MULTISCALE GAUSSIANBEAM METHOD FOR HIGHFREQUENCY
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 GAUSSIANBEAM METHOD FOR HIGHFREQUENCY
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 multiscale Gaussian beam method and its application
to seismic wave modeling and inversion.
In the ﬁrst part of thesis, we discuss the application of the multiscale Gaussian beam
method to the inverse problem. A new multiscale Gaussian beam method is introduced for
carrying out trueamplitude 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 Multiscale 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 twodimensional 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
vectorvalued 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 TrueAmplitude Migration Process
. . . . . . . . . . . . . . . . . . .
3.2.5 Motivation for Inverting the multiscale Gaussianbeam 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 GaussianBeam 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 Pwave and Swave’s Eikonal Equations . . . . . . . . . . . . . . . . .
4.2.2 Transport Equation Governing Pwave’s Amplitude Vectors
. . . . . 101
4.2.3 Transport Equation Governing Swave’s Amplitude Vectors . . . . . . 103
4.2.4
Single Beam Solution for P and Swave . . . . . . . . . . . . . . . . . 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 Pwave Reﬂecting Beams: Ray Direction . . . . . . . . . . . . . . . . 112
4.5.2 Pwave Reﬂecting Beams: The Hessian of the Phase . . . . . . . . . . 114
4.5.3 Pwave Reﬂecting Beams: Amplitude Vector . . . . . . . . . . . . . . 116
4.5.4
Swave Reﬂecting Beams: the Phase term . . . . . . . . . . . . . . . 117
Swave 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: Pwave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.8.3.1 Example 3: Pwave Reﬂection in the Constant velocity . . . 145
4.8.3.2 Example 4: Pwave Reﬂection in the Linear velocity . . . . 146
4.8.3.3 Example 5: Pwave Reﬂection in the Sinusoidal velocity . . 146
4.8.4 Reﬂection Swave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.8.4.1 Example 6: Swave Reﬂection with Orthogonal Hitting Angle 147
4.8.4.2 Example 7: Swave Reﬂection with NonOrthogonal Hitting
Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.8.4.3 Example 8: Swave Reﬂection: Linear velocity . . . . . . . . 152
4.8.4.4 Example 9: Swave 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 Nonpolluted 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 xaxis 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: Pwave Reﬂection in Constant velocity with Dirichlet Bound
ary Condition. Left: FDTD Solution Right: Gaussian Beam Solution.
146
Figure 4.10: Example 4: Pwave Reﬂection in the Linear velocity with Dirichlet
Boundary Condition. FDTD Solution (’o’), Beam Solution(’’) . . . . . . 147
Figure 4.11: Example 4: Pwave Reﬂection in Linear velocity with Dirichlet Bound
ary Condition. Left: FDTD Solution Right: Gaussian Beam Solution.
148
Figure 4.12: Example 5: Pwave Reﬂection in Sinusoidal velocity along xaxis. FDTD
Solution(’o’), Beam Solution (’’) . . . . . . . . . . . . . . . . . . . . . . 148
Figure 4.13: Example 5:Pwave Reﬂection in Sinusoidal velocity with Dirichlet Bound
ary Condition. Left: FDTD Solution, Right: Gaussian beam Solution . 149
x
Figure 4.14: Example 6: Swave Reﬂection with Orthogonal hitting Angle along
yaxis FDTD Solution (’o’), GB Solution (’’).
. . . . . . . . . . . . . . 150
Figure 4.15: Example 6: Swave Reﬂection with Orthogonal hitting Angle along
xaxis. FDTD Solution (’o’), GB Solution (’’).
. . . . . . . . . . . . . . 150
Figure 4.16: Example 7: Swave Reﬂection with Nonorthogonal 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: Swave Reﬂection in Linear velocity without Reinitializa
tion. Left: FDTD Solution Right: Gaussian Beam Solution without
Reinitialization.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Figure 4.18: Example 8: Swave Reﬂection in Linear velocity with Reinitialization.
Left: FDTD Solution Right: Gaussian Beam Solution with Reinitializa
tion.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Figure 4.19: Example 9: Swave Reﬂection in Sinusoidal velocity. GB Solution(’o’),
FDTD Solution(’’) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Figure 4.20: Example 9: Swave 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: Loglog 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 TrueAmplitude Migration . . . . . . . . . . 37
Algorithm 3 Discrete VectorValued 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
lowfrequency 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 geometricaloptics method
(WKBJ ansatz), which is required to solve a pair of equations. The ﬁrst one, which
is called eikonal equation, is a ﬁrstorder 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 multivalued [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 realvalued. The single Gaussian beam ansatz is made up of
a complexvalued phase function and a complexvalued 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
−0.4−0.200.200.20.40.60.8x1x21.1 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 oneway 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 raybased 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 ﬁrstarrival
traveltime [18, 27] instead of multivalued traveltime. Its usefulness was questioned by
3
[25, 27, 43] since the ﬁrstarrival 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 multiarrival in the general slowness is possible.
4
Second, due to the Multiscale 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 raybased
method described above. Zhang et al. [56, 57] developed trueamplitude commonshot
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 multiscale 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 singlescale 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 longterm 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 welldeﬁned. Moreover, the fact that imaginary part of
the Hessian remains to be S.P.D. leads to a welllocalized solution. The localization is
justiﬁed theoretically [49, 40, 52].
Considering its ability to resolve multivalued 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 multivalued 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
welldeﬁned around the source point and HuygensKirchhoﬀ 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
multiscale 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 Multiscale 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 onetoone
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 welllocalized 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 multiscale Gaussian
wavepacket transform [48] in Section 2.2.
In Chapter 3, we propose a new prestack inversion process based on the multiscale
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 welldesigned 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 vectorvalued 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 Swave 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 TimeDomain (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 multiscale 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 GeometricOptics 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 HamiltonJacobi 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
= iI
(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 rayansatz
methods is that beams’ phase functions τ (x, t) are complexvalued. Complexifying the
equation guarantees a welldeﬁned Hessian and a welldeﬁned 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 positivedeﬁnite imaginary part, provided that it initially does.
Transport Equation A(x, t):
With (x(t), p(t), M (t)) welldeﬁned 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 parabolicscaling principle.
This is the key to the success of our multiscale 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 welllocalized 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 π2x− 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
iξl,iτ (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,ip0,ξl,iM0, A0) as (ˆx(t), ˆp(t), ˆM (t), ˆA(t)). With respect to the ray trajectory,
dx(t)
dt
d(ξl,ip(t))
dt
= v(x(t))
ξl,ip(t)
ξl,ip(t),
= ∇v(x(t))ξl,ip(t),
x0 =
k
Ll
ξl,ip0 = 2πξl,i
19
Since (x(t),ξl,ip(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,ip(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,iM )
dt
= −∇∇v(x(t))ξl,ip(t) − ξl,iM∇v
(cid:32) ξl,ip(t)
ξl,ip(t)
M
(cid:32)
ξl,ip(t)
ξl,ip(t) − ξl,i∇v
(cid:33)
(ξl,iM )
t
xp(ˆx(t), ˆp(t)) ˆM
v(x(t))
− (ξl,iM )
ξl,ip(t) I − v(x(t))
ξl,ip(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,iM (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)p2Id − ppT(cid:17)
,
G±
pp(x, p) = ±v(x)
p3
(2.24)
where Id is the identity matrix.
The proof is easy, so we omit it here.
d ˆA
dt
= −
ˆA
2ξl,iG(x(t), p(t))
+
ˆA
2ξl,iG(x(t), p(t))
(cid:16)ξl,iGT
(v2ξl,iM (t) − ξl,iGx(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 GaussianBeam
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
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. 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 iindexes 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+ma2
=
a 2
2σ22π 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 halfspace X ≡ {x ∈
Rd : xd ≥ 0}. The boundary data D(r, s, t) used in this paper is organized as the
commonshot 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 ≥ bp,
(3.31)
for any rays hitting the surface where p is the momentum variable and pd is the dth
component of p.
30
−20200.511.522.5xrtAssumption 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), G0t<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 Gaussianbeam
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 Diracdelta function by Duhamel’s principle and the mul
tiscale Gaussian wavepacket transform can be applied to decompose the Diracdelta
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 Diracdelta function can be approx
imated by some L2 functions.
3.2.4 TrueAmplitude 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 Gaussianbeam 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 Gaussianbeam transform. Applying K∗
pq to the
boundary data D yields a singlefrequency prestack anglegather 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 TrueAmplitude 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. Anglegather 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 Gaussianbeam
transform
Hereby we provide some intuitive justiﬁcations of the inversion process of the multiscale
Gaussianbeam 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 righthand 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 Gaussianbeam solution provides a globally deﬁned asymptotic solution for
wave equations, we will carry out bruteforce calculations to justify our new methodol
ogy by heavily relying on the structure of multiscale Gaussianbeam 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 lefthand 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
4y−x2
ˆM (0) ˆK(y, p, q),
i
4y−x2
˜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
iy−x2
M0/2,
which says that the fourbeam 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.
x2
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 − ˆr02
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 − t02)
(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) ˆQ2
.
(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 socalled 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
2101x220.5x10.500Second, 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 welllocalized 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))
≤ C2ˆy(t) − ˆx(t)ˆp(t) + C1ˆp(t) − ˆξ(t),
where C1 and C2 are the upper bound of ∇v and ∇∇v respectively. Moreover,
dˆp(t) − ˆξ(t)2
dt
d(ˆp(t) − ˆξ(t))
dt
= 2
· (ˆp(t) − ˆξ(t)) ≤ 2d(ˆp(t) − ˆξ(t))
dt
ˆp(t) − ˆξ(t),
by CauchySchwartz inequality. We further get,
dˆp(t) − ˆξ(t)2
dt
≤ C1ˆp(t) − ˆξ(t)2 + C2(ˆy(t) − ˆx(t))(ˆp(t))(ˆp(t) − ˆξ(t))
ˆp(t) − ˆξ(t)2
2
.
≤ C1ˆp(t) − ˆξ(t)2 +
1
2
(C2ˆp(t)ˆy(t) − ˆx(t))2 +
50
By Gronwall inequality, we have
ˆp(t) − ˆξ(t)2 ∼ O(p),
since C2ˆp(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) ± vˆp(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 2norm.
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 welllocalized 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)2ˆp(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 realvalued 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)eiˆγω−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
2ˆy(ˆtc)−ˆx(ˆtc)2
i 1
ˆM(ˆtc)e
Re( ˆM )(ˆtc)
eiˆγω−ˆτt(ˆtc;x,ξ)− ˆϑT (r−ˆx(ˆtc))2
− 1
4ˆy(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)
4ˆy(ˆtc)−ˆx(ˆtc)2
− 1
2r−ˆx(ˆtc)2
− 1
2r−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 realvalued 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
eiˆγω−ˆτt(ˆtc;x,ξ)− ˆϑT (r−ˆx(ˆtc))2
− 1
2r−ˆy(ˆtc)2
ˆM(ˆtc)
1
× ei ˆξ(ˆtc)·(r−ˆx(ˆtc))−iˆp(ˆtc)·(r−ˆy(ˆtc))ei ˆχe
4ˆy(ˆtc)−ˆx(ˆtc)2
− 1
2r−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
42r−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
eiˆγω−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 welldeﬁned.
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
42r−ˆx(ˆtc)−ˆy(ˆtc)2
ˆM(ˆtc)
eiˆγω−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
42s−˜x(˜tc)−˜y(˜tc)2
˜M(˜tc)
ei˜γω−˜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
2ˆy(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
42r−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
2ˆy(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 realvalued 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
2ˆy(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
4ˆy(ˆtc)−ˆx(ˆtc)2
2ˆy(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
4ˆy(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 − x2
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
4y−x2
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 receiverside 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
4ˆy(ˆtc)−ˆx(ˆtc)2
dξei ˆL(x,ξ,ˆtc;y,p)e
−i 1
2y−x2
Re( ˆM )(0)
Corollary 3.3.5. The sourceside 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
2y−x2
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 lefthand 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.
4ˆy(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
2ˆy(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
2ˆy(ˆtc)−ˆx(ˆtc)2
Re( ˆM )(ˆtc) ˆB(x, ξ; y, p, q)e
− 1
4ˆy(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 − x2
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 −
2ˆy(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 realvalued 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
4ˆy(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 nondegenerate 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
4ˆy(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 nondegenerate 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 ∈
bicharacteristic of the beam ˆU x
Rd}, then after propagating to the surface, the bicharacteristic 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 bicharacteristic,
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 fullrank 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 nondegenerate Gaussian proﬁle. First,
we will use some approximations to move the lefthand 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 nondegenerate. Finally, we will use the transformation
deﬁned in Assumption 3.3.1 to move the lefthand 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 nondegenerate 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 pseudoinverse 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 nondegenerate 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 nondegenerate 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)2eT
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−x2
−y−x2
−y−x2
y−x2
ˆE12
ˆE−1
22
ˆET
12
y−x2
ˆE12
ˆE−1
22
ˆET
12
y−x2
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−ξ2
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−x2
y−x2
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
4y−x2
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
4y−x2
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
4y−x2
˜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
iy−x2
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−x2
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− ˆΦ2
e−Im( ˜β)ω− ˜Φ2
dω.
(3.158)
Here we neglect the real part of the exponent, Re( ˆβ)ω − ˆΦ2 and Re( ˜β)ω − ˜Φ2,
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− ˜Φ2
dω
D(r, s, ω)eiω(ˆtc+˜tc)e−Im( ˆβ)ω− ˆΦ2
(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−101−0.2−0.100.10.20.3x1u3.5.2 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 xaxis in each plot is ∆h , yaxis 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−0.0500.051.522.53dx1yFigure 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 singlesource 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−0.0500.05−20246dx1yFigure 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 −0.500.5−0.4−0.200.20.40.60.480.490.50.510.52Figure 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 multiplesource 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 multiplesource tests. The
4 : 1
16 : 1
dipped layer is displayed correctly in Fig. 3.12.
89
x1x2 −0.500.5−0.4−0.200.20.40.6−0.04−0.0200.020.040.06−1−0.500.51−0.06−0.04−0.0200.020.040.06x2uFigure 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 −0.4−0.200.20.40.6−0.4−0.3−0.2−0.100.10.20.30.4−0.0500.05−101−0.0500.05x2uFigure 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 −1−0.500.5−1−0.500.5−0.06−0.04−0.0200.023.5.4.3 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 multivalue 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 −0.4−0.200.20.40.600.10.20.30.40.50.60.70.620.640.660.680.70.720.740.760.780.8−0.4−0.200.200.20.40.60.8x1x2Figure 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 nonpolluted data while the
blue line is from the polluted boundary data.
93
x1x2 −1−0.500.5−1−0.8−0.6−0.4−0.200.20.40.60.8−0.08−0.06−0.04−0.0200.020.040.0600.511.522.5−0.8−0.6−0.4−0.200.20.40.60.8tuFigure 3.17: Example 7: Gaussian Slowness with polluted trace. Left: Migration result
from Nonpolluted 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 −1−0.500.5−1−0.500.5−0.1−0.0500.05−1−0.500.51−0.0200.020.040.060.080.1x1uChapter 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 initialboundary 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 vectorvalued 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 Geometricaloptics 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
initialboundary 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 Pwave and Swave’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 Pwave whose amplitude vector AP is parallel to
the ray direction ∇τ P . Another two eigenvectors are corresponding to the Swave,
( ˙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 HamiltonJacobi equation.
˙τ P,S + GP,S(x,∇τ ) = 0,
(4.11)
where the Hamiltonian of the Pwave GP (x, p) = ±√
of the Swave GS(x, p) = ±√
µp. We consider the Pwave case for the illustration,
λ + 2µp 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 xcomponent. 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 rayansatz
methods is that beams’ phase functions are complexvalued. To be more speciﬁc, its
second order derivative is complexvalued. 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 Swave with GS(x, p) = ±√
µp. One in
teresting property of the Gaussian beam solution is that it will remain welllocalized
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 positivedeﬁnite imaginary part, provided that it initially does.
Accordingly, the Hessian of beam ansatz’s phase functions is welldeﬁned at all points
even the caustics.
100
4.2.2 Transport Equation Governing Pwave’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 Pwave
and Swave.
We start with deriving the amplitude vector AP for the Pwave, which is parallel to the
ray direction ∇τ P . Therefore, the Pwave’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∇τ2
t = a∇τ2((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∇τ2
t on the left hand side of the above equation is equal to
a ˙τ∇τ2
t = 2a ˙τ (τ,k ˙τ,k)
= aτ,k( ˙τ )2
,k
= aτ,k((λ + 2µ)∇τ2),k
= a∇τ2τ,k (λ + 2µ),k + 2a(λ + 2µ)τ,jτ,jkτ,k,
(4.18)
since we are talking about the Pwave mode now and its phase τ satisﬁes the Hamiltonian
Jacobi equation ˙τ 2 − (λ + 2µ)∇τ2 = 0. Consequently, the transport equation about
the norm of the Pwave’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 welldeﬁned transport equation, while the classical GeometricalOptical
ansatz fails at the caustics region. Following [48], the ODE about the norm of ampli
102
tude can be added into the Pwave’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 Swave’s Amplitude Vec
tors
Now let’s see the Swave’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)D2 +2µaτ,kD2
,k = a¨τD2 +2 ˙a ˙τD2 +a ˙τ ˙D2
, (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 · ∇)D2 = 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 Pwave
(4.20).
Unlike the Pwave, we still need one more equation in the Swave’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 Swave
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) = iI
= − 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 Pwave and c2 = µ for the Swave. The term
G is the corresponding Hamiltonian. There is one extra equation about the direction
D in the Swave’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 singlebeam 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 vectorvalued 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 vectorvalued 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
Pwave and the Swave. 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 Pwave 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,i2
κT
l,i,kvl,i,k
vl,i,k =
κT
l,i,k
ξl,i.
(4.36)
To initialize the Swave, 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 GramSchmidt 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 preprocess 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)iξl,iG+(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 vectorversion Multiscale Gaus
sian Wavepacket is provided below.
Algorithm 3 Discrete VectorValued 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,i2
3. Generate Pwave with the amplitude vector
4. Generate Swaves 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 Pwave and the Swave,
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
∇τ2 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 vectorvalued
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
00.10.20.30.40.500.10.20.30.40.50.60.70.80.91xu4.5 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 3D 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 Pwave Reﬂecting Beams: Ray Direction
When the Pwave reﬂection happens, the total waveﬁeld is made up by three diﬀerent
sources, the original Pwave Gaussian beam, the new Pwave beam after the reﬂection
(PPwave) and the new Swave beam (PSwave). 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 Swave 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 Pwave and cS =
√
µ is the velocity of the
Swave. 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 P2 − (τ P S
y
113
)2 − (τ P S
z
)2.
(4.43)
Here we focus our derivation on the reﬂection from Pwave to Swave, 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 Pwave 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 Pwave 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 Pwave 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 PSwave’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 Swaves.
(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 Swave Reﬂecting Beams: the Phase term
Similar to the Pwave reﬂection, we have the following equation to satisfy the homo
geneous boundary condition for the Swave 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 Swave reﬂection dynamics seems
to be the same as the Pwave reﬂection developed in the last section. However, we
will ﬁnd that there will be some signiﬁcant diﬀerence between the Swave reﬂection
and the Pwave 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 Swave’s velocity cS =
√
µ is less than the Pwave’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 Pwave 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 Swave Reﬂecting Beams: the Amplitude Vector
The evanescent wave fades away quickly, however, we will still include SPwave’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 SPwave’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 SSwave.
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−1−0.500.51−1−0.500.51xu4.6 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
− ωx−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 Pwave 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 singlevariable 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
02
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 realvalued. 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
ωg
,
(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−ma2
e
2˜y−y∗
− I
02
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+y02
× e
i π
4
e
RR
2  ωp−τy
− ω
e
R +E−ma2
Substitute the expression of the term Φ into the above equation,
(cid:115)
2π
(cid:16) ωp−τy
(cid:17)−( R·E
R e
RR
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+E2
(cid:16) ωp−τy
(cid:17)−p
(cid:17)
R +E−ma
R +E
2  ωp−τy
− ω
e
R +E−ma2
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−ma2
(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
RR ei ˜B
(cid:16) ωp−τy
(cid:16) τy
ω
R +E
− ω
2 y−y0−ma2
2eiωp(y−y0−ma)e
2 ωp−τy
R −y∗
− I
2 y−y0−ma2
− 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+E2
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)
RR a
− ω
2 y−y0−ma2
4π
Re
e
2  ωp−τy
− ω
0+y0+E2
(cid:17)−p
(cid:16) ωp−τy
(cid:17)
R +E−ma
R +E−ma2
(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−ma2
e
f (p) = e
2 ωp−τy
− I
R −y∗
0+y0+E2
.
(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+E2
2  ωp−τy
− ω
(cid:17)−p
(cid:16) ωp−τy
e
R +E−ma
R +E−ma2
(cid:17)
+ R
2ω
(cid:16) ωp−τy
R +E
(cid:17)2(cid:19)
dp
(cid:18)
ω2 e
−i πR
4R eiωp0(y−y0−ma)e
(cid:18) τy−R·E
(y−y0)
(cid:19)(cid:19)
ω
2y−y∗
− I
02
(cid:16)−p0(y−y0−ma)+ R
e
− ω
2 y−y0−ma2
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
2y−y∗
− I
02
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−ωy−y0−ma2
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 1D 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) = ie−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 Swave reﬂection case. The diﬀerence is that the SPwave 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 ycomponent of the ray direction τy does not change. For the PS reﬂection,
(cSτ P S
x )2 = (cPτ 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 =
µτ 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 ) =
µτ 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 Swave’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 3D 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)
2y−y∗
− I
02
e
ei˜τy(y−y0)ei R
2 (y−y0)2(cid:90)
e−ωy−y0−ma2
dm + O
(cid:18) 1
(cid:19)
ω
,
The term ˜τy is the modiﬁed ydirection 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
−ωky−y∗
02
2y−y∗
− I
02
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
−ωky−y∗
02
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−ωy−y0−ma2
dm =
=
(cid:90)
(cid:90)
= e
0+y∗
−ωy−y∗
e
−ωy−y∗
02
e
−ωy−y∗
02(cid:90)
e
0−y0−ma2
−ωy∗
dm
0−y0−ma2
−2ω(y−y∗
e
0−y0−ma2
−2ω(y−y∗
0)(y∗
0)(y∗
e
−ωy∗
e
0−y0−ma)
0−y0−ma)dm.
(4.98)
The integral part above is
e
−ωy∗
(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−ma2
−2ω(y−y∗
e
0−y0−ma)dm =
−ωy∗
e
0−y0−ma2
0)(y∗
0−y0−ma)dm
0)(y∗
−2ω(y−y∗
e
e−ω(k+δ)a2
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−ωy−y0−ma2
is a L1 function. By monotone convergence theorem,
a
0
y0−y∗
e−ωy−y0−ma2
dm = 0
(4.100)
lim
N→∞
.
Case 1: k = 0
(cid:90) ∞
N
134
2
− 1
2
(cid:90) 1
e−ω(k+δ)a2
(cid:90) 1
(cid:90) 1
e−ωδa2
2
− 1
2
≈
e−ωδa2
≈
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−ωδa2
e2ω(y−y∗
0)(δa)dδ
dδ
(4.101)
The odd power terms in δ vanish in the last step above since e−ωδa2
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−ωδa2
e2ω(y−y∗
0)(δa)dδ ≈
≈
(cid:90) 1
(cid:90) 1
2
− 1
2
2
− 1
2
e−ωδa2(cid:16)
0)2(δa)2(cid:17)
1 + 2ω2(y − y∗
dδ
e−ωδa2
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)a2
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−ka+δa2
e2ω(y−y∗
0)(−ka+δa)dδ
e−ω−ka−˜δa2
e2ω(y−y∗
0)(−ka−˜δa)d˜δ
e−ωka+δa2
e
−2ω(y−y∗
0)(ka+δa)dδ,
(4.103)
by setting ˜δ = −δ.
When k > 0
(cid:90) 1
(cid:90) 1
e2ω(y−y∗
e−ω(k+δ)a2
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)a2
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−ωka+δa2(cid:16)
(cid:90) 1
2e−ωka+δa2(cid:16)
2
− 1
2
=
2
− 1
2
(cid:90) 1
2e−ωka+δa2(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−ma2
−ωy∗
e
(cid:90)
(cid:90) 1
e−ωka+δa2
(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)(ka+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,
02(cid:90) 1
−ωy−y∗
e
2e−ω(k+δ)2a2
e2ω2(y−y∗
0)2((k+δ)a)2
dδ ≈
−ωy−y∗
02
2e
2
− 1
2
e2ω2(y−y∗
−ωy−y∗
02
= lke
0)2(ka)2
e−ωka2
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−ωka2
,
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 Swave 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 Pwave 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
Swave 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 yvalue and
zvalue to show our reinitialization method’s correctness, The waveﬁeld in Figure 4.5
is plotted along the xaxis. 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
00.10.20.30.4020406080100xu00.10.20.30.4020406080100xu4.8.2 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 Multiscale 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 xaxis by setting z = 0.25 and y = 6
32. Now let’s
compare 2D 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 00.20.400.10.20.30.40.5−0.3−0.2−0.100.10.20.3Figure 4.7: Example 2:
General Initial Value Propagation with Periodic Boundary Condition along xaxis
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 00.10.20.30.40.500.10.20.30.40.5−0.3−0.2−0.100.10.20.34.8.3 Reﬂection: Pwave
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 Pwave 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: Pwave 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 Pwave 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:
Pwave Reﬂection in Constant velocity with Dirichlet Boundary Condition.
Left: FDTD Solution Right: Gaussian Beam Solution.
4.8.3.2 Example 4: Pwave Reﬂection in the Linear velocity
The second velocity in the Pwave 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 xaxis. Then we
set z ﬁxed as 0.25, The Pwave 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: Pwave 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 xaxis ﬁrst by ﬁxing
y = z = 0.25.
146
xy 00.20.400.10.20.30.40.5−0.2−0.100.10.2xy 00.20.400.10.20.30.40.5−0.2−0.100.10.2Figure 4.10: Example 4:
Pwave 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
Pwave reﬂection results shown above. The small error implies that the Pwave reﬂec
tion does not require the reinitialization, while it is necessary in the Swave reﬂection.
4.8.4 Reﬂection Swave
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: Swave Reﬂection with Orthogonal Hitting Angle
Firstly, we see the Swave 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
00.10.20.30.40.5−0.15−0.1−0.0500.050.10.150.2xyFigure 4.11: Example 4:
Pwave Reﬂection in Linear velocity with Dirichlet Boundary Condition.
Left: FDTD Solution Right: Gaussian Beam Solution.
Figure 4.12: Example 5:
Pwave Reﬂection in Sinusoidal velocity along xaxis.
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 00.20.400.10.20.30.40.5−0.15−0.1−0.0500.050.10.1500.10.20.30.40.5−0.2−0.15−0.1−0.0500.050.10.150.2xuFigure 4.13: Example 5:Pwave 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: Swave Reﬂection with NonOrthogonal Hitting Angle
Now if we change the ray direction to increase the width of the SPwave, 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 00.10.20.30.40.500.10.20.30.40.5−0.15−0.1−0.0500.050.10.150.2Figure 4.14: Example 6:
Swave Reﬂection with Orthogonal hitting Angle along yaxis
FDTD Solution (’o’), GB Solution (’’).
Figure 4.15: Example 6: Swave Reﬂection with Orthogonal hitting Angle along xaxis.
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
00.10.20.30.40.5−0.25−0.2−0.15−0.1−0.0500.050.10.150.20.25yu00.10.20.30.40.5−0.0200.020.040.060.080.1xuFigure 4.16: Example 7: Swave Reﬂection with Nonorthogonal 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 xaxis 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 SPwave 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 SPwave reﬂection will lose the accuracy to some degree such that adding
the reinitialization scheme is necessary.
151
00.10.20.30.40.5−0.4−0.200.20.4xU00.10.20.30.40.5−0.4−0.200.20.4xU4.8.4.3 Example 8: Swave 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: Swave 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 00.20.400.10.20.30.40.5−0.4−0.200.20.4Figure 4.18: Example 8: Swave Reﬂection in Linear velocity with Reinitialization. Left:
FDTD Solution Right: Gaussian Beam Solution with Reinitialization.
4.8.4.4 Example 9: Swave 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 00.20.400.10.20.30.40.5−0.4−0.200.20.4Figure 4.19: Example 9: Swave Reﬂection in Sinusoidal velocity. GB Solution(’o’), FDTD
Solution(’’)
Figure 4.20: Example 9: Swave 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 00.20.400.10.20.30.40.5−0.15−0.1−0.0500.050.10.15tion 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 xaxis by ﬁxing y = z = 0.25. With more general initial value, we include
P and Swave 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 00.20.400.10.20.30.40.5−0.0500.05Figure 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
00.10.20.30.40.5−0.15−0.1−0.0500.050.10.150.2xuFigure 4.24: Loglog 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 multiscale 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 multivalued 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 vectorvalued
wavepacket transform is developed to adapt to the highly oscillated vectorvalued 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 − ˆr02
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) ≥ bˆp(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 2norm.
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)
C2ˆy(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)2ˆp(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)2ˆp(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)2ˆp(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)2ˆp(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)1/2ˆp(t)−1ˆp(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)1/2ˆp(t)−1ˆp(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 realvalued 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( 1p).
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 semideﬁnite matrix as it is the sum of two
positive semideﬁnite 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 nonnegative 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)p2(cid:17)
≤ O
(cid:18) 1
(cid:19)
O
p
Here the inverse matrix is deﬁned as the pseudoinverse matrix for a rankdeﬁcient
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
42r − 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)2ˆp(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)2ˆp(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(p1/2) 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
2ˆy(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( 1p) and Im( ˜β) ∼ O( 1q). 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 realvalued 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 Swave 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 Pwave eikonal equation and ℵS follows the Swave 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 semideﬁnite matrix and
with rankthree.
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 semideﬁnite, 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 onedimensional 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 Pwave
and Swave 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 semideﬁnite
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 semideﬁnite
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 semideﬁnite matrix.
Lemma B.0.6. The imaginary part of the complete matrix Mc is a positive semi
deﬁnite matrix and with rankthree. 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
∇τ2
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 L2error on each mesh size, while
the red star line is a linear function with the slope log(1/2) for comparison.
199
00.10.20.30.40.5−1.5−1−0.500.511.5xyFigure 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. Trueamplitude 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 Gaussianbeam
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 timedependent 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 mediaGaussian 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 HamiltonJacobi 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. Trueamplitude 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, RuShan Wu, and Jinghuai Gao. Gaborframebased gaussian packet
migration. Geophysical Prospecting, 62(6):1432–1452, 2014.
[25] S. Geoltrain and J. Brac. Can we image complex structures with ﬁrstarrival
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 Gaussianbeam 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 semiclassical regime. J. Comput. Phys., 228:2951–2977, 2009.
[32] S. Leung and J. Qian. The backward phase ﬂow and FBItransformbased
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 highorder LaxFriedrichs sweep
ing schemes for pointsource 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. Higherorder schemes for 3D 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. Semiclassical 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.206248., 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 oneway wave equations.
Problems, 19(5):1113, 2003.
[57] Yu Zhang, Guanquan Zhang, and Norman Bleistein. Theory of trueamplitude
oneway wave equations and trueamplitude commonshot migration. 70, 07 2005.
205