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Edited by

Kamill Klem-Musatov Henning C. Hoeber Tijmen Jan Moser Michael A. Pelissier

SEG Geophysics Reprint Series No. 29 Sergey Fomel, managing editor Evgeny Landa, volume editor

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# 2016 by Society of Exploration Geophysicists All rights reserved. This book or parts hereof may not be reproduced in any form without permission in writing from the publisher.

Published 2016

Printed in the United States of America

ISBN 978-1-931830-00-6 (Series) ISBN 978-1-56080-322-5 (Volume)

Library of Congress Control Number: 2015951229

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We dedicate this volume to the memory Dr. Kamill Klem-Musatov. In reading this volume, you will find that the history of diffraction We worked with Kamill over a period of several years to compile theory was filled with many controversies and feuds as new theories this volume. This volume was virtually ready for publication when came to displace or revise previous ones. Kamill Klem-Musatov’s Kamill passed away. He is greatly missed. new theory also met opposition; he paid a great personal price in Kamill’s role in Classical and Modern Diffraction Theory goes putting forth his theory for the seismic diffraction forward problem. far beyond that of co-editor. He was a pioneer of the extension of To model in the subsurface, he extended the diffraction modern diffraction theory of edge and tip to the seismic theory to take semi-transparent edges into account. For a number of method. His formulation of the seismic diffraction forward and years, this approach, a departure from the theory based on non-trans- inverse problems brought a new discipline to geophysics, one we parent edges, met strong opposition and criticism. might call the seismic diffraction method. In many ways, the Working with Kamill was a great honor and privilege for us. In purpose of Classical and Modern Diffraction Theory is to communi- remembering Kamill, a quote from Newton comes to mind “If I cate the theoretical foundations and their historical context to the geo- have seen further, it is by standing on the shoulders of giants.” physical community, for it is upon this foundation that Kamill based the seismic diffraction method. H. Hoeber, T. J. Moser, M. Pelissier

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About the Editors ...... vii

Preface ...... ix

Acknowledgments ...... xi

Chapter 1: Overview ...... 1 The phenomena ...... 3 W. Arkadiew The geometrical theory of diffraction ...... 13 J. B. Keller Diffraction theory ...... 17 C. J. Bouwkamp

Developments in our concepts of diffraction phenomena (on the 130th anniversary of the death of ) ...... 57 G. D. Malyuzhinets Rays, waves and asymptotics ...... 67 J. B. Keller Rubinowicz and the modern theory of diffracted rays ...... 81 P. Ya. Ufimtsev

Chapter 2: Early Developments in Diffraction Theory ...... 89 Translation editor’s introduction ...... 107 K. Helbig A physico-mathematical treatise on , and the , Proposition 1 ...... 108 F. M. Grimaldi , Chapter 1: On the rectilinear propagation of rays ...... 117 C. The Bakerian lecture: Experiments and calculations relative to physical ...... 123 T. Young

Chapter 3: The Classical Mathematical Theory of Diffraction ...... 129 Memoir on the diffraction of light ...... 147 A. Fresnel

An essay on the application of mathematical analysis to the theories of electricity and magnetism ...... 177 G. Green Theory of air vibration in pipes with open ends ...... 187 H. Helmholtz

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On the theory of light ...... 191 G. Kirchhoff On the passage of waves through in plane screens, and allied problems ...... 205 Lord Rayleigh (J. W. Strutt)

Chapter 4: Foundations of Modern Diffraction Theory ...... 213 Reciprocal diffraction relations between circular and elliptical plates ...... 217 J. Coulson and G. G. Becknell An extension of the principle of the diffraction , and some of its structural detail ...... 223 G. G. Becknell and J. Coulson An asymptotic treatment of diffraction problems ...... 227 N. G. Van Kampen Boundary layer problems in diffraction theory ...... 235 R. N. Buchal and J. B. Keller Geometrical theory of diffraction ...... 251 J. B. Keller A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface ...... 267 R. G. Kouyoumjian and P. H. Pathak The ray method and the theory of edge waves ...... 283 K. D. Klem-Musatov and A. M. Aizenberg Resolution and super-resolution in inverse diffraction ...... 293 M. Bertero, P. Boccacci, and M. Piana

Appendix A: The Cornu Spiral ...... 303 New method for the discussion of diffraction problems in the case of a cylindrical ...... 305 M. A. Cornu

Appendix B: Babinet’s Principle ...... 315 Memoirs on meteorological optics ...... 317 M. Babinet The London and Edinburgh Philosophical Magazine and Journal of ...... 319 A. Necker

References: Reprinted Technical Papers ...... 323

References: General ...... 325

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After the defense of his candidate dissertation on mathemat- ical (Ph.D. equivalent), Kamill joined the Institute of Pet- roleum Geology and Geophysics (Novosibirsk Scientific Centre, Siberian Branch, Russian Academy of , Russian Federa- tion) in 1968 and worked there for 47 academic years, until his last working day. For the first 12 academic years, Kamill’s scientific interests were devoted to a 2D canonical forward problem: wave diffraction in a sec- torial acoustic/elastic medium with one common sharp edge and several plane interfaces. He was one of the first scientists who obtained a rigorous solution of the difficult forward problem using the Sommer- feld-Malyuzhinets technique improved by him to semitransparent interfaces. He published a series of articles on this topic in Russian scientific journals. Among them, three results were the most valuable. The first of these is a justification of the edge-diffraction law: “amplitude of acoustic, elastic, electromagnetic and other edge Kamill D. Klem-Musatov, 1930–2015 waves inside the boundary layer (vicinity of shadow boundary) can be described by the universal special function W(w).” The second “All unknown and complex is hidden in the well-known and simple.” result is verification of this law by physical (laboratory) modeling. Using the above two results, Kamill and his pupil Evgeny Landa — Kamill D. Klem-Musatov introduced a theoretical criterion of detecting small-throw faults. Kamill summarized this period in his first monograph, Theory of On the morning of 18 June 2015, Professor Kamill D. Klem-Musatov, Edge Waves and its Application in Seismic (1980, in Russian). In doctor of science in , passed away at the age of 1980, he defended his doctoral dissertation based on the monograph 85 years. After first publishing in the Geophysical Journal of the and received the degree doctor of science in mathematical physics. Royal Astronomical Society in 1984 and in the Journal of Geophysics Kamill’s pioneering work in seismic diffraction came at a high in 1985 and participating in the Liblice Workshop Meeting in 1988, personal cost. This included an almost 10-year period of hard obstruc- he became famous in the world scientific community as one of the tion of his first monograph in some Russian scientific publications. leaders in the linear theory of seismic and Different approaches to the forward problem of diffractions from diffraction. edges were at the heart of the controversy. The Saint Petersburg Kamill Davidovich Klem-Musatov was born on 3 June 1930 in school formulated the forward problem for ideal (non-transparent) Saint Petersburg (in those years, Leningrad, USSR). He went to an interfaces. Kamill’s approach included a generalization to semitran- ordinary school in Saint Petersburg. After his parents were trapped sparent interfaces. This approach was essential for the seismic in political repressions at the end of the 1930s, he continued school forward problem but not for the applications being pursued at the in Nizhniy Tagil (a town in the South Ural region). After graduation, time by the Saint Petersburg school. Opposition to Kamill’s approach he went to the Higher Arctic Marine School in Saint Petersburg, continued until publication of his second monograph, in 1994. Then where he studied for only three years. Under the pressure of political the obstruction turned to current interest. repressions, he moved to Degtyarsk (a town in the South Ural region) Mention should be made of a fortunate circumstance that pre- and worked as the surveyor in a local mine for almost two years. In ceded the second monograph. Some years after 1980, Franta Hron 1953, he went to the Moscow Mining Institute and obtained an edu- and his pupils had checked formulas from Kamill’s first monograph cation in the specialty of mining ore deposits. in quantitative comparison to synthetic seismograms modeled by After graduation in 1958, Kamill took a position as engineer the numerical solver of Boris Mikhailenko for a 2D two-layer in the acoustic laboratory of the All-Union Scientific Research Insti- elastic model with a piecewise plane reflector. The analytic edge-dif- tute of Non-ferrous Metallurgy (in the town of Ust-Kamenogorsk, in fraction approach gave an unexpected high quality of comparison what is now the Republic of Kazakhstan). For 10 years, his scientific with numerical modeling. Franta Hron published these results in study was devoted to propagation in fractured media. several reports of the University of Alberta and a tutorial on edge- He was one of the first scientists who introduced a new model of wave modeling in Studia Geophysica et Geodætica in 1995. He dis- the imperfect interfaces with slippage of contacting media. He pub- cussed these results with Larry Lines, and the two decided to rec- lished 10 articles in scientific journals and seven reports on this ommend publication of Kamill’s monograph by SEG. With their topic. All of these publications were ahead of their time, but they kind and strong recommendations, Kamill translated the monograph were unknown in the scientific community outside Russia because into English, updated the text, and published his second monograph, they were published only in Russian. Kamill’s main results of this Theory of Seismic Diffractions (1994), with SEG. period were communicated to the world scientific community (partic- During academic years 1980 through 2009, Kamill’s scientific ularly by Mike Schoenberg) much later, after the opening of Russia’s interests were devoted to a generalization of his previous results to border in 1991. 3D inhomogeneous media with curved and piece-wise curved

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interfaces. He supposed that the wave-propagation description in such Henning C. Hoeber has held various research positions with media could face problem diffraction phenomena, among them dif- CGG since 1998 and is currently research manager for reservoir pro- fraction on sharp edges and smooth concave parts of interfaces, cessing, based in London. He coordinates CGG’s development of pro- caustic phenomena, and reflection and transmission of wavefields at cessing and imaging technology for reservoir imaging and analysis. curved interfaces. To account for diffraction on sharp edges and ver- After earning degrees from Hamburg University and Edinburgh Uni- tices, in 1984, Kamill suggested modifying ART formulas by includ- versity, Hoeber became a research fellow at the ing Cauchy-type integral formulas of edge and tip waves. He obtained Supercomputing Centre in Julich, Germany, where he continued his these formulas by reducing the boundary-layer description of interfer- research on lattice gauge theory and chromodynamics. He ence of geometric (reflected/transmitted) and edge and tip waves to has contributed to various articles and short courses relating to the Sohotsky-Plemelj problem. time-lapse technology, signal processing, AVO, and seismic and To account for caustic phenomena, Kamill and his pupil Arkady GPR diffraction. A coeditor of Classics of Elastic Wave Theory,he M. Aizenberg introduced the edge-wave superposition method also coauthored Compendium of . Hoeber is (EWSM) for 2D forward problems in 1980 and the tip-wave super- Associate Editor of Geophysical Prospecting and a member of position method (TWSM) for 3D forward problems in 1985. To EAGE and SEG. account for reflection/transmission at curved interfaces, Kamill intro- duced new rigorous reflection/transmission operators at curved con- Tijmen Jan Moser received his Ph.D. from the University of tacts of acoustic media (with coauthors A. M. Aizenberg, Hans Utrecht, concentrating on the shortest-path method for seismic ray B. Helle, and Jan Pajchel, in Wave Motion, 2004, 39, 77–92; 2005, tracing. He has worked at the Institut Franc¸ais du Pe´trole, the Institut 41, 293–305). Unfortunately, Kamill did not have enough time to de Physique du Globe (Paris), and the Institute of Solid Earth Physics solve a problem of diffraction on smooth parts of curved interfaces (Bergen), and he was an Alexander von Humboldt stipendiat (Karls- and left that to his followers. ruhe). Moser consulted for Norsk Hydro and the Geophysical Institute Kamill published main results of this period in 50 articles and of Israel, subsequently joining Fugro-Jason and then Horizon Energy extended abstracts (mostly in coauthorship with Arkady M. Aizen- Partners. He then founded Zeehelden Geoservices and Moser Geo- berg). After 1991, Kamill conducted the study in close collaboration physical Services, based in The Hague. Moser is Editor-in-Chief of with Jan Pajchel (earlier at Norsk Hydro, later Statoil, Bergen, Geophysical Prospecting, a member of SEG, an honorary member Norway) and Hans B. Helle (earlier at Norsk Hydro, later Odin Pet- of EAGE, and the author of several awarded papers. In 2011, roleum, Bergen, Norway). Kamill summarized the results of this Moser taught a course on seismic and GPR diffraction for EAGE. period in the book Edge and Tip Diffractions: Theory and Appli- His main interests are seismic wave propagation and imaging, ray cations in Seismic Prospecting (with coauthors Arkady M. Aizen- methods, and diffraction analysis, modeling, and imaging. berg, Jan Pajchel, and Hans B. Helle), published in 2008 by SEG. Michael A. Pelissier is a Consulting Geophysicist with Dagang In the last several years, Kamill served as one of the chief editors Zhaodong Oil Company of PetroChina, based in Beijing. He joined of this reprint volume and its companion, Seismic Diffraction. These the industry in 1979 and has held various positions focused on two volumes are based on selected works on wave-diffraction seismic interpretation and reservoir characterization with Roc Oil, theory and its application in seismics published in the past five centu- Marathon Oil, CGG, and Phillips Petroleum. His interests include ries. This work was virtually complete, and he was editing the final quantitative interpretation workflows for field development and reser- proofs at the time of his passing. He spent a lot of time collecting valu- voir management, with an emphasis on integrating geoscience and able publications, including rarities in Russian libraries. Kamill was reservoir engineering disciplines. Pelissier received a B.S. degree in coauthor of the first key talk on the history of wave-diffraction geophysics from the State University of New York at Binghamton, theory in the past five centuries and the second key talk on modern an M.S. in geophysics from Colorado School of Mines, and a Ph.D. wave-diffraction theory presented for the first time in the “Seismic in geophysics from the University of London. He coedited Classics Diffraction” session at the 72nd EAGE Conference and Exhibition of Elastic Wave Theory and is a member of SEG and EAGE. Pelissier in Barcelona, Spain, in 2010. served on the SEG Publications Committee from 2006 to 2013. For many years, Kamill taught a course on wave-diffraction theory in the Department of Geophysics of Novosibirsk State Uni- versity (Novosibirsk Scientific Centre, Siberian Branch, Russian Academy of Sciences, Russian Federation). By invitation of the Nor- wegian Research Council, he gave lectures at the University of Bergen (Bergen, Norway) and NTNU (Trondheim, Norway) in 1993–1994. During the years, he established a school on the theory of seismic wave diffraction whose pupils were scattered across many countries. Now some of them are involved in international associations of geophysicists such as EAGE and SEG. The bright memory of the citizen, scientist, and gentleman Kamill D. Klem-Musatov will forever remain in our hearts. — Arkady M. Aizenberg Novosibirsk Scientific Centre Siberian Branch of Russian Academy of Sciences 10 July 2015

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Diffraction is the response of waves to local heterogeneity in the that can explain rectilinear propagation of light, the laws of reflection medium in which they propagate. The phenomenon applies to all and , and the laws of diffraction. Thus, the earliest scientific wave types and is the basis of applications in numerous scientific dis- solution to the diffraction problem is based profoundly on Huygens’ ciplines. The maturity level of applications varies from area to area. wave-theory construction and the understanding of interference pro- For example, in materials science, diffraction has been used for a vided by Young. Young’s second significant contribution to diffrac- long time in nondestructive testing to detect and describe fractures. tion theory is the idea of a boundary diffraction wave, the precursor to Applications in civil engineering use diffraction exten- the modern geometrical theory of diffraction (the subject of Chapter 4). sively. In aerospace and defense, diffraction plays a prominent role Chapter 3 details the classical theory of diffraction, with re- in and stealth technologies. In the oil and gas industry, diffrac- printed works by Augustin Jean Fresnel, George Green, Hermann tion to complement seismic reflection methods is an emerging tech- von Helmholtz, Gustav Kirchhoff, and Lord Rayleigh (John nology. Application of seismic diffraction methods to detect small- William Strutt). New translations of the papers by Helmholtz and scale reservoir heterogeneity (i.e., fractures, stratigraphic pinchouts, Kirchhoff from German were prepared expressly for this volume. and channel edges) is likely to increase dramatically, however. The Chapter 3 begins with Fresnel’s mathematical formalization of industry is moving toward exploiting smaller and more complex con- Huygens’ principle and definition of the Fresnel integral. Fresnel ventional reservoirs and extensive new unconventional resource modified the classical principle of Huygens and proposed that the sec- plays, where fracture-controlled permeability is the important factor ondary spherical waves interfere amongst themselves. Fresnel’s idea for recovery. influenced all subsequent developments of the theory of diffraction. Classical and Modern Diffraction Theory is concerned primarily The major step in its development was carried out by Kirchhoff in with the diffraction forward problem. We address the inverse problem 1882, based on the mathematical developments by Green and Helm- only briefly, in the context of the modern concept of superresolution. holtz. Chapter 3, therefore, includes parts of Green’s classic essay A treatment of the progression from early work on diffraction to the covering the original derivation of Green’s theorem, which served theory’s leading edge is designed to assist the reader in understanding as a foundation of the integral theorems by Helmholtz and Kirchhoff. the physics of rigorous mathematical diffraction theory. Kirchhoff showed that the Young-Fresnel principle is an approxi- Although this volume was motivated by the need to provide mate form of Helmholtz’s rigorous integral theorem and based geophysicists with an in-depth understanding of the diffraction Fresnel’s source theory on this. In the case of diffraction from the phenomena from first principles, it is suitable equally to scientists of a plane screen, Kirchhoff described the field on the in any discipline. Because classical diffraction theory and the tran- screen in the form of an integral over the aperture (Kirchhoff approxi- sition to the modern theory were associated largely with the diffrac- mation), that is, resulting from the interference of Huygens’ second- tion of light, this volume deals almost exclusively with diffraction ary sources. Rayleigh later used Kirchhoff’s integral to show that the in the context of optics. For the geophysicist, this volume provides wavefield can be determined satisfactorily by the boundary condi- a solid foundation for the companion volume Seismic Diffraction, tions on the screen. From Rayleigh’s work (and later that of Sommer- which deals with the forward and inverse problems in seismic wave feld), it became clear that the study of diffraction of waves could be propagation. reduced formally to an ordinary boundary-value problem of math- The first chapter covers the foundations of diffraction theory. ematical physics. This includes a historical review of the classical theory, a summary Chapter 4 addresses the cornerstones of modern theory. The use of the experimental results illustrating this theory, and key principles of high-frequency asymptotic approximations of the wavefield is a of the modern theory of diffraction. The paper by Arkadiew (trans- common theme in this chapter. The centerpiece of Chapter 4 is the lated from German for this volume) provides an excellent reproduc- geometrical theory of diffraction. This chapter includes defining tion of the classical experiments and introduces the similarity law works by Keller and others, along with the experimental results by of diffraction. Becknell and Coulson, which inspired Keller’s theory. Chapter 2 provides the founding papers of classical diffraction Boundary-layer approximation is a second area of focus in theory, beginning in the 17th century. The editors offer an extensive Chapter 4. This deals with wavefield behavior in the penumbra at introduction to the reprinted works by Francesco Grimaldi, Christiaan the shadow boundary where ray theory is not applicable. Here, the Huygens, and Thomas Young. A new translation from Latin of the wave phenomenon is governed by Fock’s equation of transverse dif- defining work on diffraction by Grimaldi is presented. It relates a fusion (Fock et al., 1945), describing the energy flow from the illumi- set of experiments which led Grimaldi to conclude that a new, nated zone into the shadow zone. Chapter 4 addresses the question of fourth mode of light propagation exists — he called it diffraction. resolution for the inverse problem. Traditionally, this is defined in the Still in the 17th century, Huygens developed an early form of classic paper by Rayleigh (included in Chapter 3), defining a measure wave theory, now known as Huygens’ principle (or construction). of resolution known as the Rayleigh criterion. Other papers in this This theory allowed him to derive the laws of reflection and refrac- chapter deal with the quest toward superresolution, which aims at tion, but, as he makes no reference to the phase of wave motion, his exceeding the Rayleigh resolution limit. theory is unable to explain interference and diffraction effects. In Appendix A, we include a new translation of M. Alfred Young was the first who understood that diffraction patterns were Cornu’s work, defining what is now known as the Cornu spiral. produced by wavefield interference. The combination of Huygens’ Early modeling of the diffraction response was facilitated by principle with the phenomenon of interference allows for a theory Cornu’s graphical representation of the Fresnel integral. The Cornu

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spiral is found in modern textbooks on diffraction because it provides — an intriguing phenomenon in which, for example, the diffraction an invaluable conceptual representation of the forward problem. response of an aperture is complementary to that of an opaque Appendix B features a new translation of a portion of Me´moires object of the same dimensions. Appendix B also features Louis d’optique me´te´orologique by Jacques M. Babinet. A fundamental Albert Necker, describing the author’s paradoxical observations symmetry of the diffraction response is given by Babinet’s principle that inspired Babinet’s principle.

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We are grateful to Arkady Aizenberg for many useful discussions Johannes Schmoll for their translation from Latin of a chapter and suggestions. We are grateful to Evgeny Landa for acting as on diffraction from Grimaldi’s Physico-mathesis de Lumine, volume editor, reviewing the manuscript and providing recommen- Coloribus, et Iride. We thank Klaus Helbig for his contributions dations that improved this volume. We thank Sergey Fomel for to the translation from German of papers by Arkadiew and serving as managing editor. Kirchhoff. Likewise, we thank Raphael Bornard for his contri- Several translations of historical works were prepared butions to the translations from French of papers by Babinet especially for this volume. We are grateful to Klaus Helbig and and Cornu.

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