J.-P. Fouque, J. Garnier, G. Papanicolaou

and K. Sølna

Wave Propagation and TimeReversal in Randomly LayeredMediaSPIN Springer’s internal project number, if known

April 2, 2007

Springer

Berlin Heidelberg NewYorkHongKong LondonMilan Paris Tokyo

To our families

Preface

Our motivation for writing this book is twofold: First, the theory of wavespropagating in randomly layered media has been studied extensively duringthe last thirty years but the results are scattered in many different papers.This theory is now in a mature state, especially in the very interesting regimeof separation of scales as introduced by G. Papanicolaou and his coauthorsand described in [8], which is a building block for this book. Second, we weremotivated by the time-reversal experiments of M. Fink and his group in Paris.They were done with ultrasonic waves and have attracted considerable atten-tion because of the surprising effects of enhanced spatial focusing and timecompression in random media. An exposition of this work and its applica-tions is presented in [56]. Time reversal experiments were also carried outwith sonar arrays in shallow water by W. Kuperman [113] and his group inSan Diego. The enhanced spatial focusing and time compression of signals intime reversal in random media have many diverse applications in detectionand in focused energy delivery on small targets as, for example, in the de-struction of kidney stones. Enhanced spatial focusing is also useful in sonarand wireless communications for reducing interference. Time reversal ideashave played an important role in the development of new methods for arrayimaging in random media as presented in [19]. A quantitative mathematicalanalysis is crucial in the understanding of these phenomena and for the devel-opment of new applications. In a series of recent papers by the authors andtheir coauthors, starting with [40] in the one-dimensional case and [16] in themultidimensional case, a complete analysis of time reversal in random mediahas been proposed in the two extreme cases of strongly scattering layeredmedia, and weak fluctuations in the parabolic approximation regime. Theseresults are important in the understanding of the intermediate situations andwill contribute to future applications of time reversal.

Wave propagation in three-dimensional random media has been studiedmostly by perturbation techniques when the random inhomogeneities aresmall. The main results are that the amplitude of the mean waves decreaseswith distance traveled, because coherent wave energy is converted into incoher-

VIII Preface

ent fluctuations, while the mean energy propagates diffusively or by radiativetransport. These phenomena are analyzed extensively from a physical andengineering point of view in the book of Ishimaru [90]. It was first noted byAnderson [5] that for electronic waves in strongly disordered materials thereis wave localization. This means that wave energy does not propagate, be-cause the random inhomogeneities trap it in finite regions. What is differentand special in one-dimensional random media is that wave localization alwaysoccurs, even when the inhomogeneities are weak. This means that there isnever a diffusive or transport regime in one-dimensional random media. Thiswas first proved by Goldsheid, Molchanov, and Pastur in [79]. It is thereforenatural that the analysis of waves in one-dimensional or strongly anisotropiclayered media presented in this book should rely on methods and techniquesthat are different from those used in general, multidimensional random media.

The content of this book is multidisciplinary and presents many new phys-ically interesting results about waves propagating in randomly layered mediaas well as applications in time reversal. It uses mathematical tools from prob-ability and stochastic processes, partial differential equations, and asymptoticanalysis, combined with the physics of wave propagation and modeling oftime-reversal experiments. It addresses an interdisciplinary audience of stu-dents and researchers interested in the intriguing phenomena related to wavespropagating in random media. We have tried to gradually bring together ideasand tools from all these areas so that no special background is required. Thebook can also be used as a textbook for advanced topics courses in whichrandom media and related homogenization, averaging, and diffusion approxi-mation methods are involved. The analytical results discussed here are provedin detail, but we have chosen to present them with a series of explanatory andmotivating steps instead of a “theorem-proof” format. Most of the results inthe book are illustrated with numerical simulations that are carefully cali-brated to be in the regimes of the corresponding asymptotic analysis. At theend of each chapter we give references and additional comments related to thevarious results that are presented.

Acknowledgments

George Papanicolaou would like to thank his colleagues Joe Keller and RaguVaradhan and his coauthors in the early work that is the basis of this book:Mark Asch, Bob Burridge, Werner Kohler, Pawel Lewicki, Marie Postel, PingSheng, Sophie Weinryb, and Ben White. The authors would like to thanktheir collaborators in developing the recent theory of time reversal presentedin this book, in particular Jean-Francois Clouet, for early work on time re-versal; Andre Nachbin, for numerous and fruitful recent collaborations on thesubject; and Liliana Borcea and Chrysoula Tsogka for our extended collabora-tion on imaging. We also thank Mathias Fink and his group in Paris for manydiscussions of time-reversal experiments. We have benefited from numerousconstructive discussions with our colleagues: Guillaume Bal, Peter Blomgren,

Preface IX

Gregoire Derveaux, Albert Fannjiang, Marteen de Hoop, Arnold Kim, RogerMaynard, Miguel Moscoso, Arogyaswami Paulraj, 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 ofCalifornia Irvine, Stanford University, Toulouse University, University DenisDiderot in Paris, IHES in Bures-sur-Yvette, and IMPA in Rio de Janeiro. Theauthors would like to acknowledge the hospitality of these places.

Santa Barbara, California Jean-Pierre FouqueParis, France Josselin GarnierStanford, California George PapanicolaouIrvine, California Knut Sølna

December 19, 2006

Contents

1 Introduction and Overview of the Book . . . . . . . . . . . . . . . . . . . . 1

2 Waves in Homogeneous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Acoustic Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Conservation Equations in Fluid Dynamics . . . . . . . . . . . 92.1.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.3 Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.4 The One-Dimensional Wave Equation . . . . . . . . . . . . . . . . 122.1.5 Solution of the Three-Dimensional Wave Equation by

Spherical Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.6 The Three-Dimensional Wave Equation With Source . . . 172.1.7 Green’s Function for the Acoustic Wave Equations . . . . 192.1.8 Energy Density and Energy Flux . . . . . . . . . . . . . . . . . . . . 21

2.2 Wave Decompositions in Three-Dimensional Media . . . . . . . . . . 222.2.1 Time Harmonic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.2 Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.3 Spherical Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.4 Weyl’s Representation of Spherical Waves . . . . . . . . . . . . 252.2.5 The Acoustic Wave Generated by a Point Source . . . . . . 27

2.3 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.1 Gauss–Green Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.2 Energy Conservation Equation . . . . . . . . . . . . . . . . . . . . . . 30

3 Waves in Layered Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.1 Reduction to a One-Dimensional System . . . . . . . . . . . . . . . . . . . 333.2 Right- and Left-Going Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3 Scattering by a Single Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.4 Single-Layer Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.1 Mathematical Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.4.2 Reflection and Transmission Coefficient for a Single

Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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3.4.3 Frequency-Dependent Reflectivity and AntireflectionLayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4.4 Scattering by a Single Layer in the Time Domain . . . . . . 443.4.5 Propagator and Scattering Matrices . . . . . . . . . . . . . . . . . 47

3.5 Multilayer Piecewise-Constant Media . . . . . . . . . . . . . . . . . . . . . . 483.5.1 Propagation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.5.2 Reflected and Transmitted Waves . . . . . . . . . . . . . . . . . . . 513.5.3 Reflectivity Pattern and Bragg Mirror for Periodic

Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.5.4 Goupillaud Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 Effective Properties of Randomly Layered Media . . . . . . . . . . 614.1 Finely Layered Piecewise-Constant Media . . . . . . . . . . . . . . . . . . 62

4.1.1 Periodic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.1.2 Random Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.1.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2 Random Media Varying on a Fine Scale . . . . . . . . . . . . . . . . . . . . 684.3 Boundary Conditions and Equations for Right- and

Left-Going Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3.1 Modes Along Local Characteristics . . . . . . . . . . . . . . . . . . 724.3.2 Modes Along Constant Characteristics . . . . . . . . . . . . . . . 73

4.4 Centering the Modes and Propagator Equations . . . . . . . . . . . . . 754.4.1 Characteristic Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.4.2 Modes in the Fourier Domain . . . . . . . . . . . . . . . . . . . . . . . 764.4.3 Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.4.4 The Riccati Equation for the Local Reflection Coefficient 794.4.5 Reflection and Transmission in the Time Domain . . . . . . 814.4.6 Matched Medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.5 Homogenization and the Law of Large Numbers . . . . . . . . . . . . . 824.5.1 A Simple Discrete Random Medium . . . . . . . . . . . . . . . . . 824.5.2 Random Differential Equations . . . . . . . . . . . . . . . . . . . . . . 854.5.3 The Effective Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5 Scaling Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.1 Identification of the Scaling Regimes . . . . . . . . . . . . . . . . . . . . . . . 92

5.1.1 Modeling of the Medium Fluctuations . . . . . . . . . . . . . . . . 925.1.2 Modeling of the Source Term . . . . . . . . . . . . . . . . . . . . . . . 945.1.3 The Dimensionless Wave Equations . . . . . . . . . . . . . . . . . . 955.1.4 Scaling Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.1.5 Right- and Left-Going Waves . . . . . . . . . . . . . . . . . . . . . . . 985.1.6 Propagator and Reflection and Transmission Coefficients100

5.2 Diffusion Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.2.1 White-Noise Regime and Brownian Motion . . . . . . . . . . . 1035.2.2 Diffusion Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

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5.2.3 Finite-Dimensional Distributions of the TransmittedWave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6 Asymptotics for Random Ordinary Differential Equations . 1096.1 Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.1.1 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.1.2 Infinitesimal Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.1.3 Martingales and Martingale Problems . . . . . . . . . . . . . . . . 1116.1.4 Kolmogorov Backward and Forward Equations . . . . . . . . 1136.1.5 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.2 Markovian Models of Random Media . . . . . . . . . . . . . . . . . . . . . . 1166.2.1 Two-Component Composite Media . . . . . . . . . . . . . . . . . . 1166.2.2 Multicomponent Composite Media . . . . . . . . . . . . . . . . . . 1186.2.3 A Continuous Random Medium . . . . . . . . . . . . . . . . . . . . . 120

6.3 Diffusion Approximation Without Fast Oscillation . . . . . . . . . . . 1226.3.1 Markov Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.3.2 Perturbed Test Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.3.3 The Poisson Equation and the Fredholm Alternative . . . 1246.3.4 Limiting Infinitesimal Generator . . . . . . . . . . . . . . . . . . . . 1266.3.5 Relative Compactness of the Laws of the Processes . . . . 1316.3.6 The Multiplicative-Noise Case . . . . . . . . . . . . . . . . . . . . . . 134

6.4 The Averaging and Fluctuation Theorems . . . . . . . . . . . . . . . . . . 1356.4.1 Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.4.2 Fluctuation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.5 Diffusion Approximation with Fast Oscillations . . . . . . . . . . . . . . 1396.5.1 Semifast Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.5.2 Fast Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.6 Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1456.6.1 Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1476.6.2 Ito’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1506.6.3 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . 1526.6.4 Diffusions and Partial Differential Equations . . . . . . . . . . 1536.6.5 Feynman–Kac Representation Formula . . . . . . . . . . . . . . . 155

6.7 Limits of Random Equations and Stochastic Equations . . . . . . . 1566.7.1 Ito Form of the Limit Process . . . . . . . . . . . . . . . . . . . . . . . 1566.7.2 Stratonovich Stochastic Integrals . . . . . . . . . . . . . . . . . . . . 1586.7.3 Limits of Random Matrix Equations . . . . . . . . . . . . . . . . . 160

6.8 Lyapunov Exponent for Linear Random Differential Equations 1616.8.1 Lyapunov Exponent of the Random Differential

Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1626.8.2 Lyapunov Exponent of the Limit Diffusion . . . . . . . . . . . . 169

6.9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1726.9.1 Quadratic Variation of a Continuous Martingale . . . . . . . 172

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7 Transmission of Energy Through a Slab of Random Medium1757.1 Transmission of Monochromatic Waves . . . . . . . . . . . . . . . . . . . . . 176

7.1.1 The Diffusion Limit for the Propagator . . . . . . . . . . . . . . . 1777.1.2 Polar Coordinates for the Propagator . . . . . . . . . . . . . . . . 1807.1.3 Martingale Representation of the Transmission

Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1837.1.4 The Localization Length Lloc(ω) . . . . . . . . . . . . . . . . . . . . 1857.1.5 Mean and Fluctuations of the Power Transmission

Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1877.1.6 The Strongly Fluctuating Character of the Power

Transmission Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1887.2 Exponential Decay of the Transmitted Energy for a Pulse . . . . 190

7.2.1 Transmission of a Pulse Through a Slab of RandomMedium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

7.2.2 Self-Averaging Property of the Transmitted Energy . . . . 1917.2.3 The Diffusion Limit for the Two-Frequency Propagator 193

7.3 Wave Localization in the Weakly Heterogeneous Regime . . . . . . 1967.3.1 Determination of the Power Transmission Coefficient

from a Random Harmonic Oscillator . . . . . . . . . . . . . . . . . 1967.3.2 Comparisons of Decay Rates . . . . . . . . . . . . . . . . . . . . . . . . 198

7.4 Wave Localization in the Strongly Heterogeneous White-NoiseRegime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

7.5 The Random Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 2017.5.1 The Lyapunov Exponent of the Random Harmonic

Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2027.5.2 Expansion of the Lyapunov Exponent in the Strongly

Heterogeneous Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2037.5.3 Expansion of the Lyapunov Exponent in the Weakly

Heterogeneous Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2087.6 Appendix. Statistics of the Power Transmission Coefficient . . . . 209

7.6.1 The Probability Density of the Power TransmissionCoefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

7.6.2 Moments of the Power Transmission Coefficient . . . . . . . 211

8 Wave-Front Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2158.1 The Transmitted Wave Front in the Weakly Heterogeneous

Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2168.1.1 Stabilization of the Transmitted Wave Front . . . . . . . . . . 2178.1.2 The Integral Equation for the Transmitted Field . . . . . . 2208.1.3 Asymptotic Analysis of the Transmitted Wave Front . . . 222

8.2 The Transmitted Wave Front in the Strongly HeterogeneousRegime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2258.2.1 Asymptotic Representation of the Transmitted Wave

Front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2268.2.2 The Energy of the Transmitted Wave . . . . . . . . . . . . . . . . 229

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8.2.3 Numerical Illustration of Pulse Spreading . . . . . . . . . . . . . 2308.2.4 The Diffusion Limit for the Multifrequency Propagators 2308.2.5 Martingale Representation of the Multifrequency

Transmission Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2338.2.6 Identification of the Limit Wave Front . . . . . . . . . . . . . . . 2348.2.7 Asymptotic Analysis of Travel Times . . . . . . . . . . . . . . . . 236

8.3 The Reflected Front in Presence of an Interface . . . . . . . . . . . . . 2388.3.1 Integral Representation of the Reflected Pulse . . . . . . . . . 2388.3.2 The Limit for the Reflected Front . . . . . . . . . . . . . . . . . . . 242

8.4 Appendix. Proof of the Averaging Theorem . . . . . . . . . . . . . . . . . 245

9 Statistics of Incoherent Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2499.1 The Reflected Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

9.1.1 Reformulation of the Reflection and TransmissionProblem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

9.1.2 The Riccati Equation for the Reflection Coefficient . . . . 2529.1.3 Representation of the Reflected Field . . . . . . . . . . . . . . . . 253

9.2 Statistics of the Reflected Wave in the Frequency Domain . . . . 2549.2.1 Moments of the Reflection Coefficient . . . . . . . . . . . . . . . . 2549.2.2 Probabilistic Representation of the Transport Equations 2589.2.3 Explicit Solution for a Random Half-Space . . . . . . . . . . . 2619.2.4 Multifrequency Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

9.3 Statistics of the Reflected Wave in the Time Domain . . . . . . . . . 2669.3.1 Mean Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2669.3.2 Mean Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2669.3.3 Autocorrelation and Time-Domain Localization . . . . . . . 2679.3.4 Gaussian Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

9.4 The Transmitted Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2729.4.1 Autocorrelation Function of the Transmission Coefficient2729.4.2 Probabilistic Representation of the Transport Equations 2749.4.3 Statistics of the Transmitted Wave in the Time Domain 277

10 Time Reversal in Reflection and Spectral Estimation . . . . . . 28110.1 Time Reversal in Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

10.1.1 Time-Reversal Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28310.1.2 Time-Reversal Refocusing . . . . . . . . . . . . . . . . . . . . . . . . . . 28510.1.3 The Limiting Refocused Pulse . . . . . . . . . . . . . . . . . . . . . . 28610.1.4 Time-Reversal Mirror Versus Standard Mirror . . . . . . . . 290

10.2 Time Reversal Versus Cross Correlations . . . . . . . . . . . . . . . . . . . 29110.2.1 The Empirical Correlation Function . . . . . . . . . . . . . . . . . 29210.2.2 Measuring the Spectral Density . . . . . . . . . . . . . . . . . . . . . 29310.2.3 Signal-to-Noise Ratio Comparison . . . . . . . . . . . . . . . . . . . 294

10.3 Calibrating the Initial Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

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11 Applications to Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30511.1 Detection of a Weak Reflector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30611.2 Detection of an Interface Between Media . . . . . . . . . . . . . . . . . . . 31111.3 Waves in One-Dimensional Dissipative Random Media . . . . . . . 313

11.3.1 The Acoustic Model with Random Dissipation . . . . . . . . 31311.3.2 Propagator Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31411.3.3 Transmitted Wave Front . . . . . . . . . . . . . . . . . . . . . . . . . . . 31711.3.4 The Refocused Pulse for Time Reversal in Reflection . . . 317

11.4 Application to the Detection of a Dissipative Layer . . . . . . . . . . 32011.4.1 Constant Mean Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . 32111.4.2 Thin Dissipative Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32111.4.3 Thick Dissipative Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

12 Time Reversal in Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32712.1 Time Reversal of the Stable Front . . . . . . . . . . . . . . . . . . . . . . . . . 328

12.1.1 Time-Reversal Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 32912.1.2 The Refocused Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

12.2 Time Reversal with Coda Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 33312.2.1 Time-Reversal Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 33312.2.2 Decomposition of the Refocusing Kernel . . . . . . . . . . . . . . 33512.2.3 Midband Filtering by the Medium . . . . . . . . . . . . . . . . . . . 33612.2.4 Low-Pass Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

12.3 Discussion and Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . 339

13 Application to Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . 34313.1 Review of Basic Communications Schemes . . . . . . . . . . . . . . . . . . 344

13.1.1 Nyquist Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34413.1.2 Signal-to-Interference Ratio . . . . . . . . . . . . . . . . . . . . . . . . . 34513.1.3 Modulated Nyquist Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

13.2 Communications in Random Media Using Nyquist Pulses . . . . . 34713.2.1 Direct Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35013.2.2 Communications Using Time Reversal . . . . . . . . . . . . . . . 35113.2.3 SIRs for Coherent Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . 35313.2.4 Influence of the Incoherent Waves . . . . . . . . . . . . . . . . . . . 35513.2.5 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

13.3 Communications in Random Media Using Modulated NyquistPulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35813.3.1 SIRs of Modulated Nyquist Pulses . . . . . . . . . . . . . . . . . . . 35913.3.2 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36213.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

14 Scattering by a Three-Dimensional Randomly LayeredMedium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36514.1 Acoustic Waves in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . 366

14.1.1 Homogenization Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

Contents XVII

14.1.2 The Diffusion Approximation Regime . . . . . . . . . . . . . . . . 36814.1.3 Plane-Wave Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 36914.1.4 One-Dimensional Mode Problems . . . . . . . . . . . . . . . . . . . . 37014.1.5 Transmitted-Pressure Integral Representation . . . . . . . . . 374

14.2 The Transmitted Wave Front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37414.2.1 Characterization of Moments . . . . . . . . . . . . . . . . . . . . . . . 37414.2.2 Stationary-Phase Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37614.2.3 Characterization of the Transmitted Wave Front . . . . . . 378

14.3 The Mean Reflected Intensity Generated by a Point Source . . . 38014.3.1 Reflected-Pressure Integral Representation . . . . . . . . . . . . 38014.3.2 Autocorrelation Function of the Reflection Coefficient

at Two Nearby Slownesses and Frequencies . . . . . . . . . . . 38114.3.3 Asymptotics of the Mean Intensity . . . . . . . . . . . . . . . . . . 385

14.4 Appendix: Stationary-Phase Method . . . . . . . . . . . . . . . . . . . . . . . 389

15 Time Reversal in a Three-Dimensional Layered Medium . . . 39315.1 The Embedded-Source Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 39315.2 Time Reversal with Embedded Source . . . . . . . . . . . . . . . . . . . . . 395

15.2.1 Emission from a Point Source . . . . . . . . . . . . . . . . . . . . . . . 39515.2.2 Recording, Time Reversal, and Reemission . . . . . . . . . . . 40115.2.3 The Time-Reversed Wave Field . . . . . . . . . . . . . . . . . . . . . 403

15.3 Homogeneous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40515.3.1 The Field Recorded at the Surface . . . . . . . . . . . . . . . . . . . 40615.3.2 The Time-Reversed Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

15.4 Complete Description of the Time-Reversed Field in aRandom Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41115.4.1 Expectation of the Refocused Pulse . . . . . . . . . . . . . . . . . . 41215.4.2 Refocusing of the Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

15.5 Refocusing Properties in a Random Medium . . . . . . . . . . . . . . . . 41615.5.1 The Case |zs| ¿ Lloc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41615.5.2 Time Reversal of the Front . . . . . . . . . . . . . . . . . . . . . . . . . 41715.5.3 Time Reversal of the Incoherent Waves with Offset . . . . 41715.5.4 Time Reversal of the Incoherent Waves Without Offset . 42215.5.5 Record of the Pressure Signal . . . . . . . . . . . . . . . . . . . . . . . 424

15.6 Appendix A: Moments of the Reflection and TransmissionCoefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42415.6.1 Autocorrelation Function of the Transmission

Coefficient at Two Nearby Slownesses and Frequencies . 42415.6.2 Shift Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42515.6.3 Generalized Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

15.7 Appendix B: A Priori Estimates for the Generalized Coefficients42815.8 Appendix C: Derivation of (15.74) . . . . . . . . . . . . . . . . . . . . . . . . . 430

XVIII Contents

16 Application to Echo-Mode Time Reversal . . . . . . . . . . . . . . . . . . 43516.1 The Born Approximation for an Embedded Scatterer . . . . . . . . 435

16.1.1 Integral Expressions for the Wave Fields . . . . . . . . . . . . . 43716.2 Asymptotic Theory for the Scattered Field . . . . . . . . . . . . . . . . . 439

16.2.1 The Primary Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43916.2.2 The Secondary Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

16.3 Time Reversal of the Recorded Wave . . . . . . . . . . . . . . . . . . . . . . 44216.3.1 Integral Representation of the Time-Reversed Field . . . . 44216.3.2 Refocusing in the Homogeneous Case . . . . . . . . . . . . . . . . 44416.3.3 Refocusing of the Secondary Field in the Random Case 44616.3.4 Contributions of the Other Wave Components . . . . . . . . 451

16.4 Time-Reversal Superresolution with a Passive Scatterer . . . . . . 45116.4.1 The Refocused Pulse Shape . . . . . . . . . . . . . . . . . . . . . . . . . 45116.4.2 Superresolution with a Random Medium . . . . . . . . . . . . . 453

17 Other Layered Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45717.1 Nonmatched Effective Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

17.1.1 Boundary and Jump Conditions . . . . . . . . . . . . . . . . . . . . . 45817.1.2 Transmission of a Pulse through a Nonmatched

Random Slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45917.1.3 Reflection by a Nonmatched Random Half-Space . . . . . . 464

17.2 General Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46617.2.1 Mode Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46717.2.2 Transport Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46917.2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

17.3 Medium with Random Density Fluctuations . . . . . . . . . . . . . . . . 47217.3.1 The Coupled-Propagator White-Noise Model . . . . . . . . . . 47417.3.2 The Transmitted Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47917.3.3 Transport Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48217.3.4 Reflection by a Random Half-Space . . . . . . . . . . . . . . . . . . 484

18 Other Regimes of Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48718.1 The Weakly Heterogeneous Regime in Randomly Layered

Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48718.1.1 Mode Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48818.1.2 Transport Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49018.1.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

18.2 Dispersive Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49218.2.1 The Terrain-Following Boussinesq Model . . . . . . . . . . . . . 49318.2.2 The Propagating Modes of the Boussinesq Equation . . . 49418.2.3 Mode Propagation in a Dispersive Random Medium . . . 49518.2.4 Transport Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49718.2.5 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498

18.3 Nonlinear Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49918.3.1 Shallow-Water Waves with Random Depth . . . . . . . . . . . 500

Contents XIX

18.3.2 The Linear Hyperbolic Approximation . . . . . . . . . . . . . . . 50218.3.3 The Effective Equation for the Nonlinear Front Pulse . . 50418.3.4 Analysis of the Pseudospectral Operator . . . . . . . . . . . . . 50818.3.5 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

18.4 Time Reversal with Changing Media . . . . . . . . . . . . . . . . . . . . . . . 51018.4.1 The Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51018.4.2 Convergence of the Finite-Dimensional Distributions . . . 51118.4.3 Convergence of the Refocused Pulse . . . . . . . . . . . . . . . . . 515

19 The Random Schrodinger Model . . . . . . . . . . . . . . . . . . . . . . . . . . 51919.1 Linear Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

19.1.1 The Linear Schrodinger Equation . . . . . . . . . . . . . . . . . . . . 51919.1.2 Transmission of a Monochromatic Wave . . . . . . . . . . . . . . 52119.1.3 Transmission of a Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526

19.2 Nonlinear Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52819.2.1 Waves Called Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52819.2.2 Dispersion and Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . 53119.2.3 The Nonlinear Schrodinger Equation . . . . . . . . . . . . . . . . . 53219.2.4 Soliton Propagation in Random Media . . . . . . . . . . . . . . . 53619.2.5 Reduction of Wave Localization by Nonlinearity . . . . . . . 540

20 Propagation in Random Waveguides . . . . . . . . . . . . . . . . . . . . . . . 54520.1 Propagation in Homogeneous Waveguides . . . . . . . . . . . . . . . . . . 547

20.1.1 Modeling of the Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . 54720.1.2 The Propagating and Evanescent Modes . . . . . . . . . . . . . 54820.1.3 Excitation Conditions for an Incoming Wave . . . . . . . . . . 55020.1.4 Excitation Conditions for a Source . . . . . . . . . . . . . . . . . . . 550

20.2 Mode Coupling in Random Waveguides . . . . . . . . . . . . . . . . . . . . 55120.2.1 Coupled Amplitude Equations . . . . . . . . . . . . . . . . . . . . . . 55320.2.2 Conservation of Energy Flux . . . . . . . . . . . . . . . . . . . . . . . . 55420.2.3 Evanescent Modes in Terms of Propagating Modes . . . . 55620.2.4 Propagating-Mode-Amplitude Equations . . . . . . . . . . . . . 55720.2.5 Propagator Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55820.2.6 The Forward-Scattering Approximation . . . . . . . . . . . . . . 561

20.3 The Time-Harmonic Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56220.3.1 The Coupled Mode Diffusion Process . . . . . . . . . . . . . . . . 56220.3.2 Mean Mode Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56420.3.3 Coupled Power Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 56420.3.4 Fluctuations Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566

20.4 Broadband Pulse Propagation in Waveguides . . . . . . . . . . . . . . . 56720.4.1 Integral Representation of the Transmitted Field . . . . . . 56720.4.2 Broadband Pulse Propagation in a Homogeneous

Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56920.4.3 The Stable Wave Field in a Random Waveguide . . . . . . . 569

20.5 Time Reversal for a Broadband Pulse . . . . . . . . . . . . . . . . . . . . . . 571

XX Contents

20.5.1 Time Reversal in Waveguides . . . . . . . . . . . . . . . . . . . . . . . 57120.5.2 Integral Representation of the Broadband Refocused

Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57320.5.3 Refocusing in a Homogeneous Waveguide . . . . . . . . . . . . . 57420.5.4 Refocusing in a Random Waveguide . . . . . . . . . . . . . . . . . 575

20.6 Statistics of the Transmission Coefficients at Two NearbyFrequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57920.6.1 Transport Equations for the Autocorrelation Function

of the Transfer Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57920.6.2 Probabilistic Representation of the Transport Equations 582

20.7 Incoherent Wave Fluctuations in the Broadband Case . . . . . . . . 58420.8 Narrowband Pulse Propagation in Waveguides . . . . . . . . . . . . . . 587

20.8.1 Narrowband Pulse Propagation in a HomogeneousWaveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588

20.8.2 The Mean Field in a Random Waveguide . . . . . . . . . . . . . 58820.8.3 The Mean Intensity in a Random Waveguide . . . . . . . . . . 589

20.9 Time Reversal for a Narrowband Pulse . . . . . . . . . . . . . . . . . . . . . 59020.9.1 Refocusing in a Homogeneous Waveguide . . . . . . . . . . . . . 59120.9.2 The Mean Refocused Field in a Random Waveguide . . . 59120.9.3 Statistical Stability of the Refocused Field . . . . . . . . . . . . 59220.9.4 Numerical Illustration of Spatial Focusing and

Statistical Stability in Narrowband Time Reversal . . . . . 594

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609

1

Introduction and Overview of the Book

We begin by describing the organization of the book as shown in the diagramin Figure 1.1.

The basic theory of wave propagation in one-dimensional random mediais contained in Chapters 2–9. Background for waves in deterministic, layeredmedia is given in Chapters 2 and 3. In Chapters 4 and 5 we introduce themodeling of random media and describe in detail the scaling regimes that weconsider in this book. In Chapter 6 we give a self-contained presentation ofthe asymptotic theory of random differential equations in a form that can beapplied directly to the analysis of waves in random media in the followingchapters. The asymptotic theory of reflection and transmission of waves inone-dimensional random media is presented in Chapters 7–9. Monochromaticreflection and transmission is analyzed in Chapter 7, which contains the well-known results of exponential decay of transmitted energy as the size of therandom medium increases. In Chapter 8 we analyze the propagation of wavefronts and in Chapter 9 we characterize the statistical properties of wavefluctuations in the time domain.

The theory of time reversal in one-dimensional random media, both forreflected and for transmitted waves, along with applications to detection andcommunications, is presented in Chapters 10–13.

The extension of the theory of Chapters 8 and 9 to wave propagationin three-dimensional randomly layered media is given in Chapter 14. Timereversal in such media is analyzed in Chapter 15, where we derive analyticalformulas that characterize the enhanced spatial focusing. An application toecho-mode energy refocusing on a passive scatterer is presented in Chapter16.

Chapters 17–19 contain special topics and various generalizations to otherasymptotic regimes and other types of waves. In Chapter 20 we analyze indetail wave propagation in randomly perturbed waveguides. This chapter isself-contained and could be read right after Chapter 6.

We now describe in more detail the contents of the chapters.

2 1 Introduction and Overview of the Book

Waves in layered mediaChapters 2–3

?Random mediaand scalings

Chapters 4–5

-Motivation Limit theoremsfor random

differential equationsChapter 6

?

»»»»»»»»»»»»»»9Wave reflectionand transmission

in one-dimensionalrandom mediaChapters 7–9

PPPPPPPPPPPPPPPPq

AAAAAU

RandomwaveguidesChapter 20

»»»»»»»»»9Time reversal

in one-dimensionalrandom media

Chapters 10–13

CCCW

Three-dimensionalrandomly layered

mediaChapter 14

XXXXXXXXXXXXXXzOne-dimensional

randomSchrodingerChapter 19

?

³³³³³³³³³³)Time reversal

in three-dimensionalrandomly layered

mediaChapters 15–16

PPPPPPPPPPPPq

-Other regimesof propagationChapter 18

PPPPPPPPPPPPqOther layered

mediaChapter 17

Fig. 1.1. Interdependence of the chapters.

Basic facts about wave propagation in homogeneous media are pre-sented in Chapter 2.

In Chapter 3 we consider one-dimensional piecewise constant layeredmedia, and we introduce the usual formulation of reflection and transmissionin terms of products of matrices.

Starting with Chapter 4 we consider randomly layered media. Weintroduce the linear system of acoustic equations for waves propagating inone dimension, and then carefully describe the sequence of transformations

1 Introduction and Overview of the Book 3

that will be carried out throughout the rest of the book. We pay particularattention to boundary conditions and their interpretation, and to the reflectedand transmitted waves in both the frequency and time domains. The conceptsof random media and correlation lengths are introduced in this chapter. Ourpoint of view is that randomness is closely associated with small-scale inhomo-geneities leading naturally to the regime of homogenization and the notion ofeffective medium. This is done with an application of the law of large numbers,in the context of differential equations with random coefficients. This regimecorresponds to waves propagating over distances of a few wavelengths, whichare, however, much larger than the correlation length of the inhomogeneities.

We go a step further in Chapter 5 by considering waves propagating overdistances much larger than wavelengths. The fluctuations due to the multi-ple scattering by the random inhomogeneities accumulate and create “noisy”reflected and transmitted waves. We introduce important scaling regimes inwhich diffusion approximations are valid, leading to differential equationswith random coefficients that are white noise. Even though the equations arelinear, the probability distribution of the “noisy” wave field is a highly non-linear function of the distribution of the random coefficients that model therandom inhomogeneities. For a given frequency the random differential equa-tions that enter are finite-dimensional, but in the time domain the problemsbecome infinite-dimensional. Asymptotic approximations greatly simplify theanalysis in the scaling regimes, and enable us to obtain useful informationabout the statistics of the reflected and transmitted waves.

In Chapter 6 we present concepts and results about stochastic pro-cesses needed in the modeling of one-dimensional wave propagation and itsasymptotic analysis. It is important to note that distance along the one-dimensional direction of propagation plays the role of the usual time param-eter for these stochastic processes. The physical time is transformed by goinginto the frequency domain. In this chapter we present briefly the elements ofthe theory of Markov processes used for modeling randomly layered mediaand for describing the limit processes arising in the regime of diffusion ap-proximations. A summary of the stochastic calculus is given at the end ofthe chapter, including Ito’s formula, stochastic differential equations, the linkwith parabolic partial differential equations through the Feynman–Kac for-mula, and applications to the study of Lyapunov exponents of linear randomdifferential equations.

A detailed analysis of the reflection and transmission of monochromaticwaves in a one-dimensional random medium is given in Chapter 7. In one-dimensional random media all the wave energy is eventually converted intofluctuations, giving rise to the phenomenon of wave localization. This meansthat the energy is trapped by the random medium. It is entirely reflected backin the case of a random half-space. We show that the exponential decay ofthe transmitted energy through a random slab of random medium is closelyrelated to the stability of the random harmonic oscillator, studied in thischapter. We also compute the moments of the transmitted energy, quantifying

4 1 Introduction and Overview of the Book

the exponential decay, as well as the almost-sure exponential decay that isrelated to the usual localization theory.

In Chapter 8 we study the transmitted wave front in one-dimensionalrandom media, in the regimes of the diffusion approximation introduced inthe previous chapters. A pulse is sent from one end of a one-dimensional ran-dom medium and it is observed at the other end (see Figure 1.2). When thepulse exits the slab it looks like a smeared and faded version of the originalone, followed by a noisy, incoherent coda. It is quite remarkable that in theseasymptotic regimes, the front of the transmitted pulse has a simple descrip-tion: (i) its deterministic shape is given by the convolution of the original pulsewith a deterministic kernel that depends only on the second-order statisticsof the random medium, and (ii) the transmitted wave front is centered at arandom arrival time whose probability distribution is explicitly given in termsof a single Brownian motion. In this chapter we also describe the wave frontreflected from a strong interface in a random medium.

−160 −120 −80 −40 0 40 80 120 160

Reflected Signal Transmitted Signal

Incoming Pulse

Random Medium

z

Fig. 1.2. Propagation of a pulse through a slab of random medium (0, L). A right-going wave is incoming from the left. Snapshots of the wave profile (here the pressure)at different times are plotted from bottom to top. The reflected and transmittedsignals at the last time of the numerical simulation are plotted at the top.

1 Introduction and Overview of the Book 5

In Chapter 9 we characterize the statistics of the reflected and trans-mitted waves, including the coda, in both the frequency and time domains(see the wave signals plotted at the top of Figure 1.2). This is done by acareful asymptotic analysis of the moments of the reflection and transmissioncoefficients. They satisfy a system of differential equations with random coef-ficients and are scaled so that the diffusion approximation can be applied. Thelimiting moments are obtained as solutions of systems of transport equations,which play a central role in the analysis of time reversal with incoherent waves,discussed in the following chapters. The solutions of these deterministic trans-port equations admit a probabilistic representation in terms of jump Markovprocesses, which is particularly convenient for Monte Carlo simulations and,in some cases, for deriving explicit formulas.

In Chapter 10 we analyze time reversal in reflection where the in-coherent reflected waves are recorded and sent time-reversed back into themedium. We show that stable refocusing takes place at the original sourcepoint. This is observed in physical experiments and illustrated in numericalsimulations in Figures 1.3 and 1.4. Time-reversal refocusing can be used toestimate power spectral densities of reflected waves. They contain informationabout the medium. In this chapter we also compare, with a detailed analy-sis of signal-to-noise ratios, the spectral estimation method using timereversal with a direct estimation of cross-correlations of the reflected signal.

In Chapter 11 we present two applications of time reversal to detection.In the first application, we use time reversal to detect the presence of a weakreflector buried in the many random layers. In this case the refocusing ker-nel of the time-reversal process has a jump that is related to the depth andstrength of the reflector, and we exploit this to identify the reflector. In thesecond application, we introduce absorption in the one-dimensional modeland show that refocusing still takes place after time reversal. We apply this tothe detection and characterization of a dissipative region embedded in therandom medium. In the presence of a dissipative region the refocusing kernelis modified and has a jump in its derivative. The time of this jump is relatedto the depth of the dissipative region, and its amplitude to the strength ofabsorption.

In Chapter 12 we study time reversal of waves in randomly layered mediadescribed in the previous chapters. In this chapter we analyze time reversalin transmission, which means that a pulse is emitted at one end of a randomslab, recorded at a time-reversal mirror at the other end, and then sent back.The wave refocuses at the original source point and the quality of the refo-cusing depends on how much of the transmitted wave has been recorded. Inparticular, it is shown that recording some part of the incoherent coda waveimproves refocusing.

Applications to communications are presented in Chapter 13, where weanalyze signal-to-interference ratios with and without using time reversalfor communications through a one-dimensional random channel.

6 1 Introduction and Overview of the Book

−160 −120 −80 −40 0 40 80 120 160

Random MediumRe−emitted Signal

Refocused Signal

z

Fig. 1.3. We use the same random medium as in Figure 1.2 and send back, to theright, the time-reversed reflected signal (the one plotted at the top left corner ofFigure 1.2). Snapshots of the wave profile (here the pressure) at different times areplotted from bottom to top. The refocused pulse is seen emerging from the randommedium at the top.

Starting with Chapter 14 we analyze waves propagating in a randomlylayered three-dimensional medium. By taking Fourier transforms with re-spect to time and along the layers, the problem can be formulated as infinitelymany one-dimensional problems. We model a physical source located at thesurface of the random medium. Using a stationary phase analysis, we showthat in the regime of diffusion approximations, and because of the separationof scales as in previous chapters, the stable wave front can again be describedwith an explicit formula that we derive.

Time reversal of waves propagating in three-dimensional randomly layeredmedia is discussed in Chapter 15, where we consider a time-reversal mirrorthat records the signals generated by a source embedded in the random lay-ers. We show that the time-reversed waves refocus around the original sourcepoint. We give a detailed analytical description of the refocused pulse in timeand space. We compare this refocusing with diffraction-limited refocusingin homogeneous media and show that there is superresolution from multi-

1 Introduction and Overview of the Book 7

−10 0 10

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0.2

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t

sign

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Fig. 1.4. We plot the refocused pulses generated by 10 independent simulations oftime reversal (we follow the same procedure as in Figures 1.2–1.3, and we magnifythe refocused pulse seen at the top line of Figure 1.3). The initial pulse is thesecond derivative of a Gaussian. We see here the remarkable statistical stability ofthe refocused pulse. Its shape and center do not depend on the realization of themedium, in contrast to the small-amplitude random wave fluctuations before andafter the refocusing time.

pathing. This means that the focusing is much tighter, as well as stable, inthe random medium.

In Chapter 16 we present an application of time reversal in three-dimensional randomly layered media to echo-mode energy refocusing ona passive scatterer. This means that when the reflected signals received atthe time-reversal mirror from a scatterer in a randomly layered medium aretime-reversed and suitably reemitted, they tend to focus on the scatterer.

In Chapter 17 we present an extension of the theory of wave propagationand time reversal to more general randomly layered media. We analyzemodels in which the effective parameters of the random medium do not matchthose of the adjacent homogeneous medium. We also analyze the case in whichthe effective parameters of the random medium vary smoothly at the macro-scopic scale. The case in which both the bulk modulus and the density of themedium are randomly fluctuating is analyzed in Section 17.3.

Chapter 18 is devoted to several extensions and generalizations includingthe following ones.

- We reconsider the analysis for a different regime of scale separation, inwhich the amplitude of the fluctuations of the medium parameters is smalland the typical wavelength is comparable to the small correlation lengthof the random medium.

- We extend the analysis to dispersive or weakly nonlinear randommedia. In the dispersive case, time reversal succeeds in recompressing the

8 1 Introduction and Overview of the Book

dispersive oscillatory tail as well as the incoherent part of the waves. Weanalyze the combined effect of randomness and weak nonlinearity on thefront of a propagating pulse. We show that randomness helps in preventingshock formation, so that time reversal in transmission can be done forlonger propagation distances.

- We study the effect of changes in the medium parameters before andafter time reversal. Although refocusing is affected by these changes, westill have partial refocusing. We also quantify the partial loss of statisticalstability.

In Chapter 19 we discuss the robustness of wave localization in a ran-domly layered medium when there is also nonlinearity, in the context of thenonlinear Schrodinger (NLS) equation. Using a perturbed inverse scatteringtransform, we show in this chapter that a soliton can overcome the exponen-tial decay experienced by linear waves propagating through a slab in randommedium.

Wave propagation in waveguides is analyzed in Chapter 20. We considerthe case in which the waveguide supports a finite number of propagatingmodes and the random fluctuations of the medium are three-dimensional. Weanalyze only transmitted waves through a randomly perturbed waveguide,in the forward-scattering approximation, and the space-time refocusing ofthese waves after time reversal. We show that stable refocusing does occur,especially when the number of modes is large. This chapter may be consideredas a link with the theory of wave propagation in three-dimensional randommedia.