Fractals in the Earth Sciences Fractals in the Earth Sciences Edited by Christopher C. Barton u.s. Geological Survey Denver, Colorado and Paul R. La Pointe Golder Associates Inc. Redmond, Washington Springer Science+Business Media, LLC Library of Congress Catalog1ng-1n-PublIcatlon Data Fractals In the earth sciences / edited by Christopher C. Barton and Paul R. La Pointe. p. cm. Includes bibliographical references and Index. ISBN 978-1-4899-1399-9 1. Fractals. 2. Geology—Mathematics. I. Barton, Christopher Cramer. II. La Pointe, P. R. QE33.2.F73F73 1995 55r.0r51474—dc20 94-45969 CIP Front Cover: Without an object of known size, such as a rock hammer or lens cap, it is often difficult or impossible to determine whether a photograph portrays a geological feature at the scale of centimeters, meters, or tens of meters. Back Cover: Most geologic features have heterogeneous properties that are scale invariant, i.e., have no characteristic length scale, a fundamental characteristic of fractals. Length of hammer is 0.74 meters. (Photographs by Christopher C. Barton and Sarah F. Tebbens, U.S. Geological Survey and University of South Florida, St. Petersburg, Florida) ISBN 978-1-4899-1399-9 ISBN 978-1-4899-1397-5 (eBook) DOI 10.1007/978-1-4899-1397-5 © 1995 Springer Science+Business Media New York Originally published by Plenum Publishing Corporation in 1995 Softcover reprint of the hardcover 1st edition 1995 10 98765432 All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher Dedication This volume is dedicated to Benoit B. Mandelbrot, the "father of fractal geometry," on his 70th birthday v vi DEDICATION Benoit B. Mandelbrot was born in Warsaw, Poland, November 20, 1924 and is best known as the author of the booksLes Objets Fractals, 1975, 1984, and 1989 (translated into Italian, Spanish, Portuguese, and Basque) and The Fractal Geometry of Nature, 1982 (translated into Japanese and German). He is Robinson Professor of Mathematical Sciences at Yale University and IBM Fellow emeritus at the IBM Thomas 1. Watson Research Center, Fellow of the American Academy of Arts and Sciences, Foreign Associate of the U.S. National Academy of Sciences, Member of the European Academy of Arts, Sciences, and Humanities. Dr. Mandelbrot is a graduate of the Paris Ecole Polytechnique; has aM. S. and Ae.E. in Aeronautics from the California Institute of Technology; and a Docteur d'Etat es Sciences Mathematiques from the University of Paris. He has been awarded honorary degrees of Doctor of Science from Syracuse University, Laurentian University, Boston University, State University of New York, University of Guelph, Universitat Bremen, Union College, University of Buenos Aires; and an honorary degree, Doctor of Humane Letters, from Pace University. He has received the Distinguished Service Award from the California Institute of Technology, and the Humboldt Preis from A. von Humboldt, Stiftung. His awards include the 1985 Barnard Medal granted jointly by the U.S. National Academy of Sciences and Columbia University, the 1986 Franklin Medal by the Franklin Institute of Philadelphia, the 1988 Charles Proteus Steinmetz Medal by the Institute of Electrical and Electronics Engineers, the 1988 Science for Art Prize by the Foundation Moet-Hennessy­ Louis Vuitton, the 1989 Harvey Prize for Science and Technology by the Technion-Israel Institute of Technology in Haifa, the 1991 Nevada Prize conferred by the Desert Research Institute, the 1993 Wolf Prize in Physics, and the 1994 Honda Prize. Before joining IBM, Dr. Mandelbrot's positions were with the CNRS in Paris, Philips Electronics, the Massachusetts Institute of Technology, the Institute for Advanced Study at Princeton, the University of Geneva, the University of Lille, and Ecole Poly technique, Paris. On leave from IBM, he has been an Institute Lecturer at Massachusetts Institute of Technology and a Visiting Professor of Economics, of Mathematics, of Applied Mathe­ matics and of the Practice of Mathematics at Harvard University; of Engineering at Yale University; of Physiology at the Albert Einstein College of Medicine; and of Mathematics at Universite de Paris-Sud. Mandelbrot has long been interested in earth science starting with a paper that concerned Hurst's law, published in Comptes Rendus in 1965. It was followed by "How long is the coast of Britain? Statistical self-similarity and fractional dimensions," in Science, 1967. He published four papers with 1. R. Wallis in Water Resources Research in 1968 and 1969; one of these papers is revisited in his contribution to this volume. His pioneering paper on multifractals, published in the Journal of Fluid Mechanics in 1974 was motivated in part by the spatial distribution of rare minerals. Contributors J. Angelier • Laboratoire de Tectonique Quantitative, Universite P. & M., Paris, France P. Bak • Department of Physics, Brookhaven National Laboratory, Upton, New York 11973 C. C. Barton • US Geological Survey, MS 940, Denver, Colorado 80225 S. R. Brown • Geomechanics Division, Sandia National Laboratories, Albuquerque, New Mexico 87185 K. Chen· Department of Physics, Brookhaven National Laboratory, Upton, New York 11973 A. D. Fowler • Ottawa-Carleton Geoscience Centre, University of Ottawa, Ottawa, Canada KIN 6N5 J. Huang • Department of Geological Sciences, Cornell University, Ithaca, New York 14853; present address: Exxon Production Research, Houston, Texas 77252 A. Malinverno • Lamont-Doherty Earth Observatory, Columbia University, Palisades, New York, 10964; present address: Schlumberger-Doll Research, Ridgefield, Connecticut 06877 B. B. Mandelbrot • Yale University, New Haven, Connecticut 06520 and IBM T. 1. Watson Research Center, Yorktown Heights, New York 10598 W. L. Power • Department of Geological Sciences, Brown University, Providence, Rhode Island 78712; present address: CSIRO Exploration and Mining, Nedlands, Western Austra­ lia, Australia 6009 S. A. Pruess • Department of Mathematics, Colorado School of Mines, Golden Colorado 80401 C. G. Sammis • Department of Geological Sciences, University of Southern California, Los Angeles, California 90089 J. A. Saunders • Department of Geology, Auburn University, Auburn, Alabama 36849 P. A. Schoenly • Department of Geology, Auburn University, Auburn, Alabama 36849 vii viii CONTRffiUTORS c. H. Scholz • Lamont-Doherty Earth Observatory and Department of Geological Sciences, Columbia University, Palisades, New York 10964 S. J. Steacy • Department of Geological Sciences, University of Southern California, Los Angeles, California 90089 C. Sunwoo • Korean Institute of Energy and Resources, Daejeon, Korea D. L. Thrcotte • Department of Geological Sciences, Cornell University, Ithaca, New York 14853 T. E. ThUis • Department of Geological Sciences, Brown University, Providence, Rhode Island 78712 T. Villemin • Laboratoire de Geologie Structuraleet Appliquee, Universite de Savoie, Chambery, France J. R. Wallis • IBM T. 1. Watson Research Center, Yorktown Heights, New York 10598 . Foreword Fractals have changed the way we understand and study nature. This change has been brought about mainly by the work of B. B. Mandelbrot and his book The Fractal Geometry ofNature. Now here is a book that collects articles treating fractals in the earth sciences. The themes chosen span, as is appropriate for a discourse on fractals, many orders of magnitude; including earthquakes, ocean floor topography, fractures, faults, mineral crystallinity, gold and silver deposition. There are also chapters on dynamical processes that are fractal, such as rivers, earthquakes, and a paper on self-organized criticality. Many of the chapters discuss how to estimate fractal dimensions, Hurst exponents, and other scaling exponents. This book, in a way, represents a snapshot of a field in which fractals has brought inspiration and a fresh look at familiar subjects. New ideas and attempts to quantify the world we see around us are found throughout. Many of these ideas will grow and inspire further work, others will be superseded by new observations and insights, most probably with future contributions by the authors of these chapters. It is wonderful to witness that the idea of fractal geometry has lead to such a high rate of discovery in the earth sciences. I believe that this book will also inspire others to take part in the scientific process of revealing the unseen, clarifying what is incomplete, discarding views that turn out not to stand up to the test of experiments and observations and thereby extending our understanding of the earth. Jens Feder Department of Physics, University of Oslo ix Preface It is not surprising that many earth scientists are drawn to fractal geometry. As the contributions in this volume show, fractal geometry is the natural mathematical language to describe much of what geologists observe. During their earliest field training, geologists are taught to place an object of known size, such as a rock hammer or lens cap, in photographs of outcrops in order to convey a sense of scale. Why? Without such an ob­ ject, it is not always easy to determine whether the photograph portrays a feature at the scale of centimeters, meters, or tens of meters. Likewise, the magnitude of geological events and the time scale in which they occur underpins geologists'S understanding of great floods, earthquakes, and mass extinctions. The geological