MULTISCALE GAUSSIAN-BEAM METHOD FOR HIGH-FREQUENCY

WAVE PROPAGATION AND INVERSE PROBLEMS

By

Chao Song

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Applied Mathematics — Doctor of Philosophy

2018

ABSTRACT

MULTISCALE GAUSSIAN-BEAM METHOD FOR HIGH-FREQUENCY

WAVE PROPAGATION AND INVERSE PROBLEMS

By

Chao Song

The existence of Gaussian beam solution to hyperbolic PDEs has been known to the pure

mathematics community since sometime in the 1960s [3]. It enjoys popularity afterwards

due to its ability to resolve the caustics problem and its efficiency [49, 28, 31]. In this thesis,

we will focus on the extension of the multi-scale Gaussian beam method and its application

to seismic wave modeling and inversion.

In the first part of thesis, we discuss the application of the multi-scale Gaussian beam

method to the inverse problem. A new multi-scale Gaussian beam method is introduced for

carrying out true-amplitude prestack migration of acoustic waves. After applying the Born

approximation, the migration process is considered as shooting two beams simultaneously

from the subsurface point which we want to image. The Multi-scale Gaussian Wavepacket

transform provides an efficient and accurate way for both decomposing the perturbation

field and initializing Gaussian beam solution. Moreover, we can prescribe both the region

of imaging and the range of dipping angles by shooting beams from a subsurface point in

the region of imaging. We prove the imaging condition equation rigorously and conduct

error analysis. Some numerical approximations are derived to improve the efficiency further.

Numerical results in the two-dimensional space demonstrate the performance of the proposed

migration algorithm.

In the second part of thesis, we propose a new multiscale Gaussian beam method with

reinitialization to solve the elastic wave equation in the high frequency regime with different

boundary conditions. A novel multiscale transform is proposed to decompose any arbitrary

vector-valued function to multiple Gaussian wavepackets with various resolution. After the

step of initializing, we derive various rules corresponding to different types of reflection cases.

To improve the efficiency and accuracy, we develop a new reinitialization strategy based on

the stationary phase approximation method to sharpen each single beam ansatz. This is

especially useful and necessary in some reflection cases. Numerical examples with various

parameters demonstrate the correctness and robustness of the whole method. There are two

boundary conditions considered here, the periodic and the Dirichlet boundary condition.

In the end, we show that the convergence rate of the proposed multiscale Gaussian beam

method follows the convergence rate of the classical Gaussian beam solution, i.e. O(cid:0) 1√

(cid:1).

ω

To my family

iv

ACKNOWLEDGMENTS

Firstly, I would like to express my sincere gratitude to my advisor Prof. Qian, Jianliang,

for his continuous support of my Ph.D. study and related research, for his patience, moti-

vation, and immense knowledge. His guidance helped me in all the time of research and

writing of this thesis. I could not have imagined having a better advisor for my Ph.D. study.

I would like to express my gratitude to Prof. Zhou, Zhengfang, Prof. Cheng, Yingda and

Prof. Tang, Moxun for serving on my thesis committee and critical comments and valuable

feedbacks. I also would like to thank all the professors whom I took classes from at MSU,

for their uniform excellence in teaching.

I would like to thank the support staff of Department of Mathematics for all their help

with administrative matters. I would like to acknowledge my dear friends Ruochuan Zhang,

Jun Du, Wangtao Lu and Wenbin Li for their friendship over the past six years. They made

the experience of graduate school more rewarding.

Many thanks go to my parents for supporting me spiritually throughout writing this thesis

and my life in general. Last but not the least, I would like to express my profound gratitude

to my wife Yu, who has shown me the deepest love, kindness, and patience throughout

our time together. This dissertation stands as a testament to your unconditional love and

encouragement.

v

TABLE OF CONTENTS

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

LIST OF ALGORITHMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Prestack Inversion Process . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Elastic Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
3
5
7
10

Chapter 2

Single Gaussian beam ansatz and Multiscale Gaussian wavepacket
Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Single Gaussian beam ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Multiscale Gaussian Wavepacket Transform . . . . . . . . . . . . . . . . . .
2.2.1 Multiscale Gaussian Beam Method . . . . . . . . . . . . . . . . . . .

12
12
16
19

Migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Modified Multiscale Gaussian Wavepacket Transform . . . . . . . . . . . . .
3.2 Multiscale Gaussian Wavepacket Inversion . . . . . . . . . . . . . . . . . . .
3.2.1
Setup of the True Amplitude Migration Problem . . . . . . . . . . . .
3.2.2 Born Approximation for the Trace Data . . . . . . . . . . . . . . . .
3.2.3 Multiscale Gaussian Beam Approximation of the Green’s Function . .
3.2.4 True-Amplitude Migration Process
. . . . . . . . . . . . . . . . . . .
3.2.5 Motivation for Inverting the multiscale Gaussian-beam transform . .
3.3 Theoretical Validation: The Proof about The Imaging Operator . . . . . . .
3.3.1 Road Map of The Theoretical Analysis . . . . . . . . . . . . . . . . .
3.3.2 Approximation of Gaussian Beams along the Surface . . . . . . . . .
3.3.3 Asymptotic Analysis of Two Beams’ Interaction . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . .

Chapter 3 Multiscale Gaussian-Beam Transforms for True Amplitude Prestack
21
21
29
29
31
33
34
37
39
39
42
47
3.3.3.1 Parabolic Scaling Principle
48
3.3.3.2 Difference between Two Interacted Beams’ Traveltime
51
3.3.3.3 Difference between Two Interacted Beams’ Phase and Hessians 55
58
3.3.3.4 Approximation of Two Beams’ Interaction . . . . . . . . . .
3.3.3.5
Integral about Boundary Points r and s . . . . . . . . . . .
61
63
3.3.3.6 Conclusion of Two Beams’ Interaction . . . . . . . . . . . .
66
3.3.4 Asymptotic Analysis of Four Beams’ Interaction . . . . . . . . . . . .
Integral about Wavenumber ω . . . . . . . . . . . . . . . . .
66
Integral about Momentum ξ and η: Evaluation of Real Part
of Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3.4.1
3.3.4.2

68

vi

3.3.4.3

3.3.4.4

Integral about Momentum ξ and η: Evaluation of Imaginary
Part of Phase . . . . . . . . . . . . . . . . . . . . . . . . . .
Integral about Momentum ξ and η . . . . . . . . . . . . . .
Implementation of the Prestack Imaging Operator . . . . . . . . . . . . . . .
3.4
3.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Approximation of Beams along the surface . . . . . . . . . . . . . . .
3.5.2 The Correctness of Prestack Imaging Operator . . . . . . . . . . . . .
3.5.3
Single Source Migration Test . . . . . . . . . . . . . . . . . . . . . . .
3.5.3.1 Example 1: Constant Background Slowness
. . . . . . . . .
3.5.3.2 Example 2: Multiple Flat Reflectors
. . . . . . . . . . . . .
3.5.3.3 Example 3: Linear Background Slowness . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .
3.5.4.1 Example 4: Constant Slowness with Dipped Layer
. . . . .
3.5.4.2 Example 5: Flat Layer in Lateral Background Velocity . . .
3.5.4.3 Example 6: Slowness with Caustics
. . . . . . . . . . . . .
3.5.4.4 Example 7: Polluted Trace Data . . . . . . . . . . . . . . .

3.5.4 Multiple Source Migration Test

71
76
81
85
85
86
87
87
88
89
89
89
91
92
92

Chapter 4 Fast Multiscale Gaussian Beam Method for Elastic Wave Equa-
95
tions in Bounded Domains . . . . . . . . . . . . . . . . . . . . . .
95
4.1 Asymptotic Method for the Elastic Wave equation . . . . . . . . . . . . . . .
96
4.2 The Asymptotic Ansatz Solution to the Elastic Wave . . . . . . . . . . . . .
98
4.2.1 P-wave and S-wave’s Eikonal Equations . . . . . . . . . . . . . . . . .
4.2.2 Transport Equation Governing P-wave’s Amplitude Vectors
. . . . . 101
4.2.3 Transport Equation Governing S-wave’s Amplitude Vectors . . . . . . 103
4.2.4
Single Beam Solution for P and S-wave . . . . . . . . . . . . . . . . . 105
. . . . . . . . 107
4.3.1 Multiscale Gaussian Wavepacket Transform: Vector Functions . . . . 107
4.3.1.1 Decomposition of the Single Wavepacket . . . . . . . . . . . 108
4.3.1.2 Preprocessing the Initial Condition . . . . . . . . . . . . . . 109
4.4 Multiscale Gaussian Beam Method for Periodic Boundary Value Problem . . 111
4.5 Multiscale Gaussian beam method for Homogeneous Dirichlet Boundary Con-

4.3 Multiscale Gaussian Wavepacket Transform for Elastic Waves

dition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.5.1 P-wave Reflecting Beams: Ray Direction . . . . . . . . . . . . . . . . 112
4.5.2 P-wave Reflecting Beams: The Hessian of the Phase . . . . . . . . . . 114
4.5.3 P-wave Reflecting Beams: Amplitude Vector . . . . . . . . . . . . . . 116
4.5.4
S-wave Reflecting Beams: the Phase term . . . . . . . . . . . . . . . 117
S-wave Reflecting Beams: the Amplitude Vector . . . . . . . . . . . . 119
4.5.5
4.5.6 Method of Images for Boundary Conditions
. . . . . . . . . . . . . . 120
4.6 Stationary Phase Analysis of Beams . . . . . . . . . . . . . . . . . . . . . . . 122
Stationary Phase Approximation with Respect to Spatial Variables
. 123
Stationary Phase Approximation with Respect to Momentum Variables 126
4.7 Sharpening Beams by Reinitialization . . . . . . . . . . . . . . . . . . . . . . 129
4.7.1 The First Motivation for Developing a New Reinitialization Strategy
129
4.7.2 The Second Motivation for Developing a New Reinitialization Strategy 130
4.7.3
. . . . . . . . . . 133

Sharpened Wavepackets and Convergence Analysis

4.6.1
4.6.2

vii

4.8 Numerical Examples

4.8.1.1
4.8.1.2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.8.1 Beam Reinitialization . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
SP Reflection V.S. PS Reflection . . . . . . . . . . . . . . . 139
Sharpened Beams V.S. Original Beams . . . . . . . . . . . . 141
4.8.2 Periodic Boundary Condition . . . . . . . . . . . . . . . . . . . . . . 142
4.8.2.1 Example 1: The Single Wavepacket in the Constant velocity 142
4.8.2.2 Example 2: General Initial Condition in the Constant velocity142
4.8.3 Reflection: P-wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.8.3.1 Example 3: P-wave Reflection in the Constant velocity . . . 145
4.8.3.2 Example 4: P-wave Reflection in the Linear velocity . . . . 146
4.8.3.3 Example 5: P-wave Reflection in the Sinusoidal velocity . . 146
4.8.4 Reflection S-wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.8.4.1 Example 6: S-wave Reflection with Orthogonal Hitting Angle 147
4.8.4.2 Example 7: S-wave Reflection with Non-Orthogonal Hitting

Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.8.4.3 Example 8: S-wave Reflection: Linear velocity . . . . . . . . 152
4.8.4.4 Example 9: S-wave Reflection: Sinusoidal velocity . . . . . . 153
4.8.5 General Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . 154

4.8.5.1 Example 10: The General Initial Condition in The Sinusoidal

4.8.5.2 Convergence Rate Analysis

velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
. . . . . . . . . . . . . . . . . . 157

Chapter 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Appendix A Proof in Inverse Process . . . . . . . . . . . . . . . . . . . . . . . . . 162
Appendix B Proof in Elastic Wave
. . . . . . . . . . . . . . . . . . . . . . . . . . 193

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

viii

LIST OF FIGURES

Figure 1.1: Caustics Appearance in Gaussian Velocity . . . . . . . . . . . . . . . . .

2

Figure 3.1: (cid:80)

l,i g2

l,i in different level l

. . . . . . . . . . . . . . . . . . . . . . . . . .

29

Figure 3.2: A typical source gather in Gaussian slowness . . . . . . . . . . . . . . . .

30

Figure 3.3: Real Part of Single Wavepacket φl,i,k . . . . . . . . . . . . . . . . . . . .

49

Figure 3.4: Numerical test for fast approximation of Gaussian beam along the surface.
Left: ω = -110 Right: ω = -100. . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.5: Constant Slowness: Theorem 3.3.1. Left: dx1 = dx2 Right: dx1 = 2dx2.

Figure 3.6: General Speed: Theorem 3.3.1. Left: dx1 = dx2 Right: dx1 = 2dx2.

. .

Figure 3.7: Example 1: Constant Slowness with Dipped layer. Left: True Slowness
. . . . . . . . . . . . .

Right: Migration Result with single source trace.

85

86

87

88

Figure 3.8: Example 2: True Slowness Model with Multiple Layers . . . . . . . . .

88

Figure 3.9: Example 2: Constant Slowness with Multiple Layers. Left: Migration
Result over the Whole Space Right: Migration Result V.S. True Value at
x1 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.10: Example 3: Gradient Slowness Model. Left: True Slowness, Right:
Smoothed Macro Slowness . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.11: Example 3: Gradient Slowness Model. Left: Migration Result over the
. . . .

Whole Space, Right: Migration Result V.S. True Value at x1 = 0.

89

90

90

Figure 3.12: Example 4: Constant Slowness with the Dipped Layer (Multiple Sources) 91

Figure 3.13: Example 5: Flat Layer in Lateral Background Velocity . . . . . . . . .

91

Figure 3.14: Example 6: Gaussian Slowness and its ray tracing. Left: Gaussian
. . . . . . . . . . . . .

Slowness with Flat Reflector Right: Ray Tracing

Figure 3.15: Example 6: Migration Result in the Gaussian Slowness with Caustics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(Multiple Sources)

92

93

ix

Figure 3.16: Example 7: True Trace V.S. Trace with Gaussian Error . . . . . . . . .

93

Figure 3.17: Example 7: Gaussian Slowness with polluted trace. Left: Migration

result from Non-polluted Data; Right: Migration result from Polluted data 94

Figure 3.18: Example 7: Gaussian Slowness with polluted trace.Two Migration Re-
sults at x2 = 0.65 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

Figure 4.1: Periodic Boundary Problem: The Case Wavepacket leaving the Boundary. 111

Figure 4.2: Partially Reflected beams

. . . . . . . . . . . . . . . . . . . . . . . . . . 121

Figure 4.3: Partially Reflected Beams with Odd Extension . . . . . . . . . . . . . . . 121

Figure 4.4: PS Reflection V.S. SP Reflection: Different Ratio Behaviors. PS Reflec-

tion(’o’), SP Reflection (’-’)

. . . . . . . . . . . . . . . . . . . . . . . . . 140

Figure 4.5: Sharpened Beams V.S. the Original Beam on Fixed (y, z). Left: y=0.1,
z=0.1 Right: y=0.08, z=0.13. Beam Solution after Reinitialization(’o’),
Original Beam Solution (’-’).

. . . . . . . . . . . . . . . . . . . . . . . . 141

Figure 4.6: Example 1: Single Wavepacket Propagation with the Periodic Boundary

Condition. Left: FDTD Solution Right: Gaussian Beam Solution.

. . 143

Figure 4.7: Example 2: General Initial Value Propagation with Periodic Boundary

Condition along x-axis FDTD Solution(’o’), Gaussian Beam Solution (’-’) 144

Figure 4.8: Example 2: General Initial Value Propagation with Periodic Boundary

Condition. Left: FDTD Solution Right: Gaussian Beam Solution.

. . 144

Figure 4.9: Example 3: P-wave Reflection in Constant velocity with Dirichlet Bound-

ary Condition. Left: FDTD Solution Right: Gaussian Beam Solution.

146

Figure 4.10: Example 4: P-wave Reflection in the Linear velocity with Dirichlet

Boundary Condition. FDTD Solution (’o’), Beam Solution(’-’) . . . . . . 147

Figure 4.11: Example 4: P-wave Reflection in Linear velocity with Dirichlet Bound-

ary Condition. Left: FDTD Solution Right: Gaussian Beam Solution.

148

Figure 4.12: Example 5: P-wave Reflection in Sinusoidal velocity along x-axis. FDTD

Solution(’o’), Beam Solution (’-’) . . . . . . . . . . . . . . . . . . . . . . 148

Figure 4.13: Example 5:P-wave Reflection in Sinusoidal velocity with Dirichlet Bound-

ary Condition. Left: FDTD Solution, Right: Gaussian beam Solution . 149

x

Figure 4.14: Example 6: S-wave Reflection with Orthogonal hitting Angle along

y-axis FDTD Solution (’o’), GB Solution (’-’).

. . . . . . . . . . . . . . 150

Figure 4.15: Example 6: S-wave Reflection with Orthogonal hitting Angle along

x-axis. FDTD Solution (’o’), GB Solution (’-’).

. . . . . . . . . . . . . . 150

Figure 4.16: Example 7: S-wave Reflection with Non-orthogonal Hitting Angle. Left:
Gaussian Beam Solution without reinitialization, Right: Gaussian Beam
Solution with reinitialization. FDTD Solution (’o’), GB Solution (’-’) . . 151

Figure 4.17: Example 8: S-wave Reflection in Linear velocity without Reinitializa-
tion. Left: FDTD Solution Right: Gaussian Beam Solution without
Reinitialization.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Figure 4.18: Example 8: S-wave Reflection in Linear velocity with Reinitialization.
Left: FDTD Solution Right: Gaussian Beam Solution with Reinitializa-
tion.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Figure 4.19: Example 9: S-wave Reflection in Sinusoidal velocity. GB Solution(’o’),

FDTD Solution(’-’) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Figure 4.20: Example 9: S-wave Reflection in Sinusoidal velocity with Reinitializa-
tion. Left: FDTD Solution Right: Gaussian Beam Solution with Reini-
tialization.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Figure 4.21: Example 10: General Initial Condition Propagation with Reinitializa-
tion (First Component). Left: FDTD Solution Right: Gaussian Beam
Solution with Reinitialization.

. . . . . . . . . . . . . . . . . . . . . . . 156

Figure 4.22: Example 10: General Initial Condition Propagation with Reinitialization
(Second Component). Left: FDTD Solution Right: Gaussian Beam
Solution with Reinitialization.

. . . . . . . . . . . . . . . . . . . . . . . 156

Figure 4.23: Example 10: General Initial Condition Propagation with Reinitialization

(Second Component). FDTD Solution (’o’), GB Solution(’-’).

. . . . . . 157

Figure 4.24: Log-log plot: Convergence Rate of the New Gaussian Beam

Method GB Method Error Curve(’-*’), Line with the slope = 1

2 log(1.5)(’-’)158

Figure B.1: FDTD solution justify. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Figure B.2: Convergence Rate of FDTD algorithm . . . . . . . . . . . . . . . . . . . 200

xi

LIST OF ALGORITHMS

Algorithm 1 Discrete Gaussian Wavepacket Decomposition. . . . . . . . . . . . . . . 18

Algorithm 2 Multiscale Gaussian Beam True-Amplitude Migration . . . . . . . . . . 37

Algorithm 3 Discrete Vector-Valued Gaussian Wavepacket Transform. . . . . . . . 110

xii

Chapter 1

Introduction

Hyperbolic PDEs arise in a variety of practical applications, ranging from acoustics,

elasticity, electromagnetics to geophysics. Therefore, it is desirable to develop fast

and efficient algorithms to solve this family of PDEs. Moreover, efficient algorithms

are also desirable in a lot of inverse problems modeled by hyperbolic PDEs. The

term ’high frequency’ in the high frequency wave propagation is defined relative to the

low-frequency background slowness in the model. Therefore, all medium appeared in

this thesis is smooth, if not specified.

It is very costly for the direct method, the finite difference method or the finite element

method for example, to simulate the high frequency wave propagation, since fine grid

mesh is required by these methods to capture the oscillation. Therefore, some alterna-

tive methods have been developed, such as the traditional geometrical-optics method

(WKBJ ansatz), which is required to solve a pair of equations. The first one, which

is called eikonal equation, is a first-order nonlinear PDE. The second one is the trans-

port equation relying on the differentiability of the result of eikonal equation. It yields

faithful asymptotic solutions before caustics occur. However, the amplitude function

governed by the transport equation breaks down around the caustics [6, 22, 36], where

the phase function is multi-valued [36, 39]. The appearance of caustics is inevitable

1

even in the smooth medium [54].

Figure 1.1: Caustics Appearance in Gaussian Velocity

One of the alternatives is the Gaussian beam method [49, 51, 55, 15]. To resolve the

caustics problems automatically, the Gaussian beam method relaxes the restriction

that the phase term is real-valued. The single Gaussian beam ansatz is made up of

a complex-valued phase function and a complex-valued amplitude function near its

central ray. Away from the ray path, the beam decays rapidly as a Gaussian profile.

The superposition of many single beams will be an asymptotically correct solution of

the hyperbolic PDE in the sense that both the initial condition and the boundary

condition are satisfied asymptotically as well as the PDE itself.

There are two methods based on Gaussian beam method presented in this thesis to

resolve two related problems. The first one is about the simulation of the elastic wave

in a bounded domain. The second one is using a novel Gaussian beam method to

reconstruct reflectors under the surface by the data received along the boundary.

2

−0.4−0.200.200.20.40.60.8x1x21.1 Prestack Inversion Process

In the first part, we propose a novel inversion and imaging procedure for finding the

parameter of the medium by solving the linearized inverse scatter problem. There

is a wide range of applications including seismic exploration, medical imaging and

underwater acoustics, etc.

There are many different migration methods existed to explore the geological structure.

The very early one was using the one-way wave equation [16, 17] to recover the accurate

travel time and locations of reflectors. However, the popular finite difference and finite

element method require extremely refined grid mesh in the high frequency regime to

prevent dispersion error. This is also found in the modeling process.

Kirchhoff method [7, 8, 28], which bases on the asymptotic method, is mostly investi-

gated and widely applied to resolve the high frequency pattern. The Kirchhoff method

and all other similar ray-based methods are using the asymptotic approximation (high

frequency) of the Green’s function for the acoustic wave equation.

G(x, x0, ω) = Aeiωτ (x,x0),

(1.1)

where τ (x, x0) is the traveltime from the point x0 to the point x.

There exist several other problems in the Kirchhoff migration, although it is efficient

and flexible. The first obstacle is the presence of caustics points.

It is not able to

characterize the structure in the presence of caustics, since they rely on the first-arrival

traveltime [18, 27] instead of multivalued traveltime. Its usefulness was questioned by

3

[25, 27, 43] since the first-arrival traveltime in complex media usually do not correspond

to the most energetic traveltimes crucial for imaging complex structures.

To overcome this issue, we use the Gaussian beam migration method [3, 45, 14].

The first obstacle is the way to initialize each beam solution efficiently, in other words,

how to characterize the wave propagator by beam solution in a sparse form. Another

closely related issue is the way to describe the wavefront of the perturbation. The

wavefront defines the singularities of a function not only in spatial space but also with

respect to its Fourier transform at each point. It is naturally adaptable to the high

frequency wave propagation which can be considered as the propagation of singular-

ity. From this point of view, the way to generate the decomposition should consider

both the optimal representation of the wave propagator and the sufficient condition

to reconstruct the target perturbation media. In some sense, some tradeoff should be

obtained.

The second obstacle is about the rigorous mathematical analysis of the imaging con-

dition. In this paper, the spread loss has to be proved to be compensated for in our

√
ω)
new imaging method. It is well known that the Gaussian beam solution has O(1/

convergence rate as an asymptotic solution. We will take advantage of this convergence

rate and the fact that the parabolic scaling principle is preserved along the propagation

to conduct a rigorous error analysis. Besides, we assume that a set of geodesics (rays)

have a consistent direction. The assumption simply means that the overturn rays don’t

exist in our model and this assumption is natural in the practical applications.

Our method enjoys several advantages. First, the flexibility of the Kirchhoff migration

is preserved and imaging without losing multi-arrival in the general slowness is possible.

4

Second, due to the Multi-scale Gaussian wavepacket transform and the parabolic scal-

ing principle employed, the sparse representation of the wave propagator is obtained.

This decomposition also makes the reinitialization feasible during the propagation so

that we have more control over the width of beam solution. Third, there is some re-

dundancy in the data in the view of our imaging condition. In practice, the error is

unavoidable, then this redundancy can help us to reduce the impact brought by the

noise since the average value is employed to cancel the variation caused. Fourth, our

imaging condition is performed in the time domain to avoid the extra Fourier trans-

form on the data set. This feature makes our algorithm more applicable considering the

large size of the trace dataset in the real world. The last feature is that the wavefront

set being characterized by the Gaussian wavepacket enables us to image the subsurface

partially in the sense of controlling the range of dipping angles.

There are some other true amplitude migration methods different from the ray-based

method described above. Zhang et al. [56, 57] developed true-amplitude common-shot

migration for heterogeneous media. Again, only the geometric spreading loss is re-

covered in these works. Other types of compensations, for example transmission loss

compensation, are discussed in [21].

1.2 Elastic Wave Equation

The elastic wave equation is a good model to describe the seismic wave propagation

in a uniform whole space and it has been used widely in the seismology community in

both inverse and modeling problem [1]. Similar to its simpler form, i.e. the acoustic

5

wave equation, the elastic wave equation will propagate oscillations in space and time

when the initial or boundary condition contains oscillation of the small wavelength.

We will develop a multi-scale Gaussian beam method to simulate the elastic wave in a

bounded domain.

The first problem to apply the Gaussian Beam method is how to decompose any

general initial condition to the form suitable to the beam profile.

It is resolved by

the Multiscale Gaussian Wavepacket transform developed in [48, 5] for the acoustic

wave equation and single-scale transform for the Schrodinger equation [47]. Since the

Hamiltonian of the wave equation is homogeneous of order one, a Gaussian beam

should satisfy the parabolic scaling principle at any given time if it is satisfied at the

first place. The wavepacket satisfying the parabolic scaling principle is defined as the

wavelength of the typical oscillation of this wavepacket being equal to the square of

the width of this wavepacket.

To propagate each wavepacket, the dynamics system can be obtained in the typical

way. In this paper, we extend this idea further to the decomposition of the vector-

valued initial value and preserve all the optimal properties of the Multiscale Gaussian

Wavepacket transform.

The second problem is to derive the reflection dynamics. Unlike the Cauchy problem,

we have to consider the reflected beams. Most recently in [4], a numerical method

has been proposed for the acoustic wave equation. Other discussions can be found in

[9, 49]. There will be more complex situations concerned in the elastic wave model

as there are two different types of wave modes in the process. The difference between

different wave modes requires extra efforts to preserve the accuracy.

6

The third problem is how to reinitialize Gaussian beams efficiently. Gaussian beam

solutions behave well around the caustics, however, not so well in the long-term wave

propagation. See more details illustrated in [48, 32, 42].

Therefore, there have been various methods developed to control the width of beams.

One method, for example, is to call the Multiscale Gaussian wavepacket algorithm

repeatedly during the propagation [48, 4]. A global time T is set at prior such that

beams will be summarized after propagating for T and then decompose the resulting

temporary wavefield to a new system of wavepackets. This process will be repeated

several times.

A new reinitialization method is proposed in this thesis which claims to have more

freedom and remains to be the asymptotic solution to the elastic wave equation. Instead

of decomposing the general wavefield after summation each time, we target on each

single beam in this new reinitialization method. It will give us more freedom to choose

which beams needed to be reinitialized rather than all of them. Moreover, with applying

the reinitialization strategy to each single beam, there exists the explicit expression.

1.3 Related Work

The fact that the Gaussian beam ansatz can be used to solve the wave equation has

been known to the pure mathematical community since sometime in the 1960s [3].

Then it is applied to simulate the propagation of the singularity [49, 30]. A single

Gaussian beam ansatz is an asymptotic solution concentrated on a single ray curve.

The critical point is to have a global solution to the Hessian of the phase function so

7

that the transport equation is well-defined. Moreover, the fact that imaginary part of

the Hessian remains to be S.P.D. leads to a well-localized solution. The localization is

justified theoretically [49, 40, 52].

Considering its ability to resolve multi-valued phase function automatically, the Gaus-

sian beam method was firstly introduced as a seismic imaging method by Hill in the

form of the poststack [28] and then the prestack migration procedure [29]. The perfor-

mance of the Gaussian beam migration is further tested by the common shot geometry

[26]. Most recently, a purely Eulerian computational approach was proposed in [33]

which improves the numerical method’s efficiency and its application in the semiclas-

sical quantum mechanics has been proposed in [31]. See [9, 42] for other recent works.

Besides the Gaussian beam method, there are several possible ways to construct global

asymptotic solutions for the wave equation even in the presence of caustics. The first

approach is based on Ludwigs uniform asymptotic expansions at caustics [35, 10] which

requires that the caustic structure is given. The second approach is is based on the

Maslov canonical operator theory [41], which requires to identify where the caustics

are at prior.

There is some recent advance in resolving the multi-valued traveltime problem. A

new method called fast Huygens sweeping method has been proposed in [37, 38] to

solve the Helmholtz equations in the inhomogeneous media and then it is used to solve

Schrodinger equation [34]. They take advantage of the fact that eikonal equation is

well-defined around the source point and Huygens-Kirchhoff secondary source principle.

There have been some recent advance in the optimal representation of the wave propa-

8

gator [52, 11, 50]. It is closely related to the Fourier integral operator representation of

the hyperbolic system and the special proposition of its phase function. The multiscale

Gaussian Wavepacket transform is developed [48] for the wave equation and the single

scale transform for the Schrodinger equation [47] which is also the fundamental basis

of our algorithm. This difference comes from the different Hamiltonian for these two

equations. Other paper also apply the Gaussian beam method in their true ampli-

tude migration [46, 2], however, they do not require the parabolic scaling principle for

Gaussian beams as our prestack inversion method does.

The parabolic scaling principle provides the theoretical basis for our new reinitialization

method as well as the proof of the correctness of the imaging operator in our new

multi-scale inversion algorithm. Other methods using the similar idea can be found in

[11, 50]. There are various types of such wavepackets, curvelet [13, 12] and wave atoms

[19, 20] for example. However, the Multi-scale Gaussian beam method is different

from these methods in that the single Gaussian wavepacket corresponds to the single

Gaussian beam at final time T , while the curvelet frame does not have this one-to-one

relationship.

On the other hand, the Gaussian wavepacket transform has been proved a stable and

efficient decomposition of the arbitrary function [48], equivalently, the wavepacket is a

good characterization of the wavefront set. The Gaussian window function or Gabor

frame [24] are both well-localized in the phase space as the Fourier transform of a

Gaussian profile function is again a Gaussian profile function. The size of the Gaussian

window function in the phase space can be determined by the Heisenberg Uncertainty

Principle.

9

1.4 Contents

The remainder of this thesis is organized as follows. In Chapter 2 we present a brief in-

troduction of constructing and propagating a single Gaussian beam, which is the foun-

dation to the following derivation. We then describe the original multi-scale Gaussian

wavepacket transform [48] in Section 2.2.

In Chapter 3, we propose a new prestack inversion process based on the multi-scale

Gaussian beam method. We then modify the Gaussian wavepacket transform in Section

3.1 to adapt to the imaging operator. We then develop the new imaging operator with

the help of Gaussian wavepacket transform and the Gaussian beam functions. Based

on this operator, we propose the main inversion algorithm in Section 3.2.4. The next

part is devoted to proving the correctness of this new algorithm in Section 3.3.

In

Section 3.4, we discuss the fast method to calculate the imaging operator. In the last

section of this chapter, we select several well-designed numerical examples to justify

the correctness of our analysis and the approximations mentioned earlier.

In Chapter 4 we propose the Multiscale Gaussian beam method to solve the elastic

wave equation with highly oscillated initial condition. We first extend the Multiscale

Gaussian wavepacket transform to the vector-valued initial condition in Section 4.3

and develop a new propagating dynamics for each single beam. After proposing the

decomposition scheme, the reflection dynamics for the homogeneous Dirichlet Bound-

ary value is derived in Section 4.5. The difference among various types of reflection

is analyzed in Section 4.7.2 and a new efficient reinitialization method is proposed

to resolve the problem from S-wave reflection in Section 4.6 and Section 4.7.3. The

reference solution to the elastic wave equation in general case is provided by the Finite-

10

Difference Time-Domain (FDTD) with staggered grid [53] and is justified in Section B.

In Section 4.8, several numerical experiments are conducted to show the correctness

and the convergence rate of our new multi-scale Gaussian beam method.

11

Chapter 2

Single Gaussian beam ansatz and

Multiscale Gaussian wavepacket

Transform

2.1 Single Gaussian beam ansatz

The Gaussian beam solution itself is an asymptotic solution of the acoustic wave equa-

tion even around caustics.

1

c2(x)

t u(x, t) − ∆u(x, t) = 0,
∂2

(2.1)

where x is the point coordinate in the space Rd and c(x) is smooth, positive and

bounded away from zero. Similar to the Geometric-Optics ansatz, the Gaussian beam

also assumes that the solution follows the form,

u(x, t) = A(x, t)eiωτ (x,t),

(2.2)

12

where ω is a large wavenumber, τ (x, t) is the phase function and A(x, t) is the amplitude

function. The asymptotic solutions means u(x, t) (2.2) satisfies the wave equation (2.1)

with small error when frequency ω is large. After inserting equation (2.2) into equation

(2.1) and organize all terms according to the order of ω, there will be two equations

obtained, which are eikonal and transport equations, governing τ (x, t) and A(x, t)

respectively. They come from the leading orders in inverse power of the frequency ω.

t (x, t) − c2|∇τ (x, t)|2 = 0
τ 2
2Atτt − 2c2∇A · ∇τ + A(τtt − c2trace(τxx)) = 0.

Phase function τ :

After factoring out equation (2.3), there are two branches generated,

τt ± c(x)|∇τ| = 0.

(2.3)

(2.4)

(2.5)

Equation (2.6) is a Hamilton-Jacobi equation with the Hamiltonian G±(x, p) = ±c(x)|p|.

We consider the generic situation for the eikonal equation,

τt + G(x,∇τ (x, t)) = 0.

(2.6)

13

We apply the method of the characteristics to solve the eikonal equation (2.6).

dx
dt
dp
dt

x

= Gp(x(t), p(t)),
= −Gx(x(t), p(t)),

= p0.

(2.7)

(cid:12)(cid:12)(cid:12)t=0
(cid:12)(cid:12)(cid:12)t=0

p

= x0;

where we define the ray trajectory γ = {(x(t), p(t)) : t ≥ 0}, whose initial point is
(x0, p0) in the phase space. We have that the momentum p(t) = ∇τ (x(t), t) along the

ray.

To derive the dynamics of Hessian matrices, we first differentiate the eikonal equation

(2.6) with respect to t and x:

τtx(x, t) + Gx(x,∇τ (x, t)) + τxx(x, t)Gp(x(t),∇τ (x, t)) = 0,
τtt(x, t) + Gp(x(t),∇τ (x, t)) · τxt(x, t) = 0,

(2.8)

(2.9)

Differentiating equation (4.13) with respect to x yields

τtxx + Gxx + τxxGxp + (Gxp)T τxx + τxxGppτxx + τxxxGp = 0.

(2.10)

Therefore, the Hessian M (t) = ∇∇τ (x(t), t) satisfies the following Riccati equation,

dM
dt

+ Gxx + M Gxp + GT

xpM + M GppM = 0. M

= i
I

(2.11)

(cid:12)(cid:12)(cid:12)t=0

The size parameter 
 will be given after introducing the Gaussian wavepacket trans-

form.

14

One of the most significant differences between the Gaussian beam and other ray-ansatz

methods is that beams’ phase functions τ (x, t) are complex-valued. Complexifying the

equation guarantees a well-defined Hessian and a well-defined transport equation as

a result. This is not true in general case [49]. Furthermore, the positive definite

imaginary part is always true throughout the propagation for smooth ray trajectories.

Lemma 2.1.1. If the Hamiltonian G is smooth enough, then the Hessian M(t) along

the ray path γ has a positive-definite imaginary part, provided that it initially does.

Transport Equation A(x, t):

With (x(t), p(t), M (t)) well-defined along the way, we can solve the transport equation

(2.4). Taking advantage of the fact that

dA(x(t), t)

dt

= At + ∇A · Gp(x(t), p(t)),

equation (2.4) is reduced to

dA
dt

+

A
2G

(v2trace(M ) − Gx · Gp − G T

(cid:12)(cid:12)(cid:12)t=0

p MGp) = 0 . A

= A0

(2.12)

A single beam solution is in the following form,

x0
p0 (x, t) = A(x, t)eiωτ (x,t),
U

(2.13)

and the phase function is approximated by applying Taylor expansion around the

15

central trajectory {x(t) : t ≥ 0} at time t,

τ (x, t) = p(t) · (x − x(t)) +

(x − x(t))T M (t)(x − x(t)),

1
2

(2.14)

and the amplitude is approximated by its value at trajectory γ at the same moment,

A(x, t) = A(x(t), t) = A(t).

(2.15)

2.2 Multiscale Gaussian Wavepacket Transform

After constructing a single beam solution, the next problem is how to set up the

initial condition for the ODE system. The answer is Multiscale Gaussian wavepacket

transform, which will be introduced briefly in this section. More details about this

phase space decomposition can be found in paper [48] and its single scale application

can be found in paper [47]. The wavepacket transform is applied to L2 functions f in
the Rd space.

We first partition the Fourier space Rd into several Cartesian coronae Cl for l ≥ 1 as

Cl = {ξ = (ξ1, ξ2,··· , ξd) : max
1≤i≤d

|ξi| ∈ [4l−1, 4l]}.

Now it is obvious to see that the L2 norm of ξ in Cl is O(4l). For each Cl, we can

further partition it into multiple windows with width 2l,

Bl,i =

[2lis, 2l(is + 1)],

d(cid:89)

s=1

16

where the integer multiindex (i1, i2,··· , id) is any possible choice such that the box is
in the lth layer, i.e. Bl,i ⊂ Cl. After defining these cell boxes Bl,i, we can define the

Gaussian profile function gl,i associated with the box Bl,i by the following formula,

(cid:32)|ξ−ξl,i|

(cid:33)2

−
gl,i(ξ) ≈ e

σl

,

(2.16)

where ξl,i is the center of the box Bl,i and σl = 2l is the width of the box Bl,i.

The scale listed here is designed carefully following the parabolic-scaling principle.

This is the key to the success of our multi-scale imaging process as it provides the

theoretical justification of the size of each Gaussian wavepacket. The later proof and

error analysis will rely on this conclusion heavily.

To have a partition of unity, one also needs the conjugate filters hl,i, such that

(cid:80)

gl,i(ξ)
l,i g2

l,i(ξ)

hl,i(ξ) =

,

(2.17)

The proof that the functions hl,i are well defined and well-localized can be found in

paper [48]. It is easy to see that(cid:80)

l,i gl,ihl,i = 1. By shifting the central point,

ˆφl,i,k(ξ) =

ˆψl,i,k(ξ) =

−2πi

kξ
Ll gl,i(ξ),

−2πi

kξ
Ll hl,i(ξ).

e

e

1
d/2
L
l
1
d/2
L
l

17

Taking the inverse Fourier transforms yields their definitions in the spatial domain:

2π(x− k
Ll
e

)·ξ

gl,i(ξ)dξ

2π(x− k
Ll
e

)·ξ

hl,i(ξ)dξ

(2.18)

(2.19)

(cid:90)
(cid:90)

Rd

Rd

φl,i,k(x) =

ψl,i,k(x) =

1
d/2
L
l
1
d/2
L
l

(cid:18)(cid:114) π

(cid:19)d

σl

Ll

φl,i,k(x) ≈

The approximation expression of the wavepacket φl,i,k,

2πi(x− k
Ll
e

)ξl,ie

−σ2

l π2|x− k
Ll

|2

.

(2.20)

We list the lemma from paper [48] without proof to show that our decomposition is

correct.

Lemma 2.2.1. For any f ∈ L2(Rd), we have

(cid:88)

f (x) =

(cid:104)ψl,i,k, f(cid:105)φl,i,k(x).

(2.21)

l,i,k

The idea of decomposing discrete signals into wavepackets is very similar to the contin-

uous case, therefore, we skip this part and provide the pseudo code below. The total

Algorithm 1 Discrete Gaussian Wavepacket Decomposition
1. Apply the Fast Fourier Transform(FFT) to the discrete signal f . 2. Compute hl,i(ξ) ˆf (ξ).
3. Wrap the result to the domain [−2σl, 2σl]. 4. Apply the Inverse Fourier Transform to
obtain the coefficients cl,i,k.

cost of this algorithm is O(N d log(N )).

18

2.2.1 Multiscale Gaussian Beam Method

With the Mutiscale Gaussian Wavepacket transform defined above, the initial con-

dition (x0, p0, M0, A0) for a single beam solution will be defined corresponding to a

wavepacket φl,i,k,

= Gp(x(t), p(t)),

x0 =

k
Ll

;

= −Gx(x(t), p(t)),

p0 = 2π

ξl,i
|ξl,i|,

= −Gxx − M Gxp − GT
= − A
2G

(v2trace(M ) − Gx · Gp − G T

dx
dt
dp
dt

dM
dt
dA
dt

and

xpM − M GppM, M0 = i(2π2σ2

(cid:18)(cid:114) π

(cid:19)d
l /|ξl,i|)I,

Ll N

p MGp), A0 =

σl

.

(2.22)

u(x, t) = A(x(t), t)e

i|ξl,i|τ (x,t)

,

(2.23)

Then we can use the dynamic system (2.7)-(2.12) to propagate beams. The large
wavenumber |ξl,i| serves as the key point of asymptotic methods. However, through-
out the following derivation, we will combine this constant |ξl,i| into the phase for

convenience.

We argue that there’s no difference. Denote the beam solution using initial condition

(x0, p0, M0, A0) as (x(t), p(t), M (t), At) and the one using initial condition
(x0,|ξl,i|p0,|ξl,i|M0, A0) as (ˆx(t), ˆp(t), ˆM (t), ˆA(t)). With respect to the ray trajectory,

dx(t)

dt
d(|ξl,i|p(t))

dt

= v(x(t))

|ξl,i|p(t)
|ξl,i||p(t)|,
= ∇v(x(t))|ξl,i||p(t)|,

x0 =

k
Ll

|ξl,i|p0 = 2πξl,i

19

Since (x(t),|ξl,i|p(t)) is the solution of the same Hamiltonian system equipped with the

same initial condition as (ˆx(t), ˆp(t)), by the uniqueness of the initial value problem,
ˆp(t) = |ξl,i|p(t) and ˆx(t) = x(t).
In terms of Hessian M , again we multiply |ξl,i| on both sides of equation (2.11).
(cid:33)T

d(|ξl,i|M )

dt

= −∇∇v(x(t))|ξl,i||p(t)| − |ξl,i|M∇v

(cid:32) |ξl,i|p(t)

|ξl,i||p(t)|

M

(cid:32)

|ξl,i|p(t)
|ξl,i||p(t)| − |ξl,i|∇v
(cid:33)
(|ξl,i|M )

t

xp(ˆx(t), ˆp(t)) ˆM

v(x(t))

− (|ξl,i|M )

|ξl,i||p(t)| I − v(x(t))

|ξl,i||p(t)|3 p(t)pT
= −∇∇v(ˆx(t))|ˆp(t)| − ˆM Gxp(ˆx(t), ˆp(t)) − GT
− ˆM Gpp(ˆx(t), ˆp(t)) ˆM

Again, by the uniqueness of the initial value problem, ˆMt = |ξl,i|M (t). The derivative

term Gpp is given by

Proposition 2.2.1. The second order derivative of the Hamiltonian about the momen-

tum variable is

(cid:16)|p|2Id − ppT(cid:17)

,

G±
pp(x, p) = ±v(x)
|p|3

(2.24)

where Id is the identity matrix.

The proof is easy, so we omit it here.

d ˆA
dt

= −

ˆA

2|ξl,i|G(x(t), p(t))

+

ˆA

2|ξl,i|G(x(t), p(t))

(cid:16)|ξl,i|GT

(v2|ξl,i|M (t) − |ξl,i|Gx(x(t), p(t)) · Gp(x(t), p(t)))

(cid:17)

p (x(t), p(t))M (t)Gp(x(t), p(t))

.

(2.25)

Apparently, A(t) and ˆA(t) share the same ODE and initial condition.

20

Chapter 3

Multiscale Gaussian-Beam

Transforms for True Amplitude

Prestack Migration

3.1 Modified Multiscale Gaussian Wavepacket Trans-

form

We have so far finished introducing the Gaussian wavepacket transform in [48]. To meet
our inversion algorithm’s requirements, we have to substitute (cid:104)ψl,i,k, f(cid:105) in equation
(2.21) with (cid:104)φl,i,k, f(cid:105).

Several new notations are needed in the following arguments. We first define a set for
each frequency ξ as its cover set S(ξ),

S(ξ) = {(l, i) : gl,i(ξ) > 0}.

(3.1)

21

Bl is defined as the border of the partitioning,

Bl ≡
{ξ : max
1≤s≤d

|ξs| ∈ [4l−1, 4l]} ∩ {ξ : max
1≤s≤d

|ξs| − 4l−1 ≥ 2l} ∩ {ξ : 4l − max
1≤s≤d

|ξs| ≤ 2l}.

(3.2)

The remainder part Cl \ Bl is defined as the major part of Cl. Obviously, the border

part Bl is much smaller compared with the major part. We will focus on the major

part first.

All frequencies in the major part of Cl will not interact with those from different levels,
that is, the cover sets of all frequency ξ in the major part Cl \ Bl will only contain

the compact support functions gl,i from the same level. This allows us to prove the

following claims.

Proposition 3.1.1. If ξ ∈ Cl \ Bl, we have
(cid:88)

(cid:88)

(cid:88)

g2
(cid:48)
l

,i

(cid:48) (ξ) =

(cid:48)

l

(cid:48)

,i

g2
(cid:48)
l

(cid:48) (ξ) +
,i

g2
(cid:48)
l

(cid:48) (ξ) +
,i

(cid:48)

l

(cid:48)

=l,i

(cid:48)

l

(cid:48)

<l,i

(cid:88)

(cid:48)

l

(cid:48)

>l,i

g2
(cid:48)
l

(cid:48) (ξ) =
,i

(cid:88)

(cid:48)

l

(cid:48)

=l,i

g2
(cid:48)
l

,i

(cid:48) (ξ).

(3.3)

Proof. We assume that the concerned frequency ξ = (ξ1, ξ2,··· , ξd) satisfies,

ξ1 = max
1≤s≤d

|ξs|.

(3.4)

22

On the other hand, g
l

(cid:48)

(cid:48) is a compact support function in the box centered at ξ
l
,i

(cid:48)

(cid:48)

,i

with side length 2l

(cid:48)

+1. Since

(cid:48)

2l

+1 ≤ 2l,

(3.5)

(3.6)

(3.7)

(3.8)

We first check the first term in equation (3.3), where l

(cid:88)

(cid:48)

l

(cid:48)

<l,i

g2
(cid:48)
l

(cid:48) (ξ) = 0.
,i

(cid:48)

< l,

The nearest central frequency ξ
l

(cid:48) in the lower level l

(cid:48)

,i

(cid:48)

< l should have

||ξ − ξ

l

(cid:48)

,i

(cid:48)|| ≥ |ξ1 − ξ

| ≥ 2l,

(cid:48)

(cid:48)
,i

,1

l

the frequency ξ in the level l is at most on the edge of the box B
l

(cid:48)

(cid:48) .
i

We then check the third term in equation (3.3) where l

(cid:48)

> l. We denote,

(cid:48)

ξ
l

(cid:48)

,i

,s0

= max
1≤s≤d

|ξ

l

(cid:48)

(cid:48)

,i

,s

|.

Therefore,

||ξ − ξ

l

(cid:48)

(cid:48)|| ≥ |ξs0 − ξ

,i

(cid:48)

(cid:48)
,i

,s0

| ≥ |ξ

l

(cid:48)

(cid:48)
,i

,s0

| − |ξs0| ≥ |ξ

l

(cid:48)

(cid:48)

,i

,s0

| − |ξ1| ≥ 2l+1.

(3.9)

l

To summarize, for any ξ
l

(cid:48)

,i

(cid:48) on the higher level, we have

|ξs0 − ξ

l

(cid:48)

(cid:48)

,i

,s0

| ≥ 2l+1,

23

(3.10)

along the s0 direction. We notice that the compact support area of g

,i
direction s0 has side length 2l+2, however, ξ must be on the left side of ξ
l

l

(cid:48)

(cid:48) along
,i

(cid:48)

(cid:48) along the

the s0 direction. Then, ξ will be at most on the edge of support area of the function

(cid:48)

g
l

,i

(cid:48) .

For any central frequency ξl,i ∈ Cl \ Bl, we define Jl,i

Jl,i ≡(cid:88)

(cid:48)

(cid:48)
,i

l

g2
(cid:48)
l

(cid:48) (ξl,i).
,i

g2
(cid:48)
l

,i

(cid:48) (ξl,i).

(cid:48)

By Proposition 3.1.1,

Jl,i ≡ (cid:88)

(cid:48)

l

=l,i

Proposition 3.1.2. Jl,i is independent of the index i, that is

Jl,i = Jl =

g2
(cid:48)
l

(cid:48) (ξl,i),
,i

∀ξl,i ∈ Cl \ Bl.

(cid:88)

(cid:48)

l

(cid:48)

=l,i

(3.11)

(3.12)

(3.13)

Proof. Suppose there are two different central frequencies ξl,i and ξl,i∗ in the same

major part of Cl. By Proposition 3.1.1, we should only consider gl,i from the same

level.

On the other hand, in the fixed level l, the compact support area of each box function

gl,i has side length 2l+1 along each dimension. In fact, each (l, i) will have overlapping

support with only two other i-indexes in each direction, since the central frequency ξl,i

in the level l is chosen as 2li with all possible integer multi index i.

24

(cid:32)|ξl,i,1 − ξ

l,i

|

(cid:48)

,1

σl

(cid:33)2

d

Using Proposition 3.1.1, we have

exp

(cid:48)
1=i1−1
i

 i1+1(cid:88)
−2
(cid:32)
−2

−2
d
(cid:33)2
(cid:32)|ξl,i∗,1 − ξ

(cid:48)
i
σl

l,i

σl

l

(cid:48)

(cid:48)

,i

(cid:48)
,i

g2
(cid:48)
l

(cid:88)
(cid:48) (ξl,i) ≈
 i
=1(cid:88)
 i∗
1+1(cid:88)
(cid:48)
1=i∗

1−1

=−1

exp

=

=

(cid:48)

i

i

exp

(cid:33)2

d

|

(cid:48)

,1

≈(cid:88)

(cid:48)

(cid:48)

l

,i

g2
(cid:48)
l

(cid:48) (ξl,i∗).

,i

(3.14)

Two summations are the same since Gaussian functions are only about the distance

between two frequencies.

Proposition 3.1.3. For all frequency ξ ∈ Cl \ Bl,

i

e

=

σl

−2

(cid:33)2

(cid:33)2

(cid:88)
(cid:48)∈Zd
(cid:88)
(cid:48)||∞≤4l−1
(cid:88)
(cid:48)||∞≤4l
||ξ−ξ
(cid:48)||

(cid:32)||ξ−2li
(cid:48)||
(cid:32)||ξ−2li
(cid:48)||
(cid:32)||ξ−2li
(cid:48)||
2

4l−1<||2li

−2
e

−2
e

−2

σl

σl

+

l,i
σl

||2li

+

(cid:88)

(cid:48)

i

=

e

(cid:32)||ξ−2li
(cid:48)||

(cid:33)2

σl

−2
e

(cid:88)
(cid:48)||∞>4l

||2li

(cid:33)2

+ 
,

(3.15)

where ||x||∞ ≡ max1≤s≤d |xs| and ξ

l,i

(cid:48) is the central frequency defined in the wavepacket

transform at the level l. 
 is a small number which can be ignored.

25

Proof.

−2
e

(cid:32)||ξ−2li
(cid:48)||
(cid:90) 4l−1−2l−1

(cid:88)
(cid:48)||∞≤4l−1
(cid:90)

σl

s=d(cid:89)

···

0

s=1

πerfc(3),

||2li

l

2σ−d
≤ √

(cid:33)2

≤

(cid:18)|ξs−cs|

σl

−
e

(cid:19)2

dc1dc2 ··· dcd

(3.16)

where erfc(·) is the complementary error function. Here we assume c is the central

frequency at the lower level, therefore,

c1 ≤ 4l−1 − 2l−1.

Meanwhile, we assume that ξ1 = max1≤s≤d |ξs| ≥ 4l−1 + 2l, then

ξ1 − c1 ≥ 2l + 2l−1.

With σl = 2l−1, we have the upper bound specified in equation (3.16).

(3.17)

(3.18)

Similarly, we have

(cid:88)

(cid:48)

l

>l,i

e

(cid:48)

σ−d

l

||ξ−ξ

(cid:48)
l
σl

−

2

(cid:48)||

,i

≤ √

πerfc(5).

(3.19)

26

Then

(cid:88)
(cid:48)∈Zd

i

e

(cid:32)||ξ−2li
(cid:48)||

σl

−

||ξ−ξ

(cid:48)
l
σl

2

(cid:48)||
,i

(cid:33)2

(cid:88)

(cid:48)

l

(cid:48)

=l,i

−
e

=

√
πerfc(3) +

√
πerfc(5).

+

(3.20)

Proposition 3.1.4. For any frequency ξ in the major part Cl \ Bl, we have

(cid:88)

(cid:48)

l

(cid:48)

,i

(cid:88)

(cid:48)

l

(cid:48)

=l,i

(cid:48) (ξ) ≈ Jl =

g2
(cid:48)
l

,i

g2
(cid:48)
l

(cid:48) (ξl,i),
,i

∀ξ ∈ Cl \ Bl.

(3.21)

Proof. We have already proved in Proposition 3.1.1 that any frequency ξ in the major
part Cl \ Bl satisfies,

g2
(cid:48)
l

,i

(cid:48) (ξ) =

g2
(cid:48)
l

,i

(cid:48) (ξ).

(3.22)

(cid:88)

(cid:48)

(cid:48)
,i

l

Meanwhile, according to the Poisson summation formula,

(cid:88)

(cid:48)

l

(cid:48)

=l,i

(cid:88)

(cid:19)d (cid:88)

(cid:18) a√

2π

− σ2
e

2 ||x+ma||2

=

a ||2
2σ2||2π m
− 1

e

= const.

(3.23)

m∈Zd

m∈Zd

||ξl,i−2li

σl

(cid:48)||

−2

2

.

(3.24)

Then we have,

(cid:33)2

(cid:32)||ξ−2li
(cid:48)||

σl

(cid:88)
(cid:48)∈Zd

i

−2
e

(cid:88)
(cid:48)∈Zd

i

e

=

27

Using equation (3.15),

(cid:48) (ξ) ≈ (cid:88)
||ξl,i−2li

−2

=l,i

e

(cid:48)

(cid:48)

l

σl

−2

(cid:48)||

(cid:88)

(cid:48)

(cid:48)
,i

l

=

g2
(cid:48)
l

,i

(cid:88)
(cid:48)∈Zd

i

e

2

(cid:48)||

,i

(cid:48)
l
(cid:48)

l

σ

||ξ−ξ
2

≈ (cid:88)

(cid:48)

l

=l,i

(cid:48)

−2
e

(cid:33)2

(cid:48)||

≈ (cid:88)
||ξl,i−ξ

(cid:48)∈Zd

i

e

σl

−2

(cid:48)

,i

l

σl

(cid:32)||ξ−2li
2

(cid:48)||

≈ Jl.

(3.25)

The proposition is proved.

For the frequency ξ on the border of the partitioning, their sums satisfy

Jl−1 ≤(cid:88)

l,i(ξ) ≤ Jl.
g2

(3.26)

l,i

Using Jl as an approximation will yield an overestimation, however, it is negligible

since Bl is much smaller compared with the major area away from the border.

To summarize,

Then,

Jl ≈(cid:88)

l,i

g2
l,i(ξ),

∀ξ ∈ Cl.

hl,i(ξ) ≈ 1
Jl

gl,i(ξ).

We therefore have a modified inverse wavepacket transform as the following,

Lemma 3.1.1. For any f ∈ L2(Rd), we have

f (x) ≈(cid:88)

l,i,k

(cid:104)φl,i,k, f(cid:105)φl,i,k(x).

1
Jl

28

(3.27)

(3.28)

(3.29)

To end this part, we would like to display some numerical results to justify Lemma

3.1.1. The denominator in the expression of hl,i (2.17) should be a step function

about the index l, suggested by Lemma 3.1.1. The range of the frequency concerned
is [64 : 320] × [64 : 320].

Figure 3.1: (cid:80)

l,i g2

l,i in different level l

3.2 Multiscale Gaussian Wavepacket Inversion

3.2.1 Setup of the True Amplitude Migration Problem

Let us suppose the wave propagation is governed by the scalar wave equation (2.1)

with the wave propagation velocity decomposed as,

c = v(1 + α),

(3.30)

where v is the macro velocity being responsible for the traveltime and amplitude.

Moreover, it is assumed to be smooth and does not provide the significant energy back

to the boundary data. The rapid perturbation α is small but reflects the wave signal

back to the boundary data. In our inversion model, v is known at prior and our target

29

is to image α.

We simplify our model as a constant density fluid occupying a half-space X ≡ {x ∈
Rd : xd ≥ 0}. The boundary data D(r, s, t) used in this paper is organized as the

common-shot trace, for example, Fig. 3.2, r parametrizes receiver positions on the
surface ∂X ≡ {x ∈ Rd : xd = 0}, while s parametrizes source positions. We also
assume that the sources are contained in an open set Os ⊂ ∂X and receivers are
in an open set Or ⊂ ∂X. Therefore, the boundary data D is a function defined at
Os × Or × [0, T ]. Similar to [44], we make some assumptions about rays.

Figure 3.2: A typical source gather in Gaussian slowness

Assumption 3.2.1. There are no rays leaving points in the subsurface {x ∈ X : xd >
0} and returning to graze Os or Or. Moreover, there exists an universal lower bound

b, such that

|pd| ≥ b||p||,

(3.31)

for any rays hitting the surface where p is the momentum variable and pd is the dth

component of p.

30

−20200.511.522.5xrtAssumption 3.2.2. Rays departing from a source in Os and traveling into the sub-

surface do not return to receivers in Or.

Assumption 3.2.3. There exists δ > 0, such that v(x) is a constant if 0 ≤ xd ≤ δ.

3.2.2 Born Approximation for the Trace Data

Denote the wave propagator with background velocity v as L0 and the wave propagator

with true velocity c as L. Meanwhile, the corresponding Green’s functions are written

as G0 and G, respectively,

G = −L−1; G0 = −L−1
0 ,

(3.32)

and by some formal computations,

G = G0 + G0(L − L0)G = G0 + G0V G,

(3.33)

where V = −L0 + L. The Born approximation assumes the whole scattering process

as the following. The signal is initiated from the source and travel through a smooth

medium afterwards. Then at some moment, it hits the reflector under the surface and

is reflected back. Therefore, the boundary data D(r, s, t) is the reflection data, or the

wavefield along the boundary minus the direct wave. During the whole process, we

assume that the reflection or scattering only happens once so that we ignore multiple

reflections.

31

Define the incident wavefield G0(x, t; s) generated by the source point s,

L0G0(x, t; s) = −δ(t)δ(x − s), G0|t<0 = 0.

(3.34)

Then the reflection signal is obtained by total differentiation. We assume the true

velocity c = v + αv = v + δv and the total wavefield u = u0 + δu, where L0u0 = 0,

t u − ∆u ≈ (

1
c2 ∂2

1

v2 − 2δv

v3 )∂2

t (u0 + δu) − ∆(u0 + δu).

Considering the first order term, we have

L0δu =

2δv
v3 ∂2

t u0.

Now the perturbed wavefield δG [7, 46] satisfies,

L0δG =

2α
v2

∂2G0
∂t2 .

(3.35)

(3.36)

(3.37)

This perturbed wavefield δG is the data recorded along the surface based on the Born

approximation, that is,

(cid:20)2α

(cid:21)

v2

D

(r, s, t) = δG =

(cid:90)

(cid:90)

dx

2α
v2

dh ˆG0(r, t − h; x)

∂2 ˜G0
∂t2 (x, h; s),

(3.38)

where both ˜G and ˆG are Green’s functions. The Green’s function ˆG0(r, t − h; x)
represents the perturbation received at the receiver r at the moment t − h, and its

source is the subsurface point x. ˜G is about source points s and ˆG is about receivers

32

r. The same rules are applied to other functions.

(cid:20)2α

(cid:21)

v2

D

(cid:90)

(cid:90)

(r, s, t) =

∂2
∂t2

dx

2α
v2

dh ˆG0(r, t − h; x) ˜G0(x, h; s).

(3.39)

To make things easier, we would like to develop our algorithm in the frequency domain

instead of the time domain so that we can simplify the convolution operator above.

Applying the following Fourier transform in time, we have,

Here we abuse the notation by writing the Fourier transform of the Green’s function

ˆG0(r, t; x) about the time variable t by ˆG0(r, ω; x). In equation (3.40), the reciprocity

of the Green’s function is involved as we replace ˜G0(x, ω; s) with ˜G0(s, ω; x).

3.2.3 Multiscale Gaussian Beam Approximation of the Green’s

Function

We then approximate the Green’s function in the high frequency regime by the summa-

tion of Gaussian beams so that we can define the following multiscale Gaussian-beam

transform of the perturbation of the velocity,

(cid:20)2α

(cid:21)

v2

D

(cid:90)

(cid:90)(cid:90)

(r, s, ω) = −ω2

dx

2α
v2

dξdη ˆUGB(r, ω; x, ξ) ˜UGB(s, ω; x, η),

(3.41)

(cid:20)2α

(cid:21)

v2

D

(r, s, ω) = −ω2

dx

2α
v2

ˆG0(r, ω; x) ˜G0(s, ω; x).

(3.40)

(cid:90)

33

where without confusion, we sometimes shorten D

(cid:105)

(cid:104) 2α

v2

(r, s, ω) to be D(r, s, ω) so that

we use D(r, s, ω) to denote the Fourier transform of the boundary data D with respect

to the time variable t. ˆUGB(r, ω; x, ξ) is the beam solution in the frequency domain

starting at the point x with the initial momentum ξ. From now on, the following

notation is used to the end,

ˆUGB(r, ω; x, ξ) = ˆU x

ξ (r, ω);

˜UGB(s, ω; x, η) = ˜U x

η (s, ω).

The Green’s function can be considered as the solution to the acoustic wave equation

whose initial velocity is a Dirac-delta function by Duhamel’s principle and the mul-

tiscale Gaussian wavepacket transform can be applied to decompose the Dirac-delta

function.

t G0(x, t; s) − ∆G0(x, t; s) = 0, G0
∂2

1

v2(x)

(3.42)

(cid:12)(cid:12)(cid:12)t=0

(cid:12)(cid:12)(cid:12)t=0

= 0,

∂G0
∂t

= −δ(t)δ(x − s).

Although the multiscale transform introduced in Section 2.2 and reference [48] is de-

signed to decompose any general L2 functions, the Dirac-delta function can be approx-

imated by some L2 functions.

3.2.4 True-Amplitude Migration Process

A new operator Kpq applied to the perturbation α can be defined, which is correspond-

ing to the certain pair of Gaussian beams,

(cid:18)

(cid:19)

Kpq

2α
v2

(r, s, ω) =

dx

2α
v2

ˆU x
p (r, ω) ˜U x

q (s, ω),

(3.43)

(cid:90)

34

which will be called the atomic Gaussian-beam transform. The operator Kpq maps the

subsurface information to the boundary data.

(cid:90)(cid:90)

(K∗

pqg)(y, ω) =

drds

¯ˆU y
p

¯˜U y
q g(r, s, ω),

(3.44)

which is the adjoint of the atomic Gaussian-beam transform. Applying K∗

pq to the

boundary data D yields a single-frequency prestack angle-gather imaging function Ipq,

(3.45)

(3.46)

(3.47)

Ipq(y, ω) =(cid:0)K∗

pqD(cid:1) (y, ω),

or to write it completely,

Ipq

(cid:20)2α

(cid:21)

v2

(y, ω) =

(cid:18)
K∗
pqD

(cid:20)2α

(cid:21)(cid:19)

v2

(y, ω).

Substitute equation (3.41) about the surface data D into equation (3.45),

(cid:20)2α

(cid:21)

v2

Ipq

(cid:90)(cid:90)
(cid:90)(cid:90)

(y, ω) = −ω2

(cid:90)

dx

2α
v2

drds

¯ˆU y
p (r, ω) ¯˜U y

q (s, ω)

dξdη ˆU x

ξ (r, ω) ˜U x

η (s, ω).

We will later show that the following approximation (3.47) is correct,

−

(cid:90)(cid:90)(cid:90)(cid:90)
≈ E(p, q, y)ei(p+q)(y−x)e(y−x)T iM0

dωdrdsdξdηω2 ¯ˆU y

p (r, ω) ¯˜U y

q (s, ω) ˆU x
2 (y−x),

ξ (r, ω) ˜U x

η (s, ω)

where E(p, q, y) is a constant related to the parameters of the corresponding wavepacket,

35

and M0 is a symmetric matrix with a positive definite imaginary part as defined in

the Multiscale Gaussian beam propagation. The integral (cid:82) dω should be interpreted
as(cid:82) χ(ω)dω, where χ is an arbitrary C∞ function which is zero in the low frequency

region and is 1 for the high frequency.

If we integrate (3.46) further with respect to ω, then we will have the wavepacket

transform about 2α
v2 ,

(cid:90)

2α

dx

Ipq

v2

(cid:20)2α

(cid:21)

(y, ω)dω = E(p, q, y)

(cid:90)
As we can see in equation (3.48),(cid:82) dωIpq is essentially the Gaussian wavepacket trans-
v2 in the direction of p + q. The integral (cid:82) dωIpq(y, ω) can be

v2 ei(p+q)(y−x)ei(y−x)T M0

form of the function 2α

2 (y−x).

(3.48)

considered as taking the inverse Fourier transform to obtain Ipq(y, t) at t = 0. Accord-

ing to Claerbout imaging principle [29], Ipq(y, t) at t = 0 yields the initial state of the

subsurface that we want to image.

By using the modified wavepacket transform (3.29), we can reconstruct perturbation

2α
v2 through the imaging function (3.48).

(cid:88)
(cid:88)

l,i,k

2α
v2 =

=

1
Jl

v2 , φl,i,k(cid:105)φl,i,k(x)
(cid:104)2α
(cid:104) 2α
(cid:105)
(cid:82) Ipq
(cid:88)
( k
Ll

v2

, ω)dω

l,i,k

p+q=ξl,i

EJl

Based on equation (3.49) ,

36

φl,i,k(x).

(3.49)

Algorithm 2 Multiscale Gaussian Beam True-Amplitude Migration
1.Run the multiscale Gaussian wavepacket transform to get the dictionary of central points

y and central momentum p + q.

2. Separate p+q into two different wavepackets and shoot them from the subsurface point y

to the acquisition surface.

3. Angle-gather prestack image by calculating Ipq(y, ω) =(cid:82)(cid:82) dsdr

¯ˆU y
p

¯˜U y
q D(r, s, ω).

4. Stack all partial image Ipq with the same p + q to reduce the noise.

5. Use equation (3.48) to get the coefficient of each wavepacket expansion of the perturbation

2α
v2 .
6. Run the inverse multiscale Gaussian wavepacket transform to recover the rapid pertur-

bation 2α
v2 .

3.2.5 Motivation for Inverting the multiscale Gaussian-beam

transform

Hereby we provide some intuitive justifications of the inversion process of the multiscale

Gaussian-beam transform, which may provide some theoretical guideline for carrying

out further analysis of our new methodology and extending our methodology to other

applications.

We start with considering the function b(r, s, t) defined by the linear operator D,

D[f ](r, s, t) = b(r, s, t),

(3.50)

where the function f is defined at subsurface points. To solve this linear operator

37

equation rapidly and efficiently, we would like to diagonalize the operator D by carrying

out a certain frame representation. Since the argument f of the operator D does not
sit in the same space as the right-hand side b, we first apply the adjoint operator D∗

to both sides, so that we have

D∗D[f ] = D∗b.

(3.51)

Let W be the multiscale Gaussian wavepacket transform defined in Section 2.2, which
satisfies W∗W = I. Then we apply W to both sides of equation (3.51), yielding

W(D∗D)W∗Wf = WD∗b.

(3.52)

Our results in Section 3.2.4 indicate that the above diagonalization is justified. The
operator WD∗b is essentially equation (3.46), and the operator W(D∗D)W∗ is essen-

tially captured by the diagonal factor E(p, q, y) in equation (3.47). The overall effects

of the diagonalization process are epitomized in equation (3.48).

Therefore, we have

Wf = (W(D∗D)W∗)

−1 WD∗b,

f = W∗ (W(D∗D)W∗)

−1 WD∗b,

(3.53)

which results in our fast reconstruction formula (3.49).

To establish the theoretical foundation for our new methodology in terms of the Fourier

Integral Operator (FIO) theory, we need to carry out symbolic calculus to establish

several facts by following the works in [50, 23]:

38

1. the forward operator D belonging to a certain class of FIO;
2. the conjugation process W(D∗D)W yielding a diagonal operator in the frame de-
fined by W.

On the other hand, since the FIO theory is originated from the asymptotic of geometric

optics and a Gaussian-beam solution provides a globally defined asymptotic solution for

wave equations, we will carry out brute-force calculations to justify our new methodol-

ogy by heavily relying on the structure of multiscale Gaussian-beam transform, which

is composed of Gaussian beams and multiscale Gaussian wavepacket transform.

3.3 Theoretical Validation: The Proof about The

Imaging Operator

3.3.1 Road Map of The Theoretical Analysis

We will prove the approximation (3.47) holds. Since it involves the interaction of four

beams in time, we will carry out the analysis essentially in two main steps.

The first step consists of analysis of the following two integrals dealing with the beam

interactions on the receiver side and the source side respectively,

(cid:90)(cid:90)

drdξ

¯ˆU y
p (r, ω) ˆU x

ξ (r, ω),

(cid:90)(cid:90)

dsdη ¯˜U y

q (s, ω) ˜U x

η (s, ω).

Since receivers and sources are reciprocal in wave propagation, we just need to focus

on analysis of beam interaction on the receiver side, and the analysis of the source side

39

is essentially analogous.

Our analysis of the two beams’ interaction yields the following approximations

(cid:90)(cid:90)
(cid:90)(cid:90)

drdξ

¯ˆU y
p (r, ω) ˆU x

dsdη ¯˜U y

q (s, ω) ˜U x

ξ (r, ω) ≈ eip·(y−x) ˆH(x, ξ, ω; y, p),
η (s, ω) ≈ eiq·(y−x) ˜H(x, η, ω; y, q).

where the functions ˆH and ˜H will be defined later.

Based on the first step, the second step consists of analyzing the left-hand side of

approximation (3.47) so that we have

(cid:90)(cid:90)(cid:90)(cid:90)
ξ (r, ω) ˜U x
≈ ei(p+q)(y−x) ˆH(x, ξ, ω; y, p) ˜H(x, η, ω; y, q).

¯ˆU y
p (r, ω) ¯˜U y

q (s, ω) ˆU x

drdsdξdη

η (s, ω)

After carrying out the integral about ω, the above approximation reduces to

(cid:90)(cid:90)(cid:90)(cid:90)
−
q (s, ω) ˆU x
ei(p+q)(y−x)K(p, q, y) ˆH(x, y, p, q) ˜H(x, y, p, q).

dωdrdsdξdηω2 ¯ˆU y

p (r, ω) ¯˜U y

ξ (r, ω) ˜U x

η (s, ω) =

(3.54)

(3.55)

where K(p, q, y), ˆH and ˜H are defined later.

After making an essential assumption on the invertibility of the imaging operator, we

40

are able to approximate functions ˆH and ˜H by

ˆH(x, y, p, q) ≈ e

˜H(x, y, p, q) ≈ e

i

4||y−x||2

ˆM (0) ˆK(y, p, q),

i

4||y−x||2

˜M (0) ˜K(y, p, q),

where ˆK(y, p, q) and ˜K(y, p, q) can be easily computed.

These latter approximations allow us to obtain our main theorem,

(cid:90)

(cid:90)

(cid:90)

−ω2dω

drds

¯ˆU y
p (r, ω) ¯˜U y

q (s, ω)

dξdη ˆU x

ξ (r, ω) ˜U x

η (s, ω) ≈

K(p, q, y) ˆK(y, p, q) ˜K(y, p, q)ei(p+q)·(y−x)e

i||y−x||2

M0/2,

which says that the four-beam interaction in time yields a weighted Gaussian wavepacket

centered at the scattering point y in the direction p + q, where (y, p) is the ray param-

eter for the beam from the scattering point y to the boundary receiver in the direction

p, and (y, q) is the ray parameter for the beam from the scattering point y to the

boundary source in the direction q.

The rest of the analysis will follow the above road map.

We will prove equation (3.47) in this section. Throughout the proof, the amplitude

function A is not involved as it is more smooth compared with the phase function

part. Denote the beam as (ˆy(t), ˆp(t), ˆM (t), ˆA(t)), whose initial value is (y, p, M, A),

and (ˆx(t), ˆξ(t), ˆN (t), ˆC(t)) whose initial value is (x, ξ, N, C). Moreover, ˆΞ and ˆκ are

41

defined according to a fixed beam ˆU y
p ,

ˆξ(t) = ˆκ(t; x, ξ, y, p)ˆp(t) + ˆΞ(t; x, ξ, y, p),

(3.56)

and

ˆΞ(t; x, ξ, y, p) · ˆp(t) = 0.

(3.57)

On the source side, we have similar notation (˜y(t), ˜q(t), ˜M (t), ˜A(t)), whose initial value

is (y, q, M, A), and (˜x(t), ˜η(t), ˜N (t), ˜C(t)) whose initial value is (x, η, N, C).

3.3.2 Approximation of Gaussian Beams along the Surface

The beam functions ˆU y

p (r, ω) and ˜U y

p (s, ω) are used in the inversion process to link the

data to the unknown perturbation. We would like to explore more about the beam

function’s structure to build the foundation for the future proof and calculation.

||x||2

M denotes xT M x in this paper for all vectors x and all symmetric matrices M .

For each beam ˆU y

p , we define the hitting time ˆt0 = ˆt0(y, p) and hitting point ˆr0 =

ˆy(ˆt0(y, p)) according to the arrival time of its central ray at the boundary so that
ˆr0 = ˆy(ˆt0(y, p)) ∈ {x ∈ Rd : xd = 0}.

In [49], the single beam is propagating by treating the time variable same as the spatial

components, see also [24]. The complete Hessian matrix is S.P.D. along the directions

transversal to the ray direction. Therefore, by Assumption 3.2.1, we have a Gaussian

profile if intersecting the beam solution at the surface xd = 0.

Proposition 3.3.1. The Gaussian beam ˆU y

p (r, t) along the surface for r ∈ {x =

42

(x1,··· ,d) : xd = 0} can be represented as

p (r, t) ≈ ˆA(ˆt0) exp(cid:0)i(cid:0)ˆτx(ˆt0; y, p) · (r − ˆr0) + ˆτt(ˆt0; y, p)(t − ˆt0)(cid:1)(cid:1)×
||r − ˆr0||2
i

ˆτtt(ˆt0; y, p)(t − ˆt0)2 + (t − ˆt0)ˆτ T

+

ˆU y

exp

1
2

ˆM (ˆt0)

2

tx(ˆt0; y, p)(r − ˆr0)

 .

(3.58)

where ˆt0 = ˆt0(y, p) and all partial derivatives about the phase function τ are on the

central ray. The proposition is equivalent to intersecting the complete (t, x) beam ansatz

at the surface xd = 0.

We first introduce the way to compute the terms ˆτtx and ˆτtt, which can be obtained

by inserting corresponding ray parameters into equations (4.13)-(4.14).

ˆτt(t; y, p) = − ˆG(ˆy(t), ˆp(t)),
ˆτtx(t; y, p) = − ˆGx(ˆy(t), ˆp(t)) − ˆM (t; y, p) ˆGp(ˆy(t), ˆp(t)),
ˆτtt(t; y, p) = − ˆGp(ˆy(t), ˆp(t)) · ˆτtx(t; y, p),

(3.59)

(3.60)

(3.61)

where (ˆy(t), ˆp(t)) = (ˆy(t; y, p), ˆp(t; y, p)). We also denote ˆM∗ as the complete Hessian

43

matrix about (t, x) at (t; y, p) and t = ˆt0(y, p),

 ˆτtt(t; y, p), ˆτ T

ˆτtx(t; y, p),

tx(t; y, p)

ˆM (t; y, p)

 ,

ˆM∗(t; y, p) =

ˆF (r, t; y, p) = ˆτt(t; y, p) + Re(ˆτtx(t; y, p)) · (r − ˆy(t)),
ˆθ(r, t; y, p) = ˆp(t) · (r − ˆy(t)) + ||r − ˆy(t)||2
ˆQ(r, t; y, p) = −(Im(ˆτtx)(t; y, p))T (r − ˆy(t))

Re( ˆM (t))

2

.

,

Im(ˆτtt)(t; y, p)

Similarly, we denote (ˆx(t), ˆξ(t)) = (ˆx(t; x, ξ), ˆξ(t; x, ξ)), ˆN∗(t; x, ξ), ˆF (r, t; x, ξ), ˆθ(r, t; x, ξ)

and ˆQ(r, t; x, ξ).

The following proposition is needed when taking the Fourier transform of ˆU y

p (r, t) about

time t,

Proposition 3.3.2. For any complex number γ with positive real part, i.e. Re(γ) > 0,

(cid:90) ∞
−∞ e−γt2

(cid:114) π

γ

e−iωtdt =

− ω2
4γ .

e

(3.62)

Lemma 3.3.1. The Fourier transform of ˆU y

p (r, t) with respect to time t is

ˆU y
p (r, ω) = ˆU y

p (r, ω; t)

= eiˆ
(r,t;y,p)e−iωtei ˆβ(t;y,p)|ω−ˆτt(t;y,p)−ˆζ(t;y,p)T (r−ˆy(t))|2
(cid:115)

−||r−ˆy(t)||2

ˆM(t;y,p,M0)

i2π

ˆτtt(t; y, p)

2

,

(3.63)

(cid:12)(cid:12)(cid:12)t=ˆt0(y,p)

ˆA(t)e

44

where

ˆM(t; y, p) = Im( ˆM )(t; y, p) − Im(ˆτtx)Im(ˆτtx)T
Im(ˆτtt(t; y, p))

ˆβ(t; y, p) = −

1

2ˆτtt(t; y, p)

= −

1

2iIm(ˆτtt(t; y, p)) + 2Re(ˆτtt(t; y, p))

(3.64)

,

(3.65)

ˆζ(t; y, p) = Re(ˆτtx(t; y, p)) − Re(ˆτtt(t; y, p))
Im(ˆτtt(t; y, p))

Im(ˆτtx(t; y, p)),

(3.66)

ˆ
 (r, t; y, p) = ˆθ(r, t; y, p) + ( ˆF (r, t; y, p) − ω) ˆQ(r, t; y, p)

+

Re(ˆτtt(t; y, p))

2

ˆQ(r, t; y, p)2.

(3.67)

Here t in ˆU y

p (·; t) serves as a fixed parameter, since all terms defined above is defined

at this fixed moment.

Proof. We still abbreviate ˆt0(y, p) as ˆt0 in this proof since we only concern the single

beam ˆU y

p here. From Proposition 3.3.1,

(cid:90)

p (r, ω) ≈ ˆA(ˆt0)eiˆθ(r,ˆt0;y,p)
ˆU y

−||(t−ˆt0,r−ˆy(ˆt0))||2

× e

e−iωtei ˆF (r,ˆt0;y,p)(t−ˆt0)ei 1
Im( ˆM∗(ˆt0;y,p,M0))/2dt,

2 Re(ˆτtt(ˆt0;y,p))(t−ˆt0)2

After expanding the term e

−||(t−ˆt0,r−ˆy(ˆt0))||2

Im( ˆM∗/2), we have

−||(0,r−ˆy(ˆt0))||2

Im( ˆM∗)(ˆt0;y,p,M0)

2

2 Re(ˆτtt(ˆt0;y,p))(t−ˆt0)2

− Im(ˆτtt(ˆt0;y,p))
e

2

(t−ˆt0)2

(cid:90)
p (r, ω) ≈ ˆA(ˆt0)eiˆθe−iωˆt0e
ˆU y
e−i(− ˆF +ω)(t−ˆt0)ei 1
e−(t−ˆt0)(Im(ˆτtx)(ˆt0;y,p))T (r−ˆy(ˆt0))dt.

45

(3.68)

(3.69)

To make a complete square term in equation (3.69), we have

− Im(ˆτtt)(ˆt0; y, p)
|(Im(ˆτtx)(ˆt0; y, p))T (r − ˆy(ˆt0))|2

2

2Im(ˆτtt(ˆt0; y, p))

(cid:32)

(t − ˆt0)2 − (t − ˆt0)Im(ˆτtx(ˆt0; y, p))T (r − ˆy(ˆt0)) =

− Im(ˆτtt)

2

t − ˆt0 +

(Im(ˆτtx))T (r − ˆy(ˆt0))

Im(ˆτtt)

(cid:33)2

.

Since ˆQ(r, ˆt0; y, p) = − (Im(ˆτtx)(ˆt0;y,p))T (r−ˆy(ˆt0))

Im(ˆτtt)(ˆt0;y,p)

,

p (r, ω) = ˆA(ˆt0)eiˆθe−iωˆt0ei( ˆF−ω) ˆQei 1
ˆU y
|(Im(ˆτtx)(ˆt0;y,p))T (r−ˆy(ˆt0))|2

−||r−ˆy(ˆt0)||2

2 Re(ˆτtt) ˆQ2

e

2Im(ˆτtt)(ˆt0;y,p)

(cid:90)

ei( ˆF−ω+Re(ˆτtt) ˆQ)(t−ˆt0− ˆQ)

Im( ˆM )(ˆt0)

2

e

2

2 Re(ˆτtt)(t−ˆt0− ˆQ)2
ei 1

e

− Im(ˆτtt)(ˆt0;y,p)

≈ ei ˆF ˆQei 1

2 Re(ˆτtt)( ˆQ)2

e−iω(ˆt0+ ˆQ)e

(cid:115)

(t−ˆt0− ˆQ)2
dt + O(|t − t0|2)
|(Im(ˆτtx)(ˆt0;y,p))T (r−ˆy(ˆt0))|2

2Im(ˆτtt)(ˆt0;y,p)

−||r−ˆy(ˆt0)||2

Im( ˆM )(ˆt0)

2

e

×

i2π

ˆτtt(ˆt0; y, p)

ˆA(ˆt0)eiˆθe−ˆ
|−ˆτt+ω−Re(ˆτtx)(ˆt0;y,p)·(r−ˆy(ˆt0))−Re(ˆτtt)(ˆt0;y,p) ˆQ|2

.

(3.70)

Equation (3.70) is obtained by Proposition 3.3.2 where

ˆ
(ˆt0; y, p) =

2Im(ˆτtt)(ˆt0; y, p) − i2Re(ˆτtt)(ˆt0; y, p)

1

.

(3.71)

After replacing ˆβ = iˆ
, we have the lemma proved.

46

Corollary 3.3.1. The Fourier transform of ˆU x

ξ (r, t) with respect to time t is

ˆU x
ξ (r, ω) = ˆU x

ξ (r, ω; t)

(cid:12)(cid:12)(cid:12)t=ˆt0(x,ξ)

= eiˆ
(r,t;x,ξ)e−iωteiˆγ(t;x,ξ)|ω−ˆτt(t;x,ξ)− ˆϑ(t;x,ξ)T (r−ˆx(t))|2
(cid:115)

−||r−ˆx(t)||2

ˆN (t;x,ξ,N0)

i2π

ˆC(t)e

2

,

(3.72)

ˆτtt(t; x, ξ)

where

ˆN (t; x, ξ) = Im( ˆN )(t; x, ξ) − Im(ˆτtx)Im(ˆτtx)T
Im(ˆτtt(t; x, ξ))

ˆ
 (r, t; x, ξ) = ˆθ(r, t; x, ξ) + ( ˆF (r, t; x, ξ) − ω) ˆQ(r, t; x, ξ)

+

1
2
ˆγ(t; x, ξ) = −

Re(ˆτtt(t; x, ξ)) ˆQ(r, t; x, ξ)2,

1

2iIm(ˆτtt(t; x, ξ)) + 2Re(ˆτtt(t; x, ξ))

,

(3.73)

(3.74)

ˆϑ(t; x, ξ) = Re(ˆτtx(t; x, ξ)) − Re(ˆτtt(t; x, ξ))
Im(ˆτtt(t; x, ξ))

Im(ˆτtx(t; x, ξ)).

(3.75)

We have the same conclusion for ˜U y

q (s, ω) and ˜U x

η (s, ω) as Lemma 3.3.1 and all terms

involved are defined accordingly.

3.3.3 Asymptotic Analysis of Two Beams’ Interaction

In this section, we would like to explore the interaction between two Gaussian beams,

that is

(cid:90)(cid:90)

(cid:90)(cid:90)

dsdη ¯˜U y

q (s, ω) ˜U x

η (s, ω).

drdξ

¯ˆU y
p (r, ω) ˆU x

ξ (r, ω),

47

The rest of Section 3.3.3 is organized as the following. We will first discuss the distance

between two beams ˆU y

p and ˆU x

space. In Section 3.3.3.2, we will evaluate the error caused by replacing ˆU x

ξ satisfying the parabolic scaling principle in the phase
ξ (·; ˆt0(x, ξ))
ξ (·; ˆt0(y, p)). After that, the difference between the exponents at different times

with ˆU x

is evaluated and it will allow us to map the phase term to the initial moment.

In

Section 3.3.3.4 and 3.3.3.5, we will compute the integral about r and show that there

exists a Gaussian profile centered at ˆy(ˆt0(y, p)). And the analysis can all be applied to

the source side similarly.

Here we require that there is a significant interaction between two beams concerned,

which means the distance between two central rays, (ˆy(t), ˆp(t)) and (ˆx(t), ˆξ(t)) is less

than the width of the beam ˆU y

√
p and the width of a beam is defined as 1/


, where 


is the smallest eigenvalue of Im( ˆM (t)).

3.3.3.1 Parabolic Scaling Principle

A wavepacket satisfying the so-called parabolic scaling principle means the wavelength

of the typical oscillation of the wavepacket being equal to the square of the width of

the wavepacket, and a Gaussian beam will satisfy parabolic scaling principle at any

given time if it does initially. The following graph shows a single Gaussian wavepacket
in R2 satisfying the parabolic scaling principle.

The following asymptotic analysis is needed throughout the proof.

Lemma 3.3.2. Consider two scattering beams (ˆy(t), ˆp(t), ˆM (t), ˆA(t)) and

48

Figure 3.3: Real Part of Single Wavepacket φl,i,k

(ˆx(t), ˆξ(t), ˆN (t), ˆC(t)) satisfying the parabolic scaling principle at the initial time, then

|| ˆM (t)|| ∼ O(||p||),

|| ˆN (t)|| ∼ O(||ξ||).

(3.76)

If there exists significant interaction effects between two beams, then

(cid:33)
1(cid:112)||p||
Moreover, if we have ||p − ξ|| ∼ O((cid:112)||p||) at the beginning,

||ˆy(t) − ˆx(t)|| ∼ O

(cid:32)

.

|ˆκ(t)| ∼ 1 + O(||p||− 1
2 ),

||ˆΞ(t)|| ∼ O((cid:112)||p||).

(3.77)

(3.78)

Proof. The assumption ||p− ξ|| ∼ O((cid:112)||p||) is a reasonable assumption, as we will see

later ||p − ξ|| will be controlled by a Gaussian profile, which means the value will be

exponentially decaying when p and ξ are far from each other.

First, the Hamiltonian G(x, p) = v(x)||p|| remains as a constant along the ray. This
implies that the order of the momentum ||p|| will not change as the velocity v is a
smooth function and bounded away from zero, that is ||ˆp(t)|| ∼ ||p||.

49

-2101x220.5x10.500Second, we will evaluate ˆM (t). The homogeneous of degree one Hamiltonian guarantees

that Gaussian wavepackets satisfy the parabolic scaling principle at any given time.
This implies that the size of Hessian ˆM is around O(||ˆp(t)||) and equation (3.76) is

correct.

Due to the fact that ˆU y

p is well-localized around ˆy(t) in the physical space, ˆU x

ξ will

have small interaction with ˆU y

p , if ˆx(t) is beyond this localized region. And equation

(3.77) is correct.

Third,

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)d(ˆp(t) − ˆξ(t))

dt

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)∇∇v(ˆy(t))(ˆy(t) − ˆx(t))||ˆp(t)||(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:19)T

(cid:18) ˆp(t)

||ˆp(t)||

+ ||∇v(ˆy(t))

(ˆp(t) − ˆξ(t))||

≤ C2||ˆy(t) − ˆx(t)||||ˆp(t)|| + C1||ˆp(t) − ˆξ(t)||,

where C1 and C2 are the upper bound of ||∇v|| and ||∇∇v|| respectively. Moreover,

d||ˆp(t) − ˆξ(t)||2

dt

d(ˆp(t) − ˆξ(t))

dt

= 2

· (ˆp(t) − ˆξ(t)) ≤ 2||d(ˆp(t) − ˆξ(t))

dt

||||ˆp(t) − ˆξ(t)||,

by Cauchy-Schwartz inequality. We further get,

d||ˆp(t) − ˆξ(t)||2

dt

≤ C1||ˆp(t) − ˆξ(t)||2 + C2(||ˆy(t) − ˆx(t)||)(||ˆp(t)||)(||ˆp(t) − ˆξ(t)||)

||ˆp(t) − ˆξ(t)||2

2

.

≤ C1||ˆp(t) − ˆξ(t)||2 +

1
2

(C2||ˆp(t)||||ˆy(t) − ˆx(t)||)2 +

50

By Gronwall inequality, we have

||ˆp(t) − ˆξ(t)||2 ∼ O(||p||),

since C2||ˆp(t)||||ˆy(t) − ˆx(t)|| is uniformly bounded. Consequently, equation (3.78) is

correct.

3.3.3.2 Difference between Two Interacted Beams’ Traveltime

In this part, we would like to calibrate the beam ˆU x

ξ (r, ω; ˆt0(x, ξ)) according to the

beam ˆU y

p (r, ω) by shifting time ˆt0(x, ξ) to time ˆt0(y, p).

By Corollary 3.3.1,

ξ (r, ω) = eiˆ
(r,ˆt0(x,ξ);x,ξ)e−iωˆt0(x,ξ)
ˆU x

eiˆγ(ˆt0(x,ξ);x,ξ)|ω−ˆτt(ˆt0(x,ξ);x,ξ)− ˆϑ(ˆt0(x,ξ);x,ξ)T (r−ˆx(ˆt0(x,ξ)))|2
(cid:115)

−||r−ˆx(ˆt0(x,ξ))||2

ˆN (ˆt0(x,ξ);x,ξ,N0)

i2π

ˆτtt(ˆt0(x, ξ); x, ξ)

ˆC(ˆt0(x, ξ))e

The difference |ˆt0(x, ξ) − ˆt0(y, p)| is around O

bound can be obtained using Lemma 3.3.2,

2

.

(3.79)

(cid:18)

1(cid:112)||p||

(cid:19)

. To see this, the following

|ˆyd(t) − ˆxd(t)| ≤ ||ˆy(t) − ˆx(t)|| ∼ O(

1(cid:112)||p||).

(3.80)

51

By Assumption 3.2.1, we have

|ˆt0(x, ξ) − ˆt0(y, p)| ≤ |ˆyd(t) − ˆxd(t)|

b

∼ O

(cid:32)

(cid:33)

.

1(cid:112)||p||

(3.81)

The following proof is essentially comparing each term of equation (3.79) at two dif-

ferent times ˆt0(x, ξ) and ˆt0(y, p).
First, e−iωˆt0(x,ξ) becomes,

e−iωˆt0(x,ξ) = e−iωˆt0(y,p)e−iω(ˆt0(x,ξ)−ˆt0(y,p)).

(3.82)

Second, we will discuss eiˆγ(ˆt0(x,ξ);x,ξ)|ω−ˆτt(ˆt0(x,ξ);x,ξ)− ˆϑ(ˆt0(x,ξ);x,ξ)T (r−ˆx(ˆt0))|2

.

Proposition 3.3.3.

iˆγ(ˆt0(x, ξ); x, ξ)|ω − ˆτt(ˆt0(x, ξ); x, ξ) − ˆϑ(ˆt0(x, ξ); x, ξ)T (r − ˆx(ˆt0(x, ξ)))|2 =
(r − ˆx(ˆt0(y, p)))|2

iˆγ(ˆt0(y, p); x, ξ)|ω − ˆτt(ˆt0(y, p); x, ξ) −(cid:16) ˆϑ(ˆt0(y, p); x, ξ))
(cid:17)T
(cid:18) 1

(cid:19)

(3.83)

(cid:33)

(cid:32)

1(cid:112)||p||

.

(3.84)

+ O

||p||

.

Proof. See Appendix A.

Third, we will discuss ||r − ˆx(ˆt0(x, ξ))||2
ˆN .

Proposition 3.3.4.

||r − ˆx(ˆt0(x, ξ))||2

ˆN (ˆt0(x,ξ))

= ||r − ˆx(ˆt0(y, p))||2

ˆN (ˆt0(y,p))

+ O

52

Proof. According to Lemma A.0.1, we have,

ˆN (ˆt0(x, ξ); x, ξ) = Im( ˆN )(ˆt0(x, ξ); x, ξ) − Im( ˆN ) ˆξ ˆξT Im( ˆN )

ˆξT Im( ˆN ) ˆξ

(ˆt0(x, ξ); x, ξ)).

Then,

ˆN (ˆt0(x, ξ); x, ξ) ˆξ(ˆt0(x, ξ))
= Im( ˆN )(ˆt0(x, ξ); x, ξ) ˆξ(ˆt0(x, ξ)) − Im( ˆN )(ˆt0(x, ξ); x, ξ) ˆξ(ˆt0(x, ξ)) = 0.

(3.85)

This is also correct for any vector parallel to ˆξ(ˆt0(x, ξ)). On the other hand,

r − ˆx(ˆt0(x, ξ)) = r − ˆx(ˆt0(y, p)) + ˆx(ˆt0(y, p)) − ˆx(ˆt0(x, ξ))
= r − ˆx(ˆt0(y, p)) + G±
= r − ˆx(ˆt0(y, p)) ± v(ˆx(ˆt0(y, p)))(ˆt0(y, p) − ˆt0(x, ξ))

p (ˆx(ˆt0(y, p)), ˆξ(ˆt0(y, p)))(ˆt0(y, p) − ˆt0(x, ξ))
ˆξ(ˆt0(x, ξ))
|| ˆξ(ˆt0(x, ξ))||.

(3.86)

Consequently,

||r − ˆx(ˆt0(x, ξ))||2
= ||r − ˆx(ˆt0(y, p))||2

ˆN (ˆt0(x,ξ))

ˆN (ˆt0(y,p))

= ||r − ˆx(ˆt0(y, p))||2

ˆN (ˆt0(y,p))

+ O

+ ||r − ˆx(ˆt0(y, p))||2

(cid:32)

(cid:33)

1(cid:112)||p||

= ||r − ˆx(ˆt0(y, p)) + λ ˆξ(ˆt0(x, ξ))||2

ˆN (ˆt0(x,ξ))

ˆN (ˆt0(x,ξ))− ˆN (ˆt0(y,p))

.

(3.87)

Finally, about ˆ
(r, ˆt0(x, ξ); x, ξ) defined in equation (3.74). All other terms are at

53

constant order, except for ˆξ(ˆt0(x, ξ)) · (r − ˆx(ˆt0(x, ξ))),

ˆξ(ˆt0(x, ξ)) · (r − ˆx(ˆt0(x, ξ))) = ˆξ(ˆt0(y, p)) · (r − ˆx(ˆt0(x, ξ)))

= ˆξ(ˆt0(y, p)) ·(cid:0)r − ˆx(ˆt0(y, p))(cid:1) ± v(ˆx(ˆt0(y, p)))(ˆt0(y, p) − ˆt0(x, ξ))|| ˆξ(ˆt0(y, p))||)
= ˆξ(ˆt0(y, p)) ·(cid:0)r − ˆx(ˆt0(y, p))(cid:1) ± v||ˆp(ˆt0(y, p))||(ˆt0(y, p) − ˆt0(x, ξ)) + O(1)

= ˆξ(ˆt0(y, p)) · (r − ˆx(ˆt0(y, p)) − ˆτt(ˆt0(y, p); y, p)(ˆt0(y, p) − ˆt0(x, ξ)) + O(1).

(3.88)

The asymptotic analysis in the last step comes from Lemma 3.3.2, that is ||ˆp(t) −

ˆξ(t)|| ∼ O((cid:112)||p||).

To expedite the discussion, we will use the following notations:

ˆtc = ˆt0(y, p),

∆ˆt0(x, ξ; y, p) = ˆt0(y, p) − ˆt0(x, ξ),

˜tc = ˜t0(y, q),

∆˜t0(x, η; y, q) = ˜t0(y, q) − ˜t0(x, η).

Lemma 3.3.3. By Proposition 3.3.3, Proposition 3.3.4 and equation (3.82) and (3.88),

i2π

ˆτtt(ˆtc; x, ξ)

ˆU x
ξ (r, ω; ˆt0(x, ξ)) =

ei(ω−ˆτt(ˆtc;x,ξ))∆ˆt0(x,ξ;y,p)ei ˆξ(ˆtc)·(r−ˆx(ˆtc))eiˆγ(ˆtc;x,ξ)|ω−ˆτt(ˆtc;x,ξ)− ˆϑ(ˆtc;x,ξ)T (r−ˆx(ˆtc;x,ξ))|2
−||r−ˆx(ˆtc;x,ξ)||2
e

ˆC(ˆtc)eiˆ
(r,ˆtc;x,ξ)−i ˆξ(ˆt0(x,ξ))·(r−ˆx(ˆt0(x,ξ)))e−iωˆtc
(cid:33)

ˆN (ˆtc)

2 + O

(3.89)

(cid:115)

(cid:32)

1(cid:112)||p||

.

Now, since two beams are using the same traveltime, we will abbreviate parameter ˆt0

54

in ˆU x

ξ (r, ω; ˆt0(x, ξ)). On the source side,

Corollary 3.3.2. When we calibrate the beam ˜U x

η (s, ω; ˜t0(x, η)) according to the beam

˜U y
q (s, ω) by shifting time ˜t0(x, η) to time ˜tc, we have

(cid:115)

˜U x
η (s, ω; ˜t0(x, η)) =

i2π

˜τtt(˜tc; x, η)

ei(ω−˜τt(˜tc;x,η))∆˜t0(x,η;y,q)ei˜η(˜tc)·(s−˜x(˜tc))

ei˜γ(˜tc;x,η)|ω−˜τt(˜tc;x,η)− ˜ϑ(˜tc;x,η)T (s−˜x(˜tc;x,η))|2

˜C(˜tc)ei˜
(s,˜tc;x,η)−i˜η(˜t0(x,η))·(s−˜x(˜t0(x,η)))e−iω˜tc
(cid:33)

(cid:32)

−||s−˜x(˜tc;x,η)||2
e

˜N (˜tc)

2 + O

.

1(cid:112)||q||

(3.90)

3.3.3.3 Difference between Two Interacted Beams’ Phase and Hessians

We will compare the difference between ˆM (t) − ˆN (t) in this section. We first have the

following inequality

Proposition 3.3.5. Consider two scattering beams (ˆy(t), ˆp(t), ˆM (t), ˆA(t)) and

(ˆx(t), ˆξ(t), ˆN (t), ˆC(t)), and assume that there exists significant interaction effects be-
tween these two beams. There exists two bounded positive constants C∗

1 and C∗

2 related

to the background velocity, such that

d|| ˆM (t) − ˆN (t)||

dt

≤ C∗

1

(cid:112)||p|| + C∗

2|| ˆM (t) − ˆN (t)||.

(3.91)

where || ˆM (t) − ˆN (t)|| is defined as the matrix norm induced by the vector 2-norm.

Proof. See Appendix A.

55

Lemma 3.3.4. Consider two scattering beams (ˆy(t), ˆp(t), ˆM (t), ˆA(t)) and

(ˆx(t), ˆξ(t), ˆN (t), ˆC(t)), and there exists significant interaction effects between these two

beams. Then

|| ˆM (t) − ˆN (t)|| ∼ O

(cid:16)(cid:112)||p||(cid:17)

,

∀t ∈ [0, T ],

(3.92)

Proof. First, ˆM (t)− ˆN (t) is zero at t = 0, since they satisfy the same initial condition.
Using Proposition 3.3.5 and the fact that the norm || ˆM (t)− ˆN (t)|| is positive and both
C∗
1 and C∗

2 are positive,

|| ˆM (t) − ˆN (t)|| ≤ C∗

1

C∗
2|| ˆM (s) − ˆN (s)||ds,

(3.93)

(cid:112)||p(0)||T +

(cid:90) t

0

since the boundary data D(r, s, t) is measured in the time interval [0, T ]. With Gronwall

inequality,

|| ˆM (t) − ˆN (t)|| ≤ C∗

2 t(cid:112)||p||.

1 T eC∗

(3.94)

Moreover, we have the same conclusion for other related terms,

Corollary 3.3.3.

|| ˆM(t) − ˆN (t)|| ∼ O((cid:112)||p||),
||ˆτtx(ˆtc; y, p) − ˆτtx(ˆtc; x, ξ)|| ∼ O((cid:112)||p||),
|ˆτtt(ˆtc; y, p) − ˆτtt(ˆtc; x, ξ)| ∼ O((cid:112)||p||),

|| ˜M(t) − ˜N (t)|| ∼ O((cid:112)||q||);
||˜τtx(˜tc; y, q) − ˆτtx(˜tc; x, η)|| ∼ O((cid:112)||q||);
|˜τtt(˜tc; y, q) − ˜τtt(˜tc; x, η)| ∼ O((cid:112)||q||).

Consequently, by Lemma A.0.1, both ˆU y

p (r, ω) and ˆU x

ξ (r, ω) are well-localized along

56

the boundary, and we have

1
2

||r − ˆx(t)||2

ˆM− ˆN ∼ O(||p||− 1

2 ),

which will be used in equation (3.101) when we replace ˆN with ˆM. Similarly,

||s − ˜x(t)||2

˜M− ˜N ∼ O(||q||− 1

2 ).

1
2

Lemma 3.3.5. Consider two scattering beams (ˆy(t), ˆp(t), ˆM (t), ˆA(t)) and

(ˆx(t), ˆξ(t), ˆN (t), ˆC(t)), and there exists significant interaction effects between these two

beams. Suppose the function g(t) is

g(t) = ˆp(t) · (ˆy(t) − ˆx(t)),

g(t) = g(0) + O(1),

(cid:32)

(cid:33)

.

||ˆΞ(t)||2
ˆκ(t)2||ˆp(t)||

1
2

(cid:48)

g

(t) = v(ˆx(t))

then we have

and

Proof. See Appendix A.

(3.95)

(3.96)

(3.97)

Lemma 3.3.6. Consider two scattering beams (ˆy(t), ˆp(t), ˆM (t), ˆA(t)) and

(ˆx(t), ˆξ(t), ˆN (t), ˆC(t)), and there exists significant interaction effects between these two

beams. Suppose the pure imaginary matrix ˆM (0) has a symmetric positive definite

57

imaginary part and is the initial condition of the Hessian for the beam, then

(y − x)T ˆM (0)(y − x) = (ˆy(t) − ˆx(t))T ˆM (t)(ˆy(t) − ˆx(t)) + O(1).

(3.98)

Proof. See Appendix A.

3.3.3.4 Approximation of Two Beams’ Interaction

explicit formula of(cid:82)(cid:82) dξdr

In this section, we will use the conclusion obtained in previous sections to get the

¯ˆU y
p (r, ω) ˆU x

ξ (r, ω). We first have the proposition below which

will be used in approximation,

Proposition 3.3.6. The real-valued phase terms, ˆ
(r, ˆtc; y, p)−ˆθ(r, ˆtc; y, p) and ˆ
(r, ˆtc; x, ξ)−
ˆθ(r, ˆtc; x, ξ), can be ignored since they are constant order terms with respect to the large
wavenumber ||ξl,i|| = ||p + q||.

Proof. See Appendix A.

58

According to equation (3.89), (3.63),

¯ˆU y
p (r, ω) ˆU x

ξ (r, ω) = eiO(1)

(cid:90)(cid:90)
(cid:90)(cid:90)
eiˆ
(r,ˆtc;x,ξ)−iˆθ(r,ˆtc;x,ξ)eiˆθ(r,ˆtc;x,ξ)eiˆγ|ω−ˆτt(ˆtc;x,ξ)− ˆϑ(r−ˆx(ˆtc))|2
−||r−ˆx(ˆtc)||2
e

dξdr
drdξeiωˆtc−iωˆtce−iˆ
(r,ˆtc;y,p)+iˆθ(r,ˆtc;y,p)e−iˆθ(r,ˆtc;y,p)e(i ˆβ)|ω−ˆτt(ˆtc;y,p)−ˆζ(r−ˆy(ˆtc))|2
ei(ω−ˆτt(ˆtc;y,p))∆ˆt0(x,ξ;y,p)

−||r−ˆy(ˆtc)||2

ˆM(ˆtc)

ˆN (ˆtc)

e

2

2

(cid:90)
(cid:16)

× e

where

2||ˆy(ˆtc)−ˆx(ˆtc)||2
i 1
ˆM(ˆtc)e

Re( ˆM )(ˆtc)

eiˆγ|ω−ˆτt(ˆtc;x,ξ)− ˆϑT (r−ˆx(ˆtc))|2

− 1
4||ˆy(ˆtc)−ˆx(ˆtc)||2

i ˆβ

dre

= eiO(1)

dξe−iωˆtc+iωˆtce

(cid:17)|ω−ˆτt(ˆtc;y,p)−ˆζT (r−ˆy(ˆtc))|2

(cid:90)
×
× ei ˆξ(ˆtc)·(r−ˆx(ˆtc))−iˆp(ˆtc)·(r−ˆy(ˆtc))ei(ω−ˆτt(ˆtc;y,p))∆ˆt0(x,ξ;y,p)
4||ˆy(ˆtc)−ˆx(ˆtc)||2

− 1
2||r−ˆx(ˆtc)||2

− 1
2||r−ˆy(ˆtc)||2

1

ˆM(ˆtc)e

ˆN (ˆtc)e

ˆM(ˆtc),

ei ˆχ(r,ˆtc;y,p,x,ξ)

(3.99)

||ˆy(ˆtc) − ˆx(ˆtc)||2
ˆχ(r, ˆtc; y, p, x, ξ) = −1
2
||r − ˆx(ˆtc)||2
1
2

+

Re( ˆN )(ˆtc)

Re( ˆM )(ˆtc)

− 1
2

||r − ˆy(ˆtc)||2

Re( ˆM )(ˆtc)

.

(3.100)

Here we neglect some constant order real-valued phase terms by Proposition 3.3.6,

which can be considered as a smooth residual.

59

Then we replace ˆN with ˆM, and the inner integral becomes,

(cid:90)

dre(i ˆβ)|ω−ˆτt(ˆtc;y,p)−ˆζT (r−ˆy(ˆtc))|2

eiˆγ|ω−ˆτt(ˆtc;x,ξ)− ˆϑT (r−ˆx(ˆtc))|2
− 1
2||r−ˆy(ˆtc)||2

ˆM(ˆtc)

1

× ei ˆξ(ˆtc)·(r−ˆx(ˆtc))−iˆp(ˆtc)·(r−ˆy(ˆtc))ei ˆχe
4||ˆy(ˆtc)−ˆx(ˆtc)||2

− 1
2||r−ˆx(ˆtc)||2

ˆN (ˆtc)e

× e
−i
= ei(ω−ˆτt(ˆtc;y,p))∆ˆt0(x,ξ;y,p)eiˆp(ˆtc)·(ˆy(ˆtc)−ˆx(ˆtc))e
(cid:90)

ˆξ(ˆtc)−ˆp(ˆtc)

·(2r−ˆx(ˆtc)−ˆy(ˆtc))e

ei ˆχei

2

− 1
4||2r−ˆx(ˆtc)−ˆy(ˆtc)||2

ˆM(ˆtc)

ˆM(ˆtc)ei(ω−ˆτt(ˆtc;y,p))∆ˆt0(x,ξ;y,p)

(3.101)

ˆp(ˆtc)− ˆξ(ˆtc)

2

·(ˆy(ˆtc)−ˆx(ˆtc))

e(i ˆβ)|ω−ˆτt(ˆtc;y,p)−ˆζT (r−ˆy(ˆtc))|2

eiˆγ|ω−ˆτt(ˆtc;x,ξ)− ˆϑT (r−ˆx(ˆtc))|2

dr +

(cid:32)

(cid:33)

1(cid:112)||p||

.

(3.102)

The difference of replacing Hessian matrix has been evaluated in Lemma 3.3.4 and its

Corollary. Obviously, now the integral about the receiver variable r is well-defined.

We denote its result as ˆB(x, ξ, ω; y, p),

ei ˆχei

ˆB(x, ξ, ω; y, p) =
× e(i ˆβ)|ω−ˆτt(ˆtc;y,p)−ˆζT (r−ˆy(ˆtc))|2

2

ˆξ(ˆtc)−ˆp(ˆtc)

·(2r−ˆx(ˆtc)−ˆy(ˆtc))e

− 1
4||2r−ˆx(ˆtc)−ˆy(ˆtc)||2

ˆM(ˆtc)

eiˆγ|ω−ˆτt(ˆtc;x,ξ)− ˆϑT (r−ˆx(ˆtc))|2

dr.

(3.103)

(cid:90)

(cid:90)

Similarly,

ei ˜χei

˜B(x, η, ω; y, q) =
× e(i ˜β)|ω−˜τt(˜tc;y,q)−˜ζT (s−˜y(˜tc))|2

2

˜η(˜tc)−˜q(˜tc)

·(2s−˜x(˜tc)−˜y(˜tc))e

− 1
4||2s−˜x(˜tc)−˜y(˜tc)||2

˜M(˜tc)

ei˜γ|ω−˜τt(˜tc;x,η)− ˜ϑT (s−˜x(˜tc))|2

ds.

(3.104)

We can see that the integral about r and s is accounted for in the computation of ˆB

and ˜B.

60

3.3.3.5

Integral about Boundary Points r and s

In this section, we will evaluate ˆB and ˜B defined in equation (3.103) and (3.104) to

show that they are essentially Gaussian functions. Continuing from the expression

(3.103) of ˆB(x, ξ, ω; y, p), we first simplify the exponent ˆχ(r, ˆtc; y, p, x, ξ),

||ˆy(ˆtc) − ˆx(ˆtc)||2
ˆχ = −1
2
||ˆy(ˆtc) − ˆx(ˆtc)||2

= −1
2

Re( ˆM )(ˆtc)

Re( ˆM )(ˆtc)

+

1
2

||r − ˆy(ˆtc)||2
− 1
Re( ˆM )(ˆtc)
2
(ˆy(ˆtc) − ˆx(ˆtc))T Re( ˆM )(ˆtc)(2r − ˆx(ˆtc) − ˆy(ˆtc)),

||r − ˆx(ˆtc)||2

Re( ˆM )(ˆtc)

1
2

+

so that we have

−i 1

2||ˆy(ˆtc)−ˆx(ˆtc)||2

Re( ˆM )(ˆtc)

− Re( ˆM )(ˆtc)

(ˆy(ˆtc)−ˆx(ˆtc))

2

(cid:19)
·(2r−ˆx(ˆtc)−ˆy(ˆtc))

ˆB(x, ξ, ω; y, p) = e

(cid:18) ˆp(ˆtc)− ˆξ(ˆtc)

(cid:90)

−i

2

e
− 1
4||2r−ˆx(ˆtc)−ˆy(ˆtc)||2

× e

ˆM(ˆtc) ˆF(r, x, ξ, ω; y, p)dr,

(3.105)

(3.106)

where

ˆF(r, x, ξ, ω; y, p) =
e−Im( ˆβ)|ω−ˆτt(ˆtc;y,p)−ˆζT (r−ˆy(ˆtc))|2

e−Im(ˆγ)|ω−ˆτt(ˆtc;x,ξ)− ˆϑT (r−ˆx(ˆtc))|2

eiˆg(r,x,ξ,ω;y,p),

(3.107)

61

and

ˆg(r, x, ξ, ω; y, p) = −Re( ˆβ)|ω − ˆτt(ˆtc; y, p) − ˆζT (r − ˆy(ˆtc))|2
+ Re(ˆγ)|ω − ˆτt(ˆtc; x, ξ) − ˆϑT (r − ˆx(ˆtc))|2.

(3.108)

Proposition 3.3.7. The sum of first two terms in the exponent of ˆF in (3.107) satisfy,

− Im( ˆβ)|ω − ˆτt(ˆtc; y, p) − ˆζT (r − ˆy(ˆtc))|2 − Im(ˆγ)|ω − ˆτt(ˆtc; x, ξ) − ˆϑT (r − ˆx(ˆtc))|2
= −||r − ˆx(ˆtc) + ˆy(ˆtc)

2Im( ˆβ)ˆζ ˆζT − Im( ˆβ)
||2

||ˆy(ˆtc) − ˆx(ˆtc)||2
ˆζ ˆζT

2

2

− Im( ˆβ)|ω − ˆτt(ˆtc; y, p)|2 − Im(ˆγ)|ω − ˆτt(ˆtc; x, ξ)|2 + O

,

(3.109)

(cid:32)

(cid:33)

1(cid:112)||p||

where ˆβ, ˆγ, ˆζ and ˆϑ are all defined at ˆtc. Similarly, ˆg in equation (3.108) is an O(1)

term.

Proof. See Appendix A.

Lemma 3.3.7. The integral ˆB(x, ξ, ω; y, p) defined in (3.106) has a Gaussian profile

centered at ˆy(ˆtc) and ˆp(ˆtc),

ˆB(x, ξ, ω; y, p) = e

− i
2||ˆy(ˆtc)−ˆx(ˆtc)||2

Re( ˆM )(ˆtc)−iIm( ˆβ)ˆζ ˆζT

(cid:118)(cid:117)(cid:117)(cid:116)

e−Im( ˆβ)|ω−ˆτt(ˆtc;y,p)|2−Im(ˆγ)|ω−ˆτt(ˆtc;x,ξ)|2
−||(cid:16)
(cid:17)||2

ˆp(ˆtc)− ˆξ(ˆtc)−Re( ˆM )(ˆtc)(ˆy(ˆtc)−ˆx(ˆtc))

(2π)d−1

det( ˆM(ˆtc) + Im( ˆβ)
2 Im( ˆβ)ˆζ ˆζT )−1

2

( ˆM(ˆtc)+ 1

ˆζ ˆζT )

eiO(1),

e

where we ignore the real-valued phase term ˆg.

62

Proof. Using Proposition 3.3.7, the integral ˆB in (3.106) becomes the Fourier transform

of Gaussian functions, so that we have

ˆB(x, ξ, ω; y, p) = e

−i 1

2||ˆy(ˆtc)−ˆx(ˆtc)||2

Re( ˆM )(ˆtc)−iIm( ˆβ)ˆζ ˆζT

(cid:118)(cid:117)(cid:117)(cid:116)

(2π)d−1

det( ˆM(ˆtc) + Im( ˆβ)
2 Im( ˆβ)ˆζ ˆζT )−1

2

.

( ˆM(ˆtc)+ 1

ˆζ ˆζT )

(3.110)

e−Im( ˆβ)|ω−ˆτt(ˆtc;y,p)|2−Im(ˆγ)|ω−ˆτt(ˆtc;x,ξ)|2
−||(cid:16)
(cid:17)||2

ˆp(ˆtc)− ˆξ(ˆtc)−Re( ˆM )(ˆtc)(ˆy(ˆtc)−ˆx(ˆtc))

e

Corollary 3.3.4. Similarly, on the source side,

˜B(x, η, ω; y, q) = e

−i 1

2||˜y(˜tc)−˜x(˜tc)||2

Re( ˜M )(˜tc)−iIm( ˜β)˜ζ ˜ζT

e−Im( ˜β)|ω−˜τt(˜tc;y,q)|2−Im(˜γ)|ω−˜τt(˜tc;x,η)|2
(cid:17)||2
−||(cid:16)

˜q(˜tc)−˜η(˜tc)−Re( ˜M )(˜tc)(˜y(˜tc)−˜x(˜tc))

e

(cid:118)(cid:117)(cid:117)(cid:116)

(2π)d−1

det( ˜M)(˜tc) + Im( ˜β)

2

˜ζ ˜ζT )

( ˜M(˜tc)+ 1

2 Im( ˜β)˜ζ ˜ζT )−1

eiO(1).

(3.111)

3.3.3.6 Conclusion of Two Beams’ Interaction

To summarize,

(cid:90)

(cid:90)

(cid:90)

dξe

dξ ˆU x

ξ (r, ω) ≈

− 1
4||ˆy(ˆtc)−ˆx(ˆtc)||2

2||ˆy(ˆtc)−ˆx(ˆtc)||2
i 1
ˆM(ˆtc)e

Re( ˆM )(ˆtc)

¯ˆU y
p (r, ω)

dr

ei(ω−ˆτt(ˆtc;y,p))∆ˆt0(x,ξ;y,p)eiˆp(ˆtc)·(ˆy(ˆtc)−ˆx(ˆtc))ei

ˆξ(ˆtc)−ˆp(ˆtc)

2

·(ˆy(ˆtc)−ˆx(ˆtc)) ˆB(x, ξ, ω; y, p).

(3.112)

63

By Lemma 3.3.5 and Lemma 3.3.6, equation (3.112) reduces to

(cid:90)(cid:90)

drdξ

¯ˆU y
p (r, ω) ˆU x

ξ (r, ω) = eip·(y−x) ˆH(x, ω; y, p),

(3.113)

where

ˆH(x, ω; y, p) =

(cid:90)
2 ( ˆξ(ˆtc)−ˆp(ˆtc))·(ˆy(ˆtc)−ˆx(ˆtc)) ˆB(x, ξ, ω; y, p)ei(ω−ˆτt(ˆtc;y,p))∆ˆt0(x,ξ;y,p),
ei 1

dξei( ˆψ1(ˆtc)− ˆψ1(0))ei( ˆψ2(ˆtc)− ˆψ2(0))e

− 1
4||ˆy(ˆtc)−ˆx(ˆtc)||2

ˆM(ˆtc)

(3.114)

and

ˆψ1(t; x, ξ, y, p) = ˆp(t) · (ˆy(t) − ˆx(t));

ˆψ1(0; x, ξ, y, p) = p · (y − x);

ˆψ2(t; x, ξ, y, p) =

||ˆy(t) − ˆx(t)||2

Re( ˆM )(t)

1
2

;

ˆψ2(0; x, ξ, y, p) =

||y − x||2

Re( ˆM )(0)

.

1
2

(3.115)

The extra term e

Re( ˆM )(0) = 1, since ˆM (0) is a pure imaginary matrix in

−i 1

4||y−x||2

Gaussian wavepacket transform. We denote

ˆL(x, ξ, ˆtc; y, p) =

ˆψi(ˆtc) − ˆψi(0) − 1
2

(ˆp(ˆtc) − ˆξ(ˆtc)) · (ˆy(ˆtc) − ˆx(ˆtc)).

(3.116)

(cid:88)

i=1,2

(cid:90)(cid:90)

Lemma 3.3.5 and Lemma 3.3.6 guarantees ˆL is O(1).

Lemma 3.3.8. The receiver-side beam interaction reduces to,

drdξ

¯ˆU y
p (r, ω) ˆU x

ξ (r, ω) ≈ eip·(y−x) ˆH(x, ξ, ω; y, p),

64

where

ˆH(x, ξ, ω; y, p) =

(cid:90)
ˆM(ˆtc)e
× ˆB(x, ξ, ω; y, p)ei(ω−ˆτt(ˆtc;y,p))∆ˆt0(x,ξ;y,p).

− 1
4||ˆy(ˆtc)−ˆx(ˆtc)||2

dξei ˆL(x,ξ,ˆtc;y,p)e

−i 1

2||y−x||2

Re( ˆM )(0)

Corollary 3.3.5. The source-side beam interaction reduces to,

(cid:90)(cid:90)

dsdη ¯˜U y

q (s, ω) ˜U x

η (s, ω) ≈ eiq·(y−x) ˜H(x, η, ω; y, q),

where

˜H(x, η, ω; y, q) =

(cid:90)
˜M(˜tc)e
× ˜B(x, η, ω; y, q)ei(ω−˜τt(˜tc;y,q))∆˜t0(x,η;y,q),

− 1
4||˜y(˜tc)−˜x(˜tc)||2

dηei ˜L(x,η,˜tc;y,q)e

−i 1

2||y−x||2

Re( ˜M )(0)

where ˜L is defined accordingly.

65

3.3.4 Asymptotic Analysis of Four Beams’ Interaction

Using Lemma 3.3.8 and Corollary 3.3.5, the left-hand side of equation (3.47) now

becomes,

p (r, ω) ¯˜U y

ξ (r, ω) ˜U x

η (s, ω)

dωdrdsdξdηω2 ¯ˆU y

(cid:90)(cid:90)(cid:90)(cid:90)
−
q (s, ω) ˆU x
≈ −ei(p+q)(y−x) ˆH(x, ξ, ω; y, p) ˜H(x, η, ω; y, q)
= −ei(p+q)(y−x)
(cid:90)(cid:90)
(cid:90)
dξdηei ˆL(x,ξ,ˆtc;y,p)+i ˜L(x,η,˜tc;y,q)e
ω2 ˆB ˜Bei(ω−ˆτt(ˆtc;y,p))∆ˆt0(x,ξ;y,p)ei(ω−˜τt(ˆtc;y,q))∆˜t0(x,η;y,q)dω.

4||ˆy(ˆtc)−ˆx(ˆtc)||2
− 1

ˆM(ˆtc)e

4||˜y(˜tc)−˜x(˜tc)||2
− 1

˜M(˜tc)

(3.117)

We will discuss the interaction between four beams in this subsection.

3.3.4.1

Integral about Wavenumber ω

We will evaluate the first layer of integral (3.117) about frequency ω in this subsection.

Before that, we define a function K(p, q, y),

(cid:32)

Im( ˆβ)(cid:0)ˆτt(ˆtc; y, p)(cid:1) + Im( ˜β)(cid:0)˜τt(˜tc; y, q)(cid:1)

(cid:33)2

K(p, q, y) ≡ −

Im( ˆβ) + Im( ˜β)

Im( ˆβ)Im( ˜β)(ˆτt(ˆtc;y,p)−˜τt(˜tc;y,q))2

Im( ˆβ)+Im( ˜β)

−2
e

,

(3.118)

66

e−Im( ˆβ)|ˆτt(ˆtc;y,p)−ˆτt(ˆtc;x,ξ)|2

− Im( ˆβ)
e

2

||ˆy(ˆtc)−ˆx(ˆtc)||2

ˆζ ˆζT

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2(cid:16) ˆM(ˆtc)+2Im( ˆβ)ˆζ ˆζT(cid:17)−1

,

(3.119)

e−Im( ˜β)|˜τt(˜tc;y,q)−˜τt(˜tc;x,η)|2

− Im( ˜β)

2

e

||˜y(˜tc)−˜x(˜tc)||2

˜ζ ˜ζT

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2(cid:16) ˜M(˜tc)+2Im( ˜β)˜ζ ˜ζT(cid:17)−1

.

(3.120)

a function ˆB(x, ξ; y, p, q) on the receiver side,

ˆB = e

−|∆ˆt0(x,ξ;y,p)|2
4Im( ˆβ+ ˜β)

(2π)d−1

(cid:118)(cid:117)(cid:117)(cid:116)
−(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ˆp(ˆtc)− ˆξ(ˆtc)−Re( ˆM )(ˆtc)(ˆy(ˆtc)−ˆx(ˆtc))

det( ˆM(ˆtc) + Im( ˆβ)

ˆζ ˆζT )

2

×

and a function ˜B(x, η; y, p, q) on the source side,

˜B = e

−|∆˜t0(x,η;y,q)|2
4Im( ˆβ+ ˜β)

(2π)d−1

(cid:118)(cid:117)(cid:117)(cid:116)
−(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)˜q(˜tc)−˜η(˜tc)−Re( ˜M )(˜tc)(˜y(˜tc)−˜x(˜tc))

det( ˜M(˜tc) + Im( ˜β)

˜ζ ˜ζT )

2

×

e

e

(cid:90)

−

Lemma 3.3.9. The result after taking the integral about ω can be approximated,

ei(ω−ˆτt(ˆtc;y,p))∆ˆt0(x,ξ;y,p)ei(ω−˜τt(ˆtc;y,q))∆˜t0(x,η;y,q)ω2 ˆB ˜Bdω
− i
2||ˆy(ˆtc)−ˆx(ˆtc)||2

= eiO(1)K(p, q, y) ˆB(x, ξ; y, p, q) ˜B(x, η; y, p, q)e
2||˜y(˜tc)−˜x(˜tc)||2
− i

Re( ˜M (˜tc)),

e

Re( ˆM (ˆtc))

(3.121)

where both ˆB and ˜B have phase functions with pure imaginary part only.

Proof. See Appendix A.

67

3.3.4.2

Integral about Momentum ξ and η: Evaluation of Real Part of

Phase

We can define the following integral directly from equation (3.117),

(cid:90)

ei ˆL(x,ξ,ˆtc;y,p)

ˆH(x; y, p, q) =
−i 1
e

2||ˆy(ˆtc)−ˆx(ˆtc)||2

Re( ˆM )(ˆtc) ˆB(x, ξ; y, p, q)e

− 1
4||ˆy(ˆtc)−ˆx(ˆtc)||2

ˆM(ˆtc)dξ,

where

ˆL(x, ξ, ˆtc; y, p) =

(cid:88)

i=1,2

and recall that

ˆψi(ˆtc) − ˆψi(0) − 1
2

(ˆp(ˆtc) − ˆξ(ˆtc)) · (ˆy(ˆtc) − ˆx(ˆtc)),

ˆψ1(t; x, ξ, y, p) = ˆp(t) · (ˆy(t) − ˆx(t));

ˆψ1(0; x, ξ, y, p) = p · (y − x);

ˆψ2(t; x, ξ, y, p) =

||ˆy(t) − ˆx(t)||2

Re( ˆM )(t)

1
2

;

ˆψ2(0; x, ξ, y, p) =

||y − x||2

Re( ˆM )(0)

.

1
2

(3.122)

Define functions ˆφ1 and ˆφ2 as derivatives of equation (3.122),

ˆφ1(t; x, ξ, y, p) =

d ˆψ1(t; x, ξ, y, p)

dt

, ˆφ2(t; x, ξ, y, p) =

d ˆψ2(t; x, ξ, y, p)

dt

.

(3.123)

In this subsection, we will explore the real part of the phase function in ˆH, i.e., ˆL −
2||ˆy(ˆtc) − ˆx(ˆtc)||2

1

.

Re( ˆM )

68

Proposition 3.3.8. The function ˆφ1 satisfies

ˆφ1(t; x, ξ, y, p) =

||ˆp(t) − ˆξ(t)||2

Gpp + O

1
2

(cid:32)

(cid:33)

.

1(cid:112)||p||

Proof. See Appendix A.

Define a 2d by 2d real-valued matrix ˆR,

 0,

− 1

ˆR(t; y, p) =

 .

− 1
4 I

4I,

tGpp(ˆy(t), ˆp(t))

Proposition 3.3.9. ˆH(x, y, p, q) satisfies

(cid:90)

ˆH(x, y, p, q) = eiO(1)

Proof. First,

dξ ˆB(x, ξ; y, p, q)e

− 1
4||ˆy(ˆtc)−ˆx(ˆtc)||2

ˆMe

i||(ˆy−ˆx,ˆp− ˆξ)(

(3.124)

(3.125)

2 )||2
ˆtc
ˆR

(cid:19)

(cid:18) ˆtc

2

.

ˆL − 1
2

||ˆy(ˆtc) − ˆx(ˆtc)||2

Re( ˆM )(ˆtc)

=

ˆψ1(ˆtc) − ˆψ1(0) − ˆψ2(0) − 1
2
= ˆψ1(ˆtc) − ˆψ1(0) − 1
2

(ˆp(ˆtc) − ˆξ(ˆtc)) · (ˆy(ˆtc) − ˆx(ˆtc))

(ˆp(ˆtc) − ˆξ(ˆtc)) · (ˆy(ˆtc) − ˆx(ˆtc)),

since Re( ˆM )(0) = 0.

We approximate ˆL as,

ˆψ1(ˆtc) − ˆψ1(0) =

(cid:18) ˆtc

(cid:19)

2

ˆtc.

ˆφ1(t)dt ≈ ˆφ1

(cid:90) ˆtc

0

69

(3.126)

(3.127)

(cid:18)

ˆp

(cid:18) ˆtc

2

(cid:19)

(cid:18) ˆtc

(cid:19)(cid:19)

2

(cid:18)

·

ˆy

(cid:18) ˆtc

(cid:19)

2

− ˆξ

(cid:18) ˆtc

(cid:19)(cid:19)

2

− ˆx

+ O(1).

Moreover, by using Lemma 3.3.5 twice,

(ˆp(ˆtc) − ˆξ(ˆtc)) · (ˆy(ˆtc) − ˆx(ˆtc)) =

(3.128)

To summarize,

ˆL − 1
2
ˆtc
2

ˆφ1

2

=

ˆtc
2

||ˆp

||ˆy(ˆtc) − ˆx(ˆtc)||2

(cid:18) ˆtc
(cid:19)
(cid:18) ˆtc

2

2

≈

(cid:19)

(cid:18) ˆtc
(cid:18)
(cid:18) ˆtc
(cid:19)

Re( ˆM )(ˆtc)
− ˆξ

(cid:18) ˆtc
||2
Gpp − 1

ˆp

2

2

2

2

(cid:19)

− 1
2
− ˆξ

= ||(ˆy − ˆx, ˆp − ˆξ)||2
ˆR(

ˆtc
2 )

(cid:19)(cid:19)
(cid:18)
(cid:18) ˆtc
(cid:18)

·

(cid:18) ˆtc
(cid:19)

ˆy

(cid:19)
(cid:18) ˆtc

2
− ˆξ

ˆp

(cid:18) ˆtc
(cid:19)(cid:19)

2
·

− ˆx

2

2

(cid:19)(cid:19)
(cid:18)
(cid:18) ˆtc

ˆy

2

(cid:19)

− ˆx

(cid:18) ˆtc

(cid:19)(cid:19)

2

.

(3.129)

Corollary 3.3.6.

(cid:90)

˜H(x, y, p, q) ≈

dη ˜B(x, η; y, p, q)e

− 1
4||˜y(˜tc)−˜x(˜tc)||2

˜M exp

(cid:32)

i||(˜y − ˜x, ˜q − ˜η)(

(cid:33)

.

˜tc
2

)||2
˜R(

˜tc
2 )

All terms are defined accordingly.

Now the right hand side of equation (3.117) is equal to ei(p+q)(y−x) ˆH ˜H. Therefore,
we will focus on ˆH and ˜H next.

70

3.3.4.3

Integral about Momentum ξ and η: Evaluation of Imaginary Part

of Phase

After seeing the real part of the phase function in Proposition 3.3.9 is a complete

quadratic term, we will explore more about the imaginary part in this section. Similar

to the real part, we will prove the imaginary part is a complete quadratic term as well

as a non-degenerate quadratic term.

We start with rewriting ˆB(x, ξ; y, p, q) in (3.119),

(cid:118)(cid:117)(cid:117)(cid:116)

ˆB(x, ξ; y, p, q) =

(2π)d−1

det( ˆM(ˆtc) + Im( ˆβ)

2

−||(ˆy(ˆtc)−ˆx(ˆtc),ˆp(ˆtc)− ˆξ(ˆtc))||2
e

ˆI(ˆtc;y,p)

ˆζ ˆζT )

−|∆ˆt0(x,ξ;y,p)|2
4Im( ˆβ+ ˜β)
e

e−Im( ˆβ)|ˆτt(ˆtc;y,p)−ˆτt(ˆtc;x,ξ)|2

.

where ˆI(ˆtc; y, p) is a symmetric matrix depending on the fixed beam’s parameters (y, p).

−Re( ˆM )

 ( ˆM + 2Im( ˆβ)ˆζ ˆζT )−1

(cid:20)

I

ˆI(ˆtc; y, p) =

(cid:21)

 Im( ˆβ)

2

ˆζ ˆζT , 0

0,

0

 .

−Re( ˆM ) I

+

(3.130)

Using Proposition 3.3.9, we have

71

Lemma 3.3.10.

ˆH(x; y, p, q) =

(cid:118)(cid:117)(cid:117)(cid:116)

(2π)d−1

det( ˆM(ˆtc) + Im( ˆβ)

2

ˆζ ˆζT )

(cid:90)

− 1
4||ˆy(ˆtc)−ˆx(ˆtc)||2

ˆM(ˆtc)

dξe

(cid:21)
(cid:21)

2 ) − ˆx(
ˆtc
2 ) − ˆξ(
ˆtc

ˆy(
ˆy(ˆtc) − ˆx(ˆtc)

ˆp(ˆtc) − ˆξ(ˆtc)

ˆp(

ˆtc
2 )
ˆtc
2 )




 .

ˆy(ˆtc) − ˆx(ˆtc), ˆp(ˆtc) − ˆξ(ˆtc)

ˆI

(3.131)

× e−Im( ˆβ)|ˆτt(ˆtc;y,p)−ˆτt(ˆtc;x,ξ)|2

−|∆ˆt0(x,ξ;y,p)|2
4Im( ˆβ+ ˜β)
e

2 ) − ˆx(
ˆtc

ˆtc
2 ), ˆp(

2 ) − ˆξ(
ˆtc

ˆtc
2 )

ˆy(

ˆR

i
(cid:20)
−
(cid:20)

× exp

× exp

Before proving ˆH has a non-degenerate Gaussian profile, we need one extra assumption.

Transformation U between Two Phase Spaces: Suppose there is a transfor-
mation U between two phase spaces governed by the certain Hamiltonian flow. The
ξ is initially in the phase space P1 = {(x, ξ), x ∈ Rd, ξ ∈
bi-characteristic of the beam ˆU x
Rd}, then after propagating to the surface, the bi-characteristic is in the phase space
P2 = {((t, x∗), (ω, ξ∗)), t, ω ∈ R, x∗, ξ∗ ∈ Rd−1}.
(cid:16)

(cid:17)(cid:17)

(cid:17)

(cid:16)

(ˆt0(x, ξ), ˆx∗(ˆt0(x, ξ))),

ˆx(ˆt0(x, ξ)), ˆξ(ˆt0(x, ξ))

, ˆξ∗(ˆt0(x, ξ))

,

(cid:16)−G

U(x, ξ) =

(3.132)

where ˆt0(x, ξ) is the hitting time defined before, and

ˆx(ˆt0(x, ξ)) = (ˆx∗(ˆt0(x, ξ)), 0) is the corresponding hitting point on the boundary.

G(x, ξ) is the associated Hamiltonian for the central ray and ˆξ∗ is the component of the
ray direction corresponding to the tangential direction of the surface {x ∈ Rd, xd = 0},

72

that is

ˆξ∗(ˆt0(x, ξ)) = ( ˆξ1(ˆt0(x, ξ)),··· , ˆξd−1(ˆt0(x, ξ))).

(3.133)

The component ˆξd corresponding to the normal direction of the surface can be uniquely
defined by (−G, ˆξ∗) according to eikonal equation (2.6), so the degree of freedom won’t

change. We need an extra assumption about the bi-characteristic,

Assumption 3.3.1. U is invertible.

The Gaussian profile is only related to the imaginary part of the phase function, there-
fore, we first ignore the term associated with ˆR in (3.131).

Lemma 3.3.11. There exists a full-rank 2d by 2d S.P.D. matrix ˆE, such that

− 1
4

||ˆy(ˆtc) − ˆx(ˆtc)||2

ˆM(ˆtc)

(cid:20)

−

− Im( ˆβ)|ˆτt(ˆtc; y, p) − ˆτt(ˆtc; x, ξ)|2 − |∆ˆt0(x, ξ; y, p)|2
(cid:21)

 = −||(y − x, p − ξ)||2

4Im( ˆβ + ˜β)

ˆy(ˆtc) − ˆx(ˆtc)

ˆp(ˆtc) − ˆξ(ˆtc)

ˆI

ˆE .

(3.134)

ˆy(ˆtc) − ˆx(ˆtc), ˆp(ˆtc) − ˆξ(ˆtc)

Proof. There are three steps to justify the non-degenerate Gaussian profile. First,

we will use some approximations to move the left-hand side of equation (3.134) to the
phase space P2, since ˆx(ˆtc) is not on the boundary. Second, we will prove that equation
(3.134) in the phase space P2 is non-degenerate. Finally, we will use the transformation
defined in Assumption 3.3.1 to move the left-hand side term from the phase space P2
to P1.

73

Recall that ˆtc = ˆt0(y, p),

− 1
4

||ˆy(ˆtc) − ˆx(ˆtc)||2

ˆM(ˆtc)

(cid:20)

−

ˆy(ˆtc) − ˆx(ˆtc), ˆp(ˆtc) − ˆξ(ˆtc)

− Im( ˆβ)|ˆτt(ˆtc; y, p) − ˆτt(ˆtc; x, ξ)|2 − |∆ˆt0(x, ξ; y, p)|2
(cid:21)

ˆy(ˆtc) − ˆx(ˆtc)


4Im( ˆβ + ˜β)

ˆI

ˆp(ˆtc) − ˆξ(ˆtc)

= −1
4

||ˆy(ˆt0(y, p)) − ˆx(ˆt0(x, ξ))||2

ˆM(ˆt0(y,p))

− |∆ˆt0(x, ξ; y, p)|2
4Im( ˆβ + ˜β)

−

ˆy(ˆtc) − ˆx(ˆtc), ˆp(ˆtc) − ˆξ(ˆtc)

(cid:20)

(cid:21)

− Im( ˆβ)|ˆτt(ˆt0(y, p); y, p) − ˆτt(ˆt0(x, ξ); x, ξ)|2
(cid:33)

ˆy(ˆtc) − ˆx(ˆtc)
 + O

1(cid:112)||p||

(cid:32)

ˆI

ˆp(ˆtc) − ˆξ(ˆtc)

= −1
4

||ˆy(ˆt0(y, p)) − ˆx(ˆt0(x, ξ))||2

ˆM(ˆt0(y,p))

− |∆ˆt0(x, ξ; y, p)|2
4Im( ˆβ + ˜β)

− Im( ˆβ)|ˆτt(ˆt0(y, p); y, p) − ˆτt(ˆt0(x, ξ); x, ξ)|2

−(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20)

(3.135)

(cid:32)

(cid:33)

,

1(cid:112)||p||

(3.136)

(cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2

ˆy(ˆt0(y, p)) − ˆx(ˆt0(x, ξ)) ± ∆ˆt0v

ˆξ

|| ˆξ||, ˆp(ˆt0(y, p)) − ˆξ(ˆt0(x, ξ))

ˆI + O

where v in equation (3.136) is defined at ˆy(ˆt0(y, p)). The first step (3.135) is due to

the following derivation. Using Corollary 3.3.3,

||ˆy(ˆtc) − ˆx(ˆtc)||2

ˆM(ˆtc)

= ||ˆy(ˆtc) − ˆx(ˆtc)||2

ˆN (ˆtc)

+ O

(cid:32)

(cid:33)

.

1(cid:112)|p||

Using Proposition 3.3.4,

||ˆy(ˆtc) − ˆx(ˆtc)||2

ˆN (ˆtc)

= ||ˆy(ˆt0(y, p)) − ˆx(ˆt0(x, ξ))||2

ˆN (ˆtc)

+ O

(cid:32)

(cid:33)

,

1(cid:112)|p||

74

since ˆy(ˆt0(y, p)) is on the boundary. Use Corollary 3.3.3 again,

||ˆy(ˆt0(y, p)) − ˆx(ˆt0(x, ξ))||2

ˆN (ˆt0(y,p))

||ˆy(ˆt0(y, p)) − ˆx(ˆt0(x, ξ))||2

ˆM(ˆt0(y,p))

(cid:32)

=

+ O

(cid:33)

.

1(cid:112)|p||

The second step (3.136) is due to Assumption 3.2.3 as the ray direction ˆξ remains as

a constant near the boundary.

Now we will prove equation (3.136) is non-degenerate in P2.

If ˆU y

p and ˆU x

ξ have different hitting points ˆy(ˆtc) (cid:54)= ˆx(ˆt0(x, ξ)), then equation (3.136) is
obviously negative. This is also true for the nonzero difference of travel time ∆ˆt0 and
ˆτt(ˆt0(y, p); y, p) − ˆτt(ˆt0(x, ξ); x, ξ).
If we have all difference mentioned above is zero and ˆp− ˆξ is nonzero along the tangential

directions of the boundary, then equation (3.136) becomes,

||(0, ˆp(ˆt0(y, p)) − ˆξ(ˆt0(x, ξ)))||2

ˆI = ||ˆp(ˆtc) − ˆξ(ˆtc)||2

( ˆM+2Im( ˆβ)ˆζ ˆζT )−1 > 0,

(3.137)

since ˆM and its pseudo-inverse ˆM−1 are S.P.D. if restricted to the tangential directions

by Lemma A.0.1.

Finally, due to the existence of ˆU−1, we can define a non-degenerate Gaussian profile

about (x, ξ) accordingly, that is

ˆE = ˆU−T ˆC ˆU−1,

(3.138)

75

where ˆC makes the following term

(cid:20)

||

(∆ˆt0, ˆy(ˆtc) − ˆx(ˆt0(x, ξ))),

(cid:21)
(ˆτt(ˆtc; y, p) − ˆτt(ˆtc; y, p), ˆp(ˆtc) − ˆξ(ˆt0(x, ξ)))

||2
ˆC

equal to the equation (3.136).

Remark 3.3.1. Although the real part of the phase function in equation (3.131) is

ignored in this part of the computation, it will not affect the above computation essen-

tially, especially the existence of the Gaussian profile.

3.3.4.4

Integral about Momentum ξ and η

After justifying the existence of a non-degenerate Gaussian profile centered at (y, p),
the next question is how to calculate ˆE in Lemma 3.3.11 numerically. We first define
a matrix ˆA(ˆtc; y, p, q) depending only on fixed beam’s parameters (y, p)

 1

4

ˆM(ˆtc) +

||ˆp(ˆtc)||2edeT

4(cid:0)v(ˆy(ˆtc))ˆpd(ˆtc)(cid:1)2Im( ˆβ+ ˜β)

d

(cid:19)(cid:18)

(cid:18)

, 0

(cid:19)T

0, Im( ˆβ)v2(ˆy(ˆtc))

ˆp(ˆtc)
||ˆp(ˆtc)||

ˆp(ˆtc)
||ˆp(ˆtc)||

 ,

(3.139)

ˆA(ˆtc; y, p, q) = ˆI(ˆtc; y, p) +

where ed = (0,··· , 0, 1) ∈ Rd.

Lemma 3.3.12. There exists a S.P.D. 2d by 2d matrix ˆA, such that

||(y − x, p − ξ)||2

ˆE = ||(ˆy(ˆtc) − ˆx(ˆtc), ˆp(ˆtc) − ˆξ(ˆtc))||2
ˆA,

(3.140)

and ˆA is defined in equation (3.139).

Proof. Firstly, we would like to approximate ∆ˆt0(x, ξ; y, p) and ˆτt(ˆtc; y, p)− ˆτt(ˆtc; x, ξ).

76

By the definition of hitting time and Assumption 3.2.3,

0 = ˆyd(ˆtc) − ˆxd(ˆt0(x, ξ))
= ˆyd(ˆtc) − ˆxd(ˆtc) ± v(ˆy(ˆtc))

ˆpd(ˆtc)
||ˆp(ˆtc)||∆ˆt(x, ξ; y, p).

(3.141)

Here ± is determined by the sign of the Hamiltonian. To summarize,

−|∆ˆt0(x, ξ; y, p)|2
4Im( ˆβ + ˜β)

4(cid:0)v(ˆy(ˆtc))ˆpd(ˆtc)(cid:1)2
= − ||ˆp(ˆtc)||2|eT

d (ˆy(ˆtc) − ˆx(ˆtc))|2
Im( ˆβ + ˜β)

.

(3.142)

Similarly,

−Im( ˆβ)|ˆτt(ˆtc; y, p) − ˆτt(ˆtc; x, ξ)|2 = −Im( ˆβ)v2(ˆy(ˆtc))

(cid:12)(cid:12)(cid:12)||ˆp(ˆtc)|| − || ˆξ(ˆtc)||(cid:12)(cid:12)(cid:12)2

.

(3.143)

Furthermore, it can be approximated by,

−Im( ˆβ)|ˆτt(ˆtc; y, p) − ˆτt(ˆtc; x, ξ)|2 ≈ −Im( ˆβ)v2(ˆy(ˆtc))

.

(3.144)

(cid:12)(cid:12)(cid:12) ˆp(ˆtc)
(cid:12)(cid:12)(cid:12)2
||ˆp(ˆtc)|| · (ˆp − ˆξ)

The Lemma is proved.

The integrand (3.131) now is a quadratic term about (ˆy(ˆtc) − ˆx(ˆtc), ˆp(ˆtc) − ˆξ(ˆtc)).

Furthermore, we have the following proposition.

Proposition 3.3.10. There exists a linear map ˆJ (ˆtc; y, p), such that

ˆy(ˆtc) − ˆx(ˆtc)

ˆp(ˆtc) − ˆξ(ˆtc)

 ≈ ˆJ (ˆtc; y, p)

y − x

p − ξ

 .

(3.145)

77

Proof. See Appendix A.

Similarly, the matrix ˜A(˜tc; y, p, q) and the map ˜J (˜tc; y, q) on the source side are defined

accordingly.

Now ˆH becomes,

ˆH(x; y, p, q) =

(cid:90)

dξe

(cid:118)(cid:117)(cid:117)(cid:116)

(2π)d−1

det( ˆM(ˆtc) + Im( ˆβ)

2

ˆζ ˆζT )
i|| ˆJ (

−|| ˆJ (ˆtc;y,p)(y−x,ξ−p)||2

ˆA(ˆtc;y,p,q)e

2 ;y,p)(y−x,ξ−p)||2
ˆtc

ˆR(ˆtc;y,p).

(3.146)

Therefore, ˆE under Assumption 3.2.3 is

ˆE ≈ ˆJ (ˆtc; y, p)T ˆA(ˆtc; y, p, q) ˆJ (ˆtc; y, p).

(3.147)

The above equation provides an efficient way to approximate ˆE. Now let’s compute
the integral about ξ. We suppose the S.P.D. matrix ˆE,

 ˆE11,

ˆE T
12,

 .

ˆE12
ˆE22

ˆE =

(3.148)

78

Now we would like to show that there’s a Gaussian profile about x centered at y,

−||(y−x,p−ξ)||2
ˆE

dξe

−||y−x||2

−||y−x||2

−||y−x||2

||y−x||2

ˆE12

ˆE−1

22

ˆET
12

||y−x||2

ˆE12

ˆE−1

22

ˆET
12

||y−x||2

ˆE12

ˆE−1

22

ˆET
12

ˆE11 e

ˆE11 e

ˆE11 e

(cid:90)
(cid:90)
(cid:90)

−||p−ξ+ ˆE−1

22

ˆET
12(y−x)||2
ˆE22

dξe

−||p−ξ||2

ˆE22

dξe

−||(0,p−ξ)||2
ˆE .

dξe

(3.149)

(cid:90)

= e

= e

= e

(cid:90)

ˆE11 − ˆE12 ˆE−1
displays that ˆH contains a Gaussian profile about y − x.

ˆE T
12 is S.P.D by the fact that it is the Schur complement of ˆE22 in ˆE. This

22

Together with equation (3.147), we have

−||(y−x,p−ξ)||2
ˆE

dξe

−||y−x||2

||y−x||2

ˆE12

ˆE−1

22

ˆET
12

ˆE11 e

= e

(cid:90)

−|| ˆJ (ˆtc;y,p)(0,p−ξ)||2
ˆA.

dξe

Together with equation (3.146), we obtain a way to calculate ˆH(x; y, p, q). The real
part of the phase function associated with ˆR will not affect the final result essentially

and it can be compensated by a constant order phase term.

ˆH(x; y, p, q) = eiO(1)

(2π)d−1

(cid:90)

det( ˆM(ˆtc) + Im( ˆβ)
2
i|| ˆJ (

i|| ˆJ (ˆtc;y,p)(0,p−ξ)||2
ˆAe

dξe

i

4||y−x||2

ˆM (0)

e

ˆζ ˆζT )
2 ;y,p)(y−x,p−ξ)||2
ˆtc

ˆR(ˆtc;y,p).

(3.150)

(cid:118)(cid:117)(cid:117)(cid:116)

Here we use 1

4M0 to approximate the Schur complement of ˆE.

79

Lemma 3.3.13. ˆH can be approximated by the following equation,

ˆH(x; y, p, q) ≈ ˆK(y, p, q)e

i

4||y−x||2

ˆM (0),

(3.151)

where

(cid:118)(cid:117)(cid:117)(cid:116)

(2π)d−1

det( ˆM(ˆtc) + Im( ˆβ)

2

ˆK(y, p, q) =

−|| ˆJ (ˆtc;y,p)(0,ξ−p)||2

ˆA(ˆtc;y,p,q)e

2 ;y,p)(0,ξ−p)||2
ˆtc

ˆR(ˆtc;y,p).

(3.152)

dξe

ˆζ ˆζT )
i|| ˆJ (

Similarly, ˜H can be approximated by the following equation,

(cid:90)

(cid:90)

Corollary 3.3.7.

where

˜H(x; y, p, q) ≈ ˜K(y, p, q)e

i

4||y−x||2

˜M (0)

(3.153)

(cid:118)(cid:117)(cid:117)(cid:116)

(2π)d−1

det( ˜M(˜tc) + Im( ˜β)

2

˜K(y, p, q) =

˜ζ ˜ζT )
i|| ˜J (

−|| ˜J (˜tc;y,q)(0,η−q)||2

˜A(˜tc;y,p,q)e

2 ;y,q)(0,η−q)||2
˜tc

˜R(˜tc;y,q).

(3.154)

dηe

The real part ˆR(ˆtc; y, p) and ˜R(˜tc; y, q) can be compensated by a constant phase term

and will not affect the result essentially.

Remark 3.3.2. In equation (3.151), we essentially approximate the Schur complement

ˆE11 − ˆE12 ˆE−1

22

ˆE T
12 by

M0
4 . However, it is costly to use the exact value (3.146) since

80

we have to store all matrices ˆA(ˆtc; y, p, q), ˜A(˜tc; y, p, q) generated by different pairs of
{(p, q) : p + q = ξl,i} to apply the inverse Gaussian wavepacket transform (3.49). On

the other hand, the difference between equation (3.151) and equation (3.146) will be at

constant order guaranteed by Lemma 3.3.6.

To summarize, the central direction and central point of the wavepacket will not be

affected by approximation (3.151) and the width of wavepacket is at the same scale.

The exact Hessian information (3.146) can be covered but it is costly to compute.

Theorem 3.3.1.

(cid:90)

−ω2dω

(cid:90)

(cid:90)

drds

¯ˆU y
p (r, ω) ¯˜U y

q (s, ω)

dξdη ˆU x

ξ (r, ω) ˜U x

η (s, ω) ≈

K(p, q, y) ˆK(y, p, q) ˜K(y, p, q)ei(p+q)·(y−x)e

i||y−x||2

M0/2,

(3.155)

where K(p, q, y) is defined in equation (3.118), ˆK and ˜K are defined in equation

(3.152)-(3.154). The distance between ˆτt(ˆtc; y, p) and ˜τt(˜tc; y, q) is controlled by K(p, q, y).

3.4

Implementation of the Prestack Imaging Oper-

ator

By equations (3.46), (3.47) and (3.48), we conclude that the partial imaging function

Ipq(y, ω) is related to the Gaussian wavepacket transform of 2α

v2 centered at y in the

81

direction p + q.

(cid:90)

Ipq(y, ω)dω = E(p, q, y)

(cid:90)

dx

2α

v2 ei(p+q)·(y−x)e

−||y−x||2

M0/2,

(3.156)

where E(p, q, y) = ˆK(y, p, q) ˜K(y, p, q)K(p, q, y). The numerical scheme to calculate

E(p, q, y) is given by equation (3.118) and equations (3.152)-(3.154). Therefore, this

section will be devoted to illustrating how to compute the integral of the imaging

function Ipq(y, ω) efficiently.

(cid:90)

(cid:90)

(cid:90)

dω

Ipq(y, ω)dω =

drds ¯˜U y

q (s, ω)

¯ˆU y
p (r, ω)D(r, s, ω).

(3.157)

We start with the integral about the wavenumber ω. Using Corollary 3.3.1,

ˆΦ(r, ˆtc; y, p) = ˆτt(ˆtc; y, p) + ˆζT (r − ˆy(ˆtc)),
˜Φ(s, ˜tc; y, q) = ˜τt(˜tc; y, q) + ˜ζT (s − ˜y(˜tc)).

By considering the terms containing ω in ˜U y

q (s, ω) and ˆU y

p (r, ω) only,

(cid:90)

eiO(1)

dω ¯˜U y

(cid:90)

¯ˆU y
p (r, ω)D(r, s, ω) =

q (s, ω)
D(r, s, ω)eiω(ˆtc+˜tc)e−Im( ˆβ)|ω− ˆΦ|2

e−Im( ˜β)|ω− ˜Φ|2

dω.

(3.158)

Here we neglect the real part of the exponent, Re( ˆβ)|ω − ˆΦ|2 and Re( ˜β)|ω − ˜Φ|2,

since they are constant order terms by Proposition 3.3.7. Consequently, they are small

compared with the term ω(ˆtc + ˜tc).

82

The integral now can be considered as an inverse Fourier transform about wavenumber

ω,

(cid:90)

e−Im( ˜β)|ω− ˜Φ|2

dω

D(r, s, ω)eiω(ˆtc+˜tc)e−Im( ˆβ)|ω− ˆΦ|2
(cid:90)

=
− Im( ˆβ)Im( ˜β)( ˜Φ− ˆΦ)2

e

Im( ˆβ+ ˜β)

eiS(ˆtc+˜tc)

D(r, s, ω)ei(ω−S)(ˆtc+˜tc)e−Im( ˆβ+ ˜β)(ω−S)2

dω,

(3.159)

where

S(r, s, ˆtc, ˜tc; y, p, q) =

Im( ˆβ) ˆΦ(r, ˆtc; y, p) + Im( ˜β) ˜Φ(s, ˜tc; y, q)

Im( ˜β + ˆβ)

.

(3.160)

The integral in equation (3.159) is indeed a convolution,

− Im( ˆβ)Im( ˜β)( ˜Φ− ˆΦ)2

Im( ˆβ+ ˜β)

(cid:90)

π

eiS(ˆtc+˜tc)e

(cid:114)
=
− Im( ˆβ)Im( ˜β)( ˜Φ− ˆΦ)2
e

Im( ˆβ + ˜β)

Im( ˆβ+ ˜β)

(cid:90)

eiS(ˆtc+˜tc)

D(r, s, ω)ei(ω−S)(ˆtc+˜tc)e−Im( ˆβ+ ˜β)(ω−S)2

dω

−

e

h2

4Im( ˆβ+ ˜β) eiShD(r, s, ˆtc + ˜tc − h)dh.

(3.161)

Notice the data D in the above formula is in the time domain, which means there is no

need to apply the Fourier transform to the data at prior. Moreover, the convolution

(cid:32)

(cid:113)|ˆτtt(ˆtc;y,p)|+|ˆτtt(˜tc;y,q)|

1

(cid:33)

.

integral is conducted in a small range, i.e. O

83

Naturally, use Corollary 3.3.1 and equation (3.157)

(cid:114)

(cid:90)
(cid:90)

drdse

(cid:115)

π

ˆA(ˆtc) ˜A(˜tc)

Ipq(y, ω)dω =

Im( ˆβ + ˜β)
−||r−ˆy(ˆtc)||2
ˆM
2 e

−||s−˜y(˜tc)||2
˜M
2 e

(cid:90)

× 2π

1

ˆτtt(ˆtc; y, p)˜τtt(˜tc; y, q)

− Im( ˆβ)Im( ˜β)( ˆΦ− ˜Φ)2

Im( ˆβ+ ˜β)

eiS(ˆtc+˜tc)e−i(ˆ
(r)+˜
(s))

−
e

h2

4Im( ˆβ+ ˜β) eiShD(r, s, ˆtc + ˜tc − h)dh.

(3.162)

The integral about r and s is easy due to the existence of Gaussian profiles which will

constrain the integral range. Therefore, the regular integration scheme is enough.

Here we notice that the value of the imaging function is controlled by | ˜Φ − ˆΦ| ∼
|˜τt(˜tc; y, q)− ˆτt(ˆtc; y, p)|. Therefore, an efficient way is needed to avoid computing pairs

of beams with little illumination or imaging function (cid:82) dωIpq(y, ω) (3.162) which is

small. Therefore, we should select pairs of beams (y, p) and (y, q) such that

| ˆΦ(r, ˆtc) − ˜Φ(s, ˜tc)| ≤ 2(cid:112)||p|| + 2(cid:112)||q||.

(3.163)

Fortunately, the time derivative of the phase function does not change along the ray,

which means we can estimate equation (3.163) without propagating beams.

84

3.5 Numerical Results

3.5.1 Approximation of Beams along the surface

The numerical examples in this subsection are provided to justify the approximation

in Section 3.3.2. The following tests are done by comparing our approximation results

to the results obtained by the numerical integration method.

The velocity used here is v = 0.8 + 0.4x2 and the initial beam is initiated at the

subsurface point (0, 0.5). The initial momentum is set as 30π[cos(0.1π), sin(0.1π)] and

the initial amplitude is 1 + i.

Figure 3.4: Numerical test for fast approximation of Gaussian beam along the surface. Left:
ω = -110 Right: ω = -100.

The blue line represents the result achieved from the numerical integration method

and the red star is the one from our fast approximation algorithm.

85

−101−0.0500.050.10.15x1u−101−0.2−0.100.10.20.3x1u3.5.2 The Correctness of Prestack Imaging Operator

We conduct the numerical test to justify Theorem 3.3.1.

We first fix a direction e and the point y so that x is acquired by moving along this

certain fixed direction, i.e. x = y + ∆he. We compare our proposed result with
the numerical integration result after removing the highly oscillated term eip·(y−x).

The initial subsurface point is the point y = (0, 0.5) and the initial ray direction is
p = (6π,−30π). Two different directions are picked here. The first one is e = (1, 1),

while the second is chosen as (1, 2). The x-axis in each plot is ∆h , y-axis in each plot

Figure 3.5: Constant Slowness: Theorem 3.3.1. Left: dx1 = dx2 Right: dx1 = 2dx2.

is the value of integral. As we can see in the constant slowness, the approximation

proposed in Theorem 3.3.1 has only a small amount of error.

The second velocity used for test is 1 + 0.1x2 + 0.1x1. Other setups are the same. As

we can see in Fig. 3.6, the imaging operator does not perform as well as it does in the

86

−0.0500.053.544.555.5dx1y−0.0500.051.522.53dx1yFigure 3.6: General Speed: Theorem 3.3.1. Left: dx1 = dx2 Right: dx1 = 2dx2.

constant slowness. However, the central momentum is captured correctly as stated in

Remark 3.3.2.

3.5.3 Single Source Migration Test

We will recover the reflector by the single-source data trace in this section.

3.5.3.1 Example 1: Constant Background Slowness

Fig. 3.7 is the true slowness we employ, and there is a dipped layer. The source point

here is at x = (0, 0). As the Fig. 3.7 shows, the migration result shows the ability of

our algorithm to detect the correct location and dipped angle.

87

−0.0500.05012345dx1y−0.0500.05−20246dx1yFigure 3.7: Example 1: Constant Slowness with Dipped layer. Left: True Slowness Right:
Migration Result with single source trace.

3.5.3.2 Example 2: Multiple Flat Reflectors

In this numerical example, our migration algorithm is tested by two flat reflectors at

different depth, The migration result is, The red dashed line is the true value while

Figure 3.8: Example 2: True Slowness Model with Multiple Layers

the blue line the migration result. The deeper layer is not captured as well as the first

layer. The error here is due to the Born approximation assumption.

88

x1x2  −0.500.5−0.4−0.200.20.40.60.80.780.790.80.810.82x1x2  −0.500.5−0.4−0.200.20.40.60.8−0.04−0.0200.020.040.06x1x2  −0.500.5−0.4−0.200.20.40.60.480.490.50.510.52Figure 3.9: Example 2: Constant Slowness with Multiple Layers. Left: Migration Result
over the Whole Space Right: Migration Result V.S. True Value at x1 = 0.

3.5.3.3 Example 3: Linear Background Slowness

To see the amplitude information, we plot the slowness at x1 = 0 and compare it with

the true value. And the red dashed line is the true value while the blue line is our

migration result.

3.5.4 Multiple Source Migration Test

We will use multiple-source data trace in this section.

3.5.4.1 Example 4: Constant Slowness with Dipped Layer

The background slowness is same as the one in Example 1. Sources are a series of
points along the surface − 1

4. This is applied to all multiple-source tests. The

4 : 1

16 : 1

dipped layer is displayed correctly in Fig. 3.12.

89

x1x2  −0.500.5−0.4−0.200.20.40.6−0.04−0.0200.020.040.06−1−0.500.51−0.06−0.04−0.0200.020.040.06x2uFigure 3.10: Example 3: Gradient Slowness Model. Left: True Slowness, Right: Smoothed
Macro Slowness

Figure 3.11: Example 3: Gradient Slowness Model. Left: Migration Result over the Whole
Space, Right: Migration Result V.S. True Value at x1 = 0.

90

x1x2  −0.500.5−0.4−0.200.20.40.40.450.50.55x1x2  −0.500.5−0.4−0.200.20.40.40.450.50.55x1x2  −0.4−0.200.20.40.6−0.4−0.3−0.2−0.100.10.20.30.4−0.0500.05−101−0.0500.05x2uFigure 3.12: Example 4: Constant Slowness with the Dipped Layer (Multiple Sources)

3.5.4.2 Example 5: Flat Layer in Lateral Background Velocity

We add some lateral variation to the background slowness, i.e. v = 0.8+0.1 sin(0.5πy) sin(3π(x+

0.05)). The reflector is a horizontal reflector.

Figure 3.13: Example 5: Flat Layer in Lateral Background Velocity

91

x1x2  −0.4−0.200.20.40.6−0.4−0.200.20.40.60.81−0.05−0.04−0.03−0.02−0.0100.010.020.030.040.05x1x2  −1−0.500.5−1−0.500.5−0.06−0.04−0.0200.023.5.4.3 Example 6: Slowness with Caustics

The next two examples in the section are both using the Gaussian velocity as the

macro velocity shown in Fig. 3.14 (a). This is more complex as the caustics will show

up. As we can see in Fig. 3.14, there is a caustics around the level x2 = 0.46. Our

Figure 3.14: Example 6: Gaussian Slowness and its ray tracing. Left: Gaussian Slowness
with Flat Reflector Right: Ray Tracing

flat reflector is below this caustics at x2 = 0.6. The multi-value problem caused by

caustics is resolved automatically by the Gaussian beam solution in Fig. 3.15.

3.5.4.4 Example 7: Polluted Trace Data

In the end, we would like to test our inversion process using the polluted data. We

add 5% Gaussian noise into the synthetic data. See Fig. 3.16 for more details. The

red dot line is the trace with extra Gaussian error, while the blue line is the original

trace.

92

x1x2  −0.4−0.200.20.40.600.10.20.30.40.50.60.70.620.640.660.680.70.720.740.760.780.8−0.4−0.200.200.20.40.60.8x1x2Figure 3.15: Example 6: Migration Result in the Gaussian Slowness with Caustics (Multiple
Sources)

Figure 3.16: Example 7: True Trace V.S. Trace with Gaussian Error

The migration result is displayed in Fig. 3.17. There is no much difference in resulting

images, especially around the reflector. To see more details, we compare two results at

x2 = 0.65 in Fig. 3.18, The red dot line comes from the non-polluted data while the

blue line is from the polluted boundary data.

93

x1x2  −1−0.500.5−1−0.8−0.6−0.4−0.200.20.40.60.8−0.08−0.06−0.04−0.0200.020.040.0600.511.522.5−0.8−0.6−0.4−0.200.20.40.60.8tuFigure 3.17: Example 7: Gaussian Slowness with polluted trace. Left: Migration result
from Non-polluted Data; Right: Migration result from Polluted data

Figure 3.18: Example 7: Gaussian Slowness with polluted trace.Two Migration Results at
x2 = 0.65

94

x1x2  −1−0.500.5−1−0.500.5−0.08−0.06−0.04−0.0200.020.040.06x1x2  −1−0.500.5−1−0.500.5−0.1−0.0500.05−1−0.500.51−0.0200.020.040.060.080.1x1uChapter 4

Fast Multiscale Gaussian Beam

Method for Elastic Wave Equations

in Bounded Domains

4.1 Asymptotic Method for the Elastic Wave equa-

tion

The problem considered in this paper is the initial-boundary value problem of the

elastic wave equation.

0 = ρ¨u − ∇λ(∇ · u) − ∇µ · (∇u + ∇uT ) − (λ + µ)∇(∇ · u) − µ∆u,

(4.1)

where the parameters λ and µ are known as the Lame parameters.

˙u =

∂u
∂t

;

¨u =

∂2u
∂t2

(4.2)

95

and this notation is applied to all functions. They are assumed to be smooth, positive

and bounded away from zero. The initial condition is defined as the following,

u(x, 0) = f; ut(x, 0) = g,

(4.3)

where the functions f and g are compactly supported vector-valued functions in the

space L2(Rd). We are looking for the asymptotic solution for the elastic wave equation

(4.1) with two different types of boundary conditions, the periodic boundary condition

and the homogeneous Dirichlet boundary condition, i.e.

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)∂Ω×[0,T ]

u(x, t)

= 0.

(4.4)

4.2 The Asymptotic Ansatz Solution to the Elastic

Wave

We firstly derive the eikonal and the transport equation for the elastic wave equation.

Same as the Geometrical-optics form, we consider the solution as the following series

expansion,

u(x, t) = eiωτ (t,x)

∞(cid:88)

n=0

A(n)(t, x)(iω)−n,

(4.5)

where the wavenumber ω is assumed to be a large parameter relative to the elastic

moduli λ and µ’s changing rate, i.e.

ωL

min(λ,µ) >> 1, where L is the characteristic

distance defined as the scale over which the velocity changes slowly. The asymptotic

solution for equation (4.1) is defined in the sense that both the equation itself and

96

initial-boundary conditions are satisfied approximately with a small error when ω is

large. To construct two equations governing the phase function τ and amplitude func-

tion A(0) respectively, we substitute the ansatz form A(0)eiωτ into equation (4.1). For

convenience, we write A(0) as A. The jth component of the elastic wave equation will

then be

(cid:18) 1

ω

O

(cid:19)

= eiωτ(cid:8)λ,j(Ak,k + iωAkτ,k) + µ,k[Ak,j + Aj,k + iω(τ,jAk + Ajτ,k)]
+ (λ + µ)[(cid:0)Ak,k

(cid:1) − ω2τ,jAkτ,k]

(cid:1)
,j + iω(cid:0)(Akτ,k),j + τ,jAk,k
(cid:9),

+ µ[(Aj,kk) + iω(2Aj,kτ,k + Ajτ,kk) − Ajω2τ 2
,k]
+ ρAjω2( ˙τ )2 − 2iωρ ˙τ ˙Aj − iωρAj ¨τ − ρ ¨Aj

(4.6)

here | · | denotes the Euclidean norm in Rd and

 .



∂
∂x1
...

∂
∂xd

∇ =

Other notations used frequently in this paper are · representing the inner product

between two column vectors and vT representing the transpose of vector v. Other

notations we use in the above equation (4.6) are,

τ,k = ∂τ /∂xk.

To make our derivation simpler, we let ρ = 1 without losing any generality.

97

4.2.1 P-wave and S-wave’s Eikonal Equations

To cancel out the leading term ω2 in equation (4.6), we set its coefficient to be zero,

0 = ˙τ 2Aj −(cid:0)(λ + µ)τ,jAkτ,k + µAjτ,kτ,k

(cid:1) .

After concatenating j = 1, 2,··· , d as a vector, we will get

(λ + µ)∇τ∇τ T A = ( ˙τ 2 − µτ,kτ,k)A,

(4.7)

(4.8)

so the amplitude vector A will be the eigenvector of the matrix ∇τ∇τ T . Notice that

this simple matrix is rank one matrix by some linear algebra calculations, and the

eigenvector corresponding to the single nonzero eigenvalue must be parallel to the
vector ∇τ P , while other two associated with the zero eigenvalues are orthogonal to
∇τ S. The superscript here represents the category of their wave modes. We have

( ˙τ P )2 − (λ + 2µ)(τ P

,k τ P

,k ) = 0.

(4.9)

This is the eikonal equation for the P-wave whose amplitude vector AP is parallel to
the ray direction ∇τ P . Another two eigenvectors are corresponding to the S-wave,

( ˙τ S)2 − µ(τ S

,kτ S

,k) = 0.

(4.10)

whose the amplitude vector AS is perpendicular to the ray direction ∇τ S.

With the eikonal equation at our disposal, we can apply the method of characteristics

98

to the nonlinear eikonal equations (4.9) and (4.10). These two eikonal equations (4.9)

and (4.10) are essentially the same, consequently, we consider them in the generic

situation as a Hamilton-Jacobi equation.

˙τ P,S + GP,S(x,∇τ ) = 0,

(4.11)

where the Hamiltonian of the P-wave GP (x, p) = ±√
of the S-wave GS(x, p) = ±√

µ|p|. We consider the P-wave case for the illustration,

λ + 2µ|p| and the Hamiltonian



dx
dt = GP
dt = −GP

dp

p (x(t), p(t)),

x (x(t), p(t)),

x(0) = x0;

p(0) = p0.

(4.12)

where t is the running parameter of a bicharacteristic.

Solving this ODE system yields a bicharacteristic in the phase space

{(x(t), p(t)) : t ≥ 0}

and the associated ray γ = {x(t) : t ≥ 0}, which is its x-component. Moreover, it is
noticed that we have the equation p(t) = ∇τ (t, x(t)) along the ray γ due to the method

of characteristics.

One of the most significant difference between the Gaussian beam and other ray-ansatz

methods is that beams’ phase functions are complex-valued. To be more specific, its

second order derivative is complex-valued. To derive the dynamics of the Hessian

matrix, we first differentiate the eikonal equation (4.11) with respect to t and x near

99

the ray γ:

˙τ P
x + GP

x + τ P

xxGP

p = 0,

¨τ P + (GP

p )T ˙τ P

x = 0.

Differentiating the first equation above (4.13) with respect to x again yields

˙τ P
xx + GP

xx + τ P

xxGP

xp + (GP

xp)T τ P

xx + τ P

xxGP

p = 0.

ppτ p

xx + τ P

xxxGP

(4.13)

(4.14)

(4.15)

Since the equations (4.14), (4.13) and (4.15) are all valid everywhere in the phase space,

it will still be valid if we concentrate them along the ray. Let M P (t) be the Hessian of

the phase function along the ray

dM P
dt

+ GP

xx + M P GP

xp + (GP

xp)T M P + M P GP

ppM P = 0.

(4.16)

And the same rule can be applied to the S-wave with GS(x, p) = ±√

µ|p|. One in-

teresting property of the Gaussian beam solution is that it will remain well-localized

throughout the propagation, which means the imaginary part of the Hessian M should

always be symmetric positive definite. The following lemma [49] guarantees this prop-

erty throughout the propagation for all smooth ray trajectories,

Lemma 4.2.1. If the Hamiltonian G is smooth enough, then the Hessian M(t) along

the ray path γ has a positive-definite imaginary part, provided that it initially does.

Accordingly, the Hessian of beam ansatz’s phase functions is well-defined at all points

even the caustics.

100

4.2.2 Transport Equation Governing P-wave’s Amplitude Vec-

tors

Setting the coefficient of O(ω) term in equation (4.6) equal to zero will yield the

transport equation about the amplitude vector A. We first see the jth component,

0 = (2 ˙Aj ˙τ + ¨τ Aj) − λ,j(Akτ,k) − µ,k(τ,jAk + Ajτ,k)

− (λ + µ)(cid:0)(Akτ,k),j + Ak,kτ,j

(cid:1)

− µ(2Aj,kτ,k + Ajτ,kk).

(4.17)

Although it is complex at first glance, especially compared with the transport equation

of the scalar wave equation, the complexity can be reduced by properties of the P-wave

and S-wave.

We start with deriving the amplitude vector AP for the P-wave, which is parallel to the
ray direction ∇τ P . Therefore, the P-wave’s amplitude can be separated as A = a∇τ P .

To make derivation more readable, we write τ P as τ in this part. We will yield the jth
component by inserting A = a∇τ into equation (4.17)

(cid:16)

(cid:17)
,k − 2a(µ,kτ,k)τ,j

,k + 2aτ,kτ,kj

0 = (2 ˙aτ,j ˙τ + 2a ˙τ ˙τ,j + a¨τ τ,j) − λ,jaτ 2
− (λ + µ)

− µ(cid:0)aτ,jτ,kk + 2τ,ja,kτ,k + 2aτ,kτ,kj

τ,ja,kτ,k + aτ,jτ,kk + a,jτ 2

(cid:1) .

101

Then we multiply the above equation with τ,j and then sum over j,

(2 ˙a ˙τ + a¨τ )τ 2

,kτ,kk

,k + a ˙τ|∇τ|2

t = a|∇τ|2((λ + 2µ),kτ,k) + a(λ + 2µ)τ 2

+ 2(λ + 2µ)τ 2

,k(τ,ka,k) + 2a(λ + 2µ)(τ,jτ,jkτ,k).

The term a ˙τ|∇τ|2

t on the left hand side of the above equation is equal to

a ˙τ|∇τ|2

t = 2a ˙τ (τ,k ˙τ,k)

= aτ,k( ˙τ )2
,k
= aτ,k((λ + 2µ)|∇τ|2),k
= a|∇τ|2τ,k (λ + 2µ),k + 2a(λ + 2µ)τ,jτ,jkτ,k,

(4.18)

since we are talking about the P-wave mode now and its phase τ satisfies the Hamiltonian-
Jacobi equation ˙τ 2 − (λ + 2µ)|∇τ|2 = 0. Consequently, the transport equation about

the norm of the P-wave’s amplitude vector A is

˙a +

(λ + 2µ)a,kτ,k

G

+

a
2G

((λ + 2µ)trace(M ) − ¨τ ) = 0.

(4.19)

Notice the second order derivatives of the phase function τ is involved, and the trans-

port equation will be undefined if the phase function is not smooth. Lemma 4.2.1

guarantees a well-defined transport equation, while the classical Geometrical-Optical

ansatz fails at the caustics region. Following [48], the ODE about the norm of ampli-

102

tude can be added into the P-wave’s dynamics by using equation (4.19) and dx

dt = Gp,

da(t, x(t))

dt

+

a
2G

((λ + 2µ)trace(M ) − Gx · Gp − GT

p M Gp) = 0.

(4.20)

4.2.3 Transport Equation Governing S-wave’s Amplitude Vec-

tors

Now let’s see the S-wave’s case. We abbreviate τ S as τ until the end of this section.

Again, we first separate the amplitude vector as AS = aD, where D is a vector which
is orthogonal to the ray direction ∇τ S and its norm is fixed to be a constant. After

substituting A = aD into equation (4.17), we will have the following equation for the

amplitude’s jth component,

0 = 2( ˙aDj + a ˙Dj) ˙τ + a¨τ Dj − a(µ,kDk)τ,j − a(µ,kτ,k)Dj
− (λ + µ)((aD)k,kτ,j) − 2µ(aDj),kτ,k − µaDjτ,kk,

(4.21)

for j = 1, 2, 3. After multiplying Dj with the above equation (4.21), we sum over the

index j.

(cid:0)aµ,kτ,k + aµτ,kk + 2µa,kτ,k

(cid:1)|D|2 +2µaτ,k|D|2

,k = a¨τ|D|2 +2 ˙a ˙τ|D|2 +a ˙τ ˙|D|2

, (4.22)

since D is orthogonal to ∇τ . The last term in the above equation is zero as the norm

of D is fixed, we have

(∇τ · ∇)|D|2 = 0.

(4.23)

103

Therefore, the above equation can be simplified as,

aµ,kτ,k + aµτ,kk + 2µa,kτ,k = 2 ˙a ˙τ + a¨τ .

(4.24)

We fix the norm of D to be one for convenience and the above equation (4.24) provides

the way to calculate the amplitude’s norm a.

It is not the same equation as the

transport equation in the scalar wave equation. To yield the same equation, we divide
√

µ on both sides of equation (4.24),

2 ˙a ˙τ + a¨τ√

µ

=

a
µ

= 2τ,k(

√

µ,kτ,k + a

µτ,kk + 2
√
µa)τ,kk

√
µa),k + (

√

µa,kτ,k

√

If we set ˜a =

µa as new amplitude, then

2 ˙˜a ˙τ + ˜a¨τ = µ(2τ,k˜a,k + ˜aτ,kk)

(4.25)

After yielding the same transport equation, we can obtain the same ODE as the P-wave

(4.20).

Unlike the P-wave, we still need one more equation in the S-wave’s ray system to

describe the amplitude vector AS’s direction D. To obtain the equation about the

amplitude direction D, we would like to plug equation (4.24) into equation (4.21) and

the coeeficients in front of the direction D is zeros suggested by equation (4.24),

2a ˙τ ˙Dj − 2µa(Dj,iτ,i) = τ,j

(cid:0)aµ,iDi + (λ + µ)(aD)k,k

(cid:1)

(4.26)

104

The left hand side of equation (4.26) is equal to the term

dDj (t,x(t))

dt

, since

˙τ

dDj
dt

dxi
dt

= ˙τ ˙Dj + ˙τ Dj,i
= ˙τ ˙Dj − G(x(t), p(t))DT Gp(x(t), p(t))
= ˙τ ˙Dj − µDj,iτ,i(t, x(t))

(4.27)

We know that the left hand side of equation (4.26) is 2a ˙τ
of equation (4.26) is parallel to ∇τ . Therefore, dD

dt

dDj
dt , and the right hand side
is parallel to ∇τ . Together with

the fact that the amplitude’s direction D is always perpendicular to the ray direction
p(t) = ∇τ (t, x(t)),

dDkpk(t)

0 =

dt

(cid:18) dpj(t)

dDk
0 =
dt
= −

pk(t) +

Dj

dt

Dk

dpk(t)

(cid:19) pk(t)

dt
|p(t)|2

dDk
dt

(4.28)

4.2.4 Single Beam Solution for P and S-wave

To summarize the ODE dynamics generated by the method of characteristics, we have

dx
dt
dp
dt
dM
dt
da
dt

= Gp(x(t), p(t)), x(0) = x0
= −Gx(x(t), p(t)), p(0) = p0
= −(Gxp)T M − M Gpx − M GppM − Gxx, M (0) = i
I
= − a
2G

(c2trace(M ) − Gx · Gp − GT

p M Gp, A(0) = A0

(4.29)

105

where the velocity term c2 = λ+2µ for the P-wave and c2 = µ for the S-wave. The term

G is the corresponding Hamiltonian. There is one extra equation about the direction

D in the S-wave’s dynamics,

= −

dDk
dt

(cid:18) dpj(t)

dt

Dj

(cid:19) pk(t)

|p(t)|2 , D(0) = D0.

The initial condition of the system above will be given by the Multiscale Gaussian

Wavepacket transform, which will be specified in the later section. Now the way of

propagating the phase functions τ P,S and the amplitude vectors AP,S is provided, and

it allows us to finish the construction of a single-beam asymptotic solution,

ΦP (t, x) = a(t)∇τ (t, x(t))eiωτ (t,x)

ΦS(t, x) = a(t)Deiωτ (t,x),

(4.30)

(4.31)

and the phase function is approximated by the Taylor expansion near the central ray,

τ P,S(t, x) = ∇τ P,S · (x − x(t)) +

= p(t) · (x − x(t)) +

1
2

(x − x(t))T M P,S(t)(x − x(t))
1
2
(x − x(t))T M P,S(t)(x − x(t)).

(4.32)

The Gaussian profile is offered by the imaginary part of the Hessian matrix M

(cid:16)−ω

2

(cid:17)

exp

(x − x(t))T Im(M (t))(x − x(t))

.

(4.33)

Suggested by Lemma 4.2.1, a beam ansatz will be always well localized throughout the

propagation.

106

4.3 Multiscale Gaussian Wavepacket Transform for

Elastic Waves

The initial condition of the elastic wave equation (4.3) can be any general L2 vector-

valued function, and it is not necessary to take the exact form like,

(cid:19)(cid:19)

.

(4.34)

(cid:18)

(cid:18)

A exp

iω

p(0)T (x − x0) +

1
2

(x − x0)T M (0)(x − x0)

The problem here is that how to decompose any general L2 function to multiple Gaus-

sian wavepackets like the above form (4.34) efficiently and make the total number of

beams to be calculated as small as possible.

We will provide a very brief introduction of the Multiscale Gaussian Wavepacket Trans-

form [48] for the scalar functions first in this section. More details can be found in

[48]. Then its extension designed for the vector-valued initial conditions is presented

afterwards.

4.3.1 Multiscale Gaussian Wavepacket Transform: Vector Func-

tions

After proposing the Multiscale Gaussian Wavepacket transform for the scalar function

in Section 2.2, we would like to extend this idea to the vector-valued function f . Here

we assume that each component of the vector fj is a L2 function.

107

4.3.1.1 Decomposition of the Single Wavepacket

Suppose we have already applied the wavepacket transform to each component of the

initial condition f , i.e.

f =

=





f1

f2
...

fd

(cid:88)

κl,i,kφl,i,k.

l,i,k

The idea here is to decompose each single Gaussian wavepacket into the sum of the

P-wave and the S-wave. Let the unit vector vl,i,k be

κl,i,k =

κT

l,i,kvl,i,k

vl,i,k +

κl,i,k.

(4.35)

The first term on the right hand side of equation (4.35) is the P-wave component, and

the initial condition for its amplitude vector can then be written as

(cid:16)

(cid:16)

(cid:17)

(cid:17)

ξl,i
|ξl,i|, then

(cid:16)

Id − vl,i,k(vl,i,k)T(cid:17)

(cid:32)

(cid:33)

ξl,i
|ξl,i|2

κT

l,i,kvl,i,k

vl,i,k =

κT

l,i,k

ξl,i.

(4.36)

To initialize the S-wave, we have to specify initial directions which are orthogonal to

each other and all of them are supposed to be orthogonal to vl,i,k.

108

Therefore, we choose the first direction D(1) as the unit vector of the first column

(cid:16)

Id − vl,i,k(vl,i,k)T(cid:17)

(cid:16)

Id − vl,i,k(vl,i,k)T(cid:17)

vector of the matrix
and apply the Gram-Schmidt process to
generate the rest D(m) for m = 2,··· , d − 1. And we notice that every column vector

of the matrix

is orthogonal to vl,i,k, therefore, so are their linear

combinations. The corresponding amplitude norm am,

(cid:16)

Id − vl,i,k(vl,i,k)T(cid:17)

am = κT

l,i,k

D(m), m = 1, 2,··· , d − 1.

(4.37)

4.3.1.2 Preprocessing the Initial Condition

Before applying the method described above, we pre-process the initial condition first.

Following the same technology employed in [47], there are supposed to be two different

l,i,k and κ−

branches κ+
Hamiltonian ±c(x)|p|, where c(x) is the corresponding velocity.

l,i,k for each wavepacket corresponding to different signs of the

To satisfy both the initial wavefield f and the initial velocity g, we define

l,i,k + κ−
(κ+

l,i,k)φl,i,k = κl,i,kφl,i,k = f .

Taking the derivative of the wavefield about the time variable t yields

l,i,k)(cid:0)i|ξl,i|G+(x0, p0)(cid:1) φl,i,k ≈ Ξl,i,kφl,i,k = g,

(κ−
l,i,k − κ+

(4.38)

(4.39)

where Ξl,i,k is the coefficients generated from decomposing the initial velocity g. Here

109

the left hand side of equation (4.39) is not the complete form of the beam’s time

derivative, instead, we pick the leading order term to approximate.

After solving the coefficients κ+

l,i,k and κ−

l,i,k based on equation (4.38) and equa-

tion (4.39), we can apply the decomposition described by equations (4.35) - (4.37)

to κ+

l,i,k and κ−

l,i,k respectively. The summary of the vector-version Multiscale Gaus-

sian Wavepacket is provided below.

Algorithm 3 Discrete Vector-Valued Gaussian Wavepacket Transform
1.Call the Discrete Gaussian Wavepacket Transform for each component in the discrete signal

f and g

(cid:18)
l,i,k and κ−
2. Use equations (4.38) and (4.39) to compute κ+
κ±
l,i,k · ξl,i
|ξl,i|2

3. Generate P-wave with the amplitude vector
4. Generate S-waves amD(m) by equation (4.37), for m = 1, 2,··· , d − 1.

(cid:19)

ξl,i

l,i,k

The above process defines the initial amplitude vectors for the P-wave and the S-wave,

and the initial value of the phase function and its derivatives are given by the Multiscale

Gaussian Wavepacket transform of the scalar form, that is

dx
dt
dp
dt

dM
dt
da
dt
dDj
dt

= Gp(x(t), p(t)), x(0) =

k
Ll
= −Gx(x(t), p(t)), p(0) = 2π

ξl,i
|ξl,i|,

= −(Gxp)T M − M Gpx − M GppM − Gxx, M (0) = i(2π2σ2
= − a
2G
dτ,k
dt

τ,j
|∇τ|2 D(m)(0) = D(m).

(c2trace(M ) − Gx · Gp − GT

(cid:18)(cid:114) π

p M Gp,

a(0) =

LlN

(cid:19)d
l /|ξl,i|)I.

σl

.

= (

Dk)

110

4.4 Multiscale Gaussian Beam Method for Periodic

Boundary Value Problem

In the above section, we have demonstrated the way to decompose the vector-valued

initial conditions in the L2 space. To solve the periodic boundary problem,each param-

eters and solutions are assumed to be periodic functions. Meanwhile, the central ray

in the periodic boundary problem will be smooth along the propagation in the sense

of modules.

The principle shown in Figure 4.1 will be employed to solve the periodic boundary value

problem. The red dashed line represents the wavepacket leaving the domain [0, 0.5], as

Figure 4.1: Periodic Boundary Problem: The Case Wavepacket leaving the Boundary.

the left half goes beyond the domain, while the right half still shows up. The missing

left half of the beam solution will enter from the other side with the same shape, which

is the blue line in the graph suggested by the periodic boundary condition. The cubic

region [0, 0.5]3 is chosen to test the correctness of our algorithm. We show numerical

results in Section 4.8.

111

00.10.20.30.40.500.10.20.30.40.50.60.70.80.91xu4.5 Multiscale Gaussian beam method for Homo-

geneous Dirichlet Boundary Condition

In this section, we would like to explore the solution to the homogeneous Dirichlet
Boundary condition. From now on, we use the 3-D space {x = (x, y, z) : x, y, z ∈ R}

as our model.

The wavefield on the boundary is assumed to be zero in this section. When the

reflection happens, the sum of all wavefields at time tr and the central point x(tr)

of the ray should vanish, i.e. u(tr, x(tr)) = 0. The time when the central point x(tr)

of the ray is on the boundary is defined as the reflection time. From now on, all

equations below in this section are defined on the point (tr, x(tr)), if not specified.
The Hamiltonian used in this section is assumed to be positive G = c(x)|p|, and the

negative Hamiltonian will be treated similarly.

4.5.1 P-wave Reflecting Beams: Ray Direction

When the P-wave reflection happens, the total wavefield is made up by three different

sources, the original P-wave Gaussian beam, the new P-wave beam after the reflection

(PP-wave) and the new S-wave beam (PS-wave). At the reflection point x(tr),

−aP eiτ P ∇τ P = aP P eiτ P P ∇τ P P + aP Seiτ P S

DP S.

(4.40)

Both P and S-wave will be generated after reflection.

All phase functions should have the same value at the reflection point, τ P = τ P P =

112

τ P S. Otherwise, if we change the value of large wavenumber |ξl,i|, the homogeneous

boundary condition will be violated.

The principle to derive new phase functions τ P P and τ P S is to take advantage of the

continuity conditions, i.e. the continuity of the tangential components of the first order

derivatives of τ , so

τ P
y = τ P P

y = τ P S

y

τ P
z = τ P P

z = τ P S

z

,

,

(4.41)

where we assume the reflection happens along the surface {x = (x, y, z) : x = 0}.

Besides the spatial variables, the partial derivative of the phase function with respect

to the time variable t should also follow,

˙τ P = ˙τ P P = ˙τ P S
⇒ cP|∇τ P| = cS|∇τ P S|,
cP|∇τ P| = cP|∇τ P P|.

(4.42)

√

where cP =

λ + 2µ is the velocity of the P-wave and cS =

√
µ is the velocity of the

S-wave. The partial derivatives along the tangential directions of the boundary can be

obtained directly from equation (4.41). To obtain the momentum along the reflection

direction or the normal direction of the boundary, one needs to use equations (4.42)

and (4.41) collectively.

x = −sign(τ P
τ P S
x )

(cid:115)(cid:18) cP

cS

(cid:19)2 |∇τ P|2 − (τ P S

y

113

)2 − (τ P S

z

)2.

(4.43)

Here we focus our derivation on the reflection from P-wave to S-wave, while the rule

of the reflection between the same mode can follow the same way, so we skip it here.

The only ambiguity left here is the case when a beam hits the boundary at a corner

since it causes diffraction and the above derivation does not apply any more. Here

we simply ignore the situation when a beam hits a corner of the domain since the

Gaussian method is asymptotic. The numerical accuracy will not be degraded without

those beams as those diffractions have exponentially small effects.

4.5.2 P-wave Reflecting Beams: The Hessian of the Phase

To illustrate the derivation of the second order derivative terms, we pick three entries

among six distinct entries in the Hessian for explanation, τyy, τxy and τxx, since all

other entries can be classified into one of these three types. Again, the reflection is
assumed to happen along the surface {x = (x, y, z) : x = 0} and all terms without

arguments are defined at the reflection point.

To start with the first type τyy, which is tangential component

τ P
yy = τ P S

yy = τ P P
yy .

(4.44)

τzz and τyz will also stay the same.

To derive the second type of terms τ P S

xy , we use the partial derivatives about the time

114

variable t,

˙τ P
y = cP

˙τ P S
y = cS

y |∇τ P| + cP (∇τ P
y )T∇τ P
|∇τ P|
y |∇τ P S| + cS (∇τ P S

y

|∇τ P S|

)T∇τ P S

The notation used here is ˙τy = ∂2τ

∂t∂y . We have

˙τ P
y = ˙τ P S

y

,

,

(4.45)

(4.46)

(4.47)

due to the continuity of the tangential and time component. Now substitute equation

(4.45) and equation (4.46) into equation (4.47),

cP

y |∇τ P| + cP (∇τ P
(∇τ P S

y )T∇τ P
|∇τ P|
)T∇τ P S =

y

= cS

τ P S
xy =

y

y |∇τ P S| + cS (∇τ P S
|∇τ P S|

)T∇τ P S
|∇τ P S|
y |∇τ P| + cP (∇τ P
y |∇τ P| + cP ∇τ P

(cid:32)
(cid:32)

cS

|∇τ P S|
cSτ P S

x

cP

cP

− cS

y )T∇τ P
|∇τ P|
y · ∇τ P
|∇τ P| − cS

(cid:33)
y |∇τ P S|
(cid:33)
y |∇τ P S|

− 1
τ P S
x

(τ P S

y

τ P S
yy + τ P S

z

τ P S
yz ).

(4.48)

To obtain the last type of the term τ P S

xx , we need to derive the formula of τ P S

tx first.

˙τ P = cP|∇τ P| ⇒ ¨τ P = cP (∇ ˙τ P )T∇τ P
˙τ P S = cS|∇τ P S| ⇒ ¨τ P S = cS (∇ ˙τ P S)T∇τ P S

|∇τ P|

,

|∇τ P S|

(4.49)

(4.50)

.

115

We then have the following equation from equation (4.50) and equation (4.49),

cP (∇ ˙τ P )T∇τ P
(∇ ˙τ P S)T∇τ P S =

|∇τ P|

= cS (∇ ˙τ P S)T∇τ P S
|∇τ P S|
cP (∇ ˙τ P )T∇τ P
|∇τ P S|
(cid:32)|∇τ P S|

cS

˙τ P S
x =

1
τ P S
x

|∇τ P|
cP (∇ ˙τ P )T∇τ P

|∇τ P|

cS

(cid:33)

(4.51)

− ˙τ P S

y

y − ˙τ P S
τ P S

z

τ P S
z

With the formula about the term ˙τ P S

x

given above, the term τ P S

xx can be obtained by

solving the following equation,

˙τ P S = cS|∇τ P S| ⇒ ˙τ P S

x = cS

x|∇τ P S| + cS (∇τ P S

x )T∇τ P S
|∇τ P S|

.

(4.52)

To remark, τ P S

x will not be zero as we assumed our initial conditions are compactly

supported.

4.5.3 P-wave Reflecting Beams: Amplitude Vector

We have so far already derived the initial condition of all terms involved with the

phase function for our new ODE dynamic system after P-wave reflection. Since the

phase function itself does not change after reflection in the center of the beam, we will

have the following equation about the amplitude to satisfy the homogeneous boundary

condition at the reflection point x(tr),

−aP∇τ P = aP P∇τ P P + aP SDP S.

(4.53)

116

We know that the PS-wave’s amplitude direction DP S is orthogonal to its ray direction
∇τ P S by the definition and with unit norm by our restriction. Consequently, the

following equation can be obtained if we project both sides of equation (4.53) to the
vector ∇τ P S at the same time.

−aP (∇τ P )T∇τ P S = aP P (∇τ P P )T∇τ P S,
aP P = −aP (∇τ P )T∇τ P S
(∇τ P P )T∇τ P S

.

(4.54)

Like the initial condition, aP SDP S is the summation of two S-waves.

(cid:88)

aP SDP S =

αP S
i D(i).

(4.55)

i

Similar to the initial condition, we pick the first direction D(1) to be the first column
vector of the matrix I3 − vvT and v = ∇τ P S
|∇τ P S|. Then, the second vector D2 will be
D(2) = D(1) × v, where × represents the cross product between two vectors. After
normalizing each direction, we can project the residual −aP∇τ P − aP P∇τ P P to each

direction to obtain the amplitude αP S

i

.

4.5.4 S-wave Reflecting Beams: the Phase term

Similar to the P-wave reflection, we have the following equation to satisfy the homo-

geneous boundary condition for the S-wave wavepacket at the reflection point x(tr).

−aSeiτ S

DS = aSP eiτ SP ∇τ SP + aSSeiτ SS

DSS.

(4.56)

117

After taking first glance of the above equation, the S-wave reflection dynamics seems

to be the same as the P-wave reflection developed in the last section. However, we

will find that there will be some significant difference between the S-wave reflection

and the P-wave reflection. To begin with, let us still use the case hitting the surface
{x = (x, y, z) : x = 0} as an example,

τ S
y = τ SS

y = τ SP

y

τ S
z = τ SS

z = τ SP

z

,

,

τ S
t = τ SS

t = τ SP
,
⇒ cS|∇τ S| = cP|∇τ SP|.

t

With all these equations combined, we will get

x = −sign(τ S
τ SP
x )

(cid:19)2 − (τ S

y )2 − (τ S

z )2,

(cid:115)(cid:18) cS|∇τ S|

cP

(4.57)

(4.58)

(4.59)

(4.60)

(4.61)

√

λ + 2µ. This

and the S-wave’s velocity cS =

√

µ is less than the P-wave’s cP =

leads to the possibility that the part inside the square root in equation (4.61) will be

negative, or equivalently, τ SP

x

can be in general a pure imaginary number. It will be

a disaster, since there will be some exponentially increasing wave on one side of the

boundary.

Let’s consider the regular situation first, in which case, equation (4.61) is a real number.

It is same as the one employed in the P-wave reflection as illustrated above, so we skip

it here.

118

The second case is when the term τ SP

is a pure imaginary number, i.e.

cS|∇τ S|

cP

+ (τ S

y )2 + (τ S

z )2

∇τ sp =

(4.62)

(cid:115)

−

i

x

(cid:18)



(cid:19)2

τ S
y

τ S
z

 .

This phenomenon is called the evanescent wave and the energy will fade away quickly

around the boundary in this case. Therefore, there is no need to derive its Hessian due

to its small energy.

4.5.5 S-wave Reflecting Beams: the Amplitude Vector

The evanescent wave fades away quickly, however, we will still include SP-wave’s am-

plitude vector in our derivation so as to make the derivation easier. Moreover, we need

the nonzero amplitude vector ASP to make the homogeneous boundary assumption

true at the reflection point x(tr). To summarize,

−aSDS = aSP (Re(∇τ sp) + iIm(∇τ sp)) + aSSDSS.

(4.63)

The way we get the amplitude vector ASP is still from the same idea by using the fact

that the SS wave’s amplitude direction DSS is orthogonal to its ray direction v,

(cid:0)Re(ASP )Re(∇τ SP ) − Im(ASP )Im(∇τ SP )(cid:1) · v = −Re(aS)DS · v;
(cid:0)Re(ASP )Im(∇τ SP ) + Im(ASP )Re(∇τ SP )(cid:1) · v = −Im(aS)DS · v.

(4.64)

(4.65)

119

The amplitude of the SP-wave’s amplitude ASP can be obtained by solving the above
system. Consequently, the residual −aSDS − aSP∇τ SP is now well defined. Following

the same process defined in the PS wave case, we can set up the amplitudes and

directions easily for SS-wave.

Notice that the ray after reflection is no longer smooth, which means that Lemma

4.2.1 is not applicable when reflections happen. Naturally, one needs to show that

the imaginary part of the Hessian after reflections defined above will still be positive

definite, especially for the PS wave and the SP wave case. In [4], authors have already

proved this is true for the PP wave and the SS wave, i.e. the conversion between same

wave modes. The proof about the conversion between different wave modes is provided

in Appendix B.

4.5.6 Method of Images for Boundary Conditions

In [4], authors have proposed a method to tackle the problem caused by partially

reflected beams. The partial reflection problem means the frontier part of a beam is

needed to be reflected back even when its central ray has not hit the boundary yet and

consequently the reflection dynamics has not been called. This is due to the fact that

beams have nonzero width and illustrated in the following graph.

Therefore, some modifications should be added to these partial reflection cases so that

the homogeneous boundary condition is always satisfied as well as our wavefield remains

to be continuous. Our strategy presented here is that the outer part is considered to

be reflected back, which is carried by some artificial beams. So we essentially apply

the odd extension to those beams, like what Figure 4.3 shows. The trajectory of the

120

Figure 4.2: Partially Reflected beams

Figure 4.3: Partially Reflected Beams with Odd Extension

blue dashed wavepacket in the graph is completely determined by its associates, the

red solid wavepacket in the graph. It implies that we don’t use any extra assumption

of the velocity outside the domain. The blue dashed wavepacket only serves as a

supplementary beam to satisfy the vanishing boundary condition.

121

−101−1−0.500.51xu−1−0.500.51−1−0.500.51xu4.6 Stationary Phase Analysis of Beams

To reinitialize or sharpen the single Gaussian beam ansatz, we need to apply some

stationary phase analysis to the single beam. So we first list some lemmas and com-

putations here which are needed to implement the reinitialization process in the next

section.

For any function u in L2(Rd), there is a phase space decomposition method such that,

(cid:16) ω

(cid:17)3d/2(cid:90)

2d/2eiω(p(x−x

(cid:48)

(cid:48)

)−p( ˜x−x

− ω|x−x

2

))e

(cid:48)|2

(cid:48)|2

− ω| ˜x−x

2

e

u( ˜x)d ˜xdpdx

(cid:48)

.

u(x) =

(4.66)

2π

R3d

Here x, ˜x and x

(cid:48)

are points in the spatial space Rd and p is the dual momentum

variable in the frequency space. ω is a fixed parameter determining the size of Gaussian

window functions.Here we would like to explore how to apply representation (4.66) to

Gaussian beams without considering its amplitude’s direction. Using P-wave as an

example, consider ΦP (t, x) = a(t)eiτ (t,x) instead of a(t)eiτ (t,x)p(t). Moreover, all

beam functions considered here are treated as single-variable functions by assuming a

principal variable while other variables are fixed.

In all the following derivations, the principal variable is assumed to be the variable y,

while other variables x and z are fixed.

122

4.6.1 Stationary Phase Approximation with Respect to Spa-

tial Variables

The Gaussian beams u with y as the principal variable considered in this section are

in the following form,

u(y; x, z, t) = A(x, z, t)eiΦ(y−y0)e

2 Im(τyy)(y−y∗
− 1
0)2

,

(4.67)

where (cid:0)x0, y0, z0

(cid:1) is the central point of the beam ansatz. All functions below are

defined at this point if not specified, and

Φ(y−y0) = τy(y−y0)+

Re(τyy)(y − y0)2

2

+Re(τxy)(y−y0)(x−x0)+Re(τzy)(y−y0)(z−z0).

(4.68)
τ is the phase function. To define the scalar y∗

0, we have the following equation where

all second order derivative terms are imaginary part only,

(cid:18)
x − x0, y − y0, z − z0

(cid:19)

(cid:18)
y − y0 +

= τyy



τxx τxy

τxz

τxy

τyy

τyz

τxz

τzy

τzz







x − x0
y − y0
z − z0
(cid:19)2

= τyy(y − y0)2 + 2(τxy(x − x0) + τxz(z − z0))(y − y0) + B(x, z)

τxy(x − x0) + τxz(z − z0)

τyy

+ ˜B(x, z).

(4.69)

The complete square term in equation (4.69) is defined as (y − y∗

0)2 and ˜B(x, z) is a

constant with respect to y.

123

To make the expression clear, we denote R = Re(τyy) and I = Im(τyy) throughout

this section. Applying decomposition (4.66) to u(y; x, z, t), we have

u(y; x, z, t) = A(x, z, t)

√

2eiωp(y−y

(cid:48)

− ω
2 |y−y

)e

(cid:48)|2

(cid:48)

(cid:48)

,

)dpdy

ψ(p, y

(4.70)

(cid:16) ω

(cid:17)3/2(cid:90)

2π

R2

and

(cid:90)

(cid:48)

ψ(p, y

) = ei ˜B(x,z)

e−iωp(˜y−y

(cid:48)

− ω
2 |˜y−y

)e

(cid:48)|2

eiΦ(˜y−y0)e

2|˜y−y∗
− I
0|2

d˜y.

(4.71)

We will apply the stationary phase approximation described in the following lemma to

).

calculate ψ(p, y

Lemma 4.6.1. We consider the behavior of I(ω) =(cid:82) b

a f (t)eiωg(t)dt, where f and g are

smooth enough to admit Taylor approximation near some appropriate points in [a, b],
and g is real-valued. Suppose there is some point c0 ∈ [a, b] and g
(t) (cid:54)= 0 everywhere

(cid:48)

else in the closed interval [a, b]. Moreover, g

(cid:48)(cid:48)

(c0) (cid:54)= 0. When ω >> 1, we have

(cid:48)

(cid:90) b

a

(cid:115)

(cid:18) 1

(cid:19)

ω

I(ω) =

f (t)eiωg(t)dt = f (c0)eiωg(c0)eiπδ/4

2π
(cid:48)(cid:48)

(c0)| + O

ω|g

,

(4.72)

where δ is the sign of g

(cid:48)(cid:48)

(c0).

To use Lemma 4.6.1, we first substitute y

(cid:48)

= y0 + ma into u(y; x, z, t), and the value

of a and ω will be defined later. By equation (4.71), we have

(cid:90)

(cid:18) Φ(˜y−y0)

ω

iω

e

(cid:19)

−p(˜y−y0−ma)

− ω
2 |˜y−y0−ma|2
e

2|˜y−y∗
− I
0|2
e

d˜y.

e−i ˜B(x,z)ψ(p, y0 + ma) =

(4.73)

124

To compute the critical point ˜y0 of the phase term

Φ(˜y−y0)

ω

− p(˜y − y0 − ma), we have

(˜y − y0)
ω

(cid:48)
Φ
R(˜y − y0) + Re(τxy)(x − x0) + Re(τyz)(z − z0)

− p,

ω

0 =

0 =

+

τy
ω

− p.

(4.74)

˜y0 = y0 +

ωp − τy − Re(τxy)(x − x0) − Re(τyz)(z − z0)

R

.

(4.75)

that is

We denote

−Re(τxy)(x − x0) − Re(τyz)(z − z0)

E(x, z) =

˜y0 − y0 =

ωp − τy

R

R

+ E.

,

(4.76)

Notice that E(x, z) is independent of the variable y, together with Lemma 4.6.1, the

term ψ(p, y0 + ma) is equal to

e−i ˜B(x,z)ψ(p, y0 + ma) = eiω(cid:0) Φ(

ωp−τy
R +E)
ω

−p(

ωp−τy

R +E−ma)(cid:1)
(cid:115)

2π
|R|.

2| ωp−τy
− I

R +E−y∗

0+y0|2

× e

i π
4
e

R|R|

2 | ωp−τy
− ω
e

R +E−ma|2

Substitute the expression of the term Φ into the above equation,

(cid:115)
2π
(cid:16) ωp−τy
(cid:17)−( R·E
|R| e

R|R|

i π
4

ω )

ψ(p, y0 + ma) = ei ˜Be

(cid:16) ωp−τy

(cid:16) τy

ω

R +E

× e

iω

2| ωp−τy
R −y∗
− I
0+y0+E|2
(cid:16) ωp−τy
(cid:17)−p
(cid:17)
R +E−ma

R +E

2 | ωp−τy
− ω
e

R +E−ma|2
ωp−τy

R +E)2(cid:17)

+ R
2ω (

.

(4.77)

125

We insert equation (4.77) into equation (4.70),

u(y; x, z, t) = aA(x, z, t)

(cid:17)3(cid:114) π

(cid:114)(cid:16) ω
i π
4
|R| e
2π
2 | ωp−τy
− ω
R +E−ma|2
(cid:17)2
(cid:16) ωp−τy

iω
e

R +E

≈ 2Aa

× e

× e

i R
2

(cid:114)(cid:16) ω

(cid:17)3(cid:90) √
(cid:90)
2π
R|R| ei ˜B
(cid:16) ωp−τy
(cid:16) τy

ω

R +E

− ω
2 |y−y0−ma|2

2eiωp(y−y0−ma)e
2| ωp−τy
R −y∗
− I
2 |y−y0−ma|2
− ω
eiωp(y−y0−ma)e
(cid:16) ωp−τy
(cid:16) ωp−τy
(cid:17)− R·E
(cid:17)(cid:17)
e
R +E−ma

(cid:17)−p

R +E

ψ(p, y0 + ma)dpdm
0+y0+E|2

ω

dpdm.

(4.78)

By applying the stationary phase approximation to the variable ˜y, we reduce the triple

integral (4.66) to the double integral (4.78).

Notice that in Lemma 4.6.1, the assumption R (cid:54)= 0 is required. In general, it is not

true. However, if R = 0, then the integral about the function ψ(p, y

(cid:48)

) is nothing but

the Fourier transform of a Gaussian function about y, so we omit it here.

4.6.2 Stationary Phase Approximation with Respect to Mo-

mentum Variables

Starting from the double integral (4.78), we would like to apply the stationary phase

approximation again. However, this time the variable we apply over is the momentum

126

variable p.

(cid:115)

(cid:114)(cid:16) ω
(cid:17)3
2| ωp−τy
R −y∗
− I
(cid:16) ωp−τy
(cid:17)− R·E

2π

i π
4

(cid:90)
R|R| a
− ω
2 |y−y0−ma|2
4π
|R|e
e
2 | ωp−τy
− ω
0+y0+E|2
(cid:17)−p
(cid:16) ωp−τy
(cid:17)
R +E−ma

R +E−ma|2

(cid:16) ωp−τy

+ R
2ω

R +E

e

ω

dm

R +E

u(y; x, z, t) = A(x, z, t)ei ˜B

(cid:90)

×

iω

× e

eiωp(y−y0−ma)e
(cid:18) τy

(cid:16) ωp−τy

R +E

ω

(cid:17)2(cid:19)

dp.

(4.79)

For the inner integral of equation (4.79), the phase function g(p) is

(cid:19)

+ E

(cid:18) ωp − τy
(cid:18)
(cid:19)

R
− p

E − ma +

τy
ω
ωp − τy

R

g(p) = p(y − y0 − ma) +

(cid:18)

− R · E

ω

E +

(cid:19)

(cid:18) ωp − τy

R

+

R
2ω

ωp − τy

R

(cid:19)2

+ E

,

and the smooth function f (p) is,

2 | ωp−τy
− ω

R +E−ma|2

e

f (p) = e

2| ωp−τy
− I

R −y∗

0+y0+E|2

.

(4.80)

To compute the critical point,

(cid:48)

g

(p) = y− y0− ma +

− E−

τy
R

(cid:18)

E − ma +

ωp − τy

R

(cid:19)

(cid:18) ωp − τy

R

− pω
R

+

(cid:19)

+ E

, (4.81)

127

then the critical point p0 is

ωp
R

= y − y0 − E +

(cid:0)y − y0 − E(cid:1) +

τy
R

R
ω

τy
ω

= y − y0 − E.

⇒ p0 =
⇒ ωp0 − τy

R

The second order derivative of the phase function g at the critical point p0 is

(cid:48)(cid:48)

g

(p0) = − ω
R

.

(4.82)

(4.83)

So the approximation of the inner integral of equation (4.79) is

(cid:90)

iω

× e

=(cid:114) πR

eiωp(y−y0−ma)e
(cid:18) τy
(cid:16) ωp−τy

ω

R +E

2| ωp−τy
R −y∗
− I
(cid:17)− R·E
(cid:16) ωp−τy

ω

R +E

0+y0+E|2

2 | ωp−τy
− ω
(cid:17)−p
(cid:16) ωp−τy
e
R +E−ma

R +E−ma|2
(cid:17)

+ R
2ω

(cid:16) ωp−τy

R +E

(cid:17)2(cid:19)

dp

(cid:18)
ω2 e

−i πR

4|R| eiωp0(y−y0−ma)e
(cid:18) τy−R·E
(y−y0)

(cid:19)(cid:19)

ω

2|y−y∗
− I
0|2
(cid:16)−p0(y−y0−ma)+ R

e

− ω
2 |y−y0−ma|2

2ω (y−y0)2(cid:17)

iω
e

.

iω

× e

We summarize the stationary analysis conducted in this section in the following lemma,

Lemma 4.6.2. Suppose the principal variable selected for the function u(y; x, z, t) is y

u(y; x, z, t) = A(x, z, t)eiΦ(y−y0)e

2 Im(τyy)(y−y∗
− 1
0)2

,

(4.84)

128

and R = Re(τyy) is nonzero, then the following decomposition can be obtained,

(cid:114) ω
2|y−y∗
− I
0|2
e

2π

ei˜τy(y−y0)ei R

2 (y−y0)2

dm + O

(cid:18) 1

(cid:19)

ω

(4.85)

u(y; x, z, t) = aA(x, z, t)ei ˜B(x,z)

(cid:90)

e−ω|y−y0−ma|2

where ˜τy = τy − R · E and

R · E = −Re(τxy)(x − x0) − Re(τyz)(z − z0).

(4.86)

The fixed parameter a is defined as a = 1√
I

. The value of ω will be specified later, but

its order is O(I).

4.7 Sharpening Beams by Reinitialization

In this section, we would like to propose a new reinitialization strategy based on Lemma

4.6.2 from Section 4.6. Again, we base our proof on the assumption that the variable

y is the principal variable. We first illustrate the reason why proposing a new reini-

tialization strategy is necessary.

4.7.1 The First Motivation for Developing a New Reinitial-

ization Strategy

It is necessary to add a reinitialization process into the propagation since the width

of a beam will increase exponentially in some generic medium [48]. We use the linear

velocity and 1-D problem for explanation. Suppose c(x) = α + βx where α and β are

129

constants, then the Riccati equation about the Hessian M will be

dM
dt

+ 2M β = 0, M (0) = i
.

(4.87)

Solving this simple linear ODE, we have

M (t) = i
e−2βt.

(4.88)

If the slope β > 0, then the width of the beam solution will be exponentially increasing.

As we can see, the beam solution will lose its accuracy in the simple linear velocity,

and each smooth velocity can be approximated by a linear function locally, therefore,

the same phenomenon can be expected in other situations.

4.7.2 The Second Motivation for Developing a New Reinitial-

ization Strategy

The second motivation is to resolve the problem caused by reflection beams. The idea

in Section 4.5 we have used to derive the reflection formula is theoretically correct,

however, it will cause some problems when implementing it numerically, especially in

the S-wave reflection case. The difference is that the SP-wave is more likely to be a

grazing beam.

To see this, we employ the 2D model for the illustration and let’s suppose the ray
hitting the line {x = (x, y) : x = 0}. Then, according to the analysis in Section 4.5,

130

the y-component of the ray direction τy does not change. For the PS reflection,

(cSτ P S

x )2 = (cP|τ P

x |)2 +
(λ + 2µ)|τ P

(τ P S

x )2 =

while the SP reflection follows,

(cP )2 − (cS)2(cid:17)

(cid:16)
x |2 + (λ + µ)τ 2
(cS)2

y

τ 2
y

,

(4.89)

(τ SP

x )2 =

µ|τ S

x |2 − (λ + µ)τ 2

y

(cP )2

.

(4.90)

We then compute the angle θP S and θSP between the ray direction and the reflecting
boundary {x = (x, y) : x = 0},

tan(θP S) =

=

=

τ P S
x
τ P S
y

(cid:118)(cid:117)(cid:117)(cid:116)(λ + 2µ)|τ P
(cid:115)

λ + µ

+

µ

y

x |2 + (λ + µ)τ 2
(cS)2τ 2
y
(λ + 2µ)|τ P
x |2

.

µτ 2
y

Similarly,

(cid:115)

tan(θSP ) =

µ|τ S
x |2
(λ + 2µ)τ 2
y

− λ + µ
λ + 2µ

.

(4.91)

(4.92)

We claim that for the PS reflection, the angle between the ray direction and the

boundary will be increasing after reflection, while for the SP reflection, this value will

131

be decreasing. To see this,

tan(θP S)
tan(θP )

(cid:33)2

=

=

λ+µ

µτ 2
y

µ + (λ+2µ)|τ P
x |2
(cid:19)2
(cid:18)
(cid:18) τy

λ + µ

τ P
x
τy

+

λ + 2µ

µ

µ

τ P
x

(cid:19)2

.

(cid:32)

(cid:32)

Apply the same idea to the SP reflection,

(cid:33)2

tan(θSP )
tan(θS)

=

µ

λ + 2µ

− λ + µ
λ + 2µ

(cid:18) τy

(cid:19)2

τ S
x

.

(4.93)

(4.94)

As we can see from equation (4.93) and equation (4.94), as τy is increasing or the

incidence angle is decreasing, the ratio for the PS reflection is increasing, which means

that the angle after reflection is larger than the incidence angle, while the angle for

the SP reflection is decreasing as a quadratic function of τy. It means that the angle

for the SP wave θSP will be closer to zero even when the incoming S-wave’s incidence

angle θ is away from zero. The grazing beam with larger width will interact with the

boundary. Therefore, it is needed to be sharpened to guarantee the accuracy.

To remark, there’s no problem with the reflection in the acoustic wave as the velocity

is the same. We have conducted the experiment to justify the analysis in 3-D space

and we show the result in the next section.

132

4.7.3 Sharpened Wavepackets and Convergence Analysis

By taking advantage of Lemma 4.6.2, we have

(cid:114) ω

2π

u(y) = aAei ˜B

(4.95)

2|y−y∗
− I
0|2
e

ei˜τy(y−y0)ei R

2 (y−y0)2(cid:90)

e−ω|y−y0−ma|2

dm + O

(cid:18) 1

(cid:19)

ω

,

The term ˜τy is the modified y-direction of the central ray, that is

˜τy = τy − R · E,

(4.96)

To sharpen Gaussian beams, we have the following lemma,

Lemma 4.7.1.

u(y) ≈ Aei ˜B

(cid:114) ω

2π

lka

q(cid:88)

k=0

ei˜τy(y−y0)e

−ωk|y−y∗
0|2

2|y−y∗
− I
0|2

e

2 (y−y0)2

ei R

.

(4.97)

where a = 1√
I

. Parameters ωk and q will be given in the proof.

As we can see the extra term e

−ωk|y−y∗
0|2

reduces the size of beams. And positive ωk

is obtained by choosing parameter ω appropriately.

133

Proof. To obtain equation (4.97), we first

(cid:90)

e−ω|y−y0−ma|2

dm =

=

(cid:90)
(cid:90)

= e

0+y∗
−ω|y−y∗
e
−ω|y−y∗
0|2
e
−ω|y−y∗

0|2(cid:90)

e

0−y0−ma|2
−ω|y∗

dm
0−y0−ma|2

−2ω(y−y∗
e
0−y0−ma|2

−2ω(y−y∗

0)(y∗
0)(y∗

e

−ω|y∗
e

0−y0−ma)

0−y0−ma)dm.

(4.98)

The integral part above is

e

−ω|y∗

(cid:90)
(cid:90) ˜k+ 1
(cid:88)
(cid:90) 1
(cid:88)

˜k− 1
2

˜k

2

=

k∈Z

2
− 1
2

0−y0−ma|2

−2ω(y−y∗
e

0−y0−ma)dm =

−ω|y∗
e

0−y0−ma|2

0)(y∗

0−y0−ma)dm

0)(y∗
−2ω(y−y∗
e

e−ω|(k+δ)a|2

e2ω(y−y∗

0)((k+δ)a)dδ.

(4.99)

We require k to be integers, which means the value of ˜k satisfies k = ˜k +
.
Notice first that we can truncate the above summation to finite terms, |k| ≤ q, since
e−ω|y−y0−ma|2

is a L1 function. By monotone convergence theorem,

a

0

y0−y∗

e−ω|y−y0−ma|2

dm = 0

(4.100)

lim
N→∞

.

Case 1: k = 0

(cid:90) ∞

N

134

2
− 1
2

(cid:90) 1
e−ω|(k+δ)a|2
(cid:90) 1
(cid:90) 1

e−ω|δa|2

2
− 1
2

≈

e−ω|δa|2

≈

2
− 1
2

(cid:90) 1
(cid:33)

2
− 1
2

0)((k+δ)a)dδ =

e2ω(y−y∗
(cid:32)(cid:88)
(cid:32)(cid:88)

n

n

2nωn(y − y∗
n!

(2ω)2n(y − y∗
(2n)!

0)n(δa)n

dδ

(cid:33)

0)2n(δa)2n

e−ω|δa|2

e2ω(y−y∗

0)(δa)dδ

dδ

(4.101)

The odd power terms in δ vanish in the last step above since e−ω|δa|2

is an even

function about δ and all odd power functions are odd functions. The integral of all
odd functions in [− 1

2 , 1

2] will be zero.

(cid:90) 1

2
− 1
2

e−ω|δa|2

e2ω(y−y∗

0)(δa)dδ ≈

≈

(cid:90) 1
(cid:90) 1

2
− 1
2

2
− 1
2

e−ω|δa|2(cid:16)

0)2(δa)2(cid:17)

1 + 2ω2(y − y∗

dδ

e−ω|δa|2

dδ.

(4.102)

0)2(δa)2

e2ω2(y−y∗
(cid:18) (2ω)4(y−y∗

0)4(δa)4

4!

(cid:19)

.

The leading order error from equation (4.102) is O

Case 2: k (cid:54)= 0

135

When k < 0,

(cid:90) 1

2
− 1
2

e−ω|(k+δ)a|2

e2ω(y−y∗

0)((k+δ)a)dδ =

=

=

2
− 1
2

(cid:90) 1
(cid:90) 1
(cid:90) 1

2
− 1
2

2
− 1
2

e−ω|−|k|a+δa|2

e2ω(y−y∗

0)(−|k|a+δa)dδ

e−ω|−|k|a−˜δa|2

e2ω(y−y∗

0)(−|k|a−˜δa)d˜δ

e−ω||k|a+δa|2

e

−2ω(y−y∗

0)(|k|a+δa)dδ,

(4.103)

by setting ˜δ = −δ.

When k > 0

(cid:90) 1

(cid:90) 1

e2ω(y−y∗

e−ω|(k+δ)a|2

0)((k+δ)a)dδ =

2
− 1
2
(4.104)
Given k > 0, we add (4.104) for k > 0 and (4.103) for −k < 0, so that we have

2
− 1
2

e−ω|(|k|+δ)a|2

e2ω(y−y∗

0)((|k|+δ)a)dδ

Here we use the Taylor expansion of the exponential function. Furthermore,

e2aω(y−y∗

0)(k+δ) + e

1 + 2ω2(y − y∗

−2aω(y−y∗

0)(k+δ)(cid:17)
0)2(k + δ)2a2(cid:17)

dδ

dδ

(4.105)

0)2(k + δ)2a2(cid:17)

dδ

1 + 2ω2(y − y∗

e2ω2(y−y∗

0)2((k+δ)a)2

dδ.

(4.106)

(cid:90) 1
e−ω|ka+δa|2(cid:16)
(cid:90) 1
2e−ω|ka+δa|2(cid:16)

2
− 1
2

=

2
− 1
2

(cid:90) 1
2e−ω|ka+δa|2(cid:16)
(cid:90) 1

2
− 1
2

2e−ω(k+δ)2a2

≈

2
− 1
2

136

(cid:18) (2ω)4(y−y∗

0)4(δa+ka)4
4!

The error term is O

(cid:19)

(4.102) can be summarized as an universe form, i.e, O

. The error of approximations (4.106) and

(cid:18) (2ω)4(y−y∗

(cid:19)

.

0)4(δa+ka)4
4!

Now integral (4.99) becomes

0−y0−ma|2

−ω|y∗
e

(cid:90)
(cid:90) 1
e−ω||k|a+δa|2
(cid:90) 1
(cid:88)

2
− 1
2

e

+

2
− 1
2

k>0

0)(y∗

0−y0−ma)dm =

−2ω(y−y∗
e
−2ω(y−y∗

0)(|k|a+δa)dδ

2e−ω(k+δ)2a2

e2ω2(y−y∗

0)2((k+δ)a)2

dδ.

(4.107)

To get the value lk and ωk, we start with k > 0,

0|2(cid:90) 1

−ω|y−y∗

e

2e−ω(k+δ)2a2

e2ω2(y−y∗

0)2((k+δ)a)2

dδ ≈

−ω|y−y∗
0|2

2e

2
− 1
2
e2ω2(y−y∗
−ω|y−y∗
0|2

= lke

0)2(ka)2

e−ω|ka|2

e2ω2(y−y∗

0)2(ka)2

(4.108)

First, the first step above is obtained by choosing δ = 0. Second, the value of lk is

defined by

Similarly, when k = 0,

lk = 2e−ω|ka|2

,

lk = 1.

137

(4.109)

(4.110)

The next step is to compute the value of ωk and define the parameter ω. We start

with making the exponent in equation (4.108) negative,

−ω + 2ω2k2a2 ≤ 0

ω ≤ 1

2a2k2 , k = 1, 2,··· , q

(4.111)

and a is previously defined as 1√
I

equation (4.97) is

in Lemma 4.6.2. If we choose ω = I

3q2 , then ωk in

(cid:18)3q2 − 2k2

(cid:19)

ωk =

9q4

I.

(4.112)

After defining the values lk and ωk, we will characterize the size of the error term,

(2ω)4(y − y∗
0)4(δa + ka)4
4!

=

max
k≤q

(2ω)4(y − y∗
0)4(δa + qa)4
4!

(cid:19)4(cid:18) 1√

(cid:19)4

I

(cid:18) I

≤ 2
3
≤ 2
243

3q2
1
q4

(qa)4

(4.113)

Although this is only the leading order in the series we truncated, the rest of them will

be easily controlled from the following two concerns. The first concern is the rest of

the series will have high power about q, i.e.
is that the coefficient 22n

(2n)! will decay exponentially.

1

q2n , for n = 3, 4,··· . The second concern

138

4.8 Numerical Examples

In this section, we present all numerical experiments conducted to justify the proposed

method through some complex velocity and general initial conditions. All the refer-

ence solution in this section are provided by the FDTD method with staggered grid.

Reference solution’s correctness will be examined in Appendix B.

4.8.1 Beam Reinitialization

4.8.1.1 SP Reflection V.S. PS Reflection

In the subsection, we will provide the numerical results of the analysis in Section

4.7.2 and Section 4.7.3. We firstly conduct the following experiment to illustrate the

difference between the SP reflection and the PS reflection. The experiment’s setup for

the S-wave is,



f S =

(4.114)

and

− sin(α) sin(36πx + 36π sin(α)(y − 0.25))e

sin(36πx + 36π sin(α)(y − 0.25))e

0

x2+(y−0.25)2+(z−0.25)2(cid:17)

−36π2(cid:16)
x2+(y−0.25)2+(z−0.25)2(cid:17)

−36π2(cid:16)

 ,

gS = 0.

(4.115)

139

The experiment’s setup for P-wave is,



f P =

and

 ,

(4.116)

sin(36πx + 36π sin(α)(y − 0.25))e

sin(α) sin(36πx + 36π sin(α)(y − 0.25))e

−36π2(cid:16)
x2+(y−0.25)2+(z−0.25)2(cid:17)
x2+(y−0.25)2+(z−0.25)2(cid:17)
−36π2(cid:16)

0

gP = 0

The initial width of the beams are all set to be 36π2 and λ = µ = 1. The experiment’s
result is displayed in Figure 4.4. The red star line in Figure 4.4 is the ratio sin(θP S )

,

sin(α)

Figure 4.4: PS Reflection V.S. SP Reflection: Different Ratio Behaviors.
PS Reflection(’o’), SP Reflection (’-’)

and the blue line is the ratio sin(θSP )

sin(α)

. Same as the analysis in Section 4.7.2, the angle

for the SP reflection will decrease to zero compared with the original hitting angle.

Therefore, the SP reflection wave should be modified to be more focused.

140

00.10.20.30.40.50.600.20.40.60.811.21.4sin(α)The ratio of Sin(θ)Figure 4.5: Sharpened Beams V.S. the Original Beam on Fixed (y, z).
Left: y=0.1, z=0.1 Right: y=0.08, z=0.13.
Beam Solution after Reinitialization(’o’), Original Beam Solution (’-’).

4.8.1.2 Sharpened Beams V.S. Original Beams

The second experiment in this subsection is conducted to show the fact that the pre-

cision will not be impacted after adding the new reinitialization (4.97) process. We

compare the wavefield after the reinitialization with the original one. Suppose the

S-wave hits at the boundary x = 0 with the central point (0, 0.1, 0.1) and the Hessian

equals to



36π + 36π2i

12π2i

12π2i

7π2 + 36π2i

0

0

0

0

4π2 + 36π2i

 .

Its amplitude norm is set as 100. We choose two different sets of the y-value and

z-value to show our reinitialization method’s correctness, The wavefield in Figure 4.5

is plotted along the x-axis. As we can see, the reinitialization algorithm won’t affect

the accuracy of the result, while the width of each new beam has been decreased.

141

00.10.20.30.4020406080100xu00.10.20.30.4020406080100xu4.8.2 Periodic Boundary Condition

In this subsection, we would like to display some numerical results to the periodic

boundary problem.

4.8.2.1 Example 1: The Single Wavepacket in the Constant velocity

We first test our algorithm via the constant elastic moduli λ = µ = 1, and the nonzero

initial velocity g.



f =

sin(36πx + 9πy)e−36π2(cid:0)(x−0.25)2+(z−0.25)2+(z−0.25)2(cid:1)
sin(36πx + 9πy)e−36π2(cid:0)(x−0.25)2+(z−0.25)2+(z−0.25)2(cid:1)

 ,

(4.117)

0

and the initial velocity g is

g = 0.5f .

(4.118)

We exhibit our result at the plane z = 0.25. The comparison between two results

shows that the propagation dynamics, including the eikonal equation and the transport

equation are correct. The Multi-scale Gaussian Wavepacket Transform is also justified,

although this is a single wavepacket initial value. We will test the transform further

with some more general initial condition.

4.8.2.2 Example 2: General Initial Condition in the Constant velocity

The next thing we would like to try is some general initial condition other than the

single Gaussian wavepacket to verify our initial decomposition. The initial value is

142

Figure 4.6: Example 1:
Single Wavepacket Propagation with the Periodic Boundary Condition.
Left: FDTD Solution Right: Gaussian Beam Solution.



f =

sin(72πx2)e−36π2(cid:0)(x−0.25)2+(z−0.25)2+(z−0.25)2(cid:1)
sin(72πx2)e−36π2(cid:0)(x−0.25)2+(z−0.25)2+(z−0.25)2(cid:1)

0



(4.119)

defined as

and

g = 0;

(4.120)

Let’s compare the result along the x-axis by setting z = 0.25 and y = 6

32. Now let’s

compare 2-D wavefields at z = 0.25.

For other more complex velocity, we will exhibit those results in the Dirichlet boundary

condition, since the periodic boundary problem essentially tests the correctness of

propagation dynamic and the decomposition process of the initial condition as proposed

143

xy  00.20.400.10.20.30.40.5−0.4−0.200.20.4xy  00.20.400.10.20.30.40.5−0.3−0.2−0.100.10.20.3Figure 4.7: Example 2:
General Initial Value Propagation with Periodic Boundary Condition along x-axis
FDTD Solution(’o’), Gaussian Beam Solution (’-’)

Figure 4.8: Example 2:
General Initial Value Propagation with Periodic Boundary Condition.
Left: FDTD Solution Right: Gaussian Beam Solution.

in Section 4.4. This can be examined in the Dirichlet boundary problem as well.

144

00.10.20.30.40.5−0.2−0.15−0.1−0.0500.050.10.150.2xUxy  00.20.400.10.20.30.40.5−0.3−0.2−0.100.10.20.3xy  00.10.20.30.40.500.10.20.30.40.5−0.3−0.2−0.100.10.20.34.8.3 Reflection: P-wave

We have so far used the periodic boundary condition to justify our decomposition

process and the dynamic system, now we would like to test our reflection scheme on

the pure P-wave initial condition. In this subsection, we fix our initial condition as the

following and test our algorithms over the different velocity.

−36π2(cid:16)
−36π2(cid:16)

(x−0.15)2+(y−0.25)2+(z−0.25)2(cid:17)
(x−0.15)2+(y−0.25)2+(z−0.25)2(cid:17)

sin(32πx + 8πy)e

f =

1
4 sin(32πx + 8πy)e




(4.121)

(4.122)




and the initial velocity g is

−36π2(cid:16)
−36π2(cid:16)

(x−0.15)2+(y−0.25)2+(z−0.25)2(cid:17)
(x−0.15)2+(y−0.25)2+(z−0.25)2(cid:17)

2 sin(32πx + 8πy)e

g =

1
2 sin(32πx + 8πy)e

0

0

4.8.3.1 Example 3: P-wave Reflection in the Constant velocity

Our first setting is still under the constant elastic moduli, λ = 1 and µ = 2. We

compare the wavefield at z = 0.25 and T = 0.14, after the primary reflection happens.

The P-wave reflection dynamics can be justified after this numerical experiment.

Here the mesh size of the FDTD method employed is

1
640 and this scale will be used

for all the FDTD results in the rest of the paper.

145

Figure 4.9: Example 3:
P-wave Reflection in Constant velocity with Dirichlet Boundary Condition.
Left: FDTD Solution Right: Gaussian Beam Solution.

4.8.3.2 Example 4: P-wave Reflection in the Linear velocity

The second velocity in the P-wave reflection section we use is µ = 2 and λ = 1 + 0.2x.

We first compare the result at the fixed (y, z) = (0.125, 0.25) along x-axis. Then we

set z fixed as 0.25, The P-wave reflection dynamics in some general velocity has been

justified further. As we can see in Figure 4.10, the Gaussian beam solution performs

quite well in the major region, and the tolerable error shows up near the boundary.

4.8.3.3 Example 5: P-wave Reflection in the Sinusoidal velocity

Now let’s try the sinusoidal elastic moduli λ = 1 + sin(4πx), in which there will be

some caustics points. Again, we compare the result along the x-axis first by fixing

y = z = 0.25.

146

xy  00.20.400.10.20.30.40.5−0.2−0.100.10.2xy  00.20.400.10.20.30.40.5−0.2−0.100.10.2Figure 4.10: Example 4:
P-wave Reflection in the Linear velocity with Dirichlet Boundary Condition.
FDTD Solution (’o’), Beam Solution(’-’)

Then we fix z = 0.25, To remark, the reinitialization scheme is not involved in all the

P-wave reflection results shown above. The small error implies that the P-wave reflec-

tion does not require the reinitialization, while it is necessary in the S-wave reflection.

4.8.4 Reflection S-wave

In this subsection, we will test the reinitialization process, but at first we will justify

the necessity of adding reinitialization process.

4.8.4.1 Example 6: S-wave Reflection with Orthogonal Hitting Angle

Firstly, we see the S-wave reflection with orthogonal hitting angle, that is sin(α) = 0

in Figure 4.4. It is displayed by Figure 4.14 showing that the original method without

the reinitialization is good enough.

147

00.10.20.30.40.5−0.15−0.1−0.0500.050.10.150.2xyFigure 4.11: Example 4:
P-wave Reflection in Linear velocity with Dirichlet Boundary Condition.
Left: FDTD Solution Right: Gaussian Beam Solution.

Figure 4.12: Example 5:
P-wave Reflection in Sinusoidal velocity along x-axis.
FDTD Solution(’o’), Beam Solution (’-’)

The setup of our experiment is



f =

sin(36πy)e−36π2((x−0.25)2+(y−0.15)2+(z−0.25)2)

0

0

148

 .

(4.123)

xy  00.20.400.10.20.30.40.5−0.2−0.100.10.2xy  00.20.400.10.20.30.40.5−0.15−0.1−0.0500.050.10.1500.10.20.30.40.5−0.2−0.15−0.1−0.0500.050.10.150.2xuFigure 4.13: Example 5:P-wave Reflection in Sinusoidal velocity with Dirichlet Boundary
Condition. Left: FDTD Solution, Right: Gaussian beam Solution

and there is no initial velocity, i.e. g = 0. Two velocity parameters λ and µ here are

both constants,

λ = 1; µ = 2;

Figure 4.14 compares the wave field generated by our method to the one from FDTD

method at z = 0.25, x = 0.25 and T = 0.2. Now we fix z = 0.25 and y = 5

32, As we

can see from Figure 4.4, when sin(α) = 0, the regular reflection method is expected to

be well enough and the experiment result above justifies our conclusion.

4.8.4.2 Example 7: S-wave Reflection with Non-Orthogonal Hitting Angle

Now if we change the ray direction to increase the width of the SP-wave, we will see

that the regular reflection dynamics fails in this case. To make that happen, we specify

149

xy  00.10.20.30.40.500.10.20.30.40.5−0.2−0.15−0.1−0.0500.050.10.150.2xy  00.10.20.30.40.500.10.20.30.40.5−0.15−0.1−0.0500.050.10.150.2Figure 4.14: Example 6:
S-wave Reflection with Orthogonal hitting Angle along y-axis
FDTD Solution (’o’), GB Solution (’-’).

Figure 4.15: Example 6: S-wave Reflection with Orthogonal hitting Angle along x-axis.
FDTD Solution (’o’), GB Solution (’-’).



−36π2(cid:16)
−36π2(cid:16)

(x−0.25)2+(y−0.15)2+(z−0.25)2(cid:17)
(x−0.25)2+(y−0.15)2+(z−0.25)2(cid:17)

 .

(4.124)

our initial value as,

2 sin(36πy + 18πx)e
− sin(36πy + 18πx)e

f =

0

150

00.10.20.30.40.5−0.25−0.2−0.15−0.1−0.0500.050.10.150.20.25yu00.10.20.30.40.5−0.0200.020.040.060.080.1xuFigure 4.16: Example 7: S-wave Reflection with Non-orthogonal Hitting Angle. Left:
Gaussian Beam Solution without reinitialization, Right: Gaussian Beam Solution with
reinitialization. FDTD Solution (’o’), GB Solution (’-’)

We first see that the result generated without the extra reinitialization. The above

left Figure 4.16 is plotted along the x-axis with z = 0.25, y = 0.125. As we mentioned

before, although the main pattern is captured with good accuracy, the tail region of

the Gaussian beam wavefield is not clean enough due to the fact that the SP-wave is

involved.

After adding the reinitialization, in the above right Figure 4.16, the beam solution

with the reinitialization shows the better result in the tail region without hurting the

accuracy of other parts.

The experiment shown in Figure 4.16 illustrates that the analysis in Section 4.7 is

correct. The SP-wave reflection will lose the accuracy to some degree such that adding

the reinitialization scheme is necessary.

151

00.10.20.30.40.5−0.4−0.200.20.4xU00.10.20.30.40.5−0.4−0.200.20.4xU4.8.4.3 Example 8: S-wave Reflection: Linear velocity

Let’s see the comparison under some more complicated elastic moduli µ = 2 + 0.2y

and λ = 1. We compare the wavefield without the reinitialization to the FDTD result

first at z = 0.25.

Figure 4.17: Example 8: S-wave Reflection in Linear velocity without Reinitialization.
Left: FDTD Solution Right: Gaussian Beam Solution without Reinitialization.

As we can see in the upper left corner of Figure 4.17, there is some significant per-

turbations in the FDTD result, while the beam method without the reinitialization,

which is shown in Figure 4.17, is not able to cover that part.

Now we will see the result after using the reinitialization process. As we can see

from the right Figure 4.20, the missing part in the upper left region is covered by the

reinitialization algorithm.

152

xy  00.20.400.10.20.30.40.5−0.4−0.200.20.4xy  00.20.400.10.20.30.40.5−0.4−0.200.20.4Figure 4.18: Example 8: S-wave Reflection in Linear velocity with Reinitialization. Left:
FDTD Solution Right: Gaussian Beam Solution with Reinitialization.

4.8.4.4 Example 9: S-wave Reflection: Sinusoidal velocity

In this example, we set the elastic moduli µ = 2 + 0.2 sin(x) to make the Hessian of

the velocity nonzero. And all the other components remain the same as the last one,

including the initial value, λ and the terminal time T .

Compare two wavefields on the plane z = 0.25,

In this more complex velocity, the advantage of the Gaussian beam ansatz has shown

up as the caustics problem is resolved automatically. There will be caustics in this

sinusoidal velocity as the eikonal equation will be multivalued in some region. As

Figure 4.20 suggests, the beam solution will perform well even when the caustics shows

up.

153

xy  00.20.400.10.20.30.40.5−0.4−0.200.20.4xy  00.20.400.10.20.30.40.5−0.4−0.200.20.4Figure 4.19: Example 9: S-wave Reflection in Sinusoidal velocity. GB Solution(’o’), FDTD
Solution(’-’)

Figure 4.20: Example 9: S-wave Reflection in Sinusoidal velocity with Reinitialization.
Left: FDTD Solution Right: Gaussian Beam Solution with Reinitialization.

4.8.5 General Initial Condition

Now after showing the effect of the reinitialization, especially after comparing it to the

result without the reinitialization, it is confident to say that the proposed reinitializa-

154

00.10.20.30.40.5−0.15−0.1−0.0500.050.10.150.2xyxy  00.20.400.10.20.30.40.5−0.15−0.1−0.0500.050.10.15xy  00.20.400.10.20.30.40.5−0.15−0.1−0.0500.050.10.15tion algorithm is correct and necessary in this problem. We will end our numerical

tests with some more general initial conditions.

4.8.5.1 Example 10: The General Initial Condition in The Sinusoidal ve-

locity

We test Multiscale Gaussian Wavepacket method via the general initial condition and

sinusoidal elastic moduli λ = 1 + sin(4πx), µ = 2. The initial condition is

−36π2(cid:16)
(x−0.15)2+(y−0.25)2+(z−0.25)2(cid:17)
−36π2(cid:16)
(x−0.15)2+(y−0.25)2+(z−0.25)2(cid:17)
(x−0.15)2+(y−0.25)2+(z−0.25)2(cid:17)
−36π2(cid:16)

 .

(4.125)



sin(32πx + 32πy2)e

f =

sin(32πx + 10πy)e

sin(16πz)e

Here we let our wavefield propagate until T = 0.2.

We first display the first component in the resulting wavefield. Let’s see the second

component of the resulting wavefield, and see the comparison between two methods

along the x-axis by fixing y = z = 0.25. With more general initial value, we include

P and S-wave at the same time, meanwhile, different types of the reflection happen

simultaneously, so as the different reflection modes.

Remark 4.8.1. Compared with the FDTD algorithm with parallel computing scheme,

our asymptotic algorithm has the larger time complexity. However, as the wavenumber

of the initial value is increasing, the FDTD scheme requires finer grid size, leading to

the requirement of some larger storage. This is impossible in the current GPU units,

while our method’s storage complexity is independent of the wavenumber as well as the

time complexity.

155

Figure 4.21: Example 10: General Initial Condition Propagation with Reinitialization
(First Component). Left: FDTD Solution Right: Gaussian Beam Solution with Reinitial-
ization.

Figure 4.22: Example 10:
General Initial Condition Propagation with Reinitialization (Second Component).
Left: FDTD Solution Right: Gaussian Beam Solution with Reinitialization.

156

xy  00.20.400.10.20.30.40.5−0.2−0.100.10.2xy  00.20.400.10.20.30.40.5−0.2−0.100.10.2xy  00.20.400.10.20.30.40.5−0.0500.05xy  00.20.400.10.20.30.40.5−0.0500.05Figure 4.23: Example 10: General Initial Condition Propagation with Reinitialization
(Second Component). FDTD Solution (’o’), GB Solution(’-’).

4.8.5.2 Convergence Rate Analysis

In the end, we propose the convergence rate analysis. The initial condition is set as

−42π2(cid:16)
−42π2(cid:16)

(x−0.25)2+(y−0.15)2+(z−0.25)2(cid:17)
(x−0.25)2+(y−0.15)2+(z−0.25)2(cid:17)



(4.126)



sin(η(16πx + 8πy))e

sin(η(16πx + 8πy))e

f =

0

The velocity here are all constants,

λ = 1; µ = 2;

The amplifying factor η is a geometric series, 1, 1.5, 2.25,··· , 1.55.

The blue star line is the logarithm of the L2 norm of the error at different η, while

the red line is the linear function with the slope as 1

2 log(1.5). It is well known that

the convergence rate of the Gaussian beam is

1√
ω

, and as proved in the paper [5], the

157

00.10.20.30.40.5−0.15−0.1−0.0500.050.10.150.2xuFigure 4.24: Log-log plot: Convergence Rate of the New Gaussian Beam Method
GB Method Error Curve(’-*’), Line with the slope = 1

2 log(1.5)(’-’)

Multiscale Gaussian wavepacket transform also follows O( 1√
ω

). This pattern can be

seen in Figure 4.24.

158

3.544.555.56−5.5−5−4.5−4log(wavenumber)log(error)Chapter 5

Conclusion

We propose two methods based on multi-scale Gaussian beam method in this thesis.

The first one is solving the elastic wave propagation in the bounded domain and the

second one is for the prestack inversion process, which is an inverse problem in geo-

physical applications. These two methods are both based on the multiscale Gaussian

beam method described in Chapter 2. Therefore, both methods are capable of re-

solving caustics problem automatically and they both take advantage of the parabolic

scaling principle for efficiency.

In the first part (Chapter 3), we present a new prestack inversion process, which con-

nects the boundary data to the wavefront set of the perturbation. We first modify

the multiscale Gaussian wavepacket transform [48] appropriately to suit to the imag-

ing operator. Secondly, the multi-valued traveltime information is preserved due to

the Gaussian beam function. This improves the quality of resulting migration image.

Another big advantage of our multiscale Gaussian beam inversion method is its ro-

bustness to the polluted data. Since we recover the reflector by its wavefront set, the

noise in the boundary data, which is far from the target frequency, won’t affect the

imaging result. Lastly, our imaging condition is performed in the time domain to avoid

the extra Fourier transform on the data set. This feature makes our algorithm more

159

applicable considering the large size of the trace dataset in the real world.

In the second part (Chapter 4), we present a novel Multiscale Gaussian beam method

to solve the elastic wave equation in the bounded domain. Firstly, a new vector-valued

wavepacket transform is developed to adapt to the highly oscillated vector-valued initial

condition following the parabolic scaling principle. Secondly, a novel reinitialization

strategy is added in the process to improve the efficiency and accuracy. There are

several advantages about this new reinitialization method. The first one is that the

new reinitialization is applied to the single wavepacket instead of the whole wavefield.

This will improve the efficiency greatly. The second one is that the center of each new

wavepacket after the reinitialization is same to the center before, which guarantees

all computation happening inside the domain without extra assumption outside the

domain. Although the typical FDTD (Finite Difference Time Domain) method is

faster by implementing in parallel, the requirement of large storage will still make

FDTD method unfeasible in the high frequency regime.

160

APPENDICES

161

Appendix A

Proof in Inverse Process

Hessian Matrix in Corollary 3.3.1

In Corollary 3.3.1, the quadratic term ||r − ˆr0||2
in the phase function is not
necessary to be a Gaussian profile, since ˆM is only the Schur complement of a semi

ˆM(ˆtc)

positive definite matrix.

Lemma A.0.1. For any boundary points r = {x = (x1,··· , xd) : xd = 0}, we have

(cid:32)
Im( ˆM )(ˆtc) − Im(ˆτtx)(ˆtc)Im(ˆτtx)T (ˆtc)

(cid:33)

r ≥ arT Im( ˆM )(ˆtc)r,

(A.1)

rT

Im(ˆτtt)(ˆtc)

for some constant a, if there is no grazing ray,

|ˆpd(ˆtc)| ≥ b||ˆp(ˆtc)||,

(A.2)

where b is an universal lower bound.

By equation (3.60) and (3.61), we have

Im(ˆτtx)(ˆtc; y, p) = ∓v(ˆy(ˆtc))Im( ˆM )(ˆtc)

ˆp(ˆtc)
||ˆp(ˆtc)|| ,

(A.3)

162

and

Im(ˆτtt)(ˆtc; y, p) = v2(ˆy(ˆtc))

ˆpT (ˆtc)
||ˆp(ˆtc)|| Im( ˆM )(ˆtc)

ˆp(ˆtc)
||ˆp(ˆtc)||.

(A.4)

From now on, we let all functions be defined at t = ˆtc without writing it out explicitly.

Then,

rT Im(ˆτtx)Im(ˆτtx)T

Im(ˆτtt)

r = rT Im( ˆM )ˆpˆpT Im( ˆM )

ˆpT Im( ˆM )ˆp

r.

We introduce the following notation,

(cid:48)

p

=

ˆp(cid:113)

ˆpT Im( ˆM )ˆp

.

(A.5)

(A.6)

Then equation (A.1) can be translated to the following optimization problem. We can

instead prove that the optimal value of the following optimization problem with fixed
(cid:48)

p

is a,

min

r

subject to

1 −(cid:16)

(cid:48)(cid:17)2

,

rT Im( ˆM )p

rT Im( ˆM )r = 1,

rT ed = 0,

(A.7)

(A.8)

where ed = (0,··· , 0, 1)T . It is equivalent to,

min

r

subject to

1 − rT Im( ˆM )p

(cid:48)

,

rT Im( ˆM )r = 1,

rT ed = 0,

since the optimization problem (A.7) is the square term of problem (A.8). Using

163

Lagrange multipliers,

L(r; λ1, λ2) = 1 − rT Im( ˆM )p

(cid:48)

+ λ1rT Im( ˆM )r + λ2rT ed.

(A.9)

Differentiate L with respect to r,

−Im( ˆM )p

(cid:48)

+ 2λ1Im( ˆM )r + λ2ed = 0,

(A.10)

then

2λ1r = −λ2Im( ˆM )−1ed + p

(cid:48)

.

By the second restriction rT ed = 0,

(cid:48)

eT
d p

d Im( ˆM )−1eT
eT

d

λ2 =

d Im( ˆM )−1eT

d . Therefore,

(cid:113)

We denote A =

ˆpT Im( ˆM )ˆp and D = eT

and the optimizer r(cid:63) satisfies,

(cid:48)

eT
d p
AD

.

λ2 =

r(cid:63) (cid:107) r0,

where r0 is defined as,

r0 ≡ p

(cid:48)

(cid:48) − eT
d p
AD

Im( ˆM )−1ed.

164

(A.11)

(A.12)

(A.13)

(A.14)

(A.15)

By restriction rT Im( ˆM )r = 1, we have,

(cid:113)

Equivalently, 2λ1 =

(cid:113)

r(cid:63) =

r0

0 Im( ˆM )r0
rT

,

(A.16)

0 Im( ˆM )r0. Equation (A.7) then becomes,
rT

1 −(cid:16)

min

r

(cid:48)(cid:17)2

= 1 −

rT Im( ˆM )p



0(cid:113)

rT

0 Im( ˆM )r0
rT

(cid:48)

Im( ˆM )p

2

.

(A.17)

The first term needed to be evaluated above is,

rT
0 Im( ˆM )p

(cid:48)

= (p

= (p

(cid:48) − eT
d p
AD
(cid:48)

)T Im( ˆM )p

(cid:48)

Im( ˆM )−1ed)T Im( ˆM )p

(cid:48)

(cid:48)

(cid:48) − (eT
)2
d p
A2D

.

(cid:48)

(cid:48) − eT
d p
AD

Im( ˆM )−1ed)
d Im( ˆM )−1ed − 2eT
d p
AD

eT
d p

(cid:48)

(cid:48)

The second term is

rT
0 Im( ˆM )r0 = (p

= (p

= (p

= (p

= (p

(cid:48)

(cid:48) − eT
d p
AD
(cid:48)

)T Im( ˆM )p

(cid:48)

(cid:48)

(cid:48)

)T Im( ˆM )p

)T Im( ˆM )p

)T Im( ˆM )p

Im( ˆM )−1ed)T Im( ˆM )(p

(cid:48)

(cid:48)

+

+

(cid:48)

(cid:48)

(cid:48)

)2

)2

(eT
d p
(AD)2 eT
(AD)2 D − 2eT
(eT
d p
d p
AD
(cid:48)
− 2(eT
d p
A2D

(cid:48)

)2

(cid:48)

+

(eT
)2
d p
A2D
(cid:48)
(cid:48) − (eT
)2
d p
A2D

(cid:48)

eT
d p
A

.

(A.18)

165



1 −

0(cid:113)

rT

0 Im( ˆM )r0
rT

2

(cid:48)

Im( ˆM )p

= 1 −

= 1 −

 rT
(cid:113)
(cid:32)

(cid:48)

2

(cid:48)

0 Im( ˆM )p
0 Im( ˆM )r0
rT

(cid:33)2

(p

)T Im( ˆM )p

(cid:48)

(p

)T Im( ˆM )p

(cid:48)

(cid:48) − (eT
)2
d p
A2D
(cid:48)
(cid:48) − (eT

)2
p
d
A2D

(cid:48)

=

(eT
)2
d p
A2D

.

(A.19)

To summarize,

(cid:32)

rT

and

Im( ˆM ) − (r − ˆr0)T Im(ˆτtx)Im(ˆτtx)T

Im(ˆτtt)

(cid:33)

r ≥ arT Im( ˆM )(ˆtc)r

(A.20)

a =

b2
A2D

=

b2
d (Im( ˆM ))−1ed)
(ˆpT Im( ˆM )ˆp)(eT

.

(A.21)

Proof of Proposition 3.3.5

Proposition A.0.1. Consider two scattering beams (ˆy(t), ˆp(t), ˆM (t), ˆA(t)) and

(ˆx(t), ˆξ(t), ˆN (t), ˆC(t)), and assume that there exists significant interaction effects be-
tween these two beams. There exists two constants C∗

2 related to the background

1 and C∗

velocity, such that

d|| ˆM (t) − ˆN (t)||

dt

≤ C∗

1

(cid:112)||p|| + C∗

2|| ˆM (t) − ˆN (t)||.

(A.22)

where || ˆM (t) − ˆN (t)|| is defined as the matrix norm induced by the vector 2-norm.

166

Proof. Both ˆM (t) and ˆN (t) satisfy

d ˆM
dt
d ˆN
dt

= −Gxx(ˆy(t), ˆp(t)) − ˆM Gxp(ˆy(t), ˆp(t)) − GT
= −Gxx(ˆx(t), ˆξ(t)) − ˆN Gxp(ˆx(t), ˆξ(t)) − GT

xp(ˆy(t), ˆp(t)) ˆM − ˆM Gpp(ˆy(t), ˆp(t)) ˆM ,
xp(ˆx(t), ˆξ(t)) ˆN − ˆN Gpp(ˆx(t), ˆξ(t)) ˆN .

(A.23)

Take the difference between above equations, we have

− d( ˆM (t) − ˆN (t))
(cid:16)
Gxx(ˆy(t), ˆp(t)) − Gxx(ˆx(t), ˆξ(t))

dt

=

(cid:16)

(cid:17)

(cid:16)
(cid:17) ˆN (t)

+ 2 ˆN (t)

Gxp(ˆy(t), ˆp(t)) − Gxp(ˆx(t), ˆξ(t))

(cid:17)

Gpp(ˆy(t), ˆp(t)) − Gpp(ˆx(t), ˆξ(t))

+ ˆN (t)
+ 2( ˆM (t) − ˆN (t))Gxp(ˆy(t), ˆp(t)) + ( ˆM (t) − ˆN (t))Gpp(ˆy(t), ˆp(t))( ˆM (t) + ˆN (t)).

(A.24)

The first term in equation (A.24),

||Gxx(ˆy(t), ˆp(t))−Gxx(ˆx(t), ˆξ(t))|| =

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)∇∇v(ˆy(t))||ˆp(t)|| − ∇∇v(ˆx(t))|| ˆξ(t)||(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

≤ C3(||ˆy(t) − ˆx(t)||)||ˆp(t)|| + C2(||ˆp(t) − ˆξ(t)||),

(A.25)

where C3 is the maximum value of the third order derivative of the velocity v and C2

maximum value of the second order derivative of the velocity v. According to Lemma

3.3.2, we have

||Gxx(ˆy(t), ˆp(t)) − Gxx(ˆx(t), ˆξ(t))|| ∼ O((cid:112)||ˆp(t)||).

(A.26)

167

The second term in equation (A.24),

2

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆN (t)
(cid:16)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆN (t)

= 2

≤ 2|| ˆN (t)||

Gxp(ˆy(t), ˆp(t)) − Gxp(ˆx(t), ˆξ(t))

(cid:18) ˆp(t)

∇v(ˆy(t))
(cid:32)
C2||ˆy(t) − ˆx(t)|| + C1

||ˆp(t)||

(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:32) ˆξ(t)
(cid:19)T − ∇v(ˆx(t))
(cid:33)(cid:33)
(cid:32)||ˆΞ(t)||

|| ˆξ(t)||

||ˆp(t)||

,

(cid:33)T(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(A.27)

(A.28)

(cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆN (t) ˆN (t)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:33)T ˆN (t)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12).

(A.29)

where C1 is the largest value of ||∇v||. Therefore,

(cid:16)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆN (t)

2

Gxp(ˆy(t), ˆp(t)) − Gxp(ˆx(t), ˆξ(t))

(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∼ O((cid:112)||ˆp(t)||).

The third term in equation (A.24),

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆN (t)
(cid:16)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆN (t)

+

Gpp(ˆy(t), ˆp(t)) − Gpp(ˆx(t), ˆξ(t))

v(ˆy(t))

||ˆp(t)||

(cid:18) ˆp(t)

||ˆp(t)||

(cid:19)(cid:18) ˆp(t)

||ˆp(t)||

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:32)
(cid:17) ˆN (t)
||ˆp(t)|| − v(ˆx(t))
(cid:33)(cid:32) ˆξ(t)
(cid:32) ˆξ(t)
(cid:19)T − v(ˆx(t))
|| ˆξ(t)||

v(ˆy(t))

|| ˆξ(t)||

|| ˆξ(t)||

|| ˆξ(t)||

First, we have

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)v(ˆy(t))
(cid:32)
||ˆp(t)|| − v(ˆx(t))
|| ˆξ(t)||
||ˆy(t) − ˆx(t)||

C1

||ˆp(t)||

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆN (t) ˆN (t)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤

+ v(ˆy(t))

||ˆp(t) − ˆξ(t)||

||ˆp(t)||2

(cid:33)

|| ˆN (t) ˆN (t)|| ∼ O((cid:112)||ˆp(t)||).

(A.30)

168

Second,

(cid:18) ˆp(t)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v(ˆy(t))
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)||ˆy(t) − ˆx(t)||

||ˆp(t)||

||ˆp(t)||

||ˆp(t)||

C1

(cid:19)(cid:18) ˆp(t)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + C1

(cid:32) ˆξ(t)
(cid:19)T − v(ˆx(t))
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v(ˆy(t))(1 − ˆκ(t))||ˆp(t)||2

||ˆp(t)||

|| ˆξ(t)||

|| ˆξ(t)||

||ˆp(t)||3

(cid:33)(cid:32) ˆξ(t)
(cid:33)T(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ O(||ˆp(t)||−3/2).

|| ˆξ(t)||

To summarize,

(cid:16)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆN (t)

Gpp(ˆy(t), ˆp(t)) − Gpp(ˆx(t), ˆξ(t))

(cid:17) ˆN (t)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∼ O((cid:112)||ˆp(t)||).

(A.31)

(A.32)

Insert these asymptotic analysis equations (A.26), (A.28), (A.32) into equation (A.24),

we have

= O((cid:112)||ˆp(t)||) + 2( ˆM (t) − ˆN (t))Gxp(ˆy(t), ˆp(t))

dt

− d( ˆM (t) − ˆN (t))
+ ( ˆM (t) − ˆN (t))Gpp(ˆy(t), ˆp(t))( ˆM (t) + ˆN (t))
= O((cid:112)||ˆp(t)||) + ( ˆM (t) − ˆN (t))

(cid:16)

(cid:17)

2Gxp(ˆy(t), ˆp(t)) + Gpp(ˆy(t), ˆp(t))( ˆM (t) + ˆN (t))

.

The coefficient matrix in front of ˆM (t) − ˆN (t) satisfies,

||2Gxp(ˆy(t), ˆp(t)) + Gpp(ˆy(t), ˆp(t))( ˆM (t) + ˆN (t))|| ≤ 2||∇v(ˆy(t))|| + 2

≤ O(1).

(A.33)

|| ˆM (t) + ˆN (t)||

||ˆp(t)||

(A.34)

Then, ˆM (t) − ˆN (t) satisfies,

d|| ˆM (t) − ˆN (t)||

dt

≤ C∗

1

(cid:112)||p|| + C∗

2|| ˆM (t) − ˆN (t)||,

|| ˆM (0) − ˆN (0)|| = 0.

(A.35)

169

Proof of Lemma 3.3.5

Lemma A.0.2. Consider two scattering beams (ˆy(t), ˆp(t), ˆM (t), ˆA(t)) and (ˆx(t), ˆξ(t), ˆN (t), ˆC(t)),

and there exists significant interaction effects between these two beams. Suppose the

function g(t) is

then we have

and

g(t) = ˆp(t) · (ˆy(t) − ˆx(t)),

(A.36)

g(t) = g(0) + O(1),

(A.37)

(cid:32)

(cid:33)

.

||ˆΞ(t)||2
ˆκ(t)2||ˆp(t)||

1
2

(cid:48)

g

(t) = v(ˆx(t))

(A.38)

Proof. We start with calculating the derivative dg(t)
dt
be positive, i.e. G(x, p) = v(x)||p||. Negative branch will be the same.

and assume the Hamiltonian to

(cid:18) dˆy(t)

dt

(cid:19)

− dˆx(t)
dt

(cid:32)

v(ˆy(t))

ˆp(t)

||ˆp(t)|| − v(ˆx(t))

ˆξ(t)
|| ˆξ(t)||

(A.39)

(cid:33)

.

dg
dt

=

dˆp(t)

dt

· (ˆy(t) − ˆx(t)) + ˆp(t) ·

= −∇v(ˆy(t)) · (ˆy(t) − ˆx(t))||ˆp(t)|| + ˆp(t) ·

170

Use the Taylor expansion of the velocity v at the point ˆy(t),

= ||ˆp(t)|| (v(ˆx(t)) − v(ˆy(t))) + ||ˆp(t)||v(ˆy(t)) − v(ˆx(t))

dg
dt

ˆp(t) · ˆξ(t)
|| ˆξ(t)|| + O(||ˆx(t) − ˆy(t)||2)

(cid:32)

= ||ˆp(t)||v(ˆx(t)) − v(ˆx(t))

= v(ˆx(t))

||ˆp(t)|| − ˆp(t) · ˆξ(t)
|| ˆξ(t)||

ˆp(t) · ˆξ(t)
(cid:33)
|| ˆξ(t)|| + O(||ˆx(t) − ˆy(t)||2)
+ O(||ˆx(t) − ˆy(t)||2).

Substitute decomposition (3.56) into the fraction term ˆp(t)· ˆξ(t)

|| ˆξ(t)|| . Since ˆΞ(t) is orthogonal

to ˆp(t),

ˆp(t) · ˆξ(t)
|| ˆξ(t)|| =

(cid:113)

ˆκ(t)||ˆp(t)||2

ˆκ(t)2||ˆp(t)||2 + ||ˆΞ(t)||2

ˆκ(t)||ˆp(t)||2
ˆκ(t)||ˆp(t)||

=

= ||ˆp(t)|| − 1
2

1
||ˆΞ(t)||2
||ˆκ(t)ˆp(t)||2

1 + 1
2
||ˆΞ(t)||2

|ˆκ(t)|2||ˆp(t)||.

Here we use the Taylor expansion of the square root function and Geometric series to

approximate. Then

dg
dt

= v(ˆx(t))

= O(1).

(cid:18)
||ˆp(t)|| − ˆκ(t)||ˆp(t)||2
ˆκ(t)||ˆp(t)||

(cid:19)

+

v(ˆx(t))

2

||ˆΞ(t)||2

|ˆκ(t)|2||ˆp(t)||

(A.40)

Naturally, after finite time,

g(t) = g(0) + O(1).

(A.41)

171

Proof of Lemma 3.3.6

Lemma A.0.3. Consider two scattering beams (ˆy(t), ˆp(t), ˆM (t), ˆA(t)) and (ˆx(t), ˆξ(t), ˆN (t), ˆC(t)),

and there exists significant interaction effects between these two beams. Suppose the

pure imaginary matrix ˆM (0) has a symmetric positive definite imaginary part and is

the initial condition of the Hessian for the beam, then

(y − x)T ˆM (0)(y − x) = (ˆy(t) − ˆx(t))T ˆM (t)(ˆy(t) − ˆx(t)) + O(1).

(A.42)

Proof. We denote the function g(t) as

g(t) = ||ˆy(t) − ˆx(t)||2

ˆM (t)

Throughout this proof, we use the Hamiltonian G(x, p) = v(x)||p|| and the negative

Hamiltonian will be the same. The derivative of g is,

dg(t)

= 2(Gp(ˆy(t), ˆp(t)) − Gp(ˆx(t), ˆξ(t)))T ˆM (t)(ˆy(t) − ˆx(t))

Gpx(ˆy(t), ˆp(t))(ˆy(t) − ˆx(t)) + Gpp(ˆy(t), ˆp(t))(ˆp(t) − ˆξ(t))

dt
+ (ˆy(t) − ˆx(t))T d ˆM (t)
(cid:16)
dt
= 2
+ (ˆy(t) − ˆx(t))T d ˆM (t)
dt

(ˆy(t) − ˆx(t))

(ˆy(t) − ˆx(t)).

(cid:17)T

ˆM (t)(ˆy(t) − ˆx(t))

(A.43)

From Riccati equation (2.11), we have

dM
dt

= −Gxx − M Gxp − GT

xpM − M GppM.

172

Insert the above equation into equation (A.43) and abbreviate G = G(ˆy(t), ˆp(t)), if

there’s no variable specified,

(cid:16) ˆM (t)(ˆy(t) − ˆx(t))
(cid:17)

Gpp

dg(t)

= −(ˆy(t) − ˆx(t))T Gxx(ˆy(t) − ˆx(t)) − 2(ˆy(t) − ˆx(t))T ˆM (t)Gxp(ˆy(t) − ˆx(t))

dt
+ 2(ˆy(t) − ˆx(t))T ˆM (t)Gpx(ˆy(t) − ˆx(t))

+ 2(ˆp(t) − ˆξ(t))T Gpp ˆM (t)(ˆy(t) − ˆx(t)) −(cid:16) ˆM (t)(ˆy(t) − ˆx(t))
(cid:17)T
−(cid:16) ˆM (t)(ˆy(t) − ˆx(t))
(cid:17)T

(cid:16) ˆM (t)(ˆy(t) − ˆx(t))
(cid:17)

= −(ˆy(t) − ˆx(t))T Gxx(ˆy(t) − ˆx(t)) + 2(ˆp(t) − ˆξ(t))T Gpp ˆM (t)(ˆy(t) − ˆx(t))

Gpp

= −(ˆy(t) − ˆx(t))T Gxx(ˆy(t) − ˆx(t)) − || ˆM (t)(ˆy(t) − ˆx(t)) − (ˆp(t) − ˆξ(t))||2
+ ||ˆp(t) − ˆξ(t)||2

Gpp

Gpp.

(A.44)

The first term in equation (A.44) satisfies,

(ˆy(t) − ˆx(t))T (Gxx(ˆy(t), ˆp(t)))(ˆy(t) − ˆx(t)) ∼ O(1).

(A.45)

First, Gxx(ˆy(t), ˆp(t)) ∼ O(||ˆp(t)||), since v is smooth. On the other hand, ˆy(t) − ˆx(t)’s

order is O(1/(cid:112)||ˆp(t)||) by Lemma 3.3.2.

The second term in equation (A.44) satisfies,

|| ˆM (t)(ˆy(t) − ˆx(t)) − (ˆp(t) − ˆξ(t))||2

Gpp(ˆy(t),ˆp(t)) ∼ O(||ˆp(t)||1/2||ˆp(t)||−1||ˆp(t)||1/2).

(A.46)
The term Gpp(ˆy(t), ˆp(t)) is of the order O(1/||ˆp(t)||) implied by Gpp’s expression.

Again, Lemma 3.3.2 demonstrates that ˆM (t)(ˆy(t)− ˆx(t))−(ˆp(t)− ˆξ(t)) ∼ O((cid:112)||ˆp(t)||).

173

The third term in equation (A.44) satisfies,

||ˆp(t) − ˆξ(t)||2

Gpp(ˆy(t),ˆp(t)) ∼ O(||ˆp(t)||1/2||ˆp(t)||−1||ˆp(t)||1/2) ∼ O(1).

(A.47)

It can be justified by combining Gpp’s expression and the fact that ||ˆp(t) − ˆξ(t)|| ∼

O((cid:112)||ˆp(t)||).

Proof of Proposition 3.3.3

Proposition A.0.2.

iˆγ(ˆt0(x, ξ); x, ξ)|ω − ˆτt(ˆt0(x, ξ); x, ξ) − ˆϑ(ˆt0(x, ξ); x, ξ)T (r − ˆx(ˆt0(x, ξ)))|2

≈ iˆγ(ˆt0(y, p); x, ξ)|ω − ˆτt(ˆt0(y, p); x, ξ) −(cid:16) ˆϑ(ˆt0(y, p); x, ξ))
(cid:17)T
(cid:18) 1

(cid:19)

(r − ˆx(ˆt0(y, p)))|2

(A.48)

+ O

||p||

.

Proof. The coefficient ˆγ satisfies,

ˆγ(ˆt0(x, ξ); x, ξ) = −

1

2ˆτtt(ˆt0(x, ξ); x, ξ)

.

By Assumption 3.2.3, it is safe to say that the background velocity v around ˆx(ˆt0(y, p))

is a constant function. Consequently, we have Hamiltonian satisfying Gx = 0, and

ˆGp(ˆx(ˆt0(x, ξ); x, ξ), ˆξ(ˆt0(x, ξ); x, ξ)) = ±v(ˆx(ˆt0(y, p)))

ˆξ(ˆt0(y, p)))
|| ˆξ(ˆt0(y, p))||

= ˆGp(ˆx(ˆt0(y, p); x, ξ), ˆξ(ˆt0(y, p); x, ξ)),

(A.49)

174

since ray direction does not change in the constant slowness. Then

ˆτtt(ˆt0(x, ξ); x, ξ) = ˆGT

(cid:33)
p (ˆx(ˆt0(x, ξ)), ˆξ(ˆt0(x, ξ))) ˆM (ˆt0(x, ξ); x, ξ) ˆGp(ˆx(ˆt0(x, ξ)), ˆξ(ˆt0(x, ξ)))
(ˆt0(x, ξ) − ˆt0(y, p))

p (ˆx(ˆt0(y, p)), ˆξ(ˆt0(y, p)))

ˆM (ˆt0(y, p); x, ξ) +

(cid:32)

= ˆGT

d ˆM
dt

ˆGp(ˆx(ˆt0(y, p)), ˆξ(ˆt0(y, p)))

= ˆτtt(ˆt0(y, p); x, ξ) + O((cid:112)||p||),

(A.50)

dt ∼ O(||p||) by Lemma 3.3.6. On the other hand, |ˆt0(y, p)− ˆt0(x, ξ)| ∼

since we have d ˆM
O(||p||−1/2).

We then have

ˆγ(ˆt0(x, ξ); x, ξ) =

=

≈

1

2(ˆτtt(ˆt0(y, p); x, ξ) + O((cid:112)||p||))
1 + ((cid:112)||p||)−1

2(ˆτtt(ˆt0(y, p); x, ξ)

(cid:32)

1

1

(cid:33)

1

2(ˆτtt(ˆt0(y, p); x, ξ)

= ˆγ(ˆt0(y, p); x, ξ).

(A.51)

Inside the quadratic term, we first have an invariant,

ω − ˆτt(ˆt0(x, ξ); x, ξ) = ω − ˆτt(ˆt0(y, p); x, ξ),

(A.52)

since ˆτt will be a constant along the ray.

175

The next term is ˆϑ(ˆt0(x, ξ); x, ξ).

Re(ˆτtx(ˆt0(x, ξ); x, ξ)) = −Re( ˆM )(ˆt0(x, ξ); x, ξ)Gp(ˆx(ˆt0(x, ξ); x, ξ), ˆξ(ˆt0(x, ξ); x, ξ))
= −Re( ˆM )(ˆt0(y, p); x, ξ)Gp(ˆx(ˆt0(y, p); x, ξ), ˆξ(ˆt0(y, p); x, ξ))

+

dt

(A.53)

(A.54)

(A.55)

(A.56)

(A.57)

dRe( ˆM )(ˆt0(y, p); x, ξ)

= Re(ˆτtx(ˆt0(y, p); x, ξ)) + O((cid:112)||p||).

(ˆt0(x, ξ) − ˆt0(y, p))Gp(ˆx(ˆt0(y, p); x, ξ), ˆξ(ˆt0(y, p); x, ξ))

We have the similar conclusion for imaginary part,

Im(ˆτtx(ˆt0(x, ξ); x, ξ)) = Im(ˆτtx(ˆt0(y, p); x, ξ)) + O((cid:112)||p||).

Consequently, by equation (3.75)

ˆϑ(ˆt0(x, ξ); x, ξ)) = ˆϑ(ˆt0(y, p); x, ξ)) + O((cid:112)||p||).

Next,

r − ˆx(ˆt0(x, ξ)) = r − ˆx(ˆt0(y, p)) + ˆx(ˆt0(y, p)) − ˆx(ˆt0(x, ξ))
= r − ˆx(ˆt0(y, p)) + G±
= r − ˆx(ˆt0(y, p)) ± v(ˆx(ˆt0(y, p)))(ˆt0(y, p) − ˆt0(x, ξ))

p (ˆx(ˆt0(y, p)), ˆξ(ˆt0(y, p)))(ˆt0(y, p) − ˆt0(x, ξ))
ˆξ(ˆt0(y, p))
|| ˆξ(ˆt0(y, p))||

= r − ˆx(ˆt0(y, p)) + O(1/(cid:112)||p||).

176

G± represents the different Hamiltonian by its sign. Therefore,

(cid:33)
iˆγ(ˆt0(x, ξ); x, ξ)|ω − ˆτt(ˆt0(x, ξ); x, ξ) − ˆϑ(ˆt0(x, ξ); x, ξ)T (r − ˆx(ˆt0(x, ξ)))|2 = iˆγ(ˆt0(y, p); x, ξ)

|ω − ˆτt(ˆt0(y, p); x, ξ) −(cid:16) ˆϑ(ˆt0(y, p); x, ξ) + O((cid:112)||p||)
(cid:17)T
(r − ˆx(ˆt0(x, ξ))|2 + O
≈ iˆγ(ˆt0(y, p); x, ξ)|ω − ˆτt(ˆt0(y, p); x, ξ) −(cid:16) ˆϑ(ˆt0(y, p); x, ξ))
(cid:17)T

(r − ˆx(ˆt0(y, p)))|2.

1(cid:112)||p||

(cid:32)

(A.58)

Proof of Proposition 3.3.6

Proposition A.0.3. Some real-valued phase terms, ˆ
(r, ˆtc; y, p) − ˆθ(r, ˆtc; y, p) and
ˆ
(r, ˆtc; x, ξ) − ˆθ(r, ˆtc; x, ξ), can be ignored since they are constant order terms with
respect to the large wavenumber ||ξl,i|| = ||p + q||.

Proof. The first term: ( ˆF − ω) ˆQ

(ˆτt(ˆtc; y, p) − ω + Re(ˆτtx)(ˆtc; y, p))

(cid:32)

(cid:33)

Im(ˆτtx(ˆtc; y, p))T (r − ˆy(ˆtc))
(cid:32)

Im(ˆτtt(ˆtc; y, p))

Im(ˆτtx(ˆtc; t, ξ))T (r − ˆx(ˆtc))

(cid:33)

− (ˆτt(ˆtc; x, ξ) − ω + Re(ˆτtx)(ˆtc; x, ξ))

Im(ˆτtt(ˆtc; x, ξ))

≈ (ˆτt(ˆtc; x, ξ) − ω)(ˆy(ˆtc) − ˆx(ˆtc)) + (ˆτt(ˆtc; y, p) − ˆτt(ˆtc; x, ξ))(r − ˆy(ˆtc)) + O(1).

By Corollary 3.3.1, we notice the scale of |ˆτt(ˆtc; x, ξ) is controlled by Im(ˆγ) ∼ O( 1||p||).

177

Therefore, by Lemma 3.3.2,

|ˆτt(ˆtc; y, p) − ˆτt(ˆtc; x, ξ)| =

(cid:12)(cid:12)(cid:12)ˆτt(ˆtc; x, ξ) − ω

(cid:12)(cid:12)(cid:12)||ˆy(ˆtc) − ˆx(ˆtc)|| ∼ O(1).
(cid:12)(cid:12)(cid:12)v(ˆy(ˆtc))||ˆp(ˆtc)|| − v(ˆx(ˆtc))|| ˆξ(ˆtc)||(cid:12)(cid:12)(cid:12)

≤ |∇v(ˆy(ˆy(ˆtc))|||(ˆy(ˆtc) − ˆx(ˆtc)||||ˆp(ˆy(ˆtc))||
+ v(ˆy(ˆtc))(||ˆp(ˆtc)|| − || ˆξ(ˆtc)||).

By Lemma 3.3.2, |ˆτt(ˆtc; y, p) − ˆτt(ˆtc; x, ξ)| ∼ O((cid:112)||p||), and ||r − ˆy(ˆtc)|| is controlled

by Hessian ˆM (ˆtc). So (ˆτt(ˆtc; y, p) − ˆτt(ˆtc; x, ξ))(r − ˆy(ˆtc)) ∼ O(1).
The second term: 1

2 Re(ˆτtt) ˆQ2

1
2

Re(ˆτtt) ˆQ2 ≈ 1
2

Re(ˆτtt)(r − ˆy(ˆtc))2 ∼ O(1),

(A.59)

by Lemma 3.3.2. And the same analysis can be applied to the term associated with

the beam (x, ξ).

Proof of Proposition 3.3.7

Proposition A.0.4. For the first two terms in equation (3.107), their exponents sat-

isfy,

− Im( ˆβ)|ω − ˆτt(ˆtc; y, p) − ˆζT (r − ˆy(ˆtc))|2 − Im(ˆγ)|ω − ˆτt(ˆtc; x, ξ) − ˆϑT (r − ˆx(ˆtc))|2
= −||r − ˆx(ˆtc) + ˆy(ˆtc)

2Im( ˆβ)ˆζ ˆζT − Im( ˆβ)
||2

||ˆy(ˆtc) − ˆx(ˆtc)||2
ˆζ ˆζT

2

2

(cid:32)

(cid:33)

1(cid:112)||p||

,

(A.60)

− Im( ˆβ)|ω − ˆτt(ˆtc; y, p)|2 − Im(ˆγ)|ω − ˆτt(ˆtc; x, ξ)|2 + O

178

where ˆβ, ˆγ, ˆζ and ˆϑ are all defined at ˆtc. Similarly, ˆg in equation (3.108) is an O(1)

term.

Proof. The quadratic term about r in equation (A.60) is,

−rT(cid:16)

Im( ˆβ)ˆζ ˆζT + Im(ˆγ) ˆϑ( ˆϑ)T(cid:17)

r.

(A.61)

The matrix in the parentheses is positive semi-definite matrix as it is the sum of two

positive semi-definite matrices. We apply the eigenvalue decomposition

Im( ˆβ)ˆζ ˆζT + Im(ˆγ) ˆϑ( ˆϑ)T = QT ΛQ,

(A.62)

where Λ is a diagonal matrix with non-negative entries and QT Q = I.

The cross term about r in equation (A.60) is

2Im( ˆβ)(ω − ˆτt(ˆtc; y, p))ˆζT r + 2Im( ˆβ)ˆy(ˆtc)T (ˆζ ˆζT )r
+2Im(ˆγ)(ω − ˆτt(ˆtc; x, ξ)) ˆϑT r + 2Im(ˆγ)ˆx(ˆtc)T ( ˆϑ ˆϑT )r

= 2 ˆJ T

1 (ˆtc, ω; y, p, x, ξ)r + 2 ˆJ T

2 (ˆtc, ω; y, p, x, ξ)r,

(A.63)

where

ˆJ1(ˆtc, ω; y, p, x, ξ) = Im( ˆβ)(ω − ˆτt(ˆtc; y, p))ˆζ + Im(ˆγ)(ω − ˆτt(ˆtc; x, ξ)) ˆϑ;

(A.64)

ˆJ2(ˆtc, ω; y, p, x, ξ) = Im( ˆβ)ˆζ ˆζT ˆy(ˆtc) + Im(ˆγ) ˆϑ ˆϑT ˆx(ˆtc).

(A.65)

179

Combine the cross term (A.63) and second order term (A.61),

−(Qr)T Λ(Qr) + 2 ˆJ T

1 QT Qr + 2 ˆJ T

2 QT Qr.

To make a complete quadratic term, its central point rc is,

(cid:32)
rc = QT Λ−1Q( ˆJ1 + ˆJ2)
= QT Λ−1Q ˆJ2 + O

(cid:33)

,

1(cid:112)||p||

(A.66)

(A.67)

since

and

|| ˆJ1|| ≤ ||Im( ˆβ)(ω − ˆτt(ˆtc; y, p))ˆζ|| + ||Im(ˆγ)(ω − ˆτt(ˆtc; x, ξ)) ˆϑ||

(cid:18) 1

(cid:19)

||p||

O

(cid:16)(cid:112)||p||(cid:17)

≤ O

(cid:16)(cid:112)||p||(cid:17)

,

O (||p||) ≤ O

|| ˆJ2|| ≤ ||Im( ˆβ)ˆζ ˆζT ˆy(ˆtc)|| + ||Im(ˆγ) ˆϑ ˆϑT ˆx(ˆtc)||
O(1) ≤ O(||p||).

(cid:16)||p||2(cid:17)

≤ O

(cid:18) 1

(cid:19)

O

||p||

Here the inverse matrix is defined as the pseudo-inverse matrix for a rank-deficient
matrix, that is Λ−1
(cid:18)
we set its center rc’s ith coordinate to be the same as ˆy(ˆtc)+ˆx(ˆtc)

If the diagonal term in Λ−1

ii = 0, if Λii = 0.

’s, since we have

is zero, then

(cid:19)

ii

2

in the expression of ˆB. Therefore, we carry out

exp

− 1
4||2r − ˆx(ˆtc) − ˆy(ˆtc)||2

ˆM(ˆtc)

180

calculation by assuming Λ invertible.

QT Λ−1Q = (Im( ˆβ)ˆζ ˆζT + Im(ˆγ) ˆϑ( ˆϑ)T )−1

= (2Im( ˆβ)ˆζ ˆζT + Im(ˆγ) ˆϑ( ˆϑ)T − Im( ˆβ)ˆζ ˆζT )−1.

To evaluate the order of Im(ˆγ) ˆϑ( ˆϑ)T − Im( ˆβ)ˆζ ˆζT , we first have

| ˆβ − ˆγ| =

1
2

1

(cid:12)(cid:12)(cid:12)
≤(cid:12)(cid:12)(cid:12) ˆτtt(ˆtc; y, p) − ˆτtt(ˆtc; x, ξ)

ˆτtt(ˆtc; y, p)ˆτtt(ˆtc; x, ξ)

−

ˆτtt(ˆtc; y, p)

ˆτtt(ˆtc; x, ξ)

1

(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12).

By Corollary 3.3.3, we have |ˆτtt(ˆtc; y, p) − ˆτtt(ˆtc; x, ξ)| ∼ O((cid:112)||p||). Consequently,

(A.68)

| ˆβ − ˆγ| ≤ O(||p||− 3
2 ).

On the other hand, by using Corollary 3.3.3,

||ˆζ − ˆϑ|| ≤ O((cid:112)||p||).

(A.69)

(A.70)

Then

||Im(ˆγ) ˆϑ( ˆϑ)T−Im( ˆβ)ˆζ ˆζT|| ≤ ||Im( ˆβ)

≤ O((cid:112)||p||).

(cid:16)ˆζ ˆζT − ˆϑ ˆϑT(cid:17)|| + ||Im( ˆβ − ˆγ) ˆϑ ˆϑT||

(A.71)

181

Therefore,

QT Λ−1Q = (2Im( ˆβ)ˆζ ˆζT + O((cid:112)||p||))−1 ≈ (2Im( ˆβ)ˆζ ˆζT )−1.

(A.72)

Similarly,

ˆJ2 = Im( ˆβ)ˆζ ˆζT (ˆy(ˆtc) + ˆx(ˆtc)) − Im( ˆβ)ˆζ ˆζT ˆx(ˆtc) + Im(ˆγ) ˆϑ( ˆϑ)T ˆx(ˆtc)

= Im( ˆβ)ˆζ ˆζT (ˆy(ˆtc) + ˆx(ˆtc)) + O((cid:112)||p||).

≈ Im( ˆβ)ˆζ ˆζT (ˆy(ˆtc) + ˆx(ˆtc)).

(A.73)

The central point rc becomes,

rc ≈ QT Λ−1Q ˆJ2 ≈ (2Im( ˆβ)ˆζ ˆζT )−1Im( ˆβ)ˆζ ˆζT (ˆy(ˆtc) + ˆx(ˆtc)) =

ˆx(ˆtc) + ˆy(ˆtc)

2

. (A.74)

Equation (A.60) now becomes,

− Im( ˆβ)|ω − ˆτt(ˆtc; y, p) − ˆζT (r − ˆy(ˆtc))|2 − Im(ˆγ)|ω − ˆτt(ˆtc; x, ξ) − ( ˆϑ)T (r − ˆx(ˆtc))|2
= −Im( ˆβ)|ω − ˆτt(ˆtc; y, p)|2 − Im(ˆγ)|ω − ˆτt(ˆtc; x, ξ)|2 − ||r − ˆx(ˆtc) + ˆy(ˆtc)

||2
2Im( ˆβ)ˆζ ˆζT

2

+ ˆJ T

2 (2Im( ˆβ)ˆζ ˆζT )−1 ˆJ2 − Im( ˆβ)|ˆζT ˆy(ˆtc)|2 − Im(ˆγ)|( ˆϑ)T ˆx(ˆtc)|2.

182

Using equation (A.71),

2 (2Im( ˆβ)ˆζ ˆζT )−1 ˆJ2 − Im( ˆβ)|ˆζT ˆy(ˆtc)|2 − Im(ˆγ)|( ˆϑ)T ˆx(ˆtc)|2
ˆJ T
≈ ||ˆy(ˆtc) + ˆx(ˆtc)||2

Im( ˆβ)ˆζ ˆζT − ||ˆx(ˆtc)||2

− ||ˆy(ˆtc)||2

Im( ˆβ)ˆζ ˆζT

Im( ˆβ)ˆζ ˆζT

2

= −||ˆy(ˆtc) − ˆx(ˆtc)||2

Im( ˆβ)ˆζ ˆζT

.

2

Equation (A.60) now becomes,

(A.75)

− Im( ˆβ)|ω − ˆτt(ˆtc; y, p) − ˆζT (r − ˆy(ˆtc))|2 − Im(ˆγ)|ω − ˆτt(ˆtc; x, ξ) − ( ˆϑ)T (r − ˆx(ˆtc))|2
≈ −Im( ˆβ)|ω − ˆτt(ˆtc; y, p)|2 − Im(ˆγ)|ω − ˆτt(ˆtc; x, ξ)|2
− ||r − ˆx(ˆtc) + ˆy(ˆtc)

||ˆy(ˆtc) − ˆx(ˆtc)||2

2Im( ˆβ)ˆζ ˆζT − Im( ˆβ)
||2

ˆζ ˆζT .

2

2

By applying the similar computation, the exponent of the last term in equation (3.107)

denoted as ˆg contains two O(1) terms.

Proof of Proposition 3.3.8

Proposition A.0.5. The function ˆφ1 satisfies

ˆφ1(t; x, ξ, y, p) =

||ˆp(t) − ˆξ(t)||2

Gpp + O

1
2

(cid:32)

(cid:33)

.

1(cid:112)||p||

(A.76)

Proof. Using Lemma 3.3.5 for ˆφ1(t),

ˆφ1(t; x, ξ, y, p) = v(ˆx(t))

183

(cid:32)

(cid:33)

.

||ˆΞ(t)||2
ˆκ(t)2||ˆp(t)||

1
2

We then would like to explore the term ||ˆp(t) − ˆξ(t)||2

Gpp(ˆy(t),ˆp(t)) by taking advantage

of Gpp’s expression.

||ˆp(t) − ˆξ(t)||2

Gpp(ˆy(t),ˆp(t)) = (ˆp(t) − ˆξ(t))T(cid:0)Gpp(ˆy(t), ˆp(t))(cid:1) (ˆp(t) − ˆξ(t))
||ˆp(t)||3 (ˆp(t) − ˆξ(t))T(cid:16)||ˆp(t)||2I − ˆp(t)ˆp(t)T(cid:17)
(cid:17)
||ˆp(t)|| (−ˆΞ(t))T(cid:16)

||ˆp(t)|| (− ˆξ(t) + ˆκ(t)ˆp(t))T (ˆp(t) − ˆξ(t))
(1 − ˆκ(t))ˆp(t) − ˆΞ(t)

v(ˆy(t))

v(ˆy(t))

v(ˆy(t))

=

=

=

= v(ˆy(t))

||ˆΞ(t)||2
||ˆp(t)|| .

Then,

1
2

||ˆp(t) − ˆξ(t)||2

ˆGpp

=

v(ˆy(t))

2

||ˆΞ(t)||2
||ˆp(t)|| .

(ˆp(t) − ˆξ(t))

(A.77)

(A.78)

Compare ˆφ1(t) with equation (A.77),

(cid:32)

v(ˆx(t))

(cid:33)

||ˆΞ(t)||2
ˆκ(t)2||ˆp(t)||

1
2
= O(||ˆy(t) − ˆx(t)||).

− 1
2

v(ˆy(t))

||ˆΞ(t)||2
||ˆp(t)|| ≈ − 1

2

(cid:18)

v(ˆy(t)) − v(ˆx(t))
ˆκ(t)2

(cid:19) ||ˆΞ(t)||2

||ˆp(t)||

(A.79)

This is because ˆκ(t) ∼ 1 + O(||ˆp(t)||− 1
ˆξ(t)|| ∼ O(||p||1/2) in Lemma 3.3.2.

2 ) and ˆΞ(t) ∼ O((cid:112)||ˆp(t)||) followed by ||ˆp(t) −

Then the derivative ˆφ1 becomes,

ˆφ1(t; x, ξ, y, p) ≈ 1
2

||ˆp(t) − ˆξ(t)||2

Gpp

184

Proof of Proposition 3.3.10

Proposition A.0.6. There exists a linear map ˆJ (ˆtc; y, p), such that

ˆy(ˆtc) − ˆx(ˆtc)

ˆp(ˆtc) − ˆξ(ˆtc)

 ≈ ˆJ (ˆtc; y, p)

y − x

p − ξ

 .

(A.80)

Proof. We consider (ˆy(t)−ˆx(t), ˆp(t)− ˆξ(t)) as a function about t and its initial condition
(y − x, p − ξ). We denote this initial condition as (∆x, ∆ξ) and ˆJ (t; y, p) as the

dynamical variational system,

ˆy(ˆtc) − ˆx(ˆtc; ∆x, ∆ξ)

ˆp(ˆtc) − ˆξ(ˆtc; ∆x, ∆ξ)

 ≈ ˆJ (ˆtc; y, p)

 .

∆x

∆ξ

It is well known that ˆJ (t; y, p) satisfies,

 Gxp(ˆy(t), ˆp(t))

d ˆJ (t)
dt

=

Gpp(ˆy(t), ˆp(t))

−Gxx(ˆy(t), ˆp(t)) −GT

xp(ˆy(t), ˆp(t))

 ˆJ (t),

and its initial condition is an identity matrix,

ˆJ (0) = I2d.

185

(A.81)

(A.82)

(A.83)

By solving ˆJ (t), we can get

ˆy(t) − ˆx(t; y − x, p − ξ)

ˆp(t) − ˆξ(t; y − x, p − ξ)

 = ˆJ (t; y, p)

y − x

p − ξ

 .

(A.84)

The transform ˆJ defined above is invertible due to the uniqueness of ODE system’s

solution.

Proof of Lemma 3.3.9

(cid:90)

Lemma A.0.4. The result after taking the integral about ω can be approximated,

−

ei(ω−ˆτt(ˆtc;y,p))∆ˆt0(x,ξ;y,p)ei(ω−˜τt(˜tc;y,q))∆˜t0(x,η;y,q)ω2 ˆB(x, ξ, ω; y, p) ˜B(x, η, ω; y, q)dω

= eiO(1)K(p, q, y) ˆB(x, ξ; y, p, q) ˜B(x, η; y, p, q)e
− i
2||˜y(˜tc)−˜x(˜tc)||2
e

Re( ˜M (˜tc)).

− i
2||ˆy(ˆtc)−ˆx(ˆtc)||2

Re( ˆM (ˆtc))

Proof. All terms containing ω in ˆB(x, ξ, ω; y, p) (3.110) are as the following,

(cid:16)−Im( ˆβ)|ω − ˆτt(ˆtc; y, p)|2 − Im(ˆγ)|ω − ˆτt(ˆtc; x, ξ)|2(cid:17) ≈
(cid:32)

(cid:32)

(cid:19)2

(cid:18)

ω − ˆτt(ˆtc; y, p) + ˆτt(ˆtc; x, ξ)

2

+ (ˆτt(ˆtc; y, p) − ˆτt(ˆtc; x, ξ))2

(cid:33)(cid:33)

exp

exp

≈ e

−Im( ˆβ)
(cid:18)

−2Im( ˆβ)

ω− ˆτt(ˆtc;y,p)+ˆτt(ˆtc;x,ξ)

2

e−Im( ˆβ)|ˆτt(ˆtc;y,p)−ˆτt(ˆtc;x,ξ)|2

.

(A.85)

2

(cid:19)2

186

Similarly, for ˜B,

(cid:16)−Im( ˜β)|ω − ˜τt(˜tc; y, q)|2 − Im(˜γ)|ω − ˜τt(˜tc; x, η)|2(cid:17)

exp

(cid:18)

−2Im( ˜β)

≈ e

ω− ˜τt(˜tc;y,q)+˜τt(˜tc;x,η)

2

e−Im( ˜β)|˜τt(˜tc;y,q)−˜τt(˜tc;x,η)|2

.

(A.86)

Combine the terms containing ω in equations (A.85) and (A.86) with −ω2, and denote

(cid:19)2

(cid:18)

(cid:90)

dωe

(cid:19)2

− ℵ(ˆtc, ˜tc; x, ξ, η, y, p, q) =

(cid:18)

−2Im( ˆβ)
e

ω− ˆτt(ˆtc;y,p)+ˆτt(ˆtc;x,ξ)

2

−2Im( ˜β)

ω− ˜τt(˜tc;y,q)+˜τt(˜tc;x,η)

2

(cid:19)2

ω2

ei(ω−ˆτt(ˆtc;y,p))∆ˆt0(x,ξ;y,p)ei(ω−˜τt(˜tc;y,q))∆˜t0(x,η;y,q).

(A.87)

Then the target integral becomes,

(cid:90)
−
= ℵe−Im( ˜β)|˜τt(˜tc;y,q)−˜τt(˜tc;x,η)|2

ei(ω−ˆτt(ˆt0;y,p))∆ˆt0(x,ξ;y,p)ei(ω−˜τt(˜tc;y,q))∆˜t0(x,η;y,q)ω2 ˆB(x, ξ, ω; y, p) ˜B(x, η, ω; y, q)dω

e−Im( ˆβ)|ˆτt(ˆtc;y,p)−ˆτt(ˆtc;x,ξ)|2

,

(A.88)

without considering the constant terms in ˆB and ˜B. If we can approximate ℵ by the

product of a constant K(p, q, y), functions on the receiver side and functions on the

source side, then the proposition is proved.

187

Compute the expression (A.87),

−ℵ(ˆtc, ˜tc; x, ξ, η, y, p, q) = e−iˆτt(ˆtc;y,p)∆ˆt0(x,ξ;y,p)−i˜τt(˜tc;y,p)∆˜t0(x,η;y,q)e

−2

Im( ˆβ)Im( ˜β)S2
2
Im( ˆβ)+Im( ˜β)

ω2eiω(∆ˆt0(x,ξ;y,p)+∆˜t0(x,η;y,q))e−2(Im( ˆβ)+Im( ˜β))(ω−S1)2

dω,

(cid:90)

where

S1(ˆtc, ˜tc; x, ξ, η, y, p, q) =

(cid:18)

Im( ˆβ)

ˆτt(ˆtc;y,p)+ˆτt(ˆtc;x,ξ)

2

(cid:19)

+ Im( ˜β)

Im( ˆβ) + Im( ˜β)

(A.89)

(cid:16) ˜τt(˜tc;y,q)+˜τt(˜tc;x,η)

2

(cid:17)

,

(A.90)

S2(ˆtc, ˜tc; x, ξ, η, y, p, q) =

ˆτt(ˆtc; y, p) + ˆτt(ˆtc; x, ξ)

2

− ˜τt(˜tc; y, q) + ˜τt(˜tc; x, η)

2

.

(A.91)

We then have,

ℵ = e−iˆτt(ˆtc;y,p)∆ˆt0(x,ξ;y,p)−i˜τt(˜tc;y,p)∆˜t0(x,η;y,q)e

−2

Im( ˆβ)Im( ˜β)S2
2
Im( ˆβ)+Im( ˜β)

eiS1te

d2



−

t2

2Im( ˆβ+ ˜β)

dt2

(cid:12)(cid:12)(cid:12)t=∆ˆt0(x,ξ;y,p)+∆˜t0(x,η;y,q)

.

(A.92)

Proposition A.0.7. Both S1 and S2 can be approximated as constants only related to

188

the fixed parameter (y, p) and (y, q), that is

S1(ˆtc, ˜tc; x, ξ, η, y, p, q) ≈ Im( ˆβ)(cid:0)ˆτt(ˆtc; y, p)(cid:1) + Im( ˜β)(cid:0)˜τt(˜tc; y, q)(cid:1)

,

Im( ˆβ) + Im( ˜β)

S2(ˆtc, ˜tc; x, ξ, η, y, p, q) ≈ ˆτt(ˆtc; y, p) − ˜τt(˜tc; y, q).

Proof. For S1,

S1(ˆtc, ˜tc; x, ξ, η, y, p, q) =

Im( ˆβ)(cid:0)ˆτt(ˆtc; y, p)(cid:1) + Im( ˜β)(cid:0)˜τt(˜tc; y, q)(cid:1)

Im( ˆβ) + Im( ˜β)

Im( ˆβ)(ˆτt(ˆtc; x, ξ) − ˆτt(ˆtc; y, p)) + Im( ˜β)(˜τt(˜tc; x, η) − ˜τt(˜tc; y, q))

+

2Im( ˆβ) + 2Im( ˜β)

Im( ˆβ)(cid:0)ˆτt(ˆtc; y, p)(cid:1) + Im( ˜β)(cid:0)˜τt(˜tc; y, q)(cid:1)
≈ Im( ˆβ)(cid:0)ˆτt(ˆtc; y, p)(cid:1) + Im( ˜β)(cid:0)˜τt(˜tc; y, q)(cid:1)

Im( ˆβ) + Im( ˜β)

=

Im( ˆβ) + Im( ˜β)

+ O((cid:112)||ˆpt||) + O((cid:112)||˜qt||)

,

(A.93)

since the term in the last step above is about O(||p|| + ||q||), and

2Im( ˆβ) + 2Im( ˜β)

Im( ˆβ)(ˆτt(ˆtc; x, ξ) − ˆτt(ˆtc; y, p)) + Im( ˜β)(˜τt(˜tc; x, η) − ˜τt(˜tc; y, q))

≤ |ˆτt(ˆtc; x, ξ) − ˆτt(ˆtc; y, p)| + |(˜τt(˜tc; x, η) − ˜τt(˜tc; y, q)| ≤ O((cid:112)||ˆpt|| + ||˜qt||).
S1(ˆtc, ˜tc; x, ξ, η, y, p, q) ≈ Im( ˆβ)(cid:0)ˆτt(ˆtc; y, p)(cid:1) + Im( ˜β)(cid:0)˜τt(˜tc; y, q)(cid:1)

.

Im( ˆβ) + Im( ˜β)

For S2,

S2(ˆtc, ˜tc; x, ξ, η, y, p, q) = ˆτt(ˆtc; y, p) − ˜τt(˜tc; y, q) + O((cid:112)||p||),

(A.94)

(A.95)

(A.96)

due to the fact that |ˆτt(ˆtc; y, p) − ˆτt(ˆtc; x, ξ)| = v(y)||p|| − v(x)||ξ|| and ||p − ξ|| ∼

189

O((cid:112)||p||). Then,

−2
e

Im( ˆβ)Im( ˜β)S2
2
Im( ˆβ)+Im( ˜β) ≈ e

−2

Im( ˆβ)Im( ˜β)(ˆτt(ˆtc;y,p)−˜τt(˜tc;y,q))2

Im( ˆβ)+Im( ˜β)

eO(1),

(A.97)

since Im( ˆβ) ∼ O( 1||p||) and Im( ˜β) ∼ O( 1||q||). We then approximate S2 by

S2(ˆtc, ˜tc; x, ξ, η, y, p, q) ≈ ˆτt(ˆtc; y, p) − ˜τt(˜tc; y, q).

We also obtain |ˆτt(ˆtc; y, p) − ˜τt(˜tc; y, q)| is around O(max((cid:112)||p||,(cid:112)||q||)).

(A.98)

Proposition A.0.8.

−|∆ˆt0(x,ξ;y,p)|2
4Im( ˆβ+ ˜β)

−|∆˜t0(x,η;y,q)|2
4Im( ˆβ+ ˜β)
e

−2
e

Im( ˆβ)Im( ˜β)(S2)2
Im( ˆβ)+Im( ˜β)

ℵ ≈ −S2
1e

(A.99)

Proof. The second order time derivative in equation (A.92),

(cid:32)

iS1 − ∆ˆt0(x, ξ; y, p) + ∆˜t0(x, η; y, q)

−

1

(cid:33)2



Im( ˆβ + ˜β)

Im( ˆβ + ˜β)
−|∆ˆt0(x,ξ;y,p)+∆˜t0(x,η;y,q)|2

2(Im( ˆβ+ ˜β))

.

eiS1(∆ˆt0(x,ξ;y,p)+∆˜t0(x,η;y,q))e

We now conduct some asymptotic analysis about the terms in the above equation,

S1 ∼ O(||p|| + ||q||),

1

Im( ˆβ + ˜β)

∼ O(||p|| + ||q||),

(A.100)

190

since ˆβ =

1

2ˆτtt(ˆtc;y,p)

and ˜β =

1

2˜τtt(˜tc;y,q)

∆ˆt0(x, ξ; y, p) + ∆˜t0(x, η; y, q)

Im( ˆβ + ˜β)

∼ O

(cid:32)

O(||p|| + ||q||).

(A.101)

. On the other hand,

(cid:33)

1(cid:112)||p|| +

1(cid:112)||q||

Moreover, we have

−|∆ˆt0(x, ξ; y, p) − ∆˜t0(x, η; y, q)|2

2(Im( ˆβ + ˜β))

− |∆ˆt0(x, ξ; y, p)|2
4Im( ˆβ + ˜β)

= −|∆˜t0(x, η; y, q)|2
4Im( ˆβ + ˜β)
|∆ˆt0(x, ξ; y, p) − ∆˜t0(x, η; y, q)|2
= −|∆˜t0(x, η; y, q)|2
4Im( ˆβ + ˜β)

4Im( ˆβ + ˜β)

− |∆ˆt0(x, ξ; y, p)|2
4Im( ˆβ + ˜β)

+

Consequently,

ℵ ≈ −S2

1ei(S1−ˆτt)(∆ˆt0(x,ξ;y,p))ei(S1−˜τt)(∆˜t0(x,η;y,q))e

−|∆ˆt0(x,ξ;y,p)|2
4Im( ˆβ+ ˜β)

−|∆˜t0(x,η;y,q)|2
4Im( ˆβ+ ˜β)

e

−2
e

Im( ˆβ)Im( ˜β)S2
2
Im( ˆβ)+Im( ˜β) .

+ O(1).

(A.102)

With respect to two real-valued phase terms in equation (A.102),

ei(S1−ˆτt)(∆ˆt0(x,ξ;y,p)) = e

i

Im( ˜β)(ˆτt(ˆtc;y,p)−˜τt(˜tc;y,q))

Im( ˆβ+ ˜β)

∆ˆt0(x,ξ;y,p) ∼ eiO(1),

(A.103)

and

ei(S1−˜τt)(∆˜t0(x,η;y,q)) ∼ eiO(1).

(A.104)

191

To summarize,

−|∆ˆt0(x,ξ;y,p)|2
ℵ ≈ −S2
4Im( ˆβ+ ˜β)
1e
−|∆ˆt0(x,ξ;y,p)|2
4Im( ˆβ+ ˜β)

−|∆˜t0(x,η;y,q)|2
4Im( ˆβ+ ˜β)
−|∆˜t0(x,η;y,q)|2
4Im( ˆβ+ ˜β)

−2
e

≈ K(p, q, y)e

e

Im( ˆβ)Im( ˜β)(ˆτt(ˆtc;y,p)−˜τt(˜tc;y,q))2

Im( ˆβ)+Im( ˜β)

e

.

(A.105)

This is exactly the goal (A.88) we want to achieve.

192

Appendix B

Proof in Elastic Wave

Proof of Positive Definite Hessian Matrix

Theorem B.0.1. The imaginary part of the Hessian for every beam preserves the

S.P.D property after reflection, if it is neither grazing ray nor evanescent wave after

reflection.

Proof. In this appendix, we consider the reflection happens on the surface {x =
(x, y, z) : x = 0} without the loss of generality. We denote the phase function of

both P and S-wave as τ . The same rule can be applied to the velocity c. To simplify

the presentation, all Hessian matrices mentioned below are about the imaginary part

only, if not specified. We assume the reflection point is x0 = (x0, y0, z0), and all terms

below are defined at this point, if not specified. The last simplification is that we follow

the positive Hamiltonian throughout this proof, and the negative Hamiltonian will be

treated similarly.

193

Transformation between Hessian matrices

We first define a new matrix ˜M at the reflection point



˜M =

τtt

τty

τtz

τty

τyy

τyz

τtz

τyz

τzz

 .

(B.1)

Therefore, we can define a transform between the Hessian matrix M and ˜M following

the certain eikonal equation.

ℵ( ˜M ) =



τxx τxy

τxz

τxy

τyy

τyz

τxz

τyz

τzz

 = M.

(B.2)

The terms involved with the variable x are defined by following the certain eikonal
equation and in the way shown in Section 4.5. Moreover, ℵP means that the transform
follows the P-wave eikonal equation and ℵS follows the S-wave eikonal equation.
If we can prove the transform ℵ and its inverse transform ℵ−1 preserve the S.P.D.
property, then the theorem is proved since M new = ℵP ( ˜M ) = ℵP (ℵ−1
ℵS( ˜M ) = ℵS(ℵ−1

P (M )). To prove this, instead of considering two types of matrix in

S (M )) or M new =

equation (B.2) directly, we would like to base our proof first on the complete matrix

Mc.

Mc =

 τtt ∇τ T

∇τt M

t

 ,

where ∇τt is the gradient of the phase function’s time derivative τt and M is the

194

original Hessian. Both ˜M and M are submatrix of Mc.

Proof by Contradiction: ∇c = 0

We start with the simpler case, i.e. ∇c = 0 at the point x0.
Lemma B.0.5. If ∇c vanishes at x0, then Mc is a positive semi-definite matrix and

with rank-three.

Proof. We write the complete matrix Mc first,

 τtt ∇τ T

∇τt M

t

 =

c2∇τ T

|∇τ| M ∇τ|∇τ|
c M∇τ
|∇τ|

 .

c∇τ T M|∇τ|

M

To show that the matrix Mc is positive semi-definite, we use v = (α, p)T ,

vT Mcv = α2c2∇τ T
|∇τ| M
∇τ
|∇τ| + p

(cid:18)

αc

=

∇τ
|∇τ| + 2αc

(cid:19)T

(cid:18)

M

αc

∇τ T M p
|∇τ| + pT M p,
∇τ
|∇τ| + p

(cid:19)

.

(B.3)

(B.4)

(B.5)

Equation (B.5) shows that the null space of Mc is an one-dimensional space and its

basis is ˜v,

˜v =

(B.6)



1(cid:112)
− c(cid:112)

1+c2

1+c2

∇τ|∇τ|

 .

The assumption that there are no grazing rays guarantees that τx (cid:54)= 0 for both beams

before and after reflection.

195

We now start to prove the transform ℵ and its inverse transform preserve the S.P.D.
property when ∇c = 0. In other words, if M is S.P.D, then ˜M = ℵ−1(M ) is S.P.D.
On the other hand, if we have ˜M is S.P.D, then ℵ( ˜M ) is also S.P.D. for both P-wave

and S-wave eikonal equations.

Case I: ˜M = ℵ−1(M )

There are several steps involved to prove ˜M is S.P.D. We first consider ˜M is a submatrix

of the corresponding complete matrix Mc. Then we use Lemma B.0.5 to show this

submatrix is S.P.D.

For any vector u = (u1, u2, u3) ∈ R3,

(cid:18)

(cid:19)

u1 0 u2 u3

Mc

uT ˜M u =

 ,

(B.7)

u1

0



u2

u3

if the reflection happens on the surface {x = (x, y, z) : x = 0}. Since there’s only a

single basis ˜v in the null space of the matrix Mc, all vectors in the form (u1, 0, u2, u3)T

is not parallel to ˜v. Moreover, the complete matrix Mc is a positive semi-definite
matrix. Then for any vector {u = (u1,··· , u4) : u2 = 0}, equation (B.7) will be
positive. Consequently, uT ˜M u > 0 for any u and ˜M is S.P.D.

Case 2: M = ℵ( ˜M )

Similar idea will be applied here. The Hessian M firstly is treated as a submatrix

of the complete matrix Mc and then use the fact that Mc is a positive semi-definite

196

matrix.

In terms of the transform M = ℵ( ˜M ), for any vector u ∈ R3,

(cid:18)

0 uT

(cid:19)

Mc

uT M u =

0

 .

u

(B.8)

Obviously, all the vectors concerned above (0, u)t is not in the null space of the complete

matrix Mc. In other words, for any vector (0, u),

(0, u)T (cid:54)= β ˜v,

∀β ∈ R(cid:54)=0

(B.9)

Consequently, uT M u > 0 for any u and M is positive definite.

Proof by Contradiction: ∇c (cid:54)= 0

We will follow the similar path as the constant velocity case. First, we prove the

imaginary part of the complete matrix Mc is a positive semi-definite matrix.

Lemma B.0.6. The imaginary part of the complete matrix Mc is a positive semi-

definite matrix and with rank-three. Moreover, the single basis ˜v in its null space is

˜v =

(B.10)

 1

−c ∇τ|∇τ|



197

Proof. We first prove ˜v is in the null space

Im(Mc) =

|∇τ|

c2∇τ T Im(M )∇τ

|∇τ|2
c Im(M )∇τ



∇τ T Im(M )

|∇τ|

c

Im(M )

(B.11)

Here, although the gradient of the velocity ∇c (cid:54)= 0, this will only affect the real part.

Therefore, we can apply the same argument in Lemma B.0.5 to prove.

To prove that the transform ℵ and its inverse transform will preserve S.P.D property,
we can use the same idea in the case ∇c = 0. The reason is that we only care about

the imaginary part of the matrix and their imaginary parts are exactly the same thing

as the ones in constant velocity case.

FDTD

As we mentioned previously, the reference solution is generated by the FDTD solution

with the staggered grid. Its correctness will be checked here.

To test this, we compare the FDTD solution with the exact solution in the general

boundary value problem. If this more general problem is solved correctly, then our

reference solution is justified. The parameters used here are,

λ = 2;

µ = 1;

ρ = 1;

198

while the initial condition f here is zero vector

−→
0 , and the initial velocity g is

8π cos(8πt) sin(8πx)

0

0





With the appropriate boundary condition, we know its exact solution is sin(8πt) sin(8πx).

We compare the result at T = 0.8, In Figure B.1, the blue line is the correct result,

Figure B.1: FDTD solution justify.

while the red star curve is the FDTD result with mesh size h = 0.01. Furthermore, we

display its convergence rate in Figure B.2.

We start the mesh size from 1

50 to 1

400 and each time the grid size is reduced by half.

The blue line in Figure B.2 shows the logarithm of L2-error on each mesh size, while

the red star line is a linear function with the slope log(1/2) for comparison.

199

00.10.20.30.40.5−1.5−1−0.500.511.5xyFigure B.2: Convergence Rate of FDTD algorithm

200

11.522.533.54−5.5−5−4.5−4−3.5−3−2.5number of grid pointslog(error)BIBLIOGRAPHY

201

BIBLIOGRAPHY

[1] K. Aki and P. Richards. Quantitative seismology. Freeman and Co., 1980.

[2] U. Albertin, D. Yingst, P. Kitchenside, and V. Tcheverda. True-amplitude beam
migration. In 74th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts,
pages 949–952. Soc. Expl. Geophys., 2004.

[3] V. M. Babich and V. S. Buldyrev. Asymptotic methods in short wave diffraction

problems (in Russian). Nauka, Moscow, 1972.

[4] G. Bao, J. Lai, and J. Qian. Fast multiscale Gaussian beam methods for wave

equations in bounded domains. J. Comput. Phys., 261:36–64, 2014.

[5] G. Bao, J. Qian, L. Ying, and H. Zhang. A convergent multiscale Gaussian-beam

parametrix for the wave equation. Commun. PDEs, 38:92–134, 2013.

[6] J. D. Benamou. An introduction to Eulerian geometrical optics (1992 - 2002). J.

Sci. Comp., 19:63–93, 2003.

[7] G. Beylkin and R. Burridge. Linearized inverse scattering problem of acoustics

and elasticity. Wave Motion, 12:15–22, 1990.

[8] N. Bleistein. On imaging of reflectors in the earth. Geophysics, 52:931–942, 1987.

[9] S. Bougacha, J. Akian, and R. Alexandre. Gaussian beams summation for the

wave equation in a convex domain. Commun. Math. Sci., 7:973–1008, 2009.

[10] R. Burridge. Asymptotic evaluation of integrals related to time-dependent fields

near caustics. SIAM J. Appl. Math., 55:390–409, 1995.

[11] E. Cand`es and L. Demanet. The curvelet representation of wave propagators is

optimally sparse. Commun. Pure Appl. Math., 58:1472–1528, 2005.

[12] E. Candes, L. Demanet, and L. Ying. Fast computation of Fourier integral

operators. SIAM J. Sci. Comput., 29:2464–2493, 2007.

[13] E. Cand`es and D. Donoho. New tight frames of curvelets and optimal represen-
tations of objects with piecewise C2 singularities. Commun. Pure Appl. Math.,
57:219–266, 2004.

[14] V. Cerveny. Seismic Ray Theory. Cambridge University Press, Cambridge, 2001.

202

[15] V. Cerveny, M. Popov, and I. Psencik. Computation of wave fields in inhomo-
geneous media-Gaussian beam approach. Geophys. J. R. Astr. Soc., 70:109–128,
1982.

[16] J. Claerbout. Towards a unified theory of reflection mapping. Geophysics,

36:467–481, 1971.

[17] J. F. Claerbout. Imaging the earth’s interior. Blackwell Scientific Publications,

Cambridge, MA, 1985.

[18] M. G. Crandall and P. L. Lions. Viscosity solutions of Hamilton-Jacobi equations.

Trans. Amer. Math. Soc., 277:1–42, 1983.

[19] L. Demanet and L. Ying. Wave atoms and sparsity of oscillatory patterns. Appl.

Comput. Harmon. Anal., 23(3):368–387, 2007.

[20] L. Demanet and L. Ying. Wave atoms and time upscaling of wave equations.

Numerische Mathematik, 113(1):1–71, 2009.

[21] Feng Deng and George A. McMechan. True-amplitude prestack depth migration.

GEOPHYSICS, 72(3):S155–S166, 2007.

[22] B. Engquist and O. Runborg. Computational high frequency wave propagation.

Acta Numerica, 12:181–266, 2003.

[23] D. Geba and D. Tataru. A phase space transform adapted to the wave equation.

Commun. Partial Diff. Eq., 32:1065–1101, 2007.

[24] Yu Geng, Ru-Shan Wu, and Jinghuai Gao. Gabor-frame-based gaussian packet

migration. Geophysical Prospecting, 62(6):1432–1452, 2014.

[25] S. Geoltrain and J. Brac. Can we image complex structures with first-arrival

traveltime. Geophysics, 58:564–575, 1993.

[26] S. Gray. Gaussian beam migration of common shot records. Geophysics,

70:S71–S77, 2005.

[27] S. Gray and W. May. Kirchhoff migration using eikonal equation traveltimes.

Geophysics, 59:810–817, 1994.

[28] N. Hill. Gaussian beam migration. Geophysics, 55:1416–1428, 1990.

[29] N. Hill. Prestack Gaussian-beam depth migration. Geophysics, 66:1240–1250,

2001.

203

[30] L. H¨ormander. On the existence and the regularity of solutions of linear pseudo-

differential equations. L’Enseignement Mathematique, 17:99–163, 1971.

[31] S. Leung and J. Qian. Eulerian Gaussian beam methods for Schr¨odinger equations

in the semi-classical regime. J. Comput. Phys., 228:2951–2977, 2009.

[32] S. Leung and J. Qian. The backward phase flow and FBI-transform-based
J. Comput. Phys.,

Eulerian Gaussian beams for the Schr¨odinger equation.
229:8888–8917, 2010.

[33] S. Leung, J. Qian, and R. Burridge. Eulerian Gaussian beams for high frequency

wave propagation. Geophysics, 72:SM61–SM76, 2007.

[34] S. Leung, J. Qian, and S. Serna. Fast Huygens Sweeping methods for Schrodinger
equations in the semiclassical regime. Methods Appl. Analy., 21:031–066, 2014.

[35] D. Ludwig. Uniform asymptotic expansions at a caustic. Commun. Pure Appl.

Math., 19:215–250, 1966.

[36] S. Luo and J. Qian. Factored singularities and high-order Lax-Friedrichs sweep-
ing schemes for point-source traveltimes and amplitudes. J. Comput. Phys.,
230:4742–4755, 2011.

[37] S. Luo, J. Qian, and R. Burridge. Fast Huygens sweeping methods for Helmholtz

equations in inhomogeneous media. submitted:xxx, 2012.

[38] S. Luo, J. Qian, and R. Burridge. Fast Huygens sweeping methods for Helmholtz
equations in inhomogeneous media in the high frequency regime. J. Comput.
Phys., 270:378–401, 2014.

[39] S. Luo, J. Qian, and H.-K. Zhao. Higher-order schemes for 3-D traveltimes and

amplitudes. Geophysics, 77:T47–T56, 2012.

[40] V. P. Maslov. The Complex WKB Method for Nonlinear Equations I: Linear

theory. Birkhauser Verlag, Basel, 1994.

[41] V. P. Maslov and M. V. Fedoriuk. Semi-classical approximation in quantum

mechanics. D. Reidel Publishing Company, 1981.

[42] M. Motamed and O. Runborg. Taylor expansion and discretization errors in

Gaussian beam superposition. Wave Motion, 47:421–439, 2010.

[43] D. Nichols.

Imaging complex structures using band limited Green’s functions.

PhD thesis, Stanford University, Stanford, CA94305, 1994.

204

[44] C. Nolan and W. W. Symes. Global solution of a linearized inverse problem for

the wave equation. Commun. Partial Diff. Eq., 22:919–952, 1997.

[45] M. M. Popov. A new method of computation of wave fields using Gaussian

beams. Wave Motion, 4:85–97, 1982.

[46] M.I. Protasov and Vladimir Cheverda. True amplitude imaging by inverse gen-
eralized radon transform based on gaussian beam decomposition of the acoustic
green’s function. Geophysical Prospecting, 59:197 – 209, 03 2011.

[47] J. Qian and L. Ying. Fast Gaussian wavepacket transforms and Gaussian beams

for the Schr¨odinger equation. J. Comput. Phys., 229:7848–7873, 2010.

[48] J. Qian and L. Ying. Fast multiscale Gaussian wavepacket transforms and mul-
tiscale Gaussian beams for the wave equation. SIAM J. Multiscale Modeling and
Simulation, 8:1803–1837, 2010.

[49] J. Ralston. Gaussian beams and the propagation of singularities. Studies in
partial differential equations. MAA Studies in Mathematics, 23. Edited by W.
Littman. pp.206-248., 1983.

[50] H. Smith. A parametrix construction for wave equations with C1,1 coefficients.

Ann. Inst. Fourier Grenoble, 48:797–835, 1998.

[51] N. Tanushev, B. Engquist, and R. Tsai. Gaussian beam decomposition of high

frequency wave fields. J. Comput. Phys., 228:8856–8871, 2009.

[52] N. Tanushev, J. Qian, and J. Ralston. Mountain waves and Gaussian beams.

SIAM J. Multiscale Modeling and Simulation, 6:688–709, 2007.

[53] A. Toflove and S. C. Huganess. Computational Electrodynamics: The Finite
Difference Time Domain Method, Second Editions. Artech House, Norwood,
MA, 2000.

[54] B. S. White. The stochastic caustic. SIAM J. Appl. Math., 44:127–149, 1984.

[55] B. S. White, A. Norris, A. Bayliss, and R. Burridge. Some remarks on the
Gaussian beam summation method. Geophys. J. R. Astr. Soc., 89:579–636, 1987.

[56] Yu Zhang, Guanquan Zhang, and Norman Bleistein. True amplitude wave equa-
Inverse

tion migration arising from true amplitude one-way wave equations.
Problems, 19(5):1113, 2003.

[57] Yu Zhang, Guanquan Zhang, and Norman Bleistein. Theory of true-amplitude
one-way wave equations and true-amplitude common-shot migration. 70, 07 2005.

205