Stochastic Mechanics Stochastic Modelling Random Media and Applied Probability Signal Processing and Image Synthesis (Formerly: Mathematical Economics and Finance Applications of Mathematics) Stochastic Optimization 56 Stochastic Control Stochastic Models in Life Sciences Edited by B. Rozovskii G. Grimmett Advisory Board D. Dawson D. Geman I. Karatzas F. Kelly Y. Le Jan B. Øksendal G. Papanicolaou E. Pardoux StochasticModellingandAppliedProbability formerly:ApplicationsofMathematics 1 Fleming/Rishel,DeterministicandStochasticOptimalControl(1975) 2 Marchuk,MethodsofNumericalMathematics(1975,2nd.ed.1982) 3 Balakrishnan,AppliedFunctionalAnalysis(1976,2nd.ed.1981) 4 Borovkov,StochasticProcessesinQueueingTheory(1976) 5 Liptser/Shiryaev,StatisticsofRandomProcessesI:GeneralTheory(1977,2nd.ed.2001) 6 Liptser/Shiryaev,StatisticsofRandomProcessesII:Applications(1978,2nd.ed.2001) 7 Vorob’ev,GameTheory:LecturesforEconomistsandSystemsScientists(1977) 8 Shiryaev,OptimalStoppingRules(1978) 9 Ibragimov/Rozanov,GaussianRandomProcesses(1978) 10 Wonham,LinearMultivariableControl:AGeometricApproach(1979,2nd.ed.1985) 11 Hida,BrownianMotion(1980) 12 Hestenes,ConjugateDirectionMethodsinOptimization(1980) 13 Kallianpur,StochasticFilteringTheory(1980) 14 Krylov,ControlledDiffusionProcesses(1980) 15 Prabhu,StochasticStorageProcesses:Queues,InsuranceRisk,andDams(1980) 16 Ibragimov/Has’minskii,StatisticalEstimation:AsymptoticTheory(1981) 17 Cesari,Optimization:TheoryandApplications(1982) 18 Elliott,StochasticCalculusandApplications(1982) 19 Marchuk/Shaidourov,DifferenceMethodsandTheirExtrapolations(1983) 20 Hijab,StabilizationofControlSystems(1986) 21 Protter,StochasticIntegrationandDifferentialEquations(1990) 22 Benveniste/Métivier/Priouret,AdaptiveAlgorithmsandStochasticApproximations(1990) 23 Kloeden/Platen,NumericalSolutionofStochasticDifferentialEquations(1992,corr.3rdprinting 1999) 24 Kushner/Dupuis, NumericalMethodsforStochasticControlProblemsinContinuousTime (1992) 25 Fleming/Soner,ControlledMarkovProcessesandViscositySolutions(1993) 26 Baccelli/Brémaud,ElementsofQueueingTheory(1994,2nd.ed.2003) 27 Winkler,ImageAnalysis,RandomFieldsandDynamicMonteCarloMethods(1995,2nd.ed. 2003) 28 Kalpazidou,CycleRepresentationsofMarkovProcesses(1995) 29 Elliott/Aggoun/Moore,HiddenMarkovModels:EstimationandControl(1995) 30 Hernández-Lerma/Lasserre,Discrete-TimeMarkovControlProcesses(1995) 31 Devroye/Györfi/Lugosi,AProbabilisticTheoryofPatternRecognition(1996) 32 Maitra/Sudderth,DiscreteGamblingandStochasticGames(1996) 33 Embrechts/Klüppelberg/Mikosch,ModellingExtremalEventsforInsuranceandFinance(1997, corr.4thprinting2003) 34 Duflo,RandomIterativeModels(1997) 35 Kushner/Yin,StochasticApproximationAlgorithmsandApplications(1997) 36 Musiela/Rutkowski,MartingaleMethodsinFinancialModelling(1997,2nd.ed.2005) 37 Yin,Continuous-TimeMarkovChainsandApplications(1998) 38 Dembo/Zeitouni,LargeDeviationsTechniquesandApplications(1998) 39 Karatzas,MethodsofMathematicalFinance(1998) 40 Fayolle/Iasnogorodski/Malyshev,RandomWalksintheQuarter-Plane(1999) 41 Aven/Jensen,StochasticModelsinReliability(1999) 42 Hernandez-Lerma/Lasserre,FurtherTopicsonDiscrete-TimeMarkovControlProcesses(1999) 43 Yong/Zhou,StochasticControls.HamiltonianSystemsandHJBEquations(1999) 44 Serfozo,IntroductiontoStochasticNetworks(1999) 45 Steele,StochasticCalculusandFinancialApplications(2001) 46 Chen/Yao,FundamentalsofQueuingNetworks:Performance,Asymptotics,andOptimization (2001) 47 Kushner,HeavyTrafficAnalysisofControlledQueueingandCommunicationsNetworks(2001) 48 Fernholz,StochasticPortfolioTheory(2002) 49 Kabanov/Pergamenshchikov,Two-ScaleStochasticSystems(2003) 50 Han,Information-SpectrumMethodsinInformationTheory(2003) (continuedafterindex) Jean-Pierre Fouque Josselin Garnier George Papanicolaou Knut Sølna Wave Propagation and Time Reversal in Randomly Layered Media Authors Jean-Pierre Fouque Josselin Garnier Department of Statistics and UFR de Mathématiques Applied Probability Université Paris VII University of California 2 Place Jussieu Santa Barbara, CA 93106-3110 75251 Paris Cedex 05 USA France [email protected] [email protected] George Papanicolaou Knut Sølna Mathematics Department Department of Mathemathics Stanford University University of California at Irvine Stanford, CA 94305 Irvine, CA 92697 USA USA [email protected] [email protected] Managing Editors B. Rozovskii G. Grimmett Division of Applied Mathematics Centre for Mathematical Sciences Brown University Wilberforce Road, Cambridge CB3 0WB, UK 182 George Street [email protected] Providence, RI 02912 [email protected] Mathematics Subject Classifi cation (2000): 76Q05, 35L05, 35R30, 60G, 62M40, 73D35 Library of Congress Control Number: 2007928332 ISSN: 0172-4568 ISBN-13: 978-0-387-30890-6 eISBN-13: 978-0-387-49808-9 Printed on acid-free paper. © 2007 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identifi ed as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 springer.com To our families Preface Our motivation for writing this book is twofold: First, the theory of waves propagating in randomly layered media has been studied extensively during the last thirty years but the results are scattered in many different papers. Thistheoryisnowinamaturestate,especiallyinthe veryinterestingregime of separation of scales as introduced by G. Papanicolaou and his coauthors and described in [8], which is a building block for this book. Second, we were motivatedbythetime-reversalexperimentsofM.FinkandhisgroupinParis. They weredone with ultrasonicwavesandhaveattractedconsiderableatten- tion because of the surprising effects of enhanced spatial focusing and time compression in random media. An exposition of this work and its applica- tions is presented in [56]. Time reversal experiments were also carried out with sonar arrays in shallow water by W. Kuperman [113] and his group in San Diego. The enhanced spatial focusing and time compression of signals in time reversal in random media have many diverse applications in detection and in focused energy delivery on small targets as, for example, in the de- struction of kidney stones. Enhanced spatial focusing is also useful in sonar and wireless communications for reducing interference. Time reversal ideas have played an important role in the development of new methods for array imaging in random media as presented in [19]. A quantitative mathematical analysisiscrucialintheunderstandingofthesephenomenaandforthedevel- opment of new applications. In a series of recent papers by the authors and their coauthors,startingwith [40] in the one-dimensionalcase and [16] in the multidimensional case, a complete analysis of time reversalin random media has been proposed in the two extreme cases of strongly scattering layered media, and weak fluctuations in the parabolic approximation regime. These results areimportantinthe understanding ofthe intermediate situations and will contribute to future applications of time reversal. Wave propagation in three-dimensional random media has been stud- ied mostly by perturbation techniques when the random inhomogeneities are small. The main results are that the amplitude of the mean waves de- creaseswithdistancetraveled,becausecoherentwaveenergyisconvertedinto viii Preface incoherentfluctuations,whilethemeanenergypropagatesdiffusivelyorbyra- diative transport.These phenomena are analyzedextensively from a physical and engineering point of view in the book of Ishimaru [90]. It was first noted by Anderson [5] that for electronic waves in strongly disordered materials there is wave localization. This means that wave energy does not propagate, becausetherandominhomogeneitiestrapitinfiniteregions.Whatisdifferent andspecialinone-dimensionalrandommediaisthatwavelocalizationalways occurs, even when the inhomogeneities are weak. This means that there is never a diffusive or transport regime in one-dimensional random media. This was first proved by Goldsheid, Molchanov, and Pastur in [79]. It is therefore natural that the analysis of waves in one-dimensional or strongly anisotropic layered media presented in this book should rely on methods and techniques thataredifferentfromthoseusedingeneral,multidimensionalrandommedia. Thecontentofthisbookismultidisciplinaryandpresentsmanynewphys- ically interesting results about waves propagatingin randomly layeredmedia aswellasapplicationsintime reversal.Itusesmathematicaltoolsfromprob- abilityandstochasticprocesses,partialdifferentialequations,andasymptotic analysis, combined with the physics of wave propagation and modeling of time-reversal experiments. It addresses an interdisciplinary audience of stu- dentsandresearchersinterestedintheintriguingphenomenarelatedtowaves propagatinginrandommedia.Wehavetriedtograduallybringtogetherideas and tools from all these areas so that no special backgroundis required. The book can also be used as a textbook for advanced topics courses in which random media and related homogenization,averaging,and diffusion approxi- mationmethodsareinvolved.Theanalyticalresultsdiscussedhereareproved indetail,butwehavechosentopresentthemwithaseriesofexplanatoryand motivating steps instead of a “theorem-proof” format. Most of the results in the book are illustrated with numerical simulations that are carefully cali- brated to be in the regimes of the corresponding asymptotic analysis. At the endofeachchapterwegivereferencesandadditionalcommentsrelatedtothe various results that are presented. Acknowledgments George Papanicolaouwould like to thank his colleagues Joe Keller and Ragu Varadhan and his coauthors in the early work that is the basis of this book: MarkAsch,BobBurridge,WernerKohler,PawelLewicki,MariePostel,Ping Sheng, Sophie Weinryb, and Ben White. The authors would like to thank their collaborators in developing the recent theory of time reversal presented in this book, in particular Jean-Franc¸ois Clouet, for early work on time re- versal;Andr´eNachbin,for numerousandfruitful recentcollaborationsonthe subject;andLilianaBorceaandChrysoulaTsogkaforourextendedcollabora- tiononimaging.WealsothankMathiasFinkandhisgroupinParisformany discussions of time-reversal experiments. We have benefited from numerous constructivediscussions with our colleagues:Guillaume Bal,Peter Blomgren, Preface ix Gr´egoire Derveaux, Albert Fannjiang, Marteen de Hoop, Arnold Kim, Roger Maynard,Miguel Moscoso, ArogyaswamiPaulraj,Lenya Ryzhik, Bill Symes, Bart Van Tiggelen, and Hongkai Zhao. We also would like to thank our stu- dents and postdoctoral fellows who have read earlier versions of the book: Petr Glotov, Renaud Marty, and Oleg Poliannikov. Most of this book was written while the authors were visiting the De- partments of Mathematics at North Carolina State University, University of California Irvine, Stanford University, Toulouse University, University Denis DiderotinParis,IHESinBures-sur-Yvette,andIMPAinRiodeJaneiro.The authors would like to acknowledge the hospitality of these places. Santa Barbara,California Jean-Pierre Fouque Paris,France Josselin Garnier Stanford, California George Papanicolaou Irvine, California Knut Sølna December 19, 2006 Contents 1 Introduction and Overview of the Book.................... 1 2 Waves in Homogeneous Media............................. 9 2.1 Acoustic Wave Equations................................. 9 2.1.1 Conservation Equations in Fluid Dynamics ........... 9 2.1.2 Linearization ..................................... 10 2.1.3 Hyperbolicity ..................................... 11 2.1.4 The One-Dimensional Wave Equation................ 12 2.1.5 Solution of the Three-Dimensional Wave Equation by Spherical Means................................... 14 2.1.6 The Three-Dimensional Wave Equation With Source... 17 2.1.7 Green’s Function for the Acoustic Wave Equations .... 19 2.1.8 Energy Density and Energy Flux .................... 21 2.2 Wave Decompositions in Three-Dimensional Media .......... 22 2.2.1 Time Harmonic Waves ............................. 22 2.2.2 Plane Waves...................................... 23 2.2.3 Spherical Waves................................... 24 2.2.4 Weyl’s Representation of Spherical Waves ............ 25 2.2.5 The Acoustic Wave Generated by a Point Source ...... 27 2.3 Appendix .............................................. 29 2.3.1 Gauss–Green Theorem ............................. 29 2.3.2 Energy Conservation Equation ...................... 30 3 Waves in Layered Media................................... 33 3.1 Reduction to a One-Dimensional System ................... 33 3.2 Right- and Left-Going Waves ............................. 34 3.3 Scattering by a Single Interface ........................... 36 3.4 Single-Layer Case ....................................... 39 3.4.1 Mathematical Setup ............................... 39 3.4.2 Reflection and Transmission Coefficient for a Single Layer ............................................ 41 xii Contents 3.4.3 Frequency-Dependent Reflectivity and Antireflection Layer ............................................ 43 3.4.4 Scattering by a Single Layer in the Time Domain...... 44 3.4.5 Propagatorand Scattering Matrices ................. 47 3.5 Multilayer Piecewise-Constant Media ...................... 48 3.5.1 PropagationEquations............................. 48 3.5.2 Reflected and Transmitted Waves ................... 51 3.5.3 Reflectivity Pattern and Bragg Mirror for Periodic Layers ........................................... 54 3.5.4 Goupillaud Medium ............................... 57 4 Effective Properties of Randomly Layered Media .......... 61 4.1 Finely Layered Piecewise-Constant Media .................. 62 4.1.1 Periodic Case ..................................... 63 4.1.2 Random Case..................................... 65 4.1.3 Conclusion ....................................... 68 4.2 Random Media Varying on a Fine Scale .................... 68 4.3 Boundary Conditions and Equations for Right- and Left-Going Modes ....................................... 70 4.3.1 Modes Along Local Characteristics .................. 72 4.3.2 Modes Along Constant Characteristics ............... 73 4.4 Centering the Modes and PropagatorEquations ............. 75 4.4.1 Characteristic Lines ............................... 75 4.4.2 Modes in the Fourier Domain ....................... 76 4.4.3 Propagator ....................................... 77 4.4.4 TheRiccatiEquationforthe LocalReflectionCoefficient 79 4.4.5 Reflection and Transmission in the Time Domain...... 81 4.4.6 Matched Medium.................................. 81 4.5 Homogenization and the Law of Large Numbers ............. 82 4.5.1 A Simple Discrete Random Medium ................. 82 4.5.2 Random Differential Equations...................... 85 4.5.3 The Effective Medium ............................. 88 5 Scaling Limits ............................................. 91 5.1 Identification of the Scaling Regimes ....................... 92 5.1.1 Modeling of the Medium Fluctuations................ 92 5.1.2 Modeling of the Source Term ....................... 94 5.1.3 The Dimensionless Wave Equations.................. 95 5.1.4 Scaling Limits .................................... 96 5.1.5 Right- and Left-Going Waves ....................... 98 5.1.6 Propagatorand Reflectionand TransmissionCoefficients100 5.2 Diffusion Scaling ........................................102 5.2.1 White-Noise Regime and Brownian Motion ...........103 5.2.2 Diffusion Approximation ...........................104