Seismic imaging using internal multiples and overturned waves
by
Alan Richardson
Submitted to the Department of Earth, Atmospheric and Planetary
Sciences
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Geophysics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
© Massachusetts Institute of Technology 2015. All rights reserved.
Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Earth, Atmospheric and Planetary Sciences
February 27, 2015
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Alison E. Malcolm
Associate Professor
Thesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Robert van der Hilst
Schlumberger Professor of Earth Sciences
Department Head
2
Seismic imaging using internal multiples and overturned
waves
by
Alan Richardson
Submitted to the Department of Earth, Atmospheric and Planetary Sciences
on February 27, 2015, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Geophysics
Abstract
Incorporating overturned waves and multiples in seismic imaging is one of the most
plausible means by which imaging results might be improved, particularly in regions
of complex subsurface structure such as salt bodies. Existing migration methods,
such as Reverse Time Migration, are usually designed to image solely with primaries,
and so do not make full use of energy propagating along other wave paths. In this
thesis I describe several modifications to existing seismic migration algorithms to
enable more effective exploitation of the information contained in these arrivals to
improve images of subsurface structure. This is achieved by extending a previously
proposed modification of one-way migration so that imaging with overturned waves
is possible, in addition to multiples and regular primaries. The benefit of using this
extension is displayed with a simple box model and the BP model. In the latter, the
proposed method is able to image the underside of a salt overhang when even RTM
fails, although substantial artifacts are also present. Progressing to the two-way wave
equation, I explain three new ways in which a wavefield may be separated by wave
propagation direction, and use these in proposed modifications to the RTM algorithm.
With these modifications, overturned waves and multiples can be used more effectively,
as they no longer risk subtracting from the image contributions of primaries, their
amplitude is boosted to produce greater relative amplitude accuracy, and artifacts
usually associated with the use of these arrivals are attenuated. The modifications
also provide two means of expressing image uncertainty. Among the results I show are
a demonstration of the superior image obtained using the proposed method compared
to the source-normalized imaging condition, and an improved image of a salt body
in the SEAM model. Finally, I describe another modification to RTM that further
reduces artifacts associated with the inclusion of multiples, exhibiting its effectiveness
with simple layer models, and on a portion of the SEAM model.
Thesis Supervisor: Alison E. Malcolm
Title: Associate Professor
3
4
Acknowledgments
I wish especially to thank my family and Rebecca for their love. Our short lives can
sometimes seem difficult, but loving and being loved makes it easier to enjoy the time
that we have.
Other graduate students have told me how lucky they think I am to have, in their
opinion, the best advisor in the department. Alison is not only caring, dedicated,
and approachable, but I feel that she is also very skilled at advising, managing to
naturally transition me from a newcomer to geophysics and research into someone
who is comfortable working independently in these areas. Some of my colleagues
dread meetings with their advisors, but I have enjoyed looking forward to friendly
chats with Alison, and the happiness that I felt after the abundant reassurance and
encouragement she always gave to me during our meetings.
I am also very grateful to Total. Providing my funding for over four years has
allowed me to concentrate on my work without being concerned about how I would
continue to be paid. Perhaps more importantly, it was through my connection with
Total that I met some of the other people who have helped me over the past few
years. Especially in the early years of my PhD, Henri Calandra provided a useful
industrial perspective on my work. Terrence Liao supervised me during my first
summer internship, during which I wrote the RTM code on which I based almost all
of the subsequent research I have done. A friend told me that they had never heard
me talk as highly of anyone as I do about Paul Williamson. I have been impressed
on several occasions by how quickly he has understood what I have been trying to
explain, and how he has then able to immediately make insightful observations and
share some of his wisdom. I felt privileged that he kindly agreed to serve on my thesis
committee, where he made many useful suggestions.
Taylor Perron deserves no less of my admiration and gratitude, co-advising me on
one of my General Exam projects, participating in my General Exam committee, very
patiently and generously helping me to publish my first paper, and also forming part
of my thesis committee. As a further example of his generosity, Taylor provided the
5
funding for the remainder of my time at MIT after the end of the Total sponsorship.
One of the ways in which the final member of my thesis committee, Mike Fehler,
has been instrumental in producing this thesis is very obvious, as he provided me with
the SEAM model that I used extensively to validate my ideas. Mike also made many
useful suggestions over the years on ways in which I might improve the presentation of
my work during practice sessions for SEG and ERL consortium meetings, and found
time to meet with me despite his very busy schedule.
The first research project I started working on when I came to MIT was with
Chris Hill, which became the project jointly advised by Taylor Perron. I very much
enjoyed the time that I spent with Chris, who shared my interest in high performance
computing, and he continued to provide encouragement to me even after I moved on
to working exclusively on my thesis research. Chris was a member of my General
Exam committee, and kindly worked on a General Exam project with me that was
outside his primary interest area.
One thing that struck me when I arrived at MIT was how much most of the administrative staff cared about students. Sue Turback, the administrative assistant
of ERL during most of my time, went beyond even this. She sometimes jokingly referred to herself as “mom”, but, given her concern for the wellbeing of ERL’s students,
this was quite appropriate. It would have been difficult for anyone to replace Sue,
but Natalie Counts is doing an excellent job and always greets me with a friendly
smile. I must also thank ERL’s executive director, Anna Shaughnessy, who I know
would always do anything she can to help, and thoughtfully informed me whenever
there were leftovers from meetings. I never had to worry about working out how to
get reimbursed for attending conferences thanks to Terri Macloon. The staff of the
EAPS Education Office have also always been very kind and impressed me by their
dedication.
Life at MIT is certainly not devoted exclusively to research, and the friendships
I have developed with other students over the years have greatly enhanced my time
here. Although there are many others, I mention in particular Sudhish Kumar Bakku,
Di Yang, Ahmad Zamanian, Lucas Bram Willemsen, Andrey Shabelansky, Yuval Tal,
6
Haoyue Wang, Ali Aljishi, Nasruddin Nazerali, Abdulaziz AlMuhaidib, Junlun Li,
Fuxian Song, Beebe Parker, Gabi Melo, Saleh Al Nasser, and Diego Concha, as
having been especially important parts of my life.
Another very important part of my life over the past five and a half years has
been the graduate residence known as “The Warehouse”. It has not only provided me
with the most perfect home that I could have wished for, but has also enabled me to
be part of a wonderful community outside of the department. Much of what makes
the Warehouse so nice is due to the housemasters, both the original, Steve and Lori
Lerman, and their successors, John Ochsendorf and Anne Carney.
In my first year at MIT I was a very grateful recipient of the Charles M. Vest
Presidential Fellowship, made possible by the generosity of the friends of Dr. Charles
Vest. As with the funding provided later by Total and Taylor Perron, this relieved
me from having to concern myself with anything other than my studies.
Although perhaps not as obvious a candidate for acknowledgment as the people
who have been part of my life, the creators of the software that I used extensively
in my research and thesis writing have also played a large role in making this work
possible. Particularly deserving of mention are Vim, Gnuplot, XƎLATEX, Asymptote,
and Matlab.
I was recently asked by another student what I considered to be the high point of
my time at MIT. While there are many tempting choices, such as the euphoric time
after passing my General Exam, or field work in St. Lucia with Dale Morgan, I chose
not a single experience, or even one directly related to MIT, but instead it was the
time I spent on many walks around Boston, particularly by the Charles River, that
stood out. It is a beautiful city, and one that I have very much enjoyed living in for
this portion of my life.
Finally, I am thankful to everyone who has made MIT the wonderful place that
it is, and to those who made it possible for me to be here. It has been an immense
privilege that I am unreservedly grateful for. I will cherish the memories of my time
here for the rest of my life, and am very sad that the time has come for me to leave.
7
8
Contents
1 Introduction
29
1.1
Seismic imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
1.2
Multiples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
1.3
Overturned waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
1.4
Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
2 Extending one-way migration to include multiples and overturned
waves
41
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
2.1.1
One-way migration . . . . . . . . . . . . . . . . . . . . . . . .
42
2.1.2
Attenuating multiples . . . . . . . . . . . . . . . . . . . . . .
43
2.1.3
Imaging with additional wave paths . . . . . . . . . . . . . . .
45
2.1.4
RTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
2.1.5
Proposed Method . . . . . . . . . . . . . . . . . . . . . . . . .
49
2.2
Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
2.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
2.3.1
Box model
53
2.3.2
BP salt model
2.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
57
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
3 Directional amplitude extraction during time-domain wave propagation
3.1
63
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
64
3.2
3.3
Previously proposed methods . . . . . . . . . . . . . . . . . . . . . .
65
3.2.1
Poynting vectors . . . . . . . . . . . . . . . . . . . . . . . . .
65
3.2.2
Local slowness . . . . . . . . . . . . . . . . . . . . . . . . . .
65
3.2.3
Frequency domain methods . . . . . . . . . . . . . . . . . . .
66
3.2.4
Windowed Fourier transform . . . . . . . . . . . . . . . . . .
68
New methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
3.3.1
3.4
Method 1: Plane wave decomposition followed by the Poynting
vector method . . . . . . . . . . . . . . . . . . . . . . . . . .
70
3.3.2
Method 2: Separated light cone stack . . . . . . . . . . . . . .
80
3.3.3
Method 3: Optimization . . . . . . . . . . . . . . . . . . . . .
84
3.3.4
Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
3.4.1
Crossing waves 1 . . . . . . . . . . . . . . . . . . . . . . . . .
91
3.4.2
Crossing waves 2 . . . . . . . . . . . . . . . . . . . . . . . . .
92
3.4.3
Layer over halfspace . . . . . . . . . . . . . . . . . . . . . . .
94
3.4.4
SEAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
3.5
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4 Improving RTM amplitude accuracy
4.1
4.2
105
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.1.1
RTM amplitude errors . . . . . . . . . . . . . . . . . . . . . . 107
4.1.2
Illumination compensation . . . . . . . . . . . . . . . . . . . . 109
4.1.3
Multiples and overturned waves . . . . . . . . . . . . . . . . . 111
4.1.4
Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.2.1
Uncompensated images . . . . . . . . . . . . . . . . . . . . . 113
4.2.2
Illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.2.3
Illumination compensation . . . . . . . . . . . . . . . . . . . . 122
4.2.4
Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
10
4.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.3.1
Improvement due to 𝑊 factor . . . . . . . . . . . . . . . . . . 125
4.3.2
Imaging from opposite sides . . . . . . . . . . . . . . . . . . . 127
4.3.3
Artifact attenuation . . . . . . . . . . . . . . . . . . . . . . . 129
4.3.4
Internal multiples in noisy data . . . . . . . . . . . . . . . . . 131
4.3.5
Comparison with source-normalized imaging condition . . . . 134
4.3.6
SEAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.4
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5 Single wavefield RTM: reducing artifacts and computational cost
143
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.2
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.3.1
Simple layer model . . . . . . . . . . . . . . . . . . . . . . . . 159
5.3.2
Sensitivity to errors . . . . . . . . . . . . . . . . . . . . . . . 161
5.3.3
SEAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.4
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6 Future work
6.1
6.2
6.3
169
Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.1.1
Methods 1 and 2 using LSS with variable local wave speed . . 171
6.1.2
Initial guess . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.1.3
Further performance improvements for method 3 . . . . . . . 172
6.1.4
Sparsity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.2.1
Estimating the effect of neighboring scatterers . . . . . . . . . 174
6.2.2
Displaying orientation information . . . . . . . . . . . . . . . 175
Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.3.1
Estimating the normal derivative . . . . . . . . . . . . . . . . 176
11
6.3.2
Illumination compensation . . . . . . . . . . . . . . . . . . . . 177
7 Conclusion
179
A Resolution of method 2 and the local slowness method
183
A.1 Local slowness method . . . . . . . . . . . . . . . . . . . . . . . . . . 185
A.2 Method 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
B Method 3 gradient and Hessian
191
C Method 3 implementation
193
12
List of Figures
1-1 An example of seismic imaging. (a) A synthetic model of P-wave
speed, extracted from the SEAM model (Fehler and Larner, 2008).
The circle in the center near the surface indicates the source location
used in (b). (b) Seismic data recorded at receivers 15 m below the
surface, due to a source at the location indicated in (a). The receivers
cover the full width of the model, and have a spacing of 25 m. (c) The
image produced by applying seismic imaging techniques to 120 sources
covering the width of the model. . . . . . . . . . . . . . . . . . . . . .
31
1-2 A shot gather (data recorded by all receivers for a single source location) showing the direct arrival (wave that travels directly from the
source to the receivers without reflecting), followed by three arrivals
corresponding to reflections from either three reflectors, or two with
an internal multiple between them. . . . . . . . . . . . . . . . . . . .
36
1-3 Examples of internal multiples imaging areas that are difficult to reach
using primaries. (a) A vertical structure, such as the side of a salt body,
imaged using an internal multiple reflected two times (also called a
prismatic multiple). (b) Going around a troublesome area (such as a
salt body) to image from underneath, using a triply-reflected internal
multiple. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
2-1 Different situations in which the source and receiver wavefields might
be coincident in space-time. The rightmost case does not correspond
to a reflection and so should not add to the image. . . . . . . . . . .
13
53
2-2 A demonstration of the effect of applying (a) the conventional imaging
condition (b) a Laplacian filter, and (c) the new imaging condition.
The same percentage of clipping was applied to each image. Note that
the artifacts, which obscure the box when the conventional imaging
condition is used, have been suppressed. . . . . . . . . . . . . . . . .
54
2-3 Box velocity model. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
2-4 Image of the central box in the Box model when the exact velocity
model is used with different migration algorithms. (a) Regular one-way
migration. (b) The proposed enhanced one-way migration algorithm.
(c) RTM.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
2-5 Images of the central box in the Box model when only the background
velocity model is used during migration. (a) Regular one-way migration. (b) RTM. Two images from the one-way enhanced algorithm are
shown: (c) includes all of the additional wave paths, while in (d) the
(𝑢𝑠,𝑡 , 𝑢𝑟,𝑡 ) and (𝑢𝑠,𝑚 , 𝑢𝑟,𝑚 ) wave paths are excluded. . . . . . . . . .
57
2-6 The portion of the BP velocity model used in this experiment. The
white arrow indicates the salt leg that is used as a multiples-generating
interface. The salt overhang, which is the imaging target, is identified
with a box. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
2-7 Image of the target area of the BP model when the exact velocity
model is used with different migration algorithms. (a) Regular oneway migration. (b) Enhanced one-way migration, showing only the
contributions from upgoing waves. (c) RTM. . . . . . . . . . . . . . .
59
2-8 Image of the BP model when only the background velocity model is
used during migration. (a) Regular one-way migration. (b) Enhanced
one-way migration. (c) RTM. . . . . . . . . . . . . . . . . . . . . . .
14
60
3-1 A wave propagating in the direction 𝜓 ̂ and centered at the origin at
time step 𝑡 will travel along the path A. Summing along A and dividing
by the summation length, to apply the local slowness method, will
therefore yield the value of the wave at its central peak. The circles
represent the top and bottom edges of the light cone that the wave can
travel along. The dashed lines joining the two circles indicate the shape
of the lightcone. Summing along any other line on this cone other than
A will yield zero, as long as the summation time is sufficiently long. .
67
3-2 A wave with wavefront orientation angle 𝜓 is oscillatory in the direction
𝜓 ̂ and constant in the direction 𝜓⟂̂ . To perform wavefront orientation
angle separation at the origin point in the figure, we compute the
average amplitude along lines passing through the origin. Summing
along the line B and dividing by the summation length will produce
the peak value of the wave, while summing along the perpendicular
line A will result in zero. . . . . . . . . . . . . . . . . . . . . . . . . .
72
3-3 Waves 1 and 2 have perpendicular wavefront orientations. Both are
oscillatory over the distance 𝑐𝑇 , where 𝑐 is the local wave speed. Summing along A will produce the value of wave 1 along that line with no
interference from wave 2. Summing along line B will result in zero. .
72
3-4 When the difference between the wavefront orientation angles of waves
1 and 2 is Δ𝜓, it is necessary to sum at least a distance 𝐼𝐱 along wave
2 in order to cancel contributions from wave 1, where 𝐼𝐱 is given by
Equation 3.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
3-5 A wave 𝑢 propagating in the positive 𝜓 ̂ direction (to the right) is
shown at time 𝑡 and 𝑡 + Δ𝑡. At points where the spatial derivative in
the direction 𝜓 ̂ is positive, the time derivative is negative, while the
time derivative is positive at points with a negative spatial derivative,
so 𝑃 in Equation 3.14 is positive. . . . . . . . . . . . . . . . . . . . .
15
76
3-6 (a) The point on a wave propagating in the direction π rad which is
used to investigate the effect of filter parameters in method 1. (b) With
filter parameters 𝑑 = 2 and 𝑚𝑎𝑥𝑒𝑟𝑟𝑜𝑟 = 2000, the true propagation
direction (π rad) is not a peak of absolute amplitude. (c) The location
of the maximum peak in absolute amplitude versus angle varies with
the choice of parameters for the method’s two filters. (d) As in (c),
but for relative amplitude error in the wave amplitude assigned to the
true direction of propagation. (e) As in (b) but with filter parameters
𝑑 = 100 and 𝑚𝑎𝑥𝑒𝑟𝑟𝑜𝑟 = 1000. The peak occurs at the angle of the
true propagation direction. . . . . . . . . . . . . . . . . . . . . . . . .
81
3-7 To separate waves propagating in directions differing by less than 𝜋/2,
method 1 with LSS requires a shorter summation length than the local
slowness method (measured in the plot as a multiple of the time over
which the waves are oscillatory, 𝑇 , for the local slowness method, or
𝑐(𝐱)𝑇 for method 1). For larger differences in propagation direction,
the local slowness method has better resolution for a given summation
length. The result for method 1 does not include the effect of the filters
that can be applied when using that approach. This plot is derived
from equations in Appendix A. . . . . . . . . . . . . . . . . . . . . .
82
3-8 The time needed to perform directional separation on a single time slice
of 200×200 cells, with 𝑇 = 0.085 s, and Δ𝑡 = 2.7 × 10−4 s. For method
2 it is assumed that wavefront orientation separation has already been
performed on all but the final time slice. Method 3 took approximately
31 s for Hessian construction and 32 s for the optimization. As the
Hessian does not vary over time steps, it only needs to be computed
once. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
3-9 Memory required to perform the same separation as in Figure 3-8.
Method 3 required 2.1 GB . . . . . . . . . . . . . . . . . . . . . . . .
16
88
3-10 A time slice of two waves overlapping obliquely. Directional separation
is performed at the central point (0.1 km, 0.1 km). Only the central
portion of the wavefield is shown. . . . . . . . . . . . . . . . . . . . .
92
3-11 Results of directional separation on the wavefield in Figure 3-10. Propagation angle is on the polar axis, while the radial axis represents amplitude. The amplitude range is the same for all plots except (f), in
which is it halved. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
3-12 A time slice of two waves with an overlapping region in which the
waves are propagating in opposite directions. . . . . . . . . . . . . . .
94
3-13 Results of the directional decomposition of the wavefield in Figure 3-12.
All have the same amplitude range as Figure 3-12. . . . . . . . . . . .
95
3-14 Absolute amplitude, summed over time, of the upgoing (reflected) wave
in a halfspace model. . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
3-15 P-wave speed for a 2D portion of the SEAM model, covering the region
10 km to 16 km 𝑥, 2.39 km 𝑦, 0 km to 6.25 km 𝑧. . . . . . . . . . . . .
99
3-16 Sum over time of the absolute amplitude of the backpropagated data
wavefield generated by a source at 13 km 𝑥, 15 m 𝑧. Results from the
region around the source are removed to make amplitudes in the rest
of the domain more visible. All polar plots have the same amplitude
range. Locations of discontinuities in the P-wave velocity model are
shown in the background. . . . . . . . . . . . . . . . . . . . . . . . . 101
4-1 A fraction of the backpropagated arrival from true reflector 2 will be
reflected upward from true reflector 1 if it is present in the velocity
model. This may overlap with the fraction of the forward propagated
source wave which is also reflected upward. This causes a phantom
reflector at the apex of wave path 3. As the phantom reflector is
well illuminated by large amplitude direct waves along wave path 2,
applying illumination compensation will reduce the amplitude of the
phantom reflector artifact. . . . . . . . . . . . . . . . . . . . . . . . . 109
17
4-2 (a) The component of the receiver data contributed by a scatterer at
position 𝐱′ is determined by the source wavefield 𝑢𝑠 at 𝐱′ , the scatterer amplitude 𝑚(𝐱′ ), and the Green’s function between the scatterer
location and the receiver, 𝐺+ . (b) The data wavefield 𝑢𝑑 is created by
applying the anticausal Green’s function 𝐺− to the recorded data. . . 116
4-3 Normalized image amplitude at the central point on a horizontal line of
scatterers (“reflector”), as the length of the line and the source/receiver
aperture (125 m above the scatterers, symmetric about the 𝑥 coordinate of the chosen point) vary. The amplitude should ideally be the
same in all cases. The x axis represents reflector length, while plotted lines correspond to different source/receiver aperture widths. (a)
Regular RTM. (b) Illumination compensated without the 𝑊 term in
Equation 4.20. (c) Illumination compensated with a simple approximation for 𝑊 . (d) Illumination compensated with a more sophisticated
approximation for 𝑊 . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4-4 Vertical slices through the image of a horizontal layer that is equally
illuminated from above and below. (a) The image contributions when
the layer is imaged from above and below are shown separately. (b)
The conventional RTM imaging condition does not distinguish between
image contributions from different sides of an interface, simply adding
all contributions, resulting in almost complete cancellation in this case.
(c) The proposed imaging condition reverses the sign of image contributions such that they always stack coherently, regardless of which
side the interface is imaged from, resulting in a significantly improved
image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
18
4-5 Backscatter and phantom layer artifacts are caused by reflectors in the
migration velocity model with the regular RTM imaging condition, but
are attenuated with the proposed method. (a) The velocity model, consisting of a high velocity layer sandwiched between two lower velocity
layers, that was used for receiver data generation and migration. (b)
The image obtained using the conventional cross-correlation imaging
condition. Significant backscatter artifacts are present. (c) Applying
the scattering angle filter imaging condition of Costa et al. (2009) reduces backscatter artifacts, but the phantom layer artifact remains, as
indicated by an arrow. (d) The image produced using the proposed
method. Backscatter and phantom reflector artifacts are attenuated.
130
4-6 (a) The velocity model used to generate receiver data, consisting of a
high velocity salt body (right) and salt layer (bottom), surrounded by a
smooth gradient. Source positions are indicated by circles, and receiver
positions by triangles. (b) The velocity model used for migration. The
salt body has been replaced by a sediment fill. . . . . . . . . . . . . . 132
4-7 (a) The image produced by regular RTM, focused on the region containing the salt body. The upper left corner of the salt body has been
imaged by primaries, and so is much higher amplitude than the rest of
the salt flank (indicated by an arrow), which was imaged with internal
multiples. The range of displayed amplitudes has been severely clipped
so that the internal multiple image contributions are visible. (b) The
image when the proposed method is used. The image contributions of
the internal multiples now have amplitude comparable to that of the
primaries. The true location of the salt interface is indicated by the
dotted line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
19
4-8 One means of conveying image uncertainty is by complementing the
image with a measure of illumination. (a) The image, using the proposed method, of reflectors determined during the application of the
imaging condition to be horizontal. The outline of the true location of
the salt body is shown for reference. (b) The illumination of horizontal
reflectors. Horizontal reflectors in the region of the salt body are very
poorly illuminated, indicating that if any exist there they may not be
imaged. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4-9 The image of four layers of equal scattering amplitude in a constant
background. 3D propagation was used for modeling and migration.
The amplitude of a vertical line through the center of the image is
also shown. Sources and receivers cover the top surface of the model,
with a spacing of 20 m and 10 m, respectively. (a) Regular RTM crosscorrelation imaging condition, showing significant amplitude variation.
(b) Source-normalized cross-correlation imaging condition, which is an
improvement, but amplitude variation is still noticeable. (c) Illuminationcompensated image, with significantly more consistent amplitudes than
the other two approaches. . . . . . . . . . . . . . . . . . . . . . . . . 136
4-10 (a) The P-wave velocity of the extracted 2D portion of the SEAM
model that is imaged. The high velocity structure on the right is a
salt body. (b) The smoothed velocity model used during migration. . 138
4-11 Images of the 2D portion of the SEAM model shown in Figure 4-10,
focused on the salt body. (a) Regular RTM fails to clearly image the
areas on the underside of the salt overhang indicated by arrows. (b)
The proposed method results in improved amplitude accuracy under
the salt overhang. (c) Image of weighted standard deviation of image
amplitude across shots (Equation 4.26), divided by the absolute value
of image amplitude, highlighting inconsistencies, which are primarily
artifacts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
20
5-1 Demonstration of incorrect backpropagation in RTM. (a) Forward
propagation from the source (circle) to a reflector in the subsurface,
where the wave splits into a reflected component (dashed), which returns to the surface where it is recorded by a receiver (triangle), and a
transmitted component (dotted), which continues to propagate downward. The y axis represents depth, while the x axis could be either
time or horizontal distance. (b) The backpropagated recorded data
also separates into reflected (dashed) and transmitted (dotted) components at the reflector, causing the backpropagated wavefield to not
truly represent the seismic wavefield. . . . . . . . . . . . . . . . . . . 146
5-2 Examples in which incorrect backpropagation in RTM can lead to
phantom reflector artifacts. (a) The backpropagated arrival from true
reflector 2 reflects on true reflector 1, leading to a phantom reflector
near the surface. (b) Part of the internal multiple between true reflectors 1 and 2 is incorrectly transmitted through true reflector 2, causing
a deep phantom reflector. . . . . . . . . . . . . . . . . . . . . . . . . 147
5-3 The simulation domain when using the proposed method. The interior of the domain, the shaded region 𝜏 , is where the wavefield will
be recreated from measurements on the boundary 𝛿𝜏 . Real receivers
generally only cover a portion of the boundary, so synthetic receivers
are used on the remainder. Sharp edges in 𝛿𝜏 are avoided to reduce
artifacts. The receivers must record the outward normal derivative of
the wavefield at the boundary, as indicated by 𝑛̂ ′ . . . . . . . . . . . . 151
5-4 Simplified wave amplitudes to approximately determine image amplitude contributions at a reflector. R is the reflection coefficient. (a) The
source wave when the reflector is not present in the migration model.
(b) The data wave when the reflector is not present in the migration
model. (c) The source wave when the migration model contains the reflector. (d) The data wave in regular RTM when the migration model
contains the reflector. . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
21
5-5 Even when the wave propagates along the correct path, phantom reflectors are still possible when using imaging conditions that assume
waves overlap at reflectors. . . . . . . . . . . . . . . . . . . . . . . . . 158
5-6 A demonstration of the proposed method’s ability to reduce phantom
reflector artifacts compared to regular RTM. (a) The velocity model
that is used for modeling and migration. It will produce similar wave
paths to those depicted in Figure 5-2. (b) Image produced by regular
RTM. A, B, and C indicate types of artifacts that the proposed method
can reduce. (c) The result when using the proposed method, showing
significant attenuation of artifacts. . . . . . . . . . . . . . . . . . . . 160
5-7 Sensitivity of the proposed method to errors in the model and data.
The true model is similar to that in Figure 5-6, but the velocities
have been increased so that the areas that were 1500 m/s are now
2000 m/s, and the high velocity layer has increased from 2000 m/s to
3000 m/s. (a) Misplacement of the bottom reflector in the model used
for migration so that the high velocity layer extends to 2 km+Error. (b)
Wrong velocity in the bottom region. The wave speed in the area below
the high velocity layer in the migration model is 2000 m/s+Error. (c)
Smoothing of interfaces. Instead of being sharp discontinuities, both
interfaces are smoothed over a distance of Smoothing in the migration
model. (d) Uncalibrated data. The real receiver data are scaled by
the specified multiplier and so no longer match the synthetic receiver
data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5-8 Ratio of the sum over depth of the absolute value of the deep phantom
reflector’s amplitude relative to that of the upper true reflector as the
smoothness of the interfaces varies, using the same model as that in
Figure 5-7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
22
5-9 Velocity model of the 2D portion of SEAM we use to test the proposed
method on a complicated model. (a) The true P-wave velocity model.
(b) The migration velocity model. It matches the true model at the
sea floor and at the top of the salt body, but is increasingly smoothed
below this. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5-10 The result of imaging a 2D portion of the SEAM model. (a) The
image produced by regular RTM after applying a high-pass filter. (b)
The result when the proposed method is used, after applying the same
high-pass filter as that used in (a). . . . . . . . . . . . . . . . . . . . 166
A-1 A downgoing wave at time 𝑡 + 𝐼𝑡 /2, oscillatory over the length 𝑐𝑇 . We
depict the case when 𝐼𝐱 = 𝑐𝑇 is used as the summation length for
wavefront orientation angle separation, and 𝐼𝑡 = 𝑇 is the summation
time for the local slowness spacetime slant stack. O, C, and D are
points on the wave, which move with the wave as it propagates. The
semicircle shows half of the top edge of the light cone centered on time
𝑡. At the time 𝑡 + 𝐼𝑡 /2, the LSS sum for wavefront orientation angle
will be composed of the points of the wave along the line A. At time
𝑡 − 𝐼𝑡 /2 the points of the wave will be those along the line B. It may
be useful to reexamine Figure 3-1 when considering this diagram. . . 184
A-2 Similar to Figure 3-7, but for method 2 when the waves are wave
packets of duration 𝑇 . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
23
24
List of Tables
2.1
Input wavefields in imaging condition to image using different wave
paths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
52
Computational complexity, where 𝑁𝑥 and 𝑁𝑧 are the number of cells
in the 𝑥 and 𝑧 dimensions, 𝑁𝑝 is the number of propagation directions
that we wish to separate the wavefield into, and 𝐼𝑡 is the summation
length in time. For method 2 we assume that the same summation
length (in time) is used in both the summation over time slices and
the spatial summation for wavefront orientation separation. The complexity of method 3 will depend on the choice of optimization method,
but we assume that it will be proportional to the number of elements
3.2
in the Hessian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
Memory requirements, where the symbols are described in Table 3.1.
89
25
26
List of Acronyms
ADCIG
Angle-domain common-image gather
ADR
Acquisition dip response
AVA
Amplitude versus angle
AVO
Amplitude versus offset
FFT
Fast Fourier transform
FTCS
Forward-time central-space
FWI
Full Waveform/Wavefield Inversion
LSRTM
Least-squares Reverse Time Migration
LSS
Local slant stack
MVA
Migration velocity analysis
OBH
Ocean bottom hydrophone
RTM
Reverse Time Migration
SEAM
SEG Advanced Modeling Program
SNR
Signal-to-noise ratio
SRME
Surface-Related Multiple Elimination
27
28
Chapter 1
Introduction
1.1
Seismic imaging
Seismic imaging is a geophysical method that seeks to use surface measurements to
produce an image of transitions in the properties of the Earth that affect elastic wave
propagation. The measurements are typically pressure (especially for hydrophone sensors used in marine surveys), velocity, acceleration, or displacement. The sensors may
be positioned in an array on the land surface, inside boreholes, towed in streamers
behind ships, or placed on the sea floor, among other possibilities. Seismic imaging
is common on both a global scale, where it is used to determine large-scale features
of the Earth’s interior, such as the depth of the Moho (Grad et al., 2009), and on the
kilometer scale, for more detailed studies. The latter is especially associated with hydrocarbon exploration, and so is known as exploration seismology. Although passive
approaches are possible (Draganov et al., 2004), exploration seismology is usually “active source”. An energy source, such as an air gun (marine) or vibroseis truck (land)
is used to generate the waves that are recorded. The source has a controlled location
and source signature (energy injected as a function of time). This thesis is primarily
concerned with active source exploration seismology. Although seismic measurements
are sensitive to a variety of material properties, such as bulk modulus, shear modulus, and density, in exploration seismology the velocity of P-waves (“primary” or
“pressure” waves), a combination of bulk modulus and density, is currently the most
29
frequently used quantity, occasionally with the addition of parameters to quantify
anisotropy in the velocity. Density and shear wave (S-wave) velocity may also be
used, but this is not as common, primarily because density does not generally vary
significantly over the length scales of interest for exploration, and the combination of
using sources that mainly produce P-wave energy, and recording systems that favor
P-waves (recording in the sea, where S-waves do not propagate, or only measuring
the vertical components on land, where the low velocity near surface causes S-waves
to turn such that the majority of the displacement they cause is in the horizontal
directions), means that S-waves often have a low signal-to-noise ratio (SNR). Nevertheless, they have been shown to provide information that, when correctly exploited,
can produce improved images (Stewart et al., 2002).
Active source seismic surveys consist of several (often many thousand) unique
source positions. The resulting waves are recorded by a number of receivers, usually
also in the thousands in modern surveys, for a chosen period of time after each source
wave is emitted. Receivers may be at the same locations for all source positions (a
fixed spread survey), or may move (which is especially common in towed streamer
marine surveys). The collection of data recorded by all receivers for a single source
location is called a shot record or shot gather. Each shot record can be used to
produce an image of the subsurface, and these are stacked (summed) to create the
final subsurface image. It is occasionally convenient to refer simply to a “shot”, so one
may speak of stacking over shots. The term “shot” may also be used as a synonym
for “source”. Examples of the data recorded during a seismic experiment and the
resulting image are shown in Figure 1-1.
Seismic imaging is the main geophysical technique used in hydrocarbon exploration. While it does not provide comparable detail to well logs and core samples, it
has far greater spatial extent. Although it is possible to invert for material properties using seismic data, and indeed this is the goal of a technique termed full
waveform/wavefield inversion (FWI, see Virieux and Operto (2009) for a review),
this requires considerable computational resources and the results have not yet been
judged sufficiently useful to justify the expense of inverting up to the maximum reso30
4.5
z (km)
Wave speed
(km/s)
0
6
1.5
0
x (km)
12
a
t (sec)
0
10
0
x (km)
12
b
z (km)
0
6
0
x (km)
12
c
Figure 1-1: An example of seismic imaging. (a) A synthetic model of P-wave speed,
extracted from the SEAM model (Fehler and Larner, 2008). The circle in the center
near the surface indicates the source location used in (b). (b) Seismic data recorded
at receivers 15 m below the surface, due to a source at the location indicated in (a).
The receivers cover the full width of the model, and have a spacing of 25 m. (c) The
image produced by applying seismic imaging techniques to 120 sources covering the
width of the model.
31
lution possible with the data. Instead, techniques such as FWI are currently used to
provide a background velocity model to help seismic imaging methods.
As the large, easily accessible hydrocarbon reservoirs with good permeability appear to have all already been found, exploration efforts have been directed toward
more difficult environments. This typically means deep offshore, often underneath
or on the flanks of salt bodies (Beck and Lehner, 1974; Hedberg et al., 1979), with
more exploration potentially shifting to the Arctic in the future (Gautier et al., 2009).
Drilling to prospective reservoirs in these environments is extremely expensive, so technology that can reduce the risk of not finding economic quantities of hydrocarbons
at these sites is of great importance. Improvements in seismic imaging techniques,
especially the ability to enhance images in subsalt regions, are therefore of significant
benefit, reducing exploration risk in areas of complex geology, such as the Gulf of
Mexico, the North Sea, and West Africa, which all contain salt bodies. These improvements are being derived from changes in both acquisition, where wide azimuth and
long offset (large distance between the source and the furthest receiver) surveys that
illuminate image points from a wide range of angles, are becoming common (Kapoor
et al., 2014), and from the use of more sophisticated imaging algorithms (Jones and
Davison, 2014). Such algorithms are referred to as migration methods, as they move,
or “migrate”, recorded energy into the correct location on the image to show the
subsurface structure. They have progressed from early techniques such as Hand or
Hagedoorn migration (Hagedoorn, 1954), Stolt migration (Stolt, 1978), and phase
shift or one-way migration (Gazdag, 1978), which, in their original forms, assumed
that there were no lateral velocity variations in the Earth, to Kirchhoff (Schneider,
1978) and sophisticated one-way migration methods (Gazdag and Sguazzero, 1984;
Stoffa et al., 1990; Ristow and Rühl, 1994; Xie and Wu, 1999), and finally to Reverse
Time Migration (RTM, Baysal et al. (1983)), which is currently considered to be the
most accurate technique available. Increasing accuracy is obtained through closer
adherence to the physics of finite frequency wave propagation in heterogeneous media, and has primarily been made possible by the rapid rise in available computing
resources. One such enhancement has been the possibility to use more complicated
32
arrivals (waves arriving at the receivers) than regular primaries (which simply propagate down from the source, reflect, and then propagate back up to the receivers).
Among these additional arrivals are multiples and overturned waves, which will be
described below.
RTM and one-way migration both follow similar procedures. When migrating a
shot record, two numerical simulations of wave propagation are performed. During the
“forward” propagation, the source wave is injected into a discretized domain covering
the region of the Earth to be imaged, at the location corresponding to the true source
position. An estimate of the material properties affecting wave propagation in the
domain must be supplied, which may, for simplicity, only be the P-wave velocity model.
This allows the source wave propagation into the Earth to be simulated, creating what
is referred to as the forward or source wavefield. The second simulation propagates the
recorded data back into the Earth (which, especially for RTM, is sometimes referred
to as backpropagation). This wavefield is frequently termed either the data, receiver,
or backpropagated wavefield. The final component of these migration methods is
the application of an “imaging condition”. This is a procedure for using these two
wave simulations to create an image of the subsurface. The most popular imaging
condition, the cross-correlation method (Claerbout, 1971), cross-correlates the two
wavefields with a zero time lag. This effectively makes the assumption that the
source wave and the data wavefield are coincident in time and space at the locations
of reflectors.
1.2
Multiples
Multiply reflected waves, or multiples, are waves that reflect multiple times on the
path from the source to the receivers. The simplest multiples reflect twice, once more
than primaries, but higher order multiples, which reflect more times, are also possible
(although they become progressively weaker with each reflection). Using multiples
for imaging, especially higher order multiples, may require a longer recording time
than when only primaries are used, as the additional reverberations delay the arrival
33
of the waves at the receivers. This increases the cost of acquisition and processing as
more time is needed to complete the survey, and extra computational resources are
needed. Long recording times are becoming more common, however, as more long
offset surveys are performed, and so the probability that multiples are recorded is
increasing.
Multiples are classified as internal, which reflect multiple times in the subsurface,
or surface multiples, which reflect from the Earth’s surface. Both contain useful
information, and may image parts of the subsurface that are not reached by primaries
with the acquisition geometry used. Surface multiples are generally easier to use for
imaging, since the location of one of the reflectors (the sea or land surface) is known,
and we may record the wave at the surface when it reflects there. Such arrivals
are, however, not significantly more useful than primaries, as they will image regions
that could also be accessible with primaries with an appropriate source location. They
may act like additional sources, enhancing the SNR of the image, and providing better
source sampling (both within the region covered by regular sources, and so increasing
the spatial frequency of the sources, and acting like additional sources outside this
region, and so effectively increasing the imaging aperture, as shown by Verschuur and
Berkhout (2011)). Internal multiples are more difficult to incorporate in an imaging
algorithm. An example of the problem is shown in Figure 1-2, which shows three
arrivals. These reflections could be caused by three reflectors in the subsurface, or
just two reflectors (whose location we can only approximately determine), with a
first order multiple between them. An additional difficulty with internal multiples is
that they are more sensitive to inaccuracies in the velocity model used for migration
than primaries, since they propagate over a larger distance and so accumulate errors.
Surface multiples also suffer from this problem, but the situation is less challenging
if the surface bounces are recorded, since this allows the separate surface multiple
bounces to be treated as primaries. Internal multiples are generally weaker than
surface multiples, as the surface is highly reflective, especially in marine surveys, but
the situation is sometimes reversed on land, when an unconsolidated or complicated
near surface, together with variable topography, may result in the surface being a
34
poor reflector.
Despite the difficulty of exploiting the information contained in internal multiples,
the potential image improvements that might be achievable through their use makes it
a worthwhile endeavor. One of these is their ability to image vertical structures, such
as vertical faults and the flanks of salt bodies, which primaries struggle with. Internal
multiples also provide a means of going around zones that are troublesome to image
through. It is often hard to image the base of salt bodies using primaries, for example,
as this can only be achieved by imaging through the salt body, but there is usually a
high velocity contrast between salt and the surrounding sediment, causing most of the
energy to be reflected from the upper surface. Using internal multiples, the base of salt
could be imaged by going around the salt, reflecting from a deeper layer, and imaging
the salt from below. This is depicted in Figure 1-3. The allure of using multiples
for imaging has begun to raise the standing of multiples. They were once (and still
often are) considered to be source-generated noise. This led to the development
of techniques for attenuating multiples, such as SRME (Verschuur et al., 1992) for
surface waves, the extension of Jakubowicz (1999) for internal multiples, and filters
for identifying them in the tau-p (Hampson, 1986; Lokshtanov, 1995) and f-k (Ruehle,
1983) domains. Rather than investing effort to remove data from the signal, several
efforts have been made to use multiples in imaging (Youn and Zhou, 2001; Cavalca
and Lailly, 2005; Malcolm et al., 2009, 2011; Fleury, 2013; Dai and Schuster, 2013).
RTM is even capable of naturally including multiples if the generating reflectors are
present in the velocity model, however including these reflectors may cause artifacts
in the image, and so many practitioners only provide smooth models to RTM.
1.3
Overturned waves
Overturned waves occur when a downgoing wave is refracted so that it begins propagating upward. If the wave returns to the surface without ever reflecting, it is known
as a diving wave, but we are primarily interested in waves which reflect before or after
overturning. Overturned waves occur because the wave speed of the Earth generally
35
Time (s)
0
1.3
1
Receiver number
100
Figure 1-2: A shot gather (data recorded by all receivers for a single source location)
showing the direct arrival (wave that travels directly from the source to the receivers
without reflecting), followed by three arrivals corresponding to reflections from either
three reflectors, or two with an internal multiple between them.
a
b
Figure 1-3: Examples of internal multiples imaging areas that are difficult to reach
using primaries. (a) A vertical structure, such as the side of a salt body, imaged using
an internal multiple reflected two times (also called a prismatic multiple). (b) Going
around a troublesome area (such as a salt body) to image from underneath, using a
triply-reflected internal multiple.
36
increases with depth. These waves have many of the same benefits as internal multiples: they are capable of imaging vertical structures, and, with the right acquisition
geometry, it may be possible to go around troublesome areas, such as using overturned waves to image the base of a salt body. Like multiples, they are therefore also
data present in the recorded signal containing useful information about the subsurface, and so exploiting these arrivals is one of the most plausible means of improving
seismic images. They do, however, also suffer from some of the same problems as
multiples. The velocity gradient of the Earth is such that in order to image depths
of interest to hydrocarbon exploration with overturned waves, a considerable offset
is often required. With the long offsets regularly used in modern acquisition, this
has become less of a problem and so recording overturned wave arrivals is becoming
increasingly common. The long propagation distance does mean, however, that, like
multiples, overturned waves are quite sensitive to velocity model errors. Although
they may only undergo a single reflection and so may avoid the energy loss of multiple bounces, overturned waves are still often very weak as they endure spherical
spreading losses (and attenuation) over their long propagation distance. Similarly
to multiples, a long recording time is necessary, increasing the cost of acquisition
and processing. Despite these challenges, overturned waves may be considered easier
to exploit for imaging than internal multiples as they do not require the inclusion
of reflectors in the migration velocity model. Indeed, RTM is capable of naturally
imaging with overturned waves, although their small amplitude means that they are
unlikely to make significant image contributions with the conventional imaging condition. One-way migration algorithms do not normally incorporate overturned waves,
but there have been proposals to extend these methods for this purpose (Hale et al.,
1992).
Even though overturned waves may not reflect multiple times, they will be considered to be distinct from primaries, with the latter term used to refer only to waves
that propagate down to reflectors and then up to receivers without ever overturning.
37
1.4
Outline
In this thesis I propose modifications to migration algorithms to allow the effective
and efficient incorporation of multiples and overturned waves in seismic imaging.
We begin, in Chapter 2, by further extending a previously proposed modification
of one-way migration. This modification allowed imaging with multiples, while the
extension described in this thesis also enables overturned waves to be efficiently incorporated. Results for a simple box model and the more realistic 2004 BP model
demonstrate the improvement obtained by exploiting the information contained in
the overturned waves and internal multiples. Although computationally efficient, the
one-way method, on which the proposal is based, suffers from inaccuracies due to
the approximations inherent in it. Among these are concerns about the accuracy of
propagation of overturned waves during the portion of their wave path when they are
traveling close to horizontally, inaccuracies in the image amplitudes, and a failure in
the current form of the method to account for variations in illumination. This last
point means that the image contributions of multiples and overturned waves must
be manually scaled so that they make substantial contributions to the final image
when combined with image contributions of primaries. Fortunately, this is facilitated
by the ability of the proposed method to produce separate images for image contributions from multiples and overturned waves. This may still result in there being
little relationship between image amplitude and the physical properties of the Earth.
Nevertheless, the method could be viewed as a computationally inexpensive means
of producing an image that is indicative of the gains that are possible with internal
multiples and overturned waves, and so may be used to decide whether more accurate
methods, such as those presented in subsequent chapters, are worthwhile.
Chapter 3 moves to RTM, but, rather than describing a migration method, the
chapter proposes three different methods that could be used to determine the direction in which waves are propagating during a numerical simulation (such as during
migration). These methods are presented as propagation directions will be required
in migration algorithms in later chapters, and previously proposed means of obtaining
38
this information were found to be insufficient. Propagation direction information was
available in many earlier migration algorithms, such as ray-based migration, but this
is lost in RTM. Several examples of determining the wave amplitude propagating in
different directions are shown, including demonstrations of situations in which even
the proposed methods are expected to struggle. The results are compared with previously proposed methods, and appear to produce superior outputs in the majority
of cases, although this comes at the cost of requiring additional computational resources. In addition to being integral to the success of the migration methods to be
presented, algorithms for separating wave amplitude by propagation direction, such
as those described in this chapter, also have other useful applications, such as producing ADCIGs (Sava et al., 2001), and inversion for anisotropic parameters (Li et al.,
2014).
Wave amplitude, binned by propagation direction, is a key quantity in the modifications to RTM proposed in Chapter 4. These modifications aim to improve the
accuracy of image amplitudes by reducing the effect of artifacts, avoiding a problem present in the conventional RTM imaging condition when an interface is imaged
from both sides, and applying compensation for variations in illumination so that
relative image amplitudes are more closely related to the physical properties of the
Earth. These improvements are especially beneficial for multiples and overturned
waves, allowing such waves to contribute more effectively to the image, and reducing the artifacts usually associated with their inclusion. Results presented in the
chapter show the greater accuracy obtainable with the method compared to using a
source-normalized imaging condition to compensate for illumination variations, the
significant improvement possible when an interface is imaged from both sides, robustness of the method to noise, and demonstrate the application of the method to a 2D
portion of the SEAM model (Fehler and Larner, 2008).
One of the types of image artifacts attenuated by the modifications of Chapter 4
is referred to as a phantom reflector. This is the presence of what may look like a
reflector in the image where one is not present in reality. One of the potential causes
of these artifacts is reflectors in the migration velocity model (which are necessary
39
to image with internal multiples when using RTM), and another is the presence of
internal multiples in the recorded data. While the method described in Chapter 4 may
reduce the relative amplitude of these artifacts, it is unlikely to completely eliminate
them. Such artifacts can, however, be very harmful as there is a risk that they may
be interpreted as real reflectors. When such artifacts are caused by arrivals due to
reflections from reflectors which are known and so can be included in the velocity
model, they are avoidable as they are predictable. Chapter 5 proposes a method for
avoiding these artifacts when possible. It uses synthetic receivers on image domain
boundaries where real receivers are not present. This approach has an advantage even
for regular migration of primaries in a smooth velocity model as it makes it possible
to reduce the number of wavefields which must be backpropagated during RTM. It
does, however, not completely remove all cross-talk artifacts. We compare the results
produced by this method with those of regular RTM for a simple layer model, and
on the same 2D portion of the SEAM model as in the preceding chapter. We also
analyze the sensitivity of the method to various errors in the inputs.
Finally, Chapter 6 discusses potential future work to advance the methods described in the earlier chapters. It consists of promising ideas that need additional development in order to be sufficiently robust and practical for application to industrialscale field data. This includes discussing means of improving the performance of the
optimization approach to separating a wavefield by propagation direction, extending
the other two proposed propagation direction methods so that the locally constant
velocity assumption may be relaxed, and estimating the point spread function for the
imaging operation.
40
Chapter 2
Extending one-way migration to
include multiples and overturned
waves
Abstract
Two of the most popular migration algorithms in exploration seismology are the oneway method and reverse time migration (RTM). The former is fast, but excludes
important parts of the recorded wavefield, while the computational expense of the
latter means that it can only be employed sparingly. An algorithm is proposed that
uses multiple passes to extend the one-way method to include overturned waves and
multiples. A comparison of the images of two synthetic models produced by the
regular one-way algorithm, RTM, and the new method, shows that it can significantly
improve the result in regions of interest, and in certain situations may even provide
more useful information than RTM.
2.1
Introduction
Migration algorithms tend to either be fast, with the penalty of excluding wave arrivals that provide information about potentially important areas of the subsurface,
or very computationally expensive. This paper proposes an algorithm that is a compromise between these two extremes.
41
2.1.1
One-way migration
The economic importance of reflection seismology has led to decades of research on
methods of performing migration, resulting in the proposal of many different algorithms. The large datasets involved mean that these algorithms make a variety of
approximations to reduce computational cost. One class of algorithm is known as the
one-way method. This involves propagating the source wavefield (an approximation
of the wave emitted by the source) and the recorded data (hereafter referred to as the
receiver wavefield) down into the Earth. This downward propagation implies that the
wavefields will not be calculated correctly if they contain upgoing components. An
image is then formed by applying an imaging condition to the wavefields. The most
commonly used is
𝑖𝑚𝑎𝑔𝑒(𝑥, 𝑦, 𝑧) = ∑ 𝑢𝑠,𝑧 (𝑥, 𝑦, 𝜔) × 𝑢̄𝑟,𝑧 (𝑥, 𝑦, 𝜔),
(2.1)
𝜔
where 𝑖𝑚𝑎𝑔𝑒 is the resulting representation of the subsurface reflectors, 𝑥 and 𝑦 are
the surface coordinates, 𝑢𝑠,𝑧 is the source wavefield at depth 𝑧, 𝑢̄𝑟,𝑧 is the complex
conjugate of the receiver wavefield at depth 𝑧, and 𝜔 is the frequency.
The advantage of one-way migration is that we are able to use phase shifts, easily
and computationally efficiently applied in Fourier space, to accomplish the downward
propagation (Gazdag, 1978)
𝑢𝑧+∆𝑧 (𝑘𝑥 , 𝑘𝑦 , 𝜔) = 𝑢𝑧 (𝑘𝑥 , 𝑘𝑦 , 𝜔)𝑒𝑖𝑘𝑧 ∆𝑧 ,
(2.2)
where 𝑢𝑧 is a wavefield at depth 𝑧, 𝑘𝑥 and 𝑘𝑦 are the 𝑥 and 𝑦 wavenumbers, Δ𝑧 is
the distance to downward propagate the wave, and 𝑘𝑧 = √ 𝜔𝑐22 − 𝑘𝑥2 − 𝑘𝑦2 . For each
wavefield it is only necessary to store 𝑛𝑥 × 𝑛𝑦 × 𝑛𝜔 elements in memory (where 𝑛𝑥
is the number of elements in the 𝑥 dimension, 𝑛𝑦 is the number of elements in the 𝑦
dimension, and 𝑛𝜔 is the number of frequency components to be included) as each
depth level is only accessed once.
Another advantage of the method is that it handles complex geology better than
42
ray-based algorithms, such as Kirchhoff migration (Schneider, 1978), as, unlike these
methods, it correctly accounts for multipathing.
While one-way methods were popular for a number of years, recently their dominance has been diminished, particularly in areas of complex subsurface geometry, by
the rise of other methods such as RTM (see section 2.1.4). This is largely because of
two problems with the algorithm. The first is that in its basic form it does not support lateral velocity variations. The situation was ameliorated by the introduction
of modifications such as PSPI (Gazdag and Sguazzero, 1984), split-step migration
(Stoffa et al., 1990), FFD correction (Ristow and Rühl, 1994), and the pseudoscreen
method (Xie and Wu, 1999), among many others, which enable one-way migration in
laterally heterogeneous media. With such schemes the accuracy of the propagation
decreases with angle and is still only exact for vertically propagating waves. Another
issue with the method is that it assumes waves only travel downward from the source,
reflect, and travel upward to the receivers. In reality, there are additionally many
other paths that the waves can take. One class of such paths is called multiples, and
refers to waves that reflected more than once on the path between source and receiver.
The extra reflections could take place at the Earth’s surface (the free surface; surface
multiples), which is particularly common in marine datasets, or could be between
reflectors in the subsurface, termed internal multiples. The other class of wave paths
not supported by one-way migration is overturned waves. The wave speed typically
increases with depth in the Earth. By Snell’s Law, this causes waves to turn away
from the vertical. Some waves may turn over and begin to propagate upward. This
violates the assumption of waves only traveling in one direction.
2.1.2
Attenuating multiples
If the velocity model used during migration is close to being correct, then overturned
waves will not cause artifacts as arrivals corresponding to these waves will be attenuated by the algorithm at the depth where they begin to propagate upward. This is
not the case with multiples, however, which can cause spurious reflectors in images if
not removed before migration. Thus there has been significant work on attenuating
43
multiples.
One simple technique that has been used is pattern recognition, where multiples
are predicted from a model and then arrivals matching these predictions are removed
(Guitton and Cambois, 1999). Another is differential moveout, which exploits contrasts in moveout (the variation in arrival time at different receiver positions) to distinguish primaries from multiples (Schneider et al., 1965; Foster and Mosher, 1992).
Others include frequency discrimination and tau-p deconvolution methods (Lokshtanov, 1995). None of these approaches work well in all situations, however. Another
method that has achieved popularity is surface-related multiple elimination (SRME,
Verschuur et al. (1992)). This is an iterative method for removing free surface multiples. It does not require additional information about the subsurface such as a velocity
model. The recorded data is used as a first estimate of the primaries (other methods
are frequently applied first to improve this estimate). Multiples associated with the
reflectors predicted by this data are iteratively removed. Although the method works
well, particularly for marine data, it is quite computationally expensive and requires
a good estimate of the surface reflectivity and source signature, as well as a dense
dataset.
Algorithms attempting to remove surface multiples have the advantage of knowing
the approximate location of one of the reflectors (the surface). Despite its additional
challenges, good progress has been made with removing internal multiples.
One of the key developments was the work by Jakubowicz (1999) to extend SRME.
Traditional SRME assumes that the wave path can be decomposed into two wavefield
components (if it reflects twice in the subsurface and once at the surface). It had
been suggested that to use SRME for internal multiples, the wavefields could be
downward continued into the subsurface so that the ‘surface’ reflection took place
in the subsurface (Berkhout and Verschuur, 1997). The problem with this is that
downward continuing the waves would require a velocity model for the subsurface, and
errors in this model would result in multiples attenuation occurring in the wrong part
of the data. In Jakubowicz’s method the wavefield of an internal multiple is instead
decomposed into three components. The multiple is considered as the combination
44
of two primaries (waves that only reflect once in the subsurface) minus one primary.
SRME is found to be a special case of this, when the wavefield that is subtracted
reduces to the surface reflectivity.
A similar method, described using inverse scattering series, has been proposed by
Weglein et al. (1997).
2.1.3
Imaging with additional wave paths
Multiples
The necessity to remove multiples from data before performing one-way migration is
unfortunate. This is because it is difficult to identify such waves and also because they
contain useful information about the subsurface that is lost when they are eliminated.
A particular advantage of these wave paths is their ability to image structures from
below. In areas of the subsurface where there is a strong velocity contrast, but
the exact interfaces of the velocity anomaly are unknown, waves that pass through
the region (which would be the only way of imaging the bottom of the structure
using conventional one-way migration) are unlikely to be propagated correctly by the
migration algorithm and so the image will be inaccurate. By imaging from below,
using a wave path that travels around the area and reflects off of a lower interface,
it is possible to avoid propagation through the anomalous region and so obtain a
clearer image. Another advantage is that multiply reflected waves may reflect off of
near-vertical structures, making it possible to image such features if these wave paths
are included in the algorithm. As arrivals from these wave paths represent waves
that reflected off of subsurface structures, they also hold the potential to increase
the signal-to-noise ratio (SNR) if they are used in addition to regular downgoing
singly-reflected waves, potentially yielding a clearer image. Imaging the bottom and
flanks of salt bodies and other formations is particularly important in oil and gas
exploration as these are often the locations where accumulations occur. Finally, as
multiples typically travel a greater distance than singly-reflected waves, they tend to
be more sensitive to the velocity model. This implies that multiples could be even
45
more useful than singly-reflected waves in velocity analysis.
Several algorithms for imaging with multiples and overturned waves have been
proposed. For the case of multiples, attempts can be split into two classes: imaging
with free surface multiples, and imaging with internal multiples.
An initial attempt at using free surface multiples in marine data was made by
Reiter et al. (1991). This approach assumed that the surface multiples had already
been successfully separated from the primaries. The method assumes that the ray
path of each multiple being migrated is known. Ocean bottom hydrophones (OBH)
are thus necessary to permit distinction between the two possible types of multiples
considered (referred to as ‘source’ multiples and ‘receiver’ multiples). An issue with
this approach is that it requires good knowledge of the velocity model. Since multiples
are even more sensitive to errors in the velocity model than primaries, migrating with
multiples could deteriorate the final image rather than enhance it if the velocity model
is incorrect. Nevertheless Reiter obtained an image with higher SNR and greater
lateral extent compared to using primaries alone.
An early attempt at using surface multiples in land data by Guitton (1999) also
showed promising results. This method again relied on having an existing model of
the subsurface so the path of the multiples could be determined, and furthermore
assumed that a receiver was located at the point on the surface at which the multiple
reflected. The wave recorded at this location is then used as the source signature for
the ‘primary’ between this location and the final receiver, and traditional migration is
performed. The results Guitton obtains are in fact noisier than when only primaries
are used for migration. An important discovery, however, is that when he does not
separate the primaries from the multiples and therefore uses data containing both in
the migration, the resulting image is not significantly worse, indicating that the large
effort to separate the wavefields may not be necessary when a method such as this is
used.
Another attempt at using surface multiples for imaging was made by Berkhout
and Verschuur (2006), who proposed using SRME to separate the multiples, then
transforming them into primaries and using them for imaging.
46
A recent approach proposed by Muijs et al. (2007) appears to present a viable
method of imaging using surface multiples in marine data. Similarly to the earlier
work of Reiter, an OBH array is required. In addition, a method of separating upgoing
and downgoing waves at the sea floor is necessary, such as through the use of novel
over-under acquisition strategies, or by theoretical means. Receivers must also be
located so as to record the wavefield that reflected off the surface, as it enters the earth.
The advantage provided by these additional constraints, however, is that multiples do
not need to be separated from the data by other means as this is already accomplished
by the up/down separation. A further advantage is that the multiples are not more
sensitive to the velocity model than primaries, as the multiples are simply primaries
with a different source signature (the wavefield recorded as the multiple entered the
sea floor). The method does, however, introduce a further small complication as the
source signature is no longer an impulse and so a more complicated imaging condition
must be used. The results provided appear to show that the method is successful at
imaging using multiples.
Imaging using internal multiples is a very difficult problem as each wave reflects
at several unknown locations. A method was proposed recently by Malcolm et al.
(2009). Ideally, the primaries are migrated to form an image of the reflectors reached
by primaries. The multiples and source wavefields are then downward propagated
and stored at each depth. Multiplying this data by the reflectors image ‘selects’ the
wavefield at the depths at which it would undergo reflections. Propagating the stored
data upward simulates the reflections of the waves. The stored data from each level as
it is reached is added to the upward propagating wave so that the upward propagation
will include waves reflected from all deeper reflectors. Applying the standard imaging
condition will image the underside reflections of first order multiples. Multiplying the
data by the image of the new reflectors and subsequently applying a downward pass
of the one-way propagator and imaging condition allows second order multiples to be
used for imaging. Each additional pass results in higher order multiples being used
in the imaging process. This method has the distinct advantage of not requiring any
prior knowledge of the locations of reflectors.
47
Overturned waves
Many of the advantages of multiples also apply to overturned waves, and so it is also
desirable to develop methods that are capable of imaging with these wave paths. The
first proposal for imaging with overturned waves using a one-way method was made
by Claerbout (1985). It makes the assumption that the velocity model is laterally
homogeneous and monotonically increasing with depth. A regular downward pass
is made, followed by an upward pass for regions of the wavefield that correspond
to 𝑘𝑧2 < 0 (evanescent waves). This region will grow as depth decreases (due to the
velocity decreasing), and will only include overturned waves above the depth at which
they turned. It therefore automatically takes care of upward propagating waves from
the correct depth.
One problem with this method is that it needs to propagate waves that are travelling almost horizontally. This is the reason for the restriction to laterally homogeneous media, as the accuracy of the one-way method decreases with angle from the
vertical in the case of lateral velocity variations. Zhang and McMechan (1997) realised that instead of using the one-way algorithm to propagate vertically downward,
one could instead do a horizontal pass to image with overturned waves. This will only
ameliorate the situation if the waves do not propagate vertically on their path. A
further modification was proposed which includes both vertical and horizontal passes
(Jia and Wu, 2009). This then requires the direction of the waves to be determined so
that the passes can be blended at each point, weighting in favor of whichever is likely
to be the most accurate. This method also suffers from a number of problems. The
first is that it can require many passes, especially in 3D. This can make the algorithm
computationally expensive. Another is that accurately determining the direction of
wave propagation is difficult, and the errors introduced by inaccuracies in this process
could reduce the gain over the simpler two pass method. Finally, one-way methods
typically assume that the wave speed at each depth can be decomposed into a background velocity, and a laterally-varying change to this. This is generally true for
vertical passes, since the background velocity usually increases with depth, but may
48
not be valid for horizontal axes.
2.1.4
RTM
Another migration algorithm includes overturned waves automatically: reverse time
migration (RTM). In this method both the source and receiver wavefields are propagated in time (rather than in depth). As it is not possible to implement this using a
phase shift in Fourier space, a finite difference method is used. RTM is significantly
more computationally expensive than one-way migration, but its natural ability to
include overturned waves, and to accurately propagate waves even in the presence of
laterally heterogeneous media, mean that it is frequently used in complex areas. It
is also possible to image with multiples using RTM. This will occur if the multiplegenerating interface is included as a velocity or density discontinuity. This implies
that the location of such reflectors must be known before the start of migration. Furthermore, discontinuities can lead to artifacts being introduced into the resulting image as this violates the assumption inherent in the method that only singly-scattered
waves are being migrated. Further discussion of imaging with multiples using RTM
and the attending artifacts is available in Chapter 4.
2.1.5
Proposed Method
In this paper a new migration algorithm is proposed. Employing ideas from Claerbout’s two-pass turning wave method (Claerbout, 1985) and the internal multiples
algorithm described by Malcolm et al. (2009), the proposed scheme extends the conventional one-way method to efficiently include potentially important additional arrivals.
The implementation of the new method is described in Section 2.2. As it is
anticipated that this algorithm will be of particular relevance in regions of complex
geology, the images it produces on two synthetic datasets are compared against those
of RTM. The models considered are a simple box model (Section 2.3.1), and the BP
salt model (Section 2.3.2).
49
2.2
Implementation
The proposed algorithm consists of two stages. The first is a downward pass. This
is similar to the conventional one-way method: for each shot, source and receiver
data are downward propagated and an imaging condition is applied. Images from
all shots are then stacked to produce an image of the reflectors of singly-scattered
downward propagating waves. It differs from the traditional implementation by saving
the source and receiver wavefields at each depth. As in the implementation of the
regular one-way method, memory usage may be reduced at the expense of additional
computation by checkpointing. This is achieved by only saving the wavefields every
few depth steps, and then, when necessary, downward propagating from the nearest
saved depth to compute the wavefields at depths that were not saved.
The second stage is an upward pass. As in the downward pass, each shot is
processed independently. Four new upgoing wavefields are created and initially set
to zero: waves propagating forward in time from the source that have turned over,
𝑢𝑠,𝑡 , or reflected from a multiples-generating interface, 𝑢𝑠,𝑚 , and waves propagating
backward in time from the receivers that have turned over, 𝑢𝑟,𝑡 , or reflected from a
multiples-generating interface, 𝑢𝑟,𝑚 . At each depth, the source and receiver wavefields
saved at that depth during the downward pass are loaded as 𝑢𝑠,𝑑 and 𝑢𝑟,𝑑 , respectively.
The algorithm then proceeds as follows:
1
Multiples: 𝑢𝑠,𝑚 ← 𝑢𝑠,𝑚 + 𝑢𝑠,𝑑 × 𝑖𝑚𝑎𝑔𝑒; 𝑢𝑟,𝑚 ← 𝑢𝑟,𝑚 + 𝑢𝑟,𝑑 × 𝑖𝑚𝑎𝑔𝑒;
2
ℱ𝑥,𝑦 (𝑢𝑠,𝑑 , 𝑢𝑟,𝑑 , 𝑢𝑠,𝑡 , 𝑢𝑠,𝑚 , 𝑢𝑟,𝑡 , 𝑢𝑟,𝑚 );
3
Turning: forall the 𝑘𝑥 , 𝑘𝑦 , 𝜔 do
4
5
if
𝜔2
𝑐2
< 𝑘𝑥2 + 𝑘𝑦2 then
𝑢𝑠,𝑡 ← 𝑢𝑠,𝑡 + 𝑢𝑠,𝑑 ; 𝑢𝑟,𝑡 ← 𝑢𝑟,𝑡 + 𝑢𝑟,𝑑 ;
6
phase shift(𝑢𝑠,𝑡 , 𝑢𝑠,𝑚 , 𝑢𝑟,𝑡 , 𝑢𝑟,𝑚 );
7
ℱ−1
𝑥,𝑦 (𝑢𝑠,𝑑 , 𝑢𝑟,𝑑 , 𝑢𝑠,𝑡 , 𝑢𝑠,𝑚 , 𝑢𝑟,𝑡 , 𝑢𝑟,𝑚 );
8
lateral heterogeneity correction(𝑢𝑠,𝑡 , 𝑢𝑠,𝑚 , 𝑢𝑟,𝑡 , 𝑢𝑟,𝑚 );
9
imaging condition(𝑢𝑠,𝑡 , 𝑢𝑟,𝑑 ; 𝑢𝑠,𝑑 , 𝑢𝑟,𝑡 ; 𝑢𝑠,𝑡 , 𝑢𝑟,𝑡 ; 𝑢𝑠,𝑚 , 𝑢𝑟,𝑑 ; 𝑢𝑠,𝑑 , 𝑢𝑟,𝑚 ; 𝑢𝑠,𝑚 , 𝑢𝑟,𝑚 );
In this code, 𝑖𝑚𝑎𝑔𝑒 is the reflector image from the first pass, ℱ𝑥,𝑦 is the Fourier
50
transform in 𝑥 and 𝑦, and ℱ−1
𝑥,𝑦 is its inverse. The conventional one-way phase
shift operation is applied to all of the wavefields acted on by phase shift, lateral
heterogeneity correction is the wide-angle correction used in laterally heterogeneous media, and imaging condition indicates the application of an imaging
condition on pairs of wavefields.
Multiplying a downward propagating wavefield by the reflector image in the
frequency-space domain (line 1) has the effect of selecting the wavefield at the locations of reflectors as these will be the only significantly non-zero parts of the image.
By adding this to an upgoing wavefield and propagating, reflection is simulated. This
allows multiples to be considered.
After transforming to the frequency-wavenumber domain (line 2), the portion of
the downgoing source and receiver wavefields that have turned-over (𝑘𝑧2 < 0) is added
to the corresponding upgoing turning wavefields to be propagated upward (line 5).
This includes overturned waves in migration.
On line 6, the four upgoing wavefields (the source and receiver waves that have
turned, and the source and receiver waves that have reflected) are propagated to the
next depth level above.
Returning to the frequency-space domain on line 7 allows the application of corrections to account for lateral heterogeneity in the medium on line 8.
Six additional potential wave paths are now available for imaging. These are
illustrated in Table 2.1, and correspond to the six pairs on line 9.
By loading the saved wavefield from the downward pass at each depth level, the
restriction that Claerbout’s algorithm can only be used for models with monotonically
increasing velocity is relaxed. This is because there is no longer a reliance on the
upgoing waves being automatically included only at the correct depth. The problem
of propagating through high angles in the presence of lateral velocity variations still
exists, but, when a high-accuracy propagator is used, is found empirically to not
introduce significant errors.
With one additional pass over conventional one-way migration, the proposed
method exploits more of the recorded data to augment the result with six supple51
Input
𝑢𝑠,𝑡
𝑢𝑟,𝑑
source
Path
receiver
Input
Path
𝑢𝑠,𝑑
𝑢𝑟,𝑡
𝑢𝑠,𝑡
𝑢𝑟,𝑡
𝑢𝑠,𝑚
𝑢𝑟,𝑑
𝑢𝑠,𝑑
𝑢𝑟,𝑚
𝑢𝑠,𝑚
𝑢𝑟,𝑚
Table 2.1: Input wavefields in imaging condition to image using different wave paths.
mentary wave paths. Further passes could be performed to include a greater number
of wave paths, although these are likely to yield wave paths that make a smaller
contribution to the resulting image due to the small percentage of waves taking these
paths.
Ideally, recordings corresponding to arrivals of multiples should be removed from
the data used during the first pass. Only arrivals of multiples should be used during
the propagation and imaging condition applied for the wave paths of multiples in the
second pass. Non-overturned waves should be removed from the data involved in the
imaging of overturned waves. Failure to do this can cause cross-talk that leads to
artifacts in the resulting image. Performing these manipulations of the data is errorprone and expensive, however. The proposed algorithm assumes that such separation
has not occurred. The results in subsequent sections indicate that, although artifacts
are present, they do not overwhelm the image.
A complication introduced by the inclusion of the additional wave paths can produce severe artifacts. In regular one-way migration, the source wave is always propagating downward, and the receiver wave is an upgoing wave. This results in the
simplifying assumption that the source and receiver wavefields only overlap in spacetime at the location of reflectors (Figure 2-1a). When multiples are added, however,
propagating the source wavefield up from a reflector can cause it to be coincident
52
a
b
c
Figure 2-1: Different situations in which the source and receiver wavefields might be
coincident in space-time. The rightmost case does not correspond to a reflection and
so should not add to the image.
with the upgoing wave from the reflector in the receiver wavefield. This will occur
at each depth up to the surface, resulting in long smear-like artifacts. We cannot
simply avoid imaging waves that are both propagating up or down, as these are
occasionally real reflections (Figure 2-1b). This situation also arises when RTM is
used. A solution often used in that case is to postprocess the image with a Laplacian filter. An alternative, proposed by Op ’t Root et al. (2012), is to determine
the direction of propagation in all spatial dimensions with finite differences, and suppress the images produced by waves traveling in the same direction (as such waves
obviously do not correspond to a reflection, Figure 2-1c). This is achieved with the
equation: 𝑖𝑚𝑎𝑔𝑒 =
1𝜕 𝑢 𝜕 𝑢
𝑐2 𝑡 𝑠 𝑡 𝑟
− (𝜕𝑥 𝑢𝑠 𝜕𝑥 𝑢𝑟 + 𝜕𝑦 𝑢𝑠 𝜕𝑦 𝑢𝑟 + 𝜕𝑧 𝑢𝑠 𝜕𝑧 𝑢𝑟 ). The effect of
the Laplacian and the new imaging condition is shown in Figure 2-2.
2.3
2.3.1
Results
Box model
The first comparison of the regular one-way method, enhanced one-way, and RTM
migration algorithms, is performed using the model shown in Figure 2-3, which will
be referred to as the Box model. The main feature of the model is a central box
with a velocity similar to that of a salt body. It is therefore meant to be a simplified
representation of the situation in areas such as the Gulf of Mexico, the North Sea,
and West Africa, where salt bodies are common. There is also a horizontal layer at
the bottom of the model. The purpose of this is to permit imaging of the box using
53
z (km)
0
5
0
x (km)
20
a
z (km)
0
5
0
x (km)
20
b
z (km)
0
5
0
x (km)
20
c
Figure 2-2: A demonstration of the effect of applying (a) the conventional imaging
condition (b) a Laplacian filter, and (c) the new imaging condition. The same percentage of clipping was applied to each image. Note that the artifacts, which obscure
the box when the conventional imaging condition is used, have been suppressed.
54
internal multiples. Such layers are common in the Earth. There is a vertical velocity
gradient of 0.6 m/(s m). This is higher than is typically found in the Earth, but is
designed so that all of the types of wave paths considered by the algorithm could be
0
z (km)
4
2
5
0
20
x (km)
Wave speed (km/s)
used to image the box.
Figure 2-3: Box velocity model.
Initially, the exact velocity model (including the box and horizontal layer) is used
during migration. The results are shown in Figure 2-4.
z (km)
1
z (km)
1
4
4
8
x (km)
12
8
x (km)
a
12
b
z (km)
1
4
8
x (km)
12
c
Figure 2-4: Image of the central box in the Box model when the exact velocity model
is used with different migration algorithms. (a) Regular one-way migration. (b) The
proposed enhanced one-way migration algorithm. (c) RTM.
In this situation the regular one-way method is able to clearly image the top and
bottom of the box. Since it is not possible for a downgoing source wave reflecting off
55
the vertical sides of the box to produce an upgoing wave, the regular one-way method
is not able to image the sides. When overturned waves and multiples are included,
however, these areas are easily accessible. This is demonstrated in the image from
the one-way enhanced method, which clearly images all four sides of the box. It is
slightly surprising that, although RTM images all of the box, as expected, the image
does not appear to be as clear as that produced by the one-way method. This may
be due to artifacts caused by the discontinuities in the velocity model.
In reality one is unlikely to have an exact velocity model available when migration
is performed. A more realistic test, therefore, is to only use the background velocity,
in this case the smooth gradient.
In the results, shown in Figure 2-5, the regular one-way method is now only
able to image the top of the box. This occurs because the waves will not be correctly
propagated through the area occupied by the box and so the images from the different
sources will not stack coherently. Even in RTM the bottom is not clearly resolved.
This is due to only source-receiver overturned waves, which are a very small percentage
of the total wavefield, being used to image this part of the structure. The other sides
of the box are, however, clearly imaged with this method. The image produced by oneway enhanced when all of the available wave paths are included, is shown in Figure
2-5c. The sides of the box are now clearly resolved, and the bottom is similar to that
in the RTM image, but there are significant artifacts. Since these artifacts are not
present when the exact velocity model is used, they must be due to waves not being
propagated correctly through the anomalous regions. They are only present in the
images produced when the imaging condition is applied to the pairs (𝑢𝑠,𝑡 , 𝑢𝑟,𝑡 ) and
(𝑢𝑠𝑚 , 𝑢𝑟,𝑚 ), so when these paths are excluded (Figure 2-5d), the artifacts disappear.
However, since these are the only paths by which the bottom of the box can be imaged,
this also disappears from the image.
Although only the arrivals corresponding to each wave path would ideally be used
during migration, robust techniques for performing this separation are not available.
The result is cross-talk between waves on different paths, which leads to artifacts.
Continuing developments in the field of multiples removal may make such separation
56
possible in the future. Until then, examining the images produced by the different
wave paths may provide an indication of which elements in an image are real.
z (km)
1
z (km)
1
4
4
8
x (km)
12
8
a
x (km)
12
b
z (km)
1
z (km)
1
4
4
8
x (km)
12
8
c
x (km)
12
d
Figure 2-5: Images of the central box in the Box model when only the background
velocity model is used during migration. (a) Regular one-way migration. (b) RTM.
Two images from the one-way enhanced algorithm are shown: (c) includes all of the
additional wave paths, while in (d) the (𝑢𝑠,𝑡 , 𝑢𝑟,𝑡 ) and (𝑢𝑠,𝑚 , 𝑢𝑟,𝑚 ) wave paths are
excluded.
2.3.2
BP salt model
The BP model is a synthetic velocity model that was produced to aid the testing
and comparison of algorithms (Billette and Brandsberg-Dahl, 2005). It is designed
to resemble situations encountered in certain parts of the world that are of interest
for oil and gas exploration. It contains salt bodies, layers of different velocities, and
several velocity anomalies. Only part of the model is used in this investigation (shown
in Figure 2-6). The goal of imaging in this example is primarily to image underneath
the salt overhang. This area is of particular interest as it has potential to trap
hydrocarbons.
57
0
z (km)
Wave speed (km/s)
4
2
11
0
x (km)
14
Figure 2-6: The portion of the BP velocity model used in this experiment. The white
arrow indicates the salt leg that is used as a multiples-generating interface. The salt
overhang, which is the imaging target, is identified with a box.
The exact velocity model is again used in the first test, with the results shown in
Figure 2-7. The regular one-way method images the top and legs of the salt, but the
bottom interface of the overhang is not visible. Even though the exact velocity model
is used, even RTM fails to image underneath the overhang, and the resulting image is
barely superior to that of the one-way method. Using overturned waves and multiples
generated from the legs of the salt (indicated by the white arrow in Figure 2-6), the
one-way enhanced method is able to quite accurately image the shape and location
of the area of interest, as shown in Figure 2-7b. Although this image contains many
artifacts, it results in a superior image of the overhang compared to that obtained
with RTM. The image produced by RTM must also contain the image of the overhang
from the same overturned waves and multiples, but the amplitude is too low to be
visible. The ability of the enhanced one-way method to scale the amplitude of images
from specific wave paths is the key to this successful result.
When only the background velocity model is used, which does not contain the salt
bodies, regular one-way and RTM (Figures 2-8a, 2-8c) do not image any reflectors in
the region of the overhang. There is no suggestion that this potential hydrocarbon
58
z (km)
3.2
z (km)
3.2
8
8
9
x (km)
14
9
x (km)
a
14
b
z (km)
3.2
8
9
x (km)
14
c
Figure 2-7: Image of the target area of the BP model when the exact velocity model
is used with different migration algorithms. (a) Regular one-way migration. (b)
Enhanced one-way migration, showing only the contributions from upgoing waves.
(c) RTM.
59
trap exists. When the one-way enhanced algorithm uses overturned waves and multiples from the leg in this situation, the image of the overhang underside (shown in
Figure 2-8b) is not clear, but it gives an indication of the location and shape of the
structure.
z (km)
3.2
z (km)
3.2
8
8
9
x (km)
14
9
x (km)
a
14
b
z (km)
3.2
8
9
x (km)
14
c
Figure 2-8: Image of the BP model when only the background velocity model is used
during migration. (a) Regular one-way migration. (b) Enhanced one-way migration.
(c) RTM.
60
2.4
Conclusion
An extended one-way migration algorithm is proposed. This method uses more of
the recorded data for imaging, allowing it to illuminate areas inaccessible with the
conventional implementation, such as near-vertical features and underneath salt bodies that are not included in the velocity model. The performance is examined by
comparing the results with those produced by regular one-way migration and RTM
on two synthetic models. In both cases the new algorithm is able to image important
areas of the subsurface more clearly than regular one-way. It also produces better
resolved images of certain features than even RTM, as it is possible to isolate wave
paths that illuminate these areas and scale their amplitude. It is therefore suggested
that this method could be used as a complement to RTM to provide additional information from the recorded data. It may also be used as an efficient means of
indicating whether recorded overturned waves and multiples image areas inaccessible
with primaries, and so aid in deciding whether more expensive methods for imaging
with these arrivals (such as those described in subsequent chapters) are worthwhile.
The results from the method are also likely to be improved by future developments
in techniques for isolating multiples. The presented results demonstrate that despite
its ability to automatically image with overturned waves and multiples, RTM does
not always fully exploit the information contained in these arrivals. Modifications to
rectify this are explored in subsequent chapters.
61
62
Chapter 3
Directional amplitude extraction
during time-domain wave
propagation
Abstract
Determining wave propagation direction is of critical importance in several seismic
imaging techniques and applications, including velocity analysis, AVA analysis, survey design, and illumination compensation. Current techniques, such as the Poynting
vector method, perform poorly when waves overlap, returning incorrect wave amplitude and direction at such points. We describe several new methods for separating
a wavefield by propagation direction in the time domain that can be implemented
efficiently on distributed memory computing resources. This makes it possible to determine the energy propagating in a given direction at a particular time. In contrast
to the Poynting vector method, the proposed methods are capable of separating the
wavefield even when there are overlapping waves propagating in different directions.
We evaluate the methods’ ability to separate overlapping waves in two constant velocity cases, to isolate the reflected wave in a layer over a halfspace model, and to
determine the propagation directions of the backpropagated data wavefield for one
source in a 2D slice of the SEAM model. We find that in the majority of cases the
proposed methods produce results which are superior to those of existing methods.
63
3.1
Introduction
Determining the propagation directions of waves is required in several applications,
such as constructing ADCIGs (Sava et al., 2001), which are used for velocity analysis
(Biondi and Symes, 2004) and extracting AVA information (Yan and Xie, 2012b), attenuating backscatter artifacts in RTM (Costa et al., 2009), and illumination analysis
(Yang et al., 2008). Ray tracing simulations of wave propagation naturally provide
propagation directions, but suffer from the inaccuracies resulting from the inherent
high frequency assumption (Gray et al., 2001). Methods such as finite difference
time-stepping, as used in the seismic imaging method RTM (Baysal et al., 1983),
allow closer adherence to the physics of finite-frequency wave propagation, but lack
a means of easily extracting propagation directions. In the next section we review a
selection of previously proposed methods for extracting directional information from
finite-frequency wave propagation schemes. These are principally the Poynting vector method (Yoon and Marfurt, 2006), which is computationally efficient, but makes
the assumption that the wavefield does not contain overlapping waves propagating in
different directions, and the local slowness method (Xie et al., 2005a), which is more
robust but still unreliable for overlapping waves. The image time method (Deng
and McMechan, 2007) is a very efficient means of determining propagation directions,
but is severely limited by its restriction to a single direction at each cell. Jin et al.
(2014) also compares methods for constructing ADCIGs, including the Poynting vector method, but some such methods are only capable of extracting scattering angle,
rather than propagation direction. Following the discussion of current methods, we
give a description of new methods for determining propagation direction. Such methods are needed because the assumption that there are no overlapping waves, required
for reliable operation of existing methods, is rarely true in reality. For example, this
assumption is likely to be violated when propagating a wave in a non-smooth velocity
model. Finally, we examine the effectiveness of the new methods compared to the
Poynting vector and local slowness methods.
64
3.2
3.2.1
Previously proposed methods
Poynting vectors
The Poynting vector method of determining wave propagation direction was proposed
by Yoon and Marfurt (2006) as a means of determining apparent scattering angle. It
is not limited to calculating scattering angle, and so may also be used in applications
where the propagation directions of the source and data wavefields must be known
independently, such as in illumination compensation (Yang et al., 2008).
The Poynting vector method calculates the propagation direction 𝜓 ̂ at a point 𝐱
and time 𝑡 of wavefield 𝑢 using
̂ 𝑡) = − 𝜕𝑢(𝐱, 𝑡) ∇𝑢(𝐱, 𝑡),
𝜓(𝐱,
𝜕𝑡
(3.1)
As the method assumes that there are no overlapping waves, the amplitude of the
wave propagating in the direction 𝜓 ̂ at 𝐱 is just that of the wavefield at that point.
There are several limitations of the Poynting vector approach, as discussed in Sava
and Patrikeeva (2013). Proposals have been made to reduce the effect of the method’s
weaknesses. As overlapping waves are particularly likely to occur when backpropagating the data wavefield in RTM, Zhao et al. (2012) suggest that for the purpose
of calculating scattering angle, the method only be applied to the source wavefield,
and that a migrated image then be used to estimate the scattering angle. Others,
such as Ross and Yan (2013), have proposed means of improving the robustness of
the method’s output, but are still limited to assigning a single propagation direction
to each point.
3.2.2
Local slowness
An alternative approach, called the local slowness method, was proposed by Xie
et al. (2005a). This method was initially developed to analyze near-source energy
partitioning, but it has also been applied to determining propagation directions for
illumination compensation (Xie and Yang, 2008) and constructing ADCIGs (Yan and
65
Xie, 2012a). The method sums along local slowness directions in spacetime,
𝑢𝑠 (𝐱, 𝐩, 𝑡) =
1
∑ 𝑊 (𝐱′ − 𝐱)𝑢(𝐱′ , 𝑡 − 𝐩 ⋅ (𝐱′ − 𝐱)),
𝐼𝐱 𝐱′
(3.2)
where 𝑢𝑠 is the wavefield containing only waves propagating in the direction 𝜓,̂ 𝑊 is
a space window of length 𝐼𝐱 centered on 𝐱, and 𝐩 =
𝜓̂
𝑐
is the slowness (reciprocal of
the propagation velocity). This is depicted in Figure 3-1. Summing along the local
slowness direction, as shown in the figure (line A), will sum the same part of the
waveform at each time step, while summing in other directions will sample a different
part at each time step, leading to cancellation due to the oscillatory nature of waves
and therefore a small amplitude. In contrast to the Poynting vector method, this
approach is capable, under certain conditions, of separating a wavefield even when
it contains overlapping waves propagating in different directions. The separation of
overlapping waves is exact for plane waves in a constant velocity medium if the window
𝑊 is sufficiently large, but when these assumptions aren’t satisfied overlapping waves
can still cause errors in the separation.
3.2.3
Frequency domain methods
A further step toward accurately decomposing a wavefield into different propagation
directions, even when there are overlapping waves, is made by working in the frequency domain.
If we assume that the waves passing through the point 𝐱 can be approximated by
plane waves in a region of space and time around (𝐱, 𝑡), then
𝑁(𝐱,𝑡)
𝑢(𝐱, 𝑡) = ∑ ∑ 𝐴(𝑏)𝑒𝑖(𝐤(𝑎)⋅𝐱−𝜔(𝑏)𝑡+𝜙(𝑏)) ,
(3.3)
𝑎=1 𝑏(𝑎)
where 𝑁 (𝐱, 𝑡) is the number of different directions in which waves are propagating at
(𝐱, 𝑡), 𝐴 is the wave amplitude, 𝐤 is the spatial wavevector, 𝜔 is the frequency, 𝑏 is
the index of frequency components, and 𝜙 is a phase shift.
66
Figure 3-1: A wave propagating in the direction 𝜓 ̂ and centered at the origin at time
step 𝑡 will travel along the path A. Summing along A and dividing by the summation
length, to apply the local slowness method, will therefore yield the value of the wave
at its central peak. The circles represent the top and bottom edges of the light cone
that the wave can travel along. The dashed lines joining the two circles indicate the
shape of the lightcone. Summing along any other line on this cone other than A will
yield zero, as long as the summation time is sufficiently long.
67
A simple method of extracting waves propagating in the direction 𝜓 ̂ can then
be implemented by isolating components of the Fourier transformed volume with
spatial wavevector direction close to 𝜓,̂ summing over negative 𝜔, and applying the
inverse Fourier transform. To separate the wavefield into 𝑁 propagation directions,
it is necessary to isolate the relevant portion of Fourier space and apply the inverse
Fourier transform for each direction separately.
Several other frequency domain methods exist, such as frequency domain local
slant stack and split step, which have been used for illumination compensation (Cao
and Wu, 2009), local exponential basis, another method used for illumination compensation (Mao et al., 2010), and a proposal by Benamou et al. (2004, 2012). While
these methods are suitable for applications such as illumination analysis, their lack of
temporal dependence renders them less appropriate for use in other techniques that
require separation at individual time steps, such as computing ADCIGs for velocity
analysis.
3.2.4
Windowed Fourier transform
In order to obtain the benefit of the frequency domain (the ability to distinguish
between overlapping waves), while still retaining temporal specificity, we may use
a technique referred to as either the windowed Fourier transform or the short-time
Fourier transform. Rather than transforming the wavefield for all time, as in other
frequency domain methods, this approach only transforms a window of time around
the time slice that we wish to separate into propagation directions. An implementation of the short-time Fourier transform is described by Garossino and Vassiliou
(1998), who propose to use it for noise attenuation.
While it renders frequency domain methods more useful for time domain applications, this method has a high memory requirement as the entire wavefield must
be stored for several time slices around the target time slice, including a taper in
positive and negative time to reduce artifacts. Larger tapers can be more effective,
but reduce temporal resolution, so some artifacts may be unavoidable to obtain a
sufficiently short time window. Furthermore, it is also computationally expensive, as
68
it requires one multidimensional short-time Fourier transform, and 𝑁 inverse Fourier
transforms per time slice, where 𝑁 is the number of propagation directions that we
wish to decompose the wavefield into. If the method is implemented on distributed
memory computing resources, as is often required for processing seismic surveys, and
the wavefield is split spatially over nodes so that each node contains all of the necessary time slices (including tapers) for a spatial block of the wavefield, the spatial
Fourier transforms can either be performed on the entire spatial domain, or individually on each node’s block of the wavefield. As multidimensional Fourier transforms are
poorly suited to distributed memory systems, the former may be prohibitively expensive. This problem is alleviated by transforming each node’s portion of the wavefield
independently, but such a strategy introduces the additional complications of having
to add a taper and padding to the spatial boundaries of each node’s wavefield in order
to reduce artifacts. In an effort to improve performance, Xu et al. (2011) propose
using the antileakage Fourier transform (ALFT), rather than the usual fast Fourier
transform (FFT), although this obviously makes implementation less straightforward.
3.3
New methods
In this section we propose methods which are capable of separating a wavefield into
components propagating in a given set of directions, and which may be implemented
efficiently on distributed memory computing resources.
We wish to determine the propagation directions and amplitudes of all waves
passing through the point 𝐱 at time 𝑡. We will say that there are 𝑁 (𝐱, 𝑡) such
waves. In the case when max(𝑁 ) ≤ 1, the Poynting vector method works well and is
computationally efficient, however it fails when 𝑁 > 1.
We propose three classes of methods to overcome this limitation. We describe
them in the 2D case for simplicity, however they may be extended to 3D without
difficulty. We also assume that the wave propagation occurs in an isotropic, nonattenuating medium. A simple extension to anisotropic materials is possible by simply
using the anisotropic parameters to estimate the correct angle between the propaga69
tion direction and the wavefront; an understanding of how effective this approach
would be in anisotropic media requires further research.
The first category separates the wavefield by wavefront orientation, so that the
separated components contain fewer overlapping waves than the full wavefield, ideally
none. The Poynting vector method is then applied to each of these components
separately. The second method modifies the local slowness method by separating each
time slice by wavefront orientation before performing the local slowness summation
over time slices, to enhance angular resolution. The third method formulates the
separation as an optimization problem.
3.3.1
Method 1: Plane wave decomposition followed by the
Poynting vector method
In this method we separate the wavefield into waves with different wavefront orientations, and then use Poynting vectors to determine the propagation directions.
A wavefront has orientation 𝜓,̂ when its gradient points in that direction. In
isotropic media, waves travel parallel to their wavefront orientation. A wavefront
with orientation 𝜓 ̂ must therefore belong to a wave propagating in the direction 𝜓 ̂
or −𝜓.̂ The wave amplitude is locally constant perpendicular to 𝜓,̂ and oscillatory
parallel to it. In 2D we may refer to a wavefront of orientation 𝜓 ̂ or −𝜓 ̂ as having
orientation angle 𝜓 ∈ [0, 𝜋), where 𝜓 is the angle from the positive 𝑥 axis to whichever
of 𝜓 ̂ or −𝜓 ̂ lies in the positive 𝑧 domain.
By separating the wavefield by orientation angle 𝜓, we hope that
𝑚𝑎𝑥(𝑁 ′ (𝐱, 𝜓, 𝑡)) ≤ 1,
(3.4)
where 𝑁 ′ is the number of waves passing through the point 𝐱 at time 𝑡 that have a
wavefront at point 𝐱 oriented with angle 𝜓. If this condition is satisfied, then we may
successfully apply the Poynting vector method for each direction 𝜓 ̂ to determine the
propagation amplitude in that direction. As wavefront orientation angle separation
will not separate two overlapping waves propagating in opposite directions 𝜓 ̂ and −𝜓,̂
70
since both have the same wavefront orientation angle, the condition (3.4) can never
be satisfied in this case. This method is therefore incapable of separating overlapping
waves propagating in opposite directions.
The separation into wavefront orientation angles can be accomplished by several
means. We describe three: time domain local slant stacks (LSS), the Fourier transform, and curvelets.
Local slant stack
LSS uses the fact that waves are oscillatory perpendicular to the wavefront and approximately constant along it. Summing along a wavefront in space will yield a
non-zero value. Any direction not parallel to the wavefront should sum to zero due to
the oscillatory property, if the summation length is sufficiently long. This is shown
in Figure 3-2. The shortest time over which the wave is oscillatory depends on the
source wavelet. It may be the duration of the wavelet (or half the duration if it is
symmetric, as is the case for Ricker wavelets), or the period if the wave is periodic.
Even if the wave is not periodic, if it has a single dominant frequency, the wave may
be close to oscillatory over the corresponding period. If the time period over which the
pulse is oscillatory (or almost oscillatory) is 𝑇 , then the corresponding spatial length
is 𝑐(𝐱)𝑇 , where 𝑐 is the wave speed, which we assume does not vary significantly
over this distance. To make use of this property, we therefore need to sum along a
∶
wavefront over the distance [− 𝑐(𝐱)𝑇
2
𝑐(𝐱)𝑇
2 ]
around the point 𝐱 in order to prevent
the calculated amplitude along a wavefront from being affected by a perpendicular
wavefront also centered on 𝐱. This is depicted in Figure 3-3. To avoid interference
between wavefronts not perpendicular, or not centered on 𝐱, we would need to sum
over a larger distance. Our assumptions about the planar nature of the wavefront and
the locally constant velocity are less likely to be valid at larger distances, however.
We therefore suggest ideally using the shortest summation length which will allow the
amplitude along a wavefront to be determined without interference from wavefronts
centered at 𝐱 and oriented in any of the other directions to be considered. Decomposing the wavefield into 𝑁𝑠 (𝑡) equally spaced wavefront orientation angles (which
71
will allow us to separate into 2𝑁𝑠 (𝑡) propagation directions later) therefore requires
a summation length of
𝐼𝐱 =
𝑐(𝐱)𝑇
,
sin (Δ𝜓)
(3.5)
𝜋
.
𝑁𝑠 (𝑡)
(3.6)
where
Δ𝜓 =
This is depicted in Figure 3-4.
Figure 3-2: A wave with wavefront orientation angle 𝜓 is oscillatory in the direction 𝜓 ̂
and constant in the direction 𝜓⟂̂ . To perform wavefront orientation angle separation
at the origin point in the figure, we compute the average amplitude along lines passing
through the origin. Summing along the line B and dividing by the summation length
will produce the peak value of the wave, while summing along the perpendicular line
A will result in zero.
1
վյ
z
B
x
2
A
վյ
Figure 3-3: Waves 1 and 2 have perpendicular wavefront orientations. Both are
oscillatory over the distance 𝑐𝑇 , where 𝑐 is the local wave speed. Summing along
A will produce the value of wave 1 along that line with no interference from wave 2.
Summing along line B will result in zero.
The angular resolution (Δ𝜓) obtainable with this method is approximately inversely proportional to 𝐼𝐱 when Δ𝜓 is small. The maximum possible length 𝐼𝐱 is
determined by the smoothness of the model (in smooth models the length over which
72
x
1
ဳᇐ
ժؐ
2
վյ
z
Figure 3-4: When the difference between the wavefront orientation angles of waves 1
and 2 is Δ𝜓, it is necessary to sum at least a distance 𝐼𝐱 along wave 2 in order to
cancel contributions from wave 1, where 𝐼𝐱 is given by Equation 3.5.
the approximations of the method are valid will be longer, and so a larger 𝐼𝐱 can be
used), so resolution is inversely proportional to model smoothness (a smoother model
allows the separation of waves propagating in more closely spaced directions). Resolution is approximately proportional to 𝑐(𝐱)𝑇 , the local wave speed and the shortest
oscillatory time of the waves.
To separate the wavefield into waves with different wavefront orientation angles
with LSS, we sum along different orientation angles at each point 𝐱, and divide by
the length of the sum to obtain the amplitude of the waves:
𝐼𝐱
2
𝑢𝑜 (𝐱, 𝜓, 𝑡) = ∑
𝑠=− 𝐼2𝐱
𝑢(𝐱 + 𝑠𝜓⟂̂ , 𝑡)
,
𝐼𝐱
(3.7)
where 𝜓⟂̂ is the direction along a wavefront oriented with angle 𝜓 (i.e., 𝜓⟂̂ =
(− sin(𝜓)𝑥,̂ cos(𝜓)𝑧)),
̂ 𝑢 is the full wavefield, 𝑢𝑜 is the scalar field containing the
amplitude of waves with wavefront orientation angle 𝜓 at position 𝐱 and time 𝑡, and
𝐼𝐱 is the summation length. If 𝑢 is not defined at spatial locations requested by this
summation, interpolation may be used.
Fourier transform
The wavefield can alternatively be separated by wavefront orientation angle using
the Fourier transform, in a manner that has some similarities with the frequency
73
domain propagation direction separation methods described above. A time slice of
the wavefield is Fourier transformed in space, yielding
𝑈 (𝐤, 𝑡) = ∫ d𝐱 𝑢(𝐱, 𝑡)𝑒−𝑖𝐤⋅𝐱 ,
(3.8)
ℝ𝑛
where 𝑛 is the number of spatial dimensions.
Regions of this Fourier transformed wavefield corresponding to different orientation angles are then selected and inverse Fourier transformed:
𝑢𝑜 (𝐱, 𝜓, 𝑡) =
1
∫ d𝐤 𝑈 (𝐤, 𝑡)𝑓(𝐤, 𝜓)𝑒𝑖𝐤⋅𝐱 ,
(2𝜋)𝑛 ℝ𝑛
(3.9)
where 𝑓(𝐤, 𝜓) is a filter that selects the regions where
∠𝐤
(mod 𝜋) ∈ [∠(𝜓)̂ −
𝜋
𝜋
, ∠(𝜓)̂ +
),
2𝑁𝑠 (𝑡)
2𝑁𝑠 (𝑡)
(3.10)
with 𝑁𝑠 (𝑡) being the number of wavefront orientation angles that we separate. Tapering will be required to reduce artifacts.
Achievable angular resolution depends on the frequency of the waves and the
length over which they are planar. This is because it is only possible to fully separate
waves when their wave vectors do not overlap, which will be the case when
Δ𝜙 > 2 arctan (
2
)
𝐿|𝑘|
(3.11)
where 𝐿 is the distance over which the wavefronts are planar, and we have approximated the width of the response to a finite length wavefront in frequency space as the
width of the central lobe of the sinc function resulting from the transform of a box
of width 𝐿. This demonstrates that the minimum separable angular difference (Δ𝜃)
decreases for higher frequency waves (larger |𝑘|), and when waves are planar over a
larger distance (greater 𝐿).
74
Curvelets
Curvelets (Candès et al., 2006) are one of a number of multiresolution analysis techniques that can be used to decompose a scalar field into location, spatial frequency,
and direction. Other techniques with similar objectives include wave atoms (Demanet
and Ying, 2007; Andersson et al., 2012), and contourlets (Do and Vetterli, 2005).
Application of the curvelet transform to a time slice of the wavefield produces
coefficients 𝑐(𝑗, 𝑙, 𝑘), where 𝑗, 𝑙, and 𝑘 denote the spatial frequency, angle, and location
indices, respectively. The spacing between decomposed angles decreases for features
of higher spatial frequency,
𝜃𝑙 = 2𝜋 ⋅ 2−𝑗/2 ⋅ 𝑙.
(3.12)
This implies that attainable angular resolution is again inversely proportional to
the wave frequency.
To extract the waves with wavefront orientation angle 𝜓, we apply the curvelet
transform, and isolate the coefficients corresponding to the range of angles
𝜃𝑙 (mod 𝜋) ∈ [𝜓 −
𝜋
𝜋
,𝜓 +
).
2𝑁𝑠 (𝑡)
2𝑁𝑠 (𝑡)
(3.13)
The inverse curvelet transform is applied to these coefficients, yielding the desired
result.
Poynting vector application
Performing this separation on a sufficient number of time steps to calculate a time
derivative (at least two), we calculate the Poynting vectors using:
𝜕𝑢 (𝐱, 𝜓, 𝑡) 𝜕𝑢𝑜 (𝐱, 𝜓, 𝑡)
,
𝑃 (𝐱, 𝜓,̂ 𝑡) = − 𝑜
𝜕𝑡
𝜕𝜓̂
where
𝜕
𝜕 𝜓̂
(3.14)
is the partial derivative in the direction 𝜓,̂ and 𝑢𝑜 is the wavefield of waves
with wavefront orientation angle 𝜓. This tells us which of two possible directions the
wave is propagating in (𝜓 ̂ or −𝜓,̂ as waves in isotropic media propagate parallel to
75
their wavefront orientation). If 𝑃 is positive, it indicates that the wave is propagating
in the positive 𝜓 ̂ direction, while it is traveling in the opposite direction if it has a
negative value. This is demonstrated in Figure 3-5.
Amplitude
𝑡
𝜕𝑢
𝜕𝜓̂
𝑡 + ∆𝑡
>0
𝜓̂
𝜕𝑢
𝜕𝑡
<0
Figure 3-5: A wave 𝑢 propagating in the positive 𝜓 ̂ direction (to the right) is shown
at time 𝑡 and 𝑡 + Δ𝑡. At points where the spatial derivative in the direction 𝜓 ̂ is
positive, the time derivative is negative, while the time derivative is positive at points
with a negative spatial derivative, so 𝑃 in Equation 3.14 is positive.
We therefore obtain the wavefield of waves with propagation direction 𝜓 ̂
𝑢𝑠 (𝐱, 𝜓,̂ 𝑡) =
⎧
{𝑢𝑜 (𝐱, 𝜓, 𝑡),
⎨
{0,
⎩
if 𝑃 (𝐱, 𝜓,̂ 𝑡) > 0.
(3.15)
otherwise.
Note that although the wavefront orientation angle 𝜓 ∈ [0, 𝜋), the wave propagation direction unit vector 𝜓 ̂ covers the full circle because each 𝜓 can produce a wave
propagating in 𝜓 ̂ or −𝜓.̂
The number of wavefront orientation angles into which the wavefield is separated
(𝑁𝑠 (𝐱, 𝑡)) must be at least 𝑁 (𝐱, 𝑡) for 𝑁 ′ (𝐱, 𝜓, 𝑡) ≤ 1 to be possible (as required
for the Poynting vector calculation to succeed). Without knowing the wavefront
orientation angles a priori, we choose equally separated orientation angles
𝜓∈{
(𝑘 − 12 )𝜋
∶ 𝑘 ∈ 1, … , 𝑁𝑠 (𝐱, 𝑡)} .
𝑁𝑠 (𝐱, 𝑡)
(3.16)
This indicates that we can only separate waves if their wavefront orientation angles
76
(or, equivalently, propagation directions modulo 𝜋) are separated by at least
Δ𝜓 =
𝜋
radians.
𝑁𝑠 (𝐱, 𝑡)
(3.17)
To ensure that waves are separated, 𝑁𝑠 (𝐱, 𝑡) must therefore be chosen so that Δ𝜓 is
smaller than the smallest difference in wavefront orientation angles between all waves
at (𝐱, 𝑡).
Inaccuracies in the wavefield separation can result in the computed amplitudes
being incorrect, and the detection of waves where none really exist. To reduce the
amplitude of incorrect results, we apply two filters. These reduce the amplitude
of waves that appear to be traveling with the incorrect speed, or in the incorrect
direction. To apply the first, we calculate the apparent propagation speed using
𝜕𝑢 (𝐱, 𝜓, 𝑡) 𝜕𝑢𝑜 (𝐱, 𝜓, 𝑡)
/
∣.
𝑐𝑎 (𝐱, 𝜓,̂ 𝑡) = ∣ 𝑜
𝜕𝑡
𝜕𝜓̂
(3.18)
We use the wavefields separated by wavefront orientation angle, 𝑢𝑜 , rather than
the wavefields separated by propagation direction, 𝑢𝑠 , (and, as a result, take the
absolute value) as we are not able to calculate the time derivative of 𝑢𝑠 unless the
directional separation is performed sufficiently frequently for this to be accurate.
We know that this should be 𝑐(𝐱), but errors in the wavefront separation can
result in anomalously low or high values. We will therefore penalize departures from
the expected value using
𝑓𝑖𝑙𝑡𝑐 (𝐱, 𝜓,̂ 𝑡) = 1 − min(|𝑐(𝐱, 𝑡) − 𝑐𝑎 (𝐱, 𝜓,̂ 𝑡)|/𝑚𝑎𝑥𝑒𝑟𝑟, 1),
(3.19)
where 𝑚𝑎𝑥𝑒𝑟𝑟 is the maximum permissible error in 𝑐, for example 1000 m/s. Due to
the absolute value in Equation 3.18, 𝑓𝑖𝑙𝑡𝑐 (𝐱, 𝜓,̂ 𝑡) = 𝑓𝑖𝑙𝑡𝑐 (𝐱, −𝜓,̂ 𝑡). The calculated
propagation speed will be incorrect near the peaks and troughs of the wave, as the
spatial derivative at these locations will be close to zero, making the result unstable.
We therefore smooth the calculated speed, weighted by the absolute value of the
spatial derivative of 𝑢𝑠 (𝐱, 𝜓, 𝑡).
77
To compute the propagation direction, for use in the second filter, we use Poynting
vectors:
𝑃𝑥̂ (𝐱, 𝜓, 𝑡) = −
𝜕𝑢𝑜 (𝐱, 𝜓, 𝑡) 𝜕𝑢𝑜 (𝐱, 𝜓, 𝑡)
,
𝜕 𝑥̂
𝜕𝑡
(3.20)
𝑃𝑧 (𝐱,
𝜓, 𝑡) = −
̂
𝜕𝑢𝑜 (𝐱, 𝜓, 𝑡) 𝜕𝑢𝑜 (𝐱, 𝜓, 𝑡)
,
𝜕𝑧 ̂
𝜕𝑡
(3.21)
tan(𝑃 𝑟𝑜𝑝𝐷𝑖𝑟(𝐱, 𝜓, 𝑡)) =
𝑃𝑧 (𝐱,
𝜓, 𝑡)
̂
.
𝑃𝑥̂ (𝐱, 𝜓, 𝑡)
(3.22)
Here, 𝑃 𝑟𝑜𝑝𝐷𝑖𝑟, the calculated propagation direction, should be equal to ±𝜓 ̂ if the
medium is isotropic (the wave should propagate perpendicular to its wavefront orientation). We penalize departures from this using:
𝑑
̂
𝑓𝑖𝑙𝑡𝑎𝑛𝑔 (𝐱, 𝜓,̂ 𝑡) = (1 − arccos(|𝑃𝑇 (𝐱, 𝜓, 𝑡) ⋅ 𝜓|)/𝜋)
,
(3.23)
where 𝑃𝑇 (𝐱, 𝜓, 𝑡) = 𝑃𝑥̂ (𝐱, 𝜓, 𝑡) + 𝑃𝑧 (𝐱,
𝜓, 𝑡), and 𝑑 is a parameter to adjust how
̂
severely errors are treated. This expression therefore computes the angular distance
between the calculated propagation direction and the assigned propagation direction,
derived from the wavefront orientation. If the distance is zero, the filter has value 1.
If the propagation direction is the opposite to the assigned direction, the filter has
value 0.
Multiplying 𝑢𝑠 (𝐱, 𝜓,̂ 𝑡) by these two filters reduces the amplitude of possible artifacts. The filters are based on the premise that regions where the amplitudes do not
behave as expected are considered to be artifacts and so the amplitude there should
be reduced. This means that in addition to attenuating real artifacts, applying the
filters can also worsen errors in the separated amplitudes. For example, due to incomplete separation the amplitude of a separated wavefield at a point could be lower
than it should be. As the amplitude will probably not behave as it is expected to
because of this separation error, applying the filters will further reduce the amplitude
at the point. Using the filters does ensure, however, that there are no artifacts with
large amplitudes in the result.
To obtain accurate amplitudes it is furthermore necessary that the chosen sep78
aration angles be close to the true wavefront orientation angles as the amplitudes
may fall off rapidly away from the correct directions. This is especially true when
parameters for the two filters are used which severely penalize errors. If only a small
number of angles are used it is therefore advisable to not use strong filters.
It may be possible to obtain more accurate propagation direction and amplitude
estimates by interpolating the results to a finer angle grid and selecting the values
at the peaks of absolute amplitude to reduce the effects of spreading. Only using
peak values is particularly appropriate when LSS is used to perform the wavefront
orientation angle separation, as in this case the sum over the separated wavefields
will match the full wavefield if the sum is over the amplitudes at the true orientation
angles and all other angles are excluded. This is because summing along a wavefront
and dividing by the summation length will produce the true amplitude of the wave
at that location, assuming that the amplitude is constant along the wavefront and
the summation length is sufficient to cancel contributions from other crossing wavefronts. Summing along an angle slightly away from parallel to the wavefront will not
return zero (unless the summation length is very long). Appropriate selection of filter
parameters can ensure that the amplitude away from the correct propagation angles
falls off sufficiently quickly that the true angles form peaks.
As an example of this, we consider the point on a wavefield, indicated by the
arrow in Figure 3-6a. The wave was produced by a 𝜋/3 phase shifted 20 Hz Ricker
wavelet source. Performing LSS at this point with a summation length of 0.085 s, and
filter parameters 𝑑 = 2 and 𝑚𝑎𝑥𝑒𝑟𝑟𝑜𝑟 = 2000, yields the amplitude versus angle plot
shown in Figure 3-6b. The wave is propagating in the direction π rad (180°) from
the reference angle, yet we can see that the summation length was not sufficient for
this angle to be a peak of the absolute value of the amplitude as a function of angle.
Figure 3-6c shows the effect on the location of the maximum peak of the absolute
amplitude as the parameter for the propagation angle filter (𝑑 in Equation 3.19) and
the parameter for the apparent wave speed filter (𝑚𝑎𝑥𝑒𝑟𝑟𝑜𝑟 in Equation 3.18) are
adjusted. It is apparent that using a large angle filter parameter allows the correct
angle to be the maximum peak at less stringent wave speed filter parameter values.
79
This is useful, as in Figure 3-6d we see that strong wave speed filters (only allowing
waves with a small error in apparent wave speed) can reduce amplitude accuracy.
The inaccuracies in the calculation of apparent wave speed, such as the omission of
spreading due to the use of the one-way wave equation, mean that we cannot expect
a perfect match with the actual wave speed. Using filter values of 100 and 1000 for
the angle and wave speed filter, respectively, produces the result in Figure 3-6e, which
has a peak at the correct angle, and an amplitude at the peak within 14 % of the
true value. All figures presented in the results section use parameter values of 𝑑 = 3
and 𝑚𝑎𝑥𝑒𝑟𝑟𝑜𝑟 = 2000. It is possible that further improvements to the results could
be obtained through the use of stronger filters.
3.3.2
Method 2: Separated light cone stack
A second method for separating a wavefield by wave propagation direction replaces
the Poynting vector step in method 1 with a slant stack over spacetime. It can also
be thought of as a modification of the local slowness method, where an additional
step separates time slices by wavefront orientation angle before the slant stack over
spacetime is applied, resulting in improved angular resolution.
The motivation for developing this method can be seen in Figure 3-7. Method 1
has better resolution than the local slowness method for small differences in propagation direction, but its inability to distinguish between waves propagating in opposite
directions means that it has poor resolution for large differences in propagation angle, the regime in which the resolution of the local slowness method is highest. By
combining elements of both methods we derive the benefits of method 1’s small angle
resolution while also retaining the local slowness method’s good resolution at high
angles.
If a wave passing through the point 𝐱 at time 𝑡, propagating in the direction 𝜓,̂
can be approximated by a plane wave, and the local wave speed is approximately
constant, the wave travels along the light cone path in spacetime,
′
̂
𝑤(𝑡′ , 𝐱, 𝜓, 𝑡) = 𝐱 + 𝜓𝑐(𝐱)(𝑡
− 𝑡).
80
(3.24)
0.016
Amplitude
Amplitude
0e+00
-0.013
-6e-03
z (km)
0.6
0
2π
b
Wave speed filter parameter
11
2000
Peak angle error (°)
Wave speed filter parameter
a
Angle (rad)
1000
200
0
0.36
2000
Relative amplitude error
0.4
1000
200
0.1
1 20 40 60 80 100
1 20 40 60 80 100
Angle filter parameter
Angle filter parameter
c
d
Amplitude
0e+00
-6e-03
0
Angle (rad)
2π
e
Figure 3-6: (a) The point on a wave propagating in the direction π rad which is used
to investigate the effect of filter parameters in method 1. (b) With filter parameters
𝑑 = 2 and 𝑚𝑎𝑥𝑒𝑟𝑟𝑜𝑟 = 2000, the true propagation direction (π rad) is not a peak
of absolute amplitude. (c) The location of the maximum peak in absolute amplitude
versus angle varies with the choice of parameters for the method’s two filters. (d)
As in (c), but for relative amplitude error in the wave amplitude assigned to the
true direction of propagation. (e) As in (b) but with filter parameters 𝑑 = 100 and
𝑚𝑎𝑥𝑒𝑟𝑟𝑜𝑟 = 1000. The peak occurs at the angle of the true propagation direction.
81
Minimum integration length (multiple of period)
6
Local slowness
Method 1
4
2
0
0
Propagation angle difference (rad)
π
Figure 3-7: To separate waves propagating in directions differing by less than 𝜋/2,
method 1 with LSS requires a shorter summation length than the local slowness
method (measured in the plot as a multiple of the time over which the waves are
oscillatory, 𝑇 , for the local slowness method, or 𝑐(𝐱)𝑇 for method 1). For larger
differences in propagation direction, the local slowness method has better resolution
for a given summation length. The result for method 1 does not include the effect of
the filters that can be applied when using that approach. This plot is derived from
equations in Appendix A.
82
Furthermore, wavefronts along this path that are propagating in the direction 𝜓 ̂
should have a wavefront orientation angle of 𝜓.
To determine the amplitude of this wave, we extract the waves with wavefront
orientation angle 𝜓 using a method such as one of the three described above, producing
𝑢𝑜 (𝐱, 𝜓, 𝑡) in time slices covering a period 𝐼𝑡 , sum along the path in Equation 3.24,
and divide by the number of time slices:
𝐼𝑡
2
𝑢𝑠 (𝐱, 𝜓,̂ 𝑡) = ∑
𝑡′ =− 𝐼2𝑡
𝑢𝑜 (𝑤(𝑡′ , 𝐱, 𝜓, 𝑡), 𝜓, 𝑡′ )
.
𝐼𝑡
(3.25)
As in method 1, 𝐼𝑡 is determined by the competing demands of being large enough
to provide high angular resolution, while short enough for the approximations, such
as locally constant wave speed, to remain plausible.
The summation is centered on time step 𝑡 rather than summing from 𝑡 − 𝐼𝑡 to 𝑡
so that the spatial distance from 𝐱 is minimized, reducing the likelihood of changes
in wave speed along the light cone from 𝑐(𝐱).
An advantage of this approach over method 1 is that it is able to distinguish
between overlapping waves propagating in opposite directions (which is not possible
in method 1 due to the violation of Equation 3.4). The memory requirement may
be higher, however, as it is necessary to store the entire wavefield for more than the
minimum of two time steps needed for method 1.
The local slowness method, previously proposed by Xie et al. (2005a), is similar, but does not separate the wavefield time slices by wavefront orientation angle.
The inclusion of this step increases the computational cost of the method, but it improves the ability of the method to distinguish between waves with small differences
in propagation direction. This is explored in Appendix A, where we show, for example, that separating waves with propagation directions differing by π/6 rad requires
√
√
𝐼𝑡 = (1 + 3)𝑇 ≈ 2.7𝑇 with method 2. This increases to 𝐼𝑡 = (4 + 2 3)𝑇 ≈ 7.5𝑇
when using the local slowness method. This means that for the local slowness method
to have sufficient resolution to separate such waves, the assumptions of the method
√
must hold over the distance (4 + 2 3)𝑐(𝐱)𝑇 in the direction of propagation, centered
83
on the point to be separated. The new method, on the other hand, requires that the
√
assumptions hold over a distance (1 + 3)𝑐(𝐱)𝑇 in the direction of propagation and
√
a distance (1 + 3)𝑐(𝐱)𝑇 perpendicular to the direction of propagation, a shorter distance from the point to be separated. If the waves are periodic (instead of oscillatory
wave packets), then the minimum summation distance is further reduced to 2𝑐(𝐱)𝑇 .
Although the addition of wavefront orientation angle separation improves the
angular resolution of the method, it does not provide all of the benefits of the filters
used in method 1. In the previous section we demonstrated that appropriately chosen
filter parameters could be used with method 1 to increase the probability that peaks
of amplitude versus angle occur at the true angles, allowing us to select only these
peaks as the decomposed wavefield. Without this benefit, we are less confident that
peaks occur at the correct angles. It is, however, likely that in general the peaks will
occur close to the true angles and thus have approximately the correct amplitude,
so in applications where it is important that the sum over angles of the decomposed
wavefield be close to the full wavefield, it may still be advisable to only use peak values.
The method could be extended to include filters similar to those used in method 1,
but this would further increase the computational cost of the method, which, as we
show in the performance section below, is already significant.
3.3.3
Method 3: Optimization
Summation forms a key component of the two methods already presented, as both
rely on the oscillatory nature of waves to restrict the result to plausible solutions.
The third method takes a different approach, instead performing the separation of
the wavefield by propagation direction through the use of an optimization algorithm.
This has two key advantages: allowing us to find a solution that has several desirable
properties, and giving us the ability to improve performance through the provision of
an initial guess of the solution. A further benefit of this approach is that it poses the
problem in a form that enables algorithms developed for optimization to be leveraged.
For a description of popular optimization algorithms, see Nocedal and Wright (2006).
Optimization problems seek a set of model parameters which minimize a measure,
84
known as the objective functional, of the distance between the resulting model and an
ideal solution. There are several conceivable objective functionals for the directional
separation problem. We choose one which favors solutions with plausible amplitudes.
Other characteristics of seismic waves, such as frequency band limits, may enhance
the results if included, but are not considered here.
We begin by defining two functions to make later expressions more compact,
𝐴(𝑢𝑠 , 𝑢, 𝐱, 𝑡) = ∫ d𝜓′ (𝑢𝑠 (𝐱, 𝜓′̂ , 𝑡)) − 𝑢(𝐱, 𝑡),
(3.26)
2𝜋
which measures the difference between the sum over the separated wavefields and the
full wavefield, and
𝑝𝑟𝑜𝑝(d𝑡)
𝐵(𝑢𝑠 , 𝑢, 𝐱, 𝑡) = ∫ d𝜓′ (𝑢𝑠
(𝐱, 𝜓′̂ , 𝑡)) − 𝑢(𝐱, 𝑡 + d𝑡),
(3.27)
2𝜋
which computes the difference between the sum of separated wavefields, propagated
to the next time step, and the full wavefield at the next time step.
Then, our objective functional is
energy
⎧
⎫
match 𝑡 + d𝑡
match 𝑡
{
⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞2 }
𝑓(𝑢𝑠 , 𝑢, 𝑡) = ∫ d𝐱′ ⏞⏞⏞⏞⏞⏞⏞
𝑤 𝐴(𝑢𝑠 , 𝑢, 𝐱′ , 𝑡)2 + ⏞⏞⏞⏞⏞⏞⏞
𝑤2 𝐵(𝑢𝑠 , 𝑢, 𝐱′ , 𝑡)2 + 𝑤𝑒 ∫ d𝜓′ (𝑢𝑠 (𝐱′ , 𝜓′̂ , 𝑡))
.
⎨ 1
⎬
𝐱
2𝜋
{
}
⎩
⎭
(3.28)
The first term in our objective functional favors a solution such that the sum of
the waves propagating in different directions is equal to the current time step’s full
(unseparated) wavefield at each point. This is clearly an important property for any
plausible solution, and so the weight for this term, 𝑤1 , should be large relative to the
weights for the other terms.
The second term says that the solution should still be reasonable when the full
wavefield at the next time step is considered. This is achieved by advancing the
separated waves forward by one time step in their direction of propagation, and
comparing the summation over propagation angle of the result with the full wavefield
at the next time step. The propagation is performed using the one-way wave equation.
85
A perfect match with the full wavefield is not expected, since effects present in the full
wave equation, such as spreading, are not accounted for, however we do not expect
the difference to be large after one time step. We use the Forward-Time Central𝑝𝑟𝑜𝑝(d𝑡)
Space (FTCS) method (Strang, 2007, p. 478) to calculate 𝑢𝑠
(𝐱, 𝜓′ , 𝑡). This
method is unstable, but this should not be a problem as we only use it to propagate
for one time step. Alternatives include the Lax-Friedrichs (Strang, 2007, p. 479)
and Lax-Wendroff (Strang, 2007, p. 477) methods. The former is stable for waves
propagating in any direction in the one-way wave equation, but suffers from significant
dispersion which reduces our ability to distinguish between waves propagating in
different directions after just one time step. The latter avoids dispersion problems,
but is more computationally expensive. Since the wavefield propagated using the oneway wave equation will not exactly match the two-way wave equation, the additional
computational cost may not be worthwhile. Additional description of these methods
and discussion of their stability may be found in Press et al. (2007).
With the first two terms alone the Hessian of the objective functional is found
emprically to not be positive definite, which implies that it does not have a unique
global minimum. As a simple example of this non-uniqueness, consider the situation
where the full wavefield is zero everywhere at both time steps. One solution that
matches the first two terms in the objective functional exactly is to have the separated
wavefields also be zero everywhere. Another solution, which also matches both terms
exactly, is for the wavefield for each separated direction to be a constant, such that
the sum over angles of these constants is zero. We avoid this by introducing a third
term which penalizes energy. This term should be given a low weight relative to the
other two terms so that it is used to select the lowest energy solution which satisfies
the first two terms.
The gradient and Hessian for this objective functional are presented in Appendix
B, and the implementation for 2D space is described in Appendix C.
If the velocity does not vary rapidly and the angular separation is performed
regularly during a time stepping simulation, then the output of the angular separation
at one time step can be used to calculate a good initial guess of the solution at the next
86
separation, potentially reducing the number of iterations of the optimization method
required to reach the desired accuracy. This can be done by shifting the separated
wavefields in their assigned propagation direction by the distance the waves would
have traveled in the time between separations.
3.3.4
Performance
0.6
Runtime (s)
0.5
0.4
0.3
0.2
0.1
0
ss
ne
s
w
lo
g
lS
ca
Lo
s
t
le
ve
tin
yn
ur
C
t
le
ve
S
LS
S
ur
C
LS
Po
2,
2,
1,
1,
Separation method
Figure 3-8: The time needed to perform directional separation on a single time slice
of 200 × 200 cells, with 𝑇 = 0.085 s, and Δ𝑡 = 2.7 × 10−4 s. For method 2 it is
assumed that wavefront orientation separation has already been performed on all but
the final time slice. Method 3 took approximately 31 s for Hessian construction and
32 s for the optimization. As the Hessian does not vary over time steps, it only needs
to be computed once.
Although the proposed methods are more robust than the regular Poynting vector
method, this comes at a substantial computational cost. The runtime and peak
memory needed for several of the methods discussed to decompose a 200 × 200 cell
wavefield into ten propagation directions with an oscillatory time 𝑇 of 0.085 s, are
presented in Figures 3-8 and 3-9, respectively. The separation is performed every 10
time steps. For runtime measurements we compare the following:
For method 1, the runtime is that needed to perform wavefront orientation angle
separation on two adjacent time slices and apply the Poynting vector method between
them. If we had instead performed the separation every time step then we could have
87
Peak memory usage (MB)
120
100
80
60
40
20
0
s
s
ne
w
lo
lS
ca
Lo ing
t
yn ets
Po vel
ur
C
2,
S
LS ets
2, vel
ur
C
1,
SS
L
1,
Separation method
Figure 3-9: Memory required to perform the same separation as in Figure 3-8. Method
3 required 2.1 GB
used the result from the previous separation and only needed to perform wavefront
orientation angle separation on the current time step.
For method 2 it is assumed that the wavefront orientation angle separation has
been done and the result stored for all but the final time step. The runtime for
method 2 therefore includes only the time needed for the wavefront orientation angle
separation of one time slice, and the sum over time slices. This is possible because
10Δ𝑡 is small compared to 𝑇 , with Δ𝑡 being the time step interval (2.7 × 10−4 s), and
so only using every tenth time step in the sum over spacetime still gives an accurate
approximation of the spacetime sum.
For method 3, the initial guess provided was zero everywhere. The runtime may
have been reduced if the output of the previous separation had been provided as an
initial guess.
For the Poynting vector and local slowness methods, the measured time is that
needed to produce the separated wavefields for a single time step.
These results are derived from research codes, and so have not been optimized
for production use. Additionally, method 3 was implemented in Matlab, while the
other methods were implemented in compiled languages, complicating comparisons.
Nevertheless, the results still provide some indication of the relative computational
cost of the methods. We see that, for this configuration, the first two new methods
88
require about twice the runtime of the local slowness method, and several times that
of the Poynting vector method. The curvelets variants take longer than using LSS,
while methods 1 and 2 have approximately the same runtime. Method 3 is significantly
slower. Although method 2 needs to store many time slices separated by wavefront
orientation angle, while method 1 only needs to store 2, the memory requirement of
method 2 is only about three times that of method 1. This is because a large amount
of memory (perhaps unnecessarily large) is used to construct and apply the filters in
method 1. Both methods again require several times the memory of the Poynting
vector method, however the memory requirements of the local slowness method were
comparable with those of method 1.
Indications of how these values will vary as the parameters of the separation are
changed are presented in Tables 3.1 and 3.2.
Poynting vectors
Local slowness
Method 1, LSS
Method 1, curvelets
Method 2, LSS
Method 2, curvelets
Method 3
𝑂(𝑁𝑥 × 𝑁𝑧 )
𝑂(𝑁𝑥 × 𝑁𝑧 × 𝑁𝑝 × 𝐼𝑡 )
𝑂(𝑁𝑥 × 𝑁𝑧 × 𝑁𝑝 × 𝐼𝑡 + 𝑁𝑥 × 𝑁𝑧 × 𝑁𝑝 )
𝑂(𝑁𝑥 × 𝑁𝑧 × 𝑁𝑝 × log(𝑁𝑥 × 𝑁𝑧 ) + 𝑁𝑥 × 𝑁𝑧 × 𝑁𝑝 )
𝑂(𝑁𝑥 × 𝑁𝑧 × 𝑁𝑝 × 𝐼𝑡2 + 𝑁𝑥 × 𝑁𝑧 × 𝑁𝑝 × 𝐼𝑡 )
𝑂(𝑁𝑥 × 𝑁𝑧 × 𝑁𝑝 × 𝐼𝑡 × log(𝑁𝑥 × 𝑁𝑧 ) + 𝑁𝑥 × 𝑁𝑧 × 𝑁𝑝 × 𝐼𝑡 )
𝑂(𝑁𝑥 × 𝑁𝑧 × 𝑁𝑝 )
Table 3.1: Computational complexity, where 𝑁𝑥 and 𝑁𝑧 are the number of cells in
the 𝑥 and 𝑧 dimensions, 𝑁𝑝 is the number of propagation directions that we wish to
separate the wavefield into, and 𝐼𝑡 is the summation length in time. For method 2
we assume that the same summation length (in time) is used in both the summation
over time slices and the spatial summation for wavefront orientation separation. The
complexity of method 3 will depend on the choice of optimization method, but we
assume that it will be proportional to the number of elements in the Hessian.
Poynting vectors
Local slowness
Method 1, LSS
Method 1, curvelets
Method 2, LSS
Method 2, curvelets
Method 3
𝑂(𝑁𝑥 × 𝑁𝑧 )
𝑂(𝑁𝑥 × 𝑁𝑧 × 𝐼𝑡 )
𝑂(𝑁𝑥 × 𝑁𝑧 × 𝑁𝑝 )
𝑂(𝑁𝑥 × 𝑁𝑧 × 𝑁𝑝 )
𝑂(𝑁𝑥 × 𝑁𝑧 × 𝑁𝑝 × 𝐼𝑡 )
𝑂(𝑁𝑥 × 𝑁𝑧 × 𝑁𝑝 × 𝐼𝑡 )
𝑂(𝑁𝑥 × 𝑁𝑧 × 𝑁𝑝 )
Table 3.2: Memory requirements, where the symbols are described in Table 3.1.
89
Unlike the windowed Fourier transform method, all three of the newly proposed
methods can be easily implemented efficiently on distributed memory computer clusters. This is because, particularly when LSS is used for wavefront orientation angle
decomposition in methods 1 and 2, most of the computations are spatially localized
and so little communication between computer nodes is required. If the domain is
decomposed spatially over computer nodes so that a particular node contains the
wavefield for the same portion of the spatial domain for all time slices needed to
perform the angular separation, and there is an overlap 𝐼𝐱 /2 cells wide for method
√
1 or 𝐼𝐱 / 2 cells wide for method 2, of the wavefield stored on each node, then it
is possible for methods 1 and 2 to be performed using LSS without any inter-node
communication. Using the Fourier transform or curvelet transform for wavefront orientation angle separation does require inter-node communication, however. Method
3 only requires communication of a boundary one cell thick between nodes to permit
one-way wave propagation in the second term of the objective functional, and global
reductions on the value of the objective functional, and so this, too, is well suited to
large-scale parallel computing.
It is possible to reduce the additional cost of using the proposed methods by only
applying them at locations where the regular Poynting vector method fails. This is
accomplished by first applying the Poynting vector method at all locations. Following
that, the filters described for method 1 can be evaluated to determine where the
Poynting vector method was not successful (the locations where the filters have values
below a specified threshold). In cases where the Poynting vector method is successful
over large portions of the wavefield, this enables the more expensive methods to be
reserved for problem areas.
3.4
Results
In this section we compare the previously proposed Poynting vector and local slowness
methods with the three methods described in this paper. For the first two new
methods, where different wavefront orientation separation methods can be used, we
90
use both the local slant stack (LSS) method, and the curvelet decomposition method.
To conserve space, the Fourier transform wavefront orientation separation variant of
methods 1 and 2 is not used in the comparison.
We begin with two tests of angular resolution under the ideal conditions of constant
velocity. We then examine the behavior of the methods when the constant velocity
assumption is violated. Finally, we compare the results of selected methods on a
complicated wavefield created by backpropagating receiver data through a 2D portion
of the SEAM model.
3.4.1
Crossing waves 1
In order to test the ability of the methods to separate two waves crossing obliquely,
we create a wavefield using two sources separated horizontally by a distance of 477 m.
The sources emit a 20 Hz Ricker wavelet, and the wave speed is constant everywhere
at 1500 m/s. We attempt to separate the wavefield 0.5 s after the peak source input,
as the two waves are crossing 715 m below the surface. The full wavefield at this
time is shown in Figure 3-10. As the velocity is smooth, and the waves are far
from the source, the local plane wave approximation inherent in the local slowness
method and the first two new methods is quite accurate. For these methods, we use
0.17 s, twice the duration of the source wavelet, as the summation time. This allows
the wavefield to be separated into six equally spaced propagation directions using
method 1, according to Equation 3.5.
In Figure 3-11 we show the amplitude of waves determined by different methods
to be propagating in each direction at the chosen point, where the waves from the two
sources are overlapping. To show the spread in amplitude over angle, we do not only
select the peaks as discussed previously. It is clear that had we done this, the result of
methods 1 and 2 using LSS would be very close to the true solution. Using curvelets
to separate the wavefield by wavefront orientation is also quite successful for both
methods 1 and 2, however the result is not as smooth as that obtained using LSS. As
expected, the Poynting vector method fails in this test as its assumption that waves do
not overlap is violated. The peak angle of the local slowness method is similar to that
91
0.026
z (km)
Amplitude
0
0.2
-0.026
0
x (km)
0.2
Figure 3-10: A time slice of two waves overlapping obliquely. Directional separation
is performed at the central point (0.1 km, 0.1 km). Only the central portion of the
wavefield is shown.
of the Poynting vector method, as the angular resolution with the given summation
time is not sufficient to distinguish between the two propagation directions. As the
sum over all propagation direction amplitudes for method 3, rather than the sum
over peak values, should equal the full wavefield, the amplitude in each direction
bin is smaller than it is for the other methods. Method 3 successfully separates the
two propagation directions, with a fairly small spread of amplitude around the true
solution. Furthermore, the sum of the amplitude over all angles in the result of
method 3 is close to that of the true solution: 0.0265 and 0.0244, respectively.
3.4.2
Crossing waves 2
To demonstrate the difficulties encountered by method 1 when overlapping waves
are propagating in opposite directions (and so have the same wavefront orientation),
we conduct a second test of overlapping waves created by two sources in a constant
velocity medium. This time the sources are separated vertically by a distance of
1425 m so that the waves which they emit will overlap while propagating in opposite
vertical directions halfway between the two sources after 0.475 s. The full wavefield
92
True
Method 1, LSS
Method 1, curvelets
a
b
c
Method 2, LSS
Method 2, curvelets
Method 3
d
e
f
Poynting vectors
Local slowness
g
h
Figure 3-11: Results of directional separation on the wavefield in Figure 3-10. Propagation angle is on the polar axis, while the radial axis represents amplitude. The
amplitude range is the same for all plots except (f), in which is it halved.
93
at this time is shown in Figure 3-12. We attempt to separate the wavefield into
propagation directions at this time step.
0.017
z (km)
Amplitude
0
1
-0.017
0
x (km)
1
Figure 3-12: A time slice of two waves with an overlapping region in which the waves
are propagating in opposite directions.
Figure 3-13 presents the upgoing and downgoing wavefields determined by each
method, produced by summing the amplitudes in relevant propagation directions. For
methods 1 and 2, and the local slowness method, these summations only use peak
values. As expected, method 1, using both LSS and curvelets, has trouble separating
the wavefield in the overlapping regions as the method’s assumption that isolating
different wavefront orientations will separate waves with different propagation directions is not true in this case. Method 2, which does not make this assumption, is very
successful, with a visually perfect separation. The regular Poynting vector method
again fails in the overlap regions. The local slowness method is quite successful, however some artifacts are visible in the result. Method 3 delivers a visually perfect
separation.
3.4.3
Layer over halfspace
The proposed methods all rely on the wavefield consisting of wavefronts that are
locally planar. This assumption is violated at model discontinuities, where wavefronts
94
True
up
up
Method 1, LSS
down
z (km)
0
z (km)
0
down
1
1
0
x (km)
1 0
x (km)
1
0
x (km)
a
x (km)
1
b
Method 1, curvelets
up
down
up
Method 2, LSS
down
z (km)
0
z (km)
0
1 0
1
1
0
x (km)
1 0
x (km)
1
0
x (km)
c
x (km)
1
d
Method 2, curvelets
up
down
Method 3
up
down
z (km)
0
z (km)
0
1 0
1
1
0
x (km)
1 0
x (km)
1
0
x (km)
e
x (km)
1
f
Poynting vectors
up
down
Local slowness
up
down
z (km)
0
z (km)
0
1 0
1
1
0
x (km)
1 0
x (km)
1
0
g
x (km)
1 0
x (km)
1
h
Figure 3-13: Results of the directional decomposition of the wavefield in Figure 3-12.
All have the same amplitude range as Figure 3-12.
95
may exhibit sharp changes. To examine the robustness of the methods in such a
situation, we use a model containing a 500 m thick layer with a wave speed of 1500 m/s
over a halfspace of 2100 m/s. A single source emits a 15 Hz Ricker wavelet from the
top of the layer.
A particularly interesting potential application of directional separation is the
ability to separate reflected waves from incident waves. This raises the possibility
of reducing the interference of primaries in imaging with interbed multiples. To
examine the suitability of the different methods for such an application, we plot the
sum of the absolute value of the upgoing (reflected) amplitude over all time steps,
for comparison with the theoretical Green’s function-derived values. The results are
shown in Figure 3-14. We again only sum the peak values for methods 1, 2, and
the local slowness method. A summation length of 𝑇 = 0.228 s is used, twice the
duration of the source wavelet. It may be possible to reduce the negative effects of
the velocity discontinuity by using a shorter length, but this would sacrifice angular
resolution in regions where the wave speed is locally constant.
This is a challenging task, not only violating the assumption of locally constant
velocity, but also consisting of waves propagating in opposite directions with the same
wavefront orientation, and attempting to extract the amplitude of a weak reflected
wave when, in parts of the model, it is obscured by a significantly larger incident
wave. It is therefore unsurprising that none of the methods are completely successful. Method 1 using LSS does perform fairly well. Artifacts at the top of the model
are caused by the horizontally propagating non-reflected wave interfering with the
identification of the peaks in the upgoing amplitude. At the interface, directly below the source, where the incident and reflected waves overlap while propagating in
opposite directions, the method fails, as expected. The amplitude of the result is
also erroneously low at other points close to the interface. This is likely to be caused
by the wave speed and angle filters detecting that the calculated amplitude at these
locations is incorrect, due to the violation of the locally-constant wave speed assumption, and so severely reducing the amplitude. In other locations the amplitude is very
close to the true value, and the location of most of the interface is clearly visible.
96
1
z (km)
0.5
0
0
x (km)
Method 1, LSS
0
Amplitude
z (km)
0.075
0.5
1
1
0
0
a
1
0.5
1
1
0
0
c
1
0.5
1
1
0
0
e
1
0.075
Amplitude
z (km)
0
x (km)
1
Local slowness
0
Amplitude
z (km)
0.075
0.5
0
x (km)
f
Poynting vectors
0
0.075
Amplitude
z (km)
0
x (km)
1
Method 3
0
Amplitude
z (km)
0.075
0.5
0
x (km)
d
Method 2, curvelets
0
0.075
Amplitude
z (km)
0
x (km)
1
Method 2, LSS
0
Amplitude
z (km)
0.075
0.5
0
x (km)
b
Method 1, curvelets
0
0.075
Amplitude
True
0
0.5
1
1
0
0
g
x (km)
1
h
Figure 3-14: Absolute amplitude, summed over time, of the upgoing (reflected) wave
in a halfspace model.
97
When curvelets are used instead of LSS in method 1, the upgoing amplitude is again
mostly contained above the interface, and in some locations the amplitude is close to
being correct, however the result is noisier than that obtained with LSS. Increased
noisiness of the curvelets result compared to that of LSS is also observed in the output of method 2. We also see some artifacts near the surface due to interference
with horizontally propagating non-reflected waves. The amplitude is quite accurate
in most locations, especially when LSS is used, although there is some leakage below
the interface and under estimation in several locations above. The regular Poynting vector method performs admirably well everywhere except immediately above
the interface, where, for about the half of a wavelength distance that the incident
and reflected waves overlap, the amplitude is severely underestimated. Insufficient
angular resolution at the summation length chosen again leads to overestimation by
the local slowness method. This is particularly severe away from the regions directly
below the source, where the high amplitude non-reflected waves are more likely to
leak into the upgoing angles. Method 3 has a surprisingly poor showing in this test.
At each time step the optimum solution found by the method has a small percentage
of its amplitude propagating in the opposite to correct direction. This may be an
attempt by the method to compensate for the lack of spreading in the one-way propagator used in the objective functional. The result is that this small percentage of
the downgoing non-reflected wave amplitude that is assigned to upgoing directions is
seen throughout the model. One would still expect higher amplitude above the interface, however, as the reflected wave amplitude should be added to this artifact caused
by the downgoing wave, yet this is not the case. This may be due to the reflected
wave being significantly weaker than the downgoing wave, and so the decreases in the
objective functional that could be obtained by including the reflected wave are below
the specified threshold. While this could be solved by reducing the threshold, doing
so would increase the cost of an already computationally expensive method.
98
3.4.4
SEAM
To investigate the behavior of the methods under less idealized conditions, we consider the backpropagated data wavefield for a single source in a 2D portion of the
SEAM model shown in Figure 3-15. Directionally separating the backpropagated
data wavefield like this is necessary for many applications, such as generating ADCIGs and performing illumination compensation, but the complexity of the data
wavefield means that the Poynting vector method may not be well suited to the task.
The SEAM model was designed to provide a realistic test for imaging and inversion
techniques (Fehler and Larner, 2008). The data were generated with an elastic model,
however we backpropagate them using only the P-wave velocity in an acoustic propagator. We also apply a shaping filter so that the data are approximately what would
have been recorded had the source emitted a 15 Hz Ricker wavelet.
4.48
z (km)
wave speed
(km/s)
0
6.25
1.49
10
x (km)
16
Figure 3-15: P-wave speed for a 2D portion of the SEAM model, covering the region
10 km to 16 km 𝑥, 2.39 km 𝑦, 0 km to 6.25 km 𝑧.
As we do not know the true directional decomposition of this wavefield, we can only
judge the results on how visually plausible they appear. Based on their performance in
previous tests, we select methods 1 and 2 using LSS wavefront orientation separation,
and the Poynting vector and local slowness methods for this experiment. As in
99
the previous example, we show the summation over time of absolute values of the
separated wavefields, however we now display the amplitude assigned to each direction
at a grid of points. The results are shown in Figure 3-16, with the locations of model
discontinuities in the background to aid evaluation. The output of methods 1 and 2,
the overall best performers in previous tests, is quite similar. It is apparent that the
majority of the recorded amplitude originates from reflections on sloping sediment
layers. The largest amplitude propagation directions are also largely the same in
the results of the Poynting vector and local slowness methods, indicating that either
could be used to obtain approximately the same result as that of methods 1 and 2
for these large features. The amplitudes in the Poynting vector results are larger
in most cases than those found by the other methods. This is likely to be due to
the method assigning the total amplitude at each point, including the sum over any
overlapping waves, to a single direction at each time step. The results of the local
slowness method are closer to those produced by methods 1 and 2, but a greater
spread of amplitude over angles is apparent, which is likely to be due to the method’s
poorer angular resolution.
3.5
Discussion
The results of the tests presented indicate that neither the Poynting vector method
nor the local slowness method can be used reliably to separate wavefields by propagation direction when there are overlapping waves. Although none of the newly
proposed methods performed flawlessly in all evaluations, they were on average superior, especially methods 1 and 2 using LSS wavefront orientation separation. Based
on the results of the experiments, the computational cost, and the ease of implementation, method 1 using LSS appears to be the best performer, despite its inability to
separate waves propagating in opposite directions.
Although computationally expensive, method 3 is perhaps the most elegant proposal, and performed very well on the first two tests. As discussed earlier, its result
in the third test might be improved with a lower convergence threshold. Further
100
0
5.8
10
x (km)
5.8
10
15.8
a
5.8
10
x (km)
15.8
b
0
Local slowness
z (km)
Poynting vectors
z (km)
0
Method 2, LSS
z (km)
Method 1, LSS
z (km)
0
x (km)
5.8
10
15.8
c
x (km)
15.8
d
Figure 3-16: Sum over time of the absolute amplitude of the backpropagated data
wavefield generated by a source at 13 km 𝑥, 15 m 𝑧. Results from the region around
the source are removed to make amplitudes in the rest of the domain more visible.
All polar plots have the same amplitude range. Locations of discontinuities in the
P-wave velocity model are shown in the background.
101
improvements could potentially be obtained by comparing the sum over angles after
propagation in the assigned direction with the full wavefield at additional time steps.
For example, instead of only attempting to match the wavefield at times 𝑡 and 𝑡 + d𝑡
one may also try to find a solution which is consistent with the full wavefield at time
𝑡 − d𝑡.
A variety of applications which rely on directional decomposition have already
been mentioned, however it is likely that there are other uses for this information
about the wavefield that is currently often neglected. These include using wave propagation direction to calculate particle velocity vectors for comparison with multicomponent data in seismic inversion algorithms, simultaneous source decomposition (as
suggested by the first example, although this might be problematic if there are nonhorizontal reflectors), and compensating for changes in amplitude due to the angle of
reflection in seismic imaging.
Several of the methods rely on the wavefield consisting of waves that are oscillatory
(or almost oscillatory) over a time 𝑇 . Attenuation may cause the minimum oscillatory
time to change over the range of recorded times. Applying an inverse Q filter (Wang,
2006) might avoid this problem, or alternatively the maximum 𝑇 for all recorded
arrivals could be used, even if it is longer than is necessary for certain portions of the
recorded data.
We allude on several instances to the limits on achievable accuracy, which involve
multiple factors. One of these is the frequency content of the wavefield, when the
distance over which the wavefronts are planar is finite. Equations 3.5, 3.11, and 3.12
demonstrate the diminishing ability to distinguish between wavefronts with different
orientations as the frequency of the waves decreases when using the LSS, Fourier
transform, and curvelet approaches, respectively. Similar results hold for the spacetime summation means of separation used in the local slowness method. Expressing
the resolution limits of frequency on the derivative-based approaches, method 3 and
Poynting vector analysis, is not as clear, but it is obvious that for low frequency waves,
the decreasing amplitude of the space and time derivatives will result in numerical
errors becoming more prominent. Another factor affecting accuracy is the sampling
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of the wavefield in space and time. For the methods relying on derivatives, larger
sampling intervals cause errors in the derivative calculations to grow. Sampling has
the potential to limit the accuracy of the summation components of methods using
LSS and the local slowness method since with lower frequency sampling the summation will become a less exact representation of the integral, and so summing along
oscillatory directions may not produce complete cancellation. The harmful effect of
aliasing caused by reduced sampling on results produced by the transforms is well
known. Employing more information about the wavefield, as is done in method 2, increases potential accuracy, but for discretized, finite-frequency wavefields, there will
always be limits to achievable angular resolution, regardless of the separation method
employed.
3.6
Conclusion
This paper presents three new methods, with several variants, for separating a wavefield into waves propagating in different directions. Unlike the previously proposed
Poynting vector method, these methods are capable of performing the separation even
when there are overlapping waves. The local slowness method is also able to do this,
but, as we demonstrate, it has poorer angular resolution than the new methods for an
equivalent assumed distance of constant velocity. The proposed methods are likely
to be more computationally expensive in most cases, but it is still possible to run
them on relatively large problems, such as a portion of the SEAM model, and obtain
plausible results.
103
104
Chapter 4
Improving RTM amplitude
accuracy
Abstract
We describe a time domain method for improving the relative amplitude accuracy of
Reverse Time Migration by performing illumination compensation and reversing the
sign of image contributions when an interface is imaged from below. This scheme
allows internal multiples (waves reflected multiple times in the subsurface) to be
used more effectively, by appropriately boosting their image contribution amplitude,
attenuating certain types of image artifacts caused by their inclusion, and ensuring
that they add constructively to the image. It also has similar benefits for imaging
with overturned waves. We demonstrate the method with synthetic models, including
a 2D portion of the SEAM model, yielding results with improved relative amplitude
accuracy compared to standard Reverse Time Migration.
4.1
Introduction
Seismic imaging attempts to construct an image of subsurface reflectors. Migration
methods such as Reverse Time Migration (RTM, Baysal et al. (1983)) seek to achieve
this by positioning reflected energy at the locations of the reflectors. Although it has
been shown that under idealized conditions the amplitudes in images produced with
RTM are directly related to the reflector properties (Chattopadhyay and McMechan,
2008), in general the amplitudes are also significantly affected by other factors. One of
105
these is uneven illumination. The effect of this is that reflectors in poorly illuminated
areas, such as at the edge of surveys or below salt, have lower amplitude than better
illuminated structures even when they have similar reflectivity. Contributions from
lower amplitude arrivals, such as internal multiples, may also have a negligible effect
on the image because they are so much weaker than primaries. Another problem in
accurately estimating amplitudes in RTM is the destructive stacking of the image
contributions incident from both sides of the interface. This is a particularly common issue when overturned waves and internal multiples are included in the imaging
process as these waves may image an interface from below, while regular primaries
image it from above. Having image amplitude relate to the material properties of
the Earth is important as it may allow the rock type of layers in the subsurface to
be determined. Amplitudes are also important for several hydrocarbon indicators,
such as “bright spots” (Craft, 1973) and AVO (Ostrander, 1984). In this paper we
propose modifications of the RTM algorithm to improve amplitudes by correcting for
uneven illumination, waves incident from opposite sides, and other amplitude errors
associated with internal multiples and overturned waves. This is achieved by combining illumination compensation with a modified imaging condition that alters the
sign of image contributions when appropriate so that contributions stack coherently,
regardless of which side of the interface they are incident from. We also incorporate the backscatter-attenuating method proposed by Costa et al. (2009), which can
be combined with the other operations in our proposed method for little additional
computational cost.
We begin with a description of sources of amplitude errors in RTM. This is followed by an overview of the illumination compensation method we will use, together
with the estimation of the wave propagation direction information we derive during
its application, to obtain more accurate image amplitudes. We then discuss internal multiples and overturned waves, and why our method is especially beneficial for
such waves. After a brief note on uncertainty, the method is described, followed by
results comparing RTM images produced with and without the proposed algorithm
modifications.
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4.1.1
RTM amplitude errors
Although its closer adherence to the physics of finite-frequency wave propagation
than other migration methods gives RTM a more plausible chance of producing a
true amplitude image, there are still obstacles to moving beyond a kinematically
correct (structural) image to one where the amplitudes reflect material properties.
We will describe several of these obstacles, noting which are targeted by the proposed
algorithm.
One cause of incorrect amplitudes is transmission loss. This occurs in two ways.
The first is due to losses between the wave being scattered and its recording, because
of attenuation and additional scattering. A second form of transmission loss happens
during migration if an impedance model containing reflectors is used and the backpropagated data wavefield undergoes scattering. These issues are explored in Gray
(1997). The proposed method does not address this form of amplitude error. Other
means of mitigating transmission losses are proposed by Deng and McMechan (2007).
A second important cause of incorrect amplitude is errors in the velocity model.
These errors can cause scatterers to be misplaced, resulting in their amplitudes not
stacking constructively over shots. Migration velocity analysis techniques (Sava and
Biondi, 2004) seek to produce a velocity model that positions scatterer images in
the same place across shots, so the use of such methods can reduce this source of
amplitude errors. As is the case with most seismic imaging algorithms, our proposed
method assumes that the velocity model is correct up to the oscillatory perturbations
that are to be imaged.
Even with a correct velocity model, stacking can still make amplitudes unreliable
and potentially result in misinterpretation. This is due to variations in reflected
amplitude with incidence angle. As the reflection coefficient of specular reflections
on a planar interface varies with reflection angle, the amplitude of the stacked image
of the reflector at a point will depend on the range of incidence angles illuminating
that point. The image amplitude along a reflector may therefore vary simply due to
changes in incidence angles. A more egregious amplitude error is caused when waves
107
are incident on different sides of an interface, as these waves will experience reflection
coefficients with the opposite sign, and so stacking will cause cancellation. Correcting
for variations in reflection coefficient with angle requires knowledge of the velocity
contrast across the interface, which is usually not available. As long as the source
and data wavefield propagation directions are known, however, it is possible to detect
when an interface is imaged from different sides, and so this is one of the corrections
we implement in this paper. Primary waves all tend to image reflectors from the
same side, especially when the reflectors are almost horizontal. Internal multiples and
overturned waves will often image the reflectors from the opposite side to the primaries
(Biondi and Shan, 2002), and so unless the correction we propose is implemented, such
waves may subtract from the image, rather than adding constructively.
Another source of amplitude error that is possible even when the correct velocity model is used, is image artifacts caused by reflectors in the migration velocity
model. Such reflectors are necessary in order to generate internal multiples during
migration. Multiples can be automatically used for imaging in RTM by including
discontinuities in the migration velocity model at the locations of multiple-generating
reflectors (Youn and Zhou, 2001). Two types of artifacts may be caused by this.
The most noticeable are generally the low frequency smears caused by backscatter,
which occur when the source and data wavefields reflect on the model discontinuities,
causing them to overlap over much of their propagation path. Numerous methods
of attenuating these artifacts have been proposed (Guitton et al., 2006; Costa et al.,
2009; Liu et al., 2011; Op ’t Root et al., 2012). A second type of artifact caused by
migration model reflectors is the phantom reflector. The situation that can give rise
to such an artifact is illustrated in Figure 4-1. The inclusion of the first true reflector
in the example migration model will result in the backpropagated arrival from the
second true reflector reflecting, and overlapping with the source wavefield above the
first true reflector. The method we propose will seek to reduce the effect of both of
these types of artifacts.
The finite resolution of seismic waves and limited angular illumination mean that
the impulse response (point spread function) of the imaging operation is not a point;
108
surface
wavepath 1
wavepath 2
wavepath 3
true reflector 1
true reflector 2
Figure 4-1: A fraction of the backpropagated arrival from true reflector 2 will be
reflected upward from true reflector 1 if it is present in the velocity model. This
may overlap with the fraction of the forward propagated source wave which is also
reflected upward. This causes a phantom reflector at the apex of wave path 3. As
the phantom reflector is well illuminated by large amplitude direct waves along wave
path 2, applying illumination compensation will reduce the amplitude of the phantom
reflector artifact.
the image amplitude at a point is affected by nearby scatterers. Attempts have been
made to approximate the impulse response, so that its effects can be removed from
the image, but this is a computationally expensive operation (Xie et al., 2005b; Cao,
2013). A simpler task is to compensate only for changes in illumination. Unlike resolution compensation, which seeks to reduce the image of point scatterers to points,
illumination compensation attempts only to produce a band-limited image of the scatterers, like regular RTM, but with more accurate relative amplitude. This is achieved
by removing the effects of variation in illumination from the image. Illumination compensation is the third type of amplitude correction that we perform in our modified
RTM algorithm.
4.1.2
Illumination compensation
Although variation in illumination is only one of the sources of amplitude errors we
describe above, it plays a central role in our method as information derived as part of
its application, namely the propagation directions of the source and data wavefields,
109
is used in the corrections for the other sources of amplitude errors that we target.
We assume the recorded seismic data to be
𝑑 = 𝐿𝑚,
(4.1)
where 𝐿 is a linear forward modeling operator, which is applied to 𝑚, the true model
parameters. Performing the (approximately) adjoint to the modeling operator, the
migration operator 𝐿′ , we obtain the migrated image,
𝐼 = 𝐿′ 𝑑 = 𝐿′ 𝐿𝑚.
(4.2)
Applying (𝐿′ 𝐿)−1 to the migrated image would approximately produce a least
squares estimate image of the true Earth properties, assuming we have sufficient data.
This is the goal of methods that try to compensate for resolution, although some
simplifications are necessary to make the problem tractable. Further simplifications,
to render the computational costs plausible for production use, result in an operation
termed illumination compensation.
Many attempts have been made to compensate for illumination effects in seismic
images, with varying amounts of simplification. The approximated illumination is
often taken to depend on the acquisition geometry, the migration model, and the frequency content of the source wavelet. Some attempts, such as Rickett (2003); Plessix
and Mulder (2004); Tang (2009), approximate (𝐿′ 𝐿)−1 by its diagonal, resulting in a
single correction factor for each point in space. This appears to successfully reduce
the effects of illumination variation, but its accuracy is compromised by its failure to
account for variations in illumination with reflector orientation. Illumination is, in
fact, strongly affected by reflector orientation. For example, it may not be possible
to illuminate a flat reflector at the edge of a survey, while an inclined reflector at the
same location may be well illuminated. Most recent proposals have therefore sought
to determine image amplitude and illumination as functions of position and reflector
orientation angle. The approaches are often differentiated by the proposed means of
extracting the angular dependence of the image and illumination. Several methods
110
employ forms of wavelets (Herrmann et al., 2009; Mao et al., 2010), others exploit
the pseudodifferential nature of the illumination compensation operator (Stolk, 2000;
Nammour and Symes, 2009; Demanet et al., 2012). Valenciano et al. (2006) calculate
a limited number of off-diagonals of 𝐿′ 𝐿 in a targeted area of space to reduce computational costs. Several proposals have used frequency domain methods for extracting
angular information (Cao and Wu, 2009; Mao and Wu, 2011; Ren et al., 2011), while
others work in the time domain (Yang et al., 2008). Another recent proposal, Cao
(2013), is reminiscent of the approach of Rickett (2003), but is able to extract angular
information by using a local transformation of point spread functions.
Any of the methods that calculate illumination as a function of both position and
reflector orientation can be used in the amplitude correction algorithm we propose,
however in the method section we describe a new time domain approach that is
computationally efficient, and which seeks to produce a better approximation of 𝐿′ 𝐿
for improved accuracy by describing how variations in point spread function width
could be incorporated into the correction.
4.1.3 Multiples and overturned waves
Internal multiples have long been of interest in seismic imaging. As migration methods
generally linearize the problem of positioning reflectors by making the single scattering
assumption (Bleistein et al., 2001), multiples are often seen as noise that must be
removed (Weglein et al., 1997). The tantalizing prospect of exploiting the useful
information contained in internal multiples has led many to propose schemes to extend
migration algorithms so that these waves may be used for imaging (Youn and Zhou,
2001; Cavalca and Lailly, 2005; Malcolm et al., 2009, 2011; Fleury, 2013; Dai and
Schuster, 2013). As the examples shown for many of these proposals demonstrate,
internal multiples are particularly useful for imaging vertical structure such as salt
flanks and fractures, and avoiding problematic areas such as imaging through salt by
going around them. Cao and Wu (2009) also show that internal multiples can provide
more even illumination in subsalt areas than singly-scattered primaries. Overturned
waves offer many of the same benefits as internal multiples and are usually easier
111
to incorporate into migration algorithms as they do not violate the single scattering
assumption. As they have not undergone multiple reflections, they may also have
larger amplitude than multiples. The main drawback of overturned waves is that they
require large apertures. They have nevertheless also been proposed, like multiples,
as important means of imaging in areas of complex geology (Hale et al., 1992; Zhang
et al., 2006).
Unlike most other migration algorithms, Reverse Time Migration can naturally incorporate internal multiples and overturned waves. The conventional RTM algorithm
and imaging condition do not derive the full benefit of these waves, however. This
is improved by the amplitude correction modifications that we propose. There are
several motivations for combining the inclusion of multiples with an illumination compensation method such as the one we propose. One is that multiples and overturned
waves are usually significantly weaker than regular primaries, and so illumination
compensation is necessary in order for them to contribute substantially to the image.
A second reason is that many of the amplitude errors we discuss above are caused by
overturned waves or the inclusion of reflectors in the migration model, as is necessary
to generate internal multiples. Illumination compensation, and information that is
calculated while applying it, can be used to reduce the effect of all of these amplitude errors, as we demonstrate in the results section. Applying these modifications
to RTM therefore enables the full use of subsurface information contained in internal
multiples and overturned waves to improve the image, while reducing the harmful
impacts usually associated with their inclusion.
4.1.4
Uncertainty
While illumination compensation can improve images by removing the effect of variable illumination that can impair amplitude accuracy, it may also enhance unwanted
artifacts. An artifact in a poorly illuminated area of the subsurface may have its
amplitude boosted so that it has similar prominence in the image to well illuminated
reflectors. To contend with this, we complement the images with a measure of uncertainty.
112
4.2
Method
The modified RTM algorithm that we propose to improve relative amplitude accuracy
consists of three steps, which we describe below: the creation of an image that has
reduced backscatter artifacts and consistent reflector polarity, calculating illumination, and compensating the image for variations in illumination. We also explain our
method of calculating the weighted standard deviation of the image across shots as a
means of conveying uncertainty.
4.2.1
Uncompensated images
Before illumination compensation, the images we create are similar to regular RTM
images, but with a modified imaging condition to reduce two types of amplitude errors
(backscatter and destructive stacking when an interface is imaged from both sides),
and the images are additionally binned by apparent reflector orientation. They can
therefore be produced by augmenting a regular RTM implementation to use the new
imaging condition and bin the resulting image by reflector orientation. Any of the
standard RTM imaging conditions can be modified in the way we propose; we use the
zero-lag cross-correlation condition (Claerbout, 1971).
The most important addition that needs to be made is the calculation of the
propagation directions of the source and data wavefields each time the imaging condition is applied. This could be achieved using the Poynting vector method (Yoon
and Marfurt, 2006), or, to overcome inaccuracies due to limitations of that method
(Patrikeeva and Sava, 2013), the more computationally expensive methods of Chapter
3 may be used. Knowing these propagation directions enables us to determine the
apparent reflector orientation angle, 𝜃, via
𝜃 = 𝐕𝐀 (𝛼𝑠̂ + 𝛼𝑟̂ ) ,
(4.3)
where 𝛼𝑠̂ and 𝛼𝑟̂ are the normalized propagation directions of the source and data
wavefields, respectively. The operator 𝐕𝐀 converts a vector to its angle (in the range
113
[0, 2𝜋) radians) from horizontal. For simplicity we will work in 2D; extension to 3D is
straightforward. The direction of the source wave is computed for time going forward,
while the receiver wave direction is computed with time going backward. We assume
that the Earth consists of planar reflectors that cause specular reflections. This
is violated by point diffractors, which will appear as reflectors with many different
orientations at the same point. If the source wave amplitude propagating in the
direction 𝛼𝑠̂ at position 𝐱 and time 𝑡 due to the source 𝐱𝑠 is 𝑢𝑠,𝐱𝑠 (𝐱, 𝑡, 𝛼𝑠̂ ), and the
associated data wave (backpropagated receiver data) amplitude propagating in the
direction 𝛼𝑟̂ is 𝑢𝑑,𝐱𝑠 (𝐱, 𝑡, 𝛼𝑟̂ ), then the regular RTM image using our chosen imaging
condition, binned by reflector orientation, is
𝐼(𝐱, 𝜃) = ∑ ∑ 𝑢𝑠,𝐱𝑠 (𝐱, 𝑡′ , 𝛼𝑠̂ )𝑢𝑑,𝐱𝑠 (𝐱, 𝑡′ , 𝛼𝑟̂ ).
𝐱𝑠
(4.4)
𝑡′
To enhance the image we augment this with two additional filters. The first is to
reverse the sign of image contributions arising from wave paths that reflect on the
underside of reflectors. This is accomplished by multiplying the image contribution
by −1 if 𝜃 > 𝜋. The purpose of this is to avoid destructive stacking when an interface
is imaged from both sides. With this filter, image contributions from waves incident
on both sides of a reflector of any orientation will have the same sign. Inaccuracies
in the determination of propagation direction may lead to some destructive stacking
for reflectors with orientations close to 0 or 𝜋, but this should be localized around
those angles rather than occurring for all angles as happens without this filter. For
this reason, it is advisable to choose 𝜃 = 0 to be a direction in which few reflectors
are oriented. If most reflectors have approximately vertical orientation, then this
angle could be chosen to be in a horizontal direction, for example. The second filter
we use is the reflection angle taper described by Costa et al. (2009). This uses the
propagating directions that we have calculated to determine the scattering angle, and
then reduces the amplitude of contributions due to high scattering angles (which are
likely to be artifacts caused by direct arrivals, or reflections from sharp changes in
the velocity model). This is achieved by multiplying the image by cos𝑛 𝜙, where 𝜙 is
114
the scattering angle, and 𝑛 is a real number greater than 3 (larger numbers provide
stronger filtering of high scattering angles). In our notation, the scattering angle is
calculated with quantities already available, using
cos2 𝜙 =
1
(1 + 𝛼𝑠̂ ⋅ 𝛼𝑟̂ ) .
2
(4.5)
Finally, we use the modulo operation to fold 𝜃 into the range [0, 𝜋). This is done
so that a reflector imaged from above is identified with the same reflector imaged
from below.
The source and data wavefields could have large amplitude propagating in several
different directions at the same time at a particular point. As we do not know which
incident source direction gave rise to a particular backpropagated data wavefield propagation direction, we consider all possible combinations, yielding image contributions
to several reflector orientation bins, and rely on stacking over time and shots to
strengthen the amplitude in the correct bins.
4.2.2
Illumination
It is possible, when applying the proposed algorithm, to use any method for calculating illumination that bins the result by reflector orientation, but we describe one
possibility here that can be easily implemented due to its similarity with the method
used above to produce the uncompensated image.
The forward propagated wavefield from a source at position 𝐱𝑠 , arriving at (𝐱, 𝑡)
with propagation direction 𝛼𝑠̂ is
𝑢𝑠,𝐱𝑠 (𝐱, 𝑡, 𝛼𝑠̂ ) = ∑ 𝐺+ (𝐱, 𝐱𝑠 , 𝑡 − 𝑡′ , 𝛼𝑠̂ , ∶)𝑞(𝑡′ ),
(4.6)
𝑡′
where 𝐺+ (𝐱, 𝐱𝑠 , 𝑡 − 𝑡′ , 𝛼𝑠̂ , ∶) is a partition of the causal Green’s function from 𝐱𝑠 ,
leaving in any direction (denoted by “∶”), and arriving at 𝐱 with propagation direction
𝛼𝑠̂ , after a propagation time of 𝑡 − 𝑡′ , and 𝑞 is the source wavelet.
The wave propagates in a known background medium 𝑚0 and we wish to image
115
the unknown perturbations to this, 𝑚1 (if waves in our image domain obey the
constant density acoustic, or scalar, wave equation, then the model parameters 𝑚 =
𝑚0 + 𝑚1 represent squared slowness). Although it is common for seismic imaging
methods to assume that 𝑚0 is smooth, we do not make this restriction. Indeed,
in order for internal multiples to be generated, 𝑚0 must contain sharp changes to
produce reflections. Ideally, 𝑚1 should only contain oscillatory perturbations, so
that waves propagate at the same speed in 𝑚 and 𝑚0 . Making the simplifying
assumption that the elements of 𝑚1 are sufficiently small that only single scattering
from these perturbations is non-negligible in the data (but additional scattering from
discontinuities in 𝑚0 is allowed), also known as the Born approximation, and that
direct arrivals have been removed, the recorded data at receiver location 𝐱𝑟 at time
𝑡, due to a source at 𝐱𝑠 , can be written
𝑑𝐱𝑟 ,𝐱𝑠 (𝑡) = ∑ ∑ 𝐺+ (𝐱𝑟 , 𝐱′ , 𝑡 − 𝑡′ , ∶, ∶)𝑚1 (𝐱′ )
𝑡′
𝐱′
𝐱𝑠
𝐱𝑟
𝜕2
𝑢
(𝐱′ , 𝑡′ , ∶).
𝜕𝑡′2 𝑠,𝐱𝑠
𝐱𝑠
𝐺+ (𝐱𝑟 , 𝐱′ , 𝑡 − 𝑡′ , ∶, ∶)
(4.7)
𝐱𝑟
𝐺− (𝐱′ , 𝐱𝑟 , 𝑡 − 𝑡′ , 𝛼̂ 𝑟 , ∶)
𝑢𝑑,𝐱𝑠 (𝐱, 𝑡′ , 𝛼̂ 𝑟 )
𝑢𝑠,𝐱𝑠 (𝐱, 𝑡′ , ∶)
𝐱′
𝐱′
𝛼𝑟
a
b
Figure 4-2: (a) The component of the receiver data contributed by a scatterer at
position 𝐱′ is determined by the source wavefield 𝑢𝑠 at 𝐱′ , the scatterer amplitude
𝑚(𝐱′ ), and the Green’s function between the scatterer location and the receiver, 𝐺+ .
(b) The data wavefield 𝑢𝑑 is created by applying the anticausal Green’s function 𝐺−
to the recorded data.
This is equivalent to Equation 4.1 as long as the 𝐿 operator obeys the Born
approximation. The backpropagated data wavefield is then given by
𝑢𝑑,𝐱𝑠 (𝐱, 𝑡, 𝛼𝑟̂ ) = ∑ ∑ 𝐺− (𝐱, 𝐱𝐫 , 𝑡 − 𝑡′ , 𝛼𝑟̂ , ∶)𝑑𝐱𝑟 ,𝐱𝑠 (𝑡′ ),
𝐱𝑟 ∈𝐱𝑠𝑟 𝑡′
116
(4.8)
where 𝐺− (𝐱, 𝐱𝐫 , 𝑡 − 𝑡′ , 𝛼𝑟̂ , ∶) is a partition of the anticausal Green’s function leaving
𝐱𝐫 in any direction, and arriving at 𝐱 propagating in the direction 𝛼𝑟̂ , after time 𝑡−𝑡′ ,
and 𝐱𝑠𝑟 is the set of receivers for the source 𝐱𝑠 . This is depicted in Figure 4-2.
The image using our chosen imaging condition, but not yet assuming specular
reflections, can be calculated using
𝐼𝐱𝑠 (𝐱, 𝛼𝑠̂ , 𝛼𝑟̂ ) = ∑ 𝑢𝑠,𝐱𝑠 (𝐱, 𝑡′ , 𝛼𝑠̂ )𝑢𝑑,𝐱𝑠 (𝐱, 𝑡′ , 𝛼𝑟̂ ) cos𝑛 𝜙.
(4.9)
𝑡′
This includes the scattering angle taper with power 𝑛, described above. Expanding
by substituting in Equations 4.7 and 4.8, results in
𝐼𝐱𝑠 (𝐱, 𝛼𝑠̂ , 𝛼𝑟̂ ) = ∑ 𝑢𝑠,𝐱𝑠 (𝐱, 𝑡′ , 𝛼𝑠̂ ) ∑ ∑ 𝐺− (𝐱, 𝐱𝐫 , 𝑡′ − 𝑡″ , 𝛼𝑟̂ , ∶)×
𝐱𝑟 ∈𝐱𝑠𝑟 𝑡″
𝑡′
∑ ∑ 𝐺+ (𝐱𝑟 , 𝐱′ , 𝑡″ − 𝑡‴ , ∶, ∶)𝑚1 (𝐱′ )×
(4.10)
𝑡‴ 𝐱′
2
𝜕
𝑢
(𝐱′ , 𝑡‴ , ∶) cos𝑛 𝜙
𝜕𝑡‴2 𝑠,𝐱𝑠
We rearrange to group related terms, giving
𝐼𝐱𝑠 (𝐱, 𝛼𝑠̂ , 𝛼𝑟̂ ) = ∑ ∑ ∑ ∑ ∑ 𝑚1 (𝐱′ )𝑢𝑠,𝐱𝑠 (𝐱, 𝑡′ , 𝛼𝑠̂ )
𝐱𝑟 ∈𝐱𝑠𝑟 𝐱′
𝑡′
𝑡″
𝑡‴
𝜕2
𝑢
(𝐱′ , 𝑡‴ , ∶)×
𝜕𝑡‴2 𝑠,𝐱𝑠
𝐺− (𝐱, 𝐱𝐫 , 𝑡′ − 𝑡″ , 𝛼𝑟̂ , ∶)𝐺+ (𝐱𝑟 , 𝐱′ , 𝑡″ − 𝑡‴ , ∶, ∶) cos𝑛 𝜙.
(4.11)
In preparation for simplification of this expression, we first separate it into a term
that only depends on the wavefields and model parameter at the image point (𝐱′ = 𝐱),
117
and a second term that involves other points (𝐱′ ≠ 𝐱),
𝐼𝐱𝑠 (𝐱, 𝛼𝑠̂ , 𝛼𝑟̂ ) =𝑚1 (𝐱) ∑ ∑ ∑ ∑ 𝑢𝑠,𝐱𝑠 (𝐱, 𝑡′ , 𝛼𝑠̂ )
𝐱𝑟 ∈𝐱𝑠𝑟 𝑡′
𝑡″
𝑡‴
𝜕2
𝑢𝑠,𝐱𝑠 (𝐱, 𝑡‴ , ∶)×
‴2
𝜕𝑡
𝐺− (𝐱, 𝐱𝐫 , 𝑡′ − 𝑡″ , 𝛼𝑟̂ , ∶)𝐺+ (𝐱𝑟 , 𝐱, 𝑡″ − 𝑡‴ , ∶, ∶) cos𝑛 𝜙+
∑ ∑ ∑ ∑ ∑ 𝑚1 (𝐱′ )𝑢𝑠,𝐱𝑠 (𝐱, 𝑡′ , 𝛼𝑠̂ )
𝐱𝑟 ∈𝐱𝑠𝑟 𝐱′ ≠𝐱 𝑡′
𝑡″
𝑡‴
𝜕2
𝑢
(𝐱′ , 𝑡‴ , ∶)×
𝜕𝑡‴2 𝑠,𝐱𝑠
𝐺− (𝐱, 𝐱𝐫 , 𝑡′ − 𝑡″ , 𝛼𝑟̂ , ∶)𝐺+ (𝐱𝑟 , 𝐱′ , 𝑡″ − 𝑡‴ , ∶, ∶) cos𝑛 𝜙.
(4.12)
Resolution analysis seeks to calculate both of these terms, but, to reduce computational cost, illumination studies are primarily interested in the first term. Indeed,
most previously proposed illumination calculation methods discard the second term,
however we will attempt to estimate its effect on the image. First, we make the
approximation
∑ ∑ 𝐺− (𝐱, 𝐱𝐫 , 𝑡′ − 𝑡″ , 𝛼𝑟̂ , ∶)𝐺+ (𝐱𝑟 , 𝐱, 𝑡″ − 𝑡‴ , ∶, ∶) ≈ 𝐼𝑙𝑙𝑢𝑚𝑟,𝐱𝑠 (𝐱, 𝛼𝑟̂ )𝛿(𝑡′ − 𝑡‴ ),
𝐱𝑟 ∈𝐱𝑠𝑟 𝑡″
(4.13)
where
2
𝐼𝑙𝑙𝑢𝑚𝑟,𝐱𝑠 (𝐱, 𝛼𝑟̂ ) = ∑ ∑ |𝐺+ (𝐱, 𝐱𝑟 , 𝑡″ , 𝛼𝑟̂ , ∶)| .
(4.14)
𝐱𝑟 ∈𝐱𝑠𝑟 𝑡″
Although not generally true, as it is violated when multipathing occurs, this will
tend to become more accurate with stacking (as artifacts caused by errors in this
approximation will be attenuated). It is, however, a high-frequency approximation,
as it assumes that the material is non-attenuating and non-dispersive, as it does not
consider variations in the Green’s function with frequency. If frequency dependent
effects are large over the range of frequencies in the source, it may be possible to
improve the accuracy of this approximation by performing the proposed method multiple times, using different frequency windows of the data. The effects of attenuation
could also be removed from the data by the application of an inverse Q filter (Wang,
2006). If we assume that no two receivers make significant contributions to 𝐺 at
the same time and angle (similar to the Bolker condition of Guillemin (1985)), this
118
receiver illumination may be written
2
𝐼𝑙𝑙𝑢𝑚𝑟,𝐱𝑠 (𝐱, 𝛼𝑟̂ ) = ∑ ∣ ∑ 𝐺+ (𝐱, 𝐱𝑟 , 𝑡″ , 𝛼𝑟̂ , ∶)∣ .
𝑡″
(4.15)
𝐱𝑟 ∈𝐱𝑠𝑟
This is advantageous as it avoids the necessity of computing the Green’s functions
from all receivers independently, dramatically reducing the computational cost. We
make this assumption purely to improve computational efficiency; it is not required,
and may be omitted (by computing the illumination for each receiver separately) if
its validity is in doubt. 𝐼𝑙𝑙𝑢𝑚𝑟 is calculated by propagating a unit impulse from the
receiver locations, to approximate the Green’s function (with higher accuracy when
smaller time step sizes are used). The square of this Green’s function, binned by
propagation direction 𝛼𝑟 , is then accumulated over time.
Next, we assume that the data consist of specular reflections from planar perturbations to 𝑚0 , and that we have good source and receiver coverage, which, combined
with Equation 4.3, enables us to calculate 𝜃 from 𝛼𝑠̂ and 𝛼𝑟̂ . Inserting Equation 4.13
into 4.12 and summing over scattering angle (ignoring amplitude variations due to
scattering angle), yields
𝐼𝐱𝑠 (𝐱, 𝜃) = ∑ 𝑚1 (𝐱, 𝜃) cos𝑛 𝜙𝐼𝑙𝑙𝑢𝑚𝑠,𝐱𝑠 (𝐱, 𝜃−𝜙/2)𝐼𝑙𝑙𝑢𝑚𝑟,𝐱𝑠 (𝐱, 𝜃+𝜙/2)+𝑅(𝐱, 𝐱𝑠 , 𝜃, 𝜙),
𝜙
(4.16)
where the source illumination is
𝐼𝑙𝑙𝑢𝑚𝑠,𝐱𝑠 (𝐱, 𝛼𝑠̂ ) = ∑ 𝑢𝑠,𝐱𝑠 (𝐱, 𝑡′ , 𝛼𝑠̂ )
𝑡′
𝜕2
𝑢
(𝐱, 𝑡′ , 𝛼𝑠̂ ),
𝜕𝑡′2 𝑠,𝐱𝑠
(4.17)
and where 𝑅 in Equation 4.16 is the second term in Equation 4.12, involving 𝐱′ ≠ 𝐱.
𝐼𝑙𝑙𝑢𝑚𝑠 can be calculated by forward propagating the source wave 𝑢𝑠 , and accumulating over time at each location the product of this wave with its second time derivative,
binned by propagation direction 𝛼𝑠 . With our assumption of planar reflectors comes
a new model of subsurface parameters, where we assume the scattering amplitude at
a point is the sum of the amplitude of differently-oriented reflectors passing through
119
the point,
𝑚(𝐱) = ∑ 𝑚(𝐱, 𝜃).
(4.18)
𝜃
We now introduce a new term to reduce notation, and to clarify the physical
meaning of expressions. The acquisition dip response (ADR) is the critical quantity
that tells us how well a reflector of a given orientation and position is illuminated.
It incorporates both our source and receiver geometry, and features present in our
velocity model that can affect illumination, such as salt bodies.
The ADR for a shot is given by the product of the source and receiver illuminations,
and the scattering angle filter. Unlike the source and receiver illuminations, which are
binned by propagation direction, ADR depends on reflector orientation. We therefore
need to convert using the same relations as described above for the uncompensated
images, defining a shot’s ADR as
𝐴𝐷𝑅𝐱𝑠 (𝐱, 𝜃) = ∑ 𝐼𝑙𝑙𝑢𝑚𝑠,𝐱𝑠 (𝐱, 𝛼𝑠 )𝐼𝑙𝑙𝑢𝑚𝑟,𝐱𝑠 (𝐱, 𝛼𝑟 ) cos𝑛 𝜙,
(4.19)
𝜙
where the scattering angle, 𝜙, is as defined in Equation 4.5.
Turning our attention to the second term in Equation 4.16, 𝑅, we must estimate
the effect that other points will have on the image at 𝐱. To accomplish this, we
again use our assumption of planar reflectors, which we now augment with the additional assumptions that the perturbations 𝑚1 along these planar reflectors, and
the illumination, are locally constant. von Seggern (1991) shows that with a Ricker
wavelet source, the impulse response (point spread function) of the imaging operation is a Ricker wavelet in the direction parallel to the vector sum of the propagation
directions of the source and receiver waves, and a Gaussian in the perpendicular direction. It is expected the situation will be similar for other source wavelets, since
the convolution of an oscillatory function with its second time derivative is another
oscillatory function (in the frequency domain 𝜔2 𝑈 (𝜔)𝑈 (𝜔) has zero DC component,
like the Fourier transform 𝑈 (𝜔) of oscillatory time-domain function 𝑢(𝑡)). A reflector
can be considered to be a line of point scatterers, and so the resulting image will
120
be the sum of the point scatterer impulse response functions. If the reflector normal
is oriented parallel to the vector sum of the propagation direction of the source and
receiver waves, the amplitude at a point for this orientation will be proportional to
the integral of the Gaussian. Therefore,
𝐼𝐱𝑠 (𝐱, 𝜃) ∝ 𝑚1 (𝐱, 𝜃)𝐴𝐷𝑅𝐱𝑠 (𝐱, 𝜃)𝑊𝐱𝑠 (𝐱, 𝜃),
(4.20)
where 𝑊𝐱𝑠 (𝐱, 𝜃) is proportional to the integral over the impulse response parallel to
a reflector at 𝐱 with orientation angle 𝜃. There have been several proposed equations
for approximations of the impulse response (see, for example, Vermeer and Beasley
(2012, Chapter 8) and references therein). These relate the width of the Gaussian at
a point to angular coverage of source and receiver propagation directions there. For
a 2D offset experiment, Vermeer and Beasley (2012) give the horizontal resolution
(Δ𝐻, which is related to the horizontal width of the impulse response) as
Δ𝐻(𝐱) ∝
𝜆peak (𝐱)
,
2(sin 𝜃𝑠 + sin 𝜃𝑟 )
(4.21)
where 𝜆peak is the dominant wavelength, and 𝜃𝑠 and 𝜃𝑟 are the largest angles from
the sources and receivers to 𝐱, measured from the 𝑥-axis. This assumes continuous
source and receiver coverage up to the maximum angle. In situations where this is
not the case a more sophisticated approximation for point spread function width may
be necessary. This enables us to estimate 𝑊 , and therefore to incorporate the effect
of neighboring scatterers along the reflectors passing through a point on its image
amplitude. In this way, we approximate the effect of the second term in Equation
4.12 on the image.
121
4.2.3
Illumination compensation
To remove the effect of illumination on the image, we divide the uncompensated image
by the illumination. The compensated image for a single shot is therefore:
𝐼𝐱𝑐 𝑠 (𝐱, 𝜃) =
𝐼𝐱𝑠 (𝐱, 𝜃)
𝐴𝐷𝑅𝐱𝑠 (𝐱, 𝜃)𝑊𝐱𝑠 (𝐱, 𝜃)
(4.22)
The robustness of regular RTM is significantly enhanced by stacking the output
images from different shots. We wish to obtain the same benefit, however merely
stacking the compensated images would not incorporate information about illumination: the amplitude at a point from a shot that illuminates that point poorly would
be given equal weight to a shot that illuminates it well. To account for this, we
instead stack by calculating the mean weighted by illumination. The stacked image
is therefore:
𝑐
𝐼 (𝐱, 𝜃) =
∑𝐱
𝐬
𝐼𝐱𝑠 (𝐱,𝜃)
𝐴𝐷𝑅𝐱𝑠 (𝐱, 𝜃)𝑊𝐱𝑠 (𝐱, 𝜃)
𝐴𝐷𝑅𝐱𝑠 (𝐱,𝜃)𝑊𝐱𝑠 (𝐱,𝜃)
∑𝐱 𝐴𝐷𝑅𝐱𝑠 (𝐱, 𝜃)𝑊𝐱𝑠 (𝐱, 𝜃)
(4.23)
𝐬
=
∑𝐱 𝐼𝐱𝑠 (𝐱, 𝜃)
𝐬
∑𝐱 𝐴𝐷𝑅𝐱𝑠 (𝐱, 𝜃)𝑊𝐱𝑠 (𝐱, 𝜃) + 𝜖
,
(4.24)
𝐬
where we have added a small constant 𝜖 for stability. In order to account for decreasing
image and illumination amplitudes with distance from the sources and receivers, one
may wish to make 𝜖 location-dependent.
As stated previously, this derivation assumes that scatterers occur as planar reflectors that produce specular reflections, precluding its application to diffractors. The
problem with diffractors is that Equation 4.18 will be violated, as every imaged reflector orientation at the location of the diffractor will appear to have the same scattering
amplitude. Summing over reflector orientations to produce the final image will then
scale the amplitude of the diffractor by the number of imaged reflector orientations.
This problem could be avoided by identifying diffractors before illumination compensation is applied. Rather than summing the 𝐼 𝑐 (𝐱, 𝜃) of Equation 4.24 over 𝜃 to create
a single image of all reflector orientations, for the points that contain diffractors we
122
would instead use
𝐼 𝑐 (𝐱) =
∑𝜃 ∑𝐱 𝐼𝐱𝑠 (𝐱, 𝜃)
𝐬
∑𝜃 ∑𝐱 𝐴𝐷𝑅𝐱𝑠 (𝐱, 𝜃) + 𝜖
,
(4.25)
𝐬
which sums over scattering angle before dividing by illumination, thereby only calculating a single scattering amplitude for the point. The point spread function width,
𝑊 , is set to 1 as there are now no other scatterers along the reflector to contribute
to the image amplitude at the diffractor. This issue is also a concern for very short
reflectors. Ideally, the procedure used to produce the image should transition from
Equation 4.25 to Equation 4.24 as reflectors become longer over the width of the
impulse response width, however transitioning accurately requires knowledge of the
impulse response function. Only considering the component of the image and illumination with the same orientation as the line of scatterers avoids this complication,
even for very short reflectors.
Our modifications to RTM are similar to some previous proposals, in particular
the time-domain illumination compensation method of Yang et al. (2008). We believe
our approach is unique for highlighting the importance of illumination compensation
when using multiples and overturned waves to make effective image contributions,
and reducing artifacts normally associated with these arrivals by including a technique for allowing an interface to be imaged from both sides without destructive
stacking and employing the scattering angle filter of Costa et al. (2009) in an efficient
manner. Furthermore, unlike previous proposals, we discuss the effect of neglecting
𝑅 in Equation 4.12 and describe a potential means of approximating this term. Examples demonstrating the effects of these modifications are presented in the results
section.
4.2.4
Uncertainty
Regular RTM conveys uncertainty in images by reducing amplitude. Poorly illuminated areas, or areas where different shots do not stack coherently, have low amplitude. This uncertainty is, however, mixed with the effect of reflectivity. It is not
clear whether a location in the image has low amplitude because of low reflectivity,
123
or because of one of the measures of uncertainty: low illumination or lack of agreement among shots. To clarify the uncertainty in images, we propose complementing
compensated images with images of ADR and a measure of coherence between shots.
For the latter, one possibility is to compute the standard deviation of 𝐼𝐱𝑐 𝑠 (𝐱, 𝜃) with
respect to 𝑠 (shot), weighted by ADR, using
√
2
√
∑𝐱 𝐴𝐷𝑅𝐱𝑠 (𝐱, 𝜃) (𝐼𝐱𝑐 𝑠 (𝐱, 𝜃) − 𝐼 𝑐 (𝐱, 𝜃))
√
𝐬
√
.
∑𝐱 𝐴𝐷𝑅𝐱𝑠 (𝐱, 𝜃)
𝐬
⎷
(4.26)
This expresses the lack of consistency between shots, with a small number indicating
a small spread in amplitude between shots. Some spread is expected, since different
shots will produce waves that reflect from a point at different angles, experiencing
different coefficients of reflection. This could be reduced by attempting to approximately compensate for scattering angle. Algorithms such as that proposed by West
(1979) allow the standard deviation to be computed in a single pass, which means
that shots can be added to it as they are calculated, and so images from all of the
shots do not need to be stored.
4.3
Results
In this section we present several examples which demonstrate the improvement of
image amplitude accuracy by avoiding or reducing the amplitude errors discussed
earlier. We explore the improvement in relative image amplitude due to the 𝑊 factor
in Equation 4.20. We then show the effect of imaging an interface from both sides
using regular RTM, and the improvement when the proposed method is used. This
is followed by an example of the illumination compensation component of the new
method attenuating a phantom reflector artifact caused by the use of a velocity model
with a reflector. We next demonstrate that the method can produce superior results
to conventional RTM even when the recorded data contain a substantial amount of
noise, and then compare the image of a simple layer model produced using the method
with that of the source-normalized cross-correlation imaging condition. Finally, we
124
apply the proposed method to a 2D portion of the SEAM model.
4.3.1
Improvement due to 𝑊 factor
1
900m
900m
700m
700m
500m
300m
100m
Amplitude
Amplitude
1
500m
300m
100m
0
0
Width (m)
0
965
0
a
Width (m)
965
b
1
0
900m
100m
Amplitude
Amplitude
1
0
Width (m)
0
965
c
0
Width (m)
965
d
Figure 4-3: Normalized image amplitude at the central point on a horizontal line
of scatterers (“reflector”), as the length of the line and the source/receiver aperture
(125 m above the scatterers, symmetric about the 𝑥 coordinate of the chosen point)
vary. The amplitude should ideally be the same in all cases. The x axis represents
reflector length, while plotted lines correspond to different source/receiver aperture
widths. (a) Regular RTM. (b) Illumination compensated without the 𝑊 term in
Equation 4.20. (c) Illumination compensated with a simple approximation for 𝑊 .
(d) Illumination compensated with a more sophisticated approximation for 𝑊 .
Regular illumination compensation, even when the angular dependence of illumination is considered, usually does not account for variations in the width of the
impulse response function, which may compromise the accuracy of the resulting relative amplitudes. This is addressed by the 𝑊 factor in Equation 4.20. To examine
the effect of this enhancement, we consider a horizontal line of scatterers of variable
length 𝑙, and a source and receiver aperture that is also of variable length. To avoid
the need to use Equation 4.25, we only consider the image amplitude of reflectors
with a horizontal apparent orientation (so we do not sum over reflector orientations
125
after compensation, as this would scale the amplitude of the diffractors), except for
the final example. The regular RTM amplitude of the central point on the reflector
is shown in Figure 4-3a, and Figure 4-3b shows the amplitude after compensating for
the first term in Equation 4.12, but not accounting for 𝑊 . The scattering amplitude
is the same in every case, so the amplitude of the point in the image should ideally
be constant across all cases. The images are normalized so that the maximum value
is 1. Applying illumination compensation is observed to reduce the range of variation
in normalized amplitude over the different cases by more than 4%, and by over 30%
for the longest reflector. Predicting the effect of neighboring scatterers on the image
amplitude in the various cases by approximating 𝑊 as the minimum of the impulse
response width and the actual width of the reflector, should reduce the image amplitude variations between the different cases. Making the simplifying approximation of
estimating the impulse response width to be equal to the wavelength results in Figure
4-3c. We observe that this further reduces the range of normalized image amplitudes
by almost 30%, primarily by improving the relative amplitude between very short
reflectors and longer reflectors. To further improve the image, we acknowledge that
the point spread function is not the same for all cases, or even for all scatterers in
the line. We use Equation 4.21 to estimate variations in point spread function width.
This equation calculates the horizontal impulse response width when all illumination
orientations are combined. We therefore stack the image over orientation angles before applying this compensation. Using our assumed knowledge of the reflector width
in the horizontal direction, 𝑙, and approximating the impulse response function to be
a Gaussian, the compensated image is obtained using
𝐼𝐱𝑐 𝑠 (𝐱) =
∑𝜃 𝐼𝐱𝑠 (𝐱, 𝜃)
𝑙/2
−𝑥′2
∑𝑥′ =−𝑙/2 ∑𝜃 𝐴𝐷𝑅𝐱𝑠 (𝐱 + 𝑥′ 𝑥,̂ 𝜃) exp 2∆𝐻(𝐱+𝑥
′ 𝑥)
̂ 2
.
(4.27)
Using this approach produces the result shown in Figure 4-3d, which has a minimum amplitude that is 62% of the largest amplitude. This rises to 81% for the
largest reflectors, which means there is only about 20% amplitude variation even
though the source-receiver aperture increases to up to 9 times its smallest size. Un126
like the previous attempts, this approach slightly overestimates the illumination of
long reflectors relative to the illumination of a diffractor for all apertures, resulting in
the point diffractor having the highest compensated image amplitude. It is unlikely
that significantly better results are possible without accurately determining the point
spread function. This has been attempted, such as by Valenciano et al. (2006), but
is computationally expensive.
Despite the noticeable improvement in relative amplitude accuracy obtained by
estimating 𝑊 , in the following results we neglect this term to avoid the possibility of
compromising results with the approximations inherent in Equation 4.21, and because
it requires the reflector width to be specified at each point.
4.3.2
Imaging from opposite sides
A challenge presented by the use of overturned waves and internal multiples for imaging is the tendency to image interfaces from the opposite side to primaries. This
is because they are often incident on interfaces from below, while primaries image
from above. Such a situation may also occur with primaries alone, however, where
a vertical interface may be imaged from both sides, for example. In conventional
RTM, imaging an interface from both sides would lead to reduced image quality due
to image contributions subtracting from each other, as they experience reflection coefficients with the opposite sign. This is shown in Figures 4-4a and 4-4b, where adding
image contributions incident from opposite sides of an interface results in almost
complete cancellation. By determining the apparent reflector orientation during the
application of the imaging condition, the proposed method is able to alter the sign
of image contributions when appropriate so that they stack coherently, regardless of
which side the interface is imaged from. The result of applying the proposed method
is shown in Figure 4-4c. Rather than resulting in cancellation, all of the data is now
used to more accurately estimate the amplitude of the image.
This example shows an extreme situation in which the image contributions from
above and below the interface have equal amplitude, causing significant cancellation.
When the interface is imaged from one side by primaries and the other by internal
127
0.55
z (km)
0.45
z (km)
0.45
z (km)
0.45
0.55
0.55
Amplitude
Amplitude
Amplitude
a
b
c
Figure 4-4: Vertical slices through the image of a horizontal layer that is equally
illuminated from above and below. (a) The image contributions when the layer is
imaged from above and below are shown separately. (b) The conventional RTM
imaging condition does not distinguish between image contributions from different
sides of an interface, simply adding all contributions, resulting in almost complete
cancellation in this case. (c) The proposed imaging condition reverses the sign of
image contributions such that they always stack coherently, regardless of which side
the interface is imaged from, resulting in a significantly improved image.
128
multiples, the amplitudes are unlikely to be so evenly matched, however some deterioration in image quality would still occur unless the proposed method (or something
similar) were used.
4.3.3
Artifact attenuation
Using the most accurate velocity model that is available, including sharp interfaces,
ensures that waves are propagated as accurately as possible during imaging, and enables the generation of internal multiples. The inclusion of discontinuities also results
in backscatter and phantom reflector artifacts, however. The imaging algorithm we
propose attenuates both of these types of artifacts, allowing velocity models that generate internal multiples to be used. This is demonstrated using the velocity model
shown in Figure 4-5a. The exact velocity model, which contains sharp interfaces, is
also used during migration. The result obtained using regular RTM, which is contaminated by severe backscatter artifacts, is displayed in Figure 4-5b. If the imaging
condition is replaced by the one proposed by Costa et al. (2009), the same means used
in the proposed method to reduce backscatter artifacts, the improved result shown
in Figure 4-5c is produced. While this has successfully reduced the low frequency
artifacts, a phantom layer artifact, indicated by an arrow, is still present. Although
backscatter artifacts are unwanted, as they may obscure important parts of the image,
phantom reflector artifacts also have the potential to be very damaging as they risk
being interpreted as real subsurface features. The phantom reflector in this example
is created by reflections from the upper interface of the high velocity layer. The phantom reflector is also illuminated by higher amplitude primaries, which do not see a
reflector at that location. Dividing by the illumination, as is done in the illumination
compensation component of the proposed method, therefore reduces the amplitude
of the phantom reflector artifact, as can be seen in Figure 4-5d.
Illumination compensation in fact has the ability to attenuate any type of artifact
that is only produced by certain wave paths, when other wave paths pass through
the region. The approach proposed by Fei et al. (2014) is capable of reducing certain
types of artifacts, but in doing so removes the ability to use overturned waves and
129
0
0
0
z (km)
z (km)
z (km)
0
2
2
2
z (km)
1500 m/s
2000 m/s
2
1500 m/s
0
x (km) 0.4
a
0
x (km) 0.4
0
b
x (km) 0.4
c
0
x (km) 0.4
d
Figure 4-5: Backscatter and phantom layer artifacts are caused by reflectors in the migration velocity model with the regular RTM imaging condition, but are attenuated
with the proposed method. (a) The velocity model, consisting of a high velocity layer
sandwiched between two lower velocity layers, that was used for receiver data generation and migration. (b) The image obtained using the conventional cross-correlation
imaging condition. Significant backscatter artifacts are present. (c) Applying the
scattering angle filter imaging condition of Costa et al. (2009) reduces backscatter
artifacts, but the phantom layer artifact remains, as indicated by an arrow. (d) The
image produced using the proposed method. Backscatter and phantom reflector artifacts are attenuated.
130
internal multiples for imaging in some cases. Illumination compensation may not
attenuate such artifacts as strongly as this approach, but it does not suffer from the
same limitation.
4.3.4
Internal multiples in noisy data
Boosting the amplitude of weakly illuminated areas, relying on accurate propagation
direction determination, and attempting to use weak arrivals such as internal multiples for imaging, may raise concerns about the robustness of the proposed method
to noisy data. To allay these apprehensions, we apply the method to a dataset that
has had a considerable amount of uncorrelated zero mean Gaussian noise added. The
model, shown in Figure 4-6a, consists of a salt body over a salt layer. We assume that
the location of the salt layer is known (as it is well imaged using primaries), and so
it is included in the velocity model used for migration, but the salt body is replaced
by the velocity of the surrounding sediment, as shown in Figure 4-6b. The inclusion
of the salt layer in the migration velocity model enables the generation of internal
multiples, which may be used to image the flank of the salt body. Applying regular
RTM to the data produces the result shown in Figure 4-7a. The low amplitude internal multiple contributions to the salt flank are visible when large clipping is applied
to the displayed image amplitudes, but this also increases the impact of noise on the
image. The proposed method boosts the image contributions of the weakly illuminating internal multiples relative to the noise, greatly improving the visibility of the
salt flank, as shown in Figure 4-7b. As expected, removing the effects of illumination
variations also causes the amplitude of the areas of the salt flank imaged with internal
multiples to be comparable with the upper left corner of the salt body, which was
imaged with primaries.
Illumination information also provides useful insight to the interpreter. Figure 4-8
shows the image of reflectors that have an apparent horizontal orientation. We complement this with the illumination of horizontal reflectors. This allows interpreters to
see that the reason why there are no horizontal reflectors in the image in the region
of the salt body is not necessarily because none are present, but rather because our
131
4500
z (km)
Wave speed (m/s)
0
0.9
1500
0
x (km)
1
a
4500
z (km)
Wave speed (m/s)
0
0.9
1500
0
x (km)
1
b
Figure 4-6: (a) The velocity model used to generate receiver data, consisting of a high
velocity salt body (right) and salt layer (bottom), surrounded by a smooth gradient.
Source positions are indicated by circles, and receiver positions by triangles. (b) The
velocity model used for migration. The salt body has been replaced by a sediment
fill.
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z (km)
0
0.7
0.3
x (km)
1
a
z (km)
0
0.7
0.3
x (km)
1
b
Figure 4-7: (a) The image produced by regular RTM, focused on the region containing
the salt body. The upper left corner of the salt body has been imaged by primaries,
and so is much higher amplitude than the rest of the salt flank (indicated by an
arrow), which was imaged with internal multiples. The range of displayed amplitudes
has been severely clipped so that the internal multiple image contributions are visible.
(b) The image when the proposed method is used. The image contributions of the
internal multiples now have amplitude comparable to that of the primaries. The true
location of the salt interface is indicated by the dotted line.
133
current acquisition geometry is not capable of illuminating any there.
High
0.7
Illum.
z (km)
0
z (km)
0
0.7
0.3
x (km)
1
Low
0.3
a
x (km)
1
b
Figure 4-8: One means of conveying image uncertainty is by complementing the
image with a measure of illumination. (a) The image, using the proposed method, of
reflectors determined during the application of the imaging condition to be horizontal.
The outline of the true location of the salt body is shown for reference. (b) The
illumination of horizontal reflectors. Horizontal reflectors in the region of the salt
body are very poorly illuminated, indicating that if any exist there they may not be
imaged.
4.3.5
Comparison with source-normalized imaging condition
The source-normalized imaging condition (Claerbout, 1971) has been shown to be
capable of producing images with more accurate amplitudes than the conventional
cross-correlation method (Chattopadhyay and McMechan, 2008). This is achieved
by removing some of the effects of variations in illumination. The method does not
require any additional propagation steps compared to regular RTM, as it only uses
the source illumination (Equation 4.17), which can be calculated during the forward
propagation stage. It therefore eliminates at least one backpropagation per shot compared to the proposed method (more if the assumption used in Equation 4.15 is not
used). In our implementation of the source-normalized imaging condition, we retain
the dependence on source propagation direction, but propagation directions of the
backpropagated data wavefield no longer need to be calculated. Furthermore, the illumination storage requirement is approximately half that of the proposed method as
it only needs to store the source illumination. However, the method assumes infinite
receiver aperture, which is unlikely to ever be true. As a result, it does not properly ac134
count for a finite receiver aperture, leading to incorrect image amplitudes for realistic
surveys. Although it is more computationally demanding, the illumination compensation component of the method we propose does not suffer from this limitation. This
is demonstrated by Figure 4-9, which shows the output of RTM using the regular
cross-correlation imaging condition, the source-normalized cross-correlation imaging
condition, and the proposed method, for a model consisting of four layers with equal
scattering amplitudes. Although an improvement over the regular imaging condition,
the amplitude of the layers still varies with location in the source-normalized case,
due to the changing angular range of receiver coverage, while the image made using the proposed method correctly shows very similar amplitudes for all layers. The
slight decrease in amplitude with depth is likely to be due to variations in the point
spread function width or the increasing effect of the stabilizing term in the denominator of Equation 4.24 as illumination decreases rapidly with depth. The latter could
potentially be avoided by decreasing 𝜖 with depth, as suggested previously.
4.3.6
SEAM
The above examples all use relatively simple velocity models. To demonstrate the
effectiveness of the method in a more realistic situation, we apply it to a 2D portion
of the SEAM model (Fehler and Larner, 2008). To avoid obscuring the results with
out-of-plane reflection artifacts, we use data generated using 2D modeling through the
chosen slice (Figure 4-10a). The migration velocity model, shown in Figure 4-10b, was
created by smoothing the true acoustic model, with greater smoothing horizontally
than vertically, due to seismic resolution often being higher in the vertical direction.
The regular RTM result, Figure 4-11a, images most of the subsurface structure
well, but has some weaknesses along the salt interface, particularly underneath the
overhang. Difficulties in this area are not unexpected, since it is not well illuminated
using primaries. The underside and flanks of salt bodies are, however, areas of great
interest for hydrocarbon exploration, and so these may, in fact, be the most important
regions to image clearly. Overturned waves and internal multiples are better suited
to imaging these areas. The proposed method, which is particularly appropriate
135
z (km)
0
z (km)
0
1
1
0
x (km) 0.5 Amplitude
0
x (km) 0.5 Amplitude
a
b
z (km)
0
1
0
x (km) 0.5 Amplitude
c
Figure 4-9: The image of four layers of equal scattering amplitude in a constant background. 3D propagation was used for modeling and migration. The amplitude of
a vertical line through the center of the image is also shown. Sources and receivers
cover the top surface of the model, with a spacing of 20 m and 10 m, respectively.
(a) Regular RTM cross-correlation imaging condition, showing significant amplitude
variation. (b) Source-normalized cross-correlation imaging condition, which is an improvement, but amplitude variation is still noticeable. (c) Illumination-compensated
image, with significantly more consistent amplitudes than the other two approaches.
136
for improving the image contributions from such waves, successfully produces an
improved image of the salt interface, as shown in Figure 4-11b. There continue to be
sections of the salt interface which are not well imaged, but this is likely to be because
they are not illuminated even using overturned waves and internal multiples with the
current acquisition geometry, and so may require additional data acquisition.
Due to the large size of the portion of the SEAM model considered, the Poynting
vector method (Yoon and Marfurt, 2006) was used for propagation direction determination to reduce computational cost. Inaccuracies in the results of this method when
waves overlap lead to errors in the application of the proposed method, causing image
artifacts. This is particularly obvious underneath the salt overhang, where artifacts
in the image created using the proposed method coincide with the region where waves
reflecting from the differently oriented surfaces of the salt body are likely to overlap.
Calculating the weighted standard deviation of the image across shots, as described
above, highlights image artifacts. This is because they generally have less consistency
across shot images than real subsurface features. The image of this measure of uncertainty is shown in Figure 4-11c. It is apparent that it has successfully identified
many of the artifacts visible on the illumination compensated image.
It is interesting to note that the image amplitude on the lowest horizontal surface
of the salt body (just above 6 km in depth, 10 km to 12 km horizontal) is visible but
not correctly positioned or well focused in the regular RTM image, and disappears
when illumination compensation is applied. It is likely that this feature is not well
imaged as it is inaccessible with primaries. It can be reached by internal multiples,
but these must reflect off of the vertical portion of the salt body, which has had strong
smoothing applied to it in the migration model. The poor image quality is probably
exascerbated by illumination compensation because a substantial amount of energy
from sources directly above the salt will be transmitted through the smooth upper
part of the salt body during the illumination calculation, making this horizontal
feature appear as if it is well illuminated. Dividing by the illumination therefore
futher weakens the image amplitude along this reflector.
137
4.5
z (km)
Wave speed
(km/s)
0
6
1.5
0
x (km)
12
a
4.5
z (km)
Wave speed
(km/s)
0
6
1.5
0
x (km)
12
b
Figure 4-10: (a) The P-wave velocity of the extracted 2D portion of the SEAM
model that is imaged. The high velocity structure on the right is a salt body. (b)
The smoothed velocity model used during migration.
138
z (km)
0
6
4
x (km)
12
a
z (km)
0
6
4
x (km)
12
b
z (km)
0
6
4
x (km)
12
c
Figure 4-11: Images of the 2D portion of the SEAM model shown in Figure 4-10,
focused on the salt body. (a) Regular RTM fails to clearly image the areas on the
underside of the salt overhang indicated by arrows. (b) The proposed method results
in improved amplitude accuracy under the salt overhang. (c) Image of weighted
standard deviation of image amplitude across shots (Equation 4.26), divided by the
absolute value of image amplitude, highlighting inconsistencies, which are primarily
artifacts.
139
4.4
Discussion
The proposed method is shown to improve amplitude accuracy by accounting for illumination, resulting in reflectors having correct relative amplitude, and attenuating artifacts by using the fact that other high amplitude waves were incident on the same location but did not sense a reflector there. This could potentially be extended. Rather
than averaging over scattering angle, as is done currently, illumination-corrected image amplitude as a function of scattering angle could be used to fit reflection coefficient
curves. This would allow images to be displayed as the best fitting normal incidence
reflection coefficient at each point so that image amplitude is no longer related to the
range of incidence angles. A measure of the goodness of fit of the data to the reflection coefficient curve could then serve as a measure of uncertainty. The ability to
determine reflection coefficient parameters could further be used for velocity analysis.
This approach would be challenging to implement currently, however, as it makes the
assumption that there is no more than one reflector per grid cell. This is unlikely to
be true in most geologies with current grid cell sizes and seismic wavelengths. It also
assumes specular reflections, but this constraint could be relaxed by the inclusion
of additional model parameters that allow cells to contain diffractors. Finally, good
angular coverage of every cell in the imaging domain would be needed for a reliable
inversion of the reflection parameters.
In addition to illumination compensation, another major component of the proposed method is reversing the sign of image contributions when necessary such that
contributions from waves incident on opposite sides of an interface stack coherently.
This is of particular importance when internal multiples and overturned waves are
included in the imaging process, since they often image interfaces from the opposite
side to primaries. It raises the question of the meaning of the reflector polarity in
images, which is often relied upon by interpreters. The current practice, which makes
no distinction based on the direction in which a wave is incident on an interface, has
not caused much controversy as primaries tend to always image interfaces from the
same side, usually from above, but even with primaries there are situations, such as
140
with near-vertical structures, where the interpreter must attempt to guess which side
an interface has been imaged from, and where destructive stacking of image contributions from different sides may reduce image quality. The approach we propose has
drawbacks, primarily the surprise that may be caused as the polarity of an interface
flips when its orientation turns through one of the two flipping points, but it provides
consistency and removes a little of the guesswork from interpretation.
4.5
Conclusion
In this paper we propose modifications to RTM to improve amplitude accuracy, in
particular a time-domain illumination compensation algorithm, and reversing the sign
of image contributions in such a way that contributions stack coherently even when
an interface is imaged from opposite sides. The key quantity which must be calculated for both of these is the propagation direction of waves. Once these directions
are available, they can also be used to apply the scattering angle filter of Costa et al.
(2009). Although calculating direction information is relatively computationally expensive, its reuse for these three processes results in good value being derived from
it. The modifications we propose are particularly beneficial for imaging with internal
multiples and overturned waves. They also provide means of determining two forms
of image uncertainty.
141
142
Chapter 5
Single wavefield RTM: reducing
artifacts and computational cost
Abstract
We propose a seismic imaging algorithm which can reduce image artifacts while having
lower computational requirements than RTM. This is achieved by modifying RTM to
only backpropagate a single wavefield, computed using Green’s Third Identity. We
demonstrate its effectiveness at reducing phantom reflector artifacts on a simple layer
model, analyze its sensitivity to model and data errors, and find that it is able to
produce a similar image to regular RTM on a smoothed 2D portion of the SEAM
model, at reduced computational expense.
5.1
Introduction
Active source seismic imaging is a geophysical technique that attempts to produce a
structural image of a volume from measurements of mechanical waves arriving at the
surface of the volume due to a controlled energy source. The input source wavelet,
source location, recorded data, and an estimate of wave propagation speed in the
volume are usually the prerequisites for producing an image. Many seismic imaging
algorithms to perform this operation, known as migration, have been proposed. Reverse Time Migration (RTM, Baysal et al. (1983)) is currently regarded as the most
accurate, due to its ability to closely adhere to the physics of finite-frequency wave
143
propagation.
Similarly to many other such algorithms, RTM consists of two steps. In the first
step of RTM, the source wavelet is injected as a source term in a numerical simulation
of wave propagation in the volume to be imaged.
𝐿𝑢𝑠 (𝐱, 𝑡) = 𝑠(𝑡)𝛿(𝐱 − 𝐱𝑠 ),
(5.1)
where 𝐿 is a wave operator, such as that of the constant density scalar/acoustic
wave equation (∇2𝑥 −
1 𝜕 2 ),
𝑐(𝐱)2 𝑡
𝑢𝑠 is the numerical source wavefield, 𝑠 is the source
wavelet (injected source energy as a function of time), and 𝐱𝑠 is the spatial location
of the source. As time progresses, the source wave will expand into the volume. How
faithfully it represents the evolution in time of the source wave in the real seismic experiment depends on the accuracy of the input source characteristics, velocity model,
and wave propagator. The purpose of this propagation forward in time is that it will
be necessary to recall the source wavefield at each time step in reverse order in the
second step of RTM. A means of recreating the wavefield must be employed, such as
the use of checkpointing (Symes, 2007) or saving the wavefield at the boundaries of
the simulation domain (Dussaud et al., 2008).
In the second step, the recorded data is injected at each of the receiver locations
in a time-reversed wave simulation.
𝐿𝑢𝑑 (𝐱, 𝑇 − 𝑡) = ∑ 𝑑𝑟 (𝑇 − 𝑡)𝛿(𝐱 − 𝐱𝑟 ),
(5.2)
𝑟
where 𝐿 is as in Equation 5.1, 𝑇 is the maximum recording time of the seismic
experiment, 𝑢𝑑 is the numerical backpropagated data wavefield, and 𝑑𝑟 is the data
recorded by the receiver at position 𝐱𝑟 . At each time step this backpropagated data
wavefield, and the recreated source wavefield at the same time, are used in an imaging
condition, which determines what amplitude to add to the image. The most popular
imaging condition is the zero-lag cross-correlation (Claerbout, 1971), which effectively
assumes that there is a subsurface reflector wherever the source and data waves are
144
coincident in time and space,
𝐼(𝐱) = ∑ 𝑢𝑠 (𝐱, 𝑡)𝑢𝑑 (𝐱, 𝑡).
(5.3)
𝑡
The accuracy of this assumption is partially reliant on how closely 𝑢𝑠 and 𝑢𝑑
match the true seismic wavefield. As stated above, 𝑢𝑠 can be improved through the
use of a better velocity model and propagator. The backpropagated data wavefield,
𝑢𝑑 , will, on the other hand, often be a poor representation of the true wavefield. This
is because it will not contain any portion of the wavefield that did not arrive at the
receivers. This is adequate for imaging with primary reflections, but, as we show
below, its flaws become more apparent with other types of arrivals.
Weglein et al. (2011a,b) explain that RTM makes the infinite hemisphere assumption, which states that the extent of the receiver coverage is infinite along one surface
of the simulation domain. While this assumption is clearly not satisfied, even if it
were, it is only valid for constant velocity models. Only having receivers covering a
portion (usually one side) of the domain will in fact lead to incorrect backpropagation
of the data wavefield. Even if the correct velocity model is used, the data wavefield
will not match the true wavefield in the seismic experiment. This is demonstrated in
Figure 5-1, where a source wave is forward propagated through a model consisting of
a layer over a halfspace. When the wave reaches the interface at the base of the layer,
part of its energy is reflected, while the remainder is transmitted and continues to
propagate downward. A receiver on the top surface only records the reflected component. Injecting this into the domain during a time reversed simulation, the wave
propagates down to the interface and again separates into reflected and transmitted
components. Although the reflected wave will correctly return to the source location at the right time, the backpropagated wavefield is missing the component that
was transmitted into the halfspace during the initial forward propagation, and, as a
result of this wave not returning to meet the backpropagated reflected wave at the
interface, a portion of the backpropagated wave is incorrectly transmitted into the
halfspace and so does not return to the source. Leaking energy in this way causes the
145
wave backpropagated along the correct path to have lower amplitude than it should,
reducing amplitude accuracy of image contributions.
a
b
Figure 5-1: Demonstration of incorrect backpropagation in RTM. (a) Forward propagation from the source (circle) to a reflector in the subsurface, where the wave splits
into a reflected component (dashed), which returns to the surface where it is recorded
by a receiver (triangle), and a transmitted component (dotted), which continues to
propagate downward. The y axis represents depth, while the x axis could be either
time or horizontal distance. (b) The backpropagated recorded data also separates
into reflected (dashed) and transmitted (dotted) components at the reflector, causing
the backpropagated wavefield to not truly represent the seismic wavefield.
A negative effect of this incorrect backpropagation is the presence of phantom
reflector artifacts in the output image. These are artifacts that, unlike some other
common RTM image errors, such as the low frequency smears caused by backscatter
(Yoon and Marfurt, 2006), may appear like real structure and so present a large
risk of misinterpretation. Figure 5-2 shows two ways in which these artifacts may
occur through incorrect backpropagation. In the first, the backpropagated reflection
from true reflector 2 is incorrectly reflected on true reflector 1, producing a phantom
reflector. In the second example, an internal multiple between true reflectors 1 and 2
is incorrectly transmitted through true reflector 2, forming a phantom reflector.
The phantom reflector in the first of these examples could have been avoided by
using a smooth velocity model, but this may reduce image quality by causing other
errors in forward and backward propagation, and removes the possibility of using
internal multiples for imaging. The second artifact may be avoided by removing internal multiples from the data. This is not only a difficult and error-prone operation,
but also constitutes the distasteful act of expending resources to discard useful infor146
0
Phantom reflector
z (km)
True
reflector 1
True
reflector 2
Forward
Backward
3
0
Time (s)
2
a
0
z (km)
True
reflector 1
True
reflector 2
Forward
Backward
Phantom reflector
3
0
Time (s)
3
b
Figure 5-2: Examples in which incorrect backpropagation in RTM can lead to phantom reflector artifacts. (a) The backpropagated arrival from true reflector 2 reflects
on true reflector 1, leading to a phantom reflector near the surface. (b) Part of the
internal multiple between true reflectors 1 and 2 is incorrectly transmitted through
true reflector 2, causing a deep phantom reflector.
147
mation. Indeed, internal multiples may be the only wave paths able to image certain
regions of the subsurface, particularly when the structure is complex (Malcolm et al.,
2011).
Another means of avoiding both artifacts would be to only backpropagate arrivals
in the data that are not predicted by the forward modeling step. This is the approach
suggested when RTM is considered to be one iteration of FWI (see Virieux and Operto
(2009) for an overview). Although elegant, this results in reflectors in the velocity
model not being present in the image, which can interfere with interpretation.
Weglein et al. (2011a,b) highlight the problems caused by only backpropagating
from one surface of the image volume, and propose the use of an alternative Green’s
function as a solution. Although this is an interesting approach, it has not yet been
demonstrated on a realistic dataset.
The Marchenko method (Wapenaar et al., 2014) takes a remarkably different approach to seismic imaging by abandoning the forward and backward propagation
steps of RTM, instead attempting to create data for a source and receiver at depth,
from regular surface acquisition. This enables the use of an imaging condition reminiscent of the survey-sinking approach used in one-way migration (Broggini et al.,
2013), where image amplitude at a point is determined by the recorded data at zero
time for a coincident source and receiver at the point. This method claims to avoid
phantom reflector artifacts, but is still being developed and currently has a large
computational cost.
Perhaps the most plausible approach that has been proposed for reducing phantom
reflectors in images is LSRTM (see Wong et al. (2014) for a recent discussion). Use of
this method is impeded by its substantial cost, although recent reports suggest that
it may be possible to reduce this (Tu and Herrmann, 2014).
A further problem with even regular RTM is its computational cost. Although it is
theoretically one of the simplest migration techniques and has been known for several
decades, it has only recently seen widespread use as available computing resources
have advanced to a point where it is feasible. Today it is still often implemented using
an acoustic wave propagator with simplified anisotropy. It is quite likely that image
148
improvements would be possible if propagators that more accurately represent the
physics of the Earth were used. Reducing the number of wave propagations necessary
to perform RTM may make this achievable.
In this paper we propose a modification of RTM that more effectively uses the
provided velocity model in the backpropagation step. This significantly reduces image
artifacts when the correct velocity model is used compared to regular RTM, and has
smaller computational requirements as the number of backpropagated wavefields is
reduced.
The method uses Green’s Third Identity to recreate the seismic wavefield during
the backpropagation step. The recorded wavefield is used at receiver locations, and
the forward propagated source wavefield provides the data for synthetic receivers
at the other points along the boundary of the image volume. This will cause the
backpropagated wavefield to more closely match the true seismic wavefield when more
accurate velocity models are used. We describe two imaging conditions that could be
used to produce an image of subsurface structure from this backpropagating wavefield.
One uses methods to separate the wavefield amplitude by propagation direction to
add to the image when waves propagating in different directions overlap, while the
second uses a high-pass filter.
We begin by describing the method, including two possible imaging conditions.
This section also explains how certain types of phantom reflector artifacts may remain
in the images produced with the proposed method, even when the exact velocity model
is used. This is followed by a results section in which we demonstrate the application
of the method to a simple velocity model consisting of three layers, and compare the
results with those of regular RTM. We also explore the sensitivity of the method to
errors in the model and data, and compare its output to RTM on a 2D portion of the
SEAM model (Fehler and Larner, 2008) to show its effectiveness on a model of more
realistic complexity.
149
5.2
Method
We begin by deriving Green’s Third Identity using standard Green’s function techniques. We use the constant density acoustic/scalar wave equation for simplicity,
but Green’s Third Identity can also be used for more realistic wave equations, as
Schleicher et al. (2001) and Wapenaar and Fokkema (2006) show, for example. The
definition of the Green’s function and the non-homogeneous wave equation give us
(∇2𝐱′ −
1
𝜕 2′ ) 𝐺(𝐱 − 𝐱′ , 𝑡 − 𝑡′ ) = 𝛿(𝐱 − 𝐱′ )𝛿(𝑡 − 𝑡′ )
𝑐(𝐱′ )2 𝑡
(5.4)
1
𝜕 2′ ) 𝑢(𝐱′ , 𝑡′ ) = 𝑞(𝐱′ , 𝑡′ ),
𝑐(𝐱′ )2 𝑡
(5.5)
and
(∇2𝐱′ −
where 𝑐 is wave speed, 𝐺 is a Green’s function, 𝑢 is the wavefield, and 𝑞 is the source
term, which we will assume to be non-zero only before the time 𝑇0 (the source turn-off
time).
Multiplying Equation 5.4 by 𝑢(𝐱′ , 𝑡′ ) and Equation 5.5 by 𝐺(𝐱 − 𝐱′ , 𝑡 − 𝑡′ ), and
subtracting, yields
1
𝜕 2′ ) 𝐺(𝐱 − 𝐱′ , 𝑡 − 𝑡′ )−
𝑐(𝐱′ )2 𝑡
1
𝐺(𝐱 − 𝐱′ , 𝑡 − 𝑡′ ) (∇2𝐱′ −
𝜕 2′ ) 𝑢(𝐱′ , 𝑡′ ) =
𝑐(𝐱′ )2 𝑡
𝑢(𝐱′ , 𝑡′ ) (∇2𝐱′ −
(5.6)
𝑢(𝐱′ , 𝑡′ )𝛿(𝐱 − 𝐱′ )𝛿(𝑡 − 𝑡′ ) − 𝐺(𝐱 − 𝐱′ , 𝑡 − 𝑡′ )𝑞(𝐱′ , 𝑡′ ).
We integrate Equation 5.6 over the time range [𝑡1 , 𝑡2 ] and the spatial volume 𝜏
(depicted in Figure 5-3). Using Stokes’ Theorem, we obtain
𝑡2
∫ d𝑡′ ∫ d𝐬′ 𝑢(𝐱′ , 𝑡′ )𝜕𝐧′ 𝐺(𝐱 − 𝐱′ , 𝑡 − 𝑡′ ) − 𝐺(𝐱 − 𝐱′ , 𝑡 − 𝑡′ )𝜕𝐧′ 𝑢(𝐱′ , 𝑡′ )+
𝑡1
𝛿𝜏
∫ d𝐱′ [𝑢(𝐱′ , 𝑡′ )𝜕𝑡′ 𝐺(𝐱 − 𝐱′ , 𝑡 − 𝑡′ ) − 𝐺(𝐱 − 𝐱′ , 𝑡 − 𝑡′ )𝜕𝑡′ 𝑢(𝐱′ , 𝑡′ )]𝑡𝑡2 =
1
𝜏
⎧
𝑡
{𝑢(𝐱, 𝑡) − ∫𝑡 2 d𝑡′ ∫𝜏 d𝐱′ 𝐺(𝐱 − 𝐱′ , 𝑡 − 𝑡′ )𝑞(𝐱′ , 𝑡′ ), if 𝐱 ∈ 𝜏 , 𝑡 ∈ [𝑡1 , 𝑡2 ]
1
⎨
{− ∫𝑡2 d𝑡′ ∫ d𝐱′ 𝐺(𝐱 − 𝐱′ , 𝑡 − 𝑡′ )𝑞(𝐱′ , 𝑡′ ),
otherwise,
⎩ 𝑡1
𝜏
150
(5.7)
real receivers
τ
synthetic receivers
n̂′
δτ
Figure 5-3: The simulation domain when using the proposed method. The interior
of the domain, the shaded region 𝜏 , is where the wavefield will be recreated from
measurements on the boundary 𝛿𝜏 . Real receivers generally only cover a portion of
the boundary, so synthetic receivers are used on the remainder. Sharp edges in 𝛿𝜏 are
avoided to reduce artifacts. The receivers must record the outward normal derivative
of the wavefield at the boundary, as indicated by 𝑛̂ ′ .
where 𝜕𝐧′ takes the spatial derivative in the direction outward normal to the boundary
𝛿𝜏 , and 𝑠′ is an element of 𝐱′ along 𝛿𝜏 .
During backpropagation we wish to reconstruct the wavefield within the simulation
volume from boundary values at future times. We therefore choose 𝐺 to be the
anticausal Green’s function 𝐺− , which implies that 𝐺− (𝑡) = 0 when 𝑡 > 0. This
means that 𝐺(𝐱, 𝑡 − 𝑡′ ) = 0, ∀𝑡′ < 𝑡. Taking 𝑡1 ≥ 𝑇0 , i.e., the earliest time in our
integration is at least the source turn-off time, then 𝑞(𝑡′ ) = 0, ∀𝑡′ . For simplicity,
we will also assume that the wavefield 𝑢 has returned to rest within the simulation
volume 𝜏 by the final integration time 𝑡2 = 𝑇 . This allows us to determine the
amplitude of the wavefield at any point within 𝜏 , and in the time range [𝑡1 , 𝑡2 ], using
𝑇
𝑢(𝐱, 𝑡) = ∫ d𝑡′ ∫ d𝐬′ 𝑢(𝐱′ , 𝑡′ )𝜕𝐧′ 𝐺(𝐱 − 𝐱′ , 𝑡 − 𝑡′ ) − 𝐺(𝐱 − 𝐱′ , 𝑡 − 𝑡′ )𝜕𝐧′ 𝑢(𝐱′ , 𝑡′ ).
𝑡
𝛿𝜏
(5.8)
This is Green’s Third Identity.
We now make use of the approximation described in Spors et al. (2008), which
further simplifies Equation 5.8 to
𝑇
𝑢(𝐱, 𝑡) = ∫ d𝑡′ ∫ d𝐬′ − 2𝑎(𝐱′ )𝐺(𝐱 − 𝐱′ , 𝑡 − 𝑡′ )𝜕𝐧′ 𝑢(𝐱′ , 𝑡′ ),
𝑡
𝛿𝜏
151
(5.9)
where 𝑎 is a function which only selects outgoing waves. This is not an exact solution,
as described in Rabenstein et al. (2005), but the results we show below indicate that
it does not appear to introduce significant inaccuracies.
Implementing Equation 5.9 is achieved by choosing part of the boundary of the
simulation domain, 𝛿𝜏 , to lie along the line (2D) or plane (3D) of real receivers.
Rather than requiring simply the amplitude of the wavefield at these locations, the
equation calls for the spatial derivative normal to the boundary. This would ideally
be provided by the use of an acquisition system which is capable of recording this information, such as a towed streamer survey with hydrophones at two different depths
(but only separated by a small distance). Such “over-under” surveys are already available (Moldoveanu et al., 2007). Alternatively, it may be possible to approximate the
normal derivative. On land, if the velocity near the receivers is believed to be constant,
and the recorded waves are propagating close to vertically (due to the slow weathered
layer), then the normal derivative could be determined using a time derivative
𝜕𝐧′ 𝑢(𝐱′ , 𝑡′ ) = −
1
𝜕 ′ 𝑢(𝐱′ , 𝑡′ ).
𝑐(𝐱′ ) 𝑡
(5.10)
For marine surveys, it may be possible to estimate wave propagation direction at the
receivers, and thus approximately determine the normal derivative of the wavefield,
by examining reflections from the sea surface, which is at a known height above the
receivers.
For the remainder of the boundary 𝛿𝜏 which is not covered by real receivers, we
create synthetic receivers. This is done by saving the outward normal of the forward
modeled wavefield at the boundaries at each time step during the first stage of RTM.
As with regular RTM, this wavefield will more closely represent the true seismic
wavefield when a more accurate velocity model and wave propagator are used.
The function 𝑎 for selecting only waves propagating outward through the boundary
can be easily implemented for the synthetic receivers by using methods such as those
described in Chapter 3 for determining propagation direction during the forward
modeling stage. If no waves are expected to enter 𝜏 from outside in the numerical
152
simulation (because the model is smooth beyond the boundary 𝛿𝜏 , and waves exiting
𝜏 are attenuated), then 𝑎 will select all waves, and so can be ignored. If the real
receivers are near the Earth’s surface (especially in marine surveys), 𝑎 may be applied
to the real data by using a deghosting technique (Amundsen, 1993).
As this method only requires storing the output of the forward propagation step
at cells along the boundary where real receivers are not present, it represents a substantial reduction in memory requirements. In standard RTM, several techniques
are available for recreating the forward propagating (“source”) wavefield during the
backward step, as discussed in Dussaud et al. (2008). One of these involves saving the
forward wavefield several cells deep over the entire boundary, which requires several
times the memory needed for the proposed method. Memory usage is also reduced
by only needing to store one wavefield during backpropagation, compared to two in
standard RTM.
It is necessary for the receiver spacing to be constant around the boundary. This
means that the real receiver spacing must be constant, and it must be the same as
that of the synthetic receivers. To ensure that this is the case, it may be necessary to
interpolate the real receiver data onto a regular grid. Many algorithms for achieving
this have been proposed; Stanton et al. (2012) compare three.
A further requirement is that the receiver data be calibrated such that it has
the same magnitude as would be recorded by the numerical simulation if the model
were correct. In particular, if the numerical simulation calculates wave amplitude as
displacement, for example, then the receiver data should also be displacement with
the same units. Other popular seismic methods, such as FWI, may have the same
requirement (depending on their implementation).
Since only one wavefield is backpropagated, the conventional zero-lag cross-correlation
imaging condition cannot be used. Instead, we must identify when two waves propagating in the same wavefield overlap. One means of achieving this is to separate
the wavefield amplitude by propagation direction each time the imaging condition
is applied. We may then use a regular imaging condition between wave components
traveling in different directions. With this approach, using an imaging condition such
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as that proposed by Costa et al. (2009) can be readily applied, reducing backscatter
artifacts. As the most critical time to accurately separate the wavefield is when waves
propagating in different directions overlap, the Poynting vector approach (Yoon and
Marfurt, 2006) would fail, and so more sophisticated (and thus usually more computationally expensive) separation schemes, such as those proposed in Chapter 3, must be
used. Despite requiring a more expensive imaging condition than regular RTM, it is
still possible that reducing the number of wavefields that need to be backpropagated
will result in a sufficient computational cost reduction that the proposed method provides better overall performance than regular RTM, especially if the imaging condition
is not applied every time step.
R
1
a
b
1
R
R
ɞ
R
R(1+R)
1+R
c
d
Figure 5-4: Simplified wave amplitudes to approximately determine image amplitude
contributions at a reflector. R is the reflection coefficient. (a) The source wave
when the reflector is not present in the migration model. (b) The data wave when
the reflector is not present in the migration model. (c) The source wave when the
migration model contains the reflector. (d) The data wave in regular RTM when the
migration model contains the reflector.
A potentially less accurate but significantly faster alternative imaging condition
154
relies on postprocessing images with a high-pass filter. At each time step, the square
of the backpropagated wavefield is added to the image. We use the simplified model
of the wave amplitudes involved in determining the image amplitude displayed in
Figure 5-4. When the model is smooth, the image amplitude at locations where no
real reflector is present will be
𝐼(𝐱) = ∑ 𝑢2𝑖 (𝐱, 𝑡),
(5.11)
𝑡
where 𝑢𝑖 (𝐱, 𝑡) is the wave amplitude incident on the point 𝐱 at time 𝑡. This image
amplitude will decay smoothly away from the source. At points near a real reflector,
when the migration model is smooth (i.e., it does not contain the reflector), the image
amplitude will be
𝐼(𝐱) = ∑ (𝑢𝑗 (𝐱, 𝑡) + 𝑅𝑢𝑗 (𝐱, 𝑡 −
𝑡
= ∑𝑢2𝑗 (𝐱, 𝑡) + 𝑅2 𝑢2𝑗 (𝐱, 𝑡 −
𝑡
2𝑢𝑗 (𝐱, 𝑡) 𝑅𝑢𝑗 (𝐱, 𝑡 −
2Δ𝑥
))
𝑐(𝐱)
2
(5.12)
2Δ𝑥
)+
𝑐(𝐱)
2Δ𝑥
),
𝑐(𝐱)
(5.13)
where 𝑢𝑗 is the incident wave amplitude propagating toward the reflector, 𝑅 is the
reflection coefficient of the reflector for waves incident from the current side, Δ𝑥 is the
distance of 𝐱 from the reflector, and 𝑐 is the local wave speed. This assumes normal
incidence and so ignores variation with reflection angle, and approximates the wave
speed as being locally constant. It is also reliant on the model being correct above the
reflector, and incident waves and reflectors being sufficiently separated that they do
not interfere with each other. Since 𝑢𝑖 is the amplitude incident from any direction,
it is equal to the sum of the first two terms in Equation 5.13. Applying a high-pass
filter with a cutoff frequency chosen so that it will attenuate the smooth background
amplitude created by Equation 5.11, results in a filtered image
𝐼(𝐱) = ∑ 2𝑢𝑗 (𝐱, 𝑡) 𝑅𝑢𝑗 (𝐱, 𝑡 −
𝑡
155
2Δ𝑥
),
𝑐(𝐱)
(5.14)
near the reflector, and zero elsewhere, which, when divided by two, is equivalent to
the image that would be produced by regular RTM. On the other hand, when the
reflector is present in the migration model, the image amplitude above the reflector
is
2
𝐼(𝐱) = ∑ (𝑢𝑗 (𝐱, 𝑡) + 𝑅𝑢𝑗 (𝐱, 𝑡 −
𝑡
2Δ𝑥
)) ,
𝑐(𝐱)
(5.15)
and the image below is
𝐼(𝐱) = ∑ ((1 + 𝑅)𝑢𝑘 (𝐱, 𝑡))2 ,
(5.16)
𝑡
where 𝑢𝑘 is the amplitude of waves incident on the other side of the reflector, we
make similar assumptions to above, and assume that waves are only incident on the
interface from above. The wave paths involved are shown in Figure 5-4c. After
high-pass filtering to remove wave components that are squared, the image above the
reflector becomes
𝐼(𝐱) = ∑ 2𝑢𝑗 (𝐱, 𝑡) 𝑅𝑢𝑗 (𝐱, 𝑡 −
𝑡
2Δ𝑥
),
𝑐(𝐱)
(5.17)
and below it will be
𝐼(𝐱) = 0,
(5.18)
which is the same as the smooth model case, except that it does not contain a contribution from waves incident on the other side of the interface. This implies that if the
interface is only illuminated from one side, the resulting reflector in the image will
also be one-sided. Rather than being centered on the reflector (if the source wavelet
is zero-phase), only the half of the wavelet on the side that the reflector is imaged
from will be visible. As this differs from the situation with regular RTM (as we show
below), it has the potential to initially cause some confusion for interpreters. The
image amplitude produced by regular RTM when the reflector is in the migration
model is more complicated. Using our approximations, and the wave paths depicted
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in Figures 5-4c and 5-4d, it can be expressed as
𝐼(𝐱) = ∑ (𝑢𝑗 (𝐱, 𝑡) + 𝑅𝑢𝑗 (𝐱, 𝑡 −
𝑡
2Δ𝑥
)) ×
𝑐(𝐱)
2Δ𝑥
))
(𝑅 𝑢𝑗 (𝐱, 𝑡) + 𝑅𝑢𝑗 (𝐱, 𝑡 −
𝑐(𝐱)
(5.19)
2
above the interface, and below
𝐼(𝐱) = ∑(1 + 𝑅)𝑢𝑘 (𝐱, 𝑡) (𝑅(1 + 𝑅)𝑢𝑘 (𝐱, 𝑡 +
𝑡
2Δ𝑥
)) .
𝑐(𝐱)
(5.20)
After high-pass filtering to remove squared wave components, this becomes
𝐼(𝐱) = ∑ 𝑢𝑗 (𝐱, 𝑡) 𝑢𝑗 (𝐱, 𝑡 −
𝑡
2Δ𝑥
) (𝑅 + 𝑅3 )
𝑐(𝐱)
(5.21)
2Δ𝑥
)
𝑐(𝐱)
(5.22)
above, and
𝐼(𝐱) = ∑ 𝑅(1 + 𝑅)2 𝑢𝑘 (𝐱, 𝑡) 𝑢𝑘 (𝐱, 𝑡 +
𝑡
below. After the filtered image of the proposed method above the interface, Equation
5.17, is divided by two, it matches the dominant term of the regular RTM image,
Equation 5.21. The regular RTM image also contains other contributions, and, importantly, involves waves incident on the other side of the interface, and so is two
sided even when waves are only incident on one side. This does, nevertheless, show
that the image amplitude produced when this imaging condition (including high-pass
filtering and division by two) is used with the new method, is the same as that of regular RTM with a high-pass filter when the migration model is smooth, and contains
the same dominant term (above the reflector) when the reflector is in the model. This
does not imply that the results of regular RTM have similar image quality to those
of the proposed method when the model is correct; the phantom reflector artifacts
discussed earlier do not appear in this discussion as we have neglected certain aspects
of incorrect backpropagation for simplicity.
Although using the proposed approach to migration can prevent incorrect backpropagation when the model and propagator are correct, and so makes it possible to
157
avoid the types of phantom reflector artifacts described in Figure 5-2, another form of
phantom reflector is still possible. This is caused by crosstalk involving different orders of multiples. An example of this occurring is shown is Figure 5-5. These artifacts
can also occur in regular RTM if the migration velocity model contains reflectors, but
they may have higher amplitude when using the proposed method. This is because
preventing incorrect backpropagation avoids energy leakage and so the backpropagating multiples are likely to have greater amplitude. As most of these artifacts are only
caused by high order multiples, their contributions to the image are small, and no
such artifacts are clearly visible in any of the examples we show in the results section.
In most cases, using a source-normalized imaging condition, or applying illumination
correction, would attenuate these artifacts, as shown in Chapter 4, because this would
consider the high amplitude primaries and lower order multiples that passed through
the same point without sensing a reflector there.
z (km)
0
True reflector
Phantom reflector
4
0
Time (s)
4
Figure 5-5: Even when the wave propagates along the correct path, phantom reflectors are still possible when using imaging conditions that assume waves overlap at
reflectors.
5.3
Results
In this section we examine the effect of using the proposed method by applying it first
to a simple layer model, and then to a more realistic case by using a 2D portion of the
SEAM model and comparing the results with those of regular RTM. We also explore
the sensitivity of the proposed method to four types of model and data errors. In all
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cases we use the imaging condition that employs a high-pass filter instead of needing to
separate the wavefield by propagation direction. We use two layers of receivers spaced
5 m (one grid cell) apart horizontally and vertically, allowing the normal derivative
to be calculated, and avoiding the need to interpolate between receivers along the
boundary. To allow accurate comparison, the same clipping percentage is applied to
the displayed images produced by the two methods in each case.
5.3.1
Simple layer model
We begin by verifying that the proposed method successfully removes the phantom
reflector artifacts predicted in Figure 5-2 by testing the same velocity model, which
is depicted in Figure 5-6a. In this idealized case, we use the same velocity model for
modeling to generate the “real” receiver data, and for migration. Receivers cover the
top surface of the model at two depth levels 5 m apart to enable the calculation of the
normal derivative. This is continued for the remainder of the boundary surrounding
the image domain to create the data for the synthetic receivers used in the proposed
method, as we show in Figure 5-3. We stack the images of 25 equally spaced sources
covering the surface. Following this, we apply a high-pass filter to remove features
with a wavelength greater than 120 m, attenuating the backscatter artifacts produced
by using a migration velocity model with reflectors. The resulting image when regular
RTM using the standard cross-correlation imaging condition is applied is shown in
Figure 5-6b. As expected, the phantom reflectors described in Figure 5-2 are clearly
visible, indicated by arrows A and B. Arrow C points to an additional artifact which
is caused by the finite aperture leading to diffraction artifacts at the points along
the interface where reflections are no longer recorded as they arrive at the surface
outside the real receiver array. Similar artifacts occur at the deeper interface, but,
due to the larger depth, they occur closer to the edges of the image and so are not as
substantial. All of these artifacts, which occur in regular RTM even when the exact
velocity model and wave propagator are used, significantly reduce the quality of the
image. They may obscure true structure, or be erroneously interpreted as structure
themselves.
159
When the proposed method is used, these artifacts are attenuated, as shown in
Figure 5-6c. This occurs because the backpropagated data wavefield no longer travels
along the incorrect paths illustrated in Figure 5-2. The diffraction artifacts, C, are
removed as the image volume is now surrounded by receivers (real and synthetic),
and so all reflections from the interfaces are recorded. Faint traces of artifacts A and
B remain. It is likely that these are caused by either inaccuracies in the interpolation
of the “real” receiver data from the recorded sampling rate (4 ms) to the simulation
time step interval, or by errors in the approximations used, which are discussed in
Rabenstein et al. (2005). We use the same high-pass filter as in the regular RTM
case.
0
0
1500 m/s
0
A
1500 m/s
3
z (km)
2000 m/s
z (km)
z (km)
C
B
3
0 x (km) 0.5
a
3
0 x (km) 0.5
b
0 x (km) 0.5
c
Figure 5-6: A demonstration of the proposed method’s ability to reduce phantom
reflector artifacts compared to regular RTM. (a) The velocity model that is used
for modeling and migration. It will produce similar wave paths to those depicted
in Figure 5-2. (b) Image produced by regular RTM. A, B, and C indicate types
of artifacts that the proposed method can reduce. (c) The result when using the
proposed method, showing significant attenuation of artifacts.
160
5.3.2
Sensitivity to errors
Although the preceding example demonstrates that the proposed method is capable
of significantly improving image quality, it uses the unrealistic assumption of having
a perfect migration model and data. We now examine four departures from this
idealized situation. We again use a high velocity layer sandwiched between two lower
velocity layers. To present the results, we run 2D migration using the erroneous
model or data for 25 sources, stack the results, apply a high-pass filter (the same as
that used in the previous example), and extract the central vertical slice through the
image. These vertical slices are then placed along the x-axis of the displayed images
so that changes can be easily observed.
In the first, the results of which are shown in Figure 5-7a, the location of the second
interface is misplaced to varying degrees. Regular RTM is not significantly affected
when the interface is too deep, as the velocity above both interfaces is still correct,
but additional artifacts are exhibited when the interface is too shallow (and thus
the average velocity above the bottom interface is no longer correct). The proposed
method will place a reflector in the image wherever there is a sharp discontinuity in
the migration velocity model, as the synthetic receivers will record reflections from it.
To avoid such image artifacts, sharp reflectors should therefore not be placed in the
migration velocity model unless their location is known with reasonable confidence.
Even when the interface is too deep, the image produced by the proposed method
therefore contains artifacts, as the incorrect interface location is imaged in addition to
the true reflector. Unsurprisingly, many of the phantom reflectors (with the exception
of the diffraction artifacts on the top interface) are now present, as the recordings of
the synthetic receivers do not match what would have been recorded if real receivers
were at those locations in the real seismic experiment, and so they do not prevent
incorrect backpropagation.
The second type of model error we consider is when the velocity below the second
interface is incorrect, the results of which are shown in Figure 5-7b. As the velocity
above both interfaces is correct, this model error has little effect on the output of
161
regular RTM other than to change the depth at which the deep phantom reflector
occurs. The proposed method’s image is largely similar. Diffractor artifacts on the
top interface continue to be attenuated, since they do not involve the region where
the error exists. Changing the wave speed underneath the bottom interface will have
the effect of altering the reflectivity of that interface. It is therefore not surprising
that the amplitudes of the other artifacts increase as the velocity error moves away
from zero.
A type of model error that is likely to be encountered frequently is when the
migration model is smoother than the true Earth. We reproduce this by smoothing the
migration model so that, rather than being sharp discontinuities as in the true model,
the velocity transitions are spread over up to 1 km (500 m before and after the true
interface location). The results are depicted in Figure 5-7c. In regular RTM, the only
indicator that the model has been smoothed is the slight shifting of reflector depth
as the average velocity above the reflectors changes. The shallow phantom reflector
also rapidly disappears as the increasing smoothness reduces the reflectivity of the
upper interface. With the proposed method, the deep phantom reflector appears
similarly rapidly, as the interface smoothness means that the internal multiple is
not predicted. The ratio of the deep phantom reflector’s amplitude compared to
that of the upper interface as the interfaces become increasing smoothed is displayed
in Figure 5-8. This shows that the amplitude of the phantom reflector relative to
the true interface grows smoothly with model smoothness. It grows rapidly when the
interface transition is spread over about 100 m, which is approximately the wavelength
of the source wavelet’s dominant frequency. The diffractor artifacts also appear as the
model becomes smoother. Small artifacts are visible near the true interface depths
for the first few levels of smoothing, as the interfaces are still sufficiently abrupt to
cause reflections that are recorded by the synthetic receivers. As the model becomes
increasingly smooth, its similarity with regular RTM grows. Indeed, beyond a few
tens of meters, they are almost indistinguishable, indicating that the new method
can attenuate artifacts when the model is sharp, and does not behave any worse than
RTM when it is smooth.
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Finally, we look at what happens when the real receiver data have been scaled.
This could occur because the receiver data have not been calibrated to produce amplitudes that are comparable with those predicted by the numerical simulation, or
because of inaccuracies in the numerical wave propagator. Amplitudes in the regular
RTM image all scale with the data multiplier, as expected. Also somewhat foreseeably, the reflector amplitudes in the proposed method’s image change with distance
from the central unity multiplier, but not the same extent, as reflections from them
continue to be also recorded by the unscaled synthetic receivers. The image artifacts
also grow as the amplitude discrepancy increases, due to less accurate backpropagation.
5.3.3
SEAM
The SEAM model was created to be a realistic test for imaging and inversion methods
(Fehler and Larner, 2008). It therefore provides an ideal means of investigating the
behavior of the proposed method on complicated models. We use a 2D portion of
the P-wave velocity model from the line North 23 900 m, shown in Figure 5-9a. To
avoid the results being contaminated by out-of-plane reflections, and to record the
normal derivative at the real receivers, we generate data with a 2D propagator using
the chosen portion of the model, rather than using the supplied data. To increase
the realism of the test, we smooth the model prior to migration, as displayed in
Figure 5-9b. The smoothing increases with depth, so that features just below the sea
floor remain sharp. Figures 5-10a and 5-10b show the resulting images produced by
regular RTM and the proposed method, after the application of a high-pass filter that
attenuates wavelengths greater than 120 m. The proposed method’s image contains
features such as the inclusions in the top of the salt body, and the sea floor reflector
at the edges of image, which are present in the model but not well illuminated by the
included sources and receivers, but the two images are otherwise very similar. This
indicates that the new method is able to produce almost identical results to regular
RTM on a complicated model, with reduced computational cost, while providing the
possibility of obtaining an improved image when the model is well known.
163
Regular RTM
New method
z (km)
0
Regular RTM
z (km)
New method
0
3
3
-500
0
500 -500
0
500
Error (m)
Error (m)
-0.5
0
0.5 -0.5
0
0.5
Error (km/s)
Error (km/s)
a
Regular RTM
0
0.5
1
Smoothing (km)
0
0.5
1
Smoothing (km)
0
New method
Regular RTM
z (km)
New method
z (km)
0
b
3
3
0.1
c
1
2.0 0.1
1
2.0
Multiplier
Multiplier
d
Figure 5-7: Sensitivity of the proposed method to errors in the model and data. The
true model is similar to that in Figure 5-6, but the velocities have been increased so
that the areas that were 1500 m/s are now 2000 m/s, and the high velocity layer has
increased from 2000 m/s to 3000 m/s. (a) Misplacement of the bottom reflector in
the model used for migration so that the high velocity layer extends to 2 km+Error.
(b) Wrong velocity in the bottom region. The wave speed in the area below the high
velocity layer in the migration model is 2000 m/s+Error. (c) Smoothing of interfaces.
Instead of being sharp discontinuities, both interfaces are smoothed over a distance
of Smoothing in the migration model. (d) Uncalibrated data. The real receiver data
are scaled by the specified multiplier and so no longer match the synthetic receiver
data.
164
Amplitude ratio
0.1
0
5
100
Transition length (m)
205
Figure 5-8: Ratio of the sum over depth of the absolute value of the deep phantom
reflector’s amplitude relative to that of the upper true reflector as the smoothness of
the interfaces varies, using the same model as that in Figure 5-7.
4.5
z (km)
Wave speed
(km/s)
0
6
1.5
0
x (km)
12
a
4.5
z (km)
Wave speed
(km/s)
0
6
1.5
0
x (km)
12
b
Figure 5-9: Velocity model of the 2D portion of SEAM we use to test the proposed
method on a complicated model. (a) The true P-wave velocity model. (b) The
migration velocity model. It matches the true model at the sea floor and at the top
of the salt body, but is increasingly smoothed below this.
165
z (km)
0
6
0
x (km)
12
a
z (km)
0
6
0
x (km)
12
b
Figure 5-10: The result of imaging a 2D portion of the SEAM model. (a) The image
produced by regular RTM after applying a high-pass filter. (b) The result when the
proposed method is used, after applying the same high-pass filter as that used in (a).
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5.4
Discussion
The proposed method has the advantage that, with the exception of certain artifacts
due to crosstalk related to high order multiples (and thus likely to be weak), will
produce a more accurate image when a more accurate migration model and propagator
are used. Information available from the use of expensive techniques such as FWI,
and drilling, can therefore be more effectively exploited.
Reducing phantom reflectors that are caused by the presence of reflectors in the
velocity model also has the advantage of enabling the use of internal multiples for
imaging, as they can be naturally included by RTM (and thus also the proposed
method) when their generating interfaces are present, without as many artifacts as
would normally be associated with this.
Perhaps a more widespread immediate gain from the proposed method is its reduced computational cost and lower memory requirement. The SEAM model example that we show provides especially encouraging evidence that even if the migration
model does not contain sufficient sharp reflectors to provide a noticeable improvement
in artifact reduction, the resulting image quality is comparable to regular RTM, at
reduced cost.
Only backpropagating arrivals at the real receivers which are not predicted by the
forward model, as suggested by considering RTM to be one iteration of FWI, has the
potential to avoid phantom artifacts and crosstalk associated with known reflectors
which are included in the model, and thus may seem like a good alternative to the
proposed method. As mentioned earlier, this would subsequently require additional
processing to then reintroduce the known reflectors into the image. A further problem
with this approach is that it still requires the backpropagation of two wavefields, and
so is more computationally demanding than the proposed method.
Applying the method to elastic wavefields extends the forms of phantom reflectors
attenuated to also include those created when converted phase waves are not correctly
backpropagated.
167
5.5
Conclusion
We present a modification of RTM which uses Green’s Third Identity to express the
wavefield in the interior of the simulation domain as a backpropagation of its normal
derivative on the boundaries at future times, and describe imaging conditions that
can be used to create a seismic image from this single wavefield. If the model and
propagator are accurate, this can result in fewer phantom reflector artifacts than
standard RTM. Even when this is not the case, the method still produces images
that are comparable with those of RTM, and has the further benefit of reduced
computational cost.
We examine the sensitivity of the proposed method to certain data and model
errors, and demonstrate that in many cases the resulting image deterioration is similar
to that of regular RTM, but is noticeably worse when misplaced sharp reflectors are
included in the migration velocity model. This suggests that such reflectors should
only be included if their location is well known.
168
Chapter 6
Future work
Seismic imaging has progressed significantly over the past few decades. Migration by
hand of 2D surveys, where the lateral velocity is assumed to be constant, has been
replaced by the potential to generate images of the subsurface using wide azimuth
3D surveys, together with numerical modeling that captures many of the features of
finite-frequency wave propagation in heterogeneous media. This has made it possible
to dramatically improve images in areas of complex geology, such as the Gulf of
Mexico, reducing exploration risk. There are many potential avenues of research that
could continue to yield further improvements.
An increase in available computing resources seems likely to produce the largest
increase in image accuracy, as most seismic imaging methods, including several discussed in this thesis, are currently compute limited. Faster wave propagators would
also yield a similar benefit. As numerical simulation of wave propagation is the central component of many seismic imaging algorithms, a method which dramatically
reduces the amount of computation needed to perform this task without sacrificing
accuracy for finite-frequency waves, would have similar benefits to increased computing resources. Reducing the computational restrictions might make it possible to use
more accurate wave propagators, and run iterative techniques, such as FWI, for more
iterations. These both have particular benefits for overturned waves and internal
multiples, as their long propagation paths mean that they are especially sensitive to
propagation errors. Increased computing capabilities may also potentially allow the
169
calculation of point spread functions to enable more accurate illumination compensation, which is critical for appropriately boosting the low amplitude contributions of
these wave paths. In addition to the ability to run more iterations of model building
methods, other future developments may also produce more accurate migration models, such as techniques for approximating the inverse of the Hessian for use in FWI,
or reduced drilling and well logging costs. If these improve the ability to correctly
place strong reflectors in the migration model, imaging with internal multiples will
benefit, as it requires that generating interfaces be in the model. These developments
may eventually lead to inversion replacing imaging. In the nearer term, extending
migration algorithms to extract more information from the available data, such as
by exploiting overturned waves and multiples for imaging, is likely to be a source of
image improvement.
The preceding chapters present a variety of methods for improving the images that
can be obtained using internal multiples and overturned waves. While many of these
methods could be robustly applied to industrial scale field datasets in their current
form, additional development may produce further image improvements. This chapter
describes possible future research directions for a number of the ideas contained in
this thesis. These include various improvements to the three approaches to separating
wave amplitude by propagation direction, modifications to facilitate the estimation
of the effect of neighboring scatterers on image amplitude, and better communication
of image uncertainty.
6.1
Chapter 3
This chapter proposes three new methods for decomposing a wavefield into amplitude
propagating in different directions, which has applications not only in the chapters
that follow it, but also in other techniques such as the calculation of ADCIGs. Although the methods already produce results that are generally superior to two previously proposed methods, this section describes several topics for research which might
further improve the accuracy of results and decrease computational requirements.
170
6.1.1
Methods 1 and 2 using LSS with variable local wave
speed
One weakness in the proposed methods of Chapter 3 for determining wave amplitude
as a function of propagation direction referred to as methods 1 and 2 using LSS, is the
assumption of locally constant wave speed. This assumption allows the simplification
of taking wave paths and wavefronts to be straight lines, so that the local slant
stack summations that are fundamental to both approaches can be performed along
straight lines. In reality wave speed is not generally locally constant, especially close
to sharp velocity changes, such as the boundary of salt bodies. This leads to errors
in the decomposed amplitudes, as we see in examples in Chapter 3. Under the high
frequency assumption, Snell’s Law provides a means of calculating the change in wave
propagation direction due to changes in wave speed. This could therefore be exploited
to determine the lines along which the methods should sum, when combined with local
wave speed variations. The methods also currently determine summation lengths
based on wave speed at the central point (the point being separated). Variations
in wave speed along the summation path affect the length over which the wave is
oscillatory, and so not accounting for these may result in incorrect results. Using
the average wave speed along the summation path or otherwise considering these
variations could therefore allow greater accuracy.
6.1.2
Initial guess
Due to the greater computational cost of the separation methods proposed in Chapter
3, only employing the methods in regions where the Poynting vector method (or
other low cost method) fails is suggested. Further cost reductions could be achieved
by increasing the efficiency of these methods in the regions where they are applied.
One means of obtaining such a cost reduction is to use an initial guess. This idea
is proposed in Chapter 3 for method 3, where the separation from the previous time
step in which the method was applied is propagated forward to the present time step.
Initial guesses such as this may be easily exploited with method 3, as a means of
171
providing an initial guess is naturally available for optimization-based approaches.
They could also be incorporated into methods 1 and 2 by first determining the wave
amplitude propagating in the directions with the largest amplitudes in the initial
guess. If the total amplitude found to be propagating in these directions is within
a specified tolerance value of the amplitude of the full wavefield at a point, then it
would not be necessary to examine the other directions.
Using an initial guess based on the separation from a previous time step is found
to be useful for method 3, but improvements in the guess may produce further performance improvements. One aspect to explore is the means of propagating the previous
result forward to the current time step. This could be implemented in different ways,
such as simply shifting waves in their directions of propagation by the distance suggested by the average velocity along their propagation paths, or using the one-way
wave equation to shift them in a series of potentially more accurate smaller steps.
Snell’s Law could be employed to estimate changes in the propagation directions of
the previous solution. These different approaches have varying computational costs
and levels of accuracy, so determining which results in the largest overall performance
improvement would require detailed study.
Alternatively, other means could be used to derive an initial guess, such as first
applying the separation method on a decimated grid and then interpolating the result.
This could be extended into a multigrid approach (Hackbusch, 1985). For the LSS
forms of methods 1 and 2 it may be possible to initially sum over (or otherwise examine) reduced summation lengths to determine the directions in which there appear
to be waves, and the full summation could then be performed in these initial guess
directions.
6.1.3
Further performance improvements for method 3
Although an elegant idea, method 3 is found to have significantly higher computational requirements than either of the other two proposals. The situation may be
improved by the areas of further work already suggested, but there are also other
potential sources of cost reductions.
172
The most obvious of these is the implementation of the method in a high performance compiled language such as C or Fortran, rather than Matlab, which is useful
for rapid prototyping but often has poor performance.
Another area for research which is likely to produce large performance improvements is examining means of reducing the number of model parameters. The current
approach, in which the required number of model parameters is the number of points
multiplied by the number of propagation directions considered, can result in very
large systems of equations. One means of reducing this number is through the use
of multiresolution techniques such as curvelets (Candès et al., 2006). The challenge
with such an approach is that it is likely to be more difficult to provide analytic forms
of the gradient and Hessian, and so may require numerical approaches to determine
these quantities, potentially negating any performance gains.
Another topic that should be considered for further research to improve performance of this method is the optimization algorithm that is used. Many such algorithms exist (see Nocedal and Wright (2006) for a review of the most popular) and
some are likely to be more suitable for this problem than others. As it is often difficult to determine which optimization algorithm is going to be best for a particular
situation, it may be necessary to implement and test a variety.
In addition to changing the optimization algorithm, examining the effect of varying
the objective functional is another obvious line of inquiry. Chapter 3 already suggests
that convergence may be aided by incorporating the wavefield at more than two
time steps into the objective functional. Other options include using a different oneway propagation scheme in the objective functional, and incorporating additional
components into it, such as a roughness penalty.
Finally, although using the Hessian can improve performance by reducing the
number of iterations to convergence, and despite its sparseness for this problem, computing and storing it is found to be quite computationally demanding. It may be
possible to obtain much of the benefit of the Hessian with reduced cost by only computing an approximation to it. Approximating the Hessian is a technique already in
use elsewhere in seismology, such as in FWI (Shin et al., 2001).
173
6.1.4
Sparsity
A difficulty associated with separating wavefield amplitude by propagation direction
is the large storage requirement, which, for a simple implementation, needs to store
a value for each direction considered for every point in the grid. For a 3D survey
this could become unmanageable. It is likely that waves are only propagating in a
small number of directions at a particular point at any time, and so the majority of
direction bins will have negligible amplitude. Furthermore, neighboring points along
a wavefront are likely to have similar amplitude for the propagation direction bin
that would produce a wavefront with that orientation. This invites a solution which
exploits the sparsity of the result. Significant reductions in storage requirements are
likely to be obtained by storing the result as curvelet coefficients, for example.
6.2
Chapter 4
Many of the key proposals for imaging with overturned waves and multiples are contained in Chapter 4. These include applying illumination compensation to make the
contributions of these wave paths comparable with those of primaries, and estimating
image uncertainty to reduce misinterpretation of artifacts that can be more prevalent
when these wave paths are included. Developments that could further improve these
two proposals are discussed below. As illumination compensation ideally attempts to
produce an image with correct relative amplitudes, an accurate migration model is
also important. The proposals in this chapter will therefore benefit from developments
in other areas, such as improved FWI.
6.2.1
Estimating the effect of neighboring scatterers
Results in Chapter 4 demonstrate the relative amplitude accuracy improvements that
can be achieved through estimating the effect on image amplitude of point spread
function width and reflector length. Point spread function width is estimated using
Equation 4.21, but this relies on simplifying assumptions, including continuous source
174
and receiver coverage up to the specified maximum angle, and doesn’t account for the
large variations in source and receiver illumination that are possible over the range of
sources and receivers, and which may affect point spread function width (the source at
the maximum angle may make negligible image contribution at a point, for example,
and so should not be considered when estimating the point spread function of that
point). Incorporating ADR and scattering angle information into the estimation
of this width therefore seems like a useful topic for research as it would enable the
determination of which angles make non-negligible contributions and so a more robust
estimate could be produced.
In Chapter 4, incorporating reflector length was possible as this was a controlled
quantity, but to use this information in general would require it to be estimated.
Reflector length at each point could be estimated by hand from the uncompensated
image, and this data provided to the illumination compensation process, but this is not
practical for large 3D images. Developing an algorithm for this purpose would allow
it to be performed automatically, facilitating the estimation of image contributions
at a point due to neighboring scatterers.
6.2.2
Displaying orientation information
Chapter 4 suggests that images be complemented by images of ADR and the standard
deviation of compensated image amplitude over shots, weighted by ADR, to communicate uncertainty. For greatest effectiveness, these should be viewed for each of the
possible reflector orientations, as, for example, if the ADR images for all reflector
orientations were simply summed and displayed as a single image, this would lose
the ability to show that at a particular location reflectors of one orientation may be
well illuminated while those of another may not be illuminated at all. This approach
also has drawbacks, however. One is that it is likely to make viewing and interpreting seismic images more time consuming, as multiple images would now need to be
examined to see all of the reflector orientations. Another is that it may be difficult
to visualize the whole structure and see how reflectors with different orientations are
related when they are viewed individually. A solution may be to combine the images
175
for different reflector orientations by using a different color or other visual means of
differentiation for each orientation. A naive approach of dividing the color spectrum
among the different orientations presents the problem that overlapping orientations
will combine to appear like a different orientation, but with further research it may
be possible to avoid this issue.
6.3
Chapter 5
Creating the potential for improved image accuracy through the reduction of predictable phantom reflector artifacts, while simultaneously decreasing computational
requirements, is the topic of Chapter 5. This is achieved by employing Green’s Third
Identity to more accurately recreate the seismic wavefield. Deriving maximum benefit from the method requires an accurate migration model, so research on that topic
will again yield improvements when applying this method. Another challenge with
applying the method to real data is its need for the normal derivative of the wavefield
at the boundaries of the simulation domain. It may be possible to avoid this problem
by estimation, although this might introduce errors. Many of the image artifacts
that this method can attenuate are caused by attempting to image with multiples.
The method should therefore ideally be combined with the proposals in Chapter 4,
which allow more effective use of these wave paths for imaging, and can also further
reduce the amplitude of artifacts. The method is not currently compatible with the
illumination compensation component of Chapter 4, however. Exploring means for
rectifying this is therefore important for fully exploiting sharp reflectors in migration
models to image with internal multiples.
6.3.1
Estimating the normal derivative
The principal hindrance to applying the described method to existing seismic data is
its need for the normal derivative of the wavefield along the portion of the boundary
covered by real receivers. Several means of approximating the normal derivative from
data when over-under acquisition was not performed are suggested in the chapter,
176
but further research is necessary to determine whether errors in these approximations
are sufficient to overwhelm the additional image accuracy that is possible with the
method.
An alternative to using the normal derivative of the wavefield is to use the normal
derivative of the Green’s function. This could be achieved by doubling the number
of propagations,
𝑢(𝐱′ , 𝑡′ )𝜕𝑛′ 𝐺(𝐱 − 𝐱′ , 𝑡 − 𝑡′ ) = 𝑢(𝐱′ , 𝑡′ )
𝐺(𝐱 − 𝐱′ , 𝑡 − 𝑡′ ) − 𝐺(𝐱 − (𝐱′ − 𝐧′ ), 𝑡 − 𝑡′ )
,
|𝐧′ |
(6.1)
but this reduces the advantages of the method compared to regular RTM. It may also
be possible to estimate the normal derivative of the Green’s function using similar
approaches to those proposed for estimating the normal derivative of the wavefield.
6.3.2
Illumination compensation
Applying illumination compensation to the image produced using the method proposed in this chapter would allow the benefits described in Chapters 4 and 5 to
be combined. This is complicated by the use of synthetic receivers. Including the
synthetic receivers in the illumination calculation will result in reflectors not in the
migration model being incorrectly compensated. A more logical approach may be to
only compensate for the real receivers, as this ensures that the unknown reflectors
have the correct amplitude, although it will result in incorrect amplitude for known
reflectors in the image. It may be possible, with further research, to devise a means
of correcting the amplitude at the known reflectors using our knowledge of them.
Alternatively, the FWI approach to imaging, only backpropagating the arrivals not
predicted by the model, could be used. Approximations of the point spread function
at the known reflectors, derived from illumination information, may then be employed
to insert these reflectors into the image with amplitudes that allow comparison with
the unknown reflectors.
177
178
Chapter 7
Conclusion
Imaging with overturned waves and internal multiples is more complicated than using
primaries alone, but provides the potential to image complex geological structures
that are inaccessible with primaries. This is becoming increasingly important because
of the need to reduce exploration risk in difficult areas such as around salt bodies,
which are common in many of the major oil provinces, including the Gulf of Mexico
and West Africa. This thesis aims to improve the images produced by these types
of arrivals through modifications to existing migration algorithms, allowing rapid
implementation. The proposed methods make pragmatic approximations, such as
the high frequency assumption of Equation 4.14. While these will reduce accuracy,
they make the proposals practical with existing computational resources.
The proposed modifications begin in Chapter 2, with an extension to allow oneway migration to efficiently exploit both overturned waves and multiples. Its most
compelling advantages are the computational efficiency derived from its use of the
one-way wave equation, not requiring multiple-generating reflectors to be present in
the velocity model, and its ability to isolate image contributions from different wave
paths. Although it provides good results, it is perhaps most interesting as a means of
evaluating whether multiples and overturned waves provide the potential for sufficient
image improvement to consider application of the more costly RTM-based methods
that are the focus of the remainder of the thesis.
Chapter 3 moves to the two-way wave equation, but rather than describing meth179
ods for imaging, postpones this until later chapters, and instead focuses on developing three approaches to determining the wave amplitude propagating in different
directions, including in cases when waves overlap. Although more computationally
demanding than previously proposed methods such as the Poynting vector and local
slowness methods, results indicate that the proposed methods are superior in the majority of examined situations. These methods could be used in several applications,
such as constructing ADCIGs for AVA or MVA studies, but as the next two chapters
show, they are also useful in advanced imaging techniques.
Although RTM is naturally able to image with overturned waves, and multiples
if the generating interfaces are in the migration model, results in Chapter 4 show
that it often does not use these arrivals to make effective contributions to the image.
This is because they are generally so weak compared to primaries that they have
negligible image impact. They also cause a variety of image artifacts: low frequency
backscatter artifacts when the source and data waves overlap over long portions of
their wave paths, and several forms of phantom reflector artifacts. Perhaps the most
problematic aspect of regular RTM’s use of these arrivals is that they often image
reflectors from the opposite side to primaries and so may subtract from the image
rather than adding to it, reducing image quality. These problems are all addressed
by the modifications proposed in Chapter 4, which uses a combination of illumination compensation, reversing the sign of contributions when appropriate to prevent
destructive stacking, and a scattering angle-dependent imaging condition. Chapter 4
also discusses two measures of uncertainty that can be derived from information generated by these modifications, and explores the possibility of improving illumination
compensation by approximating point spread functions.
The illumination compensation component of Chapter 4 can attenuate phantom
reflector artifacts, but it is possible to further reduce their amplitude using the approach of Chapter 5. This is achieved by using Green’s Third Identity to more effectively exploit information known about the velocity model. In doing so, it reduces
the incorrect backpropagation that gives rise to certain forms of phantom reflectors.
The method is very sensitive to sharp reflectors in the velocity model, and so these
180
should only be placed where their location is known with confidence. Reductions in
phantom reflector amplitude relative to that of real reflectors is still possible even
when the reflectors present in the migration model are not as sharp as in reality. In
addition to the ability to reduce phantom reflectors, another important benefit of the
proposed method is its reduced computational cost, due to only backpropagating a
single wavefield. Even the migration of primaries in a smooth migration model may
thus benefit from this modification.
Finally, Chapter 6 discusses potential future research directions to continue development of the ideas contained in the preceding chapters. Forming an image with
overturned waves and multiples is still difficult. It needs a good migration model, and,
except for the one-way approach of Chapter 2, even requires that sharp generating
interfaces be present in the model. With the advent of techniques such as FWI, and
especially when well log data are available, this is not very unrealistic. The proposed
methods provide a practical means of using these arrivals, which may be the only geophysical measurements capable of imaging certain areas of interest in the subsurface.
They can already produce improved images compared to regular RTM on complex
models, such as the 2D portion of the SEAM model that was tested extensively, and
following the proposed research directions is likely to further extend this difference in
image quality.
181
182
Appendix A
Resolution of method 2 and the
local slowness method
183
+D
A
ဳᇐ
ՈԿ
ɞ
C
O
վյ
-C
B
-D
Figure A-1: A downgoing wave at time 𝑡 + 𝐼𝑡 /2, oscillatory over the length 𝑐𝑇 .
We depict the case when 𝐼𝐱 = 𝑐𝑇 is used as the summation length for wavefront
orientation angle separation, and 𝐼𝑡 = 𝑇 is the summation time for the local slowness
spacetime slant stack. O, C, and D are points on the wave, which move with the wave
as it propagates. The semicircle shows half of the top edge of the light cone centered
on time 𝑡. At the time 𝑡 + 𝐼𝑡 /2, the LSS sum for wavefront orientation angle will
be composed of the points of the wave along the line A. At time 𝑡 − 𝐼𝑡 /2 the points
of the wave will be those along the line B. It may be useful to reexamine Figure 3-1
when considering this diagram.
In this appendix we investigate the differences in angular resolution between Chapter 3’s method 2 and the local slowness method with the aid of Figure A-1. This
̂ where
depicts a wave propagating toward the bottom of the page (wave 1, 𝜓1̂ = 𝜋/2,
̂ is a unit vector in the direction that makes an angle of 𝜋/2 with the positive 𝑥
𝜋/2
axis), which is oscillatory over time 𝑇 , and therefore also over the distance 𝑐𝑇 (where
𝑐 is the local wave speed). It is assumed to be a plane wave, constant parallel to the
wavefront. O is a point on the wave being separated into propagation directions at
time 𝑡. At time 𝑡 this point is at the center of the dashed semicircle. The semicircle
represents half of the top of the light cone over which the summation over spacetime
is performed. We show the location of the wave at time 𝑡 + 𝐼𝑡 /2. Summing over the
direction pointing downward would add the amplitude of the wave at the point O
at each time step, so, after division by the number of summed time steps, method
2, like the local slowness method, would correctly say that a wave with the amplitude of the point O was propagating downward, as long as there was no interference
from overlapping waves. The difference between method 2 and the local slowness
184
method becomes apparent when considering the minimum angular distance between
the propagation directions of this wave (wave 1) and an overlapping wave (wave 2),
also assumed to be oscillatory over a distance 𝑐𝑇 , such that the amplitudes assigned
to the propagation directions of the two waves in the methods’ output are correct,
when the spacetime summation is over the time range from 𝑡 − 𝐼𝑡 /2 to 𝑡 + 𝐼𝑡 /2. These
amplitudes will be correct if, when the method sums over this time along the propagation direction of one wave (which, without loss of generality, we take to be wave
2), it includes in the sum all points along the propagation direction of wave 1 over its
oscillatory distance 𝑐𝑇 (this will result in cancellation of wave 1). Here, we compute
the minimum difference in propagation direction between waves 1 and 2 for this to be
true for the local slowness method and method 2. We describe the case where LSS is
used to perform wavefront orientation angle separation in the new method, however
the Fourier transform or curvelet approaches could also be used with a similar effect.
To accomplish this, we determine the elements of wave 1 that are contained in the
summation along the propagation direction of wave 2, whose propagation direction
differs from that of wave 1 by an angle Δ𝜓 (𝜓2̂ = 𝜋/2̂
+ Δ𝜓). Although we choose
wave 1 to be propagating downward for simplicity, the results are still valid if the two
waves are rotated.
A.1 Local slowness method
For the local slowness method, we rewrite Equation 3.2, using a boxcar window
function 𝑊 ,
𝐼𝑡 /2
𝑢𝑐𝑠 [𝑥, 𝑧, 𝜓2̂ , 𝑡]
1
𝑢[𝑥 + 𝑗𝑐[𝑥, 𝑧] sin(Δ𝜓), 𝑧 + 𝑗𝑐[𝑥, 𝑧] cos(Δ𝜓), 𝑡 + 𝑗], (A.1)
𝐼
𝑡
/2
= ∑
𝑗=−𝐼𝑡
where 𝑢𝑐𝑠 [𝑥, 𝑧, 𝜓2̂ , 𝑡] is the amplitude determined to be propagating in the direction
𝜓2̂ at position (𝑥, 𝑧) and time 𝑡, and the other symbols are as defined previously, in
particular 𝐼𝑡 is the number of time steps in the summation.
This sum will include the same value of wave 2 at each time step, but different
185
elements of wave 1. We may therefore separate it into two terms,
𝐼𝑡 /2
𝑢𝑐𝑠 [𝑥, 𝑧, 𝜓2̂ , 𝑡]
1
(𝑢𝑠 [𝑥, 𝑧, 𝜓2̂ , 𝑡]+
𝐼
/2 𝑡
= ∑
𝑗=−𝐼𝑡
̂ 𝑡 + 𝑗])
𝑢𝑠 [𝑥 + 𝑗𝑐[𝑥, 𝑧] sin(Δ𝜓), 𝑧 + 𝑗𝑐[𝑥, 𝑧] cos(Δ𝜓), 𝜋/2,
(A.2)
=𝑢𝑠 [𝑥, 𝑧, 𝜓2̂ , 𝑡]+
𝐼𝑡 /2
1
̂ 𝑡 + 𝑗],
𝑢𝑠 [𝑥 + 𝑗𝑐[𝑥, 𝑧] sin(Δ𝜓), 𝑧 + 𝑗𝑐[𝑥, 𝑧] cos(Δ𝜓), 𝜋/2,
𝐼
/2 𝑡
∑
𝑗=−𝐼𝑡
(A.3)
̂ 𝑡] is the true
where 𝑢𝑠 [𝑥, 𝑧, 𝜓2̂ , 𝑡] is the true amplitude of wave 2, and 𝑢𝑠 [𝑥, 𝑧, 𝜋/2,
amplitude of wave 1.
In order for the calculated amplitude to be correct, we require that
𝑢𝑐𝑠 [𝑥, 𝑧, 𝜓2̂ , 𝑡] = 𝑢𝑠 [𝑥, 𝑧, 𝜓2̂ , 𝑡].
(A.4)
1
̂ 𝑡 + 𝑗] = 0.
𝑢𝑠 [𝑥 + 𝑗𝑐[𝑥, 𝑧] sin(Δ𝜓), 𝑧 + 𝑗𝑐[𝑥, 𝑧] cos(Δ𝜓), 𝜋/2,
𝐼
𝑡
/2
(A.5)
For this to be true,
𝐼𝑡 /2
∑
𝑗=−𝐼𝑡
To relate this equation more directly to wave 1, we will replace the fixed reference
frame with a reference frame moving downward with wave 1. Since wave 1 is assumed
to be constant perpendicular to the propagation direction, we will also disregard the
𝑥 coordinates, giving
𝐼𝑡 /2
1
̂ 𝑡] = 0.
𝑢𝑠 [𝑥, 𝑧 + 𝑗𝑐[𝑥, 𝑧] cos(Δ𝜓) − 𝑗𝑐[𝑥, 𝑧], 𝜋/2,
𝐼
/2 𝑡
∑
𝑗=−𝐼𝑡
(A.6)
This sum is over values of the downgoing wave from the point -C (at time 𝑡 − 𝐼𝑡 /2)
186
to C (at time 𝑡 + 𝐼𝑡 /2) in Figure A-1, where
C=O+
𝑐𝐼𝑡
(1 − cos Δ𝜓)𝑧.̂
2
(A.7)
The points C and -C are symmetric about the point O rather than the center of the
semicircle (even though summation along any direction on the lightcone will indeed
be symmetric about the center of the semicircle) because the figure shows the points
on the downgoing wave that are included in the summation and the downgoing wave
moves with time.
We know that wave 1 is oscillatory over time 𝑇 , and therefore distance 𝑐[𝑥, 𝑧]𝑇 ,
which implies that
𝑐[𝑥,𝑧]𝑇 /2
̂ 𝑡] = 0,
𝑢𝑠 [𝑥, 𝑧 − 𝑖, 𝜋/2,
∑
(A.8)
𝑖=−𝑐[𝑥,𝑧]𝑇 /2
where we have assumed that wave 1 is either a wave packet of length 𝑐[𝑥, 𝑧]𝑇 and 𝑧
is in the center of it, or is periodic.
Comparing Equations A.6 and A.8 indicates that for A.4 to be true, we must have
𝐼𝑡 (cos(Δ𝜓) − 1) = −𝑇
⇒ Δ𝜓 = arccos (1 −
𝑇
).
𝐼𝑡
(A.9)
(A.10)
This is equivalent to the condition
2C ≥ 𝑐𝑇
(A.11)
If the waves have propagation directions separated by Δ𝜓 = 𝜋/6, we therefore
require that
√
𝐼𝑡
=4+2 3
𝑇
187
(A.12)
A.2
Method 2
We will now follow a similar approach for method 2 using LSS for wavefront orientation angle separation. This involves a summation over space, to apply LSS, and the
results of this are then summed in spacetime along the light cone. This gives
𝐼𝑡 /2
𝑢𝑐𝑠 [𝑥, 𝑧, 𝜓2̂ , 𝑡]
𝑐[𝑥,𝑧]𝐼𝑡 /2
= ∑
∑
𝑗=−𝐼𝑡 /2 𝑖=−𝑐[𝑥,𝑧]𝐼𝑡
1
×
𝑐[𝑥, 𝑧]𝐼𝑡2
/2
𝑢[𝑥 + 𝑗𝑐[𝑥, 𝑧] sin Δ𝜓 + 𝑖 cos Δ𝜓, 𝑧 + 𝑗𝑐[𝑥, 𝑧] cos Δ𝜓 − 𝑖 sin Δ𝜓, 𝑡 + 𝑗]
(A.13)
for the calculated amplitude, 𝑢𝑐𝑠 , which is equivalent to
𝐼𝑡 /2
𝑢𝑐𝑠 [𝑥, 𝑧, 𝜓2̂ , 𝑡]
= ∑
𝑐[𝑥,𝑧]𝐼𝑡 /2
∑
𝑗=−𝐼𝑡 /2 𝑖=−𝑐[𝑥,𝑧]𝐼𝑡
1
(𝑢𝑠 [𝑥, 𝑧, 𝜓2̂ , 𝑡]+
2
𝑐[𝑥,
𝑧]𝐼
𝑡
/2
𝑢𝑠 [𝑥 + 𝑗𝑐[𝑥, 𝑧] sin Δ𝜓 + 𝑖 cos Δ𝜓,
̂ 𝑡 + 𝑗]),
𝑧 + 𝑗𝑐[𝑥, 𝑧] cos Δ𝜓 − 𝑖 sin Δ𝜓, 𝜋/2,
(A.14)
where, as in the local slowness calculation, we split the wavefield 𝑢 into components
from waves 1 and 2.
In order for 𝑢𝑐𝑠 [𝑥, 𝑧, 𝜓2̂ , 𝑡] = 𝑢𝑠 [𝑥, 𝑧, 𝜓2̂ , 𝑡] to be true, we therefore require that
𝐼𝑡 /2
𝑐[𝑥,𝑧]𝐼𝑡 /2
∑
∑
𝑗=−𝐼𝑡 /2 𝑖=−𝑐[𝑥,𝑧]𝐼𝑡
1
̂ 𝑡] = 0.
𝑢 [𝑥, 𝑧 + 𝑗𝑐[𝑥, 𝑧] cos Δ𝜓 − 𝑖 sin Δ𝜓 − 𝑗𝑐[𝑥, 𝑧], 𝜋/2,
2 𝑠
𝑐[𝑥,
𝑧]𝐼
𝑡
/2
(A.15)
At time 𝑡 + 𝐼𝑡 /2 this spatial summation is along the line A, as shown in the figure.
At time 𝑡 − 𝐼𝑡 /2 it will be along the part of the downgoing wave covered by the
line B. As with the labeled points, these lines use the (moving) downgoing wave as
a reference frame, rather than a fixed point in space. If a fixed point in space had
instead been used, B would be the mirror of A through the center of the semicircle.
The combination of these two summations will therefore cover the range of values of
188
the downgoing wave from −D to D, where
D=O+
𝑐𝐼𝑡
(1 − cos Δ𝜓 + sin Δ𝜓)𝑧.̂
2
(A.16)
While Equation A.15 cannot be directly related to Equation A.8 due to the double
summation, we see that for small values of Δ𝜓,
𝑗𝑐[𝑥, 𝑧] cos Δ𝜓 − 𝑗𝑐[𝑥, 𝑧] ≈ 0.
(A.17)
In this small angle regime, we can match Equation A.8 when
𝐼𝑡 sin Δ𝜓 = 𝑇
(A.18)
𝑇
⇒ Δ𝜓 = arcsin ( ) .
𝐼𝑡
(A.19)
This is, in fact, the minimum difference in wavefront orientation angle between
two waves that the LSS method can resolve, for a given 𝑇 and 𝐼𝑡 . It is therefore the
resolution of method 1 using LSS when no filters are applied. As shown in Figure 3-7,
below 𝜋/2 this angle is smaller for a given 𝐼𝑡 (as a multiple of 𝑇 ) than the minimum
angle resolvable with the local slowness method.
When Δ𝜓 is not small, the 𝑗𝑐[𝑥, 𝑧](cos Δ𝜓 − 1) term is non-negligible. If wave 1
is periodic, with period 𝑇 , then as long as the LSS sum covers the distance 𝑐𝑇 in the
direction of periodicity, shifting the sum in space, as occurs in Equation A.15, does
not affect the output. The condition in Equation A.19 therefore still holds.
When wave 1 is not periodic, the sum in Equation A.15 will in general include
different 𝑧 values of the wave different numbers of times. While some cancellation
may take place, there is likely to be a nonzero remainder. If wave 1 is a wave packet
which is only nonzero over a length 𝑐𝑇 in 𝑧, then we may ensure that the sum equates
to zero by choosing 𝐼𝑡 such that all of the nonzero 𝑧 elements are included an equal
number of times in the sum. By inspection of Figure A-1, this can be achieved for
189
Δ𝜓 ≤ 𝜋/2 by requiring that
𝐼𝑡
1
≥
,
𝑇
cos Δ𝜓 + sin Δ𝜓 − 1
(A.20)
and for Δ𝜓 > 𝜋/2 with the condition
𝐼𝑡
1
≥
,
𝑇
− cos Δ𝜓 − sin Δ𝜓 + 1
(A.21)
These equations are plotted in Figure A-2. The method is able to attain similar
resolution to method 1 for small Δ𝜓, and to the local slowness method near Δ𝜓 = 𝜋.
It has difficulty when waves are propagating in directions separated by right angles,
however. This is because when Δ𝜓 = 𝜋/2, any choice of 𝐼𝑡 will result in the value of
wave 1 at O being included in the summation twice as many times as the values at
the edges of the wave packet, and so full cancellation is not possible (but the result
should still be small). Nevertheless, we see from the results section that the method
still appears to work effectively in many cases.
Minimum integration length (multiple of period)
6
4
2
1/(-cos(x)-sin(x)+1)
1/(cos(x)+sin(x)-1)
0
0
Propagation angle difference (rad)
π
Figure A-2: Similar to Figure 3-7, but for method 2 when the waves are wave packets
of duration 𝑇 .
190
Appendix B
Method 3 gradient and Hessian
191
In this appendix we present the gradient and Hessian for the objective function
of method 3, Equation 3.28 of Chapter 3. These are necessary to efficiently compute
updated model parameters in the optimization method. The symbols are the same
as those in Equation 3.28, including 𝐴, defined by Equation 3.26, and 𝐵, defined by
Equation 3.27.
Gradient
𝑔(𝑢𝑠 , 𝑢, 𝐱, 𝜓,̂ 𝑡) =𝜕𝑢
̂
𝑠 (𝐱,𝜓,𝑡)
𝑓(𝑢𝑠 , 𝑢, 𝑡)
=2𝑤1 𝐴(𝑢𝑠 , 𝑢, 𝐱, 𝑡)+
2𝑤2 ∫ d𝐱′ {𝐵(𝑢𝑠 , 𝑢, 𝐱′ , 𝑡)𝜕𝑢
𝐱
̂ 𝐵(𝑢𝑠 , 𝑢, 𝐱
𝑠 (𝐱,𝜓,𝑡)
′
, 𝑡)} +
(B.1)
2𝑤𝑒 𝑢(𝐱, 𝜓,̂ 𝑡)
Hessian
𝐻(𝑢𝑠 , 𝑢, 𝐱, 𝜓,̂ 𝐱′ , 𝜓′̂ , 𝑡) =𝜕𝑢
′
′̂
𝑠 (𝐱 ,𝜓 ,𝑡)
𝑔(𝑢𝑠 , 𝑢, 𝐱, 𝜓,̂ 𝑡)
=2𝑤1 𝛿(𝐱 − 𝐱′ )+
2𝑤2 ∫ d𝐱″ {𝜕𝑢
𝐱
′
′̂
𝑠 (𝐱 ,𝜓 ,𝑡)
𝐵(𝑢𝑠 , 𝑢, 𝐱″ , 𝑡)𝜕𝑢
̂
𝑠 (𝐱,𝜓,𝑡)
𝐵(𝑢𝑠 , 𝑢, 𝐱″ , 𝑡)} +
2𝑤𝑒 𝛿(𝐱 − 𝐱′ )𝛿(𝜓 ̂ − 𝜓′̂ )
(B.2)
192
Appendix C
Method 3 implementation
193
This appendix describes the implementation of the objective function, gradient,
and Hessian for Chapter 3’s method 3. We assume that the wavefield to be separated,
𝑢, is 2D.
Objective function
We begin with the functions 𝐴 and 𝐵, defined for continuous spacetime in Equations 3.26 and 3.27, respectively. As stated above, we use the FTCS finite difference
scheme to propagate the separated wavefields forward one time step. Δ𝑥 and Δ𝑧 are
the grid cell sizes in the 𝑥 and 𝑧 directions, while Δ𝑡 is the time step interval. We
implement directional derivatives using
̂
𝜕𝜓̂𝑢 = 𝜕𝑥̂ 𝑢 ⋅ 𝜓𝑥̂ + 𝜕𝑧 𝑢
̂ ⋅ 𝜓𝑧 ,
(C.1)
where 𝜓𝑥̂ and 𝜓𝑧̂ are the 𝑥 and 𝑧 components of the direction unit vector 𝜓.̂
𝐴[𝑢𝑠 , 𝑢, 𝑥, 𝑧, 𝑡] = ∑(𝑢𝑠 [𝑥, 𝑧, 𝜓′̂ , 𝑡]) − 𝑢[𝑥, 𝑧, 𝑡]
(C.2)
𝜓′̂
𝐵[𝑢𝑠 , 𝑢, 𝑥, 𝑧, 𝑡] = ∑(𝑢𝑠 [𝑥, 𝑧, 𝜓′̂ , 𝑡]+
𝜓′̂
Δ𝑡 𝑐[𝑥, 𝑧]𝜓𝑥′̂
𝑢𝑠 [𝑥 + 1, 𝑧, 𝜓′̂ , 𝑡]−
2Δ𝑥
Δ𝑡 𝑐[𝑥, 𝑧]𝜓𝑥′̂
𝑢𝑠 [𝑥 − 1, 𝑧, 𝜓′̂ , 𝑡]+
2Δ𝑥
Δ𝑡 𝑐[𝑥, 𝑧]𝜓𝑧′̂
𝑢𝑠 [𝑥, 𝑧 + 1, 𝜓′̂ , 𝑡]−
2Δ𝑧
Δ𝑡 𝑐[𝑥, 𝑧]𝜓𝑧′̂
𝑢𝑠 [𝑥, 𝑧 − 1, 𝜓′̂ , 𝑡]) − 𝑢[𝑥, 𝑧, 𝑡 + 1]
2Δ𝑧
(C.3)
When computing the gradient and Hessian we will require the partial derivative
of 𝐵 with respect to the model parameters, so for clarity we provide it here,
194
𝜕𝑢
𝑠 [𝑥
′ ,𝑧′ ,𝜓′̂ ,𝑡]
𝐵[𝑢𝑠 , 𝑢, 𝑥, 𝑧, 𝑡] =𝛿[𝑥 − 𝑥′ ]𝛿[𝑧 − 𝑧 ′ ]+
Δ𝑡 𝑐[𝑥, 𝑧]𝜓𝑥′̂
𝛿[𝑥 + 1 − 𝑥′ ]𝛿[𝑧 − 𝑧 ′ ]−
2Δ𝑥
Δ𝑡 𝑐[𝑥, 𝑧]𝜓𝑥′̂
𝛿[𝑥 − 1 − 𝑥′ ]𝛿[𝑧 − 𝑧 ′ ]+
2Δ𝑥
Δ𝑡 𝑐[𝑥, 𝑧]𝜓𝑧′̂
𝛿[𝑧 + 1 − 𝑧 ′ ]𝛿[𝑥 − 𝑥′ ]−
2Δ𝑧
Δ𝑡 𝑐[𝑥, 𝑧]𝜓𝑧′̂
𝛿[𝑧 − 1 − 𝑧 ′ ]𝛿[𝑥 − 𝑥′ ].
2Δ𝑧
(C.4)
Our objective function (defined for continuous spacetime in Equation 3.28) then
becomes
𝑓[𝑢𝑠 , 𝑢, 𝑡] = ∑ ∑ 𝑤1 𝐴[𝑢𝑠 , 𝑢, 𝑥′ , 𝑧 ′ , 𝑡]2 +𝑤2 𝐵[𝑢𝑠 , 𝑢, 𝑥′ , 𝑧 ′ , 𝑡]2 +𝑤𝑒 ∑(𝑢𝑠 [𝑥′ , 𝑧 ′ , 𝜓′̂ , 𝑡])2 .
𝑥′
𝑧′
𝜓′̂
(C.5)
Gradient
Discretizing Equation B.1 gives the gradient in a form that can be implemented,
𝑔[𝑢𝑠 , 𝑢, 𝑥, 𝑧, 𝜓,̂ 𝑡] =2𝑤1 𝐴[𝑢𝑠 , 𝑢, 𝑥, 𝑧, 𝑡]+
2𝑤2 (∑ ∑ 𝐵[𝑢𝑠 , 𝑢, 𝑥′ , 𝑧 ′ , 𝑡]𝜕𝑢
𝑥′
̂
𝑠 [𝑥,𝑧,𝜓,𝑡]
𝑧′
𝐵[𝑢𝑠 , 𝑢, 𝑥′ , 𝑧 ′ , 𝑡]) +
2𝑤𝑒 𝑢𝑠 [𝑥, 𝑧, 𝜓,̂ 𝑡].
(C.6)
195
Hessian
The Hessian for this objective function is sparse, as we show below.
𝐻[𝑢𝑠 , 𝑢, 𝑥, 𝜓,̂ 𝑥′ , 𝜓′̂ ] = 2𝑤1 𝛿[𝑥 − 𝑥′ ]𝛿[𝑧 − 𝑧 ′ ]+
2𝑤2 (∑ ∑ (𝜕𝑢
𝑥″
𝑧″
𝑠 [𝑥
′ ,𝑧′ ,𝜓′̂ ,𝑡]
𝐵[𝑢𝑠 , 𝑢, 𝑥″ , 𝑧 ″ , 𝑡]) (𝜕𝑢
̂
𝑠 [𝑥,𝑧,𝜓,𝑡]
𝐵[𝑢𝑠 , 𝑢, 𝑥″ , 𝑧 ″ , 𝑡])) +
2𝑤𝑒 𝛿[𝑥 − 𝑥′ ]𝛿[𝑧 − 𝑧′ ]𝛿[𝜓 ̂ − 𝜓′̂ ]
(C.7)
Expanding, using Equation C.4,
𝐻[𝑢𝑠 , 𝑢, 𝑥, 𝜓,̂ 𝑥′ , 𝜓′̂ ] = 𝛿[𝑥 − 𝑥′ ]𝛿[𝑧 − 𝑧 ′ ] (2𝑤1 + 2𝑤𝑒 𝛿[𝜓 ̂ − 𝜓′̂ ]) +
2𝑤2 (∑ ∑
𝑥″
𝑧″
(𝛿[𝑥″ − 𝑥′ ]𝛿[𝑧 ″ − 𝑧 ′ ]+
Δ𝑡 𝑐[𝑥″ , 𝑧 ″ ]𝜓𝑥′̂
𝛿[𝑥″ + 1 − 𝑥′ ]𝛿[𝑧 ″ − 𝑧 ′ ]−
2Δ𝑥
Δ𝑡 𝑐[𝑥″ , 𝑧 ″ ]𝜓𝑥′̂
𝛿[𝑥″ − 1 − 𝑥′ ]𝛿[𝑧 ″ − 𝑧 ′ ]+
2Δ𝑥
Δ𝑡 𝑐[𝑥″ , 𝑧 ″ ]𝜓𝑧′̂
𝛿[𝑧″ + 1 − 𝑧 ′ ]𝛿[𝑥″ − 𝑥′ ]−
2Δ𝑧
Δ𝑡 𝑐[𝑥″ , 𝑧 ″ ]𝜓𝑧′̂
𝛿[𝑧″ − 1 − 𝑧 ′ ]𝛿[𝑥″ − 𝑥′ ])×
2Δ𝑧
(C.8)
(𝛿[𝑥″ − 𝑥]𝛿[𝑧 ″ − 𝑧]+
Δ𝑡 𝑐[𝑥″ , 𝑧 ″ ]𝜓𝑥̂
𝛿[𝑥″ + 1 − 𝑥]𝛿[𝑧 ″ − 𝑧]−
2Δ𝑥
Δ𝑡 𝑐[𝑥″ , 𝑧 ″ ]𝜓𝑥̂
𝛿[𝑥″ − 1 − 𝑥]𝛿[𝑧 ″ − 𝑧]+
2Δ𝑥
Δ𝑡 𝑐[𝑥″ , 𝑧 ″ ]𝜓𝑧̂
𝛿[𝑧″ + 1 − 𝑧]𝛿[𝑥″ − 𝑥]−
2Δ𝑧
Δ𝑡 𝑐[𝑥″ , 𝑧 ″ ]𝜓𝑧̂
𝛿[𝑧″ − 1 − 𝑧]𝛿[𝑥″ − 𝑥])).
2Δ𝑧
We rearrange this to isolate the different components, making the sparsity more
196
obvious. The Hessian has 13 non-zero entries for each propagation direction that the
wavefield is separated into at each position.
197
𝐻[𝑢𝑠 , 𝑢, 𝑥, 𝜓,̂ 𝑥′ , 𝜓′̂ ] = 𝛿[𝑥′ − 𝑥]𝛿[𝑧 ′ − 𝑧](2𝑤1 + 2𝑤2 (1+
(
Δ 𝑐[𝑥′ + 1, 𝑧′ ] 2 ′̂ ̂
Δ𝑡 𝑐[𝑥′ − 1, 𝑧 ′ ] 2 ′̂ ̂
) 𝜓𝑥 𝜓𝑥 + ( 𝑡
) 𝜓𝑥 𝜓𝑥 +
2Δ𝑥
2Δ𝑥
(
Δ𝑡 𝑐[𝑥′ , 𝑧 ′ − 1] 2 ′̂ ̂
Δ 𝑐[𝑥′ , 𝑧 ′ + 1] 2 ′̂ ̂
̂
) 𝜓𝑧 𝜓𝑧 + ( 𝑡
) 𝜓𝑧 𝜓𝑧 ) + 2𝑤𝑒 𝛿[𝜓′̂ − 𝜓])−
2Δ𝑧
2Δ𝑧
𝛿[𝑥′ − 1 − 𝑥]𝛿[𝑧 ′ − 𝑧](2𝑤2
Δ𝑡
(𝑐[𝑥′ , 𝑧 ′ ]𝜓𝑥̂ − 𝑐[𝑥′ − 1, 𝑧 ′ ]𝜓𝑥′̂ ))+
2Δ𝑥
𝛿[𝑥′ + 1 − 𝑥]𝛿[𝑧 ′ − 𝑧](2𝑤2
Δ𝑡
(𝑐[𝑥′ , 𝑧 ′ ]𝜓𝑥̂ − 𝑐[𝑥′ + 1, 𝑧 ′ ]𝜓𝑥′̂ ))−
2Δ𝑥
𝛿[𝑥′ − 𝑥]𝛿[𝑧 ′ − 1 − 𝑧](2𝑤2
Δ𝑡
(𝑐[𝑥′ , 𝑧 ′ ]𝜓𝑧̂ − 𝑐[𝑥′ , 𝑧 ′ − 1]𝜓𝑧′̂ ))+
2Δ𝑧
𝛿[𝑥′ − 𝑥]𝛿[𝑧 ′ + 1 − 𝑧](2𝑤2
Δ𝑡
(𝑐[𝑥′ , 𝑧 ′ ]𝜓𝑧̂ − 𝑐[𝑥′ , 𝑧 ′ + 1]𝜓𝑧′̂ ))−
2Δ𝑧
𝛿[𝑥′ − 2 − 𝑥]𝛿[𝑧 ′ − 𝑧](2𝑤2 (
Δ𝑡 𝑐[𝑥′ − 1, 𝑧′ ] 2 ′̂ ̂
) 𝜓𝑥 𝜓𝑥 )−
2Δ𝑥
𝛿[𝑥′ + 2 − 𝑥]𝛿[𝑧 ′ − 𝑧](2𝑤2 (
Δ𝑡 𝑐[𝑥′ + 1, 𝑧′ ] 2 ′̂ ̂
) 𝜓𝑥 𝜓𝑥 )−
2Δ𝑥
𝛿[𝑥′ − 𝑥]𝛿[𝑧 ′ − 2 − 𝑧](2𝑤2 (
Δ𝑡 𝑐[𝑥′ , 𝑧 ′ − 1] 2 ′̂ ̂
) 𝜓𝑧 𝜓𝑧 )−
2Δ𝑧
𝛿[𝑥′ − 𝑥]𝛿[𝑧 ′ + 2 − 𝑧](2𝑤2 (
Δ𝑡 𝑐[𝑥′ , 𝑧 ′ + 1] 2 ′̂ ̂
) 𝜓𝑧 𝜓𝑧 )−
2Δ𝑧
𝛿[𝑥′ − 1 − 𝑥]𝛿[𝑧 ′ − 1 − 𝑧]2𝑤2 ((
Δ 𝑐[𝑥′ , 𝑧 ′ − 1] 2 𝜓𝑧′̂ 𝜓𝑥̂
Δ𝑡 𝑐[𝑥′ − 1, 𝑧′ ] 2 𝜓𝑥′̂ 𝜓𝑧̂
)
+( 𝑡
)
)+
2
Δ𝑥 Δ𝑧
2
Δ𝑧 Δ𝑥
𝛿[𝑥′ + 1 − 𝑥]𝛿[𝑧 ′ − 1 − 𝑧]2𝑤2 ((
Δ𝑡 𝑐[𝑥′ + 1, 𝑧′ ] 2 𝜓𝑥′̂ 𝜓𝑧̂
Δ 𝑐[𝑥′ , 𝑧 ′ − 1] 2 𝜓𝑧′̂ 𝜓𝑥̂
)
)
+( 𝑡
)+
2
Δ𝑥 Δ𝑧
2
Δ𝑧 Δ𝑥
𝛿[𝑥′ − 1 − 𝑥]𝛿[𝑧 ′ + 1 − 𝑧]2𝑤2 ((
Δ 𝑐[𝑥′ , 𝑧 ′ + 1] 2 𝜓𝑧′̂ 𝜓𝑥̂
Δ𝑡 𝑐[𝑥′ − 1, 𝑧′ ] 2 𝜓𝑥′̂ 𝜓𝑧̂
)
)
+( 𝑡
)−
2
Δ𝑥 Δ𝑧
2
Δ𝑧 Δ𝑥
𝛿[𝑥′ + 1 − 𝑥]𝛿[𝑧 ′ + 1 − 𝑧]2𝑤2 ((
Δ𝑡 𝑐[𝑥′ + 1, 𝑧′ ] 2 𝜓𝑥′̂ 𝜓𝑧̂
Δ 𝑐[𝑥′ , 𝑧 ′ + 1] 2 𝜓𝑧′̂ 𝜓𝑥̂
)
)
+( 𝑡
)
2
Δ𝑥 Δ𝑧
2
Δ𝑧 Δ𝑥
(C.9)
198
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