ro and height h,; thus No = 1. At first order we have r, = rd4 so that we will use rectangular boxes with width rd4 and height ho14. We wish to determine how many of these boxes are required to cover the first-order fractal illustrated in Figure 7.2(b). This determination is illustrated in Figure 7.3, where we find N, = 8. Using the standard technique given in Chapter 2 we find from (2.2) that

We find that the fractal dimension is between 1 and 2, intermediate between D = 1 for the straight line at zero order and D = 2 for the entire box considered. For the higher-order constructions illustrated in Figure 7.2(b) and (c) 1 1 we have r, = 16, N2 = 64 and r3 = 64, N3 = 512. For this example the Hausdorff measure Ha = and the fractal dimension D = This is consistent with the relation Ha = 2 - D, which we will derive in general terms.

i.

Figure 7.3. Box-counting technique applied to the firstorder self-affine fractal given in Figure 7.2(b). To determine its fractal dimension, rectangular boxes with width r , = r d 4 and height h , = h d 4 are used. We find that the shaded N, = 8 boxes (out of 16) cover the fractal construction. Noting that No = 1 for the single box of width ro and height h,, we

find^=+.

136

SELF-AFFINE FRACTALS

7.2Time series Before continuing our discussion of self-affine fractals, it is appropriate to introduce some of the fundamental concepts of time series (Chatfield, 1989). As the name implies, a time series is the set of numerical values of any variable that changes with time. Our consideration of time series is a direct extension of our consideration of probability and statistics in Chapter 3. Just as we had continuous and discrete data, we have continuous and discrete time series. A continuous time series would be a set of values y (t) that are continuous in time over the interval T. An example would be a continuous record of the atmospheric temperature at a specified point. Another example would be the discharge down a river measured at a gauging station (Salas,'1993). A discrete or noncontinuous time series consists of a set of values that are not continuous. A discrete time series can be obtained from a continuous time series by sampling it at specified time intervals T, or by integrating the continuous time series over a specified time interval T. Values are usually specified at equal increments of time T so that we have values of y (t) given at t = nT, n = 1, 2, . . . ,N with T = NT.An example of a discrete time series would be the sequence of daily rainfall totals at a measuring station or the maximum flood discharge at a gauging station each year. Time series are good representations of other data sets. For example, the heights of topography along linear tracks as illustrated in Figure 7.1 are entirely equivalent to continuous time series. Another example would be the concentration of a mineral (i.e., gold) as a function of depth in a drill core. The actual concentration would be continuous, with possibly a few exceptions, but from practical considerations measurements of concentrations would be carried out at discrete intervals, giving a discrete time series. Well logs are another example of a time series in a geological context. Digitized measurements of density, porosity, and/or permeability at prescribed depth intervals represent discrete time series. Time series have a wide range of applications (Box et al., 1994). Time series may be characterized by discontinuities, a trend component, one or more periodic components, and a stochastic component. The trend component is a long-term increase or decrease in the series. The rotational period of the earth (length of day) can be considered to be a time series. Over long periods of time the length of day is increasing due to tidal friction, which is a trend component. Many time series have periodic components; the atmospheric temperature time series will have strong daily and yearly periodicities. The stochastic component includes the fluctuations not included in either the trend or periodic components. An important aspect of the stochastic component is whether it is persistent, random, or antipersistent. If adjacent values in the time series are uncorrelated with each other, then the stochastic component is random. If adjacent values are positively correlated, then adjacent values are on average closer than for a random time series, and

SELF-AFFINE FRACTALS

the stochastic component is persistent. If adjacent values are negatively correlated, then adjacent values are on average further apart than for a random time series, and the stochastic component is antipersistent. A continuous time series is, by definition, persistent. An important question regarding time series is ergodicity. If a time series is ergodic, an average at a given time over a large number of realizations of the time series is entirely equivalent to the average in time of a single realization. In general, the ergodicity of practical time series is assumed, although it is often difficult to prove. The values of the stochastic component of a time series will have a statistical distribution of values and the discussion given in Chapter 3 is applicable. Typical distributions are Gaussian (normal) and log normal. The mean, variance, and coefficient of variation of the distribution of values can be determined. The persistence (or antipersistence) of the stochastic component can be quantified by using the autocorrelation function r. For a continuous time series we have

with

and

The time s is the lag; with s = 0 we have cS= co = V (the variance) and rs = 1. As s increases, rs generally decreases as the statistical correlations of y (t + s) with y(t) decrease. The plot of rs versus s is known as a correlogram. A rapid decay of the correlogram indicates weak persistence (short memory), and a slow decay indicates strong persistence (long memory). Since the time series is continuous, it is required that rs + 1 as s + 0. For a discontinuous time series the autocorrelation function is given by

with

137

138

SELF-AFFINE FRACTALS

and

with increasing values of k corresponding to increasing lag. For a random stochastic time series the values of r, will be near zero. Positive values of r, indicate persistence and negative values indicate antipersistence. In terms of self-affine fractals, one of the most important aspects of time series is the question of stationarity. A time series is stationary if the statistical properties of the time series are independent of its length T. If either the mean or the variance is a function of T, then the time series is nonstationary. A measure of long-range correlations that is valid for both stationary and nonstationary time series is the semivariance y. For a continuous time series the semivariance is given by

Note that neither the mean j nor the variance V is used in this definition. For a discontinuous time series we have

with increasing values of k corresponding to increasing lag. For a random stochastic time series the values of y, will approach the variance V. The plot of ys versus s or y, versus k is known as a semivariogram (Gallant et al., 1994). For a stationary time series the autocorrelation function r and the semivariance y are related by

An alternative measure of long-range correlations is rescaled-range (WS) analysis, which will be introduced after we consider some examples of time series. The classic example of a stationary, discontinuous time series is a white noise. Consider a random variable E~with a Gaussian (normal) distribution; : . If a time series is conthe distribution has zero mean and a variance VE= a

SELF-AFFINE FRACTALS

139

structed with a set of y, = E ~ i, = 1, 2, . . . , n, then adjacent values are uncorrelated and this is known as a white noise. Four examples of white noises are given in Figure 7.4(a). In each case V , = 1 and a different set of random numbers has been used. The classic example of a nonstationary time series is the

Figure 7.4. (a) Four examples of Gaussian white noises. Adjacent values are not correlated. (b) The four white noises in (a) are summed to give four Brownian walks. In each case the standard deviation after n steps, given by (7.13), is included.

140

SELF-AFFINE FRACTALS

Brownian walk. A Brownian walk is simply obtained by summing a Gaussian white-noise sequence; the values in a Brownian-walk time series yBw are given by

The white-noise sequences shown in Figure 7.4(a) have been summed to give the four Brownian walks illustrated in Figure 7.4(b). The variance of a Brownian walk after n values have been summed is given statistically by

where u,2 is the variance of the white-noise sequence. The corresponding standard deviation of the walk is given by

This result is compared with each of the four Brownian walks illustrated in Figure 7.4(b). A Brownian walk is an example of a statistical self-affine fractal. The association of white noises and Brownian walks is the basis for the kinetic theory of gases. The distribution of distances in a specified direction that a molecule in a gas travels between collisions is Gaussian. Thus the sequence of distances that a molecule travels in a gas is a Gaussian white noise. The sum of these distances, the distance the molecule diffuses in the gas, is a Brownian walk. This is the basic reason that the distance that a contaminant diffuses in a gas scales with the square root of time. Several empirical models have been developed to produce persistent (correlated) noises (Bras and Rodriguez-Iturbe, 1993). We first consider the moving average model (MA). In this model the discrete times series is given by

where E~ is again the random variable described above and the 0, (O,, 0,, . . . , Oq) are q prescribed coefficients relating yi to the q previous values of ei.The parameters in this model are the mean j, the variance of the white noise u,2, and O,, €I,.,. . , Oq.Taking q = 1 the MA model simplifies to

SELF-AFFINE FRACTALS

The mean for this correlated noise is j and its variance is

The autocorrelation function is

rk = - for 1 + 0; r, = 0

for

k = l

k >1

(7.18)

The correlation is very short since only the adjacent point has a non-zero autocorrelation function. Examples of this time series with 0, = 0,0.2, 0.5,0.9 are given in Figure 7.5. An alternative empirical model for a persistent (correlated) time series is the autoregressive model (AR). This time series is given by

+,

where ei is again the random variable and ( j= 1,2, . . . ,p) are prescribed coefficients relating y i to the p previous values of yi - y. Clearly the MA and AR models are closely related. In the MA model q previous values of the random variable ei are included, and in the AR model p previous values of the deviation of the time series from the mean y, - j are included. Taking p = 1 the AR model simplifies

L

The mean for this correlated noise is again and its variance is

The autocorrelation function is

The AR model has longer range correlations than the MA model, but the correlations remain short range. Examples of this time series with 4, = 0,0.2,0.5,

141

142

SELF-AFFINE FRACTALS

and 0.9 are given in Figure 7.6. There is clearly much greater smoothing of the time series in the AR than in the MA model. Correlograms for these time series are given in Figure 7.7. The agreement with (7.22) is excellent; the correlations increase systematically with increasing values of as expected.

+,

2 Yn 0

Figure 7.5. Examples of moving average (MA)time series from (7.15) with (a) 8, = 0, (b) 8, = 0.2, (c) 0 = 0.5,and (d) 0, = 0.9; in each casej=Oandu,2= 1 .

-2 -4

-6 0

128

256

n

(4

384

512

SELF-AFFINE FRACTALS

143

The autoregressive moving average model (ARMA) combines the two models given above. The ARMA time series can be written as

256

0

384

512

n 6 4

2 Yn

0 -2

-4 -6 0

128

256

n fd)

384

512

Figure 7.6. Examples of autoregressive (AR) time series from (7.20) with (a) 4, = 0, (b) 4, = 0.2, (c) 4, = 0.5, and (d) 4, = 0.9; in each case j = 0 and o: = 1 .

144

SELF-AFFINE FRACTALS

The variables in this model have been discussed above. With p = 0 the ARMA model reduces to the moving average (MA) model, with q = 0, the ARMA model reduces to the autoregressive (AR) model. Taking p = q = 1 the ARMA model simplifies to

Figure 7.7. Correlograms for the four AR time series given in Figure 7.6; excellent agreement with the theoretical prediction (7.22) is found.

SELF-AFFINE FRACTALS

The mean of this noise is again j and the variance is given by

The autocorrelation function is given by

The time-series models considered above are based on normal (Gaussian) statistics with both positive and negative values. Each of these time series can be converted into a skewed, positive only, time series by using the transformation from a normal (Gaussian) distribution to a log-normal distribution given in (3.30). Just as a stationary white noise can be summed to give a nonstationary Brownian walk, a stationary ARMA model can be summed to give a nonstationary autoregressive integrated moving average model (ARIMA).

7.3 Self-affine time series Although time series are defined to be sets of values as a function of a single variable, that is, y(t), time series concepts can be extended to two, three, and four dimensions. An example in two dimensions is topography h(x,y) as a function of the two horizontal coordinates x and y. Topography has a statistical distribution (the hypsometric curve), a mean value relative to sea level, and a variance. Topography clearly has horizontal persistence (i.e., adjacent values of topography are correlated). Consider the difference in elevation Ah between two points separated by a horizontal distance L. Self-similarity of topography implies that

where Ha is again the Hausdorff measure. Ahnert (1984) found that actual topography is in excellent agreement with (7.28), taking Ha = 0.635 0.105. Similar results were obtained by Dietler and Zhang (1992). Topography is an excellent example of a self-affine fractal.

+

145

146

SELF-AFFINE FRACTALS

An example of a "time series" in three dimensions is the surface of the ocean. The surface height h (x, y, t) is a function of the horizontal coordinates x and y but is also a function of time t. Another example would be mineral grade C(x, y, z) in an ore deposit as a function of position x, y, z. An example in four dimensions is the temperature in the atmosphere T (x, y, z, t) as a function of position coordinates x, y, z and time t. In terms of the time-series analysis introduced above, a time series is a self-affine fractal if its variance scales according to

where Ha is again the Hausdorff measure. Comparing (7.29) with (7.13) we find that Ha = for a Brownian walk. Thus the four Brownian walks given in Figure 7.2(b) are self-affine fractals. An equivalent definition of a self-affine fractal is the requirement that the semivariogram scales such that

Another expression for a self-affine fractal is

where F(y) is the cumulative distribution function of the basic distribution as defined in (3.10). The dependence of y on t in a Brownian walk is similar to the shape of the Koch island illustrated in Figure 2.5. There are variations in the time series at all scales. Neither the length of the track defined by the time series nor the local derivatives (slopes) are defined. Thus it is appropriate to consider y(() to be a fractal. Just as in the case of the triadic Koch island and other fractal constructions on a plane, the fractal dimension is between 1 and 2 and the appropriate Euclidean dimension is 2. With d = 2, (6.12) can be written Pr

;--D = constant where the appropriate scale for the time series is T. In (7.31) the time series diverges with the interval T according to the power law THO. Comparing (7.31) and (7.32) we define

SELF-AFFINE FRACTALS

This is the basic definition of the fractal dimension for a self-affine fractal. Below we give an alternative derivation that gives the same result. For the deterministic self-affine fractal illustrated in Figure 7.2 we found Ha = $ and D = in agreement with (7.33). For a Brownian walk we also have Ha = a n d ~ = i . ~ 1o
4

However, from (7.29) we have

so that we can write

and combining (7.34) and (7.36) gives

This is basically a fractal relation if we associate T,, with r,,. Comparing (7.37) with the definition of the fractal dimension in (2.1) we have the relation 2 - Ha = D, which is identical to (7.33).

147

148

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A standard approach to the analysis of a time series is to carry out a Fourier transform of the series. A time series can be prescribed either in the physical domain as y(t) or in the frequency domain in terms of the spectrum Y ( j T ) , where f is the frequency. The quantity Y ( j T ) is generally a complex number indicating both the amplitude and the phase of the signal. To obtain Y ( j T ) we use the Fourier transform of y(t) in the interval 0 < t < T; it is given by

The complementary equation relating y(t) to Y ( j 7') is the where i = fi. inverse Fourier transform

A time series with a single periodic component will have a single spike in its spectrum at that frequency. A time series with several components will have spikes in its spectrum at those frequencies. A white noise has no embedded frequencies and its spectrum is flat. The quantity I Y ( j T)12 df is the contribution to the total energy of y(t) from those components with frequencies between f and f + d$ The vertical bars in I YI refer to the absolute value of the complex quantity. The power is obtained by dividing by T. The power spectral density of y(t) is defined by

in the limit T + -. The product S( f)df is the power in the time series associated with the frequency range between f and f + d$ For a time series that is a self-affine fractal, the power spectral density has a power-law dependence on frequency:

We now obtain a relationship between the power P, the Hausdorff measure Ha, and the fractal dimension D. We consider two time series y , ( t ) and y2(t) that are related by

The fundamental property of a self-affine fractal time series is that y , ( t ) has the same statistical properties as y,(t). The Fourier transform of y,(t) is given by

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Substituting (7.42) and making the change of variable t' = rt, we obtain

and comparing (7.44) with (7.38) we have

From the definition of the power spectral density given in (7.40) we obtain

Since y2 is a properly rescaled version of y,, their power spectral densities must also be properly scaled. Thus we can write

From (7.41) and (7.33) it follows that p=2Ha+1=5-20 For a self-affine fractal (0 < Ha < 1, 1 < D < 2) we have 1 < P < 3. For a Brownian walk with Ha = $ (D = 23 ) we have P = 2.

7.4 Fractional Gaussian noises and fractional Brownian walks We next introduce the concept of fractional Gaussian noises. These will be generated synthetically from Gaussian white noises using the following steps: (1)

A Gaussian white noise sequence is generated. Four examples have been given in Figure 7.4(a).

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(2)

(3)

(4)

A discrete Fourier transform is taken of the random values. The Fourier coefficients are given by

This transform maps N real numbers (the y n ) into N complex numbers (the Ym) Because the transform is taken of a Gaussian white noise sequence the Fourier spectrum will be flat, that is p = 0 in (7.41). Except for the statistical scatter the amplitudes of the lYml will be equal. The resulting Fourier coefficients Ynlare filtered using the relation

The power PI2 is used because the power spectral density is proportional to the amplitude squared. The amplitudes of the small-m coefficients correspond to short wavelengths Am and large wave numbers km = 2n/Am. The large-m coefficients correspond to long wavelengths and small wave numbers. An inverse discrete Fourier transform is taken of the filtered Fourier coefficients. The sequence of points is given by

These points constitute the fractional Gaussian noise. To remove edge effects (periodicities)only the central portion should be retained. Several examples of fractional Gaussian noises are given in Figure 7.8. In each case the Gaussian white noise sequence with p = 0 has been filtered using the steps given above. Fractional Gaussian noises are given for P = - 1.0, -0.5, 0.5, and 1.0. Note that the range of ps corresponding to fractional Gaussian noises is - 1 < p < 1. Just as a Gaussian white noise (p = 0) can be summed to give a Brownian walk (P = 2), a fractional Gaussian noise can be summed to give fractional Brownian walks. In each case we have Pmw= 2 + P,". Fractional Brownian walks are self-affine fractals and are restricted to the range 1 < p < 3 as discussed above. The white and fractional Gaussian noises in Figure 7.8(a) with (3 = - 1.0, -0.5, 0, 0.5, and 1.0 have been summed using (7.1 1) to give the fractional Brownian walks illustrated in Figure 7.8(b) with P = 1.0, 1.5, 2.0, 2.5, and 3.0. Each fractional noise and walk given in Figure 7.8

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151

has been rescaled to have zero mean y = 0 and unit variance V = 1. The fractional Gaussian noise in Figure 7.8(a) with P = 1.0 is statistically identical to the fractional Brownian walk in Figure 7.8(b) with P = 1.O. As the value of p is increased the contribution of the short wavelength (large wave number) terms is reduced. The result is that adjacent values in

Figure 7.8. (a) The white Gaussian noise, p = 0, has been filtered to give fractional Gaussian noises with p = - 1.0, -0.5,0.5, and 1 .O. (b) Each of these fractional Gaussian noises with pGnhas been summed using (7.11) to give fractional Brownian walks with PfBu,= 2 + PfGn,P = 1.0, 1.5, 2.0,2.5, 3.0. The fractional Brownian walks are self-affine fractals with D = (5 - P,,")/2.

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the time series become increasingly correlated and profiles are smoothed. The persistence in the time series is increased. This is clearly illustrated in Figure 7.8 as P is increased from 0 to 3.0. With P = -0.5 and - 1.0 the shortwavelength contributions dominate over the long-wavelength contributions. These time series are antipersistent, and adjacent values are less correlated than for the random white noise (p = 0). An alternative method for the direct generation of fractional Brownian walks is the method of successive random additions (Voss 1985a, 1988). Consider the time interval 0 5 t I 1 as illustrated in Figure 7.9. Random values of y are generated based on the Gaussian probability distribution given in (3.15) with zero mean j = 0 and unit variance V, = 1. Three of these values are placed at t = 0, 1 as shown in Figure 7.9(a). Two straight lines are drawn between these three points. The midpoints of these two line segments are taken as initial values of y at t = $ and as illustrated in Figure 7.9(b). The five points are now given random additions. These random additions are also based on the Gaussian probability distribution (3.15) with zero mean 7= 0 but with a reduced variance given by (7.29). Since the interval has been reduced by a factor of two, the variance is given by V2 = ( f ) ~ . . For

i,

our example we take Ha = so that V2 = $.The five resulting random additions are given in Figure 7.9(c). After addition to the five values of y, in Figure 7.9(b), the resulting five values of y, are given in Figure 7.9(d). Again the five points are connected by four straight-line segments and the four midpoints are taken as initial values of y at t = Q, and g7 as illustrated in Figure 7.9(e). All nine points are now given random additions using a Gaussian probability distribution (3.15) with zero mean but a further reduced vari1 ance from (7.28) V3 = Again taking Ha = f we have V3 = 4. The nine random additions are given in Figure 7.9(f). After addition to the nine values of y2 given in Figure 7.9(e), the resulting nine values of y, are given in Figure 7.9(g). The process is repeated until the desired number of points is obtained. Our example with 4097 points is given in Figure 7.9(h). With Ha = and (3 = 2 this is a Brownian walk and strongly resembles the Brownian walks given in Figures 7.4 and 7.8. A sequence of fractional Brownian walks generated by the method of successive random additions is given in Figure 7.10. Fractional Brownian walks are given for Ha = 0 (P = l), Ha = 0.25 (P = 1.5)- Ha = 0.50 (P = 2), same as Figure 7.9(h), Ha = 0.75 (P = 2.5) and Ha = 1.00 ( p = 3); in each case 4097 points are given. As expected, these noises closely resemble those generated by the filtering technique given in Figure 7.8. A detailed comparison of fractional Gaussian noises and fractional Brownian walks using the Fourier filtering technique and the method of successive random additions has been given by Gallant et al. (1994). These authors also considered a third method using Weierstrass- Mandelbrot functions.

i,i,

i

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153

- . 0.0

(a)

Figure 7.9. Illustration of the generation of a fractional Brownian walk using the method of successive random additions. (a) Three random numbers are generated using a Gaussian distribution with zero mean and unit variance; these areplacedatt=O,f, 1. (b) Values at t = $ and are obtained by linear interpolation. (c) Assuming Ha = five random numbers are generated using a Gaussian distribution with zero mean and V = (: )2Ha = f . (d) The random numbers in (c) are added to the values in (b). (e) values at t = i,{, are obtained by linear interpolation. ( f ) Nine random numbers are generated using a Gaussian distribution with zero mean and v=(:)~H.=:. (g) The random numbers in (f ) are added to the values in (d). (h) The construction has been continued to 4097 points; the result is a Brownian walk.

5

4

-2 0.000

I (f) 0.125

0.250

0.375

0.500

t

0.625

0.750

0.875

1.000

i,

154

Figure 7.10. A sequence of fractional Brownian walks generated by the method of successive random additions. (a) H a = 0 (p = 1). (b) H a = 0.25 (P = 1.5). (c) H a = 0.5 (p = 2, a Brownian walk). (d) H a = 0.75 (P = 2.5). (e) H a = 1.OO (p = 3).

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155

Just as the fractional Gaussian noises generated using the filtering technique with - I < p I 1 can be summed to give fractional Brownian walks with 1 I p I 3, the fractional Brownian walks generated using the method of successive random additions with 1 < P < 3 can be differenced to give fractional Gaussian noises with - 1 6 p I 1. Extended fractional Gaussian noises with -3 I p I - 1 can be obtained by differencing the fractional Gaussian noises with - 1 I p < 1. Similarly extended fractional Brownian walks with 3 I p I 5 can be obtained by summing fractional Brownian walks with 1 I (3 I 3. Although the mathematical definition of self-affine fractals restricts the applicable range of P to 1 I P I 3, naturally occurring time series with a power-law dependence of the power spectral density on frequency have values of p outside this range. Just as naturally occurring self-similar powerlaw distributions may or may not fall within the range of D values prescribed by mathematical constraints, so too naturally occurring self-affine time series may or may not fall within the range of f3 values prescribed by mathematical constraints. Using the definition of the semivariance y, given in (7.9), semivariograms for several of the fractional Gaussian noises and fractional Brownian

1

10

100

Z

1000

10000

1

10

100

Z

1000

10000

Figure 7.11. Semivariograms for several of the fractional Gaussian noises and fractional Brownian walks illustrated in Figure 7.8, T = 4 , 8 , 16,. . . ,2048.The triangles given for P = 0, 1 are for the filtered fractional Gaussian noises from Figure 7.8(a). The circles given for p = I, 2, 3 are for the summed fractional Brownian walks given in Figure 7.8(b). The squares given for P = 3 are for the summed fractional Brownian walk with p = 1. The straight-line correlations are with (7.30).

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walks illustrated in Figure 7.8 are given in Figure 7.11. For the uncorrelated Gaussian white noise, p = 0, the semivariance scatters statistically about y = 1 as expected since V = 1. For P = 1, 2, and 3 excellent correlations are obtained with the fractal relation (7.30). For P = 2 we find H a = 0.47 compared to the expected value Ha = 0.50. The values of Ha obtained for the best fit of (7.30) to the semivariograms in the range - 1 I p 5 5 are given in Figure 7.12. The straight-line correlation is with the self-affine fractal relation (7.48). Quite good agreement is found in the range 1 < P < 3, where the fractional Brownian walks are expected to be self-affine fractals. From Figure 7.12 it is seen that Ha .= 0 for fractional Gaussian noises in the range - 1 < p < 1. From (7.29) and (7.35) we conclude that the variance V and standard deviation a are not dependent on the length of the signal T. Thus these fractional noises are stationary even though adjacent values may be correlated or anticorrelated. The fractional Brownian walks in the range 1 < p < 3 are clearly nonstationary from (7.29) and (7.35) since H a varies from 0 to 1 and V and a have a power-law dependence on the length of the signal T.

Figure 7.12. The dependence of the Hausdorff measure Ha on p as obtained from the best fit of (7.30) to semivariograms. The triangles given for - 1 5 p < 1 are for the filtered fractional Gaussian noises given in Figure 7.8(a). The circles given for 1 5 p 5 3 are for the summed fractional Brownian walks given in Figure 7.8(b). The squares given for 3 5 p < 5 are for the extended fractional walks obtained by summing the fractional Brownian walks with 1 I P I 3. The straight-line correlation is with the selfaffine fractal relation (7.48) for 1 5 p S 3 .

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157

7.5 Fractional log-normal noises and walks

The fractional Gaussian noises and fractional Brownian walks we have considered have both been based on a Gaussian distribution of values. Thus the resulting time series have both positive and negative values. Many naturally occurring time series have only positive values. For example, the volumetric flow in a river Q ( t ) is always positive. Another example is the density or porosity in a well, which is also always positive. The coefficient of variation cVis the ratio of the standard deviation of the signal to its mean (3.33). If

(a) c , = 0.2

0

128

256 t 384

512

0

128

256 t 384

512

(b) C , = 0.5

(c) c , = 1.0

(d) c , = 2.0

Figure 7.13. Examples of fractional log-normal noises and walks. In each case a fractional Gaussian noise or a fractional Brownian walk has been converted to a fractional log-normal noise or walk, examples are given for p = 0, 1.2. The conversions were made using (3.31) to (3.33). Examples are given for: (a) ~,,=0.2,(b)c,,=O.5,(c)c~= 1 .O, and (d) cV= 2.0. In all cases the mean of the time series is unity f = 1.

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cV< < 1 it may be appropriate to consider Gaussian statistics. In many cases, however, this will be a poor approximation. The most widely used positive distribution is the log-normal distribution discussed in Chapter 3. A normal distribution can be converted to a log-norma1 distribution using the relation y = In x where x is a log-normal distribution and y is a normal distribution as given in (3.30). However, a log-normal distribution can be specified only if its coefficient of variation is given as defined in (3.33). A Gaussian white-noise sequence can then be converted to a log-normal, white-noise sequence using (3.31) and (3.32). Log-normal white-noise sequences, p = 0, with unit mean, 2 = 1, are given in Figure 7.13 for cv = 0.2, 0.5, 1.0, and 2.0. With cv = 0.2 the standard deviation is small compared with the mean, the distribution is nearly symmetric, and it closely resembles a Gaussian white noise. With cv = 2 the variance is large compared with the mean and the distribution is strongly asymmetrical. Just as a Gaussian white-noise sequence can be converted to a log-normal, white-noise sequence, so too can fractional Gaussian noises and fractional Brownian walks be converted to fractional log-normal noises and walks using (3.31) to (3.33). Several examples are given in Figure 7.13. In each case the mean is unity i = I . This is a two-parameter family of noises and walks. The values of (3 are a measure of the persistence of the time series. The values of cv are a measure of the skew of the distribution of values. Extensive studies of fractional log-normal noises and walks have been given by Mandelbrot and Wallis (1968). These authors referred to the dependence on cV as the Noah effect and the dependence on P as the Joseph effect. The time series in Figure 7.13 resemble typical river flow time series. Increasing cV,the Noah effect, is indicative of a climate where there is large variability in river flow. Increasing P, the Joseph effect, is indicative of more strongly correlated values. With higher values of P a year of flood is more likely to follow a previous year of flood, and a year of drought is more likely to follow a previous year of drought.

7.6 Resealed-range (R/S)analysis An alternative approach to the quantification of correlations in time series was developed by Henry Hurst (Hurst et al., 1965). Hurst spent his life studying the hydrology of the Nile River, in particular the record of floods and droughts. He considered a river flow as a time series and determined the storage limits in an idealized reservoir. On the basis of these studies he introduced empirically the concept of rescaled-range ( R B ) analysis. His method is illustrated in Figure 7.14. Consider a reservoir behind a dam that never overflows or empties; the flow into the reservoir is the flow in the river up-

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159

stream of the dam Q(t). The flow out of the reservoir Q(T) is assumed to be the mean of the flow into the reservoir for a period T

The volume of water in the reservoir as a function of time V(t) is given by V(t) = V(0)

+

1

Q(tf)dt'

-

tQ

(7.53)

where V(0) is the volume of water at t = 0. Taking t = T and substituting (7.52) into (7.53), we have V(T) = V(0). The range R(T) is defined to be the difference between the maximum volume of water Vmaxand the minimum volume of water Vminduring the period T

The rescaled range is defined to be R/S where S is the standard deviation of the flow Q(t) during the period T

This is identical to the definition of a introduced in (3.3). We use S here to maintain the standard R/S nomenclature. Hurst et al. (1965) found empirically that many data sets in nature satisfy the power-law relation

Figure 7.14. Illustration of how resealed-range (RB) analysis is camed out. The flow into a reservoir is Q(t) and the flow out is The maximum volume of water in the reservoir during the period T is Vmax( T ) and the minimum Vmln ( T ) ;the difference is the range R (7') = Vmax( T ) - Vmin (TI.

a

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where Hu is known as the Hurst exponent. Examples included river discharges and lake levels, thicknesses of tree rings and vanes, atmospheric temperature and pressure, and sunspot numbers. They generally found that 0.70 < Hu < 0.80. The WS analysis is easily extended to discrete time series. The running sum of the series relative to its mean is

The range is defined by

with

and and a, are obtained from (3.1) and (3.3). The Hurst exponent, Hu,is obtained from

For example, if 64 values of yi (i = 1, 2, . . . , 64) are available for a time series, the R, and S, for N = 64 are obtained. Then the data are broken into two parts (i = 1, 2, . . . , 32 and i = 33, 34, . . . , 64) and values for R,, and S,, are obtained for the two parts. The two values of R,,IS,, are then averaged to give (R,,IS,,),,. The data set is then broken into four parts (i = 1, 2 , . . . , l 6 ; i = l 7 , 1 8 , . . . , 3 2 ; i = 3 3 , 3 4 , . . . , 4 8 ; a n d i = 4 9 , 5 0 , . . . ,64)the values of R,gS,, are obtained for the four parts and are averaged to give (RIJSl,)aV.This process is continued for N = 8 and N = 4 to give (R,ISJaV and (R41S4)aV. For N = 2 we have R, = S, so that R21S2= 1. The values of log (R,IS,),, are plotted against log (Nl2) and the best-fit straight line gives Hu from (7.60). In practice there is generally some curvature of values for small N and the values of (R,IS,),, for small values of N are omitted. The running sum of a Gaussian white noise with P = 0 is a Brownian walk with P = 2 and Ha = 0.5. This would imply that

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161

And from (7.48) that

Since a white noise p = 0 is a random process it would be appropriate to conclude that Hu = 0.5 implies randomness. It follows that 0.5 < Hu < 1.0 implies persistence and that 0 < Hu < 0.5 implies antipersistence. The Hurst resealed-range analysis was first applied to Gaussian fractional noises and fractional Brownian walks by Mandelbrot and Wallis (1969a). The on log 7 (log N) for several of the fractional noises dependence of log (WS)," and walks illustrated in Figure 7.8 are given in Figure 7.15. For P = 0, 1, and 2 excellent correlations with the Hurst relation (7.60) are obtained. For P = 0 we find H u = 0.56 compared with the expected value of 0.5 for the random white Gaussian noise.

Figure 7.15. Hurst WS analyses for several of the fractional Gaussian noises and fractional Brownian walks illustrated in Figure 7.8. Average va'lues of WS are given as a function of the interval N for N = 4, 8, 16, . . . ,4096. The triangles given for p = - 1,0, 1 are for the filtered fractional Gaussian noises from Figure 7.8(a). The diamonds given for p = - 1 are for the extended fractional Gaussian noise obtained by differencing the filtered fractional Gaussian noises with p = 1. The circles given for p = 1.2areforthe summed fractional Brownian walks given in Figure 7.8(b). The straight-line correlation is with (7.60).

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The values of Hu obtained for the best fit to the Hurst relation (7.60) in the range -3 I 5 3 are given in Figure 7.16. The straight-line correlation is with (7.61). Reasonably good agreement is found in the range - 1 < P < 1. The Hurst exponent provides a quantitative measure of persistence and antipersistence for fractional Gaussian noises. Extensive R/S analyses of fractional Gaussian noises and fractional Brownian walks have been carried out by Bassingthwaighte and Raymond (1994). Several closely related techniques have been introduced to quantify the self-affine properties of observed time series. Malinverno (1990) introduce the roughness-length method. In this method the local trend is determined as a function of window length. Ivanov (1994a, b) introduced counterscaling. Two types of counterscaling were considered. In the first the variance was determined for different window lengths. In the second the means were obtained for various window lengths and the variances of these means were obtained as a function of window length. Gomes Da Silva and Turcotte (1994) have applied the counterscaling technique to fractional noises and walks.

Figure 7.16. The dependence of the Hurst exponent Hu on p as obtained for the best-fit correlations with the Hurst relation (7.60). The triangles given for - 1 I p I 1 are for the filtered fractional Gaussian noises given in Figure 7.8(a). The circles given for 1 < fiI 3 are for the summed fractional Brownian walks given in Figure 7.8(b). The diamonds given for -3 I p S1 are for the extended fractional Gaussian noises obtained by differencing the filtered fractional Gaussian noises with - 1 I p I l . T h e straight-line correlation is with (7.61).

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7.7 Applications of self-affine fractals

Three examples of elevation along linear tracks were given in Figure 7.1. These are equivalent to time series and are examples of naturally occurring self-affine fractals. Many authors have carried out Fourier spectral analyses of topography and bathymetry along linear tracks (Bell, 1975, 1979; Berkson and Mathews, 1983; Barenblatt et al., 1984; Fox and Hayes, 1985; Gilbert and Malinverno, 1988; Fox, 1989; Gilbert, 1989; Malinverno, 1989, 1995; Mareschal, 1989). In general a power-law dependence of the power spectral density on wave number was found with P = 2 (D = 1.5). Similar results were obtained for Venus by Kucinskas et al. (1992). Thus the elevation of topography is approximately a Brownian walk. Twenty-four examples of the dependence of the power spectral density on wave number for linear topographic profiles from three different parts of Oregon are given in Figure 7.17. One-dimensional Fourier spectral analyses were obtained using

Figure 7.17. Plots of onedimensional power spectral density versus wave number selected from three regions with different tectonic and geomorphic settings in Oregon. The profiles are offset vertically so as not to overlap. (a) Willamette lowland, (b) Wallowa Mountains, (c) Klamath Falls.

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the periodogram method. Three different regions were considered with different geomorphic and tectonic settings. The Willamette lowland is dominated by sedimentary processes, the Wallowa Mountains are associated with a major tectonic uplift, and the Klamath Falls area belongs to the basin and range tectonic regime. The topography was digitized along lines of latitude and longitude at seven points per kilometer. For each of the three regions, 20 equally spaced one-dimensional profiles of length 5 12 points were analyzed in both the latitudinal and longitudinal directions. Log-log plots of the spectral power density versus wave number show a good power-law dependence in all three regions, as shown in Figure 7.17. Eight typical examples are given for each of the three regions. The best-fit fractal dimension for each profile is obtained using (7.41). The mean fractal dimensions for three regions are given in Table 7.1. The mean values are close to D = 1.5, indicating that the spectral power density corresponds to a Brownian walk to a good approximation. A variety of previous studies have found values for D near 1.5.

Power Spectra track 8 track 7 track 6 track 5

track I

Klarnath

Figure 17. (cont.)

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Two implications of this result will be discussed. The first is the comparison with the value of D for topography obtained in Chapter 2 using the ruler method. As illustrated in Figure 2.7 the ruler method generally gives fractal dimensions near D = 1.2. These are systematically lower than the values near D = 1.5 obtained using the spectral method. Fundamentally there is no reason why the two fractal dimensions should be equal. Elevation profiles are not necessarily related to the shape of contours. The correspondence of topography and bathymetry to a Brownian walk also implies, importantly, that they are truly self-similar. For a Brownian walk the amplitude coefficients are directly proportional to the corresponding wavelengths. Thus the height-to-width ratios of mountains and hills are the same at all scales. It should also be noted that the power-law spectra given in Figure 7.17 provide further information beyond the fractal dimension. The spectra are characterized by the amplitude in addition to the slope. A quantitative measure of the amplitude is the intercept (value of S) at a specified wave number (k = 1 cycle km-1).These reference amplitudes are a measure of the roughness of the topography. The mean intercepts for the latitudinal and longitudinal directions for the three regions in Oregon are given in Table 7.1. Another application of spectral techniques is to well logs. It is common practice to make a variety of measurements as a function of depth in oil wells. Typical measurements include the local acoustic velocity, the electrical conductivity, and neutron activation. The measured quantities are obtained as a function of depth so that they are equivalent to a time series and spectral techniques can be applied.

Table 7.1. Regional averages over one-dimensional profiles of Oregon topography Area Willarnette lowland Latitude Longitude Wallowa Mountains Latitude Longitude Klamath Falls Latitude Longitude

Fractal Dimension

Roughness

1.436 1SO7

5.948 6.354

1.499 1.485

6.549 6.830

1.492 1SO0

5.825 5.963

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The power spectral densities obtained from porosity logs for eight wells in the Gulf of Mexico are given in Figure 7.18 (Pelletier and Turcotte, 1996). At spatial scales greater than 10 ft a good correlation is obtained with the fractal relation (7.41) taking P in the range 1.31-1.6. Below this scale the variability decreases significantly in most wells. This may be attributed to increased homogeneity within beds. Todoeschuck et al. (1990) have also considered the fractal behavior of well logs. Leary and Abercrombie (1994) attribute the shear-wave source and coda-wave displacement spectra obtained from seismograms in the Cajon Pass borehole to scattering that was observed to obey power-law spectra from well logs. There are many other examples of measurements in geology and geophysics that yield power-law spectra. Brown and Scholz (1985) have carried out spectral studies of natural rock surfaces. They generally find fractal behavior with a relatively large range of variability between 1 < D < 1.6. Similar studies have been carried out by Power and Tullis (1991, 1995) and by Pyrak-Nolte et al. (1995). The observation that fracture surfaces are selfaffine fractals can be used to scale the fluid permeability associated with fractures. An interesting question is whether climate obeys fractal statistics (Nicolis and Nicolis, 1984). Fluigeman and Snow (1989) have shown that the spa-

Figure 7.18. Dependence of the spectral power density on wave number for porosity logs from eight wells in the Gulf of Mexico (Pelletier and Turcotte, 1996). The straight-line correlations are with (7.41).

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tial distribution of oxygen isotope ratios in sea floor cores obey fractal spectral statistics. Since it is generally accepted that the isotope ratios are proportional to the local temperatures, these results can be taken as evidence that climate obeys fractal statistics. Hsui et al. (1993) have shown that variations of sea level with time are a self-affine fractal. This is consistent with the fractal distribution of sedimentary hiatuses discussed in Chapter 2. An important application of power-law spectra is in interpolating between measured data sets. Consider the bathymetry of the oceans. Bathymetry is typically measured from ships along linear tracks and must be interpolated to make bathymetric charts. This interpolation can make use of the fact that the bathymetry has a power-law spectrum. The amplitude coefficients are determined from the applicable fractal relation and the data are used to determine the phases in a two-dimensional Fourier expansion of the bathymetry. This method can also be used to interpolate airborne magnetic surveys. Hewett (1986) has used fractal techniques to interpolate well-log porosity data from production wells to obtain the full three-dimensional porosity distribution in an oil field. The horizontal variations in porosity are treated as a Brownian walk in analogy to the fractal behavior of topography, and the fractal behavior of the vertical variations are obtained directly from the well logs. Molz and Boman (1993) have used this technique to interpolate welllog data to predict the ground water movement and pollutant dispersion adjacent to a waste disposal site. In some cases a time series x(t) will have a well-defined correlation dimension (Grassberger and Procaccia, 1983a, b). A vector for the time series at t = t, is defined by the quantities x(t,), x(t, + T), x (t, + 27), . . . , x (t, + n ~ )At . a later time t = t, another vector is defined by the quantities x (t,), x (t, + T),x (t, + 27), . . . ,x (t2 + "7). As long as the signals at t, and t, are uncorrelated, the delay T can be small. This process is known as forming ntuples and n is the embedding dimension. The cumulative number of pairs of points N separated by a distance less than r is plotted against r for embedding dimensions n = 2, 3, 4, . . . . If a straight-line correlation is obtained such that N (r, n) rd and if d becomes independent of n, for all values of d greater than dc,then dc is the correlation dimension. Smith and Shaw (1990) have applied this technique to sea-floor bathymetry. Cortini and Barton (1993) analyzed the inflation-deflation patterns of an active volcanic caldera (Campi Flegrei, Italy) as a self-affine time series and were able to make successful forward predictions. Nicholl et al. (1994) have applied this approach to the sequence of intervals between eruptions of Old Faithful geyser in Yellowstone National Park, Wyoming. They conclude that the sequence of eruption intervals is a chaotic time series. Osbourne and Provenzale (1989) have obtained values of the correlation dimension for fractional Brownian walks. These authors find a systematic dependence with dc = 1.140 0.005

-

+

167

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for p = 3, d c = 2.053 + 0.009 for f3 = 2, dc = 3.749 5 0.015 for P = 1.5, and dc> 10forp=1. We now turn to two-dimensional spectral studies. Of particular interest is the earth's topography and bathymetry. It is common practice to expand data sets on the surface of the earth in terms of spherical harmonics; examples include topography and geoid. Using topography as an example the appropriate expansion for the radius r of the earth is

+

where a, is a reference earth radius, 8 is latitude, is longitude, Cc, and Cs, are coefficients, and P , are associated Legendre functions fully normalized so that

The variance of the spectra for order 1 is defined by

and the power spectral density is defined by

where ko is the wave number and A, = Ilko is the wavelength over which data are included in the expansion. With A, = 2naOwe have

A fractal dependence can be defined if S, has a power-law relation to the wave number k,. The power spectral density of the earth's topography and bathymetry as a function of wave number is given in Figure 7.19. This is based on the

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169

Figure 7.19. Power spectral density S as a function of wave number k for spherical harmonic expansions of topography (degree I ) for (a) the earth (Rapp, 1989) and (b) Venus (Kucinskas and Turcotte, 1994). Correlations are with (7.41) taking P = 2.

170

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spherical harmonic expansion to order 1 = 180 given by Rapp (1989). Except for the low-degree harmonics an excellent correlation is obtained with (7.41) taking P = 2 (D = 1.5). The spectral dependence of topography corresponds to a Brownian walk as previously noted. The power spectral density of topography on Venus (Kucinskas and Turcotte, 1994) is given in Figure 7.19(b), and again a good correlation with a Brownian walk (P = 2) is obtained. The spectral power density on Venus is about an order of magnitude less than on the earth for the same wave number. This is attributed to the higher surface temperature and, thus, weaker lithosphere on Venus. Nevertheless, it is somewhat surprising that topography on both the earth and Venus are Brownian walks since erosion and deposition are dominant in the evolution of many landforms on the earth, whereas these processes are essentially absent on Venus so that tectonic processes are dominant. This suggests that the tectonic processes that build topography and the erosional and depositional processes that destroy topography both give Brownian walk statistics. Once again the fractal dimension of topography does not appear to be diagnostic. The power spectral density of the earth's geoid is given as a function of wave number in Figure 7.20; this was compiled by Turcotte (1987) from the data given by Reigber et al. (1985). Except for the low-degree harmonics an

lo8

10' Sg m2

cycles km

lo6

1 o5 Figure 7.20. Power spectral density of the earth's geoid as a function of wave number. The circles represent a compilation of the data of Reigber et al. (1985). The solid line represents (7.41) with P = 3.5 (D = 0.75).

1o4

103

J

1

1

1

1

1 o4

1

.

I

1

cycles

I

I

,

,

,

1o

-~

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171

excellent correlation is obtained with (7.41) taking P = 3.5 (D = 0.75). This is another example that falls outside the range of fractional Brownian walks and the formal limits for fractal behavior. The Fourier spectral approach to fractal analysis for one-dimensional profiles discussed previously can be extended to two-dimensional image analysis (Dubuc et al., 1989b). Consider an N X N grid of equally spaced data points in a square with linear size L. The W data points are denoted by hrImwith (n,m)specifying the position in the x- and y-directions respectively. A case with N = 8 is illustrated in Figure 7.2 1. The first step is to carry out a two-dimensional discrete Fourier transform on the W set of data points hnm.An N X N array of complex coefficients HF,is obtained by the usual definition

m

18

28

38

48

58

68

78

88

17

27

37

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86

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n (a) The 64 nm coefficients for an 8 x 8 sub-set of raw data. 8

8

8

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7

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7

8

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S

( b ) Equivalent radial coefficients r for various coefficients s and

r in spatial frequency space.

Figure 7.21. Illustration of subscript arrangement in two-dimensional spectral analysis.

172

SELF-AFFINE FRACTALS

where s denotes the transform in the x-direction (s = 0, 1,2, . . . ,N - 1) and t denotes the transform in the y-direction (t = 0, 1, 2, . . . ,N - 1). Then each transform coefficient Hstis assigned an equivalent radial number using the relation

The two-dimensional mean power spectral density S2j for each radial wave number k, is given by

where N,is the number of coefficients that satisfy the condition j < r <j + 1 and the summation is carried out over the coefficients Hs, in this range. The coefficients assigned to each interval for the example given in Figure 7.21(a) are illustrated in Figure 7.21(b). The dependence of the mean power spectral density on the radial wave number k, for a fractal distribution is (Voss, 1988)

instead of (7.41). The addition of minus one to the power is required because of the radial coordinates that are used in phase space. The dependence of V(L) on L given in (7.29) is still valid but with the additional dimension the "box" derivation that follows now gives

for the fractal dimension of the surface instead of (7.33). Similarly, the derivation of the relationship between P and Ha must be reexamined but

remains valid. Combining (7.73) and (7.74) gives

for the two-dimensional case.

SELF-AFFINE FRACTALS

Synthetic images can be generated using the same technique used to generate a synthetic fractional Gaussian noise. The method used to generate images is as follows. We consider an N X N square grid consisting of NZ equally spaced points. Each point is given a random value hmnbased on the Gaussian probability distribution defined in (3.15). A typical example is illustrated in Figure 7.22a. This is Gaussian white noise so that adjacent points are totally uncorrelated and P = 0. A two-dimensional discrete Fourier transform is taken using (7.69) generating an N X N array of complex coefficients Hs,. A fractal dimension D, is specified and the corresponding value for p is obtained from (7.75). A new set of complex coefficients are obtained from the relation

An inverse two-dimensional discrete Fourier transform is carried out to generate a new image. Five examples of synthetically generated images are given in Figure 7.22. In (b) P = 1.2 (D2=2.9), in (c) P = 1.6 (D,= 2.7), in (d) P =2.0(D2= 2.5). in (e) P = 2.4 (D, = 2.3), and in (f) P = 2.8 (D, = 2.1). The synthetic result for D, = 2.5 looks quite realistic for a typical topographic map. We have carried out one-dimensional spectral decompositions of linear profiles of our synthetic data. The results for synthetic topography with D, = 2.6,2.7 and 3.0 are given in Table 7.2. For realistic topography with D, = 2.6 we find that the corresponding one-dimensional profiles give D, = 1.58. This is consistent with the previously published results for one-dimensional bathymetric and topographic profiles where values near D = 1.5 have been

Table 7.2. Summary of mean fractal dimensions estimated by onedimensional and two-dimensional spectral analysis for the topography of Oregon and for synthetic images Average D Data Oregon topography Synthetic topography

Two-dimensional analysis

One-dimensional analysis

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Figure 7.22. Synthetic fractal images on a 256 X 256 grid. (a) White noise without fractal filtering. (b) Filtered white noise with f3 = 1.2 and D = 2.9. (c) Filtered white noise with p= 1.6 and D = 2.7. (d) Filtered with p = 2.0 and D = 2.5. (e) Filtered with p=2.4andD=2.3 (f) Filtered with p = 2.8 and D = 2.1.

At the time of going to press a color version of this figure was available for download from http:llwww.cambridge.org,978052 1567336

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175

found as discussed above. In general the relation D,= D, + 1 is a good approximation. The results given above can be extended to carry out fractal mapping of digitized images. As a specific example we consider the digitized topography of the state of Oregon (Huang and Turcotte, 1990b). Combining Defense Mapping Agency (DMA) 1" X 1" data with topographic maps, the U.S. Geological Survey (Flagstaff) has produced digitized topography on a grid scale of about seven points per kilometer. The topography for Oregon is illustrated in Figure 7.23. The fact that fractal statistics are a good approximation for topography allows us to make fractal maps of a region of diverse tectonics. Using the digitized topography of Oregon, plots of power spectral density versus wave number are made for subregions. From these plots a fractal dimension (slope) and unit wave number amplitude are obtained for each subregion. The amplitude is a measure of roughness. We are basically carrying out a texture analysis using the fractal statistics as a basis.

Figure 7.23. Map of the digitized topography for Oregon. Data resolution is about seven points per kilometer. The width of the state of Oregon is about 375 miles.

At the time of going to press a color version of this figure was available far download from http://www.cambridge.0'g/978052l567336

176

SELF-AFFINE FRACTALS

Fractal dimensions and roughness amplitudes are obtained using suhregions of 32 X 32 data points. Thus fractal dimensions and roughness amplitudes are obtained for each 4.5 km X 4.5 km subregion in the state; maps are generated. The 32 X 32 set was chosen because it generally gives welldefined fractal spectra; for smaller regions the errors in fractal dimension and roughness become substantially larger. For larger regions, the spatial resolution of the map is degraded. The following technique is used to obtain a fractal dimension and roughness amplitude for each subregion. A 32 X 32 set of digitized elevations is chosen to form each subregion (N = 32). The mean and linear trends for each subset of data are removed. A two-dimensional discrete Fourier transform is carried out, and an N X N array of complex Fourier coefficients H,,is obtained using (7.69). is assigned an equivalent radial wave number r Each coefficient Hs, using (7.70). The two-dimensional mean spectra energy density SZj is obtained for each radial integer wave number k, using (7.72). The mean slope on a log-log plot of +j versus kj obtained by a leastsquares regression yields a fractal dmension D, using (7.72) and

Figure 7.24. Plots of mean power spectral density versus radial wave number for four typical 32 x 32 point subregions in Oregon. The linear trend on the Log-log plot indicates a power-law (fractal) distribution.

-0.-

-w.s

V.V

V.L

v.-

J.6

log(k, krn-l) At the time of going to press a color version of this figure was available for download fmm http:l/www.cambridge.org/9780521567336

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177

(7.75); the intercept at kj = 1 cycle kmm' yields a roughness amplitude. Examples for four randomly selected subregions in Oregon are given in Figure 7.24. The mean two-dimensional fractal dimension for all of Oregon is D, = 2.586 (Table 7.2). This is remarkably close to the mean value D, = 2.59 that was obtained for the state of Arizona (Huang and Turcotte, 1989). It is seen that the results are in good agreement with the relation D,,= 1 D. Maps of fractal dimension and roughness amplitude are given in Figure 7.25. As expected there is relatively little variation in the fractal dimension about the mean value, although the range is from about 2.4 < D, < 2.9. The variation in the roughness amplitude in Figure 7.25(b) is much more impressive. The sedimentary Willamette lowland shows low overall roughness while the erosion system associated with the nearby mountain ranges and the Wallowa Mountains in the northeast stand out as regions of high roughness.

+

Figure 7.25. Maps of (a) fractal dimension and (b) roughness amplitude for Oregon. There is generally limited systematic variation in the fractal dimension: however, the roughness amplitude is sensitive to texture changes.

At the time of going to press a color version of this figure was available for download from hftp://ww.cambridgeeorg/s78O52I567336

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SELF-AFFINE FRACTALS

The roughness contrasts in the southern basin and range region are also quite remarkable. The fractal analysis gives a quantitative measure of roughness. In this chapter we have shown that we typically have D, = 1.5 and D, = 2.5 for self-affine topography. In Chapter 2 we found that the self-similar fractal dimension of topography is near Dm = 1.25. Kondev and Henley (1995) have generated synthetic two-dimensional topography and have studied the self-similar fractal behavior of topographic contours. They argue that

Thus for two-dimensional Brown topography with D2 = 2.5, they would obtain Dss = 1.25, in good agreement with many observations.

Figure 25. (conr.)

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179

Problems

Problem 7.1. Consider the deterministic first-order self-affine fractal construction illustrated in Figure 7.26(a). The vertical scale is divided into three equal parts and the horizontal scale is divided into nine equal parts. (a) Extend the construction to second order. (b) How many boxes with dimensions hd9 by rd9 are required to cover the first-order construction? What is the corresponding fractal dimension? Problem 7.2. Consider the deterministic first-order self-affine fractal construction illustrated in Figure 7.26(b). The vertical scale is divided into three equal parts and the horizontal scale is divided into five equal parts. (a) Extend the construction to second order. (b) How many boxes with

Figure 7.26. Four examples of deterministic self-affine fractal constructions.

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dimensions hd5 by rd5 are required to cover the first-order construction? What is the corresponding fractal dimension? Problem 7.3. Consider the deterministic first-order self-affine fractal construction illustrated in Figure 7.26(c). The vertical scale is divided into three equal parts and the horizontal scale is divided into seven equal parts. (a) Extend the construction to second order. (b) How many boxes with dimensions hd7 by rd7 are required to cover the first-order construction? What is the corresponding fractal dimension? Problem 7.4. Consider the deterministic first-order self-affine fractal construction illustrated in Figure 7.26(d). The vertical scale is divided into three equal parts and the horizontal scale is divided into six equal parts. (a) Extend the construction to second order. (b) How many boxes with dimensions hd6 by rd6 are required to cover the first-order construction? What is the corresponding fractal dimension? Problem 7.5. Consider the 16-year record of annual rainfall totals for Miami, FL, given in Table 7.3. (a) Determine the mean, variance, standard deviation, and coefficient of variation for these values. (b) Determine (RNISN)aV for N = 4, 8, and 16. Problem 7.6. Consider the 16-year record of annual rainfall totals for New York, NY, given in Table 7.3. (a) Determine the mean, variance, standard deviation, and coefficient of variation for these values. (b) Determine (RNISN),V for N = 4, 8, and 16.

Table 7.3. Annual rainfall totals in millimeterslyear for the 16 years between 1969 and 1984 for Miami, New York, Seattle, and Phoenix Miami

New York

Seattle

Phoenix

SELF-AFFINE FRACTALS

Problem 7.7. Consider the 16-year record of annual rainfall totals for Phoenix, AR, given in Table 7.3. (a) Determine the mean, variance, standard deviation, and coefficient of variation for these values. (b) Determine (RNISN),,for N = 4,8, and 16. Problem 7.8. Consider the 16-year record of annual rainfall totals for Seattle, WA, given in Table 7.3. (a) Determine the mean variance, standard deviation, and coefficient of variation for these values. (b) Determine (RNISN)aV for N = 4, 8, and 16. Problem 7.9. Derive (7.16). It is appropriate to assume E,E,-,

--x 1"

&Zi

n .i Z.l

= uzand

-x, 1 "

n,,.

-

= 0 since adjacent values of ei are uncorrelated.

Problem 7.10. Derive (7.17) and (7.18), use the relations given in Problem 7.9. Problem 7.1 1. Obtain an expression for the semivariance y, for the MA model given in (7.15). Problem 7.12. The following set of random numbers have a Gaussian distribution with zero mean: -0.4287, -0.0541,0.6224, -0.9545, -0.3745, 0.0455, -1.0512, 0.3431, 0.1318, -0.6346, 0.4436, 0.3743, 0.4589, 1.3667, -0.403 1,O.1154. Use (7.15) to determine a MA time series with 0 = 0.5, j = 0, using these random numbers. Determine the variance of the time series and compare the result with the predicted value from (7.16). 1" Problem 7.13. Derive (7.21). It is appropriate to assume that -CE~(Y;-, ni,l j ) = 0 and

-x 1"

(y,- 1 - j)2 = 02 as well as the relations given in ni,l Problem 7.9. Problem 7.14. Obtain an expression for the semivariance y, for the AR model given in (7.20). Problem 7.15. Using the set of random numbers given in Problem 7.13, determine an AR time series from (7.20) with = 0.5 and j = 0. Determine the variance of the time series and compare the result with the predicted value from (7.21). Determine the auto correlation function for k = 1 , 2 and compare the results with the predicted values from (7.22). Problem 7.16. Consider the 16 random numbers given in Problem 7.13. Determine (RNISJaVfor N = 4, 8, and 16. Determine the best fit value of Hu from (7.60). Problem 7.17. The definition of red noise is P = 1. What is the fractal dimension? How do the variance V and standard deviation a depend upon the interval T? Is red noise an example of a fractional Brownian walk; if so, why? Problem 7.18. Determine the aspect ratio (height-to-width ratio of the mountains and valleys) using the correlation line from Figure 7.18. From this

+,

181

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case the spectral power density is defined as the amplitude coefficient squared times the circumference of the earth. Problem 7.19. Determine the ratio of the height of topography on Venus to earth using the data in Figure 7.18. Note that the spectral power density is defined as the product of the square of the amplitude and the circumference of the body. Problem 7.20. In W S analysis for N = 2 show that R, = S,.

Chapter Eight

GEOMORPHOLOGY

8.1 Drainage networks

In the previous chapters we concluded that landscapes generally obey fractal statistics. Analyses of shore lines and topographic contours using ruler or box-counting methods provide statistical correlations with the self-similar fractal relation (2.1). Spectral studies of topography and bathymetry correlate well with the self-affine fractal relation (7.41). We will show in this chapter that drainage networks are fractal trees. Landforms evolve as a result of the tectonic processes that produce them and the erosional processes that destroy them. Landforms are a classic example of a complex phenomenon that can be quantified using fractal concepts (Turcotte 1993, 1994b, 1995). We will begin our study of geomorphic processes by considering drainage networks. Drainage networks are a universal feature of landscapes on the earth. Over a large fraction of land areas, water eventually flows into an ocean. The surface area that drains into an ocean through a river defines the drainage basin of that river. The streams and rivers that drain into the Mississippi River and eventually drain into the Gulf of Mexico define the Mississippi River basin. Small streams merge to form larger streams, large streams merge to form rivers, and so forth. A typical example of a drainage network is given in Figure 8.1, from the Volfe and Bell Canyons in the San Gabriel Mountains near Glendora, California. We will show that drainage networks are classic examples of fractal trees (Tarboton et al., 1988; Beer and Borgas, 1993; Garcia-Ruiz and Otalora, 1992). Long before the concept of either fractals or fractal trees was introduced, a quantitative stream-ordering system was introduced by Horton (1945) and Strahler (1957). The Strahler (1957) ordering system is illustrated in Figure 8.2. Streams on a standard topographic map with no upstream tributaries are defined to be first-order (i = 1) streams. When two first-order streams combine they form a second-order (i = 2) stream. When two second-order streams combine they form a third-order (i = 3) stream, and so forth. Note that first-order (i = 1) streams can also join second-order (i = 2), third-order

184

Figure 8.1. The drainage network in the Volfe and Bell Canyons, San Gabriel Mountains, near Glendora, California, obtained from field mapping (Maxwell, 1960).

Figure 8.2. Illustration of the Strahler ( 1957) streamordering system.

GEOMORPHOLOGY

GEOMORPHOLOGY

(i = 3), and all other higher-order streams. Similarly second-order (i = 2)

streams can join third-order (i = 3) streams and so forth. We will refer to these as side tributaries. Horton (1945) defined the bifurcation ratio Rb according to

where N represents the number of streams of order i. It is the ratio of the number of streams of one order to the number of streams of the next higher order. For example, if there are 20 third-order streams and 5 fourth-order streams we have Rb = ?$! = 4. Horton also introduced the length-order ratio defined by

where ri represents the mean length of streams of order i. Empirically, both R, and Rr are found to be nearly constant for a range of stream orders in any given drainage basin. These are known as Horton's laws. Taking the definition of the fractal dimension D to be

the substitution of (8.1) and (8.2) gives the fractal dimension of a drainage network as

Standard stream-ordering parameters are directly related to the fractal dimension of the network. The validity of Horton's laws implies that drainage networks are fractal trees. The dependence of the number of streams of various orders on their mean length for the drainage network illustrated in Figure 8.1 is given in Figure 8.3. The smallest streams on this map based on field studies are one order lower than the smallest streams on the topographic map of this region; we refer to these streams as 0-order streams. The highest order streams in this region are fourth order. It is seen that the results correlate well with (8.3) taking D = 1.81. The equivalent number-length statistics for the entire United States are given in Figure 8.4. The single tenth order river is the Mississippi. Again a good correlation with the fractal relation (8.3) is obtained with D = 1.83.

185

186

Figure 8.3. Dependence of the number of streams N of various orders 0-4 on their mean length r for the example illustrated in Figure 8.1. The power-law, straightline correlation is with (8.3) takingD= 1.81.

Figure 8.4. Dependence of the number N of rivers with a specified order on their mean length r. The correlation with (8.3) gives D = 1.83.

GEOMORPHOLOGY

GEOMORPHOLOGY

187

It is seen that stream networks are, to a good approximation, fractal and have fractal dimensions near 1.80, clearly somewhat less than the space filling D = 2. This is consistent with the example illustrated in Figure 8.1. Over short distances, fluxes of water act diffusively, flowing over the surface or through the near-surface without incising channels (streams).

8.2 Fractal trees

We now turn our attention to deterministic fractal trees. Three examples are given in Figure 8.5. To specify the geometry of a deterministic fractal tree, three quantities must be given: the bifurcation ratio R,, the length-order ratio R , and the angle of divergence 0. And these three quantities are independent of order. For the example given in Figure 8.5(a), R, = 3, Rr = 3 , O = 30"; and from (8.4) D = 1 for this fractal tree. For the example given in Figure 8.5(b), R, = 2, Rr = 2, 0 = 60°, and again D = 1. And for the example in Figure 8.5(c), R, = 2, Rr = fi,0 = 90°, and D = 2. In all cases the constructions can be extended to infinite order without overlap. If the construction in Figure 8.5(c) is extended to infinite order, the plane is entirely covered by the construction but with no overlap. Thus, this construction is an example of a self-similar (identical at all scales), deterministic network that can drain every point on a surface at as small a scale as is specified. This is the implication of D = 2, the dimension of a plane. Comparing the drainage network in Figure 8.1 with the fractal trees illustrated in Figure 8.5 shows an important discrepancy. The drainage network has side tributaries whereas the fractal trees do not. First-order streams intersect other first-order streams to form second-order streams. But other first-order streams intersect second-order, third-order, and all higher-order streams. Similarly second-order streams intersect other second-order streams to form third-order streams. But other second-order streams intersect third-order, fourth-order, and all higher-order streams.

Figure 8.5. Three examples of fractal trees. (a) R, = N,INi+I= 3, R,= ri+llri= 3. 8=30°,D= l.(b)R,= N,INitI =2,R,=ri+l/ri=2, 8=60°, D = l . ( c ) R , = NiIN,,, = 2, Rr= ri+,lri= fi,0 = 90°, D = 2.

188

Figure 8.6. (a) Binary, selfsimilar fractal tree with N , = 8 ( N , , = 8 , N I 2= N I 3= N,, = O), N2 = 4 (N22= 4 , N23= N2, = 0 ) . N3 = 2 (N33= 2 , N3, = O), and N, = 1 (N,, = 1). Thus we have R, = R, = 2 and D = 1. (b) Self-similar fractal tree with side branches. We have N , = 27 ( N , , = 18, N12= 9, N13= N,, = 0 ) . N2 = 9 (Nz2= 6 , NZ3= 3, N2, = 0 ) . N3 = 3 (N,, = 2 , N3, = l ) , and N, = 1 ( N , = I). This gives R, = 3, R, = 2, and D = In 3lln 2 = 1 S 8 5 . (c) Addition of more side branches. We have N , = 43 ( N , , = 22, N12= 1 1 , N I 3= 6 , N l , = 4 ) , N 2 = 11 (N2,=6, N23= 3, N2, = 2)' N 3 = 3 (N,, = 2 , N,, = 1) and N, = 1 ( N , = 1). This fractal tree becomes statistically selfsimilar in the limit of infinite order with R, 4 4 and D + 2.

GEOMORPHOLOGY

To account for side tributaries Tokunaga (1978, 1984) introduced the family of fractal trees illustrated in Figure 8.6. His approach has been considered in detail by Peckham (1995). In Figure 8.6(a) we have a standard binary fractal tree with R, = 2, Rr = 2, and 0 = 45". From (8.4) we have D = 1. There are no side branches on this tree. In Figure 8.6(b) a first-order branch has been added to each second-order branch, a second-order branch to each third-order branch, and a third-order branch to the fourth-order branch. We have N , = 27, N, = 9, N , = 3, N4 = 1 so that R, = 3 and D = In 3fln 2 = 1.585. This is clearly also a self-similar fractal tree. In Figure 10.6(c) further side branches have been added. l h o first-order branches have been added to each third-order branch and two second-order and four first-order branches have been added to the fourth order branch. We now have N , = 43, N2 = 11, N, = 3, N4 = 1. This tree is not a deterministic fractal tree but becomes statistically self-similar in the limit of infinite order; we will show that in this limit R, + 4 and D + 2. To quantify this more general class of fractal trees, a first-order branch joining a first-order branch is denoted "1 1" and the number of such branches is N , , , a first-order branch joining a second-order branch is denoted "12"and the number of such branches is N,,, a second-order branch joining a secondorder branch is denoted "22" and the number of such branches is N2,, and so forth. This classification of branches is illustrated in Figure 8.6. The branch numbers Nu for a fractal tree of order n constitute a square, upper-triangular matrix. The branch-number matrices for the three fractal trees illustrated in Figure 8.6 are given in Figure 8.7. The total number of streams of order i, N is related to the Ni by

for a fractal tree of order n.

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This class of fractal trees can also be quantified in terms of branching ratios To. These are the average numbers of branches of order i joining branches of order j. Branching ratios are related to branch numbers by

If the primary branching is binary, (8.5) and (8.6) can be combined to give

Again the branching ratios Ti constitute a square, upper-triangular matrix. The branching-ratio matrices for the three fractal trees illustrated in Figure 8.6 are given in Figure 8.8. We now define self-similar trees to be the subset of trees for which Ti,i + k = Tk where Tkis a branching ratio that depends on k but not on i. From Figure 8.8 we see that the three fractal trees illustrated in Figure 8.6 have (a) Tl = T2= T3 = 0, (b) T , = 1, T2 = T3 = 0, and (c) Tl = 1, T2 = 2, T, = 4 . Tokunaga(1978, 1984) introduced a more restricted class of self-similar fractal trees by requiring

This is now a two-parameter family of fractal trees. For the fractal tree illustrated in Figure 8.6(b) we have a = 1, c = 0 and for the fractal tree illustrated in Figure 8.6(c) we have a = 1 and c = 2. Substitution of (8.8) into (8.7) gives n -i

n-i

If we divide (8.9) by N and introduce the branching ratios from (8.2) we obtain 2

n-i

Rb

k=l

Ck-l

1=-+ax--

8 0 0 0 4 0 0 2 0 1

(a)

Rkb

189 0 0 6 3 0 2 1

22116 4 6 3 2 2 1

1

1

(b)

(c)

N1l N12 N13 N14 N22 N23 N24 N33 N34 N44

(dl

Figure 8.7. The branch-number matrices for the fractal trees illustrated in Figure 8.6(a), (b), (c) are given in (a), (b), and (c). The nomenclature is illustrated in (d).

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which can be written as

And if n

-

i - 1 is large we can approximate (8.11) with

which becomes the quadratic

with the solution

Thus large Tokunaga trees have branching ratios that are independent of order. For the tree illustrated in Figure 8.6(c) with a = 1 and c = 2 we have Rb + 4 as n + =. And when the length-order ratio Rr is specified the fractal dimension is given by

For the tree illustrated in Figure 8.6(c) we have Rr = 2 and D = 2. The Tokunaga model can be further constrained if the number of side branches is assumed to be proportional to the length of the branch they enter. For this case a = Rr- 1 and c = R , thus the Tokunaga branching ratio is given by Figure 8.8. The branchingratio matrices for the fractal trees illustrated in Figure 10.6(a), (b), (c) are given in (a), (b), and (c). The nomenclature of the branching ratios is given in (d). In (a) we have TI = T, = T3 = 0, in (b) we have TI = 1, T2=T3=Oora=l , c = O from (8.8), and in (c) we have T, = 1, T, = 2, T3 = 4, or a = 1, c = 2 from (8.8).

Tk = ( R , - 1)R y

(8.16)

The fractal tree illustrated in Figure 8.6(c) is a deterministic example of this class of tree with Rr = 2, a = 1, and c = 2. Substitution of (8.16) into (8.14) gives

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191

Substitution of (8.16) into (8.15) gives

For this subclass of Tokunaga fractal trees the branching ratio R, and the fractal dimension D can be obtained from (8.17) and (8.18) if the length-order ratio Rr is specified. A related quantification of side branching has been given by Vannimenus and Viennot (1989) and Ossadnik (1992). We now address the question whether the statistics of actual drainage networks are represented by Tokunaga fractal trees. Peckham (1995) has determined branching-ratio matrices for the Kentucky River basin in Kentucky and the Powder River basin in Wyoming. Both are eighth-order basins with the Kentucky River basin having an area of 13,500 km2 and the Powder River basin an area of 20,18 1 km2. For the Kentucky River basin the bifurcation ratio is R, = 4.6 and the length-order ratio is Rr = 2.5; for the Powder River basin the bifurcation ratio is R, = 4.7 and the length-order ratio is Rr = 2.4. The dependence of the number of streams of various orders on their mean length for the two basins are given in Figure 8.9. Again the results correlate well with the fractal relation (8.4) taking D = 1.85. The branching-ratio matrices for the two river basins are given in Figure 8.10. We now deterover i mine values for Tkby averaging the values of

lo5 lo4 -

0 Kentucky River x Powder River

lo3 N

lo2

-

10

8 v n

1

V

1 6'

1

10 r

lo2

I lo3

Figure 8.9. Dependence of the number of streams N of various orders 1-8 on their mean length r for the Kentucky River basin in Kentucky and the Powder River basin in Wyoming. The straight-line correlation is with (8.4) taking D = 1.85.

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1 Tk =

n-k

(n - k ) i = l

Ti,i

+k

For example, we find that T, = 18 for the Kentucky River basin by taking the average of T,, = 15.6, T,, = 20.3, T,, = 16.0, and T,, = 20.0. The values of T, for the two basins are given in Figure 8.11 as a function of k. It is seen that

Figure 8.10. Branching-ratio matrices for (a) the Kentucky River basin and (b) the Powder River basin as obtained by Peckham (1995).

0 Kentucky River x Powder River DLA synthetic network

Figure 8.1 1. Dependence of the mean branching ratios T, on k for the Kentucky River basin and the Powder River basin. The straight-line correlation is with (8.8) taking a = 1.2 and c = 2.5. Also included are the mean branching ratios for a fifthorder diffusion limited aggregation (DLA) network.

GEOMORPHOLOGY

the results correlate well with (8.8) taking a = 1.2 and c = 2.5. These results are also tabulated in Table 8.1. At least for these two basins, in quite different geological settings, good agreement with Tokunaga fractal trees is obtained with these values of the parameters a and c. It is also of interest to compare these values with those given in (8.16). With Rr = 2.5 for the Kentucky River basin and Rr = 2.4 for the Powder River basin, the values from (8.16) are a = 1.5, c = 2.5 and a = 1.4, c = 2.4 respectively. These are in quite good agreement with the best-fit values of a = 1.2 and c = 2.5. Empirically, actual drainage basins appear to be well approximated by Tokunaga fractal trees. There are a number of other applications of fractal trees in geology and geophysics. River deltas are one obvious choice. Another is the upward migration of magma beneath a volcano. Partial melting in the earth's mantle occurs on grain boundaries. Because the melt, the magma, is lighter than the residual solid, it drains upward eventually reaching the earth's surface, resulting in a volcanic eruption. One approach to the magma ascent problem is to treat it as a flow in a uniform porous media (Turcotte and Schubert, 1982, pp. 413-416). An alternative is to treat the magma paths like a drainage network. Rivelets of magma combine to form ascending magma streams, and ascending streams of magma combine to form magma rivers. Hart (1993) has proposed a fractal tree model for the ascending magma and has considered its implications on magma composition. The concepts of fractal trees also have a wide variety of other applications. Examples include the growth of actual trees and other plants, as well as the cardiovascular distribution of veins and arteries and the bronchial system. Returning to drainage networks, another fractal correlation to drainage patterns is obtained if the length of the principal river in a drainage basin P is plotted against the area of the basin A. Data for several basins in the northeastern United States are given in Figure 8.12 (Hack, 1957). The applicable fractal relation is

Table 8.1 .Branching statistics for a variety of fractal trees R,

Rr

D

a

c

8

2 4 4.6

2 2 2.5

1 2 1.67

0 1 1.2

0 2 2.5

8

4.7

2.4

1.77

1.2

2.5

10 7

5.15 3.98

2.87 2.09

1.56 1.87

1.5 1.2

2.70 2.5

Order Idealized tree Figure 8.6(a) Idealized tree Figure 8.6(c) Kentucky River basin (Peckham, 1995) Powder River basin (Peckham, 1995) DLAa cluster (Ossadnick, 1992) DLA drainage network model

"DLA, diffusion limited aggregation.

00

193

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and good agreement with the data in Figure 8.12 is obtained taking D = 1.22. Robert and Roy (1990) have discussed this fractal relation between mainstream length and drainage area. It is clear that drainage networks are self-similar and fractal to a good approximation. But this is basically an empirical statement that does not address how drainage networks evolve. It is clear that in most terrains drainage networks are a direct consequence of erosion. In young terrains tectonic processes play an important role; however, erosional processes may still be dominant. Consider the Hawaiian chain of volcanic islands. A young island such as Hawaii is made up of deterministic conical structures associated with shield volcanoes. These are not fractal. However, sufficient erosion has occurred on Maui and Oahu in a few million years to develop an irregular, scale-invariant morphology that exhibits fractal statistics. The erosional evolution of landscapes is a problem that has fascinated natural scientists for centuries. The forms of mature landscapes evolve through processes of erosion and deposition. An essential question is whether it is possible to develop a basic theory of landscapes or whether it is necessary to consider only statistical aspects of the problem. A variety of models were proposed in the 1960s to describe the statistics and origins of drainage networks (Smart, 1972). Descriptive models were introduced by Shreve (1966, 1967) and Schreidegger (1967) in which drainage networks were considered as infinite topologically random networks (i.e., no

Figure 8.12. Dependence of the length P of the principle river on the area A of the drainage basin for several drainage basins in the northeastern United States (Hack, 1957).

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195

one distribution of network links is preferred over any other). They showed that the statistics of real drainage networks matched the most probable number-order distribution of a topologically random network. Snow (1989) has shown that the sinuosity of streams exhibits fractal behavior, and Nagatani (1993) has shown that meander patterns are fractal. Although these models have proven useful as a way to describe drainage networks, they contain little information on the dynamical processes that form them. Other workers have proposed random growth models to explain the planform organization of drainage networks. Leopold and Langbein (1962) and Schenck (1963) proposed models in which the streams themselves followed random walks. Thus the network was not headward growing, but propagated laterally from the most central "trunk" stream. In addition, the network grew by the addition of entire stream segments, rather than by gradual expansion (accretion). Howard (1971) introduced an accretionary headward growth model, a site adjacent to the existing network was chosen randomly, and the network propagated to this site. Thus, all sites on the network had an equal probability for growth.

8.3 Growth models A growth model that has been applied to stream networks as well as a wide variety of other applications is diffusion limited aggregation (DLA). However, before considering DLA we illustrate a deterministic fractal growth model based on the Koch snowflake. This model is illustrated in Figure 8.13. An initial unit square at zero order [(Figure 8.13(a)] grows at first order by Figure 8.13. Illustration of a deterministic fractal growth (accretion) model based on the Koch snowflake. (a) A single unit-square seed particle at zero order. (b) At first-order, four unit-square particles are added to the corners of the seed particle. This is the generator for the higher-order fractal constructions. (c) At secondorder, four of the first-order structures are added at the comers of the first order structure. (d) At third-order, four of the second-order structures are added at the comers of the second-order structure.

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the addition of four unit squares at the four corners of the original "seed" particle, as illustrated in Figure 8.13(b). At second order, four of the firstorder structures are added as shown in Figure 8.13(c). At third order, four of the second-order structures are added as shown in Figure 8.13(d). We have No= 1, r o = 1; N, = 5 , r, = 3; N2=25, r 2 = 9 ;N3= 125, r,=27 and from (2.2) we have D = In 5Iln 3 = 1.465. One approach to quantifying the growth of an aggregate such as that illustrated in Figure 8.13 is to determine the number of particles as a function of size. At zero order the number of particles is No = 1 and a circle with radius ro = l l f i coven the particle, at first order the number of particles is N, = 5 and a circle with radius r, = 3 1 f i coven the particles, at second order the number of particles is N2 = 25 and a circle with radius r, = 9 1 f i covers the particles, and at third order the number of particles is N3 = 125 and a circle with radius r, = 2 7 1 f i . Noting that in this case N rD we again find D = In 51ln 3 = 1.465 just as above. However, in applications to statistical growth models and to natural phenomena, it is generally preferable to use the "radius of gyration" rather than the radius of a circle (sphere) that covers the growing aggregate. The definition of the radius of gyration for an aggregation of N particles growing from a seed particle in two dimensions is

-

where ri is the radial distance of particle i from the seed particle. A fractal relation is defined by

where a is the particle size. For the example given in Figure 8.13 we find that at first order we have the centers of the four accreted particles at a distance fifrom the seed particle. Thus from (8.21) we have

Similarly at second order we have

And from (8.22) the corresponding fractal dimension is given by

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197

slightly less than the value D = 1.46 obtained for the basic construction. For the third-order construction illustrated in Figure 8.13(d) we obtain rg4/a= 12.06 and from (8.22)

indicating a fractal dependence.

8.4 Diffusion-limited aggregation (DLA)

The concept of diffusion-limited aggregation was introduced by Witten and Sander (1981). They considered a two-dimensional grid of cells and placed a seed cell near the center of the grid. An accreting cell was randomly introduced on a "launching circle" as illustrated in Figure 8.14. The accreting cell was allowed to follow a random path, that is, it was a random walker, until (1) it accreted to the growing cluster of cells by entering a cell adjacent to the cluster or (2) it wandered across the "killing circle" in which case a new cell (random walker) was introduced on the "launching circle." An illustration of a cluster created by DLA is given in Figure 8.15. The resulting sparse dendritic structure results because the walkers are more likely to accrete near the tips of the cluster rather than in the deep interior. Halsey and Leibig

"Killing Circle"

n

Random Path

"Launching Circle"

Figure 8.14. illustration of diffusion-limited aggregation (DLA) growth of a cluster. An accreting cell is introduced at a random point on the "launching circle." The cell carries out a random walk until it either accretes to the growing cluster (as shown) or crosses the "killing circle," in which case the walk is terminated.

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sulting fractal structure often resembles DLA clusters (Chen and Wilkinson, 1985; Malay et al., 1985; Nitmann et al., 1985, 1986; Van Damme et al., 1986; Feder and Jossang, 1995). Sornette et al. (1990) have suggested that the fractal distributions of faults and joints discussed in Chapter 4 are the result of a DLA random growth. Two-dimensional surface exposures of fractures and joints are generally fractal with D = 1.7 in agreement with the values obtained for DLA clusters. DLA models for crack propagation have also been used to model fragmentation, and power-law number-size statistics for fragments were obtained (Gomes and Sales, 1993). 8.5 Models for drainage networks

A modified version of the diffusion-limited aggregation (DLA) technique has been used to generate realistic drainage networks (Masek and Turcotte, 1993). To apply DLA to drainage networks several modifications are necessary. Random walkers are introduced randomly over the entire grid and are allowed to walk until they either intersect the evolving network or are lost from the grid. The random walkers can be viewed as unit water fluxes (rainfall and overland flow) that migrate over a relatively flat surface until they find a gully (network) in which to flow. When the flux joins the gully, it erodes and expands the network. The growth of a synthetic drainage network based on the DLA model is illushated in Figure 8.16. A square grid of 15 X 15 cells is used in this illushation. Five seed cells are introduced at random points on thc lower boundary. The evolving network must grow from these seed cells. From the exam-

Seed Cells I I I

1 1 1 1

I

1 1 1

I I 1 I I I I

I I IWI IU-hI

I I I I I

Accreted Cells

+

0

Newly Added Cell Random Walk Prohibited Sltes Other allowed sites for accretion

Figure 8.16. Illustration of the mechanism for network growth in the DLA model for drainage network evolution. A random walker is randomly introduced to an unoccupied cell. The random walk proceeds until a cell is encountered with one (and only one) of the four nearest neighbors occupied (hatched). The new cell is accreted to the drainage network. If a random walker enters a prohibited cell or wanders off the grid it is terminated.

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ple shown, 16 cells have been accreted to the seed cells. Cells are allowed to accrete if one (and only one) of the four nearest neighbor cells is part of the preexisting network. Prohibited sites that already have two neighboring sites occupied are identified by stars. Sites available for accretion to the network are indicated by open circles. A random walker is introduced at a random cell on the grid and the resulting random-walk path is traced by the solid line. After 28 random walks it accretes to the network at the cross-hatched cell. A random walk proceeds until the walker (1) accretes to the network, (2) exits the grid, or (3) lands on a prohibited cell. In cases (2) and (3) the walk is terminated and a new walker is introduced on a new, randomly selected site. The iteration of this basic procedure results in a branching network composed of linked drainage cells. This "self-avoiding" algorithm prevents local clumping of drainage cells. Although the model is highly schematic, the mechanics outlined here are analogous to the mechanics operating in real drainage systems. The accretionary nature of network growth produces a headward evolving drainage

Figure 8.17. Illustration of a synthetic drainage network generated using a diffusion limited aggregation (DLA) model on a 256 X 256 grid using 20,000random walkers.

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201

pattern similar to patterns of headward erosion seen in nature. The accretion process itself is analogous to a flux of water (e.g., overland flow) intersecting an existing drainage, thereby initiating a new first-order channel. The self-avoiding algorithm prevents drainages from becoming locally space filling at the finest scales. In nature, this limit may be controlled by a threshold transition from diffusive slope processes to advective channelization processes. An illustration of a DLA drainage network simulation grown on a 256 X 256 grid of cells is illustrated in Figure 8.17. One-third of the bottom row of cells were tagged as seed cells and 20,000 random walkers were introduced. There is considerable visual similarity between this simulated network and the actual drainage network illustrated in Figure 8.1. The dependence of the number of streams of various orders in the simulation on their mean length is given in Figure 8.18. The highest-order streams in the simulation are seventh order. It is seen that the results correlate well with (8.3) taking D = 1.85. This result is in good agreement with actual drainage networks. The branch-number matrix and the branching-ratio matrix for a fifth order network obtained using the DLA model are given in Figure 8.19. Using (8.19) the correspond-

Figure 8.18. Dependence of the number of streams N of various orders 1-7 on their mean length r for the DLA simulation given in Figure 8.17. The correlation is with (8.3) taking D = 1.85. Figure 8.19. (a) Branchnumber matrix and (b) branching-ratio matrix for a fifth-order synthetic drainage network generated using the DLA model illustrated in Figure 8.17.

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ing values of mean-branching ratios T, are determined and are compared with those for actual drainage networks in Figure 8.1 1. Good agreement between the synthetic DLA network and the actual drainage networks is found. These results are also tabulate in Table 8.1.

8.6 Models for erosion and deposition To totally model landforms it is also necessary to include erosion with the evolution of drainage networks. A number of theories have been proposed for erosion and have been summarized by Scheidegger (1991).The simplest theory hypothesizes that erosion is proportional to the elevation h so that

where

T

is a characteristic time for erosion. With the initial condition that

h = h, at t = 0 the solution is

This gives an exponential decay of topography with a characteristic time T but no information on the form of the evolving landscapes. Culling (1960, 1963, 1965) hypothesized that the horizontal flux of eroded material m, is proportional to the slope:

where J is the transport coefficient and we consider only the one-dimensional problem h = h(x,t).With the conservation of mass relation

this gives

which is the linear diffusion equation.

GEOMORPHOLOGY

Solutions to the diffusion equation take a variety of forms but in many cases reduce to the error function. The form of many alluvial fans and prograding deltas can be approximated quite accurately with the error function (Kenyon and Turcotte, 1985). In addition, a number of authors have used the Culling model to estimate the age of faults and shoreline scarps (Wallace, 1977; Buckman and Anderson, 1979; Nash, 1980a, b; Mayer, 1984; Hanks et al., 1984; Hanks and Wallace, 1985; Andrews and Hanks, 1985). Typical values for J are in the range 10-2-10-3 m2 yr-I. However, solutions to a linear equation such as the diffusion equation cannot produce fractal, selfsimilar solutions. A variety of stochastic models have been proposed for deposition. The simplest model of random deposition was presented by Schwarzacher (1976). In his model the rate of deposition on a two-dimensional cross-section of the landscape is Gaussian white noise. Once sediment is deposited, it cannot be transported along the surface. Although this model produces a rough surface, the model does not produce self-similar (self-affine) topography. A two-dimensional model may be considered to be an idealized model of deposition in three dimensions with channels that are parallel and of uniform capacity in the direction of flow. Because the depositional channels that evolve at the surface are uniform in the flow direction, the function representing the surface will be uniform in that direction. As such, we can describe the evolution of the two-dimensional surface by the one-dimensional cross section. We will assume that the surface of the landscape has a constant mean growth rate. With this assumption, the variations in the surface in time and depth are equivalent. Pelletier and Turcotte (1997) proposed an extension of Schwarzacher's model. A site on a one-dimensional lattice is chosen at random. A unit of sediment is deposited at that site or at one of its nearest neighbors depending on which site has the lowest elevation. This model is illustrated in Figure 8.20. The local elevation is the total number of units of sediment that have been deposited at the site. This model of surface growth was first analyzed by Family (1986) with applications to the growth of atomic surface layers. He reported the results of computer simulations that showed that the model produces scale-invariant variations of the surface in space and time. He found that the standard deviation, a , of the surface obeys the relation

where L is a length scale and T is a time scale. Surfaces with scale-invariant standard deviations a(L,T) a LHa TK have a power-law dependence of the power spectral density, S(k), on wave number k of the form S(k) a k-2Ha-I (i.e., a k-2 for Ha = $,) and a power-law dependence on frequency of the form S(f) ~ l f - 2 ~ - 1 (i.e., cx f-3i2 for K =

i).

203

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GEOMORPHOLOGV

Equilibrium topography produced by this model with 1024 lattice sites is illustrated in Figure 8.21. The resulting dependence of the spectral power density on wave number is given in Figure 8.22. The data points are the average of 50 simulations. The correlation with the self-affine fractal relation (7.41) is excellent taking fl = 2.01, and the topography is basically a Brownian walk. Variations in the rate of growth at a specified lattice point are given in Figure 8.23. Deviations of the elevation from the mean elevation are given as a function of time. The resulting dependence of the spectral power density on frequency is given in Figure 8.24. The data points are again the average of 50 simulations. The correlation with the self-affine fractal relation (7.41) is excellent taking fl = 1.5 1. This value is in reasonable agreement with the values given for the porosity logs in Figure 7.10. Although the analysis given above is for deposition, it can be easily modified to model erosion. Instead of randomly adding units, units are randomly removed. A random unit is eroded either at the chosen site or on one of its nearest neighbors depending on which site has the highest elevation.

Figure 8.20. Illustration of the sediment deposition model. The dotted block is the unit of sediment being added randomly to the surface. The arrows point toward the site upon which the unit of sediment will be deposited. In (a) the randomly chosen site has a lower elevation than either of its nearest neighbors, so the sediment is deposited at the chosen site. In (b) one of the nearest neighboring sites has a lower elevation and the sediment is deposited at that lower site. If both adjacent sites have lower but equal elevations, one of the two sites is chosen randomly, as illustrated in (c).

b)

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256.0

51 2.0

1024.0

768.0

lattice site

I

t

-2.8

-2.4

-2.0

-1.6

log k

- 1.2

-0.8

-0.4

205

Figure 8.21. Depositional topography obtained using the model illustrated in Figure 8.20 with 1024 lattice sites.

Figure 8.22. Dependence of the spectral power density S on wave number k for the synthetic depositional topography illustrated in Figure 8.21. The data points are the average of 50 simulations. An excellent correlation with the selfaffine fractal relation (7.41) is obtained taking P = 2.01.

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Figure 8.23. Deviations from the mean rate of growth as a function of time at a chosen site.

Figure 8.24. Dependence of the spectral power density S on frequency f for the time dependence of the deposition illustrated in Figure 8.23. Again the data points are the average of 50 simulations. An excellent correlation with the self-affine fractal relation (7.41) is obtained taking p = 1.51.

12.0

15.0

18.0

timesteps ( ~ 1 0 0 0 0 0 )

I

-6.3

-6.0

-5.7

-5.4

-5.1

log f

-4.8

-4.5

-4.2

GEOMORPHOLOGY

Family (1 986) showed that a continuous version of this discrete model is provided by the diffusion equation with a Gaussian white noise term

where q ( X J ) is the Gaussian white noise. This is one form of the linear Langevin equation and is also known as the Edward-Wilkinson equation. Comparing (8.29) and (8.27) shows that the Culling model gives Brownianwalk topography if erosion or deposition occurs randomly. An extensive discussion of growth processes has been given by Barabasi and Stanley (1995). A wide variety of models have been proposed to simulate scale-invariant (fractal) topography and/or river networks. Chase (1992) has combined a cellular-automata advection model with diffusion and has generated reasonably realistic topography with drainage networks. Similar models have been given by Takayasu and Inaoki (1992) and Lifton and Chase (1992). Leheny and Nagel (1993) introduced an avalanche model and derived both topography and river networks. Meakin et al. (1991) have applied a DLA approach, and Willgoose et al. (1991) an advective-diffusive model. Kramer and Marder (1992) have modeled the development of drainage networks on a water-covered landscape assuming that the erosion is proportional to the product of velocity and pressure. Barzini and Ball (1993) use a similar model to develop synthetic braided rivers. They point out that both the simulations and real braided rivers have a fractal distribution of island sizes with D = 1. Stark (1991) used an invasion percolation technique, in which the growing network was superposed on a fixed random field (analogous to a substrate with variable erodibility). At each time step, the network propagated to the adjacent site having the highest erodibility value over the entire perimeter. Although all sites on the network had differing probabilities for growth, these probabilities did not change through time since the random field was fixed from the start. Stark (1994) modeled patterns of erosion using invasion percolation, Eden growth, and DLA models. Liu (1992) has utilized percolation clusters. Takayasu (1993) used random self-affme tiling to explain the power-law distribution of drainage basin sizes. Nikora and Sapozhnikov (1993) have presented a model based on random walk simulations. Minimum energy dissipation and/or entropy methods have been applied to landforms and river networks by Rinaldo et al. (1993) and by Sun et al. (1993,1994,1995). Sornette et al. (1994) have presented a model for the tectonic generation of fractal topography utilizing statistical distributions of displacements on fault arrays.

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It is also desirable that models of topography exhibit the Brownian-walk spectrum discussed in the previous chapter (Newman and Turcotte, 1990). It has been shown that the three-dimensional nonlinear Langevin equation can yield topography with the observed Brownian-walk spectrum (Sornette and Zhang, 1993).

8.7 Floods The volumetric flow Q(t) in a river constitutes a time series. However, the flow is strongly asymmetric so that Gaussian statistics are generally a poor approximation. Large values of Q for relatively short periods of time constitute floods, whereas low values of Q for relatively long periods constitute droughts. Most rivers also have a strong annual periodic component. Floods present a severe natural hazard; to assess the hazard and to allocate resources for its mitigation it is necessary to make flood-frequency hazard assessments. The integral of the flow in a river is required for the design of reservoirs and to assess available water supplies .during periods of drought. An important question in geomorphology concerns which floods dominate erosion. Is erosion dominated by the 10 year, the 100 year, or the very largest floods? The answer to this question depends upon whether extreme flood probabilities have an exponential or power-law dependence on time. One estimate of the severity of a flood is the peak discharge at a station Q,. The magnitude of the peak discharge is affected by a variety of circumstances including (1) the amount of rainfall produced by the storm or storms in question, (2) the upstream drainage area, (3) the saturation of the soil in the drainage area, (4) the topography, soil type, and vegetation in the drainage area, and (5) whether snow melt is involved. In addition dams, stream channelization, and other man-made modifications can affect the severity of floods. To estimate the severity of future floods, historical records are used to provide flood-frequency estimates. Unfortunately, this record generally covers a relatively short time span and no general basis has been accepted for its extrapolation. Quantitative estimates of peak discharges associated with paleofloods are generally not sufficiently accurate to be of much value. A wide variety of geostatistical distributions have been applied to flood-frequency forecasts, often with quite divergent predictions. Examples of distributions used include power law (fractal), log normal, gamma, Gumbel, log Gumbel, Hazen, and log Pearson. It is standard practice to use the annual peak discharges in flood-frequency analyses. In the United States this is the peak discharge during a water year, which extends from October 1 of the preceding year to September 30. There are serious problems with this approach and alternatives will be

GEOMORPHOLOGY

discussed. However, the basic question we wish to consider is whether floods satisfy power-law (fractal) scaling (Turcotte and Greene, 1993; Turcotte, 1994c; Wu and Lye, 1994). If floods are fractal, then it would be expected that the peak discharge Q, in a time interval T is related to the interval by

where the Hausdorff measure H a plays a role similar to that in (7.29). The Hausdorff measure is related to the fractal dimension by (7.33). Since river discharges have a strong annual variability, the interval T is generally taken as an integer number of years when floods are considered. This scale-invariant distribution can also be expressed in terms of a flood-frequency factor F, which is the ratio of the peak discharge over a 10-year period to the peak discharge over a 1-year period. With self-similarity the flood-frequency factor F is also the ratio of the 100-year peak discharge to the 10-year peak discharge and the ratio of the 1000-year peak discharge to the 100-year peak discharge. In terms of H a and D we have

The flood-frequency factor is a measure of the severity of great floods. We now turn to the analysis of flood-frequency records. As our first example, the ten benchmark stations considered by Benson (1968) will be studied. Benson (1968) applied a variety of geostatistical distributions to the data from these stations; these will be compared with the fractal approach discussed above. The maximum annual floods for two stations are given in Figure 8.25. Values for station 1- 1805 on the Middle Branch of the Westfield River in Goss Heights, Massachusetts, are given in Figure 8.25(a) for the period 1911-1960 (Green, 1964) and values for station l 1-0980 in the Arroyo Seco near Pasadena, California, are given in Figure 8.25(b) for the period 1914-1965 (Young and Cruff, 1967). Carrying out statistical studies on an annual flood series systematically underestimates the number of smaller floods. The reason is that the second or third largest flood in some water years is larger than the largest flood in other water years. The alternative is to consider a partial-duration series, which may include several floods in a given water year. However, the definition of two statistically independent floods is arbitrary. We assume that two floods are statistically independent if they are separated by more than two months. The largest floods for each record are ordered: The largest flood is assigned a period equal to the length of the record T,,, the second largest flood a period td2, the third largest flood a period 743, and so forth. The log of the

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peak discharge for each flood is plotted against the log of its assigned period. This is the same technique that was used for earthquakes in Chapter 4. Results for station 1-1805 (Goss Heights, MA) are given in Figure 8.26(a). The solid line is the least square fit of (8.30) with the data over the range 50 < Q, < 200 m31s; large floods are omitted from the fit because of their small number. The solid line corresponds to Ha = 0.5 1 and from (8.3 1) we have F = 3.3. Results for station 11-0980 (Pasadena, CA) are given in Figure 8.26(b), the solid line is the best fit of (8.30) with the data over the range 10 < Q, < 100 m31s. The solid line corresponds to H a = 0.87, and from (8.31) we have F = 7.4. In both cases the fit to the power-law (fractal) relation is quite good. The values of Ha and F in California are considerably larger than in Massachusetts. Large floods are relatively more probable in the arid climate than in the temperate climate. Many statistical distributions have been applied to historical records of floods. Benson (1968) has given six statistical correlations for each of his

Figure 8.25. Maximum annual floods Qm for (a) station 1 - 1805 on the Middle Branch of the Westfield River, Goss Heights, Massachusetts, and (b) station 11 -0980 in the Arroyo Seco near Pasadena, California.

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211

ten benchmark stations. His results for the 2-parameter gamma (Ga), Gumbe1 (Gu), log Gumbel (LGu), log-normal (LN), Hazen (H) and log Pearson type I11 (LP) are given in Figure 8.26(a) for station 1-1805 and in Figure 8.26(b) for station 11-0980. For large floods the fractal prediction (F)correlates best with the log Gumbel (LGu), whereas the other statistical techniques predict longer recurrence times for very serious floods. The fractal and log Gumbel are essentially power-law correlations, whereas the others are essentially exponential.

Figure 8.26. The points are the observed floods at the measuring stations during the periods considered (a) for Station 1-1805 and (b) for Station 11-0980. The peak discharge Qm is given as a function of recurrence intervals T. The power-law (fractal) prediction, F, is compared with the six statistical predictions given by Benson (1 968). 2parameter gamma (Ga), Gumbel (Gu), log Gumbel (LGu), log-normal (LN), Hazen (H), and log Pearson type I11 (LP). For the fractal correlation (F)the corresponding values of Ha and Fare given.

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The values of Ha,D, and F are given for all ten benchmark stations in Table 8.2. The correlations with the fractal relation (8.30) in Figure 8.26 are typical of the ten stations. The parameter F is a measure of the relative severity of flooding. The higher the value of F, the more likely that severe floods will occur. Our results show that there are clear regional trends in values of F. The values in the southwest including Nevada (F = 4.13) and New Mexico ( F = 4.27) as well as California ( F = 7.4) are systematically high. These high values can be attributed to the arid conditions and the rare tropical (monsoonal) storm that causes severe flooding. Central Texas (F = 5.24) is also high and Georgia (F = 3.47) is intermediate. These areas are influenced by hurricanes. The northern tier of states including Massachusetts (F = 3.26), Minnesota (F = 2.95), Nebraska ( F = 3.47), and Wyoming ( F = 3.31) range from low values in the east to intermediate values in the west. Washington (F = 2.04) has the lowest value of the stations considered; this low value is consistent with the maritime climate where extremes of climate are rare. We have also determined the Hurst exponent Hu for the ten benchmark stations. Values of R/S for T = 5, 10,25, and 50 years (WS = 1 for T = 2 by definition) are given in Figure 8.27(a) for station 1-1805 (Goss Heights, MA) and in Figure 8.27(b) for station 11-0980 (Pasadena, CA). Good correlations are obtained with (7.59) taking Hu = 0.67 for station 1-1805 and Hu = 0.68 for station 11-0980. Values of Hu for all ten stations are given in Table 8.1. The values are nearly constant, with a range from 0.66 to 0.73, indicating moderate persistence. It is not surprising that the values of the Hausdorff measure Ha differ from the values of the Hurst exponent Hu since the former refers to the statistics of the flood events and the latter to the statistics of the running sum. The results indicate that there is considerable variation of F but very little variation in Hu. Mandelbrot and Wallis (1968, 1969b) introduced the Table 8.2. Values of the Hausdorff measure Ha, fractal dimension D, flood intensity factor F, and Hurst exponent Hu for the ten benchmark stations Station

River

(State)

Ha

D

F

Hu

1-1805 2-2 185 5-3310 6-3440 6-8005 7-2 165 8-1500 10-3275 1 1-0980 12-1570

Westfield Oconee Mississippi Little Missouri Elkhorn Mora Llano Humboldt Arroyo Seco Wenatchee

(MA) (GA) (MN) (wy) (NE) (NM)

0.513 0.540 0.470 0.520 0.540 0.630 0.719 0.616 0.870 0.310

1.39 1.46 1.53 1.48 1.46 1.37 1.28 1.38 1.13 1.69

3.26 3.47 2.95 3.31 3.47 4.27 5.24 4.13 7.40 2.04

0.67 0.72 0.72 0.72 0.67 0.73 0.70 0.66 0.68 0.72

(TX) (NV) (CAI (WA)

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213

Noah and Joseph effects. The Noah effect is the skewness of the distribution of flows in a river (or of a non-Gaussian distribution) and the Joseph effect is the persistence of the flows. It is reasonable to conclude that the variations in F can be attributed to the Noah effect and the constancy of the Hurst exponent can be attributed to the Joseph effect. An important conclusion is that R/S analysis is not relevant to flood-frequency hazard assessments.

'T yrs

lo

r-

-

5

-

R s

2

-

I

1

2

I 5

I

I

I

I

I

I

10

20

'$ yrs

I

I

I 50

Figure 8.27. The rescaled range (WS) for several intervals T. (a) Station 1 - 1805. (b) Station 11 -0980. The correlations are with (7.59) and the Hurst exponents Hu are given.

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8.8 Wavelets Fourier transforms have a long history of applications to a wide variety of problems. They have great utility in terms of obtaining the frequency content of a time series. Despite the many advantages of Fourier transforms, there are also disadvantages. To overcome some of these disadvantages Grossman and Morlet (1984) introduced the wavelet transform. This transform has a fractal basis and is particularly useful when applied to local, nonperiodic, multiscaled phenomena such as stream flows. The wavelet transform is essentially a filter that is passed over a time series, the width of the filter being generally increased by powers of two. The generalized form of the wavelet transform is given by

whereflt') is the time series and g [(t' - t)la] is the filter. The filter is centered at t and a is a measure of the width of the filter. The quantity g (t') is known as the "mother wavelet." Other wavelets are rescaled versions of the mother wavelet. The area of each wavelet must sum to zero so that

When a is increased by powers of two, a suite of wavelets is generated that can accommodate a wide range of scales in the signalflt). A commonly used mother wavelet is the "Mexican hat" wavelet, which is the second derivative of the Gaussian distribution and takes the form

Figure 8.28. Mother Mexican hat wavelet g ( r ) from (8.34).

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215

This wavelet satisfied the conditions given in (8.33). Substitution of (8.34) into (8.32) gives

W ,a ) =

(

)

[I

Jrn

-

( 1

(

e x - 2

a

1

t

)dt

(8.35)

The mother Mexican hat wavelet is illustrated in Figure 8.28. For more details of the wavelet transform, the reader is referred to Schiff (1992), Daubechies (1988), and Young (1992). To illustrate the application of the wavelet transform we will consider two streamflow time series (Smith et al., 1997). We first consider a sevenyear, daily discharge record for the Ammonoosuc River, Maine. The record of daily discharges is given in Figure 8.29(a). The record is characterized by

0

500

1000

1500

t (days)

2000

2 500

Figure 8.29. Wavelet transforms for a seven-year, daily discharge record for the Ammonoosuc River, Maine. (a) Discharge record. (b) Wavelet scalogram. (c) Wavelet transform magnitudes for a = 1, 2.4, 8, 16.32.64, and 128 days.

216

GEOMORPHOLOGY

long-period high flows associated with the annual spring snow-melt events and by short-period high flows associated with rainstorm events throughout the year. Wavelet-transform magnitudes of the time series W(a,t) are given in Figure 8.29(c) for scales a = 1, 2, 4, 8, 16, 32, 64, and 128 days. The transform magnitudes are contoured using a single threshold W= 5.5 X 10-6 m3/s in Figure 8.29(b) giving a wavelet scalogram. Regions of this a versus t plot in which the magnitude threshold is exceeded appear in black. The seven annual snow-melt events are clearly illustrated. As a second example we consider the hourly discharge record for the Forrest Kerr Creek in northwestern British Columbia, Canada. The 31 1 km2 drainage area contains several large glaciers, and we consider the 100-day summer 1992 record given in Figure 8.30(a). The record has a strong diurnal variation associated with the daily melting cycle. Wavelet transform magnitudes of the time series W(t,a) are given in Figure 8.30(c) for scales a = 1,2, 4,8, 16,32,64, and 128 hours. The corresponding contoured wavelet scalogram is given in Figure 8.30(b). The strong diurnal signal appears in the a = 8 and 16 hour wavelet transform magnitudes and are clearly illustrated in the wavelet scalogram. In addition there are melt episodes with a 5-20-day periods associated with the regional climate.

Figure 8.30. Wavelet transforms for a 100-day, hourly discharge record for the Forrest Kerr Creek, British Columbia, Canada. (a) Discharge record. (b) Wavelet scalogram. (c) Wavelet transform magnitudes for a = 1 , 2 , 4 , 8 , 16,32,64, and 128 hours.

t (hours)

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217

Wavelet transforms have been applied to a variety of geophysical time series including seismograms (Lou and Rial, 1995), the length of day (Gambis, 1992), the geomagnetic field (Alexandrescu et al., 1995), bathymetric profiles (Little et al., 1993; Little, 1994), the earth's geoid (Cazenave et al., 1995), and solar irradiance (Gambis, 1992). Flandrin (1993) has applied wavelet analysis to fractional Brownian walks. Wavelet analyses are particularly useful when a time series has strong fluctuations over a wide range of time scales. The application of Fourier transforms to this class of problems can easily lead to spurious results.

Problems Problem 8.1. Consider the fractal tree illustrated in Figure 8.31(a). The length-order ratio is Rr = 2. (a) Determine the bifurcation ratios for 1st to 2nd, 2nd to 3rd, and 3rd to 4th order branches. What are the corresponding fractal dimensions? (b) Write the branch-number matrix and the branch-ratio matrix for this tree. (c) Determine T, for k = 1, 2, 3. (d) Determine a and c as defined in (8.8). (e) Determine the asymptotic bifurcation ratio and fractal dimension for large trees of this form from (8.14) and (8.15). Problem 8.2. Consider the fractal tree illustrated in Figure 8.31(b). The length-order ratio is Rr = 3. (a) Determine the bifurcation ratios for 1st to 2nd, 2nd to 3rd, and 3rd to 4th order branches. What are the corresponding fractal dimensions? (b) Write the branch-number matrix and branch-ratio matrix for this tree. (c) Determine T, for k = 1,2, 3. (d) Determine a and c as defined in (8.8).

Figure 8.3 1. Illustration of several fractal trees.

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Problem 8.3. The asymptotic branching relation (8.14) is not valid for the fractal tree illustrated in Figure 8.31(b) because it has triple branching rather than double (binary) branching. (a) Show that the correct relation is

(b) Determine the asymptotic branching ratio for the fractal tree illustrated in Figure 8.3 1(b) and the corresponding fractal dimension. Problem 8.4. Consider the fractal tree illustrated in Figure 8.31(c). The length-order ratio is Rr = 3. (a) Determine the bifurcation ratios for 1st to 2nd, 2nd to 3rd, and 3rd to 4th order branches. What are the corresponding fractal dimensions? (b) Write the branch-number matrix and branch-ratio matrix for this tree. (c) Determine T, fork = 1 , 2 , 3 . (d) Determine a and c as defined in (8.8). (e) Determine the asymptotic bifurcation ratio and fractal dimension for large trees of this form using the result given in Problem 8.3. Problem 8.5. Consider the deterministic fractal growth model illustrated in Figure 8.32. An initial equilateral triangle at zero order (a) grows at first order (b) by the addition of three triangles at the three corners of the original seed particle. At second order (c), three of the first-order construction are added as shown. (a) What is the fractal dimension of this construction? (b) What is the radius of gyration at first and second order? What is the corresponding fractal dimension from (8.22)?

Figure 8.32. Illustration of a deterministic fractal growth model based on equilateral triangles.

Chapter Nine

DYNAMICAL SYSTEMS

9.1 Nonlinear equations We now turn our attention to some examples of deterministic chaos that have applications in geology and geophysics. There are two requirements for a solution that exhibits deterministic chaos. The first requirement is that we are solving deterministic equations with specified initial andlor boundary conditions. Thus the applicable equations are deterministic, not statistical. The second requirement is that solutions that have initial conditions that are infinitesimally close diverge exponentially as they evolve. However, before we consider solutions that are chaotic, some necessary introductory material is presented. Chaotic behavior is found only for nonlinear systems of equations. In this chapter some of the standard nomenclature for the study of nonlinear equations is presented (Verhulst, 1990). Probably the simplest nonlinear total differential equation is the logistic equation

This equation is a simple model for population dynamics with x representing population and t time; the first term on the right-hand side accounts for births and the second deaths. The parameter T is the characteristic time and xe is a representative population. It is standard procedure in treating nonlinear equations to introduce nondimensional variables. This procedure reduces the number of parameters that must be varied to study the solutions. For the logistic equation it is appropriate to introduce

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Substitution of (9.2) into (9.1) gives

and there are no governing parameters. An exact solution of (9.3) is

where Z = Z, at t = 0 is the initial condition. Solutions of (9.4) for various values of Xo are given in Figure 9.1. Although (9.1) is solved by writing (9.4), it is illustrative to examine the of solutions in somewhat more detail. We first examine the fixed points ip (9.3). As the solutions of (9.3) evolve in time, the time dependence dies out and i approaches a particular value. This value is known as a stable fixed point. The fixed points are formally obtained by setting dildl = 0 in (9.3) with the result that

If the initial condition is either of these values, then no time dependence is obtained. It is also of interest to examine the stability of the fixed points. To do this the applicable equation is linearized about the fixed point. In the vicinity of the fixed point ip = 0, it is appropriate to neglect the quadratic term in (9.3) with the result

Figure 9.1. Solutions of the logistic equation (9.1) for several initial conditions go. All solutions converge to the fixed point X = 1.

DYNAMICAL SYSTEMS

d x- -X dt The solution is T = &,el where i , is the value of 2 at t = 0 and is assumed to be finite but small. Thus solutions in the immediate vicinity of the fixed point 3, = 0 diverge with time and are unstable; this fixed point is unstable. To examine the fixed point = 1 it is appropriate to introduce the new variable 2,:

Substitution into (9.3) gives

In the vicinity of the fixed point Z, = 0 it is appropriate to neglect the quadratic term in (9.9) with the result

which has the solution

where i,,is again assumed to be small but finite. As time evolves i , approaches zero. Thus the solutions in the immediate vicinity of the fixed point x = 1 are stable. The stability of the fixed point 3p= 1 is clearly illustrated in dgure 9.1. For all initial conditions iothe solut~ons"flow" in time toward the stable fixed point 2, = 1. Also, adjacent solutions tend to converge toward each other. These solutions are not chaotic. To discuss further the behavior of nonlinear equations we consider the van der Pol equation

221

222

DYNAMICAL SYSTEMS

If a = p = 0 this is the equation of motion of a spring-mass oscillator system with M the mass, k the spring constant, and x the extension of the spring. The coefficients ci and p represent the linear and nonlinear damping terms, respectively. Again it is appropriate to introduce nondimensional variables. The frequency of the harmonic oscillator o = (klM)lQintroduces a natural time to the problem. The relative amplitudes of the damping terms a and P introduce a natural length scale (a/P)112/o to the problem. Using these time and length scales we introduce the nondimensional variables t and i ;we also define the nondimensional parameter E according to

Substitution of (9.13)into (9.12)gives

with E the only parameter governing the behavior. In considering the solutions of this second-order nonlinear equation, it is standard practice to introduce the definition of velocity

and to rewrite (9.14)as

Dividing (9.16)by (9.15)gives

The E-space is known as the phase space or phase plane. Solutions of (9.17) follow phase trajectories in this two-dimensional plane with time t as a parameter. We first consider the solution of (9.17) when E = 0. In this case it becomes

DYNAMICAL SYSTEMS

which integrates to give

Simple harmonic motion is a circle in the phase plane. The radius of the circle is determined by the initial nondimensional amplitude 2, (or the initial velocity). The relation (9.19) also represents conservation of energy in this nondissipative system; it is the sum of the potential and kinetic energies. For this system the fixed point at x = j = 0 is known as a center. The behavior is illustrated in Figure 9.2(a). For finite values of E it is necessary to solve (9.17) numerically. The result for E = 1 is given in Figure 9.2(b). solutions for all initial conditions converge toward a limit cycle; this limit cycle is independent of the initial conditions. The physical reason for this behavior can be seen in the original van der Pol equation (9.12). For small amplitudes the negative linear damping term dominates and the amplitude increases. For large amplitudes the positive cubic damping term dominates and the amplitude decreases. The result is that all solutions converge on the same limit cycle at large times. Many sets of equations that produce deterministic chaos for a range of parameter values produce limit cycles for other parameter values. Before considering chaotic solutions we will present some further introductory material on singular points. Consider the pair of linear total differential equations

Figure 9.2. (a) Solutions of the nondimensional van der Pol equation (9.14) in the phase plane with E = 0.The solution is a circle representing simple harmonic motion. The position of the circle is dependent upon initial conditions. (b) Solution of the van der Pol equation (9.14) in the phase plane with E = 1. The solutions approach a limit cycle independent of initial conditions. Solutions for initial conditions inside the limit cycle spiral out to it and solutions for initial conditions outside the limit cycle spiral into it.

224

DYNAMICAL SYSTEMS

where a, b, c, and f are constants. Dividing (9.20) by (9.21) gives

If b = - c and a =f = 0 this reduces to (9.18); the solution is given by (9.19) and is a circle in the xy-plane as illustrated in Figure 9.2(a). If a = a/f and b = c = 0 we can write (9.22) as

This equation has the solution

y = yx" where y is the constant of integration. If a > 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 <1 and adjacent solutions converge. We next consider the triangular or tent map defined by

Figure 10.8. Illustration of iterations of the linear map (10.28). (a) With a = 0.6 and x,, = 0.8, the iteration converges on stable fixed point x, = 0. (b) With a = 1.2 and x,, = 0.8, the iteration diverges to infinity.

242

LOGISTIC M A P

with 0 < a < 1 and 0 < x < 1 . This map can also be defined by x n + , = 2a xn

for

0

1

< xn < 2

(10.33)

For 0 < a < $ the only fixed point is at x, = 0 and it is stable. For $ < a < 1 2a ;both fixed points are unstable. there are fixed points at x, = 0, x, = (1 + 2a) Illustrations of the iteration of this map for a = 0.4, 0.8, and x, = 0.8 are given in Figure 10.9. With a = 0.4 we have x , = 0.16, x2 = 0.128, x3 = 0.1024, and x4 = 0.08192, and the solution iterates to the stable fixed point x, = 0. With a = 0.8 we have x , = 0.32, x2 = 0.512, x, = 0.7808, and x4 = 0.35072. The iteration for a = 0.8 is chaotic. All iterations with 0.5 < a < 1 are

Figure 10.9. Illustration of iterations of the triangular map (10.33). (a) With a = 0.4 and x,, = 0.8, the iteration converges to the stable fixed point x, = 0. (b) With a = 0.8 and x, = 0.8, the iteration is chaotic in the range 0.32 c x < 0.8.

Figure 10.10. Bifurcation diagram for the triangular map (10.33) as a function of the parameter a.

LOGISTIC M A P

chaotic; this can be demonstrated by noting that the Lyapunov exponents for the triangular map are easily obtained from the Lyapunov exponents for the linear map given in (10.31); the result is A = - log 2a log 2

Thus A is negative for x < a <0.5 and positive for 0.5 < a < 1. In the chaotic regime, the range of values of x is 2a (1 - a ) < x < a. The resulting bifurcation diagram for the triangular map is given in Figure 10.10. For 0 < a < 0.5 the solutions converge to the stable fixed point x, = 0. For 0.5 < a < 1.0 chaotic behavior is found between the limits given above. The recursive maps considered in this chapter are clearly very simple models with limited direct applicability to problems in geology and geophysics. However, the complex chaotic behavior exhibited by the simple models is strongly indicative that many natural systems can be expected to behave chaotically. Some natural systems exhibit behavior that closely resembles the behavior of recursive maps. As an example Sornette et al. ( 1991) and Dubois and Cheminee (1991) have treated the return periods for eruptions of the volcanoes Piton de la Fournaise on Reunion Island and Mauna Loa and Kilauea in Hawaii as return maps. The results appear to resemble the chaotic maps considered in this chapter.

Problems Problem 10.1. Determine x i , x,, x,, and x, for the logistic map (10.1) taking a = 0.5 and xo = 0.5. What is the value of x,? Problem 10.2. Determine x , , x,, x,, and x, for the logistic map (10.1) taking a = 0.9 and x, = 0.75. What is the value of x,? Problem 10.3. Determine x , , x,, x,, and x, for the logistic map (10.1) taking a = 2 and xo = 0.2. What is that value of x,? Problem 10.4. Determine x , , x,, x,, and x, for the logistic map (10.1) taking a = 2.5 and xo = 0.3. What is that value of x,? Problem 10.5. Determine x,, and xn for the logistic map (10.1) taking a = 3.2. Problem 10.6. Determine x,, and x, for the logistic map (10.1) taking a = 3.4. Problem 10.7. For a = 3.7 the logistic map (10.1) is fully chaotic. What are the maximum and minimum values of xn? Problem 10.8. For a = 3.8 the logistic map (10.1) is fully chaotic. What are the maximum and minimum values of xn?

243

244

LOGISTIC M A P

Problem 10.9. Determine x,, x,, x,, x,, and x, for the logistic map (10.1) takinga = 4 and p = ( 2 ~ ) - 1 . Problem 10.10. Determine x,, x,, x,, x,, and x, for the logistic map (10.1) takinga = 4 and p = ( 3 ~ ) - 1 . Problem 10.1 1. Show that x, = 0 is a fixed point of the linear map (10.28). Determine the value of r defined in (10.6) for this map and determine the stability of the fixed point. Problem 10.12. Determine the fixed points for the triangular points for the triangular map (10.32). Determine the values of r defined in (10.6) for the fixed points and determine their stability. Problem 10.13. Consider the iterative map

This map is also used for population dynamics (May and Oster, 1976). (a) Determine the fixed points and the range of positive values for a that are stable. (b) Determine x,, x,, x,, x,, and x, taking a = 3 and x, = 0.5.

(c) For a = 3, what are the maximum and minimum values of xn?

Chapter Eleven

SLIDER-BLOCK MODELS

We next turn to a low-order example of deterministic chaos that has somewhat more direct applications to geology and geophysics. As discussed in Chapter 4, we accept the hypothesis that earthquakes occur repetitively on preexisting faults. A simple model for the behavior of a fault is a slider block pulled by a spring as illustrated in Figure 11.1 (Burridge and Knopoff, 1967). The block is constrained to move smoothly along the surface. It interacts with the surface through friction; this friction prevents sliding of the block until a critical value of the pulling force is reached. The block sticks and the force in the spring increases until it equals the frictional resistance to sliding on the surface, and then slip occurs. The extension of the spring is analogous to the elastic strain in the rock adjacent to a fault. The slip is analogous to an earthquake on a fault. This is stick-slip behavior. The stored elastic strain in the spring is relieved; this is known as elastic rebound on a fault. The behavior of this simple spring-block model will now be studied quantitatively. A constant velocity driver moving at velocity v extends the spring with spring constant k until the pulling force ky equals the frictional static resisting force Fs. The static condition for the onset of sliding is thus

Figure 1 1 . 1 . Illustration of the slider-block model for fault behavior. The constant velocity driver extends the spring until the force ky exceeds the static friction force Fs.

246

SLIDER-BLOCK MODELS

Once sliding begins the equation of motion for the block is

where rn is the mass of the block and Fd is the dynamic or sliding friction. The sliding is analogous to an earthquake and relieves the stress in the spring in analogy to elastic rebound. The further assumption is made that the loading velocity of the driver, v, is sufficiently slow so that we may assume it to be zero during the sliding of the block. This is reasonable since an earthquake lasts only a few tens of seconds, whereas the interval between earthquakes on a fault is typically hundreds of years or more. The static-dynamic friction law is the simplest that generates stick-slip behavior. A necessary and sufficient condition for stick-slip behavior is that the static friction exceeds the dynamic friction, F, > F,. A variety of empirical velocity-weakening friction laws are in agreement with laboratory observations and also generate stick-slip behavior. Dynamic instabilities associated with complicated friction laws are well known from single-block models (Byerlee, 1978; Dieterich, 1981; Ruina, 1983; Rice and Tse, 1986). Slider-block models have been used to simulate foreshocks, aftershocks, pre- and post-seismic slip, and earthquake statistics (Dieterich, 1972; Rundle and Jackson, 1977; Cohen, 1977; Cao and Aki, 1984, 1986). Gu et al. (1984) found some chaotically bounded oscillations; Nussbaum and Ruina (1987) used a two-block model with spatial symmetry and found periodic behavior. Huang and Turcotte (1990a, 1992) and McCloskey and Bean (1992) studied the same system without spatial symmetry and obtained classic chaotic behavior. We first consider the solution for the behavior of the single block shown in Figure 11.1. It is convenient to introduce the nondimensional variables

In terms of these variables, the sliding condition (11.1) becomes

and the equation of motion (1 1.2) becomes

SLIDER-BLOCK MODELS

247

We assume sliding starts at T = 0 with Y = 1 (1 1.4) and dY1d.r = 0. The applicable solution is

Y = - +

4

(1 - -t)

COST

Sliding ends at T = .rr when dY1d.r is again zero. When the velocity is zero the friction jumps to its static value, preventing further sliding. The position of the block at the end of sliding is Y = (2/+) - 1 so that the slip during sliding is

The dependence of Y and d Y l d ~on T during sliding are given in Figure 11.2 for 4 = 1.25. For this case Y drops from 1 to 0.6 during sliding and AY = -0.4. After sliding is completed, the spring extends due to the velocity of the driver until Y again equals unity and the cycle repeats. With a single slider block periodic behavior is obtained. The variables Y and dY1d.r define a phase plane for the solution. We next consider the behavior of a pair of slider blocks as illustrated in Figure 11.3. We will show that the behavior of the blocks can be a classical example of deterministic chaos. The blocks are an analog of two interacting faults or two interacting segments of a single fault. A constant velocity driver drags the blocks over the surface at a mean velocity v. The two blocks are coupled to each other and to the constant velocity driver with springs whose constants are kc, k , , and k,. Other model parameters are the block masses m , and m, and the frictional forces F, and F,. The position coordinates of the blocks relative to the constant velocity driver are y , and y,. The static conditions for the onset of sliding are the force balances

Figure 11.2. Dependence of Y and dY1d.r on T during a sliding episode assuming = 1.25.

+

248

SLIDER-BLOCK MODELS

Once sliding begins the applicable equations of motion are (11.10)

(11.11) To simplify the model we assume rn, = rn2 = rn, k, = k2 = k, and Fs,IFD, = Fs21FD2 = In addition to the nondimensional variables introduced in (1 1.3), we write the nondimensional parameters

+'

(11.12) The parameter a is a stiffness parameter. If a = 0 the blocks are completely decoupled and each will exhibit the periodic behavior described above for a single block. As a + the blocks become locked together and act as a single block that again exhibits the periodic behavior described above. If P = 1 there is complete symmetry between the two blocks, P # 1 introduces an asymmetry. In terms of the nondimensional variables (1 1.3) and parameters (1 1.12), the sliding conditions (1 1.8) and (1 1.9) become

-

and the equations of motion (1 1.10) and (11.11) become

Figure 11.3. Illustration of the two-block model. The constant velocity driver extends the springs until sliding of a block commences. In some cases sliding of one block induces the sliding of the second block.

SLIDER-BLOCK MODELS

249

The blocks are expected to exhibit stick-slip behavior for c$ > 1. The first block will begin sliding if (11.13) is satisfied, and the second block will begin to slide if (11.14) is satisfied. Together (11.13) and (11.14) define a failure envelope in the Y, Y,-plane. The sliding behavior is governed by (1 1.15) and (1 1.16). In some cases the sliding of one block induces the sliding of the second block. The solutions for the behavior of the blocks can be represented in a fourdimensional phase plane consisting of Y,, Y2, dY,ld~,and dY,ld~.For simplicity we consider the projection of the solution onto the Y, Y2 plane. We first consider the symmetric case in which both blocks have the same frictional behavior, that is, P = 1. An example with a = 3 and y = 1.25 is given in Figure 11.4. The diagonal lines converging at Y, = Y2 = 1 are the failure envelope given by (1 1.13) and (1 1.14). A periodic orbit is given by abcd in Figure 11.4. At point a block 2 fails with Y, = 0.780 and Y2 = 0.835. During the slip of block 2, block 1 remains fixed and the slip of block 2 is represented by the vertical line ab in the Y,Y2 phase plane. The termination of the sliding of block 2 is obtained from (11.16). Sliding of block 2 termi-

Figure 11.4. Behavior of the symmetric p = 1 two-block model illustrated in Figure 1 1 . 3 w i t h a = 3 a n d + = 1.25. Cyclic behavior is obtained with alternating slip of the two blocks.

250

SLIDER-BLOCK MODELS

nates at point b where Y, = 0.780 and Y2 = 0.735. From point b to point c the blocks stick and the springs extend due to the movement of the constant-velocity driver. The increments in Y, and Y, are equal and the strain accumulation phase is represented by the diagonal line bc, which has unit slope. The termination of this strain occurs when this line intercepts the failure envelope. This occurs at point c, where Y, = 0.865 and Y2 = 0.820. During the slip of block 1, block 2 remains fixed and the slip of block 1 is represented by the horizontal line cd. The termination of the sliding of block 1 is obtained from (1 1.15). Sliding of block 1 terminates at point d, where Y, = 0.765 and Y2 = 0.820. Between points d and a the blocks stick and the springs again extend at equal rates due to the movement of the constant-velocity driver. The increments in Y, and Y2 are equal and the strain accumulation phase is represented by the diagonal line ad, which has unit slope. The termination of this strainaccumulation phase occurs when this line intercepts the failure envelope. This occurs at point a, and the cycle repeats. The behavior of this symmetrical two-block model is periodic, with first one block sliding and then the other. We next consider an asymmetric case with p = 2.5, a = 3.49, and = 1.25. The results are given in Figure 11.5. The behavior is similar to that given in Figure 10.5 and is fully chaotic. The curves that fall outside the failure envelope are cases in which both blocks are sliding simultaneously. When a diagonal strain-accumulation line intercepts the upper failure envelope, block 2 begins to slide. Because P is relatively large, the failure force

+

Figure 11.5. Behavior of an asymmetrical two-block model with f3 = 2.25, a = 3.49, and 4 = 1.25. Chaotic behavior is obtained.

SLIDER-BLOCK MODELS

251

for block 2 is considerably larger than the failure force for block 1. Thus the vertical failure path for block 2 crosses the failure envelope of block 1 and it begins to slide. The sliding of both blocks results in S-shaped curves. The next strain-accumulation phase intercepts the lower failure envelope and block 1 begins to slide. Because of the large force required to induce the failure of block 2, these horizontal failure paths do not cross the upper failure envelope. The result is a chaotic sequence of failures of both blocks together, followed by a failure of the weaker block 1 . To study this behavior further a bifurcation diagram is given in Figure 11.6. The values of Y, - Y , at the termination of slip are given for various values of a with p = 2.25 and = 1.25. A detailed illustration of the behavior in the range 3.2 < a < 3.5 is given in Figure 11.7. Solutions evolve to an asymptotic, large-time behavior independent of the initial conditions. As illustrated in Figures 11.6 and l l .7, the system may evolve to limit cycle behavior or chaotic behavior. A series of period-doubling pitchfork bifurcations are clearly illustrated in Figure 11.7. The cyclic behavior for the n = 2 limit cycle obtained for a= 3.25 is given in Figure 11.8. The cyclic behavior for the n = 4 limit cycle obtained for a = 3.38 is given in Figure 11.9. These limit cycles evolve into the type of chaotic behavior illustrated in Figure 11.5. The behavior of the asymmetric two-block model is remarkably similar to that of the logistic map. The values of the Lyapunov exponents corresponding to the points given on the bifurcation diagram in Figure 11.6 are given in Figure 1 1.10. The windows of chaotic behavior are clearly illustrated.

+

'I.

.4l::...l!:...

Figure 11.6. Bifurcation diagram for an asymmetric two-block model with f3 = 2.25 and = 1.25. The values of Y, = Y, are given as a function of a. Singular point, limit cycle, and chaotic types of behavior are found.

+

252

SLIDER-BLOCK MODELS

A modification to the analysis of a pair of slider blocks is to allow only one slider block to slip at a time. When a slider block becomes unstable, it is allowed to move according to (1 1.6). The stability of the second slider block is determined, and if it is unstable, it is allowed to move according to (1 1.6). The primary advantage of this modification is that slider-block displacements are given by an analytical expression (11.6) and numerical solutions

Figure 11.7. Details of the bifurcation diagram given in Figure 11.6 for the region 3.2 < a < 3.5. The pitchfork bifurcation illustrated evolves to deterministic chaos.

Figure 11.8. Behavior of the asymmetric two-block model with p = 2.25, a = 3.25, and c$ = 1.25. An n = 2 limit cycle is obtained.

SLIDER-BLOCK MODELS

253

of differential equations are not required. Studies of this modified sliderblock problem (Narkounskaia and Turcotte, 1992; Narkounskaia et al., 1992) yield a variety of behaviors including periodic solutions, limit cycles, and chaotic solutions. Huang and Turcotte ( 1 9 9 0 ~ )have applied the chaotic behavior of the asymmetric two-block system to two examples of interacting fault segments. The Pacific plate descends beneath the Asian plate, resulting in the forma-

Figure 11.9. Behavior of the asymmetric two-block model with p = 2.25, a = 3.38, and = 1.25. Ann = 4 limit cycle is obtained.

+

Figure 11.10. Dependence of the Lyapunov exponent on a corresponding to the bifurcation diagram given in Figure 11.6.

254

SLIDER-BLOCK MODELS

tion of the Nankai trough along the coast of southwestern Japan. The relative motion between the plates has resulted in a sequence of great earthquakes that have been documented through historical records for the period AD 684-1946. The sequence is marked by an irregular but somewhat repetitive pattern in which whole section failures occur following several alternate failures of single segments. In the two-block model the simultaneous slip of both blocks corresponds to an earthquake that ruptures the entire section, and single-block failures correspond to an earthquake on a single segment. ) chaotic Taking P = 1.05 and a = 0.81, Huang and Turcotte ( 1 9 9 0 ~ found model behavior that strongly resembled the observed sequence of earthquakes in the Nankai trough. Another example is the interaction between the Parkfield segment and the rest of the south central locked segment of the San Andreas fault in California. A sequence of magnitude-six earthquakes occurred on the Parkfield segment in 1881, 1901, 1922, 1934, and 1966. The last great earthquake on the locked segment to the south occurred in 1857 and is also associated with a rupture on the Parkfield segment. Taking P = 2 and a = 1.2, Huang and Turcotte ( 1 9 9 0 ~ )found chaotic model behavior similar to that described above. A sequence of slip events on the weaker block often preceded the simultaneous slip of the weaker and stronger blocks. The model simulation suggested two alternative scenarios for a great southern California earthquake following a sequence of Parkfield earthquakes. In the first case a Parkfield earthquake will transfer sufficient stress to trigger the great southern California earthquake; the Parkfield earthquake is thus essentially a foreshock for the great earthquake. In the second case a small additional strain after a Parkfield earthquake will trigger an earthquake on the southern section and this will result in an additional displacement on the Parkfield section. The evolution of the system is chaotic: its evolution is not predictable except in a statistical sense. Spring-block models are a simple analogy to the behavior of faults in the earth's crust. However, the chaotic behavior of low-dimensional analog systems often indicates that natural systems will also behave chaotically. Thus it is reasonable to conclude that the interaction between faults that leads to the fractal frequency-magnitude statistics discussed in Chapter 4 is an example of deterministic chaos. The prediction of earthquakes is not possible in a deterministic sense. Only a probabilistic approach to the occurrence of earthquakes will be possible.

Problems

+

Problem 11.1. Consider a single slider block with = 1.5. (a) At what value of Y does slip occur? (b) What is the value of Y after slip?

SLIDER-BLOCK MODELS

+

Problem 11.2. Consider a single slider block with = 3. (a) At what value of Y does slip occur? (b) What is the value of Y after slip? Problem 11.3. For a single slider block determine the dependence of V = dYldT on Y during slip. Problem 11.4. Consider a pair of slider blocks with a = 0, P = 1, and = 2. Assume that initially Y, = 0.5, Y2 = 0. (a) What are the values of Y, and Y, when block 1 first slips? (b) What are the values of Y, and Y, after block 1 slips? (c) What are the values of Y, and Y2 when block 2 first slips? (d) What are the values of Y, and Y2 after block 2 slips? (e) Draw the behavior of the system in the Y, Y2 phase plane. Problem 11.5. Consider a pair of slider blocks with a = 0, P = 1, and 4 = 413. Assume that initially Y, = 0.75, Y2 = 0.5. (a) What are the values of Y, and Y, when block 1 first slips? (b) What are the values of Y, and Y2 after block 1 slips? (c) What are the values of Y, and Y, when block 2 first slips? (d) What are the values of Y, and Y2 after block 2 slips? (e) Draw the behavior of the system in the Y, Y2 phase plane.

+

255

EQUATIONS

Sets of coupled nonlinear differential equations can also yield solutions that are examples of deterministic chaos. The classic example is the Lorenz equations. Lorenz (1963) derived a set of three coupled total differential equations as an approximation for thermal convection in a fluid layer heated from below. He showed that the solutions in a particular parameter range had exponential sensitivity to initial conditions and were thus an example of deterministic chaos. This was the first demonstration of chaotic behavior. The Lorenz equations have been studied in detail by Sparrow (1982). Because of their historical significance and because thermal convection in the earth's mantle drives plate tectonics, we will consider the Lorenz equations in some detail. When a fluid is heated its density generally decreases because of thermal expansion. We consider a fluid layer of thickness h that is heated from below and cooled from above; the cool fluid near the upper boundary is dense and the fluid near the lower boundary is light. This situation is gravitationally unstable. The cool fluid tends to sink and the hot fluid tends to rise. This is thermal convection. Appropriate forms of the continuity, force balance, and energy balance equations are required for a quantitative study of thermal convection. We will restrict our attention to two-dimensional flows in which the velocities are confined to the xy-plane. Continuity of fluid requires that

where u is the horizontal component of velocity in the x-direction and v is the vertical component of velocity in the y-direction, measured downward from the upper boundary. In writing (12.1) the density is assumed to be constant. Force balances in the x and y directions require that

LORENZ EQUATIONS

where p is density, Ap the density difference, p pressure, CI, viscosity, and g the acceleration due to gravity. The fractional density difference Aplp is assumed to be small so that Ap can be neglected except in the buoyancy term Ap g of the vertical force equation (12.3). This is known as the Boussinesq approximation. The terms on the left-hand side of (12.2) and (12.3) are the inertial forces associated with the acceleration of the fluid. The first terms on the right-hand sides of the equations are the pressure forces and the second terms are the viscous forces. The energy balance requires that

where T is temperature, k thermal conductivity, and cpis the specific heat at constant pressure. The terms on the left-hand side of (12.4) account for the convection of heat and the terms on the right-hand side account for the conduction of heat. A full derivation of these equations has been given by Turcotte and Schubert (1982, pp. 240-74). In the absence of convection, that is, u = v = 0, the temperature Tc is a linear conduction profile given by (12.5) where T , is the constant temperature of the upper boundary ( y = 0) and T, is the constant temperature of the lower boundary ( y = h). The incompressible continuity equation in two dimensions (12.1) is satisfied if we introduce a stream function I+!J defined by

It is also convenient to introduce the temperature difference 0 between the actual temperature T and the temperature Tc in the absence of convection:

The density difference Ap in the buoyancy term of the vertical force the equation (12.3) is related to this temperature difference by

257

258

LORENZ EQUATIONS

where a is the volumetric coefficient of thermal expansion. Substitution of (12.6)-(12.8) into (12.1)-(12.4) gives

The problem has been reduced to the solution of two partial differential equations for q!~and 8. To better understand the roles of various terms, it is appropriate to introduce the nondimensional variables

where K = k/(pcp) is the thermal diffusivity. With these nondimensional variables two nondimensional parameters govern the behavior of the equations:

where Ra is the Rayleigh number and Pr is the Prandtl number. The Rayleigh number is a measure of the strength of the buoyancy forces that drive convection relative to the viscous forces that damp convection. The higher the Rayleigh number the stronger the convection. The Prandtl number is the ratio of the momentum diffusion to the heat diffusion. It is instructive to estimate these two parameters for the earth's mantle. Due to solid-state creep, the earth's mantle has a mean viscosity of around = 1021 Pa s, its thickness is h = 2880 krn, and the temperature increase across it is estimated to be T2 TI = 3000 K. For the rock properties we taken K = 1 mm2s-1 and a = 3 X 10-5K-1. We assume g = 10 m s-2 and an average density p = 4000 kg m-3

LORENZ EQUATIONS

and find Ra = 8.6 X lo6 and Pr = 2.5 X 1023, both very large values. The behavior of the earth's mantle will be discussed further in the next chapter. We now return to the basic equations. Substitution of (12.11) to (12.13) into (12.9) and (12.10) gives

The solution is determined by the two parameters Ra and Pr and the boundary conditions. For small values of the Rayleigh number, the viscous forces are sufficiently strong to prevent any flow. Thus there is a critical minimum value of the Rayleigh number for the onset of thermal convection. We next consider a linearized stability analysis for the onset of convection as given by Rayleigh (1916). Only terms linear in 8 and 3 are retained and the marginal stability problem is solved by setting dlat = 0. Thus (12.14) and (12.15) become

It is appropriate to assume solutions that are periodic in the horizontal coordinate 3. Solutions that satisfy (12.17) are

t j= 0=

sin

275 (T) sin

sin rrj

The flow is assumed to be periodic in the horizontal direction with a wavelength A; the nondimensional wavelength is h = hlh. The flow consists of counter-rotating cells; each cellular flow has width A12, as illustrated in Fig-

259

260

LORENZ EQUATIONS

ure 12.1. The temperature boundary conditions e = 0 at 7 = 0, 1 are satisfied by (12.19). The requirement of no flow through the walls corresponds to i = = 0 at 7 = 0, 1 and these conditions are satisfied by (12.18). If the boundaries of the layer are solid surfaces, we would require E = - a@ay = 0 at j = 0, 1. These are the no-slip conditions requiring that there be no relative motion between a viscous fluid and a bounding solid surface at the solidfluid interface. If the boundaries of the layer are free surfaces, that is, if there is nothing at the boundaries to exert a sheer stress on the fluid, we would require that the shear stress be zero at L = 0, 1. These free surface boundary conditions can be written as aiilaj = -a2$/ay2 = 0 at 7 = 0, 1. The stream function given in (12.18) satisfies the free surface boundary conditions. In order that (12.18) and (12.19) also satisfy (12.16) we require that

When the nondimensional wavelength h is specified, this is the minimum value of the Rayleigh number at which convection will occur. As the Rayleigh number for a fluid layer is increased, flow will occur at the wavelength for which the Rayleigh number given by (12.20) is a minimum. This value of the Rayleigh number is given by

Ra,

27 7r4 4

= ---- =

657.5

This is the critical Rayleigh number for the onset of thermal convection in a fluid layer heated from below. At Rayleigh numbers less than that given by (12.21), thermal convection will not occur. The nondimensional wavelength corresponding to (12.21) is

Figure 12.1. Illustration of two-dimensional cellular convection in a fluid layer heated from below.

LORENZ EQUATIONS

This is the wavelength of the initial convective flow that takes the form of counter-rotating, two-dimensional cells. Each cell has a width 21/2h, one-half the wavelength of the initial disturbance. A pitchfork bifurcation occurs at the critical Rayleigh number. If Ra < Rac the only solution is the conduction solution, which is stable. Above the critical Rayleigh number the conduction solution remains a solution of the governing equations but it is now unstable. Above the critical Rayleigh number there are two stable convective solutions corresponding to cellular rolls rotating either clockwise or counterclockwise. This is identical to the pitchfork bifurcation illustrated in Figure 9.4(b). Because the applicable equations are linear, the stability analysis does not predict the amplitude of the convection. It is not possible to specify the value of $, in (12.18) and (12.19). To determine the amplitude of the thermal convection, it is necessary to retain nonlinear terms. One approach to the solutions of the full nonlinear equations (12.14) and (12.15) is to expand the variables and 8 in double Fourier series in i and j with coefficients that are functions of time. Lorenz (1963) strongly truncated these series and retained only three terms of the form

1

$ = j q (4 + i 2 ) ~ ( rsin ) 7r3 4~~

= -(4

2 6 (T) sin ~7

+ X2l3 C ( I ) sin 27rJ - 21i2Z3(r) cos

(9) 2

-

sin ~ j ] (12.24)

where

with Rac given by (12.20). These equations satisfy the same set of boundary conditions that (12.18) and (12.19) satisfy. The expansion of the stream function (12.23) is essentially identical to the form used in the linear stability analysis (12.18). However, the expansion of the temperature (12.24) includes an additional term that is not dependent on i . It is necessary to derive differential equations for the time dependence of This is done by substituting the expansions the coefficients A(T), B(r), C(T). (12.23) and (12.24) into the governing equations (12.14) and (12.15). Coefficients of the Fourier terms are equated to obtain the necessary equations.

261

262

LORENZ EQUATIONS

All nonlinear terms in the stream function equation (12.14) are neglected. When (12.23) and (12.24) are substituted into this equation the result is

As expected this is a linear equation and no further approximations have been made in writing it. To maintain a consistent approximation in the energy equation (12.15), it is necessary to retain several nonlinear terms. Substitution of (12.23) and (12.24) into (12.15) gives several terms that are not in the form required. These products of trigonometric functions are reduced to the standard form using the multiple angle formulas

cos (271~1%) sin .rrj cos (271~1% ) cos T%= sin 2-

(12.29)

with higher-order terms neglected. Equating coefficients gives

where

The three first-order total differential equations (12.27), (12.30), and (12.31) are the Lorenz equations. These equations would be expected to give accurate solutions to the full equations when the Rayleigh number is slightly supercritical, but large errors would be expected for strong convection because of the extreme truncation. Solutions of the Lorenz equations represent cellular, two-dimensional convection. Because only one term is retained in the expansion of the stream function, the particle paths are closed and represent streamlines even when the flow is unsteady. The time dependence of the coefficient A determines the velocity of a fluid particle. But the fluid particle follows the same closed trajectory independent of its time variation. The coefficient B represents temperature variations associated with the stream function mode A. The co-

LORENZ EQUATIONS

efficient C represents a horizontally averaged temperature mode. A detailed discussion of the behavior of the Lorenz equations has been given by Sparrow (1982). To examine the behavior of the Lorenz equations, we first determine the allowed steady-state solution. Obviously the steady-state solution A = B = C = 0 corresponds to heat conduction without flow. An additional pair of allowed solutions is

These solutions correspond to an infinite set of two-dimensional cells as illustrated in Figure 12.1. Adjacent cells rotate in opposite directions; the choice of sign given in (12.33) determines whether a specified cell rotates clockwise or counterclockwise. A stability analysis for the conduction solutions shows that it is stable for r < 1 and unstable for r > 1. Thus the Lorenz equations exhibit the same type of pitchfork bifurcation at r = 1 (Ra = Rat) that the full equations do. This is expected since the linearized form of the Lorenz equations is identical to the linearized form of the full equations. The stability of the steady solution given in (12.33) and (12.34) can also be examined. Expanding about this solution with

and substituting into (12.27), (12.30), and (12.31) with linearization gives the characteristic equation A3

+ (Pr + b + l)A2 + ( r + Pr)bA + 2b Pr(r - 1) = 0

(12.38)

This equation has one real negative root and two complex conjugate roots when r > 1. If the product of the coefficients of A2 and A equals the constant term we obtain

At this value of r the complex roots of (12.38) have a transition from negative to positive real parts. This is the critical value of r for the instability of steady convection and represents a subcritical Hopf bifurcation. If Pr > b + 1

263

264

LORENZ EQUATIONS

the steady solutions given by (12.33) and (12.34) are unstable for Rayleigh numbers larger than those given by (12.39). To examine further the behavior of the Lorenz equations it is necessary to carry out numerical solutions. Following Lorenz (1963)we consider h, = 8112, the critical value from (12.22), so that b = For these values the steadystate solution given by (12.23), (12.24), (12.33), and (12.34) becomes

i.

= 2 [24(r -

3 = E 4{

( r

-

I)]''~sin

(Z) [y -

I) sin 2?rj

sin 71.y

T -

r (

-

1)

cOs (.Dli) 21/2

}

Ty

(12.41)

This steady-state solution is valid if r > 1 and is less than the critical value given by (12.39). As the Rayleigh number increases above r = 1, the strength of the convection increases, as indicated by (12.40). This results in larger transport of heat by convection, and as a result the thermal gradients at the upper and lower boundaries increase. The Nusselt number is a measure of the efficiency of the convective heat transfer across the layer. The Nusselt number Nu is the ratio of the heat transferred by convection to the conductive value without convection. In terms of our nondimensional variables it is given by

where ( ), indicates an average across either the upper or lower surface. These averages must be equal since the mean heat flux into the layer across the lower boundary must be equal to the mean heat flux out of the layer through the upper boundary. For the steady-state solution given by (12.41) the Nusselt number is

This relation is not in good agreement with experiment when r is significantly larger than unity. This is clearly due to the extreme nature of the truncation, which is expected to be valid only near the stability limit r = 1. Nevertheless, it is of interest to explore the behavior of the Lorenz equations for larger values of r. The critical Rayleigh number for stability of the steady-state solution is r = 24.74 for h = 8112 from (12.39). For values of r greater than this, unsteady

LOREN2 EQUATIONS

solutions are exposed. A numerical solution of the Lorenz equations for r = 28 is given in Figure 12.2. The time dependence of the three variables A(T), B(T), C(T) is obtained. It is convenient to study the solution in the threedimensional ABC phase space; the time T is a parameter. The projection of the solution onto the BA-plane is given in Figure 12.2(a) and the projection onto the BC-plane is given in Figure 12.2(b). These are known as phase portraits. The dependence of the variable B on time is given in Figure 12.2(c). The solution randomly oscillates between cellular rolls with clockwise rotation for B > 0 and with counterclockwise rotation for B < 0. The unstable fixed points from (12.33) and (12.34), A = B = 272'12, C = 27, are the crosses in Figures 12.2(a) and 12.2(b). This solution is chaotic in that adjacent solutions diverge exponentially in time. The solution oscillates about a fixed point with growing amplitude until it flips into the other quadrant, where it oscillates about the other fixed point. The fixed points behave as "strange attractors." In Figure 12.2(d) the fixed points are projected onto the rB-plane. The solid lines represent stable fixed points and the dashed lines represent unstable fixed points; the solid circle is a pitchfork bifurcation and the open circles are Hopf bifurcations. The pitchfork bifurcation at r = 1 corresponds to the onset of thermal convection; the Hopf bifurcations at r = 24.74 correspond to the onset of chaotic flows. An essential feature of the solution illustrated in Figure 12.2(b) is that the value of C is always positive and oscillates aperiodically around a positive value. It is the C term that gives an approximation to a thermal boundary layer structure. The growth and decay of C around a positive value is an approximation to the growth and separation of the thermal boundary layers at the top and bottom of the convection cell. It is the buoyancy forces in the boundary layers that drive the flow. The direction of the trajectory near B = 0 is always toward C = 0; the direction of the trajectory for large B is always in the positive C-direction. When C is at its largest the thermal boundary layers are thin, the buoyancy forces are small, and the flow decelerates (B decreases). This quiescence of the convective flow allows the thermal boundary layers to thicken and C decreases. The thickening thermal boundary layers become unstable and the resultant buoyancy accelerates the flow, increasing B; depending on where the instability occurs, either clockwise or counterclockwise flow results. As the convective motion increases, the thermal boundary layers again thin, and the value of C grows. The amplitude of the convective motion consequently slows down, and the absolute values of A and B once again move toward zero. Once convective motion has slowed enough, the boundary layer can grow again. Every time the convective motion stops to let the thermal boundary layer grow, the cell is presented with a choice of whether to convect in a clockwise or a counterclockwisedirection. It is this freedom of choice in a completely deterministic system that produces the chaotic, aperiodic behavior of the Lorenz attractor.

265

266

LORENZ EQUATIONS

The essential feature of the solution of the Lorenz equations in this parameter range is deterministic chaos. One consequence of the deterministic chaos of the Lorenz equations is that solutions that begin a small distance apart in phase space diverge exponentially. With essentially infinite sensitiv-

Figure 12.2. A numerical solution of the Lorenz equations (12.27), (12.30). (12.31) with Pr = 10, b = 813, r = 28. The solution in the ABC phase space is shown projected (a) into the BAplane and (b) onto the BCplane.

LORENZ EQUATIONS

267

60

40

time (c)

Figure 12.2. (conr.) (c) Time dependence of the coefficient B. (d) The loci of the fixed points are projected onto the &plane. The solid lines represent stable fixed points, the broken lines represent unstable fixed points, the solid circle is a pitchfork bifurcation, and the open circles are Hopf bifurcations.

268

LOREN2 EQUATIONS

ity to initial conditions, the zero-order behavior of a solution is not predictable. It should be emphasized that the numerical solution of the Lorenz equations given above is not a valid solution for thermal convection in a fluid layer. The Rayleigh number is outside the range of validity of the Lorenz truncation. Nevertheless, this solution has a great significance in that it was the first solution to exhibit the mathematical conditions for chaotic behavior. But the significance goes beyond this. Experimental studies and numerical simulations of thermal convection in a fluid layer heated from below at high Rayleigh numbers and intermediate values of Prandtl number are generally unsteady and "turbulent." There is good reason to accept that the solutions to the full set of equations are also chaotic in this range of parameters. Thus weather is taken to be chaotic, and deterministic predictions are not possible. It is essential to treat weather as a statistical problem with uncertainties increasing with forward extrapolations.

Problems Problem 12.1. Show that the critical Rayleigh number given by (12.20) has the minimum value as given by (12.21) and (12.22). Problem 12.2. For the steady-state solution of the Lorenz equations given in (12.40) and (12.41), determine an expression for the mean horizontal velocities on the boundaries at y = 0, 1.

ChapterThirteen

IS MANTLE CONVECTION CHAOTIC? The Lorenz equations are a low-order expansion of the full equations applicable to thermal convection in a fluid layer heated from below. For the range of parameters in which chaotic behavior is obtained, the low-order expansion is not valid; higher-order terms should be retained. Nevertheless, the chaotic behavior of the low-order analog is taken as a strong indication that the full equations will also yield chaotic solutions. Numerical solutions of the full equations are strongly time dependent for high Rayleigh number flows; these solutions appear to be turbulent or chaotic. It is generally accepted that thermal convection is the primary means of heat transport in the earth's mantle. Heat is produced in the mantle due to the decay of the radioactive isotopes of uranium, thorium, and potassium. Heat is also lost due to the cooling of the earth. The surface plates of plate tectonics are the thermal boundary layers of mantle convection cells. The plates are created by ascending mantle flows at ocean ridges. The plates become gravitationally unstable and founder into the mantle at ocean trenches (subduction zones). Intraplate hot spots such as Hawaii are attributed to mantle plumes that ascend from the hot unstable thermal boundary layer at the base of the convection mantle. An important question with regard to the earth is whether mantle convection is chaotic. The earth's solid mantle behaves as a fluid on geological time scales because of thermally activated creep. The discussion in the previous chapter considered only a constant viscosity. This is a poor approximation for the earth's mantle because the dependence of strain on stress is almost certainly nonlinear and is an exponential function of temperature and pressure. Also, the Boussinesq approximation is not applicable because of the significant increase in density with depth (i.e., pressure). Nevertheless, calculations assuming a linear stress-strain relation and constant fluid properties can provide important insights. In the previous chapter we estimated that the Rayleigh number for mantle convection is near 107 and the Prandtl number is larger than 1023. The latter is such a large value that it is appropriate to assume that the mantle has an infinite Prandtl number. Because the

270

IS MANTLE CONVECTION CHAOTIC?

Prandtl number for the mantle is so large, the momentum terms on the lefthand side of the momentum equations (12.2) and (12.3) can be neglected. Thus the only nonlinear terms are those in the energy equation (12.4). The question is whether these terms can generate chaotic behavior and thermal turbulence. The first question we address is whether the Lorenz equations yield chaotic solutions in the limit Pr -. In this limit (12.27) requires

The substitution of this result into (12.30) and (12.31) gives

~ 0, solution A = B = C = 0 Again these equations have the steady-state, d / d = corresponding to conduction. These equations also have the same fixed points as the Lorenz equations; these fixed points are given in (12.33) and (12.34) and correspond to cellular rolls rotating either in the clockwise or counterclockwise directions. A stability analysis again shows that the conduction solution is stable for r < 1 and unstable for r > 1. However, the steady solution consisting of cellular rolls is now stable for the entire range r > 1. In the limit of infinite Prandtl number the Lorenz equations do not yield chaotic solutions. This has been taken as evidence that mantle convection is not chaotic. To study this problem further Stewart and Turcotte (1989) considered a higher-order expansion than the Lorenz equations. The variables and 8-are and Om,n expanded in double Fourier series in F and j. The coefficients represent the terms sin (2rrrnilh) or cos (2-/h) and sin ( m y ) in the ex- the left-hand side of (12.14) pansions. In the limit of infinite Prandtl number, is zero, resulting in a linear equation between @m,n and Om,, that can be written

3 qm,

This result is then substituted into (12.15). The lowest consistent order of truncation beyond that used by Lorenz is m = 2 for the expansion in x (m = 0, 1,2) and n = 4 for the expansion in y (n = 1,2,3,4). This truncation yields a set of 12 ordinary differential equations for the time dependence of the temthat can be written perature coefficients,.,

IS MANTLE CONVECTION CHAOTIC?

It is necessary to take the resolution in the vertical direction twice compared with the horizontal direction to resolve the convection terms in the energy equation. The time evolution of the 12 coefficients 8 8 8,,,, 8,,,, 8 8,,,, el,,, 8 a,,,, 8,,,, 8,,? is found by integrating numerically the 12 equations given by (13.5). The time evolution can be thought of as trajectories in a 12dimensional phase space. It is convenient to project the 12-dimensional trajectories onto the two-dimensional phase space consisting of 8,,, and 8,,,; these correspond to the fundamental mode and the first subharmonic. There are two parameters in this problem, the Rayleigh number, Ra or r, and the wavelength. In this discussion solutions are given only for the critical value of the wavelength h = 2312. At the subcritical Rayleigh numbers 0 < r < 1 (0 C Ra < 657.512), the only fixed point of the solution is at the origin and it is stable; there is no flow. For higher Rayleigh numbers, the two fixed points corresponding to clockwise and counterclockwise rotations in the fundamental model o , , , be-

,,,, ,,,,

,,,,

Table 13.1. Numerical values of Fourier coefficients of the fixed points of the 12-mode equations (13.5)

271

272

IS MANTLE CONVECTION CHAOTIC?

come stable. The steady-state solution for Ra = lo4 (r = 15.21) is given in Table 13.1. It is seen that only six of the 12 coefficients are nonzero: go,?, O,,, a,,,, 8 j,,,, and 0,:,. This solution was obtained by specifying an in;the 12 coeftial condition near the origin and studying the time evolution-of ficients using (13.5). This time evolution projected onto the 02JOl,l-planeis given in Figure 13.l(a). Although the subharmonic coefficient O,,, is zero at the fixed point, it is nonzero during the time evolution. The steady-state solution for Ra = 3 X 104 (r = 45.62) is also given in Table 13.1. It is seen that only four of the 12 coefficients are nonzero: e,,?, O,,, , , ! . At this Rayleigh number the fundamental mode and its associated coefficients 8,,,, 8,,,, and 8,, are zero at the stable fixed point. The time evolution of the solution projected onto the ~ , , 8 , , , - p l a nis e given in Figure 13.l(b). Finally, the steady-state solution for Ra = 4.3 X 104 (r = 65.39) is given- in Table of the 12 coefficients are - 13.1. - It -is seen - that - eight now nonzero: O,,,, O,,, O,,,, O,,,, O,,,, O,,,,&,,, 19,,~.All of the Ole coefficients are zero including the fundamental mode O,,,.The time evolution of the solution projected onto the 8,,,e,,,-plane is given in Figure 13.l(c). It is seen that - fixed point is much more complex; the evolution prior to entering the stable the solution oscillates in the positive O,., quadrants before entering the negative quadrants. The time evolution of the solution for Ra = 4.5 X lo4(r = 68.44) is given in Figure 13.l(d); it is fully chaotic and no fixed points are stable. The flow alternates between aperiodic oscillations about the two fundamental modes (clockwise and counterclockwise) and the two subharmonic modes (clockwise and counterclockwise). All 12 coefficients are nonzero and are time dependent. The time dependences of the and coefficients are shown in Figure 13.2. The resemblance to the time behavior of the Lorenz attractor illustrated in Figure 12.2(c) is striking. Oscillatory behavior of the 8,,, mode amplifies until the flow undergoes a turbulent burst in the fundamental el,, mode, where it is briefly trapped before flipping into the 8,,, mode with the opposite sense of rotation. To better understand the transitions in the behavior of the time-dependent solutions, Figure 13.3 gives two projections of the loci of fixed points as a function of the Rayleigh number of the system. The solid lines denote stable fixed points, the dashed lines denote unstable fixed points, and the open circles denote Hopf bifurcations. Figure 13.3(a) shows a projection onto the ~a,8,,,-plane. The origin representing the conduction solution becomes unstable and bifurcates at Ra = 657.5 12, giving two stable symmetric solutions that do not contain the o,,, mode. One branch of this solution appears in the positive quadrant and is labeled "8,,,(pure)" in Figure 13.3(a) to distinguish it from the mixed-mode solution, which contains a contribution from the 8,,, mode. Each branch becomes unstable and undergoes a subcritical pitchfork bifurcation at Ra = 3.802 X 104, producing four unstable mixed-mode solutions, labeled "a,,,(mixed)". Each 8,,,mixed-mode branch

a,,, al,,

-

IS MANTLE CONVECTION CHAOTIC?

sweeps back to a saddle bifurcation at Ra = 1.909 X lo4.This type of bifurcation configuration (subcritical pitchfork plus two saddles) typically produces hysteresis effects when the saddle has one stable branch and one unstable branch. Here, the %,,,(mixed)solution has one stable manifold (out of 12) on one side of the saddle, and two unstable manifolds on the other. The second bifurcation of the conduction solution is at Ra = 1315.023, where two unstable symmetric fixed points dominant in the subharmonic 8,,, mode appear. Since these fixed points contain no component in the 8,,, mode, we call these unstable solution branches. Each of these branches becomes stable and undergoes a pitchfork bifurcation at Ra = 2041.918, resulting in the branching solution labeled "%2,,(mixed)'' in Figures 13.3(a) and 13.3b(b). The 8,,, mixed-mode branches nearly connect with the %,,,mixed-mode branches. Both the fundamental and the subharmonic pure-mode solutions are stable between Ra = 2369 and Ra = 3.802 X lo4. The trajectories in Figures 13.1(a) and 13.1(b) have the same initial condition, yet the trajectory in Figure 13.l(a) converges to the fundamental subharmonic stable fixed point. Presumably this is because the unstable mixed-mode branches disrupt the separatrix between the basin of the attraction of the fundamental and subharmonk pure-mode solutions. Note that the transition from Figure 13.l(a) to Figure 13.1(b) occurs at a Rayleigh number above the stability limit of the fundamental mode. The third bifurcation of the conduction solution is at Ra = 4.140 X 104, where two unstable symmetric solutions in the fundamental mode 8,,, appear. These are labeled "new 8,.,(pure)" in Figure 13.3(a). Each of these undergo Hopf bifurcations at Ra = 5.23 X lo4 and Ra = 5.53 X 104. At no point does the origin itself undergo a Hopf bifurcation, nor does the conduction solution bifurcate to mixed-mode solutions. No Hopf bifurcations were detected for the conduction or fundamental harmonic solutions; however, the stable subharmonic branch undergoes two Hopf bifurcations (Figure 13.3(b)), one at Ra = 4.37 X lo4 and one at Ra = 6.36 X 104. Each mixed-mode saddle (%,,,(mixed))undergoes two Hopf bifurcations, at Ra = 4.491 X 104 and at Ra = 5.039 X 104. Each mixed-mode saddle (j,,,(mixed)) undergoes six Hopf bifurcations. Each of these Hopf bifurcations sheds stable or unstable periodic orbits that are responsible for the oscillations of the trajectory at Ra = 4.5 X lo4 shown in Figure 13.l(d). In Figure 13.4, the-first - 7000 points of the trajectory at Ra = 4.5 X 104 are projected onto the 8,,,8,,,-plane (dotted line) and are shown superimposed on the central portion of the branches of the fixed points (solid lines). Note that the trajectory weaves aperiodically around several Hopf bifurcations (open circles). Physically, infinite Prandtl number, high Rayleigh number convection becomes time dependent through boundary layer instabilities that generate thermal plumes. In terms of spectral expansions, these instabilities result from the nonlinear coupling in the convective terms of the heat equation.

273

274

IS MANTLE CONVECTION CHAOTIC?

Rayleigh Number = 40000

Figure 13.1. Numerical solutions of the 12-mode infinite Prandtl number equations projected onto the 8,,,-plane of the 12-dimensional phase space. (a) Ra = 10 000(r = 15.21), (b) Ra = 30 000(r = 45.62),

a,,,

IS MANTLE CONVECTION CHAOTIC?

275

-Rayleigh Number = 43000

r

I-0 0

--

Rayleigh Number = 45000

'-4000.0

- 2000.0

0.0

2000.0 4.1

(4

4000.0

Figure 13.1. (cont.) (c) Ra = 43 000(r = 65.39), (d) Ra = 45 000(r = 68.44).

IS MANTLE CONVECTION CHAOTIC?

276

Figure 13.2. Time dependences of the and (b) coefficients (a) 8,., for the solution given in Figure l3.l(d); Ra = 45 000.

a,,,

IS MANTLE CONVECTION CHAOTIC?

_ new ' 4 - --- '-'V

-

'

Q2, (mixed)

d

/

-'-- ' M e - - - - - - -

277

I

$ 1 (pure

8

I

\

-.

%

RAYLElGH NUMBER (x 10')

(a)

20 4.0 RAYLEIGH NUMBER (x 1P)

(b)

Figure 13.3. Bifurcation diagram for the 12-mode infinite Prandtl number equations. The fixed points of (13.5) are projected (a) onto the Ra, a,,,-plane,(b) onto the Ra, a,,,-plane. Stable branches are shown as solid lines, unstable branches as dashed lines, pitchfork bifurcations as solid circles, and Hopf bifurcations as open circles.

278

IS MANTLE CONVECTION CHAOTIC?

Figure 13.4. Loci of the fixed points from Figure 13.3 projected onto the 8,,, 8,,,-plane.Superimposed as a dotted line is the time evolution of the chaotic solution from Figure 13.l(d).

Problems Problem 13.1. Show that the temperature coefficient B in the Lorenz expansion (12.24) is related to TI,,, by

For Ra = 104 and h = 8112 compare the value of B from the three-mode (Lorenz) expansion with the value from the 12-mode expansion. Problem 13.2. Show that the temperature coefficient C in the Lorenz expansion (12.24) is related to %o,2 by

A

For Ra = 104 and = 8112 compare the value of C from the three-mode (Lorenz) expansion with the value from the 12-mode expansion.

Chapter Fourteen

RIKITAKE DYNAMO

The earth's magnetic field is attributed to the electrically conducting outer core, which acts as a dynamo. The liquid outer core is primarily composed of iron, which is an excellent electrical conductor at core conditions. Electrical currents in the core generate a magnetic field. Buoyancy forces in the core, due to either temperature or composition, drive a fluid flow. The flowing electrical conductor in the magnetic field induces an electric field. This is a self-excited dynamo. Paleornagnetism is the study of the earth's past magnetic field from the records preserved in magnetized rocks. Rocks containing small amounts of ferromagnetic minerals such as magnetite and hematite can acquire a weak permanent magnetism when they are formed. This fossil magnetism in a rock is referred to as natural remanent magnetism. Many volcanic rocks at the surface of the earth can be magnetized because of the presence of minerals such as magnetite. When these volcanic rocks were cooled through the Curie temperature, they acquired a permanent magnetism from the earth's field at the time of cooling. Paleomagnetic studies of remanent magnetism have provided a variety of remarkable conclusions. These studies have traced the movement of the rocks due to plate tectonics and continental drift over periods of hundreds of millions of years. They have shown that the magnetic field at the surface of the earth has been primarily a dipole, as it is today, and has remained nearly aligned to the earth's axis of rotation. These studies have also shown that the earth's magnetic field has been subject to random reversals in which the north magnetic pole becomes the south magnetic pole and vice versa. The observed polarities of the earth's magnetic field for the last 10 million years (Myr) are given in Figure 14.1. Measurements indicate that for the last 720,000 years the

Figure 14.1. Observed polarity of the earth's magnetic field for the last 10 Myr; the scale is in Myr before the present (BP). The solid bands are the normal (present) polarity and the open bands are reversed polarity. The last polarity reversal occurred 720,000 years ago.

280

RlKlTAKE DYNAMO

magnetic field has been in its present (normal) orientation; between 0.72 and 2.5 Myr ago there was a period during which the orientation of the field was predominantly reversed. Clearly one characteristic of the core dynamo is that it is subject to spontaneous reversals. A question that can be asked is whether the reversals give a fractal distribution of polarity intervals. This question has been addressed by Gaffin (1989) and his results are given in Figure 14.2. The number of polarity intervals N ( T ) of length greater than T is given as a function of T. For intervals between 300,000 yr and 50 Myr a good correlation with the fractal relation (2.6) is obtained taking D = 1.43. No detailed theory exists for the behavior of the core dynamo. The viscosity of the liquid outer core is sufficiently small that the flow is undoubtedly turbulent. Thus the patterns of flow, electrical currents, and magnetic fields are very complex. Because of this complexity, relatively simple disk dynamos have been proposed as analog models. Rikitake (1958) proposed the symmetric two-disk dynamo illustrated in Figure 14.3. It is composed of two symmetric disk dynamos in which the current produced by one dynamo energizes the other. Equal torques G are applied to the two dynamos in order to overcome ohmic losses. Rikitake (1958) found that this dynamo was subject to random reversals of the magnetic field, but it was much later (Cook and Roberts, 1970) that it was demonstrated that the Rikitake dynamo behaved in a chaotic manner.

Figure 14.2. Cumulative number of reversal polarity intervals N ( T ) with length greater than T as a function of T as given by Gaffin (1989).The Cretaceous superchron has been excluded. The correlation line is the fractal relation (2.6)with D = 1.43.

RlKlTAKE DYNAMO

281

The steady-state behavior of the Rikitake dynamo is relatively easy to understand, as illustrated in Figure 14.3. The current I, passes through the current loop on the right in a clockwise direction. This current loop generates a magnetic field B, that is positive in the downward direction. This magnetic field also passes through the rotating, electrically conducting disk on the right. Because of the applied torque G, the disk is rotating in the counterclockwise direction with an angular velocity R,. The induced electric field in the disk Ei = u X B is the inward radial direction as illustrated. This induced electrical field drives the electric current I,. In the steady state the applicable circuit equation is

where R is the resistance in either circuit and M is the mutual inductance between the current loops and the electrically conducting disks. An electrical current I in a magnetic field B results in the electromotive force F = I X B per unit length of current path. The interaction between the magnetic field B, and the radially inward electric current I, results in a torque in the clockwise direction. In the steady state this torque balances the applied torque G and is given by

The discussion given above is also applicable to the current loop and rotating disk on the left in Figure 14.3.

Figure 14.3. Illustration of the Rikitake two-disk dynamo. The applied torques G drive the shafts and electrically conducting disks to rotate in the counterclockwise direction. The currents in the coils generate magnetic fields that interact with the rotating disks to generate the currents.

282

RlKlTAKE DYNAMO

The equations governing the unsteady behavior of the two-disk Rikitake dynamo are

where L is the self-inductance in either circuit and C the moment of inertia of each dynamo. Subtracting (14.5) and (14.6) and integrating gives

where ROis a constant. This can replace either (14.5) or (14.6). Again it is appropriate to introduce nondimensional variables and parameters according to

y2 = ( g ) 1 / 2 R 2 A , = ( g ) 1 / 2 R o ,A ,

=

~'2 (---) GLM

Substitution into (14.3)-(14.5) and (14.7) gives

1/2

RlKlTAKE DYNAMO

283

This is a set of three coupled, nonlinear differential equations that determine the time evolution of the Rikitake dynamo. Setting the time derivatives equal to zero the steady-state solutions are obtained:

where

The plus and minus signs refer respectively to the normal and reversed states of the magnetic field. Stability calculations (Cook and Roberts, 1970) have shown that the singular points given above are unstable for all parameter values. Their numerical solutions for p. = 1 and K = 2 are given in Figures 14.4 and 14.5. The sin-

Figure 14.4. Numerical solution of the Rikitake twodisk dynamo equations (14.9)(14.11) in the X,X,Y, phase space projected onto the X,Y, phase plane. The singular points corresponding to normal and reversed polarizations of the magnetic field, X , = 50.5, Y , = 4 are shown.

284

RlKlTAKE DYNAMO

gular points X2 = ?;, Y, = 4 are shown in the X2Y, phase plane illustrated in Figure 14.4. The strange-attractor behavior of the solution is very similar to that of the Lorenz equations given in Figures 12.l(a), (b). The time evolution of the solutions, given in Figure 14.5, is also similar to that of the Lorenz equations given in Figure 12.l(c). Oscillations grow in one polarity of the field until it flips into the other polarity. Extensive studies of the behavior of the Rikitake dynamo equations have been carried out by Ito (1980) and by Hoshi and Kono (1988). The behavior is found to be periodic or chaotic. A map of these two behaviors in the K - p parameter space is given in Figure 14.6. The transition from periodic to chaotic behavior follows the period-doubling route to chaos previously obtained for the logistic map. A bifurcation diagram for K = 2 illustrating the period doubling is given in Figure 14.7. Marzocchi et al. (1995) have studied the reversal statistics of the Rikitake dynamo and found that they are not similar to the reversals of the earth's magnetic field. This is not surprising since the Rikitake dynamo is a low-order system and the earth's dynamo must be a very high-order system.

Figure 14.5. Time evolution of X, and X, for the solution given in Figure 14.4.

RlKlTAKE DYNAMO

285

An alternative low-order model for the generation of the earth's mag-

netic field was proposed by Robbins (1977). The governing equations take the form

CHAOS

Figure 14.6. Map of the K - p parameter space giving regions in which periodic and chaotic behaviors are found (Ito, 1980).

Figure 14.7. Bifurcation diagram illustrating the period doubling route to chaos of the Rikitake dynamo equations as the parameter p, is varied with K = 2 (Ito, 1980).

286

RlKlTAKE DYNAMO

where a, R, and v are adjustable parameters. Chillingworth and Holmes (1980) have obtained extensive solutions to these equations and found behavior similar to that of the Rikitake dynamo. Cortini and Barton (1994) have determined the correlation dimensions from the reversal sequences of the Rikitake and Chillingworth-Holmes models and have compared the results with values obtained from the actual reversal sequence. Krause and Schmide (1988) have proposed a third-order recursive map as a model for the reversals for the earth's magnetic field

The fixed points of this equation are x, = 0 and x, = 2 (1 - a-1)lt2.The fixed point at x, = 0 is stable for 0 < a < 1 and unstable for a > 1. The fixed point at x, = 2 (1 - a-l)"2 is real only for a > 1; it is stable for 1 < a < 2 and unstable for a > 2. If 0 < a < 3 f i 1 2 , values of x do not change sign and the behavior is similar to that of the logistic map considered in Chapter 10. If a > 3, the map is not confined and iterations cannot be carried out. In the range 3 f i 1 2 < a < 3 reversals in the sign of x can occur. Chaotic oscillations occur within the range -2013 fi < x < 2 a 1 3 f i . As an illustration of the behavior of this iterative map with reversals we consider an example with a = 2.75 as shown in Figure 14.8. The curve represents the function

Taking x, = 0.2 we draw a vertical line; its intersection withflx) gives x, = 0.528. A horizontal line drawn from this intersection to the diagonal line of unit slope transfers x ~ to+ x,,. ~ A vertical line is drawn to flx) giving x, = 1.0472056. Because x > 1 the next iteration results in a change of sign (a reversal) and we obtain x, = 0.2783043. Further iterations give x, = -0.7060590 and x, = - 0.9737061 as illustrated. One hundred iterations of this map are given in Figure 14.9. The sequence of chaotic reversals is clearly illustrated. The models considered above are clearly gross simplifications of the complex fluid flows that occur in the earth's core. Nevertheless, the models produce patterns of random reversals that are remarkably similar to the re-

RlKlTAKE DYNAMO

287

versals of the earth's magnetic field. Again this can be taken as evidence that the dynamo action in the core is chaotic. It is certainly desirable to consider higher-order systems that better simulate the "turbulent" interactions between electrical currents and flows of the electrically conducting fluid. A start in this direction has been given by Glatzmaier and Roberts (1995).

Figure 14.8. Illustration of the iteration of the third-order recursive map (14.21) with a = 2.75 and x, = 0.2. A reversal in sign occurs on the third iteration.

yo 1

Figure 14.9. Illustration of100 iterations of the third-order recursive map (14.21) with a = 2.75 and x, = 0.2. The sequence of chaotic reversals is clearly shown.

288

RlKlTAKE DYNAMO

Problems Problem 14.1. Consider a single-disk dynamo. Determine the steady-state current I and angular velocity i2 in terms of resistance R, mutual inductance M, and applied torque G. Problem 14.2. Determine the fixed points of the Chillingworth-Homes dynamo equations given in (14.18)-(14.20). Problem 14.3. Determine x,, x,, x,, x,, and x, for the third-order recursive map (14.21) with a = 3 and xo = 0.8. What is the range of values of x?

Chapter Fifteen

RENORMALIZATION GROUP METHOD

15.1 Renormalization In the first eight chapters of this book we considered the fractal behavior of natural systems. This behavior was generally statistical and the physical causes were generally inaccessible. In the six chapters that followed we considered low-order dynamical systems that exhibit chaotic behavior. Because of the low order, the examples are generally quite far removed from natural systems of interest. In this chapter and the next we take a collective view of natural phenomena and consider some applications in geology and geophysics. Thermodynamics represents the standard approach to collective phenomena. System variables are defined, that is, temperature, pressure, density, entropy; and the evolution of these variables is determined from the first law of thermodynamics (conservation of energy) and the second law of thermodynamics (variation of entropy). Statistical mechanics provides the rational microscopic basis for much of thermodynamics. In general, neither thermodynamics nor statistical mechanics yields fractal statistics or chaotic behavior. Exceptions include critical points and phase changes. A characteristic feature of a phase change is a discontinuous (catastrophic) change of macroscopic parameters of the system under a continuous change in the system's state variables. For example, when water freezes its viscosity changes from a very small value to a very large value with no change in temperature. The renormalization group method has been used successfully in treating a variety of phase change and critical-point problems (Wilson and Kagut, 1974). This method often produces fractal statistics and explicitly utilizes scale invariance. A relatively simple system is considered at the smallest scale; the problem is then renormalized (rescaled) to utilize the same system at the next larger scale. The process is repeated at larger and larger scales. This is very similar to our renormalization models for frag-

RENORMALIZATION GROUP METHOD

mentation and the concentration of ores given in Chapters 3 and 5 respectively. We consider three specific applications to illustrate the use of the renormalization group method. We first consider a model for the flow of a fluid through a porous medium. Such problems have wide applicability in groundwater hydrology and petroleum engineering. The porous media may either be a granular material such as a sandstone or a matrix of rock fractures. In either case Darcy's law is generally applicable and the fluid velocity is proportional to the pressure gradient; Darcy's law is a linear relation.

15.2 Percolation clusters

We apply the renormalization group method to understand the onset of fluid flow through a porous medium. We first consider the two-dimensional model illustrated in Figure 15.l(a). A square array is made up of a matrix of square boxes that are referred to as elements; each box (element) may be either permeable or impermeable. For the example given in Figure 15.l(a) there are n = 256 elements. The probability that an element is permeable is p,, the probability it is impermeable is 1 - p,. The question is whether the square array is permeable or impermeable. The array is defined to be permeable if there is a continuous permeable path from the left side of the array to the right side of the array. This is clearly a statistical question since the actual distribution of permeable and impermeable elements is random. For a specified value of p, there is a probability P that the array with n elements is permeable. For large arrays it is found that P is very small ( P 4 1) if 0 < p, < p* where p* is the critical probability for the percolation threshold and P is near unity (P = 1) if p* < p, < 1 (Stinchcombe and Watson, 1976).

Figure 15.1.(a)A16 X 16 array of square elements. The probability p, that an element is permeable is 0.5; either the dark or the light elements can be assumed to be permeable. For either case, no permeable path across the array is found. (b) Illustration of the renormalization group method; four square elements are considered at each of the four scales.

RENORMALIZATION GROUP METHOD

Thus there is a critical value of p,, p*, for the onset of flow through the grid of elements. If p, is less than this critical value p*, a large square grid will almost certainly be impermeable to flow. If p, is greater than this critical value p*, a large square grid will almost certainly be permeable to flow. It may be easier to visualize this problem if one considers a forest made up of a square grid of trees. The probability that a grid point has a tree is p,. The question is whether a forest fire can burn through the forest if a tree can ignite only its nearest neighbors. If there are no nearest neighbors the fire does not spread. This forest-fire problem is mathematically identical to the percolation problem considered above. We now turn to the percolation problem and consider in some detail the 16 X 16 array of square elements illustrated in Figure 15.l(a). The total number of elements n = 256. Taking p, = 0.5, it has been randomly determined whether each element is permeable or impermeable. With p, = 0.5 either the dark squares or the light squares can be taken to be permeable. In either case no continuous permeable path is found either horizontally or vertically. Using the Monte Carlo approach a large number of random realizations would be carried out and the probability P(p,) that the array is permeable would be determined. For the two-dimensional square array with large n, numerical simulations find that the critical probability for the onset of flow in the array is p* = 0.59275 (Stauffer and Aharony, 1992). It is also of interest to consider the number-size statistics for percolation clusters at the critical limit p, = p*. The size of a percolation cluster is defined to be the number of permeable elements in contact with each other when the array first becomes permeable. The number of elements in the percolation cluster n: has been determined numerically as a function of the array size n. For two-dimensional arrays it is found that (Stauffer and Aharony, 1992)

This result can be compared with the deterministic Sierpinski carpet assuming that the remaining squares illustrated in Figure 2.3 represent a percolation cluster. For the Sierpinski carpet ne = 8 when n = 9 and ne = 64 when n = 8 1, thus

Comparison of (15.1) and (15.2) shows that the fractal dimension for the percolation cluster at the critical limit is D = 91/48 = 1.896, which is very close to the value for a Sierpinski carpet D = In 81111 3 = 1.893. At the onset of percolation, the sites through which flow takes place are known as the percolation backbone. Sahimi et al. (1992, 1993) explained the

291

292

Figure 15.2. A cell consisting of four elements is considered. A cell is defined to be permeable if there is a continuous permeable path from left to right. In (a) all four elements are impermeable, in (b) one element is permeable and the four possible configurations are given, in (c) two elements are permeable and the six possible configurations are given, in (d) three elements are permeable and the four possible configurations are given, and in (e) all four elements are permeable. The configurations that are permeable from left to right are denoted by (+) and the configurations that are impermeable from left to right are denoted by (-).

RENORMALIZATION GROUP METHOD

fractal distribution of seismicity illustrated in Figure 4.12 in terms of the percolation backbone of a three-dimensional percolation cluster. The direct statistical approach to the percolation problem becomes extremely time consuming if the number of elements is large since many random realizations must be considered. An alternative approximate approach is illustrated in Figure 15.l(b). At the lowest order a square array of four elements is considered. The probability p, that the first-order cell is permeable is determined in terms of the probability p, that an individual first-order element is permeable. The cell is defined to be permeable if there is a continuous permeable path from left to right. The problem is then renormalized and four first-order cells become the four second-order elements in a second-order cell. The probability p, that the second-order cell is permeable is then determined in terms of the probability p, that a second-order element (first-order cell) is permeable. The process is repeated at larger and larger scales (higher and higher orders). This is the renormalization group method and is essentially the same as the fractal fragmentation model illustrated in Figure 3.6. We now consider the first-order cell and determine the probability that it is permeable. All possible configurations are illustrated in Figure 15.2. The probability that all four elements are impermeable is (1 - p0)4 and there is only one configuration, as shown in Figure 15.2(a). This configuration is clearly not permeable (-). The probability that one element is permeable and three elements are not is p, (1 - p0)3 and there are four configurations for the cell as illustrated in Figure 15.2(b). The permeable element can be in any of the four positions shown but the cell is not permeable in any of them (-). The probability that two elements are permeable and two elements are not is pi(1 - po)2 and there are six independent configurations as shown in Figure 15.2(c). The first and sixth configurations result in the cell being permeable for flows from left to right (+) and the other four configurations are impermeable (-). The probability that three elements are permeable and one element is not is pi(l - p,) and there are four independent configurations as shown in Figure 15.2(d).All four cell configurations are permeable (+). The

RENORMALIZATION GROUP METHOD

293

probability that all four elements are permeable is p; and there is only one configuration as shown in Figure 15.2(e): it is permeable (+). Taking into account all possible configurations the probability that the first-order cell is permeable is given by

The first-order probability includes the two configurations with two permeable elements, the four configurations with three permeable elements, and the single configuration with four permeable elements. Renormalization is carried out and four first-order cells become second-order elements. After renormalization exactly the same statistics are applicable to the secondorder cell with the result

This result can be applied to the nth-order cell with the result

This recursive relation for the probability is quite similar to the logistic map considered in Chapter 10. Figure 15.3 shows the dependence of p,, on pn given in (15.5). To consider the fixed points it is appropriate to rewrite (15.5) as

,

Figure 15.3. The dependence of the probability p,,, that an ( n + 1)th-order cell is permeable on the probability p,, that an nth-order cell is permeable. The critical probability for the onset of permeability is p* = 0.618. The iteration for p, = 0.4 is illustrated; this iteration is to the impermeable fixed point p_ = 0.The iteration for p, = 0.8 is also illustrated; this iteration is to the permeable fixed point p_ = 1.

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The fixed points are obtained by settingf ( x ) = x with the result

In the range 0 < x < 1 there are three fixed points x = 0,0.618, 1. The corresponding values of A = dfldx are 0, 1S28, 0. The fixed points at x = 0 and 1 are stable since IAl < 1 but the fixed point at x = 0.618 is unstable. To illustrate further the iteration of the probabilities given by (15.5) we consider two specific cases. For po = 0.4 we find p , = 0.294, p, = 0.166, and p, = 0.054 as illustrated in Figure 15.3. The construction is the same as that used in Figures 10.1-10.5. As the iteration is continued to large n the probability p,, approaches the stable fixed point p,, = 0. A large two-dimensional array is impermeable forp, = 0.4. Forp, = 0.8 we findp, = 0.870, p, = 0.941, and p, = 0.987 as illustrated in Figure 15.3. As the iteration is continued to large n the probability p,, approaches the stable fixed point pn = 1. A large two-dimensional array is permeable for p, = 0.8. The unstable fixed point at p* = 0.618 is a critical point. At the critical point, p,, = p* for all values of n; the probability that a cell is permeable is scale invariant. For probabilities smaller than the critical value 0 < p, < p* the iteration is to the impermeable limit pn = 0. For probabilities greater than the critical value p* < p, < 1 the iteration is to the permeable limit pn = 1. The value p* = 0.618 obtained by the renormalization group method compares with p* = 0.59275 obtained by the direct numerical simulations for large arrays. The two-dimensional renormalization group method can be based on a larger lowest-order array. Consider a 3 X 3 array with nine elements. Taking into account all possible configurations, the probability p,,, that the (n 1)th-order cell is permeable is related to the probability pn that the nth-order cell is permeable by

+

The behavior of this relation is essentially similar to the behavior of (15.5). The critical value for the onset of permeability is p* = 0.609. This is essentially identical to the result obtained using the 2 X 2 renormalization group result given above. Extensive studies have also been carried out on three-dimensional cubic arrays made up of n cubic elements. In this case the individual cubic elements are taken to be either permeable or impermeable. The array is defined to be permeable if there is a continuous permeable path from one side of the array to the other. Numerical studies of large cubic arrays show that the crit-

RENORMALIZATION GROUP METHOD

ical probability for the onset of flow in the array is p* = 0.3 117 (Stauffer and Aharony, 1992). Again, a fractal relation of the form (15.2) is obtained between the number of permeable elements ne in the critical percolation cluster and the total number of elements n with D = 2.5. This result can be compared with the deterministic Menger sponge illustrated in Figure 2.4(a), assuming that the remaining cubes represent a percolation cluster. For the Menger sponge ne = 20 when n = 27 and ne = 400 when n = 729; the value D = 2.727 for the Menger sponge is somewhat higher than the value for three-dimensional percolation clusters. The simplest renormalization group model for the array of cubic elements is a 2 X 2 X 2 cubic array of eight elements. Taking into account all possible configurations, the probability that the (n + 1)th-order cell is permeable is related to the probability that the nth-order cell is permeable by

The critical value for the onset of permeability is p* = 0.282. This is in reasonably good agreement with the numerical results considering the simplicity of the model. For fluid flow through rocks the two measurable quantities are the porosity (the degree to which void space becomes filled with fluid) and the permeability (the ability of the fluid to flow through the rock under fluid pressure). The highly idealized model considered above predicts that there will be a sudden onset of permeability at a critical value of the porosity. Although rocks with low porosity have essentially zero permeability, the sudden onset of permeability at a universal critical value of the porosity is not observed. This is attributed to the variety of aperture sizes and lengths occurring in natural systems, which is not fully described by the idealized model. Problems associated with electrical conduction through a matrix of elements are essentially identical to the percolation problems considered above. Madden (1 983) has applied the renormalization group method to the onset of electrical conductance through a grid of electrical conductors and insulators.

15.3 Applications to fragmentation Our second application of the renormalization group method is to fragmentation (All&greet al., 1982; Turcotte, 1986a). We again consider fragmentation in terms of the scale-invariant model illustrated in Figure 3.6 with fragments that differ in size by factors of two. However, there is a fundamental

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difference. In Chapter 3 we considered a large zero-order cell and made smaller and smaller elements. Here we consider the first-order elements to be the smallest fragments and make larger and larger cells following the renormalization group method. At the lowest order a cubic array of eight elements is assumed to constitute a first-order cell. Following Allbgre et al. (1982), each element in a cell is hypothesized to be either fragile (f) if it is permeated with microfractures or sound (s) if it is not. The probability that a first-order cell is fragile is p, and is determined in terms of the probability that an individual element is fragile, p,. It is necessary to specify a condition for the fragility (soundness) of the first-order cell. Allkgre et al. (1982) hypothesized that a cell was sound if a "pillar" of sound elements linked two faces of the cell. Having prescribed the probability p, that an element is fragile, it is necessary to determine the probability p, that a cell is fragile. To do that it is necessary to consider all alternative configurations. In each cell there can be zero to eight fragile elements so that there are 28 = 256 possible combinations. Excluding multiplicities, there are 22 topologically different configurations; these are illustrated in Figure 15.4. The numbers in parentheses are the multiplicities of each configuration. The fragile elements are indicated by solid dots at the comers. Configurations 4f, 5c. 6b, 6c, 7, and 8 are fragile and are indicated by solid underlining in Figure 15.4. Examples of sound and fragile cells are illustrated in Figure 15.5; 5b is a sound cell with the pillar of strength illustrated by heavy lines, and 5c is a fragile cell. The probability that all eight elements are fragile is pi, the probability that seven elements are fragile and one is sound is p:(l - p,), and so forth. Taking into account the configurations that are fragile and their multiplicFigure 15.4. Illustration of 22 topologically different configurations. The "fragile" elements are indicated by solid dots on the comers. The first number beneath a configuration indicates the number of fragile elements, the letters indicate the various configurations with the same number of fragile elements, and the number in parentheses gives the multiplicity of that configuration. Using the AlDgre er al. (1982) "pillar of strength" condition, the fragile cells are underlined.

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297

ities the probability p , that a first-order cell is fragile is related to the probability po that a first-order element is fragile by

After renormalization exactly the same statistics are applicable at higher orders. Thus we can write

If the characteristic size of the first-order cell is 2h, then the characteristic size of the nth-order cell is 2nh. Figure 15.6 shows the dependence of pn+!on pn given in (15.11). To consider the fixed points it is appropriate to rewrite (15.1 1) as

The fixed points are obtained by settingflx) = x with the result

In the range 0 < x < 1 there are three fixed points x = 0,0.896, 1. The corresponding values of A = dfldx are 0, 1.766, 0. The fixed points at x = 0 and 1 are stable since IAl < 1 but the fixed point at x = 0.896 is unstable.

Figure 15.5. Eight cubic elements with dimension h form a cell with dimension 2h. Configurations 5b and 5c are illustrated; the five fragile elements are shaded, and the three sound elements are unshaded. Cell 5b is sound because the unshaded (outlined) "pillar" of sound elements links two faces. Cell 5c is fragile because no pillar of sound elements links two faces.

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To illustrate further the iteration given by (15.1l), we consider a specific case. Forpo=0.6 wefindp, =0.2736,p,=0.0118,andp,=3.91 X 10-gas illustrated in Figure 15.6. The construction is again the same as that used in Figures 10.1-10.5 and in Figure 15.3.As the iteration is continued to large n, the probability p,, approaches the stable fixed point pm = 0. Fragmentation does not occur forp, = 0.6. The unstable fixed point atp* = 0.896 is a critical point corresponding to catastrophic fragmentation. For probabilities smaller than the critical value 0 < po < p* the iteration is to pm= 0 and fragmentation does not occur. At the critical limit p* the probability of fragmentation is the same at all orders. Thus it is appropriate to set p* equal to the probability f that a cell will fragment into eight elements, which was introduced in (3.70). Settingf = p * = 0.896 and substituting into (3.76) we find that the resulting fractal dimension is D = 2.84. It is of interest to relate the renormalization group approach to fragmentation as given above to the discussion of fragmentation given in Chapter 3. The renormalization group approach provides a rational basis for assuming that the fragmentation probability f is scale invariant. The applicable value off is model dependent. For the "pillar of strength" model the derived fractal dimension for fragments is D = 2.84. For the comminution model proposed in Chapter 3 the derived fractal dimension for fragments was D = 2.6. The latter is in somewhat better agreement with observed distributions.

Figure 15.6. Probability pn+, of fragility at order n + 1 as a function of the probability p,, of fragility at order n from (15.1 1). The critical probability p* corresponding to catastrophic fragmentation is 0.896. The renormalization group iteration is illustrated for p, = 0.6, giving p , = 0.2736, p, = 0.01 18, and p, = 3.91 X 10-8.

RENORMALIZATION GROUP METHOD

15.4 Application to fault rupture

As our final application of the renormalization group method we consider a fractal-tree model for the rupture of a fault. When a fault ruptures the result is an earthquake. An earthquake is in many ways analogous to a critical phenomenon. At a critical value of the stress a catastrophic event, the earthquake, occurs. As a result of the earthquake there is a discontinuity in the stress and strain. A fault is generally considered to be a planar structure and can be modeled using a velocity-weakening coefficient of friction. An actual fault in the earth is generally more complex, being imbedded in a matrix that is fragmented by faults and joints. It appears appropriate to assume that a fault contains asperities on a wide range of scales. At a small scale an asperity can be an interacting roughness on the fault. At a large scale an asperity can be a bend in the fault. An asperity has a limiting strength and when this limiting strength is reached it will rupture. As a result of the rupture the stress on the asperity will be transferred to adjacent asperities and they may or may not rupture. In many ways this rupture is analogous to the failure of a stranded cable. If one strand fails, the stress carried by that strand is transferred to adjacent unbroken strands. Extensive numerical calculations of this failure problem have been carried out (Harlow and Phoenix, 1982). Two observations are of interest. The first is that a stranded cable will fail after only a few strands have failed. The second is that the load that can be carried by a stranded cable is, on average, less than the load that could have been carried on a singlestrand cable of the same mass. Both of these observations appear to be applicable to faults. There is seldom any seismic indication of small ruptures on a fault that is about to fail producing a large earthquake. Also, the failure stress on faults is generally less than that predicted by simple frictional considerations. Numerical simulations of the failure of stranded cables and the problems relation to fault rupture have been given by Newman and Gabrielov (1991), Gabrielov and Newman (1994), and Newman et al. (1994). To model fault rupture Smalley et al. (1985) considered the failure of a fractal tree; they applied the renormalization group approach. Their basic model is illustrated in Figure 15.7(a). The force F is carried by each of the two first-order elements that make up the first-order cell. If one element fails, the force on it is transferred to the other element and it must carry the force 2F. A third-order fractal tree is illustrated in Figure 15.7(b). It includes four first-order cells each of which carries a force 2F. These each become second-order elements in the two second-order cells each of which carries a force 4F. The two second-order cells become third-order elements in the single third-order cell, which carries a force SF. The objective is to determine the probability of failure of a cell in terms of the probabilities of failure of its elements. We assume that the probability p,(F) of failure of an element is given by the quadratic Weibull distribution (3.55):

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where Fo is a reference strength. For a cell containing two elements that are either broken (failed) or unbroken, four states are possible: (1) [bb],(2) [bu],(3) [ub],(4) [uu],where b represents a broken element and u represents an unbroken element. Note that states (2) and (3) are equivalent and can be combined into a single state with a multiplicity of two. The probabilities of these states in terms of the probability of failure po are:

In writing (15.15) to (15.17) the transfer of the force (stress) when an element fails has not been considered. If one element fails and the other is unbroken it is necessary to determine whether the second element will fail when the stress from the first element is transferred to it. We introduce the conditional probability p,, that an unbroken element already supporting a force F will fail when an additional force F is transferred to it. This mechanism for stress transfer leads to induced failures. The probabilities that the [ub] state will be broken or unbroken under stress transfer are given by

Figure 15.7. Illustration of the fractal tree model for the renormalization group approach to an earthquake rupture. (a) The basic model; (b) a third-order construction.

RENORMALIZATION GROUP METHOD

From (15.14) and (15.17) the probability that a zero-order cell fails, p , , is given by

It is now necessary to determine the conditional probabilityp,,; it is given by

where p0(2F) is the probability of failure under a force 2F. For the quadratic Weibull distribution given in (15.14) we have po(2F) = 1 - exp

[

-

(31

The substitution of (15.14) into (15.22) gives

Combining (15.21) and (15.23) the conditional probability for the quadratic Weibull distribution is given by

Substitution of (15.24) into (15.20) gives the probability that a cell fails, p , , in terms of the probability that the element fails, p,:

After renormalization exactly the same statistics are applicable to the second-order cell, with the result

This result can be applied to the nth-order cell, with the result

Again this recursive relation resembles the logistic map considered in Chapter 10. Figure 15.8 shows the dependence of p,,, on pn given in (15.27). To consider the fixed points it is appropriate to rewrite (15.27) as

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The fixed points are obtained by settingf(x) = x with the result

Figure 15.8. Dependence of the probability pn+,of failure at order n + 1 on the probability pn of failure at order n from (15.27) for cells containing two asperities with a quadratic Weibull distribution of strengths. The procedure described in the text for determining the probability of cell failure for successive iterations is illustrated for p, = 0.6,O. 1. The critical probability of failure p* gives the bifurcation point for catastrophic failure of the system. If 0 < p, < p*, the solution iterates to pn = 0 and no failure occurs. If p* < p, < 1 , the solution iterates to pn = 1, and the system has failed.

In the range 0 < x < 1 there are three fixed points x = 0,0.2063, 1. The corresponding values of A = dfldx are 0, 1.6 l9,O. The fixed points at x = 0 and 1 are stable since IAl = 1 but the fixed point at x = 0.2063 is unstable. To illustrate further the iteration of the probabilities given by (15.27), we consider two specific cases. For p, = 0.1 we find p , = 0.05878, p, = 0.02184, and p, = 0.00322 as illustrated in Figure 15.8. The construction is the same as used in Figures 10.1-10.5. As the iteration is continued to large n, the probability p,, approaches the stable fixed point p _ = 0. The large fractal tree does not fail for p, = 0.1. For p, = 0.6 we find p , = 0.8093, p, = 0.9615, and p, = 0.9985 as illustrated in Figure 15.6. As the iteration is continued to large n, the probability p,, approaches the stable fixed point pm = 1. The large fractal tree fails for p, = 0.6. The unstable fixed point at p* = 0.2063 is a critical point. For probabilities smaller than the critical value 0 < p, < p* the iteration is to the unbroken limit pm = 0. For probabilities greater than the critical value p* < p, < 1 the iteration is to the broken limit p_ = 1 . The value p* = 0.2063 corresponds to the catastrophic failure of the fractal tree. Because of the transfer of stress, all elements fail when the probability of failure of an individual element is only 0.2063. From (15.14) this corre-

RENORMALIZATION GROUP METHOD

sponds to FIFO = 0.4807. Considered individually half the elements will have failed when p, = 0.5 and FIFO = 0.8326. Thus the transfer of stress results in a lower failure stress than if a single element was considered. Clearly this renormalization group approach is an idealization of actual faults. However, some of the results are directly related to field observations. A failure of a few elements cascades into a catastrophic total failure. Actual faults do not have enhanced seismicity prior to a major rupture. This observation coupled with the model studies indicates that stress transfer is important in fault ruptures. The relatively low failure stress is also in agreement with field observations.

15.5 Log-periodic behavior In the renormalization group approach to failure considered in the previous section, each element was assigned a prescribed probability of failure as a function of the stress on the element. For many materials it is preferable to assign a statistical distribution of times to failure as a function of the stress on the element. If a wire is placed under a stress a it does not fail instantaneously, but has a statistical distribution of lifetimes (Coleman, 1958). The mean lifetime decreases as the applied stress is increased. In many cases an exponential cumulative distribution of a failure times rf is assumed

where p(r,) is the probability that failure will occur in a time less than tf and v(u) is known as the "hazard rate" under stress a. One-half of a large collection of wires under stress a will have failed when t,,, = (In 2)lv. It is often appropriate to assume that the dependence of the hazard rate on stress is given by

where v, is the hazard rate under stress a, and p is typically in the range 2-5. Combining (15.30) and (15.31) gives

This is a modified form of the Weibull distribution given in (3.55). To illustrate the application of the time-to-failure approach, we consider the failure of a stranded cable composed of N, elements each initially carrying a load a,. When an element fails, the load on that element is transferred

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uniformly to all the remaining elements; this is known as a global load-sharing model. This approach is also an example of a mean-field approximation. When n, elements have failed, the stress s on the n = No - n, surviving elements is given by

The rate at which elements fail is assumed to be given by the rate law

The solution of this deterministic equation with v a constant and n, = 0 at t = 0 basically corresponds to the probabilistic distribution given in (15.30). Substituting the dependence of the hazard rate on stress from (15.31), we combine (15.33) and (15.34) to obtain

Integrating with the condition that n, = No when t = t, we obtain

The number of surviving elements has a power-law dependence on the time to failure with the exponent p-1. The route to failure is scale invariant and exhibits fractal behavior. The time to failure is given by t, = (p vo)-1. An increase in the power p decreases the time to failure as expected. The occurrence of aftershock sequences following earthquakes can be explained in terms of the time-to-failure approach discussed above. The displacement on the primary fault associated with the initial earthquake results in an increased stress in some adjacent regions. Earthquakes subsequently occur on preexisting faults in these regions, resulting in aftershocks. The delay in the occurrence of the aftershocks can be attributed to the applicability of (15.32). It is interesting to note that a power-law decay in the rate of aftershocks R of the form

is found to be universally applicable with p == 1. This relation is known as Omori's law (Utsu, 1961).

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It is also of interest to consider whether there is a precursory change in seismicity that occurs prior to a major earthquake that can be associated with the time-to-failure approach. Varnes (1989), Bufe and Varnes (1993), and Bufe et al. (1994) have argued that there is a power-law increase in the regional cumulative Benioff strain release prior to an earthquake. The Benioff strain B is the square root of the seismic moment M defined in (4.3):

The cumulative Benioff strain in northern California prior to the October 17, 1989, Loma Prieta earthquake is given in Figure 15.9(a) (Bufe and Varnes, 1993). The increase in the Benioff strain illustrated in this figure fits the exponential scaling given in (15.36) quite well, but there also appears to be a periodic component. Sornette and Sammis (1995) have also considered these data and concluded that there is an excellent fit to a log-periodic increase in seismic activity. A simple power-law (fractal) increase in the cumulative Benioff strain B is given by

where t, - t is the time prior to the earthquake and the constant a is negative. To obtain log-periodic behavior, we assume that exponent a is complex: a = 5 + iq. In this case we obtain

=

(1 -

8' [. $1 cos

1.(1 -

where 2 stands for the real part. This is log-periodic behavior. Combining (15.39) and (15.40) a generalized self-similar expression for the cumulative Benioff strain takes the form

where C, specifies the amplitude and 6 the phase of the log-periodic component. This result is fully self-similar.

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The sequence of positive maxima and/or minima in a log-periodic sequence can be used to predict the time to failure t,. The positive maxima in ( 15.40)correspond to the sequence

1920

1930

1940

1950

1960

Date (a)

Figure 15.9. (a) The data points are the cumulative Benioff strain release in magnitude 5 and greater earthquakes in the San Francisco Bay area prior to the October 17, 1989, Lorna Prieta earthquake (Bufe and Varnes, 1993). The solid line is the log-periodic correlation with the data. (b) Predicted dates for the Loma Prieta earthquake based on the data applicable prior to the cutoff date (Sornette and Sammis, 1995).

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where n = 1, 2, 3, . . . If three successive values of the maxima are specified, t , , t,, t,, the failure time t, is given by

This result is also valid for the more general form of log-periodic behavior given in (15.41). Because of the basic scale invariance, the value of t, obtained from (15.43) is independent of the origin of the time scale used. Sornette and Sammis (1995) used the generalized log-periodic relation (15.41) to fit the strain accumulation data of Bufe and Varnes (1993); the result is given in Figure 15.9(a). The data appear to exhibit a series of log-periodic fluctuations. An important question is whether this type of strain accumulation data can be used to predict earthquakes. Sornette and Sammis (1995) used the log-periodic fit to the data available prior to a cutoff date in Figure 15.9(a) to predict when an earthquake would be expected. Their results are given in Figure 15.9(b). The predictions became increasingly accurate as the cutoff times approached the date of the Loma Prieta earthquake, October 17, 1989. Although the ability to make this retrospective prediction is encouraging, it remains to be demonstrated that this technique can be used successfully to predict earthquakes. As a further illustration of log-periodic behavior, we now consider a hierarchical model for failure similar to that discussed above but including the time-to-failure approach (Newmann et al., 1995). An array of stress-carrying elements is considered analogous to the strands of an ideal, frictionless cable. Each element has a time-to-failure that is dependent on the stress the element carries and has a statistical distribution of values. When an element fails, the stress on the element is transferred to a neighboring element; if two adjacent elements fail, stress is transferred to two neighboring elements; if four adjacent elements fail, stress is transferred to four neighboring elements; and so forth. The hierarchical model for failure is illustrated in Figure 15.10. At the lowest order in this example there are 128 zero-order elements. These elements are paired to give 64 first-order elements, the 64 first-order elements are paired to give 32 second-order elements, and so forth. A statistical distribution of lifetimes is assigned to the lowest-order elements. When one of these elements fails, the stress on the element is transferred to the neighboring element, increasing the stress on it. If a pair of zero-order elements fail, that is, a first-order element, the stress is transferred to the adjacent pair of zero-order elements, that is, to the adjacent first-order element, and so forth. To illustrate the stress transfer consider the second-order (n = 4) example given in Figure 15.1 1. Each element is given a probabilistic "lifetime" and two examples of failure are illustrated. At time t = 0 the stress a, is ap-

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plied to the four elements. In both examples element "2" has the shortest lifetime and it is the first to fail. The stress uo on element "2" is transferred to element "1" placing a stress 2u0 on this element as illustrated in Figure 15.11 (ii). The question now is whether the enhanced stress on element "1" will cause it to fail prior to elements "3" or "4." In example (a) element "1" is the next to fail and the stress 2u, on this element is transferred to elements

Figure 15.10. Illustration of a seventh-order (N = 128) example of the hierarchical model of failure.

Figure 15.1 1. Illustration of stress transfer in a secondorder (N = 4) example of the hierarchical model. Each element is given a statistical "lifetime." In example (a) element "2" fails, transfemng stress to element "1"; element "1" then fails and stress is transferred to elements "3" and "4"; element "4" fails, transferring stress to element "3," which subsequently fails. In example (b) element "2" fails, transfemng stress to element "1"; element "4" fails, transfemng stress to element "3"; element "3" fails, transferring stress to element "1 ," which subsequently fails.

(ii)

(iii)

(b)

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"3" and "4" placing a stress 2a0 on both of these elements. Element "4" is the next to fail and the stress 20, on it is transferred to the last surviving element "3," which has a stress 4a0. In example (b) element "4" is the second element to fail and the stress a, on this element is transferred to element "3" placing stress 2a0 on this element. Element "3" is the next to fail and the stress 20, is again transferred to the last surviving element "1," which has a stress 4a0. The zone of stress transfer is equal in size to the zone of failure. This local load-sharing model simulates the elastic redistribution of stress adjacent to a rupture. If elements are subjected to a constant stress a at t = 0, (15.32) gives the statistical distribution of failure times t,. However, with stress transfer the stress is not necessarily constant. To accommodate the increase in stress caused by local load sharing from failed elements, we introduce a reduced time to failure for each element Ti, given by

Each element i is assigned a random time to failure ti, under stress a, based on (15.30). The actual time to failure of element i, namely Ti, is reduced below t, if stress is transferred to the element. The time Ti, is obtained by requiring that (15.44) be satisfied. Consider the example illustrated in Figure 15.11(a). The four elements i = 1, 2, 3, 4, carrying stress a, are assigned failure times t,,, t,,, t,,, and t4, using the probability distribution (15.30). Element "2" has the shortest failure time so that

Upon failure the stress a, carried by this element is transferred to element "1 ," as illustrated in Figure 15.11(a) (ii). Element "1" is the next element to fail, and its failure time TI, is given by

Upon failure of "1," the stress 2ao is transferred to elements "3" and "4," as illustrated in Figure 15.11(a) (iii). Element "4" is the next element to fail and its failure time T4, is given by

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Upon the failure of "4," the stress a, is transferred to element "3," as illustrated in Figure 15.11(a) (iv). The time to failure of element "3" is given by

Figure 15.12. Failure sequence for a 16th-order (N = 65,536) realization of the model. (a) Entire failure sequence (failure is completed at t = 0.0480266). (b) Expansion of the final sequence of partial failures. (c) Further expansion of two partial failures.

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311

Alternative failure sequences are also possible, one example of which is illustrated in Figure 15. l l(b). Again "2" is the first element to fail; however, in this case the second element to fail is "4," then "3" fails, and finally "1" fails. Results of a numerical calculation using a 16th-order (n = 65,536) realization of this model are given in Figure 15.12. The total failure sequence is given in Figure 15.12(a). The nondimensional time is taken to be T = v,t and failure in this case occurs at T = 0.048027. It is interesting that failure occurs at a nondimensional time that is more than an order of magnitude shorter than the mean time to failure of an individual element T,,, = 0.61315. The lifetime of the composite material is much shorter than the mean lifetime of individual elements. This is in agreement with the results obtained above using the renormalization group approach. The failure sequence between T = 0.0445 and total failure is expanded in Figure 15.12(b). There is a well-defined sequence of partial failures prior to the total failure at T~ = 0.048027. Well-defined partial failures occur at T , = 0.047965, T~ = 0.047799, T~ = 0.047487, T~ = 0.047 162, and T , = 0.046124. The failure sequence between T = 0.04745 and T = 0.04785 is further expanded in Figure 15.12(c) to show the structure of the partial failures at T = 0.047792 and T = 0.047487. In each case there is a nested sequence of higher-order partial failures. Further expansion would show higher orders of nesting. The structure is basically self-similar or fractal. There is a scaleinvariant sequence of precursory failures at all levels. Because of the stochastic nature of the model, the embedding is not always clear, and a partic-

Figure 15.12. (cont.)

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ular partial failure may be a part of a sequence or may be precursory to another failure in the sequence. But this is also the problem with distributed seismicity. It is also of interest to determine whether the sequence of partial failures can be inserted into the predictive log-periodic relation (15.43) to predict the time of the total failure. Taking the sequence of partial failures T,, T, and T,, we obtain the prediction T, = 0.047636 from (15.43); taking the sequence T,, r3, and T,, we obtain T, = 0.054774 from (15.43); and taking the sequence T,, r2, and T,, we obtain T, = 0.048154 from (15.43). The results are summarized in Table 15.1. There is clearly considerable scatter in the predictions. Other realizations give similar results. Although the embedded sequences of precursory failures are a ubiquitous feature of all realizations, there is considerable stochastic variability of the timing. In Figure 15.13 the logarithm of the number of unfailed elements is given as a function of the logarithm of the time to failure for a realization with 4096 points and p = 4. The power-law fit shown by the dashed line has a slope of 0.24, which compares with the power p = 0.25 predicted by the global load-sharing relation (15.36). Although there is considerable scatter, the power-law relation does appear to be a reasonable predictor for this model, just as Bufe and Varnes (1993) found for regional seismicity. It is im-

Table 15.1. Successive precursory failure events illustrated in Figure 15.12 are used to predict the final total failure time in the simulation. Three successive precursory failures are substituted into (15.43) to obtain the estimated failure times. The actual failure time is also given

3

0.047636 1st estimate 0.048627 0.054775 2nd estimate

0.047799 0.047487 0.047799 0.047965

3

0.048154 3rd estimate

True 3

failure time

RENORMALIZATION GROUP METHOD

313

portant to note, however, that the quality of the fit deteriorates as complete failure is approached. The global analysis employed in the derivation of (15.36) deteriorates owing to the increasing importance of localization in the evolution of the cascade of failures. The sequence of failures as a function of position on the linear array of elements is shown in Figure 15.14 for the above realization. The precursory

I o4 .

,

. , .... . . , ,.,, . . .... . , ,..,, . , .,..,. . . .,.,. . .....,.

1

,,

i

i

,

I

I

-------y = 1.391e+04 * xA(0.2445)R= 0.913 1

I

'

,.

, ,,

.........I .... .

Figure 15.13. Failure sequence for a 12th-order (N= 4096) realization of the model. The dashed line is a power-law fit to the failure sequence based on ( 1 5.36).

Figure 15.14. Sequence of failures as a function of position. First 512 elements.

3 14

RENORMALIZATION GROUP METHOD

cascades of failure are clearly illustrated. This figure illustrates the growing importance of localization in failure events as criticality is approached.

Problems Problem 15.1. A unit square is divided into 16 smaller squares of equal size and the four central squares are removed; the construction is repeated. Assume that the remaining squares represent a percolation cluster and determine ne for n = 16 and 256. Problem 15.2. Determine the equivalent expression to (15.2) for a cubic array; use the Menger sponge as an example. Problem 15.3. Consider the 3 X 3 renormalization group approach to the two-dimensional array of square elements. For this array the tabulation (permeable elements, impermeable elements, alternative configurations, and permeable configurations) is: (0,9, 1, O), (1,8,9, O), (2,7,36, O), (3, 6, 8431, (4,5, 126,22), (5,4, 126,591, ( 6 3 , 84,67), (7,2,36,36), (8, 1,9,9), (9,0, 1, 1). Derive equation (15.8). Problem 15.4. Consider the 2 X 2 X 2 renormalization group approach to the three-dimensional array of cubic elements. For this array the tabulation (permeable elements, impermeable elements, alternative configurations, and permeable configurations) is: (0,8, 1, O), (1,7,8, O), (2,6,28, 41, (3,5,56,24), (4,4,70,54), (5,3,56,56), (6,2,28,28), (7, 1, 8, 81, (8,0, 1, 1). Derive equation (15.9). Problem 15.5. Assuming the dark elements in Figure 15.1 are permeable, what is the size of the largest percolation cluster? Problem 15.6. Assuming the light elements in Figure 15.1 are permeable, what is the size of the largest percolation cluster? Problem 15.7. Derive the conditional probability given by (15.21). Problem 15.8. Derive (15.23) from (15.14) and (15.22). Problem 15.9. Consider the third-power Weibull distribution given by

instead of (15.14). Show that (15.23) should be replaced by

Show that the recursive failure relation becomes

RENORMALIZATION GROUP METHOD

Find the value of the unstable fixed point p* and the corresponding value of FIFO. Problem 15.10. Consider the fourth-power Weibull distribution given by

instead of (15.14). Show that (15.23) should be replaced by

Show that the recursive failure relation becomes

Find the value of the unstable fixed point p* and the corresponding value of FIFO. Problem 15.11. Consider the rate law for failure given in (15.34). Assume v = v, a constant. Show that the time where n, = 31 NOis T,,, = (In 2)lv0. Problem 15.12. Consider the failure relation given in (15.36). Show that the time to failure is given by t, = (pvo)-P. Problem 15.13. Derive (15.43) from (15.42). Problem 15.14. Show that (15.43) is invariant to a change in the origin of time. Substitute t, = ti + to, t, = t; + to, etc. and show that the primed times also satisfy (15.43). Problem 15.15. If the first three maxima in a log-periodic sequence are t,, t, and t,, (a) show that the fourth maxima in the sequence t, is given by

(b) Show that this result is invariant to a change in the origin of time, substitute t, = ti + to, t, = ti + to, etc. and obtain the same result. Problem 15.16. Assume that a series of events satisfy log-periodic behavior leading to a catastrophic event. The first three events occur at t, = 0, t, = 15 days, and t, = 25 days. Determine t, and t, (use the result obtained in Problem 15.15). Problem 15.17. Assume that three earthquakes that occurred in 1956.2, 1980.7, and 1994.5 are precursors to a great earthquake. Also assume that log-periodic behavior is applicable. Determine when the fourth earthquake in the precursory sequence will occur and when the great earthquake will occur.

315

Chapter Sixteen

SELF-ORGANIZED CRITICALITY

16.1 Sand-pile models

In the last chapter we considered the renormalization group method for treating large interactive systems. By assuming scale invariance a relatively small system could be scaled upward to a large interactive system. The approach is often applicable to systems that have critical point phenomena. In this chapter we consider the alternative approach to large interactive systems. This approach is called self-organized criticality. A system is said to be in a state of self-organized critically if it is maintained near a critical point (Bak et al., 1988). According to this concept a natural system is in a marginally stable state; when perturbed from this state it will evolve naturally back to the state of marginal stability. In the critical state there is no longer a natural length scale so that fractal statistics are applicable. The simplest physical model for self-organized criticality is a sand pile. Consider a pile of sand on a circular table. Grains of sand are randomly dropped on the pile until the slope of the pile reaches the critical angle of repose. This is the maximum slope that a granular material can maintain without additional grains sliding down the slope. One hypothesis for the behavior of the sand pile would be that individual grains could be added until the slope is everywhere at an angle of repose. Additional grains would then simply slide down the slope. This is not what happens. The sand pile never reaches the hypothetical critical state. As the critical state is approached additional sand grains trigger landslides of various sizes. The frequency-size distribution of landslides is fractal. The sand pile is said to be in a state of self-organized criticality. On average the number of sand grains added balances the number that slide down the slope and off the table. But the actual number of grains on the table fluctuates continuously. The principles of self-organized criticality are illustrated using a simple cellular-automata model. As in the previous chapter we again consider a square grid of n boxes. Particles are added to and lost from the grid using the following procedure.

SELF-ORGANIZED CRITICALITY

(1)

(2)

(3)

(4)

A particle is randomly added to one of the boxes. Each box on the

grid is assigned a number and a random-number generator is used to determine the box to which a particle is added. This is a statistical model. When a box has four particles it is unstable and the four particles are redistributed to the four adjacent boxes. If there is no adjacent box the particle is lost from the grid. Redistributions from edge boxes result in the loss of one particle from the grid. Redistributions from the corner boxes result in the loss of two particles from the grid. If after a redistribution of particles from a box any of the adjacent boxes has four or more particles, it is unstable and one or more further redistributions must be carried out. Multiple events are common occurrences for large grids. The system is in a state of marginal stability. On average, added particles must be lost from the sides of the grid.

This is a nearest neighbor model. At any one step a box interacts only with its four immediate neighbors. However, in a multiple event interactions can spread over a large fraction of the grid. The behavior of the system is characterized by the statistical frequency-size distribution of events. The size of a multiple event can be quantified in several ways. One measure is the number of boxes that become unstable in a multiple event. Another measure is the number of particles lost from the grid during a multiple event. When particles are first added to the grid there are no redistributions and no particles are lost from the grid. Eventually the system reaches a quasiequilibrium state. On average the number of particles lost from the edges of the grid is equal to the number of particles added. Initially, small redistribution events dominate, but in the quasi-equilibrium state the frequency-size distribution is fractal. This is the state of self-organized criticality. There is a strong resemblance to the renormalization group approach considered in the last chapter. In the renormalization group approach the frequency-size statistics are fractal only at the critical point. In the cellular automata model the frequency-size statistics are fractal only in the state of self-organized criticality. The behavior of a sand pile and the behavior of the cellular automata model have remarkable similarities to the seismicity associated with an active tectonic zone. The addition of particles to the grid is analogous to the addition of stress caused by the relative displacement between two surface plates, say, across the San Andreas fault. The multiple events in which particles are transferred and are lost from the grid are analogous to earthquakes in which some accumulated stress is transferred and some is lost. There is a strong similarity between the frequency-magnitude statistics of multiple events and the Gutenberg-Richter statistics for earthquakes. Before consid-

317

318

SELF-ORGANIZED CRITICALITY

ering the analogy further, we will describe the behavior of the cellular automata model in some detail. As a specific example we consider the 3 x 3 grid illustrated in Figure 16.1. The nine boxes are numbered sequentially from left to right and top to bottom as illustrated in Figure l6.l(a). The cellular automata model has been run for some time to establish a state of self-organized criticality. The further evolution of the model is as follows and is illustrated in Figure 16.1(b). Step I A particle has been randomly added to box 8. The number of particles in this box has been increased from two to three. Step 2 A particle has been randomly added to box 6, increasing the number of particles from one to two. This addition is illustrated in the change between steps 1 and 2 in Figure 16.l(b). Step 3a A particle has been randomly added to box 5, increasing the number of particles from three to four and making it unstable; the four particles are redistributed to the four adjacent boxes, increasing the number of particles in box 2 from three to four, the number of particles in box 4 from three to four, the number of particles in box 6 from two to three, and the number of particles in box 8 from three to four. Boxes 2, 4, and 8 are now unstable. No particles are lost from the

Figure 16.1. Illustration of the cellular automata model for a 3 X 3 grid of boxes. The boxes are numbered 1 to 9 as shown in (a). Particles are randomly added to boxes in (b) as shown in steps 1 and 2. In step 3a an added particle in box 5 gives four particles and these are redistributed to the adjacent boxes. Nine more redistributions are required in steps 3b to 3j before the grid is stabilized. The first number below the grid is the number of boxes that have been unstable in the sequence of redistributions. The second number is the cumulative number of particles that have been lost from the grid in the sequence of redistributions.

7

8 9

1

0

2

1

3

3

4

4

5

5

8

10 10

9

12 lla

1

0

2

1

3

2

4

3

1

2

2

3

3

3

4

5

6

7

1

2

SELF-ORGANIZED CRITICALITY

grid. This redistribution is illustrated in step 3a in Figure 16.l(b). The numbers below the grid are, on the left, the cumulative numbers of boxes subject to redistribution and, on the right, the cumulative number of particles lost from the grid. Step 3b Since several boxes are now unstable, an arbitrary choice must be made about which box will be considered first for further redistribution. The choice does not have a significant effect on the statistical evolution of the system. The four particles in box 2 are redistributed. One is lost from the grid and box 3 becomes unstable with four particles. Boxes 3, 4, and 8 remain unstable. In this sequence of redistributions two boxes have been made unstable and one particle has been lost from the grid. Step 3 c The four particles in box 3 are redistributed. Two are lost from the grid and box 6 becomes unstable with four particles. Boxes 4 , 6 , and 8 remain unstable. In this sequence of redistributions three boxes have been made unstable and three particles have been lost from the grid. Step 3d The four particles in box 4 are redistributed. One is lost from the grid and box 1 becomes unstable with four particles. Boxes 1,6, and 8 remain unstable. In this sequence of redistributions four boxes have been made unstable and four particles have been lost from the grid. Step 3e The four particles on grid point 8 are redistributed. One is lost from the grid and boxes 7 and 9 become unstable with four particles. Boxes 1, 6, 7, and 9 remain unstable. In this sequence of redistributions five boxes have been made unstable and five particles have been lost from the grid. Step 3f The four particles in box 9 are redistributed. Two are lost from the grid and box 6 is now unstable with five particles. Grid points 1, 6, and 7 remain unstable. In this sequence of redistributions six boxes have been made unstable and seven particles have been lost from the grid. Step 3g Four of the five particles in box 6 are redistributed. One is lost from the grid and box 5 is now unstable. Boxes 1, 5, and 7 remain unstable. In this sequence of redistributions seven boxes have been made unstable and eight particles have been lost from the grid. Step 3h The four particles in box 5 are redistributed for the second time. No particles are lost and no boxes are made unstable. Boxes 1 and 7 remain unstable. In this sequence of redistributions seven boxes have been made unstable and eight particles have been lost from the grid. Step 3i The four particles in box 7 are redistributed and two are lost from the grid. No boxes are made unstable so that 1 is the only remaining unstable box. In this sequence of redistributions eight boxes have been made unstable and ten particles have been lost from the grid.

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Step 3j The four particles in box 1 are redistributed and two are lost from the grid. No boxes remain unstable so that the sequence of 10 redistributions has completed step 3. During step 3 all nine boxes were unstable and 12 particles were lost from the grid. Step 4 A particle has been randomly added to box 5, increasing the number of particles from zero to one. Step 5 A particle has been randomly added to box 6, increasing the number of particles from two to three.

This relatively simple example illustrates how the cellular automata model works. To develop significant statistics larger grids must be considered. Kadanoff et al. (1989) have carried out extensive studies of the behavior of this model as a function of grid size. One statistical measure of the size of an event is the number of grid points that become unstable. The results for a 50 X 50 grid of boxes are given in Figure 16.2. The number of events N in which a specified number of boxes A participated is given as a function of the number of boxes. A good correlation with a fractal power law is obtained, with a slope of 1.03. Since the number of grid points is equivalent to an area, the equivalent fractal dimension is D = 2.06. This statistical behavior appears to resemble that of distributed seismicity. However, the statistics in Figure 16.2 are not cumulative. In fact a fractal relation is not obtained for the cumulative statistics.

2

: C9 r(

2 V bD

0

-e

4

y=2.99-1.03~

Figure 16.2. Statistics for a cellular-automatamodel on a 50 X 50 grid. The number N of events in which a specified number A of boxes became unstable is given as a function of A.

2 8

Incremental Distribution Grid Size: 50x50

0.0

0.4

0.8

1.2

1.6

log(Area)

2.0

2.4

2.8

SELF-ORGANIZED CRITICALITY

A number of groups have studied the frequency-size statistics of avalanches on real sand piles, and in some cases fractal distributions have been found (Evesque, 1991; Nagel, 1992; Puhl, 1992). Held et al. (1990) found fractal statistics applicable for small avalanches but not for large avalanches. Bretz et al. (1992) and Rosendahl et al. (1993) found near-periodic large avalanches and a fractal distribution of small avalanches. Segre and Deangeli (1995) have developed a more realistic cellular-automata model for actual landslides. The fractal statistics of actual landslides have been considered by Yokoi et al. (1995). Turbidite deposits are associated with slumps off the continental margin. These avalanche-like events can be considered a natural analog for sand slides and thus for the cellular-automata model considered above. Turbidite deposits are generally composed of a sequence of layers, each layer representing a distinct event (slump). Each layer is composed of an upward gradation from coarse-grained sediments to fine-grained sediments, and individual layers are generally separated by well-defined bedding planes. Several studies of the thickness statistics of turbidite deposits have been carried out. Rothman et al. (1994) carried out direct measurements on an outcrop of the Kingston Peak Formation near the southern end of Death Valley, California. Their results are given in Figure 16.3(a); an excellent correlation with the fractal relation (2.6) is obtained taking D = 1.39. Hiscott et al. (1992) have studied a volcaniclastic turbidity deposit in the Izu-Bonin forearc basin off the shore of Japan. Layer thicknesses were obtained from formation-microscanner images from well logs in the middle to upper Oligocene part of the section. Results for two DSDP holes located 75 km apart are given in Figure 16.3(b); a good correlation with (2.6) is obtained taking D = 1.12. It is difficult to make a direct comparison between the thickness statistics of the sedimentary layers and the volume statistics of sand piles. However, the layer statistics appear to be scale invariant to a good approximation. It is interesting to note that the fractal dimensions of the thickness statistics are greater than one. For such a one-dimensional sequence it would appear that this would be impossible considering the examples given in Figure 2.1. The constructions illustrated in Figure 16.4 show that D can in fact be greater than one. The standard Cantor set is illustrated in Figure 16.4(a); one layer, N, = 1, with thickness r , = two layers, N2 = 2, with thickness r2 = $, four layers, N3 = 4, with thickness r3 = &. From (2.1) D = In 2lln 3 = 0.6309. In Figure 16.4(b) a stretched Cantor set is illustrated. At each step the remaining segments are stretched by a factor of two before being further subdivided. This gives N, = 1 with r , = 3, N, = 2 with r2 = $,N3 = 4 with r3 = &. From (2.1) D = In 2lln (312) = 1.710. The length L of the set is unbounded, L + = as r + 0. For real data sets with both upper and lower bounds on r, this construction illustrates that values of D greater than one are acceptable.

4,

321

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To model crustal seismicity, Barriere and Turcotte (1994) introduced a cellular-automata model in which the boxes have a scale-invariant distribution of sizes. The objective was to model a scale-invariant distribution of fault sizes. When a redistribution from a box occurs, it is equivalent to a characteristic earthquake on the fault. A redistribution from a small box (a

Log layer thickness h (m) (a)

Figure 16.3. Cumulative frequency-thickness statistics for turbidite sequences of sedimentary layers. (a) Kingston Peak Formation near the southern end of Death Valley, CA. (b) Izu-Bonin forearc basin off the shore of Japan. The roll-off for thin layers is attributed to loss of resolution. The straight-line correlations with the fractal relation (2.6) give D = 1.39 in (a) and D = 1.12 in (b).

-2.5

kL _ -2.0

I-

-1.5

_ 1 L -1.0 -0.5 0.0 0.5 Log layer thickness h (m)

_.

(b)

. J

1.0

';4 J

SELF-ORGANIZED CRITICALITY

323

foreshock) may trigger an instability in a large box (the main shock). A redistribution from a large box always triggers many instabilities in the smaller boxes (aftershocks). As a specific example we again consider the surface exposure of the fractal fragmentation model given in Figure 3.3. A fifth-order realization of this construction is given in Figure 16.5. We have N , = 1 box with r , = $, N2 = 3 boxes with r, = $, N , = 9 boxes with r, = N4 = 27 boxes with r4 = h, and N, = 108 boxes with r, = Except for N, the N are related to the ri by the fractal relation (2. I) with D = In 3lln 2 = 1.5850.

2.

i,

Figure 16.4. (a) Cantor set N,=l,r,=:;~,=2,r,=a; N,=4,r,=h;~=ln2nn 3 = 0.6309. (b) Stretched CantorsetN, = l , r , = f ; N, = 2, r,= ~ , = 4 , r, = $, ; D = In 21111(312) = 1.710.

i,

Figure 16.5. Illustration of the fractal cellular model corresponding to the discrete model for comminution illustrated in Figure 3.7 carried to fifth order, D = In 311112 = 1 S85.

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The standard cellular-automata rules are applied to this model: (1)

(2) (3)

(4)

Particles are added one at a time to randomly selected boxes. The probability that a particle is added to a box is proportional to the area A, = r;of the box. A box becomes unstable when it contains 4A. particles. Particles are redistributed to immediately acfjacent boxes or are lost from the grid. The number of particles redistributed to an adjacent box is proportional to the linear dimension ri of that box. If, after a redistribution of particles from a box, any of the adjacent boxes are unstable, one or more further redistributions are carried out. In any redistribution, the critical number of particles is redistributed. Redistributions are continued until all boxes are stable.

The cumulative frequency-magnitude statistics for main shocks of a seventh-order (1 28 X 128) version of the model are given in Figure 16.6. We find an excellent correlation with the fractal relation (2.6) taking D = 2.50 (b = 1.25). This is significantly higher than the observed values for distributed seismicity. Evernden (1970) has obtained b-values for regional seismicity and concludes that b = 0.85 0.20. It was also found that 31.5% of the largest events had foreshocks. This is in reasonable agreement with studies of actual earthquakes; von Seggern et al. (1981) found that 21%of the earthquakes studied had foreshocks and Jones and Molnar (1979) found that 44%

+

Figure 16.6. Cumulative frequency-magnitude statistics for unstable events. The number of events Nc in boxes equal to or smaller than r is divided by the total number of events N, and given as a function of r. The correlation is with (2.6) taking D = 2.5.

Box Size r

SELF-ORGANIZED CRITICALITY

of larger shallow earthquakes that could be recorded teleseismically had foreshocks. The aftershocks also correlate well with (2.6) taking D = 2.02 (b = 1.01). A similar model has been proposed by Henderson et al. (1994).

16.2 Slider-block models Slider-block models that exhibit self-organized criticality can also be constructed. In Chapter 11 we showed that a pair of interacting slider blocks can exhibit deterministic chaos. This model is easily extended to include large numbers of slider blocks. Carlson and Langer (1989) considered long linear arrays of slider blocks with each block connected by springs to the two neighboring blocks and to a constant-velocity driver. They used a velocityweakening friction law and considered up to 400 blocks. Slip events involving large numbers of blocks were observed, the motion of all blocks involved in a slip event were coupled, and the applicable equations of motion had to be solved simultaneously. Although the system is completely deterministic, the behavior was apparently chaotic. Frequency-size statistics were obtained for slip events and the events fell into two groups: smaller events obeyed a power-law (fractal) relationship but there was an anomalously large number of large events that included all the slider blocks. This model was considered to be a model for the behavior of a single fault, not a model for distributed seismicity. The large events were associated with characteristic earthquakes on the fault and smaller events with background seismicity on the fault between characteristic earthquakes. Nakanishi (1990, 1991) proposed a model that combined features of the cellular-automata model and the slider-block model. A linear array of slider blocks was considered but only one block was allowed to move in a slip event. The slip of one block could lead to the instability of either or both of the adjacent blocks, which would then be allowed to slip in a subsequent step or steps, until all blocks were again stable. Brown et al. (1991) proposed a modification of this model involving a two-dimensional array of blocks. Other models of this type have been considered by Bak and Tang (1989), Takayasu and Matsuzaki (1988), Ito and Matsuzaki (1990), Sornette and Sornette (1989, 1990), Langer and Tang (1991), Carlson (1991a, b), Carlson et al. (1991, 1993a, b, 1994), Matsuzaki and Takayasu (1991), Rundle and Brown (1991), Feder and Feder (1991), Chen et a1. (1991), Shaw et al. (1992), Huang et al. (1992), Christensen and Olami (1992), Olami and Christensen (1992), Olami et al. (1992), Vasconcelos et al. (1992), de Sousa Vieira (1992), Cowie et al. (1993), Shaw (1993a, b, 1994, 1995), Rundle and Klein (1993, 1995), de Sousa Vieira et al. (1993), Schmittbuhl et al. (1993), Knopoff et al. (1993), Ding and Yu (1993), Lu et al. (1994), Senatorski (1994), Xu and Knopoff (1994), Pepke and Carlson (1994), Pepke et al.

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(1994), Rubio and Galeano (1994), Robinson (1994), Espanol (1994), and Lin and Taylor (1994). McCloskey (1993), and McCloskey and Bean (1994) considered arrays of slider blocks connected to two driver plates, and these driver plates were treated as a pair of interacting slider blocks. The standard two-dimensional array of slider blocks is illustrated in Figure 16.7. In the cellular-automata approximation it is assumed that during the sliding of one block, all other blocks are stationary; this requirement limits the system to nearest neighbor interactions, which is characteristic of cellular-automata systems. To minimize the complexity we considered a discontinuous static-dynamic friction law. After non-dimensionalization of the governing equations, the governing parameters are a = kclkl (kc is the spring constant of the connector springs, k, is the spring constant of the puller springs), a is a measure of the stiffness of the system, = FJF, (the ratio of the static friction F, to dynamic friction F,), and N the number of blocks considered. In this model the parameter can be eliminated by rescaling. Thus for large systems (N very large) the only scaling parameter is the stiffness a. Frequency-size statistics for a 50 X 50 (N = 2500) array are given in Figure 16.8 for several values of the stiffness parameter a . A good correlation is obtained with the fractal relation (2.6) with D = 2.72. The frequency-size relation shows a roll-off from the power law near the larger end of the scaling region. This deviation is reduced as the parameter a increases. Frequency-size statistics for several different size arrays are given in Figure 16.9. When the parameter alNlJ2 is greater than one, we observe an excess number of catastrophic events that include the failure of all blocks. The failure statistics of these multiple-block systems clearly indicate a self-organized critical behavior and are remarkably similar to distributed seismicity.

+

+

Figure 16.7. Illustration of the two-dimensional slider block model. An array of blocks each with mass m is pulled across a surface by a driver plate at a constant velocity V. Each block is coupled to the adjacent blocks with either leaf or coil springs with constant kc, and to the driver plate with leaf springs with spring constant k,. The extension of the (i, j ) pulling spring is xii

SELF-ORGANIZED CRITICALITY

327

The frequency-size distribution of events associated with self-organized criticality certainly resembles the regional distribution of earthquakes in a zone of active tectonics. This suggests that interactions between faults play an essential role in the behavior df such zones.

Figure 16.8. The ratio of the number of events N with size Nf to the total number of events No is plotted against Nf (Nf is the number of blocks that participate in an event and is a measure of the area of an event). Results are given for 4 = 1.5 and a = 10, 15,20, 30, and 40. The solid line is the correlation with the power-law (fractal) relation (2.6); the corresponding fractal dimension is D = 2.72.

I

0.0

0.5

1.0

1.5 2.0 log(N~)

2.5

3.0

3.5

Figure 16.9. The ratio of the number of events N with size Nf to the total number of events No is plotted against N,. Results are given for systems of size 20 X 20.30 X 30.40 X 40, and 50 X 50 with parameters 4 = 1.5 and a = 50. The peaks at log N, = 2.60,2.95, and 3.20 correspond to catastrophic events involving the entire system.

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An important consequence of a critical state in the crust is the large range of interactions. A basic question is whether an earthquake on one part of the planet, say Mexico, is correlated with an earthquake at a large distance, say Japan. The classical approach to earthquakes would say that this is impossible. The stresses associated with seismic waves are too small to trigger an earthquake and there is no conclusive observational evidence for correlated events on this spatial scale. The stress changes associated with the fault displacement are localized and are damped by the athenospheric viscosity. However, interactions at large distances are a characteristic of critical phenomena. The interactions are not through the direct transmission of stress but through the interactions of faults with each other. Scholz (1991) has argued that the entire earth's crust is in a state of self-organized criticality. He sites as direct supporting evidence the induced seismicity associated with dams and other sources. Whenever a reservoir is filled behind a large new dam, extensive swarms of earthquakes are generally triggered. This is evidence that the crust is at the brink of failure even at large distances from plate boundaries. This action at a distance may help to explain the apparent success of the earthquake-prediction algorithms developed at the International Institute for the Theory of Earthquake Prediction and Theoretical Geophysics in Moscow under the direction of Academician V. I. Keilis-Borok. This approach is based on pattern recognition of distributed regional seismicity (Keilis-Borok, 1990; Keilis-Borok and Rotwain, 1990; Keilis-Borok and Kossobokov, 1990). The pattern recognition includes quiescence (Schreider, 1990), increases in the clustering of events, and changes in aftershock statistics (Molchan et al., 1990). In reviewing regional seismicity after a major earthquake it is often observed that the regional seismicity in the vicinity of the fault rupture was anomalously low for several years prior to the earthquake (Kanamori, 1981; Wyss and Haberman, 1988). This is known as seismic quiescence. The problem has been to provide quantitative measures of quiescence prior to the major earthquake. We discussed the fractal clustering of earthquakes in Chapter 6. The clustering of regional seismicity appears to become more fractal-like prior to a large earthquake. There also appears to be a systematic reduction in the number of aftershocks associated with regional intermediate-sized earthquakes prior to a major earthquake. Pattern-recognition algorithms were developed to search earthquake catalogs for anomalous recursory behavior. Premonitory seismicity patterns were found for strong earthquakes in California and Nevada (algorithm "CN") and for earthquakes with M > 8 worldwide (algorithm "M8"). When a threshold of the anomalous behavior was reached, a warning of the time of increased probability (TIP) of an earthquake was issued. On a worldwide basis TIPS were triggered prior to 42 of 47 events. TIPS were released prior to the Armenian earthquake on December 7, 1988, and to

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329

the Loma Prieta earthquake on October 17, 1989. These are illustrated in Figure 16.10. The TIP issued for region 3 in the Caucasus during January 1987 was still in effect when the Armenian earthquake occurred in this region on December 7, 1988. TIPS were issued for region 5 in California during October 1984 and for region 6 during January 1985. These warnings were still in effect when the Loma Prieta earthquake occurred within these overlapping regions on October 17, 1989. The fault rupture of the Loma Prieta earthquake extended over about 40 km. However, the prediction algorithms detected anomalous seismic behavior over two regions with diameters of 500 km. Self-organizedcriticality can explain anomalous correlated behavior over large distances. This approach is certainly not without its critics. Independent studies have established the validity of the TIP for the Loma Prieta earthquake; however, the occurrence of recognizable precursory patterns prior to the Landers earthquake are questionable. Also, the statistical significance of the size and time intervals of warnings in active seismic areas has been questioned. Nevertheless, seismic activation prior to a major earthquake certainly appears to be one of the most promising approaches to earthquake prediction. I . THE CAUCASUS. M 2 6.5

44

42

40

38

b. CALIFORNIA-NEVADA,M 2 7

50 1975 1

45

1980

2 40

3 '4

35

-

5

6-

30

25 -135 -130

-125

-120

-115 -110

1985

Figure 16.10. Illustrations of the Armenian (December 7, 1988) and Lorna Prieta (October 18, 1989, Moscow time) earthquakes by 'KeilisBorok (1990). In (a) the Caucasus region is broken up into 10 areas with diameters of 500 km; two warnings (for region 3 and 9) are shown on !he right. The locations and times of four earthquakes are also given. In (b) the California-Nevada region is broken up into eight areas with diameters of 500 km. Four warnings (for regions 4-6.8) and the locations and times of four earthquakes are given.

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Although long-distance correlations between earthquakes are a subject of considerable controversy, such correlations have been widely accepted in China and Russia (as well as the former Soviet Union). A striking example was a sequence of five earthquakes that occurred in China between 1966 and 1976. These were the m = 7.2 Shentai (1966), m = 6.3 Hijien (1967), m = 7.4 Bo Sea (1969), m = 7.3 Haicheng (1975), and the m = 7.8 Tangshan (1976) earthquakes. These earthquakes spanned a distance of some 700 km, and the Haicheng earthquake was successfully predicted by the Chinese, at least partially on the basis of seismic activation (Scholz, 1977). However, the Tangshan earthquake was not predicted and estimates of fatalities in this earthquake range from 250,000 to 450,000. Seismic activation has been previously recognized in association with an increase in seismicity that occurred in the San Francisco Bay area prior to the 1906 earthquake (Sykes and JaumC, 1990). Earthquakes with estimated magnitudes between 6.5 and 7.0 occurred in 1865 (Santa Cruz Mountains), 1868 (Hayward), 1892 (Vacaville), and 1898 (Mare Island). There is a serious concern that a similar seismic activation is now underway in southern California. A number of intermediate-size earthquakes have occurred in southern California in the last 45 years. These include the m = 7.4 Kern County earthquake on July 21, 1952, the m = 6.4 San Fernando earthquake on February 9, 1971, the m = 7.6 Landers earthquake on June 28, 1992, and the m = 6.6 Northridge earthquake on January 17, 1994. The Landers earthquake provided direct evidence that faults interact with each other over large distances (Hill et al., 1993). The Landers earthquake triggered earthquakes at 14 distant sites scattered over the western United States. The farthest site was Yellowstone National Park in Wyoming, 1250 km from Landers. Just how information is transmitted over these distances is uncertain. One hypothesis is that the surface waves of the Landers earthquake were responsible. However, the stress levels associated with surface waves at this distance are no larger than the daily variations in stress associated with the earth tides. It is known from statistical mechanics that near a critical point spatial correlations extend to large distances. To better understand the statistical mechanics of slider-block models, we consider a two-dimensional array of slider blocks without a driver plate. Each block is connected to its four neighbors with springs (spring constant k) and is confined to move in the x-direction. The slider blocks interact frictionally with a surface; however, to conserve energy the dynamic friction is assumed to be zero. The problem is specified by the static friction and the initial total energy in the system. The force on a block (i, j) is

SELF-ORGANIZED CRITICALITY

A block slides if lF;,.,l > Fs,where Fsis the prescribed static friction force. To simplify the analysis and simulations, only one block in the array is updated during each microscopic time step. Before the update there are two possible states:

(I)

IF;, ,In+

The block was stuck after the previous update < Fs. However, the forces on the block have changed because of subsequent updates on neighboring blocks, there are now two possibilities: (a) The block is still stable, l ~ ~ , ~-, ,<, ,Fs, and the update is terminated. - > Fs. In this case motion of (b) The block is still unstable, the single slider block is given by

IF, ,In+,

The slipping block executes one-half of an harmonic cycle and sticks when the velocity is again zero. The change in the position of block (i, j), A x ,,, is related to the initial force on the block (FO;,,), + 1 - by

The new net force on the block (Firj)n+l+is determined, again there are two possibilities: < F, the block remains stuck until the next up(i) If date, (ii) If IF^,^^,, > Fsthe block slips until the next update. The block was slipping after the previous update > Fs. But again the forces on the block have changed because of subsequent updates on neighboring blocks. There are two possibilities: (a) The block is now stable, IF;,]^+^- < Fs, and the step is terminated. (b) The block is still unstable > Fs, and then (16.3) is used to determine the new position of the block and the new net force on the block (F,,j)n+ is determined. Again there are two possibilities: < Fs the block remains stuck until the next up(i) If (F;,]~+, date. I > Fs, the block slips until the next update. (ii) If IF^,

IF;,

+

(2)

IF;, ,In+

IF;, , I n + , -

,+

+

jln+

+

331

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The slider blocks are considered sequentially using a checker-board algorithm to sweep across the two-dimensional array. It is convenient to introduce the nondimensional variables Ti, = Fi,,IFs. Z = kxlF,, i = t m , and = k E k l qwhere E, is the energy in spring k. The nondimensional force on a block is

zk

and the motion of a block is given by

If

IFi]- > 1, block (i, j) is unstable and its nondimensional slip is given by

At t = 0 the blocks are given a random distribution of displacements of the nondimensional energy in spring k is The only parameter in this problem is the mean energy per spring introduced at t = 0, E, which is given by

&.

Since no energy is dissipated, this value remains constant and we use it as a control parameter for the model. If E is large, very few blocks will stick and we would expect that the system should behave like a set of harmonic oscillators with a Maxwell-Boltzmann (Gaussian) distribution of displacements. If the distribution of displacements of individual block (i,j) is Gaussian, the distribution of spring displacements will also be Gaussian. If this is the case, the probability distribution function for the energies in the springs & will be given by

One of the questions we address is whether the system evolves to this Maxwell-Boltzmann distribution. The corresponding probability distribution function for the forces on the springs is

SELF-ORGANIZED CRITICALITY

However, the slip condition for a block is determined by the statistical distribution of forces on the blocks. From (16.4) it is seen that the random force on a block is the sum of four random forces on the neighbor springs. These forces are not independent as one can see from (16.4) and the Gaussian distribution of forces on the block is

A block can slip if [FBI> 1. Using (16.10) the probability that a block will be slipping Psis

We will show that our results satisfy this condition. We have carried out a series of simulations on square arrays of up to 2000 X 2000 blocks. Springs on the boundaries of the array are attached to fixed walls. Random initial displacements were given to the blocks corresponding to specified values of E. Various initial distributions of energy (non-Gaussian) were used, and in all cases the system evolved to the Maxwell-Boltzmann distribution (16.8). A typical example is given in Figure 16.11 with E = 1. The statistical distribution of forces on the blocks was also determined and was found to be in excellent agreement with (16.10). The fraction of the blocks that are slipping Ps are given for several values of the mean energy E in Figure 16.12. Good agreement with the equilibrium prediction (16.11) is found. As the fraction of slipping blocks increases with increasing values of E, a path of slipping blocks across the array is eventually established. This strongly resembles the percolation threshold for the site-percolation model considered in Chapter 15. Both are critical points and the critical value of E is 0.213 with the corresponding fraction of slipping blocks Ps = 0.583. This value can be compared with the critical point for the site percolation model where the probability that a lattice percolates is p* = 0.5927. The small discrepancy between

333

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the two values is attributed to correlations between blocks in the slider-block model. Figure 16.13 shows the frequency-size distribution at this critical point (E = 0.213) and, as a reference, the frequency-size distribution for the site-percolation model with p = 0.5927; the two distributions are virtually identical power laws. A typical slider-block configuration with a continuous path of slipping blocks across the array is shown in Figure 16.14. It is clearly very similar to the site-percolation distribution given in Figure 15.1l(a).

Figure 16.11. T_he probability distribution p(E,)of the nondimensional energies E k in the springs of a multiple slider-block model. The crosses are the result for a 2000 X 2000 array of slider blocks with E = 1. The solid line is the MaxwellBoltzmann distribution of energies given in (16.8).

Figure 16.12. The fraction of the blocks that are slipping PI (E)is given as a function of the mean energy E. The crosses are results for a 1000 X 1000 array of slider blocks, and the soiid line is the prediction from (16.1 1).

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335

This simple energy-conserving system exhibits behavior that is quite similar to the seismicity in active tectonic regions such as California. In southern California the seismic activity level in the magnitude range 2 < M < 5 not only satisfies the fractal Gutenberg-Richter frequency-magni-

p=0.213 site percolatm

I

i

Figure 16.13. Number of clusters ns of size s as a function of s. The solid line is the distribution of percolating clusters for a 2000 X 2000 array with the critical percolation probability p = 0.5927. The dashed line is the distribution of slipping clusters of blocks on our 2000 X 2000 array of slider blocks at the critical point E = 0.2 13.

Figure 16.14. Illustration of a typical configuration of sliding blocks at the critical point E = 0.2 13 for a 64 X 64 array. White blocks are stuck and black blocks are sliding. A continuous path of sliding blocks across the array is present.

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tude scale, but also, the level of activity does not vary from year to year (see Figure 4.3). Earthquakes in this magnitude range strongly resemble the statistical fluctuations of the slider-block array near its critical point. Further evidence supporting the applicability of the "percolationw-like model comes from the spatial distribution of seismicity in southern California. The distributions given in Figure 4.12 appear to correspond to the fractal dimension of the percolation "backbone" of a critical three-dimensional percolation cluster. It appears that the earthquakes on a complex array of faults form a connected path across the zone of crustal deformation in direct analogy to the "percolation backbone" of slipping blocks in the array. Rundle et al. (1995) found that the block energy distribution for a driven slider-block model is a Maxwell-Boltzmann distribution as the model approaches the mean field where fluctuations are minimal.

16.3 Forest-fire models We next consider a class of models that are referred to as forest-fire models. These models generally exhibit self-organized criticality. We consider a square grid of sites, with each site designated by two numbers ij, where i is the row and j is the column. At each time step either a tree is randomly planted on a site or a match is dropped on the site. The sparking frequency f indicates how many trees are planted before a match is dropped. If r = 99 trees are planted (or are attempted to be planted) before a match is dropped. If a match is dropped on an empty site, nothing happens. But if a match is dropped on a tree, it ignites and all immediately adjacent trees bum. As a specific example of our forest-fire model we consider the 10 X 10 grid illustrated in Figure 16.15. The model has been run for some time to establish a state of self-organized criticality and its initial state is given in Figure 16.15(a). We take f = so that four trees are planted before a match is dropped. Between Figures 16.15(a) and (b) there are 5 time steps and the randomly selected grid points were 71, 76, 56, 81, and e5L Trees were planted on 7 1,56, and 8 1; 76 already had a tree, and a match was dropped on 95. This tree ignited and 35 adjacent trees also burned. Note that only trees immediately above, below, or to the sides of a burning tree also ignite. Following this forest fire 10 additional time steps are camed out to reach the distribution illustrated in Figure 1 6 . 1 3 ~ ) The . 10 randomly selected grid points are 72, 36, 00, 88, 48, 65, 44, 30, 45, and 44. Trees were planted on 72, 36, 88, 65, 44, 30, and 45; 00 already had a tree. The match dropped on 08 did not ignite because there was no tree on the grid point. The match dropped on 44 ignited this tree and burned the adjacent tree on 45. Following this small fire 25 additional time steps are carried out to reach the distribution illustrated in Figure 16.15(d). The 25 randomly selected grid points are

&,

:

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337

56,36,05, 15,25,68,40,52, 1 8 , 8 1 , 0 3 , 7 9 , 3 5 , 3 5 , 9 5 , 5 6 , 5 9 , 8 0 , 5 1 , 0 7 , 20,56,86,46, and 30. Trees were already planted on 36,05,79,35,56, and 56. The matches dropped on 25, 81, 95, and 07 did not ignite because there were no trees on these grid points. The match dropped on 30 ignited this tree and burned six adjacent trees. Frequency-size statistics for forest fires can be determined. Two examples for a 100 X 100 tree forest are given in Figure 16.16. The number of 1 burning clusters N is given as a function of their size A, forf = &j and f = m. For the larger value f = & j , fires consume the forest before large clusters can form. A reasonably good correlation with the fractal relation (2.2) is obtained taking D = 2.00. The roll-off from the power law near the larger end of the scaling region is very similar to that illustrated for the slider-block we model in Figure 16.8. When the sparking frequency f is reduced to observe an excess number of catastrophic fires that consume all or nearly all of the 10,000 trees. Again this is very similar to the behavior found for slider blocks when the stiffness parameter is large as illustrated in Figure 16.9.

&,

0

0

0

X

X X

X

X X

X X

0

X

X

0

X

0

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X X

X

X

x

X

x

m

x

0

0

Figure 16.15. Illustration of the forest-fire model on a 10 X 10 grid of points. Each point is identified by its ij coordinates (i-row,j-column). We take f = &.(a) Grid points with trees are indicated by circles. Between (a) and (b) trees have been planted on points 71.56, and 81; a match was then dropped on point 95, igniting the tree (indicated by m) and burning 35 adjacent trees (indicated by xs). Between (b) and (c) trees were planted on points 72,36, 88,65,44, 31, and 45; a match was dropped on point 44, igniting this tree (indicated by m ) and burning the adjacent tree at 45 (indicated by an x). Between (c) and (d) trees were planted on points 56, 15.68.40.52. 18,03,35,59, 80,20, 86, and 46; a match was dropped on 30 (indicated by m) and burned 6 adjacent trees (indicated by xs).

338

SELF-ORGANIZED CRITICALITY

i

Best fit line

Best fit line

1

\

lo'

N io3

1' 0

Figure 16.16. The number of forest fires N with size Af (Af is the number of trees that bum in a fire) is given as a function of A, Results are for a 100 X 100~orest-grid with sparking parameters f = &j and

A.

10'

o

10

+-I

1o0

*-. Best fit line) ,&lope =--0.59) - -

Figure 16.17. Cumulative number per year N o f fires with burned areas greater than Af given as a function of AT The data is for the Australian Capital Temtory for the period 1926-199 1. The best-fit straight line is with the fractal relation (2.6) . . taking D = 1.18.

0.1

0.01 0.01

0.1

1

A,, km2

10

100

loo0

SELF-ORGANIZED CRITICALITY

It is also of interest to compare the results of forest-fire models with the frequency-size statistics for actual forest fires. Data for forest and brush fires in the Australian Capital Territory for the period 1926-1991 are given in Figure 16.17. A reasonably good straight-line fit with the fractal relation (2.6) is obtained taking D = 1.18. It should be emphasized that this is the cumulative number, whereas the model results given in Figure 16.16 are noncumulative. The model results given in Figure 16.16 clearly illustrate the "Yellowstone Park" effect. After a massive forest fire covered a significant fraction of Yellowstone National Park, it was argued that if smaller fires had been allowed to burn, the massive forest fire could have been prevented. Allowing small fires to burn is equivalent to having a larger sparking frequency. The results given in Figure 16.16 illustrate how the small fires prevent the occurrence of catastrophic fires that burn essentially the entire model forest. A variety of authors have studied forest-fire models, including Drossel and Schwabl (1992a, b, 1993a, b, 1994), Mosner et al. (1992), Bak et al. (1992), Drossel et al. (1993), Henley (1993), Christensen et al. (1993), and Clar et al. (1994), and Strocka et al. (1995). Johansen (1994) has applied the forest-fire model to the spread of diseases.

Problems Problem 16.1. Consider the evolution of the cellular-automata model illustrated in Figure 16.l(b). (a) Which boxes have an additional particle in steps, 6 , 7 , 9 and lo? (b) Which boxes are unstable and how many particles are lost from the grid in steps 8, I la, I lb, I lc, and 1Id? Problem 16.2. Consider the evolution of the cellular-automata model illustrated in Figure 16.l(b). (a) Which boxes have an additional particle in steps 12, 13, 14, and 15? (b) Which boxes are unstable and how many particles are lost from the grid in steps 16a, 16b, 16c, 16d, 17a, and 17b? Problem 16.3. Consider the evolution of the cellular-automata model illustrated in Figure 16.l(b). (a) Which boxes have an additional particle in steps 18 and 19? (b) Which boxes are unstable and how many particles are lost from the grid in steps 20a, 20b, 20c, 20d, and 20e? Problem 16.4.

Consider a 2 X 2 grid of four boxes as illustrated above in (a). Also given above in ( 6 ) is a sequence of random numbers in the range 1-4.

339

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SELF-ORGANIZED CRITICALITY

Use the random numbers to assign particles to boxes and carry out the cellular automata model described in this chapter. Problem 16.5.

Consider the linear grid of four boxes illustrated above. Use the sequence of random numbers given in Problem 16.4 to assign particles to the four boxes. Use the following rules: When a box has two particles it is unstable and they are redistributed to the two adjacent boxes. If either of these boxes has two elements, they are again redistributed. Particles are lost from the ends of the linear grid. Problem 16.6. Consider the evolution of the forest-fire model illustrated in Figure 16.15. Consider the configuration given in (d) and determine its subsequent evolution using the random number sequence 96,09,35,67, 13, 33, 94,44, 66, 37. (a) How many trees are planted? (b) How many forest fires occur and how many trees are burned in them? Problem 16.7. Consider the evolution of the forest-fire model illustrated in Figure 16.15. Consider the configuration given in (d) and determine its subsequent evolution using the random number sequence 15,81,55,25, 53, 65, 29, 17, 7 3 , s . (a) How many trees are planted? (b) How many forest fires occur and how many trees are burned in them? Problem 16.8. Consider a linear (one-dimensional) forest-fire model using a grid of 10 points numbered sequentially from 0 to 9. Consider p = 4 so that after three trees are planted on random points, a match is dropped on a random point. Assume initially that trees are planted on points 1,3, and 5 and consider the random sequence 0, 1,7,2,3,2,6,4,0,7,7,4,9,4,7, 6. (a) Which points have trees after these 16 time steps? (b) How many forest fires occurred and how many trees burned in each fire?

Chapter Seventeen

WHERE DO WE STAND?

The concepts of fractals and chaos were introduced in 1967 (Mandelbrot, 1967) and in 1963 (Lorenz, 1963), respectively. Unlike many advances in science, the attribution is in both cases quite clear. In both cases it took more than 10 years before either concept received wide attention. Today, most scientists take these concepts for granted, although their utility may be questioned. The concept of self-organized criticality was introduced in 1988 (Bak et al., 1988). Again the attribution is not open to question, but the definition of the concept remains somewhat unclear, particularly in.regard to classical problems in criticality. There is no question that fractals are a useful empirical tool. They provide a rational means for the extrapolation and interpolation of observations. The normal distribution and the power-law (fractal) distribution have very wide applicability. Just as the central limit theorem provides a broad base for the application of the normal distribution, scale invariance provides a basis for the application of the power-law distribution. Some statisticians argue that fractals are a trivial extension of the Pareto distribution. But in its general form, the Pareto distribution is not scale invariant and historically it has been applied empirically without justification. A strong case can certainly be made for the wide applicability of powerlaw (fractal) distributions, but does this applicability have a more fundamental basis? Fractality appears to be fundamentally related to chaos and selforganized criticality. The chaotic behavior of deterministic nonlinear maps and sets of differential equations has been demonstrated beyond question. Chaotic behavior has been quantified in terms of the Lyapunov exponent. Toy models have been constructed that experimentally verify the basic concepts of chaos. It can be argued convincingly that the chaotic behavior of low-order systems demonstrate that weather, climate, seismicity, and so on are chaotic phenomena. But the low-order examples of chaos are far removed from practical problems. There are large gaps between the Lorenz equations and fluid tur-

342

WHERE DO WE STAND?

bulence, and between the Rikitate dynamo and the generation of the earth's magnetic field. The introduction of models that exhibit self-organized criticality has been a major advance in extending concepts of chaos to higher-order systems. In this regard slider-block models play a central role. Slider-block models were introduced as simple analog models for earthquakes. Distributed seismicity is taken to be a type example of self-organized critical behavior. The behavior of slider-block models is deterministic. Two slider blocks exhibit the classical chaotic behavior of a low-order system. Large numbers of slider blocks are self-organized critical. By systematically increasing the number of blocks, the transition from chaotic to self-organized critical behavior can be studied. One of the present frontiers of research is to examine the relationship between models that exhibit self-organized criticality and the basic aspects of statistical mechanics. For example, can earthquakes be better understood in terms of the statistical fluctuations of a quasi-equilibrium system? Another recent development is the recognition that complex fractal dimensions lead to log-periodic behavior. It has been suggested that log-periodic behavior may lead to a viable earthquake-prediction strategy. Some would argue that the concepts covered in this book fall under the broad umbrella of "complexity." But complexity is so broad a term that it defies any all-encompassing definition. Certainly, many aspects of geology and geophysics are complex; just as many problems in biology, economics, and human behavior are complex. There are also links between important problems in all these areas. This has led a number of scientists to propose a new science of complexity. The science would include fractals, chaos, and self-organized criticality. This is a major feature of the activities at the Santa Fe Institute. But there has also been a strong reaction against "complexity" with regard to its generality and a failure to deliver on promises made by some of its practitioners. The entire area of fractals, chaos, self-organized criticality, and complexity remains extremely active, and it is impossible to predict with certainly what the future holds.

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Abe, K. (1981). Magnitudes of large shallow earthquakes from 1904 to 1980, Phys. Earth Planet. Intr. 27,72-92. Ahnert, F. (1984). Local relief and the height limits of mountain ranges, Am. J. Sci. 284, 1035-55. Aki, K. (1981). A probabilistic synthesis of precursory phenomena, in Earthquake Prediction, D . W . Simpson & P. G. Richards, eds., pp. 566-74, American Geophysical Union, Washington, D.C. Alexandrescu, M., Gibert, D., Hulot, G., Le Mouel, J. L. & Saracco, G. (1995). Detection of geomagnetic jerks using wavelet analysis, J. Geophys. Res. 100, 12,557-72. Allain, C. & Cloitre, M. (1991). Scaling rules in rock fracture and possible implications for earthquake prediction, Nature 297,47-9. Allbgre, C. J., Le Mouel, J. L. & Provost, A. (1982). Scaling rules in rock fracture and possible implications for earthquake prediction, Nature 297,47-9. Allkgre, C. J. & Lewin, E. (1995). Scaling laws and geochemical distributions, Earth Planet. Sci. Lett. 132, 1 - 1 3. Alkgre, C. J. & Minster, J. F. (1978). Quantitative models of trace element behavior, Earth Planet. Sci. Lett. 38, 1-25. Allkgre, C. J., Treuil, M., Minster, J. F. Minster, B. & Albarede, F. (1975). Systematic use of trace element in igneous process, Contrib. Min. Petrol. 60,57-75. An, L. J. & Sammis, C. G. (1994). Particle size distribution of cataclastic fault materials from southern California, Pure Appl. Geophys. 143,203-27. Anderson, J . G. (1986). Seismic strain rates in the central and eastern United States, Seis. Soc. Am. Bull. 76, 273-90. Anderson, J. G. & Luco, L. E. (1983). Consequences of slip rate constraints on earthquake occurrence relations, Seis. Soc. Am. Bull. 73,47 1-96. Andrews, D. J. & Hanks, T. C. (1985). Scarp degraded by linear diffusion: inverse solution for age, J. Geophys. Res. 90, 10193-208.

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Wu, Q., Borkovee, M. & Sticher, H. (1993). On particle-size distributions in soils, Soil Sci. Soc. Am. J. 57,883-90. Wyss, M. & Haberman, R. E. (1988). Precursory seismic quiescence, Pure Appl. Geophys. 126,3 19-32. Xu, H. J. & Knopoff, L. (1994). Periodicity and chaos in a one-dimensional dynamical model of earthquakes, Phys. Rev. E50,3577-8 1 . Yokoi, Y., Carr, J. R. & Watters, R. J. (1995). Fractal character of landslides, Env. Eng. Geosci. 1,75-8 1 . Young, I. E. & Cruff, R. W. (1967). Magnitude and frequency of floods, Part 1 1 , U.S. Geological Survey, Water Supply Paper 1685, pp. 714-5. Young, R. K. (1992). Wavelet Theory and Its Applications, Kluwer, Dordrecht, 223 pp. Youngs, R. R. & Coppersmith, K. J. (1985). Implications of fault slip rates and earthquake recurrence models to probabilistic seismic hazard estimates, Seis. Soc. Am. Bull. 75,939-64. Zipf, G. P. (1949). Human Behavior and the Principle of Least EfSort, Addison-Wesley, Reading, Mass.

Appendix A

GLOSSARY OFTERMS

A point in phase space toward which a time history evolves as transients die out. BAS IN O F ATT R ACT1 ON Some dynamical systems have more than one fixed point. The region in phase space in which solutions approach a particular fixed point is known as the basin of attraction of that fixed point. The boundaries of a basin of attraction are often fractal. B 1 FUR C AT I 0 N A change in the dynamical behavior of a system when a parameter is varied. B R 0 W N I A N WALK The running sum of a sequence of random values usually obtained from a normal distribution. C A N TOR D U S T A fractal set generated by subdividing a line into parts. C H A O S Solutions to deterministic equations are chaotic if adjacent solutions diverge exponentially in phase space; this requires a positive Lyapunov exponent. C L U S T E R A group of particles with nearest-neighbor links to other particles in the cluster. D E T E R M I N IS T I C A dynamical system whose equations and initial conditions are fully specified and are not stochastic or random. D I F F E R E N C E E Q U AT I 0 N An equation that relates a value of a function x,, to a previous value x,,. A difference equation generates a discrete set of values of the function x. D I F F U S I O N - L I M I T E D A G G R E G A T I O N ( D L A ) Diffusing (randomwalking) particles accrete to a seed particle to form a dendritic structure. D I M E N S I O N The usual definition of dimension is the topological dimension. The dimension of a point is zero, of a line is one, of a square is two, of a cube is three. In this book we have introduced the concept of fractional (noninteger) dimensions, or fractals. F E I G EN B A U M C 0 N S TA NT The ratios of successive differences between period-doubling bifurcation parameters approach this number (F = 4.699 202). ATTR ACTOR

372

A P P E N D I X A: GLOSSARY OF T E R M S

A point in phase space toward which a dynamical system approaches as transients die out. F R A C T A L An object that is quantified by a fractional dimension. A measure of a scale-invariant object. In some cases a power-law relation between the number of objects and their linear size. F R A C T A L D I M E N S I O N The fractional dimension of a fractal. The power in the power-law fractal scaling. HA U S D 0 R F F M E AS U R E The power law in the scaling of a self-affine fractal. H 0 P F B I FU R C AT I 0 N A bifurcation from a fixed point to a limit cycle. H U R S T E X P 0 N E NT Scale-invariant correlation used in rescaled range (WS)analysis. L A C U N A R I T Y A quantitative measure of clustering. L I M I T C Y C L E A periodic orbit in phase space toward which a dynamic system approaches as transients die out. L 0 G - PER I 0 D I C B E H AV I 0 R Power-law (fractal) behavior when the power is complex. L 0 R E N Z E Q U AT I 0 N S A set of three first-order differential equations derived from the equations governing thermal convection. Historically this was the first example of deterministic chaos. LYA P U N 0 V E X P O N E N T Solutions to deterministic equations are chaotic if adjacent solutions diverge exponentially in phase space; the exponent is known as the Lyapunov exponent. Solutions are chaotic if the Lyapunov exponent is positive. M A P A mathematical relation that translates one or more points into other points. M U LT I FR A CTA L S An infinite sequence of fractal dimensions obtained from the moments of a statistical distribution. N O D E A fixed point toward which solutions evolve. P E R C O L AT1 ON C LU S T E R A grid of sites in two or three dimensions. The probability that a site is permeable is specified, and there is a sudden onset of flow through the grid at a critical value of the probability. P E R I O D D O U B L I N G A sequence of periodic oscillations in which the period doubles as a parameter is varied. P H A S E S PACE A coordinate space defined by the state variables of a dynamical system. P I T C H F O R K B I F U R C A T I O N A bifurcation in which the period doubles. P 0 I N c A ~ f S iE C T I 0 N The sequence of points in phase space generated by the penetration of the evolving trajectory through a specified planar surface. PR 0 B A B I L I T Y The likelihood that a particular event will occur. The probability that the next flip of a coin will be heads is 0.5. FIXED POINT

A P P E N D I X A: GLOSSARY OF T E R M S

A choice that is determined by pure chance. For example, a flip of a coin. RENORMALIZATION The transformation of a set of equations from one scale to another by a change of variables. R E S C A L E D R A N G E ( R / S ) A N A L Y S I S An empirical technique for quantifying correlations in time series. S A D D L E POINT A fixed point that attracts only a singular set of trajectories. S C A L E IN VA R I A N C E The phenomenon whereby an object appears identical at a variety of scales. S ELF - A FFI N E Under an affine transformation the different coordinates are scaled by different factors. If an object is scaled using an affine transformation, the object is described as being self-affine. S ELF - S I M I L A R I T Y A property of a set of points if their geometrical structure at one length scale is the same as at another length scale. S TOC HA ST1 C A process determined by chance. S T R A N G E ATTRACTOR A fixed point in a phase space in which the orbits are chaotic. T I M E S E R I E S Values of a quantity as a function of time. WAVELET A scale-invariant filter used to quantify time series. WHITE N 0IS E A sequence of random values, often obtained from a normal distribution. RANDOM

373

Appendix B

UNITS AND SYMBOLS Table B 1. SI units Quantity

Unit

Basic units Length Time Mass Temperature Electric current

meter second kilogram kelvin ampere

Derived units Force Energy Power Pressure Frequency Charge Electric potential Magnetic field

newton joule watt pascal hertz coulomb volt tesla

Multiples of 10 lo-' 10-6 lo-' 10-l2 I 0' 1O6 I o9 10l2

milli micro nano pic0 kilo mega giga tera

Symbol m s

kg K A

N J W Pa Hz C V

T

Equivalent

APPENDIX

B: U N I T S A N D S Y M B O L S

Table B2. Symbols Symbol

Quantity

Equation introduced

Power Parameter Radius of earth Frequency of earthquakes Area Lorenz variable b-value for earthquakes Lorenz variable Benioff strain Constant Specific heat at constant pressure Coefficient of variation Constant Concentration Pair correlation distribution Lorenz variable Moment of inertia Constant Euclidean dimension Fractal dimension Energy Probability Probability distribution function Probability of fragmentation Fraction Frequency Cumulative distribution function Flood frequency factor Feigenbaum constant Force Wavelet filter Acceleration due to gravity Applied torque Elevation Layer thickness Hausdorff measure Hurst exponent Electrical current Transport coefficient Boltzmann constant Constant Wave number Spring constant Thermal conductivity Partition coefficient Rikitake parameter Length Lacunarity Self-inductance

(3.40) (10.1) (7.63) (4.1) (3.67) (12.23) (4.1) (12.24) (15.38) (4.4) (12.4) (3.33) (2.1) (5.1) (6.13) (12.24) (14.3) (4.4) (6.12) (2.1) (4.2) (3.5) (3.12) (3.70) (6.22) (7.41) (3.10) (8.31) (10.11) (11.1) (8.32) (12.3) (14.2) (7.28) (12.5) (7.1) (7.56) (14.1) (8.25) (3.16) (3.40) (7.66) (9.12) (12.4) (5.29) (14.13) (2.9) (6.20) (14.3)

SI units

m s-I m2

Jl/?

J kg-' K

kg m2

J

-- I

N ms- ? Nm m m A m2s-I JK-' m-I Nm-I Wm-I K-' m VsA-1 (continued)

376

A P P E N D I X B: U N I T S A N D S Y M B O L S

Table B2. (cont.) Symbol

Quantity Mass Earthquake magnitude Mass Earthquake moment Moment Mutual inductance Number Number Nusselt number Number per unit time Pressure Probability Perimeter Prandtl number Power Volumetric flow Linear dimension Autocorrelation function Radius Ratio of Rayleigh numbers Rate Range Resistance Rayleigh number Bifurcation ratio Length-order ratio Information entropy Entropy Power spectral density Standard deviation Time Temperature Time interval Branching ratio Variable Horizontal velocity coordinate Velocity Vertical velocity Volume Variance Wavelet transform Variable Horizontal coordinate

Equation introduced

SI units

m (continued )

APPENDIX B: U N I T S A N D SYMBOLS

Table B2. (cont.)

Symbol

Quantity

Y

Variable Vertical coordinate Fourier transform Nondimensional position Constant Mass ratio Stiffness parameter Volume coefficient of thermal expansion Power Constant Symmetry parameter Constant Coefficient of skew Semivariance Slope Displacement across fault Strain Small quantity Parameter Latitude Polar coordinate Temperature difference Thermal diffusivity Wavelength Lyapunovexponent Shear modulus Viscosity Rikitake parameter Power Hazard rate Density Polar coordinate Standard deviation Stress Time interval Nondimensional time Porosity Enrichment factor Longitude Friction parameter Stream function Angular velocity

Y a

P

B Y

IS E

0

K

A

CL v P

u T

dJ

$

a

Equation introduced (3.18) (2.7) (7.38) (11.3) (4.5) (5.13) (11.12) (12.8) (7.41) (9.12) (11.12) (4.6) (3.4) (7.8) (10.6) (4.3) (4.27) (5.39) (9.13) (7.63) (9.28) (12.7) (12.11) (7.66) (10.19) (4.3) (12.2) (14.8) (3.48) (15.30) (3.82) (9.28) (3.3) (15.44) (4.23) (11.3) (3.81) (5.1) (7.63) (1 1.3) (12.6) (14.1)

SI units m

K-'

m

K m2 s-I m

Pa Pa s t-I kg m-' Pa s

m2s-I s-I

377

ANSWERS TO SELECTED PROBLEMS N3 = 8 , r 3 = 1/27,N4 = 16,r4 = 1181. N3 = 27, r3 = 11125, N4 = 81, r, = 11625. ( b ) N l = 2, N2 = 4 , N , = 8 , r , = 115,r2 = 1125,r, = 11125. ( c ) D = 0.4307. ( b ) N , = 3 , N 2 = 9 , N 3 = 27, r , = 117, r2 = 1149, r, = 11343. (c) D = 0.5646. ( b ) N l = 4 , N, = 16, N3 = 6 4 , r , = 117,r2 = 1149,r, = 11343. ( c ) D = 0.7124. N3 = 512, N4 = 4096, r3 = 1/27, r4 = 1/81. ( b )N , = 2, N, = 4 , N, = 8, r , = 112,r2 = 114, r3 = 118. ( c )D = 1. ( b )N , = 5 , N, = 25, N3 = 125, r , = 113, r2 = 119, r, = 1/27. (e) D = 1.46. ( b ) N , = 5 , N2 = 25,N3 = 125,rl = 113,r2 = 119,r3 = 1/27. ( c ) D = 1.46. ( b )N , = 12, N, = 144, N, = 1728, r , = 114, r2 = 1116, r3 = 1164. ( c ) D = 1.79. ( b )N , = 24, N2 = 576, N3 = 13 824, r , = 115, r2 = 1/25, r, = 11125. ( c ) D = 1.9746. ( b ) N , = 17,N2 = 289, N, = 4 9 1 3 , r , = 115,r2 = 1125,r3 = 11125. ( c ) D = 1.76. D = 2.975. ( b )A , = 112, A, = 1212, A, = 12012. ( c ) No. ( b )No = 4 , N l = 32, N2 = 256, ro = 1, rl = 114, r2 = 1/16, Po = 4 , P I = 8, P, = 16. ( c ) D = 1.5. ( a ) N , = 1,N2 = 8, N, = 6 4 , r l = 113,r2 = 119, r, = 1127. ( b )Yes. ( c ) D = 1.89. Yes, 1. 0.577 Myr, 1.208 Myr.

ANSWERS TO SELECTED PROBLEMS

(a) 3 km, 6 km, 5 Myr, 15 Myr, 10 Myr, 20 Myr. (b) 1 km, 2 km, 4 km, 5 km, 7 km, 8 km, 1 Myr, 3 Myr, 11 Myr, 13 Myr, 2 1 Myr, 23 Myr, 2 Myr, 4 Myr, 12 Myr, 14 Myr, 22 Myr, 24 Myr. (d) R (25 Myr) = 0.36 m d y r , R (5 Myr) = 0.6 m d y r , R (1 Myr) =

0.112, 0.108,0.096, 0.079. 0.240,0.760. 1.65, 2.16, 1.31. 0.894. 0.707.

vz.

0.523. 7 = vrO/(v I), V = v(v + 3)(v - I)ril(v + 2)(v + 1)2. 7, 140,2800; 2.727. Nl=4,N,= 16,N3=64,D=2. 6, 126,2646; 2.77 12. 4, = 0.0385, p, = 0.9627p0, 4, = 0.0727, p, = 0.9273p0, D = 2.9656. 4, = 0.2857, p, = 0.7778p0, 4, = 0.395 1, p, = 0.6049p0, D = 2.7712. E, = 2.1 X 1015 J, M = 4 X lO19J,A = 530km2, ae = 2.5 m. E, = 7.1 X 1013J, M = 1.3 X 1018J,A = 54km2, ae = 0.8m. 100. 11.9 yr. 103 yr. (a) a = 2 X 106yr-1. (b) re = 158 yr. (a) a = 7 X lo4 yr-1. (b) re = 22 yr. 400 yr. 1600 yr. 1, 8,64; 1.446. 1.292. 1,o. 1000 yr. 113 yr.

+

379

380

ANSWERS TO SELECTED PROBLEMS

1.57. 1.26. 113. 213. (a) (8 - 4,)Cd7. (b) (8 - $8)+8Cd7' ( c ) 1 < < 8. (d)O < D < 3. 107 kg. 3.18 X 108 kg. 88 kg. 4.85 X 106 kg. 8.37 X 1010 kg. 3.9 X 107 kg. 2.55. 0.5. 1, 113, 119, 1/27. 1, 1, 1, 1. 1,315,9125. 1,417, 16149,641343. 1,317,9149,271343. 1, 113, 119, 1127. 1, 112, 114, 118. 1, 17125,2891625. 1, 618, 36164, 2161512. 1,26127, 6761729. c (112) = 4, C (1) = 2. C (1) = 6. C (1) = 24, C ( f i ) = 2 4 1 f i , C ( f i ) = 8 1 f i .

ANSWERS TO SELECTED PROBLEMS

381

382

ANSWERS TO SELECTED PROBLEMS

1.513; 0.75, 2.0194, 1.40. y = clear,x = c2efi. = clept(diverges), x, = ~ ~ e - (converges), 2~' x, = c,e-2~' (converges). x = 1 (stable), x = - l (unstable). (a) - 1, 0, 1. (b) stable, unstable, stable. x = 2n7r (stable), x = 1 + 2nn (stable), n = 0, 1, 2, . . . 0, - p, p < 0: 0 stable, - p unstable. p > 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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