Wave propagation and time reversal in randomly layered media:
Gespeichert in:
| Format: | Elektronisch E-Book |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
Berlin
Springer New York
2007
Berlin Springer Bln |
| Ausgabe: | 1. Ed. |
| Schriftenreihe: | Stochastic Modelling and Applied Probability
56 |
| Schlagworte: | |
| Online-Zugang: | BTU01 TUM01 UBA01 UBR01 UBT01 UPA01 Volltext Inhaltsverzeichnis |
| Beschreibung: | 1 Online-Ressource (XX, 612 S.) Ill. |
| ISBN: | 9780387498089 |
| DOI: | 10.1007/978-0-387-49808-9 |
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| 245 | 1 | 0 | |a Wave propagation and time reversal in randomly layered media |c Jean-Pierre Fouque ... |
| 250 | |a 1. Ed. | ||
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| 264 | 1 | |a Berlin |b Springer Bln | |
| 300 | |a 1 Online-Ressource (XX, 612 S.) |b Ill. | ||
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Datensatz im Suchindex
| _version_ | 1804136632062836736 |
|---|---|
| adam_text | 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 Propagator and Scattering Matrices 47
3.5 Multilayer Piecewise Constant Media 48
3.5.1 Propagation Equations 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 Propagator Equations 75
4.4.1 Characteristic Lines 75
4.4.2 Modes in the Fourier Domain 76
4.4.3 Propagator 77
4.4.4 The Riccati Equation for the Local Reflection Coefficient 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 Propagator and Reflection and Transmission Coefficients 100
5.2 Diffusion Scaling 102
5.2.1 White Noise Regime and Brownian Motion 103
5.2.2 Diffusion Approximation 104
Contents xiii
5.2.3 Finite Dimensional Distributions of the Transmitted
Wave 106
6 Asymptotics for Random Ordinary Differential Equations . 109
6.1 Markov Processes 110
6.1.1 Semigroups 110
6.1.2 Infinitesimal Generators Ill
6.1.3 Martingales and Martingale Problems Ill
6.1.4 Kolmogorov Backward and Forward Equations 113
6.1.5 Ergodicity 115
6.2 Markovian Models of Random Media 116
6.2.1 Two Component Composite Media 116
6.2.2 Multicomponent Composite Media 118
6.2.3 A Continuous Random Medium 120
6.3 Diffusion Approximation Without Fast Oscillation 122
6.3.1 Markov Property 123
6.3.2 Perturbed Test Functions 124
6.3.3 The Poisson Equation and the Fredholm Alternative . . . 124
6.3.4 Limiting Infinitesimal Generator 126
6.3.5 Relative Compactness of the Laws of the Processes .... 131
6.3.6 The Multiplicative Noise Case 134
6.4 The Averaging and Fluctuation Theorems 135
6.4.1 Averaging 135
6.4.2 Fluctuation Theory 136
6.5 Diffusion Approximation with Fast Oscillations 139
6.5.1 Semifast Oscillations 139
6.5.2 Fast Oscillations 142
6.6 Stochastic Calculus 145
6.6.1 Stochastic Integrals 147
6.6.2 Ito s Formula 150
6.6.3 Stochastic Differential Equations 152
6.6.4 Diffusions and Partial Differential Equations 153
6.6.5 Feynman Kac Representation Formula 155
6.7 Limits of Random Equations and Stochastic Equations 156
6.7.1 ltd Form of the Limit Process 156
6.7.2 Stratonovich Stochastic Integrals 158
6.7.3 Limits of Random Matrix Equations 160
6.8 Lyapunov Exponent for Linear Random Differential Equations 161
6.8.1 Lyapunov Exponent of the Random Differential
Equation 162
6.8.2 Lyapunov Exponent of the Limit Diffusion 169
6.9 Appendix 172
6.9.1 Quadratic Variation of a Continuous Martingale 172
xiv Contents
7 Transmission of Energy Through a Slab of Random Medium 175
7.1 Transmission of Monochromatic Waves 176
7.1.1 The Diffusion Limit for the Propagator 177
7.1.2 Polar Coordinates for the Propagator 180
7.1.3 Martingale Representation of the Transmission
Coefficient 183
7.1.4 The Localization Length Lioc(u ) 185
7.1.5 Mean and Fluctuations of the Power Transmission
Coefficient 187
7.1.6 The Strongly Fluctuating Character of the Power
Transmission Coefficient 188
7.2 Exponential Decay of the Transmitted Energy for a Pulse .... 190
7.2.1 Transmission of a Pulse Through a Slab of Random
Medium 190
7.2.2 Self Averaging Property of the Transmitted Energy .... 191
7.2.3 The Diffusion Limit for the Two Frequency Propagator 193
7.3 Wave Localization in the Weakly Heterogeneous Regime 196
7.3.1 Determination of the Power Transmission Coefficient
from a Random Harmonic Oscillator 196
7.3.2 Comparisons of Decay Rates 198
7.4 Wave Localization in the Strongly Heterogeneous White Noise
Regime 199
7.5 The Random Harmonic Oscillator 201
7.5.1 The Lyapunov Exponent of the Random Harmonic
Oscillator 202
7.5.2 Expansion of the Lyapunov Exponent in the Strongly
Heterogeneous Regime 203
7.5.3 Expansion of the Lyapunov Exponent in the Weakly
Heterogeneous Regime 208
7.6 Appendix. Statistics of the Power Transmission Coefficient.... 209
7.6.1 The Probability Density of the Power Transmission
Coefficient 209
7.6.2 Moments of the Power Transmission Coefficient 211
8 Wave Front Propagation 215
8.1 The Transmitted Wave Front in the Weakly Heterogeneous
Regime 216
8.1.1 Stabilization of the Transmitted Wave Front 217
8.1.2 The Integral Equation for the Transmitted Field 220
8.1.3 Asymptotic Analysis of the Transmitted Wave Front . . . 222
8.2 The Transmitted Wave Front in the Strongly Heterogeneous
Regime 225
8.2.1 Asymptotic Representation of the Transmitted Wave
Front 226
8.2.2 The Energy of the Transmitted Wave 229
Contents xv
8.2.3 Numerical Illustration of Pulse Spreading 230
8.2.4 The Diffusion Limit for the Multifrequency Propagators 230
8.2.5 Martingale Representation of the Multifrequency
Transmission Coefficient 233
8.2.6 Identification of the Limit Wave Front 234
8.2.7 Asymptotic Analysis of Travel Times 236
8.3 The Reflected Front in Presence of an Interface 238
8.3.1 Integral Representation of the Reflected Pulse 238
8.3.2 The Limit for the Reflected Front 242
8.4 Appendix. Proof of the Averaging Theorem 245
9 Statistics of Incoherent Waves 249
9.1 The Reflected Wave 249
9.1.1 Reformulation of the Reflection and Transmission
Problem 249
9.1.2 The Riccati Equation for the Reflection Coefficient 252
9.1.3 Representation of the Reflected Field 253
9.2 Statistics of the Reflected Wave in the Frequency Domain .... 254
9.2.1 Moments of the Reflection Coefficient 254
9.2.2 Probabilistic Representation of the Transport Equations 258
9.2.3 Explicit Solution for a Random Half Space 261
9.2.4 Multifrequency Moments 262
9.3 Statistics of the Reflected Wave in the Time Domain 266
9.3.1 Mean Amplitude 266
9.3.2 Mean Intensity 266
9.3.3 Autocorrelation and Time Domain Localization 267
9.3.4 Gaussian Statistics 269
9.4 The Transmitted Wave 272
9.4.1 Autocorrelation Function of the Transmission Coefncient272
9.4.2 Probabilistic Representation of the Transport Equations 274
9.4.3 Statistics of the Transmitted Wave in the Time Domain 277
10 Time Reversal in Reflection and Spectral Estimation 281
10.1 Time Reversal in Reflection 283
10.1.1 Time Reversal Setup 283
10.1.2 Time Reversal Refocusing 285
10.1.3 The Limiting Refocused Pulse 286
10.1.4 Time Reversal Mirror Versus Standard Mirror 290
10.2 Time Reversal Versus Cross Correlations 291
10.2.1 The Empirical Correlation Function 292
10.2.2 Measuring the Spectral Density 293
10.2.3 Signal to Noise Ratio Comparison 294
10.3 Calibrating the Initial Pulse 302
xvi Contents
11 Applications to Detection 305
11.1 Detection of a Weak Reflector 306
11.2 Detection of an Interface Between Media 311
11.3 Waves in One Dimensional Dissipative Random Media 313
11.3.1 The Acoustic Model with Random Dissipation 313
11.3.2 Propagator Formulation 314
11.3.3 Transmitted Wave Front 317
11.3.4 The Refocused Pulse for Time Reversal in Reflection. . . 317
11.4 Application to the Detection of a Dissipative Layer 320
11.4.1 Constant Mean Dissipation 321
11.4.2 Thin Dissipative Layer 321
11.4.3 Thick Dissipative Layer 324
12 Time Reversal in Transmission 327
12.1 Time Reversal of the Stable Front 328
12.1.1 Time Reversal Experiment 329
12.1.2 The Refocused Pulse 331
12.2 Time Reversal with Coda Waves 333
12.2.1 Time Reversal Experiment 333
12.2.2 Decomposition of the Refocusing Kernel 335
12.2.3 Midband Filtering by the Medium 336
12.2.4 Low Pass Filtering 337
12.3 Discussion and Numerical Simulations 339
13 Application to Communications 343
13.1 Review of Basic Communications Schemes 344
13.1.1 Nyquist Pulse 344
13.1.2 Signal to Interference Ratio 345
13.1.3 Modulated Nyquist Pulse 346
13.2 Communications in Random Media Using Nyquist Pulses 347
13.2.1 Direct Transmission 350
13.2.2 Communications Using Time Reversal 351
13.2.3 SIRs for Coherent Pulses 353
13.2.4 Influence of the Incoherent Waves 355
13.2.5 Numerical Simulations 357
13.3 Communications in Random Media Using Modulated Nyquist
Pulses 358
13.3.1 SIRs of Modulated Nyquist Pulses 359
13.3.2 Numerical Simulations 362
13.3.3 Discussion 363
14 Scattering by a Three Dimensional Randomly Layered
Medium 365
14.1 Acoustic Waves in Three Dimensions 366
14.1.1 Homogenization Regime 366
Contents xvii
14.1.2 The Diffusion Approximation Regime 368
14.1.3 Plane Wave Fourier Transform 369
14.1.4 One Dimensional Mode Problems 370
14.1.5 Transmitted Pressure Integral Representation 374
14.2 The Transmitted Wave Front 374
14.2.1 Characterization of Moments 374
14.2.2 Stationary Phase Point 376
14.2.3 Characterization of the Transmitted Wave Front 378
14.3 The Mean Reflected Intensity Generated by a Point Source .. . 380
14.3.1 Refiected Pressure Integral Representation 380
14.3.2 Autocorrelation Function of the Reflection Coefficient
at Two Nearby Slownesses and Frequencies 381
14.3.3 Asymptotics of the Mean Intensity 385
14.4 Appendix: Stationary Phase Method 389
15 Time Reversal in a Three Dimensional Layered Medium.. . 393
15.1 The Embedded Source Problem 393
15.2 Time Reversal with Embedded Source 395
15.2.1 Emission from a Point Source 395
15.2.2 Recording, Time Reversal, and Reemission 401
15.2.3 The Time Reversed Wave Field 403
15.3 Homogeneous Medium 405
15.3.1 The Field Recorded at the Surface 406
15.3.2 The Time Reversed Field 407
15.4 Complete Description of the Time Reversed Field in a
Random Medium 411
15.4.1 Expectation of the Refocused Pulse 413
15.4.2 Refocusing of the Pulse 414
15.5 Refocusing Properties in a Random Medium 416
15.5.1 The Case |zs| L oc 416
15.5.2 Time Reversal of the Front 417
15.5.3 Time Reversal of the Incoherent Waves with Offset .... 417
15.5.4 Time Reversal of the Incoherent Waves Without Offset. 422
15.5.5 Record of the Pressure Signal 424
15.6 Appendix A: Moments of the Reflection and Transmission
Coefficients 424
15.6.1 Autocorrelation Function of the Transmission
Coefficient at Two Nearby Slownesses and Frequencies . 424
15.6.2 Shift Properties 425
15.6.3 Generalized Coefficients 426
15.7 Appendix B: A Priori Estimates for the Generalized Coefficients428
15.8 Appendix C: Derivation of (15.74) 430
xviii Contents
16 Application to Echo Mode Time Reversal 435
16.1 The Born Approximation for an Embedded Scatterer 435
16.1.1 Integral Expressions for the Wave Fields 437
16.2 Asymptotic Theory for the Scattered Field 439
16.2.1 The Primary Field 439
16.2.2 The Secondary Field 440
16.3 Time Reversal of the Recorded Wave 442
16.3.1 Integral Representation of the Time Reversed Field .... 442
16.3.2 Refocusing in the Homogeneous Case 444
16.3.3 Refocusing of the Secondary Field in the Random Case 446
16.3.4 Contributions of the Other Wave Components 451
16.4 Time Reversal Superresolution with a Passive Scatterer 451
16.4.1 The Refocused Pulse Shape 451
16.4.2 Superresolution with a Random Medium 453
17 Other Layered Media 457
17.1 Nonmatched Effective Medium 457
17.1.1 Boundary and Jump Conditions 458
17.1.2 Transmission of a Pulse through a Nonmatched
Random Slab 459
17.1.3 Reflection by a Nonmatched Random Half Space 464
17.2 General Background 466
17.2.1 Mode Decomposition 467
17.2.2 Transport Equations 469
17.2.3 Applications 471
17.3 Medium with Random Density Fluctuations 472
17.3.1 The Coupled Propagator White Noise Model 474
17.3.2 The Transmitted Field 479
17.3.3 Transport Equations 482
17.3.4 Reflection by a Random Half Space 484
18 Other Regimes of Propagation 487
18.1 The Weakly Heterogeneous Regime in Randomly Layered
Media 487
18.1.1 Mode Decomposition 488
18.1.2 Transport Equations 490
18.1.3 Applications 490
18.2 Dispersive Media 492
18.2.1 The Terrain Following Boussinesq Model 493
18.2.2 The Propagating Modes of the Boussinesq Equation . . . 494
18.2.3 Mode Propagation in a Dispersive Random Medium . . . 495
18.2.4 Transport Equations 497
18.2.5 Time Reversal 498
18.3 Nonlinear Media 499
18.3.1 Shallow Water Waves with Random Depth 500
Contents xix
18.3.2 The Linear Hyperbolic Approximation 502
18.3.3 The Effective Equation for the Nonlinear Front Pulse . . 504
18.3.4 Analysis of the Pseudospectral Operator 508
18.3.5 Time Reversal 509
18.4 Time Reversal with Changing Media 510
18.4.1 The Experiment 510
18.4.2 Convergence of the Finite Dimensional Distributions . . . 511
18.4.3 Convergence of the Refocused Pulse 515
19 The Random Schrodinger Model 519
19.1 Linear Regime 519
19.1.1 The Linear Schrodinger Equation 519
19.1.2 Transmission of a Monochromatic Wave 521
19.1.3 Transmission of a Pulse 526
19.2 Nonlinear Regime 528
19.2.1 Waves Called Solitons 528
19.2.2 Dispersion and Nonlinearity 531
19.2.3 The Nonlinear Schrodinger Equation 532
19.2.4 Soliton Propagation in Random Media 536
19.2.5 Reduction of Wave Localization by Nonlinearity 540
20 Propagation in Random Waveguides 545
20.1 Propagation in Homogeneous Waveguides 547
20.1.1 Modeling of the Waveguide 547
20.1.2 The Propagating and Evanescent Modes 548
20.1.3 Excitation Conditions for an Incoming Wave 550
20.1.4 Excitation Conditions for a Source 550
20.2 Mode Coupling in Random Waveguides 551
20.2.1 Coupled Amplitude Equations 553
20.2.2 Conservation of Energy Flux 554
20.2.3 Evanescent Modes in Terms of Propagating Modes .... 556
20.2.4 Propagating Mode Amplitude Equations 557
20.2.5 Propagator Matrices 558
20.2.6 The Forward Scattering Approximation 561
20.3 The Time Harmonic Problem 562
20.3.1 The Coupled Mode Diffusion Process 562
20.3.2 Mean Mode Amplitudes 564
20.3.3 Coupled Power Equations 564
20.3.4 Fluctuations Theory 566
20.4 Broadband Pulse Propagation in Waveguides 567
20.4.1 Integral Representation of the Transmitted Field 567
20.4.2 Broadband Pulse Propagation in a Homogeneous
Waveguide 569
20.4.3 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 571
20.5.2 Integral Representation of the Broadband Refocused
Field 573
20.5.3 Refocusing in a Homogeneous Waveguide 574
20.5.4 Refocusing in a Random Waveguide 575
20.6 Statistics of the Transmission Coefficients at Two Nearby
Frequencies 579
20.6.1 Transport Equations for the Autocorrelation Function
of the Transfer Matrix 579
20.6.2 Probabilistic Representation of the Transport Equations 582
20.7 Incoherent Wave Fluctuations in the Broadband Case 584
20.8 Narrowband Pulse Propagation in Waveguides 587
20.8.1 Narrowband Pulse Propagation in a Homogeneous
Waveguide 588
20.8.2 The Mean Field in a Random Waveguide 588
20.8.3 The Mean Intensity in a Random Waveguide 589
20.9 Time Reversal for a Narrowband Pulse 590
20.9.1 Refocusing in a Homogeneous Waveguide 591
20.9.2 The Mean Refocused Field in a Random Waveguide ... 591
20.9.3 Statistical Stability of the Refocused Field 592
20.9.4 Numerical Illustration of Spatial Focusing and
Statistical Stability in Narrowband Time Reversal 594
References 599
Index 609
|
| adam_txt |
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 Propagator and Scattering Matrices 47
3.5 Multilayer Piecewise Constant Media 48
3.5.1 Propagation Equations 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 Propagator Equations 75
4.4.1 Characteristic Lines 75
4.4.2 Modes in the Fourier Domain 76
4.4.3 Propagator 77
4.4.4 The Riccati Equation for the Local Reflection Coefficient 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 Propagator and Reflection and Transmission Coefficients 100
5.2 Diffusion Scaling 102
5.2.1 White Noise Regime and Brownian Motion 103
5.2.2 Diffusion Approximation 104
Contents xiii
5.2.3 Finite Dimensional Distributions of the Transmitted
Wave 106
6 Asymptotics for Random Ordinary Differential Equations . 109
6.1 Markov Processes 110
6.1.1 Semigroups 110
6.1.2 Infinitesimal Generators Ill
6.1.3 Martingales and Martingale Problems Ill
6.1.4 Kolmogorov Backward and Forward Equations 113
6.1.5 Ergodicity 115
6.2 Markovian Models of Random Media 116
6.2.1 Two Component Composite Media 116
6.2.2 Multicomponent Composite Media 118
6.2.3 A Continuous Random Medium 120
6.3 Diffusion Approximation Without Fast Oscillation 122
6.3.1 Markov Property 123
6.3.2 Perturbed Test Functions 124
6.3.3 The Poisson Equation and the Fredholm Alternative . . . 124
6.3.4 Limiting Infinitesimal Generator 126
6.3.5 Relative Compactness of the Laws of the Processes . 131
6.3.6 The Multiplicative Noise Case 134
6.4 The Averaging and Fluctuation Theorems 135
6.4.1 Averaging 135
6.4.2 Fluctuation Theory 136
6.5 Diffusion Approximation with Fast Oscillations 139
6.5.1 Semifast Oscillations 139
6.5.2 Fast Oscillations 142
6.6 Stochastic Calculus 145
6.6.1 Stochastic Integrals 147
6.6.2 Ito's Formula 150
6.6.3 Stochastic Differential Equations 152
6.6.4 Diffusions and Partial Differential Equations 153
6.6.5 Feynman Kac Representation Formula 155
6.7 Limits of Random Equations and Stochastic Equations 156
6.7.1 ltd Form of the Limit Process 156
6.7.2 Stratonovich Stochastic Integrals 158
6.7.3 Limits of Random Matrix Equations 160
6.8 Lyapunov Exponent for Linear Random Differential Equations 161
6.8.1 Lyapunov Exponent of the Random Differential
Equation 162
6.8.2 Lyapunov Exponent of the Limit Diffusion 169
6.9 Appendix 172
6.9.1 Quadratic Variation of a Continuous Martingale 172
xiv Contents
7 Transmission of Energy Through a Slab of Random Medium 175
7.1 Transmission of Monochromatic Waves 176
7.1.1 The Diffusion Limit for the Propagator 177
7.1.2 Polar Coordinates for the Propagator 180
7.1.3 Martingale Representation of the Transmission
Coefficient 183
7.1.4 The Localization Length Lioc(u ) 185
7.1.5 Mean and Fluctuations of the Power Transmission
Coefficient 187
7.1.6 The Strongly Fluctuating Character of the Power
Transmission Coefficient 188
7.2 Exponential Decay of the Transmitted Energy for a Pulse . 190
7.2.1 Transmission of a Pulse Through a Slab of Random
Medium 190
7.2.2 Self Averaging Property of the Transmitted Energy . 191
7.2.3 The Diffusion Limit for the Two Frequency Propagator 193
7.3 Wave Localization in the Weakly Heterogeneous Regime 196
7.3.1 Determination of the Power Transmission Coefficient
from a Random Harmonic Oscillator 196
7.3.2 Comparisons of Decay Rates 198
7.4 Wave Localization in the Strongly Heterogeneous White Noise
Regime 199
7.5 The Random Harmonic Oscillator 201
7.5.1 The Lyapunov Exponent of the Random Harmonic
Oscillator 202
7.5.2 Expansion of the Lyapunov Exponent in the Strongly
Heterogeneous Regime 203
7.5.3 Expansion of the Lyapunov Exponent in the Weakly
Heterogeneous Regime 208
7.6 Appendix. Statistics of the Power Transmission Coefficient. 209
7.6.1 The Probability Density of the Power Transmission
Coefficient 209
7.6.2 Moments of the Power Transmission Coefficient 211
8 Wave Front Propagation 215
8.1 The Transmitted Wave Front in the Weakly Heterogeneous
Regime 216
8.1.1 Stabilization of the Transmitted Wave Front 217
8.1.2 The Integral Equation for the Transmitted Field 220
8.1.3 Asymptotic Analysis of the Transmitted Wave Front . . . 222
8.2 The Transmitted Wave Front in the Strongly Heterogeneous
Regime 225
8.2.1 Asymptotic Representation of the Transmitted Wave
Front 226
8.2.2 The Energy of the Transmitted Wave 229
Contents xv
8.2.3 Numerical Illustration of Pulse Spreading 230
8.2.4 The Diffusion Limit for the Multifrequency Propagators 230
8.2.5 Martingale Representation of the Multifrequency
Transmission Coefficient 233
8.2.6 Identification of the Limit Wave Front 234
8.2.7 Asymptotic Analysis of Travel Times 236
8.3 The Reflected Front in Presence of an Interface 238
8.3.1 Integral Representation of the Reflected Pulse 238
8.3.2 The Limit for the Reflected Front 242
8.4 Appendix. Proof of the Averaging Theorem 245
9 Statistics of Incoherent Waves 249
9.1 The Reflected Wave 249
9.1.1 Reformulation of the Reflection and Transmission
Problem 249
9.1.2 The Riccati Equation for the Reflection Coefficient 252
9.1.3 Representation of the Reflected Field 253
9.2 Statistics of the Reflected Wave in the Frequency Domain . 254
9.2.1 Moments of the Reflection Coefficient 254
9.2.2 Probabilistic Representation of the Transport Equations 258
9.2.3 Explicit Solution for a Random Half Space 261
9.2.4 Multifrequency Moments 262
9.3 Statistics of the Reflected Wave in the Time Domain 266
9.3.1 Mean Amplitude 266
9.3.2 Mean Intensity 266
9.3.3 Autocorrelation and Time Domain Localization 267
9.3.4 Gaussian Statistics 269
9.4 The Transmitted Wave 272
9.4.1 Autocorrelation Function of the Transmission Coefncient272
9.4.2 Probabilistic Representation of the Transport Equations 274
9.4.3 Statistics of the Transmitted Wave in the Time Domain 277
10 Time Reversal in Reflection and Spectral Estimation 281
10.1 Time Reversal in Reflection 283
10.1.1 Time Reversal Setup 283
10.1.2 Time Reversal Refocusing 285
10.1.3 The Limiting Refocused Pulse 286
10.1.4 Time Reversal Mirror Versus Standard Mirror 290
10.2 Time Reversal Versus Cross Correlations 291
10.2.1 The Empirical Correlation Function 292
10.2.2 Measuring the Spectral Density 293
10.2.3 Signal to Noise Ratio Comparison 294
10.3 Calibrating the Initial Pulse 302
xvi Contents
11 Applications to Detection 305
11.1 Detection of a Weak Reflector 306
11.2 Detection of an Interface Between Media 311
11.3 Waves in One Dimensional Dissipative Random Media 313
11.3.1 The Acoustic Model with Random Dissipation 313
11.3.2 Propagator Formulation 314
11.3.3 Transmitted Wave Front 317
11.3.4 The Refocused Pulse for Time Reversal in Reflection. . . 317
11.4 Application to the Detection of a Dissipative Layer 320
11.4.1 Constant Mean Dissipation 321
11.4.2 Thin Dissipative Layer 321
11.4.3 Thick Dissipative Layer 324
12 Time Reversal in Transmission 327
12.1 Time Reversal of the Stable Front 328
12.1.1 Time Reversal Experiment 329
12.1.2 The Refocused Pulse 331
12.2 Time Reversal with Coda Waves 333
12.2.1 Time Reversal Experiment 333
12.2.2 Decomposition of the Refocusing Kernel 335
12.2.3 Midband Filtering by the Medium 336
12.2.4 Low Pass Filtering 337
12.3 Discussion and Numerical Simulations 339
13 Application to Communications 343
13.1 Review of Basic Communications Schemes 344
13.1.1 Nyquist Pulse 344
13.1.2 Signal to Interference Ratio 345
13.1.3 Modulated Nyquist Pulse 346
13.2 Communications in Random Media Using Nyquist Pulses 347
13.2.1 Direct Transmission 350
13.2.2 Communications Using Time Reversal 351
13.2.3 SIRs for Coherent Pulses 353
13.2.4 Influence of the Incoherent Waves 355
13.2.5 Numerical Simulations 357
13.3 Communications in Random Media Using Modulated Nyquist
Pulses 358
13.3.1 SIRs of Modulated Nyquist Pulses 359
13.3.2 Numerical Simulations 362
13.3.3 Discussion 363
14 Scattering by a Three Dimensional Randomly Layered
Medium 365
14.1 Acoustic Waves in Three Dimensions 366
14.1.1 Homogenization Regime 366
Contents xvii
14.1.2 The Diffusion Approximation Regime 368
14.1.3 Plane Wave Fourier Transform 369
14.1.4 One Dimensional Mode Problems 370
14.1.5 Transmitted Pressure Integral Representation 374
14.2 The Transmitted Wave Front 374
14.2.1 Characterization of Moments 374
14.2.2 Stationary Phase Point 376
14.2.3 Characterization of the Transmitted Wave Front 378
14.3 The Mean Reflected Intensity Generated by a Point Source . . 380
14.3.1 Refiected Pressure Integral Representation 380
14.3.2 Autocorrelation Function of the Reflection Coefficient
at Two Nearby Slownesses and Frequencies 381
14.3.3 Asymptotics of the Mean Intensity 385
14.4 Appendix: Stationary Phase Method 389
15 Time Reversal in a Three Dimensional Layered Medium. . 393
15.1 The Embedded Source Problem 393
15.2 Time Reversal with Embedded Source 395
15.2.1 Emission from a Point Source 395
15.2.2 Recording, Time Reversal, and Reemission 401
15.2.3 The Time Reversed Wave Field 403
15.3 Homogeneous Medium 405
15.3.1 The Field Recorded at the Surface 406
15.3.2 The Time Reversed Field 407
15.4 Complete Description of the Time Reversed Field in a
Random Medium 411
15.4.1 Expectation of the Refocused Pulse 413
15.4.2 Refocusing of the Pulse 414
15.5 Refocusing Properties in a Random Medium 416
15.5.1 The Case |zs| L\oc 416
15.5.2 Time Reversal of the Front 417
15.5.3 Time Reversal of the Incoherent Waves with Offset . 417
15.5.4 Time Reversal of the Incoherent Waves Without Offset. 422
15.5.5 Record of the Pressure Signal 424
15.6 Appendix A: Moments of the Reflection and Transmission
Coefficients 424
15.6.1 Autocorrelation Function of the Transmission
Coefficient at Two Nearby Slownesses and Frequencies . 424
15.6.2 Shift Properties 425
15.6.3 Generalized Coefficients 426
15.7 Appendix B: A Priori Estimates for the Generalized Coefficients428
15.8 Appendix C: Derivation of (15.74) 430
xviii Contents
16 Application to Echo Mode Time Reversal 435
16.1 The Born Approximation for an Embedded Scatterer 435
16.1.1 Integral Expressions for the Wave Fields 437
16.2 Asymptotic Theory for the Scattered Field 439
16.2.1 The Primary Field 439
16.2.2 The Secondary Field 440
16.3 Time Reversal of the Recorded Wave 442
16.3.1 Integral Representation of the Time Reversed Field . 442
16.3.2 Refocusing in the Homogeneous Case 444
16.3.3 Refocusing of the Secondary Field in the Random Case 446
16.3.4 Contributions of the Other Wave Components 451
16.4 Time Reversal Superresolution with a Passive Scatterer 451
16.4.1 The Refocused Pulse Shape ' 451
16.4.2 Superresolution with a Random Medium 453
17 Other Layered Media 457
17.1 Nonmatched Effective Medium 457
17.1.1 Boundary and Jump Conditions 458
17.1.2 Transmission of a Pulse through a Nonmatched
Random Slab 459
17.1.3 Reflection by a Nonmatched Random Half Space 464
17.2 General Background 466
17.2.1 Mode Decomposition 467
17.2.2 Transport Equations 469
17.2.3 Applications 471
17.3 Medium with Random Density Fluctuations 472
17.3.1 The Coupled Propagator White Noise Model 474
17.3.2 The Transmitted Field 479
17.3.3 Transport Equations 482
17.3.4 Reflection by a Random Half Space 484
18 Other Regimes of Propagation 487
18.1 The Weakly Heterogeneous Regime in Randomly Layered
Media 487
18.1.1 Mode Decomposition 488
18.1.2 Transport Equations 490
18.1.3 Applications 490
18.2 Dispersive Media 492
18.2.1 The Terrain Following Boussinesq Model 493
18.2.2 The Propagating Modes of the Boussinesq Equation . . . 494
18.2.3 Mode Propagation in a Dispersive Random Medium . . . 495
18.2.4 Transport Equations 497
18.2.5 Time Reversal 498
18.3 Nonlinear Media 499
18.3.1 Shallow Water Waves with Random Depth 500
Contents xix
18.3.2 The Linear Hyperbolic Approximation 502
18.3.3 The Effective Equation for the Nonlinear Front Pulse . . 504
18.3.4 Analysis of the Pseudospectral Operator 508
18.3.5 Time Reversal 509
18.4 Time Reversal with Changing Media 510
18.4.1 The Experiment 510
18.4.2 Convergence of the Finite Dimensional Distributions . . . 511
18.4.3 Convergence of the Refocused Pulse 515
19 The Random Schrodinger Model 519
19.1 Linear Regime 519
19.1.1 The Linear Schrodinger Equation 519
19.1.2 Transmission of a Monochromatic Wave 521
19.1.3 Transmission of a Pulse 526
19.2 Nonlinear Regime 528
19.2.1 Waves Called Solitons 528
19.2.2 Dispersion and Nonlinearity 531
19.2.3 The Nonlinear Schrodinger Equation 532
19.2.4 Soliton Propagation in Random Media 536
19.2.5 Reduction of Wave Localization by Nonlinearity 540
20 Propagation in Random Waveguides 545
20.1 Propagation in Homogeneous Waveguides 547
20.1.1 Modeling of the Waveguide 547
20.1.2 The Propagating and Evanescent Modes 548
20.1.3 Excitation Conditions for an Incoming Wave 550
20.1.4 Excitation Conditions for a Source 550
20.2 Mode Coupling in Random Waveguides 551
20.2.1 Coupled Amplitude Equations 553
20.2.2 Conservation of Energy Flux 554
20.2.3 Evanescent Modes in Terms of Propagating Modes . 556
20.2.4 Propagating Mode Amplitude Equations 557
20.2.5 Propagator Matrices 558
20.2.6 The Forward Scattering Approximation 561
20.3 The Time Harmonic Problem 562
20.3.1 The Coupled Mode Diffusion Process 562
20.3.2 Mean Mode Amplitudes 564
20.3.3 Coupled Power Equations 564
20.3.4 Fluctuations Theory 566
20.4 Broadband Pulse Propagation in Waveguides 567
20.4.1 Integral Representation of the Transmitted Field 567
20.4.2 Broadband Pulse Propagation in a Homogeneous
Waveguide 569
20.4.3 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 571
20.5.2 Integral Representation of the Broadband Refocused
Field 573
20.5.3 Refocusing in a Homogeneous Waveguide 574
20.5.4 Refocusing in a Random Waveguide 575
20.6 Statistics of the Transmission Coefficients at Two Nearby
Frequencies 579
20.6.1 Transport Equations for the Autocorrelation Function
of the Transfer Matrix 579
20.6.2 Probabilistic Representation of the Transport Equations 582
20.7 Incoherent Wave Fluctuations in the Broadband Case 584
20.8 Narrowband Pulse Propagation in Waveguides 587
20.8.1 Narrowband Pulse Propagation in a Homogeneous
Waveguide 588
20.8.2 The Mean Field in a Random Waveguide 588
20.8.3 The Mean Intensity in a Random Waveguide 589
20.9 Time Reversal for a Narrowband Pulse 590
20.9.1 Refocusing in a Homogeneous Waveguide 591
20.9.2 The Mean Refocused Field in a Random Waveguide . 591
20.9.3 Statistical Stability of the Refocused Field 592
20.9.4 Numerical Illustration of Spatial Focusing and
Statistical Stability in Narrowband Time Reversal 594
References 599
Index 609 |
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| any_adam_object_boolean | 1 |
| building | Verbundindex |
| bvnumber | BV022524530 |
| classification_rvk | SK 820 UF 5100 SK 540 |
| classification_tum | MAT 000 |
| collection | ZDB-2-SMA |
| ctrlnum | (OCoLC)873429750 (DE-599)BVBBV022524530 |
| discipline | Physik Mathematik |
| discipline_str_mv | Physik Mathematik |
| doi_str_mv | 10.1007/978-0-387-49808-9 |
| edition | 1. Ed. |
| format | Electronic eBook |
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| id | DE-604.BV022524530 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:04:27Z |
| indexdate | 2024-07-09T20:59:29Z |
| institution | BVB |
| isbn | 9780387498089 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015731223 |
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| physical | 1 Online-Ressource (XX, 612 S.) Ill. |
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| publishDate | 2007 |
| publishDateSearch | 2007 |
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| publisher | Springer New York Springer Bln |
| record_format | marc |
| series | Stochastic Modelling and Applied Probability |
| series2 | Stochastic Modelling and Applied Probability |
| spelling | Wave propagation and time reversal in randomly layered media Jean-Pierre Fouque ... 1. Ed. Berlin Springer New York 2007 Berlin Springer Bln 1 Online-Ressource (XX, 612 S.) Ill. txt rdacontent c rdamedia cr rdacarrier Stochastic Modelling and Applied Probability 56 Zeitumkehr (DE-588)4190632-9 gnd rswk-swf Wellenausbreitung (DE-588)4121912-0 gnd rswk-swf Geschichtetes Medium (DE-588)4232784-2 gnd rswk-swf Wellenausbreitung (DE-588)4121912-0 s Zeitumkehr (DE-588)4190632-9 s Geschichtetes Medium (DE-588)4232784-2 s DE-604 Fouque, Jean-Pierre Sonstige oth Erscheint auch als Druck-Ausgabe, Hardcover 978-0-387-30890-6 Stochastic Modelling and Applied Probability 56 (DE-604)BV035421331 56 https://doi.org/10.1007/978-0-387-49808-9 Verlag Volltext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015731223&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wave propagation and time reversal in randomly layered media Stochastic Modelling and Applied Probability Zeitumkehr (DE-588)4190632-9 gnd Wellenausbreitung (DE-588)4121912-0 gnd Geschichtetes Medium (DE-588)4232784-2 gnd |
| subject_GND | (DE-588)4190632-9 (DE-588)4121912-0 (DE-588)4232784-2 |
| title | Wave propagation and time reversal in randomly layered media |
| title_auth | Wave propagation and time reversal in randomly layered media |
| title_exact_search | Wave propagation and time reversal in randomly layered media |
| title_exact_search_txtP | Wave propagation and time reversal in randomly layered media |
| title_full | Wave propagation and time reversal in randomly layered media Jean-Pierre Fouque ... |
| title_fullStr | Wave propagation and time reversal in randomly layered media Jean-Pierre Fouque ... |
| title_full_unstemmed | Wave propagation and time reversal in randomly layered media Jean-Pierre Fouque ... |
| title_short | Wave propagation and time reversal in randomly layered media |
| title_sort | wave propagation and time reversal in randomly layered media |
| topic | Zeitumkehr (DE-588)4190632-9 gnd Wellenausbreitung (DE-588)4121912-0 gnd Geschichtetes Medium (DE-588)4232784-2 gnd |
| topic_facet | Zeitumkehr Wellenausbreitung Geschichtetes Medium |
| url | https://doi.org/10.1007/978-0-387-49808-9 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015731223&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV035421331 |
| work_keys_str_mv | AT fouquejeanpierre wavepropagationandtimereversalinrandomlylayeredmedia |