Full Wavefield Inversion Methods for Monitoring
Time-lapse Subsurface Velocity Changes
by
OF TECHNOLOGY
Di Yang
OCT 16 204
B.S, M.S., Nanjing University (2009)
LIBRARIES
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
September 2014
@ Massachusetts Institute of Technology 2014. All rights reserved.
Signature redacted
A uthor .......
..............................
Department of Earth, Atmospheric and Planetary Sciences
June 2, 2014
Signature redacted
Certified by
Signature redacted
Alison E. Malcolm
Assistant Professor
Thesis Supervisor
Certified by.
Signature redacted
Accepted by...
1~
Michael Fehler
Senior Research Scientist
Thesis Supervisor
.. .. . .
. v. . . .Hilst
Schlumberger Professor of Earth Sciences
Head, Department of Earth, Atmospheric and Planetary Sciences
2
Full Wavefield Inversion Methods for Monitoring Time-lapse
Subsurface Velocity Changes
by
Di Yang
Submitted to the Department of Earth, Atmospheric and Planetary Sciences
on June 2, 2014, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Geophysics
Abstract
Quantitative measurements of seismic velocity changes from time-lapse seismic experiments provide dynamic information about the subsurface that improves the understanding of the geology and reservoir properties. In this thesis, we propose to
achieve the quantitative analysis using full wavefield inversion methods which are
robust in complex geology. We developed several methodologies in both the data
domain and image domain to handle different time-lapse seismic acquisitions. In the
data domain, we implemented double-difference waveform inversion (DDWI), and investigated its robustness and feasibility with realistic acquisition non-repeatabilities.
Well-repeated time-lapse surveys from Valhall in the North Sea are used to compare
DDWI and conventional time-lapse full waveform inversion (FWI) schemes. An FWI
approach that uses the baseline and monitor datasets in an alternating manner is
proposed to handle time-lapse surveys without restrictions on geometry repeatability, and to provide an uncertainty analysis on the time-lapse changes. In the image
domain, we propose time-lapse image domain wavefield tomography (IDWT) that
inverts for P- and S-wave velocity changes by matching baseline and monitor images
produced with small offset reflection surveys. This method is robust to survey geometry non-repeatabilities and baseline velocity errors. A low velocity zone caused
by local CO 2 injections in SACROC, West Texas is found by IDWT with time-lapse
walkaway vertical seismic profile surveys. The methods in this thesis combined, allow
for an integrated velocity inversion to achieve high-resolution subsurface monitoring
with various types of acquisitions in complex geology.
Thesis Supervisor: Alison E. Malcolm
Title: Assistant Professor
Thesis Supervisor: Michael Fehler
Title: Senior Research Scientist
3
4
Acknowledgments
Time flies incredibly fast for the past five years, with my memory of my first day
at MIT still fresh. I came to this place with the expectation of making an impact.
Thanks to all the people I met these years, this expectation has not died out, but
becomes grounded.
MIT opened a door to a world of opportunities for me, but I could not have
grasped any without the support from my advisors. I was lucky to have the chance
to work with the great people at MIT, to whom I cannot be more grateful.
The first time I got up the courage to talk to Dr. Michael Fehler about potential
projects, I had little understanding about the Earth sciences. Since then, he became
the person I ask for advice whenever I have problems in research. In addition to
his scientific insights, Mike connects me with the experts in national labs and energy
companies, who become our collaborators and good friends. Conversations with Mike
are always enjoyable. I learn not only science, but also communication, culture and
life from him. The glass of wine at the 7500 feet altitude that easily made me drunk,
is a memory worth cherishing.
Professor Alison Malcolm is my double mentor for the past five years. She is an
incredible resource of both seismology and parenting to me. We often finish up our
scientific discussions with a progress update of my daughter Miya. What I deeply
appreciate about Alison, is the freedom and support she generously offers on my research. Without her encouragement, my thesis could not have gone this far. Besides
research, the most awkward but extremely helpful class I took at MIT is the presentation training, also by Alison. The unique experience of pretending the annoying
audience for each other brought me laughters and the skills that I will carry on for
my future career.
The uniqueness about MIT, is the freedom of exploration. I could not have imagined that someday I would work on a significant space mission with NASA, if I was
not at MIT. Professor Maria Zuber, introduced me to a brand new world of space
exploration and the NASA MESSENGER geophysics team. The experience of work5
ing with the top-notched scientists from around the globe, brought me to a different
level of understanding on both what science is, and why we do science.
Maria is
an amazing advisor, from whom I learned how to look at big pictures, and how to
prioritize science questions.
During my years at MIT, many people helped me overcome the challenges of transferring from Electronic Engineering to Earth Science. The instructive conversations
with Prof. Nafi Toksoz pointed out directions when I was struggling with my understanding of the field. Excellent classes given by Prof. Robert van der Hilst, Prof.
Dale Morgan, Prof. Stephane Rondenay, and Prof. Laurent Demanet prepared me a
grounded knowledge base for my research. Dr. William Rodi was always available for
discussions on inverse problems. I want to give my sincere thanks to all the experts
for the education.
Without the tremendous support from my collaborators, I could not have finished
the thesis work. Dr. Lianjie Huang from Los Alamos National Laboratory offered
me my first internship, which opened the first page of my thesis research on full
waveform inversion. Dr. Scott Morton and Dr. Faqi Liu, provided precious resources
and help at Hess Corporation that made our 3D real data application possible and
successful. Dr. Mark Meadows, Dr. Phil Inderweisen, and Dr. Jorge Landa from
Chevron Corporation helped me with their rich experience investigate the practical
issues of time-lapse seismic experiments. I also want to thank Dr. Yong Ma from
Conoco Philips, and Dr. Hyoungsu Baek from Aramco Research Center for their
insightful opinions and constructive communications on the theoretical developments
of my thesis. Special thanks should also go to Prof. Oliver Jagoutz for his precious
time serving on my thesis committee.
People at ERL, are like in a big family. The cohesive bonding between us made
my years in the US a lot easier. Xinding Fang and Xuefeng Shang, are my brothers.
Although they are both younger than me, I always sought for help from them when
I was in trouble with research and life. Fuxian Song brought me to the leadership
team of the MIT Energy Club, which significantly extended my experience on the
business and policy side of the energy sector. The discussion with Yingcai Zheng and
6
Yang Zhang, inspired many research ideas, and improved my understanding of the
academia and industrial careers. I also would like to thank all the members of "The
Chinese Mafia", including Junlun Li, Hui Huang, Chunquan Yu, Tianrun Chen, Chen
.Gu, Ning Zhao, Haoyue Wang, Jing Liu, Dr. Zhenya Zhu and many others for their
generous help and the joyful company.
Leaning and living in the English world is not easy for a grown-up Chinese. Thanks
should go to my friends Sudhish Bakku, Deigo Concha, Alan Richardson, Ahmed Zamanian, Andrey Shabelansky, Yulia Agramakova, Abdul-Aziz Al-Muhaidib and Gabi
Melo for helping me explore the culture, and sharing the knowledge and experience.
We also had a lot of fun together as the members of the gym team and the thesiswriting team.
The most beautiful thing that happened to me in the past five years, is my lovely
wife Misako AiBa. She devoted herself to our family and brought me my precious
daughter Miya. I want to thank her for the unselfish support, caring and love that
motivated me to get this far.
Last but not least, I want to thank my parents and grandparents, who raised me
with all their hearts. They gave me the best education in my life about how to be a
person with integrity, courage and kindness.
This thesis is dedicated to my grandmother Qingsheng Liu in heaven.
7
8
Contents
2
Introduction
31
Objective
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
31
1.2
Data Domain vs. Image Domain. . .
. . . . . . . . . . . . . . . . .
33
1.3
Challenges and Contributions
. . . .
. . . . . . . . . . . . . . . . .
34
1.4
Thesis Outline . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
36
.
.
.
1.1
.
1
Double Difference Waveform Inversion: Method, Feasibility and Robustness Study
Introduction ......................
. . . .
44
2.2
Methodology
. . . . . . . . . . . . . . . . .
.
. . . .
45
2.3
Scheme Comparison with Acoustic Inversion
.
. . . .
46
2.4
Baseline Model Dependence . . . . . . . . .
. . . .
49
2.5
Survey Non-repeatability . . . . . . . . . . .
. . . .
50
2.5.1
Random Noise . . . . . . . . . . . . .
. . . .
51
2.5.2
Source and Receiver Positioning Error
. . . .
52
2.5.3
Source Wavelet Discrepancy . . . . .
. . . .
54
2.5.4
Overburden Velocity Changes . . . .
. . . .
56
2.6
Discussion and Conclusion . . . . . . . . . .
. . . .
58
2.7
Acknowledgments . . . . . . . . . . . . . . .
. . . .
59
.
.
.
.
.
.
.
2.1
Time-lapse Full Waveform Inversion with Ocean Bottom Cable Data:
Application on Valhall Field
83
3.1
84
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
3
43
9
Theory . . . . . . . . . . . . . . . . . . . . . . . . .
86
3.3
Examples Using Synthetic Data . . . . . . . . . . .
88
3.4
Time-lapse Full waveform Inversion on Valhall . . .
89
3.4.1
Acquisition, Repeatability and Preprocessing
90
3.4.2
Inversion Setup . . . . . . . . . . . . . . . .
91
3.4.3
Initial Velocity Model . . . . . . . . . . . . .
92
3.4.4
Baseline Inversion Result . . . . . . . . . . .
92
3.4.5
Time-lapse Inversion Result . . . . . . . . .
93
3.5
Discussion . . . . . . . . . . . . . . . . . . . . . . .
94
3.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . .
96
3.7
Acknowledgments . . . . . . . . . . . . . . . . . . .
96
.
.
.
.
.
.
.
.
.
.
.
3.2
4 Alternating Time-lapse Full Waveform Inversion with Different Sur109
vey Geometries
Introduction . . . . . . . . . . . . . . . . . .
. . . . . .
110
4.2
T heory . . . . . . . . . . . . . . . . . . . . .
. . . . . .
111
4.3
Synthetic Examples with Marmousi Model .
. . . . . .
113
4.3.1
Surveys with Shifted Sources . . . . .
. . . . . .
114
4.3.2
Surveys with Different Illuminations
... .
4.3.3
Surveys with Strong Random Noise
. . . . . .
.
.
.
.
4.1
115
116
Discussion . . . . . . . . . . . . . . . . . . ...
. ... . . . 117
4.5
Conclusion . . . . . . . . . . . . . . . . . . .
. . . . . .
117
4.6
Acknowledgments . . . . . . . . . . . . . . .
. . . . . .
118
.
.
4.4
5 Time-Lapse Walkaway VSP Monitoring for CO 2 Injection at the
131
SACROC EOR Field: A Case Study
Introduction . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 132
5.2
Methodology
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 135
.
.
5.1
5.2.1
Reverse-Time Migration ........
. . . . . . . . . . . . . 135
5.2.2
Full-Waveform Inversion ........
. . . . . . . . . . . . . 135
5.2.3
Image-Domain Wavefield Tomography
. . . . . . . . . . . . . 136
10
6
. . . . . . .
.
138
5.3.1
Geology and Injection History
5.3.2
Well Logs and Reservoir Properties
Seismic Imaging and Inversions
138
139
140
5.4.1
Data Acquisition and Processing
.
140
5.4.2
Initial Velocity Model . . . . . . . .
141
5.4.3
Reverse-Time Migration . . . . . .
142
5.4.4
Full-Waveform Inversion . . . . . .
143
5.4.5
Image-Domain Wavefield Tomography
144
.
.
.
.
. . . . . .
.
5.4
Site Background of SACROC
.
5.3
Discussion . . . . . . . . . . . . . . . . . .
146
5.6
Conclusions . . . . . . . . . . . . . . . . .
149
5.7
Acknowledgment
149
.
.
5.5
.
. . . . . . . . . . . . . .
Using Image Warping for Time-lapse Image Domain Wavefield Tomography
167
Introduction . . . . . . . . . . . . . . . .
. . . . . . . .
168
6.2
T heory . . . . . . . . . . . . . . . . . . .
. . . . . . . .
170
6.3
Examples Using Synthetic Data .......
. . . . . . . .
173
6.3.1
Three-layer Model ...........
. . . . . . . .
174
6.3.2
Multi-layer Model ..........
. . . . . . . .
176
6.3.3
Baseline Velocity Errors .......
. . . . . . . .
176
6.3.4
Source Geometry Non-repeatability
. . . . . . . .
177
6.3.5
Marmousi Model ............
. . . . . . . .
180
.
.
6.1
6.4
Discussion ......................
. . . . . . . .
181
6.5
Conclusion ......................
. . . . . . . .
183
6.6
Acknowledgments .................
. . . . . . . .
184
7 Image Registration Guided Shear Wave Velocity Model Building
197
. . . . . . . . .
198
7.2
Theory . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .
200
7.2.1
. . . . . . . . .
200
11
.
.
.
.
Elastic Reverse Time Migration . . . . .
.
Introduction . . . . . . . . . . . . . . . . . . . .
.
7.1
7.3
8
. . . . . . . . . . 201
7.2.2
Dynamic Image Warping for Elastic Images
7.2.3
Elastic Image Domain Wavefield Tomography . . . . . . . . . 203
7.2.4
Multi-level Optimization . . . . . . . . . . . . . . . . . . . . . 205
Synthetic Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
. . . . . . . . . . . . . . . . . . . . . . . . 207
7.3.1
Three-layer Model
7.3.2
Modified Marmousi . . . . . . . . . . . . . . . . . . . . . . . . 209
7.4
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
7.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
7.6
Acknowledgment
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Conclusion and Future Directions
227
227
8.1
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2
Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
A Adjoint-state Method for Time-lapse IDWT
233
B Adjoint-state Method for Elastic IDWT
237
12
List of Figures
1-1
Schematic diagrams of time-lapse FWI methods. (a) FWI is applied to
the baseline and monitor datasets in parallel. The subtraction between
the final models generates the velocity changes.
(b) DDWI inverts
the difference between the baseline and monitor datasets for velocity
changes. (c) In AFWI, FWI is applied to the baseline and monitor
datasets in an alternating manner: the baseline FWI generates the
starting model for the monitor FWI, and the monitor FWI generates
the starting model for the subsequent baseline FWI. The process provides a confidence map of time-lapse velocity changes in the end.
1-2
. .
40
Schematic diagrams of the image domain methods for time-lapse velocity inversion. (a) Migration velocity analysis is applied to the baseline
and monitor datasets in parallel. The subtraction between the final
models generates the velocity changes. (b) Time-lapse IDWT inverts
for velocity changes by matching the monitor migrated image with the
baseline migrated image.
2-1
. . . . . . . . . . . . . . . . . . . . . . . .
41
Scheme I: Two independent FWI are conducted for the baseline and
monitor datasets, respectively.
The model changes are obtained by
subtracting the inverted baseline model from the inverted monitor model. 60
2-2
Scheme II: The baseline model is found by FWI with the baseline
dataset.The monitor inversion starts from the baseline inversion result. The model updates are considered to be model changes between
baseline and monitor.
. . . . . . . . . . . . . . . . . . . . . . . . . .
13
61
2-3
Scheme III: The time-lapse inversion starts from the baseline inversion result, and inverts the baseline and monitor datasets jointly. The
model updates are considered to be model changes between baseline
and m onitor.
62
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-4 The true baseline P-wave velocity model (a), and the true baseline
density model (b) that are used for generating synthetic 'real' data for
the baseline survey.
2-5
The normalized power spectrum of the source wavelet we used.
2-6
The shot gather generated by the source in the middle of the model on
the water surface.
2-7
63
. . . . . . . . . . . . . . . . . . . . . . . . . . .
64
. . .
65
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
The starting P-wave velocity model (a) and density model (b) for baseline FWL. The models are obtained by averaging the true models in
Figure 2-4(a) horizontally.
2-8
The final baseline p-wave velocity model (a) and density model (b)
. . . . . . . . . . . . . . . . . . . . . . . . .
67
The true time-lapse changes in P-wave velocity (a) and density (b). .
68
after 60 FW I iterations.
2-9
66
. . . . . . . . . . . . . . . . . . . . . . . .
2-10 (a) The true time-lapse changes in P-wave velocity, saturated in color
to +50 m/s for comparison; (b), (c), (d) are the time-lapse P-wave ve69
locity changes recovered by inversion scheme I, II and III, respectively.
2-11 (a) The true time-lapse changes in density, saturated in color to +40kg/m
3
for comparison; (b), (c), (d) are the time-lapse density changes recovered by inversion scheme I, II and III, respectively.
. . . . . . . . . .
70
2-12 Curve: The cost function curve of the baseline inversion; Dots: The
selected iterations: 1, 5, 10, 20, 50, 99
. . .... . . . . . . . . . . . .
71
2-13 (a) The true baseline P-wave velocity model; (b) - (f) are the baseline
P-wave velocity models after 1, 5, 10, 20, and 99 iterations. .....
14
72
2-14 P-wave velocity changes obtained with incorrect baseline velocity models. (a) The true time-lapse changes in P-wave velocity, saturated in
color to t50 m/s; (b) - (f) are the recovered time-lapse P-wave velocity changes by DDWI starting from the baseline models shown in
Figure 2-13b to 2-13f. The recovery of the velocity changes is clearly
improved with better starting baseline models.
. . . . . . . . . . . .
73
2-15 Normalized power spectra of a sample trace with different noise contamination levels. The random noise spectrum obeys a uniform distribution from 0 to 15 Hz. Six noise levels are tested.
. . . . . . . . . .
74
2-16 (a) A near offset monitor trace with 1% noise energy. The amplitude
of the noise is about the same level as that of the coda waves. (b)
Difference between noise-free monitor and baseline traces (red) and
between noisy monitor and baseline traces (blue). Note small waveform
changes shown in red trace between about 3 to 5 seconds are obscured
by noise in the blue trace.
. . . . . . . . . . . . . . . . . . . . . . . .
2-17 Baseline models obtained by FWI on noisy data.
75
(a) - (f) are the
baseline P-wave velocity models recovered by FWI starting from the
same layered model shown in Figure 2-7a. The recovery of the dominant structure is very robust to random noise. As the noise energy
increases, the details in the model are more contaminated.
. . . . . .
76
2-18 P-wave velocity changes obtained with noisy data. (a) - (d) are the
recovered time-lapse P-wave velocity changes obtained from DDWI
starting from the baseline models shown in Figure 2-17a to 2-17d, respectively.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
2-19 P-wave velocity changes obtained using monitor data with random
source and receiver positioning errors. (a) The true time-lapse changes
in P-wave velocity, saturated in color to
50 m/s; P-wave velocity
changes resolved by DDWI with the monitor survey (b) randomly perturbed source positions; (c) randomly perturbed receiver positions; (d)
randomly perturbed source and receiver positions.
15
. . . . . . . . . .
78
2-20 Effects of systematic shifts in source positions.
(a) The true time-
lapse changes in P-wave velocity, saturated in color to
50 m/s; P-
wave velocity changes resolved by DDWI with the source positions
systematically perturbed in the monitor survey. In (b), (c) and (d),
the sources are divided into 1, 2, and 4 groups. Each group of sources
is shifted 1 grid (6.25 m) in one direction.
. . . . . . . . . . . . . . .
79
2-21 Effects of source wavelet discrepancies between baseline and monitor
surveys. (a) The true time-lapse changes in P-wave velocity, saturated
in color to t50 m/s; P-wave velocity changes resolved by DDWI with
the source wavelet in the monitor survey shifted in phase by (b) 2
degrees; (c) 5 degrees; (d) 10 degrees; (e) 20 degrees and (f) 30 degrees,
for all frequencies.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
2-22 P-wave velocity changes resolved by DDWI with the water velocity
in the monitor survey increased by (a) 8 m/s maximum; (b) 40 m/s
maximum.
3-1
. . . . . . . . . . . . . . .. . . . . . . . . . . .
. . . . .
81
(a) True P-wave velocity baseline model. The reservoir is located in
the anticine below the salt layers (white wedges) that have the highest
velocities. Five stars mark the source locations that are used in both
baseline and monitor acquisitions. (b) True time-lapse P-wave velocity
changes.
The layer is located in the reservoir, and has an uniform
velocity increase of 200 m/s, simulating a hardening effect when the
reservoir is compacting.
3-2
. . . . . . . . . . . . . . . . . . . . . . . . .
97
(a) The starting velocity model for FWI. The model is obtained by
smoothing the true velocity model with a Gaussian window. (b) The
velocity model obtained after 90 iterations of FWI. Details of the layers
are significantly improved.
same.
The color-scales in both figures are the
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
98
3-3
Time-lapse velocity changes recovered by Scheme I (a), Scheme II (b)
and Scheme III (c). The differences are obtained by subtracting the
final baseline inversion models from the final time-lapse inversion models for each scheme. The final baseline inversion models are the same
model that is recovered by the baseline inversion. Both (a) and (b)
contain strong artifacts, while (c) is clean and localized.
3-4
. . . . . . .
99
Layout of the LoFS survey. White points denote the positions of common shots used in the acquisitions in LoFS10 and LoFS12. Blue dots
denote the common receiver positions. The missing shot lines are those
with low quality in either survey. The holes in the shot map are the
locations of platforms.
3-5
. . . . . . . . . . . . . . . . . . . . . . . . . .
100
The shot positioning error distributions of survey LoFS 10 (circled line)
and LoFS 12 (solid line). The error is between the designed positions
and the actual positions measured by GPS. Both distributions have
mean values close to zero. LoFS 12 acquisition is improved with a
much smaller standard deviation of less than 2 meters.
3-6
. . . . . . . .
101
Traces from LoFS 10 (white line) and LoFS 12 (yellow line) are plotted
together to show their similarity. All traces are from the same commonreceiver gather. The pair from a near offset shot is plotted in (a), and
the pair from a far offset shot is plotted in (b). The strong phases
like the diving waves and direct waves, and the coda waves match well
between surveys.
3-7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
102
(a) Initial model for baseline FWI obtained by smoothing the model
built by [53] using a combination of FWI and tomography. (b) Baseline
model obtained after 200 iterations starting from (a). The shallow
structures are improved with higher resolution. The black arrow points
to the gas cloud area. The low velocity layer beneath the gas cloud
that is not visible in the starting model is recovered.
17
. . . . . . . . .
103
3-8
Data residuals of one common receiver gather (a) before the baseline
inversion and (b) after the baseline inversion are compared to show the
convergence of FWI. The traces are ordered by the shot index. Residuals in far offset diving waves (marked by the white dashed circles)
and near offset reflected waves (marked by the black dashed circles)
are both reduced significantly.
3-9
. . . . . . . . . . . . . . . . . . . . . 104
3D view of time-lapse P-wave velocity changes resolved by Scheme I
(a), II (b) and III (c). The slices are at the same coordinates as those
in Figure 3-7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
105
3-10 X-Y slice at the depth where the maximum time-lapse changes occur.
Time-lapse P-wave velocity changes resolved by Scheme I (a), II (b)
and III (c) are compared. Note that the color-scale in (b) is larger
than those in (a) and (c) meaning stronger magnitudes. Black squares
show the locations of platforms. Note the better focusing of time-lapse
changes with Scheme III.
. . . . . . . . . . . . . . . . . . . . . . . . 106
3-11 Y-Z slice at the location where maximum time-lapse changes occur
along the X-axis.
Time-lapse P-wave velocity changes resolved by
Scheme I (a), II (b) and III (c) are compared. The Scheme I result
(a) shows changes of similar magnitude at both shallow and deep locations. The Scheme II result (b) has fewer shallow changes but contains
strong and broad changes in the deeper part. The Scheme III result
(c) shows localized changes in the layer underneath the gas cloud. The
gas cloud region is marked with a black dashed circle.
18
. . . . . . . . 107
3-12 The decomposition of the monitor dataset. The monitor data can be
separated into two branches by the modeling capability. The parts
that can be simulated by the modeling engine are considered as signal,
while the rest is treated as noise. In the signal branch, part of the
baseline signal can not be explained by the current baseline model due
to the imperfection of the baseline inversion. This part would generate
artificial time-lapse changes in Scheme I and II, but will be canceled
in DDWI. In the noise branch, these non-repeatable components will
remain in all schemes, but the repeatable components will be canceled
in D DW I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-1
108
(a) Illustration of the inversion with different source locations in baseline (yellow) and time-lapse (red) surveys. (b) Cartoon of the convergence curves of the model parameters inside (blue) and outside (red)
of the time-lapse change region.
4-2
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
121
(a) Inverted time-lapse changes by subtracting two independent inversions. (b) The time-lapse changes recovered by the joint inversion.
4-5
120
(a) Starting P-wave velocity model. (b) The baseline P-wave velocity
model inverted by FWI.
4-4
119
(a) The true baseline P-wave velocity model. (b) The true time-lapse
P-wave velocity changes.
4-3
. . . . . . . . . . . . . . . . . . . .
.
122
(a) The confidence map 3 obtained by AFWI. (b) The convergence
curves of the parameters marked as stars in (a). To better compare
the curves of different parameters, we subtract a reference value from
the parameter estimates for each curve. The curve of the parameter
within the time-lapse changes (red star) exhibits strong oscillations.
The curve of the parameter outside the time-lapse changes (white star)
is m onotonic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
123
4-6
The true time-lapse changes with three anomalies.
The stars mark
the positions in each anomaly for which convergence comparisons are
shown later. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4-7
(a) The gradient of the baseline cost function. The right side of the
model is not illuminated. The black stars show the locations of the
baseline sources, and the blue triangles mark the width of the receiver
array. (b) The gradient of the monitor cost function. The left side of
the model is not illuminated. The black stars show the locations of the
monitor sources, and the blue triangles mark the width of the receiver
array. Black lines outline the time-lapse anomalies. Only the center
anomaly is illuminated by both surveys.
4-8
. . . . . . . . . . . . . . . .
125
(a) The recovered model with the baseline dataset. The smooth model
in Figure 4-3a is used as the starting model. Due to the limited illumination, the right side of the model is not inverted. (b) The recovered
model with both the baseline and monitor datasets. The whole model
is resolved.
4-9
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
(a) The confidence map
f
obtained with AFWI. The white lines outline
the locations of the time-lapse changes. Only the area of the center
anomaly exhibits high confidence. (b) Convergence curves of the parameters marked by the corresponding colored stars in Figure 4-6. Only
the curve of the center anomaly (yellow stars) shows strong oscillations. 127
4-10 (a) One baseline shot gather with random noise. The black dashed line
marks the location of the trace shown in (b). In (b), the noisy trace is
compared to the clean trace. The amplitude of the noise is as strong
as the reflections.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
20
4-11 (a) The confidence map 3 obtained by applying AFWI on the noisy
datasets. The white lines outline the locations of the time-lapse changes.
The area of the center anomaly exhibits high confidence. Another area
marked by the black star shows relatively high confidence, but is not
an area of time-lapse changes. (b) Convergence curves of the parameters marked by the corresponding colored stars in (a) and Figure 4-6.
The curve of the center anomaly (yellow stars) shows strong oscillations. The black star curve is also oscillatory, because different noise
between surveys causes confficting parameter estimates. The red star
curve decreases monotonically. The white star curve shows very weak
oscillations due to the noise.
5-1
. . . . . . . . . . . . . . . . . . . . . . 129
Schematic illustration of walkaway VSP surveys and CO 2 injection and
monitoring wells at the SACROC EOR field. The red dots denote the
wells with logging records. The green squares denote the two CO 2
injection wells. The blue star marks the VSP monitoring well where
downhole receivers are installed. The black circle has a radius of 1 km.
The blue dashed line is the walkaway VSP source line.
5-2
. . . . . . . .
Gamma ray logs from three wells: 37-11, 59-2a, and 56-23.
150
Green
blocks mark the interval of the Wolfcamp shale formations that have
high Gamma ray values.
5-3
. . . . . . . . . . . . . . . . . . . . . . . . . 151
The resistivity, porosity and sonic velocity profiles from the logging
record at well 59-2a. Green blocks mark the interval of the Wolfcamp
shale formation. The carbonate reservoir is beneath the shale formation. It is clear that the interface between the shale and the carbonate
is at 1900 m eters.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
152
5-4
The schematic configuration of a VSP survey. The injection well is
slightly out of the plane. Black and red dashed lines illustrate the
downgoing (black) and upgoing (red) portions of paths for waves propagating from sources to receivers. The blue dashed line sketches the
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
reservoir location.
5-5
153
The processed common-receiver gathers of the data in 2008 (blue) and
2009 (red). The receiver is at 1585 meters in well 59-2. The datasets are
balanced in amplitude and traveltime using their first reflections. The
traveltime differences in the later arrivals are the time-lapse-change
signals.
5-6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
154
Black line: the sonic velocity profile from logging records in well 592a. Red line: the initial model built using the zero-offset VSP and the
sonic velocity profile.
5-7
. . . . . . . . . . . . . . . . . . . . . . . . . .
155
(a) RTM image produced with data from 2008; (b) RTM image produced with data from 2009. Both images show the local layered structures. The shorter reflector is at 1900 meters that is the top of the
reservoir. For the 2009 image, the reflector below the reservoir is shifted
slightly downwards compared to the baseline image.
5-8
. . . . . . . . . 156
Sample traces from the RTM images of 2008 (blue) and 2009 (red). The
lower reflectors in the 2009 image are shifted downwards compared to
the 2008 im age.
5-9
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
157
The image difference by subtracting the RTM image of 2008 from that
of 2009. The changes at the deeper reflector are stronger than those in
the reservoir layer.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
158
5-10 (a) The P-wave velocity model reconstructed using FWI with data
from 2008; (b) The P-wave velocity model reconstructed using FWI
with data from 2009. Both models contain similar structures.
The
2009 model is shifted slightly downwards compared to the 2008 model. 159
22
5-11 The P-wave velocity model difference obtained by subtracting the model
of 2008 from that of 2009. The changes in the reservoir layer is comparable in amplitude with those at the deeper reflector. The changes
are oscillating rather than smooth.
. . . . . . . . . . . . . . . . . . .
160
5-12 The P-wave velocity changes reconstructed using IDWT. A smooth
low-velocity zone is resolved within the reservoir. Some scattered velocity changes caused by image noise are also observed.
. . . . . . .
161
5-13 A synthetic layered model with the same geometry as the SACROC
model. The blue layer is the shale formation, below which is the reservoir layer (red).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
162
5-14 The synthetic P-wave velocity change caused by a fluid injection into
a borehole located on the right side of the model.
. . . . . . . . . . .
163
5-15 The baseline RTM image obtained using one common-receiver gather.
164
5-16 The P-wave velocity changes reconstructed using IDWT with the synthetic data. The low-velocity zone is confined within the reservoir and
limited in width by the with of the reflectors in the image.
6-1
. . . . . . 165
(a) The three-layer density model for both baseline and monitor surveys. Red Stars denote the locations of the shots, and blue triangles
denote the receiver locations. (b) Differences in the P-wave velocities
between baseline and monitor surveys. Maximum velocity change is
800m/s. (c) The baseline image Io obtained using one shot gather and
the constant velocity model. (d) The monitor image I, obtained using one shot gather and the constant velocity model. The center part
of the second reflector is vertically shifted due to the absence of the
velocity anomaly in (b).
6-2
. . . . . . . . . . . . . . . . . . . . . . . . .
185
The image warping function w(x, z) calculated from Figure 6-1c and 6Id. Units on the color scale are image points. Positive values indicate
upwards shifts. The maximum warping is 4 grid points (i.e. 40 meters). 186
23
6-3
(a) The velocity changes found by IDWT with 5 sources. The anomaly
is correctly positioned. However, the limited aperture of the acquisition
makes the waves travel primarily in the vertical direction, so the recovered velocity anomaly is smeared vertically. (b) The monitor migration
image obtained using one shot gather and the velocity model inverted
by IDWT. The second reflector is correctly positioned.
(c) The ve-
locity changes refined by FWI after IDWT. The amplitude differences
and subtle phase shifts between data and simulation are minimized to
resolve the fine details in the velocity model. FWI has significantly
reduced the vertical smearing observed in Figure 6-3a.
(d) The ve-
locity changes obtained with standard FWI applied to the monitor
data, starting from the baseline constant background velocity model.
The Gaussian anomaly is barely visible. An artificial reflector is erroneously created to account for data differences. This failure is due to
the combined effects of cycle skipping and limited survey geometry.
6-4
. 187
Cost function curves for IDWT, FWI after IDWT, and FWI only. The
cost functions are normalized by their values before the 1st iterations.
IDWT converged within 10 iterations. FWI after IDWT converged
much slower. The cost function of FWI starting from the constant
velocity plateaued after 10 iterations.
6-5
. . . . . . . . . . . . . . . . .
(a) The six-layer baseline and time-lapse density model.
188
Layers in
the center are smaller in thickness than the size of time-lapse velocity
anomaly (white circle). (b), (c) and (d) show the IDWT results with 1
shot, 10 shots and 20 shots, respectively. As we include more shots, the
amplitude distribution within the anomaly is corrected. The vertical
smearing is well constrained by the reflector. The maximum velocity
change is closer to the true value as the changes are confined to a
sm aller area.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
189
6-6
(a) True baseline velocity model with a Gaussian anomaly with peak
velocity change of 200 m/s. We assume the anomaly is not known,
and use a constant velocity model for the baseline migrations.
(b)
True time-lapse velocity changes with peak value of 200 m/s. (c) True
time-lapse velocity model I with two Gaussian anomalies ((a) plus (b)).
(d) The time-lapse velocity changes found using IDWT. (e) True timelapse velocity model II. We increase the peak amplitude of the Gaussian anomaly in the baseline velocity model to 800 m/s, and use the
same time-lapse velocity changes as in (b). (f) The time-lapse velocity changes inverted by IDWT. The shape of the anomaly is distorted
because of the large error in the baseline velocity model, but the basic
location and amplitude is preserved.
6-7
. . . . . . . . . . . . . . . . . . 190
This figure shows robustness tests of IDWT to random source positioning errors and baseline velocity errors. The sources in the monitor
survey are randomly shifted
10 meters from their baseline positions.
The baseline velocity error for each case has maximum value of 0 (a),
200 (b) and 800 m/s (c). Compared to the case where there is no
mispositioning in Figures 6-5d, 6-6d, and 6-6f, the random source positioning error has little effect on the performance of IDWT.
6-8
. . . . .
191
Robustness tests of IDWT against source positioning error plus baseline velocity error. In the 3x3 plot, the monitor survey sources are
systematically shifted 10, 20 and 50 meters from their correct positions for each column, respectively.
The baseline velocity error for
each row has maximum value of 0, 200 and 800 m/s. Black dotted
circles mark the areas where false velocity changes are resolved due to
the baseline velocity error, which is at the same location as shown in
Figure 6-6e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
25
6-9
Migrated images for one baseline shot and one shifted monitor shot.
Dotted lines show the wave paths along which velocities are updated.
Portions of the monitor migrated image marked as unconstrained image
(dashed circles), have no corresponding image points from the baseline
image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
193
6-10 (a) The center part of the original Marmousi model is used as the
true baseline velocity model. The maximum source-receiver offset is 2
km. Five shots (red stars) are used to generated synthetic data. (b)
True time-lapse velocity model with a negative velocity change marked
with a black dashed line. The black arrow points to the area where
the boundary of the changes is located in the middle of the layer.
We designed this half-layer velocity change intentionally to show how
IDWT would smear the changes within a layer.
. . . . . . . . . . . .
194
6-11 (a) A smoothed version of the Marmousi model is used as the baseline
model for migration. (b) Migrated image for one shot (red star). Areas
pointed to by arrows are poorly imaged due to the limited receiver
aperture.
Dashed lines mark the boundary of the velocity changes.
The interfaces above and below the anomaly are well-imaged.
. . . .
195
6-12 (a) The true time-lapse velocity changes. The anomaly is smooth at
its boundary (dashed lines). (b) The inverted time-lapse changes using IDWT with 5 shots. The black arrow points to the area where
the inverted velocity changes diffuse across the boundary of the true
changes (dashed lines), and are both smeared towards and bounded by
the lower interface of this layer. . . . . . . . . . . . . . . . . . . . . .
7-1
196
(a) True P-wave velocity model. (b) True S-wave velocity model. The
S-wave velocities in three layers are 1767, 2060 and 2150 m/s respectively.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
214
7-2
(a) Starting model with constant S-wave velocity 1900 m/s. (b) Comparison between PP (left half) and PS (right half) images. Both images
are formed with all 8 sources. The first reflector in the PS image is
shifted downwards for about half of the wavelength.
7-3
. . . . . . . . . 215
(a) Polarity-corrected PS image formed with one shot gather. Black
star marks the location of the source. (b) The warping function calculated by DIW. It describes how much the depth shift is for each image
point in the PS image in (a).
7-4
. . . . . . . . . . . . . . . . . . . . . . 216
(a) Original PS image without polarity correction. The same source is
used as in Figure 7-3a. (b) The zoom-in view of the reflectors marked
by black dashed line in (a).
The original image (red wiggles) and
the fractionally warped image (blue wiggles) are shown together. The
differences between them are used to generate the adjoint sources.
7-5
217
(a) The gradient for the source marked by the black star. Dominant
energy is along the receiver wave-path. (b) Total gradient by stacking
partial gradients from each source. Positive value indicates the current
velocity is too high.
7-6
. . . . . . . . . . . . . . . . . . . . . . . . . . .
(a) The recovered S-wave velocity model after 20 iterations.
218
Both
the low and high velocity layers are resolved. The recovery is limited
by the illumination of the survey. (b) The PS image formed with the
recovered S-wave velocity model in (a) is compared with the PP image.
Both reflectors are aligned. The alignment is poor on the edges due to
the same illumination limits.
7-7
. . . . . . . . . . . . . . . . . . . . . . 219
The modified elastic Marmousi model. (a) True P-wave velocity model.
(b) True S-wave velocity model. Velocity ranges are modified to be
smaller to allow larger grid size and time step in finite difference. The
top layer is solid instead of water.
7-8
. . . . . . . . . . . . . . . . . . . 220
(a) Smooth P-wave velocity model. It is assumed to be obtained with
P-wave velocity model building. (b) The PP RTM image produced
with all 18 sources.
. . . . . . . . . . . . . . . . . . . . . . . . . . . 221
27
7-9
(a) The PS image produced with one shot gather. The polarities are
corrected. The black star marks the location of the source. (b) The
warping function that registers the PS image in (a) to the PP image in
Figure 7-8b. The shifts are all negative (upwards) because the current
S-wave velocity is lower than all velocities in the true S-wave velocity
model.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
7-10 (a) The normalized gradient calculated with the same source. The
black star marks the source location. (b) The normalized total gradient
by summing the gradients from all 18 sources. Negative values indicate
that the current velocities need to be increased.
. . . . . . . . . . . . 223
7-11 The final S-wave velocity model after 50 iterations. The bottom part of
the model is poorly resolved because the converted S-waves are outside
of the acquisition surface due to the tilted reflectors.
. . . . . . . . . 224
7-12 (a) The comparison between PP and PS images before the inversion.
The image is divided into seven sections in the horizontal direction.
Odd sections are PP images, and even sections are PS images. The
mismatch is clear on the section interfaces. In the circled area, the
reflectors are not well imaged due to the incorrect velocity. (b) The PS
image based on the inverted S-wave velocity model is compare to the
PP image in the same setup as in (a). The coherency at the section
interfaces is markedly improved. The reflectors in the circled area are
all well resolved and aligned with those in the PP image.
. . . . . . 225
7-13 (a) The P-wave reflectivity model resolved by FWI starting from the
smooth model in Figure 7-8a. (b) The S-wave reflectivity model resolved by FWI starting from the smooth model in Figure 7-11. Except
the areas outside of illumination, the recovery of the S-wave model is
of a similar quality to that of the P-wave model in (a). It also proves
the success of the S-wave velocity model building with RG-EIDWT.
28
226
List of Tables
5.1
Rock and fluid properties derived from well logs. Symbols are defined
as in Equations 5.9 and 5.10. . . . . . . . . . . . . . . . . . . . . . . .
29
148
30
Chapter 1
Introduction
1.1
Objective
Human activities like hydrocarbon production and CO 2 sequestration, induce significant changes in the subsurface. Large volume fluid extraction and injection change
not only the fluid content and saturation of the rocks, but also the pore pressure,
temperature and porosity. Resulting changes in elasticity and density lead to changes
in seismic response [60, 103]. Therefore, repeated seismic experiments can be used to
extract the time-lapse (also often called 4D) information about the subsurface.
For hydrocarbon reservoirs, monitoring temporal changes are critical for production optimization. Early in a field's life, 4D seismic helps calibrate the effectiveness
of the initial reservoir-management and depletion plan [16]. Later surveys are used
to identify undrained areas and to optimize infill-wells [67, 88]. For fields undergoing
enhanced recovery, 4D seismic also has a large impact. Thermal recovery methods
are often used in viscous, heavy-oil reservoirs where 4D seismic is used to find the
bypassed oil and to maximize enhancement efficiency [28, 48, 99]. In cases where
miscible-gas (e.g. C0 2 ) is injected to reduce the oil's viscosity and the interfacial tension between in situ and injected fluids, 4D seismic can detect free gas and indicate
potential premature gas break-through [41, 36, 73]. For carbon capture and sequestration (CCS), 4D seismic is required for both project process control and regulatory
purposes [22, 5,19]. CO 2 leakage through caprock and potential surface hazards have
31
to be monitored. Monitoring data are also expected to provide information to verify
the subsurface inventory for CCS [47].
Despite the diverse applications of time-lapse seismic, only two major types of
differences in the data are used: amplitude and time-shift. For thin reservoirs with
only two fluids (water and oil), amplitude change or impedance change in full-stack
seismic sections gives an adequate interpretation. For more complicated reservoirs,
multiple attributes are needed to distinguish different mechanisms. Time-shifts can
help constrain the interpretation of amplitude changes [29]. Time-strain (derivative
of time shift) is an indicator of relative velocity changes in non-compacting reservoirs [17].
Beyond the zero-offset assumption, partial-angle stacks motivate elas-
tic inversion methods to obtain changes in P-wave and S-wave impedance quantitatively [78]. The analysis of amplitude versus offset (AVO) using comparisons of nearand far-angle stack differences or impedance changes, help separate fluid-saturation
effects from pressure effects [95]. For compacting reservoirs, time strain together with
geomechanical modelling are used to analyze stress field changes in the overburden
structure [42].
However, most of these existing 4D seismic methods are not looking for the actual
property changes (i.e. P and S-wave velocity changes, and density changes). Some of
the methods are quantitative, but these quantities are only indicators of the actual
changes. For example, relative velocity changes derived from time strain are not the
actual velocity changes because of the zero-offset assumption. In addition, changes in
a given parameter may be measured in different ways and combined analysis of these
measures can lead to better estimates of changes. For example, impedance inversion
based on amplitude variations is not accurate if velocity model is not updated with
time-shift information.
Another limitation of these methods is that simple earth
models (e.g. two-layer model: overburden and reservoir layer) and high-frequency
approximations are used to simulate wave propagation, whereas in complex geology
these models and approximations are not sufficient.
The primary objective of this thesis is to develop full wavefield inversion methods
32
.
that translate changes in seismic data directly into changes in P- and S-wave velocities.
Our methods are distinct from previous studies in a few aspects. First, we solve wave
equations to simulate seismic wave propagation which makes no assumptions about
the earth structures or the wave-paths.
Second, the estimated quantities are the
actual physical parameters that would cause the observed seismic changes, not just
indicators. Third, the inversions integrate all pre-stack data (near- and far- offset)
rather than analyzing them separately or using the full-stack or partial-stack data in
which some 4D signals are already removed. Fourth, our methods discriminate the
real temporal subsurface changes from the inversion induced deviations of parameter
estimations.
1.2
Data Domain vs. Image Domain
Full wavefield inversion methods have been actively studied for seismic velocity model
building for many years. They are computationally heavier compared to ray-based
methods because of the calculation of finite-frequency wavefield propagation, however,
they are becoming feasible with increasing computer power. They can be conducted
in either data domain or image domain. By convention, the data domain method is
called full waveform inversion (FWI), and the image domain method is called waveequation migration velocity analysis (WEMVA).
FWI was first proposed by [93] in 1980s. The objective of FWI is to estimate
subsurface model m by matching synthetic data F(m) and recorded data d, where
F is the modeling operator. A commonly used objective function is a least-squares
measure: - 1 F(m) - d 12, which takes all the information in the waveforms into account. Gradient-based inversion strategies are used to solve the optimization problem
efficiently. Compared to traveltime tomography and migration velocity analysis, the
model obtained by FWI has higher resolution, and may include multiple physical parameters like velocity, density and attenuation factor, all of which are quantitatively
measured. FWI is an ideal tool for 4D analysis. Successive models can be obtained
by FWI with successive time-lapse surveys. The estimated property changes can
be directly input to rock physics models to generate engineering measures like pore
33
pressure and fluid saturation.
Although FWI seems superior for its advantages, its feasibility is restricted by data
acquisition. Low frequency data are needed to increase the convexity of the inverse
problem, and to mitigate cycle-skipping effect. Long offset acquisition is required
to recover low wavenumber components of the velocity model [101]. However, many
seismic acquisitions are of limited frequency content and limited offset. With such
limited data, FWI suffers from the depth-velocity ambiguity (i.e., a reflector depth
shift and a local velocity change can cause the same data differences). To resolve
the ambiguity, WEMVA measures flatness of events in angle gathers or misfocusing
in differential semblance [81, 91].
It has no requirements on data frequency and
acquisition offset. However, the resolution of the resulting models is lower. Recent
developments attempt to combine the two methods to form joint inversions that would
potentially overcome individual disadvantages [15, 90].
This thesis explores methods in both the data domain and the image domain.
For long offset acquisitions, we develop 4D inversion methods in the data domain,
and obtain high resolution time-lapse velocity changes in both synthetic tests and in
real data applications (Chapter 2, 3, and 4). For narrow offset acquisitions, image
domain methods are developed to obtain velocity changes with limited resolution in
both synthetic and real data cases (Chapter 5, 6, and 7). The methods in the two
domains can be combined to form joint inversions, or applied strategically to achieve
high resolution with limited acquisitions.
1.3
Challenges and Contributions
We develop methods based on FWI and WEMVA, however, they are more than
straightforward extensions. Even if direct applications of FWI or WEMVA on individual time-lapse surveys can generate good velocity models, the subtraction between
models is not guaranteed to produce reliable time-lapse velocity changes. It is because
of the nonlinearity of the inverse problems. Time-lapse inversions using either FWI
or WEMVA on individual datasets are likely to fall into different local minima, which
34
may lead to erroneous estimates of which parameters changed and did not change
between surveys. Since 4D responses are generally weak, the model differences due
to different local minima might saturate the real velocity changes. This is one of the
major challenges that we want to conquer in this thesis. Chapter 2, 4 and 6 present
new methodologies. In Chapter 2, we introduce double-difference waveform inversion (DDWI) which inverts data differences between the baseline and monitor surveys
for velocity changes. In Chapter 4, we propose an alternating FWI (AFWI) strategy
to build the confidence map of the time-lapse changes. In Chapter 6, we introduce
image domain wavefield tomography (IDWT) which finds interval velocity changes by
matching images from time-lapse surveys. All methods are designed to focus on the
relative changes between surveys, and mitigate the effects of non-equivalent estimates
for parameters that are actually not changed.
Another major challenge for time-lapse inversions is the effect of survey nonrepeatability.
Seismic surveys acquired at different time contain different random
noise, and are likely subjected to different acquisition conditions. For example, source
and receiver positions are difficult to be perfectly repeated in marine environments.
Source time functions could deviate between surveys because of source types (e.g. air
gun types in marine, source coupling on land). Overburden conditions could also be
different (e.g. water table depth change, surface weathering and subsidence). Significant data differences can be generated by subtle non-repeatability in addition to real
time-lapse signals. From a practical perspective, we thoroughly discuss the impact of
different types of non-repeatability on the performance of DDWI in Chapter 2. In
Chapter 3, we use real time-lapse datasets from Valhall, North Sea, to demonstrate
the performance of DDWI with well-repeated ocean bottom cable surveys. In Chapter 5, we use synthetic examples to show the robustness of IDWT to source and
receiver position perturbations. For extreme situations where two surveys are not
designed to be repeated (e.g. ocean bottom nodes placed very differently between
surveys), we develop AFWI in Chapter 4, to build a confidence map of time-lapse
changes which is used to regularize the inversion for changes.
To quantify the changes in S-wave velocity is often more challenging than that
35
for P-wave velocity. Although in elastic FWI, P- and S-wave inversions are naturally coupled in theoretical derivation, it is difficult to build a good S-wave velocity
model as a starting model. The inverse problem for S-waves is more ill-posed because S-waves are mostly converted phases in active seismic surveys. The inverse
problem nonlinearity is strong due to the dependence of S-waves on P-waves. From a
computational point of view, S-wave propagation requires longer recording time and
finer model grid sizes and time steps in finite difference modeling. Since we show
that P-wave velocity inversion is reliable (synthetics in Chapter 2 and real data
in Chapter 3), we propose an efficient S-wave velocity model building tool based
on P-wave velocity model. It does not require large offset and low frequency data
acquisition, and can start from any arbitrary S-wave velocity model (e.g. constant
velocity). With trivial modifications, the method can be easily extended to invert for
S-wave velocity changes similar to the method presented in Chapter 6.
We think a good estimate of time-lapse velocity changes should have the following
qualities: 1. localized in space; 2. correct in magnitude; 3. focused on relative
changes. The three qualities are inter-dependent. The contribution of this thesis is
to provide a systematic framework that handles various kinds of data to achieve this
type of estimate.
1.4
Thesis Outline
Here we briefly describe the structure and content of the thesis:
In Chapter 1, we introduce the problem we want to solve by this thesis work,
and explain the challenges both in theory and in practice. We then summarize how
we conquer these challenges from different angles, and outline the thesis.
From Chapter 2 to Chapter 4, we focus on data domain methods. In Chapter
2, we describe the methodology, implementation and mechanism of double-difference
waveform inversion (DDWI). This method was not first proposed by us (see e.g., [24
and [119]).
Our work adds greatly to the understandings of the applicability of
the method. Instead of doing independent inversions (Figure 1-la), DDWI inverts
36
differenced data for model changes starting from a baseline velocity model (Figure 11b).
We compare the performance of DDWI with those of independent inversion
schemes, and show DDWI's advantage in producing clean and interpretable time-lapse
changes. Two practical aspects that determine the success of DDWI are discussed
and tested: the dependence of DDWI on the quality of the baseline model, and the
robustness of DDWI to survey non-repeatability. With realistic synthetic examples,
we show that DDWI is reliable within a wide range of baseline models, and is robust
to mild realistic survey non-repeatability.
In Chapter 3, we apply both independent inversions and DDWI on real time-lapse
datasets from Valhall, North Sea. A synthetic time-lapse velocity model that mimics
a hardening reservoir layer is used to compare the results from different inversion
schemes as a benchmark. We briefly introduce the Valhall field, and the life of field
seismic (LoFS) system that is used to collect the time-lapse data. The repeatability
of the surveys is within the range where DDWI is tested to be robust. The inversion
results are consistent with the synthetic benchmark, and confirms the conclusions we
draw in Chapter 2.
In Chapter 4, we focus on situations where subtracting datasets is not possible.
An alternating inversion strategy is developed for time-lapse surveys that are far apart
in geometry: baseline and monitor datasets are inverted independently (one dataset at
a time) but alternatively in a sequential manner (a few iterations with baseline data,
then a few iterations with monitor data) (Figure 1-1c). For each parameter, we build
the convergence curve (model estimates vs. iteration). A confidence map of timelapse changes is constructed based on the oscillations of the curves. The underlying
principle is that for parameters outside of the time-lapse region and within illuminations of both baseline and monitor surveys, both inversions approach their true values
consistently; for parameters within time-lapse region and within both illuminations,
the estimates from baseline and monitor datasets conflict (oscillations in convergence
curves); for parameters within only one of the illuminations, one inversion inverts
for them consistently, and the other inversion has little impact. The confidence map
regularizes a joint inversion with both datasets which better resolves the time-lapse
37
velocity changes. Synthetic examples with different acquisition scenarios are used to
test this method.
Chapter 5 is another real data application and serves as a transition from the
data domain to the image domain. In this chapter, we study the time-lapse walkaway
Vertical Seismic Profile (VSP) monitoring for CO 2 injection in SACROC, West Texas.
We describe the basic geology of the field and the production and injection history.
Well-logs are used to determine the location of the caprock and the reservoir formation. Density logs and sonic logs are used to build the density model and the initial
P-wave velocity model. Reverse time migration (RTM) and FWI are both used and
proved not sufficient in locating and quantifying the velocity changes induced by local
injections. The failure of FWI is mainly because of the limited acquisition aperture
which reduces FWI to least-squares migration. Velocity changes are not resolved, but
transformed to depth shifts of reflectors instead. IDWT (Figure 1-2b) is introduced
to solve this problem by matching the reflectors in monitor images to those in baseline
images. We find a low velocity zone right beneath the low-permeability shale formation, which indicates that the CO 2 migrated from the injection points upwards and
accumulated under the caprock. A synthetic test designed to simulate the scenario is
used to validate the result.
Chapter 6 modifies the IDWT method in Chapter 5 using the L-2 norm of
image shifts as the objective function. The modified IDWT method purely inverts
for kinematics which better separates time-shift from amplitude changes. It is helpful
to distinguish the velocity changes from density changes that are often ambiguous
in FWI as discussed in Chapter 2. The image shifts are estimated by dynamic
image warping (DIW) which is not affected by cycle-skipping. Therefore, IDWT with
DIW can handle large velocity changes in cases like steam and CO 2 injections. We
first introduce the methodology and explain how to use the adjoint-state method to
calculate the gradient. Simple synthetic examples are used to show the characteristics
of the method.
One important advantage of IDWT is that the inverted velocity
changes are bounded by the interfaces above and below the changes. Interpretations
and leakage monitoring are much easier and reliable with such results. The robustness
38
of IDWT to baseline velocity errors and survey positioning errors are demonstrated
by synthetic examples. The Marmousi model is used to show that IDWT works with
complex geology and narrow offset acquisitions.
Chapter 7 extends the methodology in Chapter 6 from acoustic to elastic.
Elastic IDWT (EIDWT) inverts for the shear wave velocity model by matching P-S
images to P-P images, assuming that the P-wave velocity model is obtained. Image
registration is also accomplished by DIW, but here we do not use the image shifts
as the objective function. Instead, we present an alternative approach: registration
guided IDWT (RG-IDWT). The P-S images are fractionally warped to the directions
of the estimated shifts but within one half wavelength to avoid cycle-skipping. The
L-2 norm of the image differences between PS images and their warped versions are
used as the cost function. The two-level optimization consistently minimize the image
differences in the inner loop and the image shift in the outer loop. This method can
start from an arbitrary constant S-wave velocity and recover a smooth S-wave velocity
model for which the final P-S images aligned with P-P image. The recovered model
can serve as a starting model for elastic FWI. The extension of the method to timelapse S-wave velocity inversion is straightforward. We use a simple two-layer model to
demonstrate how the method functions, and show its advantages and limitations. A
modified Marmousi model is used to show that the method works well with complex
geology.
In Chapter 8, we summarize our innovations and experience on time-lapse velocity inversion. Potential improvements to the methods and possible future research
directions are discussed.
39
Parallel FWI
DDWI
Baseline Data
Monitor Date
Baseline FWl
Monitor FW
Baseline FWl
Monitor FW1
Baseline Datal
Baseline
Beocity
Monitor Data
FW1
Data Difference
DW
Moel
Val ocity Changes
Velocity Changes
(a) Parallel FWI
(b) DDWI
AFWI
Baseline Data
Monitor Data
IBaseline
Velocity Model
FWI
FWI
LJ
Confidence Map of Chang
(c) AFWI
Figure 1-1: Schematic diagrams of time-lapse FWI methods. (a) FWI is applied
to the baseline and monitor datasets in parallel. The subtraction between the final
models generates the velocity changes. (b) DDWI inverts the difference between the
baseline and monitor datasets for velocity changes. (c) In AFWI, FWI is applied
to the baseline and monitor datasets in an alternating manner: the baseline FWI
generates the starting model for the monitor FWI, and the monitor FWI generates
the starting model for the subsequent baseline FWI. The process provides a confidence
map of time-lapse velocity changes in the end.
40
Time-lapse IDWT
Parallel MVA
Baseline Data
Monitor Dae
;Baseline Data
MVA
BBaselin
Image
e
Monitor Data
rMVA
oi
V,= n
VeoIy Model
mg
Monitor
velocity Model
IDWT
Miration
Migraton
B seline Image
Image
Vel)e Changes
)
Velocity Changes
(a) Parallel MVA
(b) Time-lapse IDWT
Figure 1-2: Schematic diagrams of the image domain methods for time-lapse velocity
inversion. (a) Migration velocity analysis is applied to the baseline and monitor
datasets in parallel. The subtraction between the final models generates the velocity
changes. (b) Time-lapse IDWT inverts for velocity changes by matching the monitor
migrated image with the baseline migrated image.
41
42
Chapter 2
Double Difference Waveform
Inversion: Method, Feasibility and
Robustness Study
Summary
Full waveform inversion has been proposed as a potential tool for retrieving subsurface
properties like P and S-wave velocities, and density by fitting simulated waveforms to
seismic data. An extension of this method to time-lapse applications seems straightforward but in fact requires more tailored processes like double-difference waveform
inversion (DDWI) which uses the baseline and monitor datasets jointly by inverting
data differences for velocity changes. In this chapter, we use realistic synthetic pressure data examples to compare the performance of DDWI with that of two other
inversion schemes with pressure data acquisitions. P-wave velocity changes are reliably recovered in the inversion, and DDWI is shown to deliver the best results. To
further investigate the feasibility of using DDWI in practice, the dependence of DDWI
on the quality of the baseline models, and its robustness to survey non-repeatability
are studied with numerical tests. Various types of non-repeatability are considered
separately in the synthetic tests, including random noise, acquisition geometry mis43
match, source wavelet discrepancy, and overburden velocity changes. The correlation
between the levels and types of non-repeatability and the resulting contamination
of the inversion results is explored.
With pressure data, DDWI is capable of in-
verting reliably for the P-wave velocity changes under realistic conditions of survey
non-repeatability.
2.1
Introduction
Full-waveform inversion (FWI) aims to estimate the subsurface density and elastic
parameters directly from seismic records [93, 101]. Ideally, the extension of FWI to
4D is straightforward. Two FWI runs can be conducted for the baseline and monitor
datasets, and the difference of the two resulting models should reveal the reservoir
property changes. Nevertheless, nonlinear artifacts arising from the nonlinear nature
of the inverse problem introduce differences between the inverted models in addition
to the real time-lapse changes. To address this problem, [107] applied differential
waveform tomography in the frequency domain to cross-well time-lapse data during
gas production and showed that the results are more accurate for estimating velocity
changes in small regions than those obtained using conventional inversion schemes.
[66] applied a similar strategy to conduct differential travel-time tomography using
cross-well surveys. [24] proposed a double-difference waveform inversion algorithm
using time-lapse reflection data in the time domain and demonstrated, with synthetic
data, that the method has the potential to produce reliable estimates of reservoir
changes. A successful real data application of DDWI is reported in [112].
In this work, we apply the DDWI methodology of [24], to a synthetic data set,
and investigate the feasibility, advantages and limitations of DDWI when applied
to realistic time-lapse data acquisition scenarios. In the following sections, we first
describe the mathematics of DDWI and its implementation. We then compare the
performance of DDWI with that of two other inversion schemes using (I): the same
initial model for both baseline and monitor inversions; (II): the baseline inversion
result as a starting model for time-lapse monitor inversion, with both schemes using
44
an acoustic model and noise-free pressure data. The dependence of DDWI on the
quality of the baseline model is discussed using several baseline models of increasing accuracy and levels of convergence obtained by applying more iterations of FWI.
The success of 4D seismic analysis also relies on the repeatability of the time-lapse
surveys. Co-processed data, in which the non-reservoir related differences are minimized, give rise to much improved time-lapse results [77, 18]. In practice, however,
it is impossible to correct perfectly for the lack of repeatability between surveys [55].
To investigate the effect of survey non-repeatability on DDWI, several common acquisition mismatches are discussed separately, including contamination with random
noise, survey source and receiver positioning errors, source wavelet discrepancies, and
seasonal water velocity changes. For simplicity, all our numerical tests are conducted
with a 2D acoustic finite-difference model.
2.2
Methodology
Standard full-waveform inversion can be expressed as a minimization problem with
the cost function:
Estanda,.(m) = -Iu - d1 2 ,
2
(2.1)
where u is the modeled synthetic waveform, d is the acquired field data, and m is
the model parameter that is inverted for (see e.g., [101] for a recent review of FWI).
Many successful real data applications have also been published [83, 71, 106], which
establish FWI as a tool for quantitative subsurface property estimation.
To extend FWI to the time-lapse case, the most straightforward strategy is to
execute two inversions for the baseline and monitor datasets, respectively. We call
this strategy Scheme I in this paper. As shown in Figure 2-1, the subtraction of the
final models should give the property changes between surveys. One could argue that
it is wasteful to start the time-lapse monitor inversion in Scheme I from an initial
model that is independent of the baseline model, and that it would be more reasonable
to start from the final model of the baseline inversion. We refer to such a strategy
as Scheme II. As shown in Figure 2-2, the updated part of the model in the monitor
45
inversion from Scheme II should show the correct property changes, provided that
the baseline inversion has converged. Scheme III, as shown in Figure 2-3, is DDWI.
In DDWI the monitor inversion also starts from the final baseline inversion model,
but the cost function is changed to:
EDDWI(m)=
1 (Ubaseline -
Umonitor) -
(dbasetine
-
dmonitor) 2,
(2.2)
where us.eline and umonit, are simulated waveforms from the baseline inverted model,
mo, and the monitor model, m, respectively, with m being iteratively updated. The
field data are dbaeline and dmonito from the baseline and monitor surveys, respectively.
The name "double difference" comes from the two differences in Equation 2.2, one
between baseline and monitor field datasets and one between modeled datasets baseline and monitor datasets. As E(m) is minimized, the property changes (m - mo)
corresponding to the data differences are recovered. In practice, before starting the
time-lapse inversion, we invert the baseline dataset for the baseline model, mo. Using mo, we then generate a synthetic dataset, ubele. To allow the use of standard
FWI algorithms, a synthesized monitor dataset, dn,, is created by adding the data
difference (dmonito - daeiine) to usbeline. The inverse problem is then reduced to a
standard FWI with cost function:
E (m)
= 1u n,,tc,(mo +6m) - dy 12,
(2.3)
and can be solved by regular FWI solvers. The only extra step is the synthesis of
dyn, which requires a trivial amount of computation in the overall process.
2.3
Scheme Comparison with Acoustic Inversion
Figure 2-4 shows the synthetic P-wave velocity and density models used for the numerical study. The dominant geologic structure is the anticline in the center, which
lies underneath a sloping water bottom. The layering of the model is very detailed in
order to simulate a realistic sedimentary environment. The synthetic data are gener-
46
ated by finite-difference modeling using 64 sources (250 m spacing) and 680 receivers
(25 m spacing), all evenly placed on the water surface. The frequency band of the
source wavelet is shown in Figure 2-5. The low frequency components are included
to enhance the recovery of the low-wavenumber part of the model. An example shot
gather from the simulated baseline survey is shown in Figure 2-6. The direct arrivals
are not filtered out, however, they have no contributions to FWI because we use the
correct water velocity in the initial model.
For all listed schemes, we first apply FWI to the baseline data to obtain the
baseline velocity model. To make the test realistic, we use the horizontally layered
models shown in Figure 2-7 for the initial P-wave velocity and density models. The
inverted models from FWI are shown in Figure 2-8. The anticline is well-recovered
and the layers are resolved with a resolution limited by the frequency band of the
data. The lower right and lower left parts of the model are not as well-recovered as
the center part because of the limited illumination of the survey.
For constructing the monitor models, we implanted realistic changes in P-wave
velocity and density that are typically observed in practice into the baseline models,
as shown in Figure 2-9. To test FWI's capability of distinguishing such parameter changes during the inversion, some of the density and P-wave velocity changes
have different signs in the three reservoirs. P-wave velocity changes in the shallower
reservoirs are smaller than those in the deepest reservoir. By contrast, the relatively
shallow density changes are stronger than the deeper changes, and there are no timelapse density decreases. In order to compare fairly the results of all three schemes
in later difference plots, we clipped the color scale of the true velocity changes at
50 m/s and the true density changes at
40 kg/M 3 , and will use the same color-bar
for all subsequent inversion results.
As stated in the Methodology section, for Scheme I we conduct an independent
FWI with the monitor data starting from the same initial layered model used for the
baseline inversion shown in Figure 2-7. The inverted P-wave velocity and density
changes from the subtraction between the inverted monitor and baseline models are
shown in Figures 2-10b and 2-11b, respectively. Major features of the P-wave velocity
47
changes are recovered, including both velocity increases and decreases. Amplitudes of
the velocity changes are only partially recovered
(~ 50%), however.
In addition to the
true changes, other changes that follow the structures in the background are visible,
albeit weak in amplitude. The pattern of inverted density changes is basically the
same as that of the P-velocity changes. However, the inversion adds some reductions
in density, which do not exist in the real density model. It is difficult to differentiate
density and P-velocity changes from pressure data because they have very similar
radiation patterns, even over a large offset range [94, 71]. P-wave velocity is better
constrained by the P-wave kinematic information in the data. Density is estimated
from the amplitude information which is also affected by P-wave velocity. Hence,
density changes are not accurately recovered.
In Scheme II, we start the time-lapse inversion from the final model that we
obtained from the baseline inversion (Figure 2-8). Figures 2-10c and 2-11c show the
inverted P-wave velocity and density models, respectively.
It is obvious that the
background structures, including the seafloor properties, are updated together with
the reservoir changes. The resolved changes are also weaker in amplitude compared
to those from Scheme I.
Scheme III (DDWI) results are shown in Figures 2-10d and 2-11d. The results are
cleaner than those of the other two schemes, and the amplitude is better recovered as
well. However, the density ambiguity is still not resolved. To invert for the density
more accurately, a different parametrization, such as velocity-impedance, would be
helpful, as discussed in [94] and [71]. We will not further address this issue here, as it
is not directly associated with our analysis of different time-lapse inversion schemes.
The major improvement in Scheme III, as compared to Schemes I and II, is the
removal of the coherent background structures (model residual) that are not related
to the actual reservoir changes. These structures correspond to the residuals in the
baseline inversion
Rbaseline
Rbseline = Ubseline
- dbase.ine.
Since there is no perfect inversion,
is never exactly zero. For Scheme I, the nonlinearity of the inverse problem
leads to different levels of convergence for baseline and monitor inversions at different locations in the model where Ra,eine
48
$
R
nit,.
Accordingly, the subtraction
between models gives non-zero contributions over the entire model. For Scheme II,
Rbseiine gets injected into the model together with the real time-lapse signals due to
reservoir changes, and generates model perturbations in regions where there are no
time-lapse changes. As a result, the monitor inversion is trying to recover further
the background model as well as finding the reservoir changes. In Scheme III, the
common data residuals are subtracted out in the cost function in Equation 2.2. This
can be shown by taking the first derivative of Equation 2 with respect to umonitor:
ME(m)= (Umonitor - dmonitor) - (Ubaseline - dbaseiine) = Rmonitor
- Rbaseline.
umonitor
From this, we see that when
E(m)
aUmonitor
(2.4)
0, the cost function E(m) reaches its minimum
where the baseline and monitor data residuals are equal. As a result, the inverted
changes are free of background structures because the residuals cancel out perfectly,
at least in theory.
2.4
Baseline Model Dependence
With the acoustic synthetic example, we observed that DDWI delivers cleaner and
better inversion results, at least for the time-lapse P-wave velocity changes. In this
section, we investigate how the quality of baseline models affects the performance
of DDWI, since the baseline model from the baseline inversion is a pre-requisite for
DDWI.
As we described in the Methodology section, the baseline model is not updated
in DDWI, which only focuses on the time-lapse changes. The dependence of DDWI
on the accuracy of baseline models must be investigated to help decide if the baseline
model is reliable enough in practice. As shown in Figure 2-12, we selected six baseline
models corresponding to different convergence levels along the cost function curve for
the FWI of the baseline data. With more iterations, the model improves as the
predicted data get closer to the recorded data.
For each of the baseline models,
we generate a synthetic data set d,,, (Equation 2.3), and run DDWI to invert for
49
the P-wave velocity changes.
Figure 2-13 shows all the selected baseline models.
The corresponding velocity changes resolved by DDWI starting from each one of the
baseline models are shown in Figure 2-14.
It is clear that DDWI gives an improved result with a better baseline model. In
Figure 2-13b, the baseline model has the correct water depth, but is far from the true
velocity model. In this case, DDWI fails to invert for the changes correctly, as shown
in Figure 2-14b. Some of the changes have the wrong sign, and the overall recovery in
amplitude is poor (by
15 m/s or more). This is because the baseline model controls
the kinematics when the data differences are back-projected. With a poor baseline
model, DDWI will project the data differences into the wrong locations, where the
signals cannot be correctly stacked. With a better background velocity model, such
as those in Figures 2-13c and 2-13d, DDWI is able to invert for the velocity changes
with the correct sign (Figures 2-14c and 2-14d), however, side-lobes remain. From
Figures 2-14e to 2-14f, the amplitude recovery improves as the baseline model contains
more and more details, which better match the reflections and scattering when the
data differences are back propagated.
From these observations, the performance of DDWI certainly depends on the
quality of the baseline velocity model. With a good background model like a smooth
migration velocity model without the details, we expect DDWI to be able to invert
correctly for at least the locations of the velocity changes. Including more details in
the baseline model improves the recovery of the amplitude changes. Within a wide
range of convergence levels for baseline inversion (Figure 2-12), DDWI is robust and
capable of delivering reliable results.
2.5
Survey Non-repeatability
Survey repeatability is a common issue in time-lapse seismic analysis. The successful
acquisition of individual surveys does not guarantee quality time-lapse signals. Small
deviations in acquisitions can cause significant signal differences between datasets.
Because the seismic time-lapse response to reservoir changes is relatively subtle, the
50
true seismic differences are easily overwhelmed by data differences caused by survey
non-repeatability.
In this section, we discuss the major causes of non-repeatable
noise, including random noise, source and receiver positioning errors, source wavelet
discrepancies and overburden velocity changes (modeled as static shifts), and their
impacts on the performance of DDWI.
2.5.1
Random Noise
Although the sources of random noise vary within and .across surveys, they can be
effectively characterized by a random distribution of power spectrum and phase. In
our study, we aimed to investigate the impact of random noise on both baseline
inversion and DDWI, so we use uniformly distributed random power and phase spectra
to generate random noise. Such an approach more strongly influences portions of the
spectrum where signals are weak, such as the low-frequency signals that are known
to be crucial for obtaining reliable velocities from FWI [101]. This is illustrated in
Figure 2-15, which shows the power spectra of signal plus noise for the six noise
levels we investigated. The black dotted curve shows the power spectrum of the clean
trace. The colored curves from red to black show the spectra of the noisy traces with
different levels of noise contamination. To quantify the noise level, we use the overall
energy ratio between the noisy and clean signals of one entire shot gather:
r = E 1 n 2 * 100%,
where ni2 is the noise energy of the ith trace, s
2
(2.5)
is the signal energy of the ith
trace, and 1is the number of traces in a gather. In Figure 2-16b, the data difference
between the noisy monitor and baseline surveys for the same sample trace location in
Figure 2-16a is plotted together with the clean data difference trace. Due to the weak
reservoir response, the ratio between noise and the real time-lapse signal is extremely
large, even for only 1% noise energy.
Figures 2-17a-(f) show the baseline FWI P-wave velocity results for each noise
level shown in Figure 2-15. All the inversion results capture the dominant structural
51
features as well as the fine stratigraphic layers, but they are contaminated by random
noise in proportion to the noise level. Although both the noise and the primary
waves have the same energy level in Figure 2-17f, FWI still gives a reasonable result.
Figures 2-18a to (d) show the DDWI results starting from the corresponding baseline
FWI results in Figures 2-17a-(d) for each noise level. Only four cases are included
because above 64% noise energy, the reservoir changes cannot be identified from the
image. DDWI is able to deliver reasonable results in which reservoir changes are
clearly distinguishable even with relatively high noise levels.
We attribute the success of both baseline FWI and DDWI in the presence of
noise to the coherency in seismic data and their constructive interference during wave
propagation through a good velocity model. In FWI, as the data are injected into the
model, most of the random noise cancels during propagation, while the real signals
constructively interfere and produce coherent model updates. This also explains the
pattern of the noisy structures in the results. The random noise we use here is not
completely random in space (across-traces) because it is generated with a spatial
correlation, as is often observed in reality. The spatial correlation leads to coherent
stacking in model space to some extent. In the cases we tested, DDWI is very robust
to random noise even when the real signal is not directly observable from the data
difference.
2.5.2
Source and Receiver Positioning Error
If the baseline and monitor surveys use the same acquisition geometry, it is straightforward to apply DDWI because the two data vintages can be differenced trace by
trace. However, even with advanced GPS guidance during acquisition, the positioning
of sources and receivers still contains errors. A small positioning error is expected in a
well-repeated monitor survey [12, 112]. Nonetheless, small positioning deviations can
generate huge data differences over the entire dataset; these differences are normally
stronger in amplitude than the real 4D signals. In this section, we focus on the impact
of this type of error on the performance of DDWI. We assume the baseline survey
positioning is known, and use a perturbed survey geometry to generate monitor data.
52
The resulting data differences are input directly into DDWI without correcting for
these positioning errors. Due to the inherent limitations of finite-difference modeling,
we are only able to perturb the source and receiver positions by an integral number
of grid points (e.g., for a grid spacing of 6.25 m). Two types of perturbations are
studied: random perturbations and systematic perturbations. Both types are applied
to sources and receivers simultaneously.
For random source and receiver perturbations, we generate a random sequence
of numbers with zero mean that determines if the source or receiver position is perturbed by one grid point to the right or to the left. Figure 2-19 shows the P-wave
velocity changes resolved by DDWI with random perturbations in source positions
only, receiver positions only, and combined source and receiver positions for the monitor survey. Despite mild contamination in the background and at the seafloor, the
reservoir changes are recovered with an acceptable quality when compared to the
clean-data case in Figure 2-10d. The receiver-only perturbation case (Figure 2-19c)
appears cleaner than the source-only perturbation case (Figure 2-19b) because the
number of receivers (680) is 10 times the number of sources (64); hence, artifacts
induced by positioning errors are better canceled out in Figure 2-19c.
Although
not attempted here, we expect that including more shots will improve the image in
Figure 2-19b. In Figure 2-19d where both sources and receivers are perturbed, the
artifacts from DDWI show the combined effects of those in Figures 2-19b and (c).
For systematic source perturbations, we divide the 64 sources into groups, and
perturb each group by the same shift. Figure 2-20 shows the DDWI results with
source positions perturbed in groups, with the 64 evenly-spaced sources numbered
sequentially from left to right along the top of the model. In Figure 2-20b, all sources
in the monitor survey are shifted one grid point to the right. The recovered velocity
changes are slightly weaker in amplitude than those in Figure 2-19b, but the image
is relatively clean. In Figure 2-20c, source 1 to 32 are perturbed one grid point to
the right, and source 33 to 64 are perturbed one grid point to the left. In Figure 220d, source 1 to 16 and source 33 to 48 are shifted one grid point to the right, and
sources 17 to 32 and 49 to 64 are shifted one grid point to the left. Similar velocity
53
changes are recovered for the last two cases (c and d), and background artifacts follow
the true structures, although they are weaker in amplitude compared to the resolved
reservoir changes. The pattern of these artifacts reflects the number of source groups
that were perturbed. In particular, the amplitude polarity flip of the artifacts at
the seafloor is directly correlated with the position of the shifted sources. As in the
random perturbation case, the results for systematically perturbing the receivers are
similar to those for source perturbations, although the artifacts are smaller due to
more effective stacking.
In practice, source and receiver position errors due to limited GPS accuracy
(~1 m) and streamer feathering effects can be larger than what we have tested here
( 6.25 m).
However, the mismatch between post-processed baseline and monitor
surveys can be reduced to a much lower level by data binning, interpolation and
regularization.
In highly repeatable acquisitions like ocean-bottom cable systems,
the source positioning mismatch in the raw data can be even smaller than 6.25 meters [12, 112]. In practice, errors are likely to arise from a combined effect of both
systematic and random perturbations. From all the tests above, it is expected that
DDWI will be able to deliver good results with mild source and receiver positioning
discrepancies. To some extent, the randomness of this error helps to mitigate the
artificial patterns seen in the inversion results.
2.5.3
Source Wavelet Discrepancy
Source wavelets are likely to be different between surveys in real acquisitions. Both acquisition conditions (e.g., air-gun types) and initial data processing can introduce discrepancies in source wavelets. These errors are commonly minimized by co-processing
the baseline and monitor datasets. The source wavelet can be shaped by applying
matched filtering in the cross-equalization process [57]. However, after all such optimization steps are applied, the resulting wavelets are still likely to have small discrepancies (e.g., a few degrees of phase rotation). In this section, we focus on the impact
of phase differences between baseline and monitor source wavelets on the performance
of DDWI.
54
We use a standard zero-phase Ricker wavelet for the baseline survey, and phaserotated Ricker wavelets for the monitor surveys. It is apparent that a small discrepancy between source wavelets will cause significant data differences across the entire
survey. To simulate the situation in which we cannot further shape the wavelets, d,,,
in Equation 2.3 is synthesized by directly subtracting baseline and monitor datasets.
In DDWI, we use the same standard Ricker wavelet as that in the baseline inversion
to simulate the synthetic data differences.
Figure 2-21 shows all the DDWI results with increasing levels of phase rotation
in the monitor source wavelet. The inverted P-wave velocity changes are as accurate
as those of previous inversions for all the cases tested in terms of location, shape
and amplitude. However, the trend that larger phase rotations gives rise to stronger
artifacts in the model is also clearly observed. Up to 10 degrees, reservoir changes can
be easily distinguished from the incorrectly determined background structures. With
larger phase rotations, however, source wavelets in the monitor survey are markedly
shifted from the baseline wavelet, and the corresponding data differences are large
enough to produce significant model changes that overwhelm the real changes. In
practice, a phase difference of less than 10 degrees is generally achievable, in which
case DDWI appears to be robust.
It is important to point out that all the inversions in this section are masked
(i.e., no water layer was involved). Even for a small phase rotation (e.g., 2 degrees),
DDWI cannot converge when the entire model is included in the inversion. This
is because the dominant signal, i.e., the major contributor in the L-2 norm cost
function of Equation 2.2, is the direct arrival. A trivial phase rotation in the source
wavelet will generate huge data differences, especially for the dominant phases (e.g.,
direct waves and water bottom reflections). These data differences are non-physical,
and cannot be explained by the wave equation without attenuation. For example, a
delayed direct arrival indicates a decrease in water velocity between the source and
receiver; however, the phase-rotation-induced delay is frequency dependent, which
means only a dispersive velocity can explain the travel-time delay. In addition, the
data differences are not random enough to cancel each other. As a result, DDWI
55
is not able to find a perturbation in the shallow part of the model (i.e., the water
layer) that makes the cost function decrease. When the model is masked, these data
differences are not activated in the cost function, allowing DDWI to focus on the
reservoir responses.
2.5.4
Overburden Velocity Changes
In terms of the effect of model changes on FWI, areas outside the reservoir may
be even more important than those inside. In particular, overburden structures may
change between surveys. For example, compaction within the reservoir can change the
stress field and velocity above and below it [87]. Water velocity also varies seasonally.
All such overburden changes will affect the entire dataset and cause data differences
unrelated to reservoir changes.
In this section, we use water velocity changes to
represent this type of survey non-repeatability.
For the synthetic model in this study, the water depth can reach 2000 meters.
Deep-water production areas like the Gulf of Mexico (GoM) have water depths of up
to 3000 meters [59]. In such water depths, seismic amplitudes and travel-times can
be perturbed significantly even with small variations in water velocity. Factors that
influence water velocity include temperature, salinity and depth. We adopt Medwin's
Equation [62] to describe their relationship:
v = 1499.2 + 4.6T - 0.055T2 + 0.00029T3 + 1.34 - 0.01T(S - 35) + 0.016D, (2.6)
where T is temperature (*C), S is salinity (in parts per thousand, or ppt), and D
is water depth (in m). For our purpose, it is not necessary to discuss these factors
separately. We assume that salinity (S = 35ppt) and sea level stay constant, while
temperature changes. Typically, water temperature in the GoM varies at the surface
from 30'C in the summer to 15*C in winter, and it decreases to 4*C below 1000m.
Water velocity can change by up to 40 m/s at the surface seasonally according to
Equation 2.6.
From the water surface down to 1000 meters, we assume a linear
temperature gradient, and compute the water velocity with Equation 2.6. Water
56
velocity is assumed constant (= 1500 m/s) below 1000 m, and acquisition of the
baseline survey is assumed to be in winter (with a surface temperature 15*C).
We use 18'C and 30*C for surface water temperatures of two monitor surveys
acquired in the spring and the summer, respectively. According to Equation 2.6, the
maximum water velocity changes are
-
9 m/s and
-
39 m/s, respectively, at the
surface. For both cases, we directly difference the monitor and baseline datasets to
generate deyn, assuming that no corrections have been made in data processing to
account for the water velocity changes. The DDWI results are shown in Figure 2-22.
The reservoir changes are well recovered together with the water velocity changes. As
expected, larger water velocity changes contains stronger background noise (Figure 222b) than that for smaller velocity changes (Figure 2-22a).
In fact, background noise will exist even if the exact water layer velocity model is
used in DDWI. We can write the data difference as:
Jd = G(m) * Smwate, + G(m) * Smreservoir +
(2.7)
... ,
where * denotes a convolution operator, and G(m) is the Green's function in the true
model m; mwater and mreservoir are model perturbations in the water layer and reservoir, respectively. Neglecting higher order terms, the major contributor to the data
differences is first-order scattering caused by water velocity and reservoir changes. If
we managed to obtain both the exact water velocity and reservoir changes, the data
residual would be
Umonito,
-
dy =
(G (mo)
-
G (m)) * Smwater + (G (mo)
-
G (M)) * 6 mremvcir +
...
,
(2.8)
where G(mo) is the Green's function based on the inverted baseline model mo. Since
we cannot obtain the exact baseline velocity model m, the data residual does not
go to zero even with correct water velocities and reservoir properties. Instead, the
data difference in the cost function tries to update the background model to minimize
the misfit, generating spurious model changes in the background that are not from
time-lapse effects.
From Equation 2.8, we would expect this type of background
57
noise in DDWI even without water velocity changes.
However, the second-order
-
scattering caused by the reservoir changes and imperfect baseline model, ((G(mo)
G(m)) * 6mreervoir), is significantly weaker than that caused by the water velocity
changes, (G(mo) - G(m)) * 3 mwater), and the first-order scattering from the reservoir,
(G(m) * 6 mreservoir). Therefore, the second order reservoir scattering is not strong
enough to contaminate the result.
Although the results of the two cases presented here are of good quality and
interpretable, DDWI is not able to overcome water velocity differences by itself because, once the data difference is taken, the inversion does not differentiate between
the sources of these signal changes. DDWI could be improved if we processed the
time-lapse dataset carefully with a calibrated water velocity before taking the data
difference. After this, DDWI would function as if there were no water velocity changes.
2.6
Discussion and Conclusion
As we observed from the mathematical derivations and the synthetic tests, the advantage of DDWI over the other two time-lapse inversion schemes discussed here is that
the common data residuals are subtracted out and do not generate background velocity updates that are unrelated to reservoir changes. It is important for interpreters to
make decisions based on clean and meaningful images in which the reservoir information is not contaminated by background noise. However, in practice data subtraction
is intuitively dangerous whenever at least some of the differences between datasets
do not originate from the reservoir response. When these non-reservoir signals are
included in the cost function, it is reasonable to expect that if DDWI will produce
artifacts in the inverted images by attempting to fit such data. What we observe, however, from our synthetic study does not obey this intuition. Neither strong random
noise nor mild survey non-repeatability severely harms the performance of DDWI. It
is worth clarifying that the mechanism of this robustness is not only due to DDWI,
but also to the merits of full-waveform inversion itself. Unlike linear imaging methods
(e.g., reverse-time migration), FWI or DDWI does not directly map all of the data
58
into the model domain. Instead, these methods look for a model perturbation that can
explain the data via the wave equation. It is not difficult to understand that random
noise has little effect on DDWI because most of the noise energy is stacked out during
back-propagation. Data differences caused by survey non-repeatability are strong and
coherent, and, therefore, are not fully stacked out during back-propagation. However,
these differences do not produce strong velocity updates because they cannot easily
be generated by wave-equation-based velocity perturbations. While reservoir changes
are generally well resolved by DDWI, data differences arising from non-repeatability
effects have relatively small contributions to the final results. Obviously, when the
non-repeatability becomes severe, some of the data differences will lead to spurious
model perturbations. Therefore, to achieve a successful time-lapse waveform inversion, baseline and monitor datasets still need to be carefully co-processed to mitigate
non-repeatability effects before applying DDWI. If any combination of noise effects
we have tested in this study applies to the same dataset, the aggregated effect will
deteriorate the performance of DDWI.
In summary, our synthetic examples show that DDWI gives better results than
conventional inversion schemes by suppressing background model updates. The investigation of non-repeatable noise shows that within a practical range of data quality
(e.g., a few degrees of phase rotation, a few meters of positioning error, etc), DDWI
is robust enough to give a reliable estimate of the time-lapse P-wave velocity changes
within the reservoir.
2.7
Acknowledgments
This work was supported by the MIT Earth Resources Laboratory Founding Members
Consortium, and Chevron Energy Technology Company. We would like to especially
thank Chevron Corporation for the permission to publish this work. We also want
to thank our collaborators: Mark Meadows, Phil Inderwiesen and Jorge Landa from
Chevron for their efforts in this project.
59
Baseline Data
Monitor Data
I~
'I
I
Figure 2-1: Scheme I: Two independent FWI are conducted for the baseline and
monitor datasets, respectively. The model changes are obtained by subtracting the
inverted baseline model from the inverted monitor model.
60
Baseline Data
Monitor Data
I
I1
I
I
Figure 2-2: Scheme II: The baseline model is found by FWI with the baseline
dataset.The monitor inversion starts from the baseline inversion result. The model
updates are considered to be model changes between baseline and monitor.
61
Baseline Data
Monitor Data
I
Il
I
I
Figure 2-3: Scheme III: The time-lapse inversion starts from the baseline inversion
result, and inverts the baseline and monitor datasets jointly. The model updates are
considered to be model changes between baseline and monitor.
62
True P-velocitv Model
0
Velocity (m/s)
4500
1
4000
2
N
3500
3
3000
4
2500
5
2000
6
2
1500
4
6
8 X (km)10
12
14
(a) true pvel
True Density Model
0
16
Den
(kg/m3)
2400
1
2200
2
N
2000
3
1800
4
1600
1400
5
1200
6
2
4
6
8
10
X (km)
(b) true dens
12
14
16
Figure 2-4: The true baseline P-wave velocity model (a), and the true baseline density
model (b) that are used for generating synthetic 'real' data for the baseline survey.
63
a)
Power Spectrum
1
4-
E
-00.5
N
0
z
0
2
4
6 8 10 12
Frequency (Hz)
14
Figure 2-5: The normalized power spectrum of the source wavelet we used.
64
Baseline Data
0
2
3
4
E
6
7
8
9
100
200
300
400
Trace Number
500
600
Figure 2-6: The shot gather generated by the source in the middle of the model on
the water surface.
65
-ed P-velocity Model
0
Velocity (m/s)
4500
1
4000
E
N
2
13500
3
3000
4
2500
5
2000
6
2
4
6
1500
8
10
X (kin)
(a)
Lavered Densitv Model
0
12
14
16
Density (kg/m3)
2400
1
2200
N
''2
I l 2000
3
1800
4
1600
1400
5
1200
6
2
4
6
8
X (k)
(b)
10
12
14
16
Figure 2-7: The starting P-wave velocity model (a) and density model (b) for baseline FWI. The models are obtained by averaging the true models in Figure 2-4(a)
horizontally.
66
Velocity (m/s)
4500
Baseline Inverted P-velocity Model
0
1
N
4000
2
3500
3
3000
4
2500
5
2000
6
1500
2
4
6
8
X (ki)
10
12
14
16
(a)
Baseline Inverted Densitv Model
0
E
Density (kg/m3)
2400
1
E2200
2
2000
3
1800
1600
4
1400
5
1200
6
2
4
6
8 X (km)10
12
14
16
(b)
Figure 2-8: The final baseline p-wave velocity model (a) and density model (b) after
60 FWI iterations.
67
Velocity (mis)
100
80
60
40
20
0
-20
-40
-60
-80
-100
True P-velocity Change
0
1
2
E
3
4
5
6
2
4
24
0
10
8
8X (kin)
(a)
True Density Change
6
6
12
14
16
Density kg/m3)
40
35
30
25
20
15
10
5
0
1
2
N
3
4
5
6
2
4
6
8
10
X (kin)
(b)
12
14
16
Figure 2-9: The true time-lapse changes in P-wave velocity (a) and density (b).
68
6
2
4
6
8 10 12 14 16 -50
X (km)
2
4
6
(a)
8 10 12 14 16
X (km)
150
(b)
P-velocit Chan e: Scheme II m/s)
518
P-velocit Chan e: Scheme Ill(m/s)
51
E
E
0
6
2
4
6
-8 10 12 14 16
X (km)
0
50
6
2
4
6
8 10 12 14 16
X (km)
-50
(d)
(c)
Figure 2-10: (a) The true time-lapse changes in P-wave velocity, saturated in color
to i50 m/s for comparison; (b), (c), (d) are the time-lapse P-wave velocity changes
recovered by inversion scheme 1, 11 and III, respectively.
69
True Densi
Chan
1
/mn)
Density Change: Scheme I (kg/r)
1
00
0
-4r
2
4
6
-30
8 10 12 14 16
X (km)
2
(a)
Densit Chan e: Scheme I (k/m3 )
4
6
8 10 12 14 16
X (km)
(b)
Densit Chan e: Scherne III (k W)
1
1
0
2
4
6
8 10 12 14 16
X (km)
0
1-30
2
4
6
8 10 12 14 16
X (km)
-30
(d)
(c)
Figure 2-11: (a) The true time-lapse changes in density, saturated in color to
+40kg/M 3 for comparison; (b), (c), (d) are the time-lapse density changes recovered by inversion scheme I, II and III, respectively.
70
Cost Function Curve
CA)
0
0.5
E
0
z
0
20
40
60
80
Iteration Number
100
Figure 2-12: Curve: The cost function curve of the baseline inversion; Dots: The
selected iterations: 1, 5, 10, 20, 50, 99
71
True P-veloci
E
Model
km/s)
.
1
3 .5
Iteration 1
0
1
E2
Velocity (km/s)
4.5
3.5
3'-3
N 4
5
N-5
* 2
4
6
8 10 12 14 16
X (kin)
.5
6
2.5
2
4
6
(a)
Iteration 5
0
1
E2
8 10 12 14 16
X (kin)
1.
(b)
Veloci
kn/s)
4.5
.3.5
N
1
E
Iteration 10
0
km/s)
4.5
Velo
3.5
N
2
4
6
8 10 12 14 16
X (km)
5
6
1.5
2
4
6
0
Velocity km/s)
4.5
E2
N 3
4
3.5
Iteration 99
0
E2
N 3
4
2.5
5
6
1.5
(d)
(c)
Iteration 20
8 10 12 14 16
X (km)
Veloci
km/s)
4.5
35
71a
-
.
5
6
5
2
4
6
8 10 12 14 16
X (kin)
1.5
6
(e)
2
4
6
8 10 12 14 16
X (kin)
1.5
(f)
Figure 2-13: (a) The true baseline P-wave velocity model; (b) - (f) are the baseline
P-wave velocity models after 1, 5, 10, 20, and 99 iterations.
72
DDWI from Iter 1
0
1
E2
E2
N
N4
4
Veloci
5
6
2
4
6
3
8
10
X (km)
12 14
1 6
8 10 12 14 16
X (km)
(a)
(b)
DDWI from Iter 5
05
Veloci
(m/s)
25
1e
0
1
N
5
2
4
6
8 10 12 14 16
X (km)
DDWI from Iter 10
0
4
5
1-25
6
ie
4
b
5
10
X (km)
12
14 lb
-30
(d)
Velocity (m/s)
1
M0
0
DDWI from Iter 99
Veloci
(m/s)
~~1
E2
E2
_d3
50
-3
N4
5
6
(m/s)
30
0
(c)
nnwi frnm Ii r 20
Veloci
E2
-- 3
0
N4
6
115
6
16
(m/s)
15
N4
10 4
Z
4
b
0t lU
X (km)
5
6
12 14 lb
4
b
U
14
50
X (km)
(e)
(f)
Figure 2-14: P-wave velocity changes obtained with incorrect baseline velocity models.
(a) The true time-lapse changes in P-wave velocity, saturated in color to 50 m/s; (b)
- (f) are the recovered time-lapse P-wave velocity changes by DDWI starting from the
baseline models shown in Figure 2-13b to 2-13f. The recovery of the velocity changes
is clearly improved with better starting baseline models.
73
Power Spectrum
--1%
Energy
-4%
Energy
-16%
Energy
-64%
Energy
256% Energy
- -1024%
Energ
Noise Free
E
-)
ND0.5
E
0
z
0
5
10
Frequency (Hz)
15
Figure 2-15: Normalized power spectra of a sample trace with different noise contamination levels. The random noise spectrum obeys a uniform distribution from 0 to 15
Hz. Six noise levels are tested.
74
Sample Trace
_0
-5-
1
E
N
-1
0
0
z
1
2
3
4
C
6
5
Time (s)
8
7
9
0
(a)
Sample Trace
---
Data Difference with Noise
ifU..A~i~L~p:I~jA~
.~1.
9AAAii.AM~ALA~~A
-Clean
~N~AJ.A
0
VYI-
)
-5-
F
E -0.5 - 1
-1[*
)
Cz
0
z
Data Difference
1
2
3
4
5
Time (s)
6
7
8
9
10
(b)
Figure 2-16: (a) A near offset monitor trace with 1% noise energy. The amplitude of
the noise is about the same level as that of the coda waves. (b) Difference between
noise-free monitor and baseline traces (red) and between noisy monitor and baseline
traces (blue). Note small waveform changes shown in red trace between about 3 to 5
seconds are obscured by noise in the blue trace.
75
P-velocitv: 40% Noise
(m/st
3.5
N
(m/s
3.5
E
2.5
6
6
M11.5
8 10 12 14 16
X (km)
2
4
6
(a)
10
8
X (km)
Ise
II
(b)
elocitv: 16% Noise
P-velocitv: 640 Noise
(m/s)
%
N
2.5
I6
FE
m/s
3.5
N
4
X (km)
N
(c)
P-vplneitv- PR 0 A Nni-A
0
2.5
I
b
b
1U
X (km)
12 14
6 11.5
(d)
M/s)
P-vInitv- lr4O- ,*
Nni-Q
4.5
4.5
3.5
3.5
2.5
2.5
11.5
0
X (km)
X (km)
(e)
(f)
Il
1rO1.5
Figure 2-17: Baseline models obtained by FWI on noisy data. (a) - (f) are the
baseline P-wave velocity models recovered by FWI starting from the same layered
model shown in Figure 2-7a. The recovery of the dominant structure is very robust
to random noise. As the noise energy increases, the details in the model are more
contaminated.
76
E
E
0
0
6
2
4
6
8 10 12 14 16
X (km)
6
1-50
2
4
6
P-veloci
m/s
E
2
4
Chan e: 64% Noise (m/s
E
[50
6
1-50
(b)
(a)
P-velocit Chan e: 16% Noise
8 10 12 14 16
X (km)
6
6
8 10 12 14 16
X (km)
2
4
6
8 10 12 14 16
X (km)
-5
(d)
(c)
Figure 2-18: P-wave velocity changes obtained with noisy data. (a) - (d) are the
recovered time-lapse P-wave velocity changes obtained from DDWI starting from the
baseline models shown in Figure 2-17a to 2-17d, respectively.
77
True P-veloci
Chan e
11
E
(m/s)
58
4
6
(m/s)
58
E
0
N
2
Random Source Error
6-50
8 10 12 14 16 -50
X (km)
(a)
Random Receiver Error
0
N
X (km)
(b)
Random S & R Error
rm/s)
~~.1
58
E
(m/s)
58
E
0
N
2
4
0
-50
6
N
8 10 12 14 16
X (km)
2
4
6
8 10 12 14 16
X (km)
-50
(d)
(c)
Figure 2-19: P-wave velocity changes obtained using monitor data with random source
and receiver positioning errors. (a) The true time-lapse changes in P-wave velocity,
saturated in color to t50 m/s; P-wave velocity changes resolved by DDWI with the
monitor survey (b) randomly perturbed source positions; (c) randomly perturbed
receiver positions; (d) randomly perturbed source and receiver positions.
78
(m/s
5
1 Source GroUD
lM/s)
150
0
N
N
2
-50
-50
4
b
1U
6
X (km)
12
14
z
]b
4
0
0
X (km)
IU
1z
(a)
(b)
2 Soure. Grouns
4 Source GrouDs
MQ
10
(r /s
N
N
d
1U
X (km)
12
14
2
lb
-50
4
b
6
1U
X (km)
(d)
(c)
Figure 2-20: Effects of systematic shifts in source positions. (a) The true time-lapse
changes in P-wave velocity, saturated in color to 50 m/s; P-wave velocity changes
resolved by DDWI with the source positions systematically perturbed in the monitor
survey. In (b), (c) and (d), the sources are divided into 1, 2, and 4 groups. Each
group of sources is shifted 1 grid (6.25 m) in one direction.
79
Phase Rotation: 2 degree
(rn/s
N
N
4
X (km)
6
(a)
8 10 12 14 16
X (km)
1-50
(b)
Phasp Rotation: S din
ie (m/s
Phase Rotation: 10 degree (m/s)
N
N
50
2
4
Phn-q
b
6
1U
X (km)
(c)
12
14 1b
Rntntinn- 90 danro
-50
2
4
6
8
X (km)
10 12
50
(d)
(m/s
Rntntinn- fA ri
Phcan
N
ee (m/s
N
1 50
4
X (km)
(e)
6
-1050
8 10 12 14 16
X (km)
(f)
Figure 2-21: Effects of source wavelet discrepancies between baseline and monitor
surveys. (a) The true time-lapse changes in P-wave velocity, saturated in color to
50 m/s; P-wave velocity changes resolved by DDWI with the source wavelet in the
monitor survey shifted in phase by (b) 2 degrees; (c) 5 degrees; (d) 10 degrees; (e) 20
degrees and (f) 30 degrees, for all frequencies.
80
Water Veloci
Chan e (9 m/s)
(m/s)
Water Velocit Change 39 m/s) (m/s)
50
50
N
N
2
4
6
8 10 12 14 16
X (km)
-50
(a)
6
2
4
6
8 10 12 14 16
X (kin)
-50
(b)
Figure 2-22: P-wave velocity changes resolved by DDWI with the water velocity in
the monitor survey increased by (a) 8 m/s maximum; (b) 40 m/s maximum.
81
82
Chapter 3
Time-lapse Full Waveform
Inversion with Ocean Bottom
Cable Data:
Application on Valhall Field
Summary
Knowledge of changes in reservoir properties resulting from extracting hydrocarbons
or injecting fluid is critical to future production planning. Full waveform inversion
(FWI) of time-lapse seismic data provides a quantitative approach to characterize
these changes by taking the difference of the inverted baseline and monitor models.
The baseline and monitor datasets can be inverted either independently or jointly as
discussed in Chapter 2. Time-lapse seismic data collected by ocean bottom cables
(OBC) in the Valhall field in the North Sea are suitable for such time-lapse FWI tests
because the acquisitions are long-offset and the surveys are well-repeated. We apply
both independent and joint FWI schemes to two time-lapse Valhall OBC datasets
which were collected one year apart.
The joint FWI scheme is double-difference
waveform inversion (DDWI) discussed in Chapter 2, which inverts differenced data
83
for model changes. We find that DDWI gives a cleaner and more easily interpreted
image of the reservoir changes, as compared to that obtained with the independent
FWI schemes. A synthetic example is used to demonstrate the advantage of DDWI
in mitigating spurious estimates of property changes, and to provide cross-validation
for the Valhall data results.
3.1
Introduction
Time-lapse seismic monitoring is widely used in reservoir management in the oil industry to obtain information about reservoir changes caused by fluid injection and
subsequent production. The seismic responses change according to the fluid saturation and pressure variations in the reservoir. The optimal goal of time-lapse seismic is
to track fluid flow in areas without well logs [57]. Conventional analysis of time-lapse
seismic data gives either qualitative information, like seismic amplitude, or indirect
parameters like image shifts and traveltime differences. This information needs to be
transferred to reservoir properties by matching reservoir modeling [56]. Quantitative
4D techniques are used to estimate reservoir compaction and velocity changes using
time shift and time strain in the data [51, 118]. Amplitude versus offset analysis
inverts partial-angle stacks for elastic impedance changes [78, 95]. However, these
methods assume simple subsurface structures, and often involve manual event picking.
Full waveform inversion has the potential to estimate density and elasticity parameters quantitatively [93, 101]. Subsurface properties are updated iteratively by
fitting data with modeled waveforms which are generated by solving wave equations.
Ideally, by subtracting the models inverted from each dataset in a series of time-lapse
surveys, the geophysical property changes over time can be quantified. Instead of
analyzing small-offsets and large-offsets separately as in [118], FWI naturally takes
all types of waves into account, including diving waves, supercritical reflections, and
multi-scattered waves. Both structural depth changes and velocity changes can be well
represented in FWI inverted models, therefore separate analyses are not necessary,
84
as in conventional time-lapse methods [51]. In addition, FWI makes no assumption
about the subsurface structures, and involves less manual interaction.
However, the convergence levels of waveform inversions for individual datasets
are affected by data quality and computational parameters used in the inversion,
which may differ between surveys. Model differences caused by different local minima
between inversions may generate misleading time-lapse images. To mitigate these
problems, [107] applied a differential waveform tomography in the frequency domain
for crosswell time-lapse data during gas production, and showed that the results are
more accurate for estimating velocity changes in small regions than those obtained
using the conventional method. [66] applied a similar strategy to conduct differential
traveltime tomography using crosswell surveys.
[24] developed a double-difference
waveform inversion (DDWI) algorithm using time-lapse reflection data in the time
domain and demonstrated, using synthetic data, that the method has the potential to
produce reliable estimates of reservoir changes. Similar approaches are also reported
in [119] and [120]. Nonetheless, to our best knowledge, very few field data applications
of DDWI have been reported.
The major obstruction to successful field data applications of both FWI and DDWI
is data acquisition. To recover a model having a broad wavenumber spectrum, lowfrequency and long-offset data are required, but are often not available in legacy
seismic experiments. Advanced technologies like wide-aperture and wide azimuth
acquisitions make FWI more feasible nowadays. However, DDWI requires pre-stack
data subtractions which imposes a higher standard on time-lapse survey repeatability.
One way to obtain such data is with 4D ocean bottom cable (OBC) acquisitions using
receiver cables installed on the seafloor. Source and receiver positioning discrepancies
between surveys are significantly reduced compared to streamer acquisitions. Signal
quality is also improved because of better receiver coupling. The repeatability of 4D
OBC acquisitions appears promising for DDWI application.
Since 1998, OBC data have been collected in the Valhall field, in the North Sea [39].
A permanent OBC system was installed in 2003 to enable frequently repeated timelapse surveys to help manage the field. Due to the wide aperture and high quality
85
of the surveys, numerous studies on 2-D and 3-D FWI use the Valhall data as the
field example (e.g.
[70, 71, 82, 54]). [10] discussed the potential business impact of
FWI and time-lapse FWI at Valhall, but technical details and comparisons between
time-lapse FWI approaches are not presented.
In this chapter, we first introduce three time-lapse inversion schemes: (I), use
the same initial model for both baseline and time-lapse inversions; (II), use the final
model from baseline inversion as the starting model for time-lapse inversion; (III),
DDWI which inverts the differenced data for model changes starting from the final
baseline inversion model. A 2D synthetic example using the Marmousi model is used
to demonstrate how DDWI can improve the inversion quality in terms of suppressing
spurious model perturbations.
We then apply all three schemes to two datasets
collected one year apart, one as baseline and the other as monitor, collected by the
OBC from the Valhall field. We compare the results obtained from all schemes, and
show that DDWI produces a cleaner and more interpretable image of the reservoir
changes.
The mechanism causing the differences between the results of different
inversion schemes is discussed for both synthetic and real data.
Cross-validations
between synthetic studies and the Valhall application enhance the credibility of the
DDWI result.
3.2
Theory
FWI for individual surveys minimizes a cost/objective function of the difference between modeled data u and observed data d:
E(m) =
2
d - u(m) 12,
(3.1)
where m is the model parameter (e.g. density, P-wave and S-wave velocities) to be
recovered. Gradient based methods such as nonlinear conjugate gradient and the
Gauss-Newton method have been adopted in many studies to solve this optimization
problem efficiently [65, 69, 101].
86
The most straightforward manner for time-lapse FWI is to repeat the process on
each individual dataset. One can choose to use the same starting model for each of the
individual inversions. For example, a smooth velocity model derived from tomography
can be used for the inversions of both the baseline and monitor datasets. We label this
as Scheme I. It is also reasonable to choose the final model of the baseline inversion as
the starting model for inverting monitor datasets to achieve faster convergence. This
is labeled Scheme II. We want to clarify here that if, starting from a model obtained
from a previous inversion of the baseline dataset, both baseline and monitor datasets
are further inverted for more model improvements, this is considered as Scheme I.
Other than individual inversions, the datasets can be used jointly. One efficient
way to do a joint inversion without doubling the computation is to apply DDWI.
Similar to Scheme II described above, DDWI starts from a model obtained from the
baseline inversion. To include both datasets, the cost function is modified to:
E(m)
=
(dmonitor - daseine) - (umonito,(m) - uaneline(mo)) 12,
(3.2)
where dmoit, and dbaseline are monitor and baseline data respectively, and umonit, is
the synthetic data calculated from the model m that is updated in every iteration. We
denote by Ubseline the synthetic data calculated from the starting model mo that is the
final model from the baseline inversion. Because mo is not updated in DDWI, ubaeine
does not change throughout the inversion process. Equation 3.2 can be rewritten as:
E(m) = 1
umonito, - dayn 12,
(3.3)
where d,yn = Ubaseline + (dmonitor - dbaseline). DDWI looks for the changes in the model
that can explain the waveform changes between time-lapse datasets. It reduces the
effects of uncertainties in the baseline model. The mechanism and implementation of
the method are well-explained in [119] and [116].
87
3.3
Examples Using Synthetic Data
In this section, we use the Marmousi model to illustrate the different behaviors of the
inversion schemes introduced above, and to provide context for interpreting our real
data results in later sections. Figure(3-la) shows the true baseline P-wave velocity
model. In the time-lapse velocity model, a thin layer of P-wave velocity increase is
placed in the second anticline under the salt layers (bright wedges) to simulate a
hardening reservoir as shown in Figure 3-1b. The maximum magnitude of velocity
change is 200 m/s. We use five shots, marked by white stars in Figure(3-1a), on the
water surface and 400 receivers evenly spaced at the water bottom to cover the entire
area. The same source and receiver geometry is used for both baseline and monitor
surveys to mimic a time-lapse ocean bottom cable acquisition. Synthetic baseline
and monitor data are generated with a finite difference acoustic wave equation solver.
The source time function is a standard Ricker wavelet centred at 6 Hz.
We use a smoothed version of the Marmousi model (Figure 3-2a) as the starting
model for the baseline inversion. The conjugate gradient method is used to invert
for the P-wave velocity model. After 90 iterations, we obtain the recovered baseline
model shown in Figure(3-2b). It is slightly blurred compared to the true model due
to the limited resolution of the data. The dominant features of the structures are
well-recovered, while some of the deeper layers underneath the salt are less resolved
because of lower energy penetration.
Following Scheme I, we can invert the monitor dataset using the same initial
model (Figure 3-2a) for the same number of iterations. Figure 3-3a shows the model
difference between the final time-lapse and baseline models. The reservoir change is
recovered to some extent, however, model differences also exist almost everywhere
outside of the reservoir layer. Some of the false changes (e.g.
in the salt layers)
are as strong as the real changes. The nonlinear behavior of the inversion makes it
difficult to avoid such false changes between two inversions. The model subtraction
is not able to differentiate between the differences caused by time-lapse effects, and
the differences caused by these false changes.
88
We can also choose to invert the monitor dataset starting from the recovered
baseline model (Figure 3-2b) as described in Scheme II. Figure 3-3b shows the model
difference between the final time-lapse model and the baseline model in Figure 3-2b.
The non-reservoir related differences are stronger than those in Figure 3-3a because
effectively more iterations are applied to update these parameters. Therefore, parameters that are less well estimated in the previous baseline inversion would exhibit
larger magnitudes in the model difference. This explains why the real changes in the
reservoir layer are saturated by the strong updates nearby in Figure 3-3b.
Starting from the same baseline model (Figure 3-2b), DDWI (Scheme III) is applied to find the time-lapse changes. Figure 3-3c shows the time-lapse changes recovered by subtracting the baseline model from the final time-lapse model. The image
is almost free of contamination. The clearest feature is the velocity increase within
the reservoir layer. Both the shape and magnitude of the velocity changes are well
recovered. Neither the coherent structures in the shallow part nor the salt layers have
any footprint in the image. This is because, as we stated in the methodology section,
DDWI only finds the velocity perturbations that caused the data differences. Therefore the parameters that are not completely recovered from the baseline inversion are
not updated at all in DDWI.
Comparing the three images in Figure 3-3, it is easier to make an interpretation
with the DDWI result. Without the interference from background structures, tracking
the locations of changes is easier. In addition, because the magnitude of the changes
is more accurately recovered, the reservoir properties inferred from this information
are also more reliable.
3.4
Time-lapse Full waveform Inversion on Valhall
Valhall field sits in the southern part of the Norwegian North Sea, and has been
producing hydrocarbons since 1982. Recently approved plans could potentially extend
its life to 2048. The reservoir layer is at a depth of about 2400 m, and its thickness
ranges from 10 to 70 m. The reservoir formation consists primarily of high porosity,
89
low permeability Cretaceous chalk. Pressure depletion of the highly porous rocks leads
to significant reservoir compaction which both drives the production and induces the
subsidence of the overburden structures [9]. Significant 4D seismic time shifts due
to reservoir compaction has been observed in a previous study by cross-matching
of 3D streamer data collected in 1992 and 3D ocean-bottom-cable data collected in
1998 [39].
Acoustic impedance changes that reflect the depletion of the reservoir,
have been derived from amplitude differences by comparing marine streamer surveys
in 2002 and 1992 [9].
To allow for more detailed and frequent analyses of induced 4D seismic changes,
a permanent array, life of field seismic (LoFS), was installed in 2003 [9, 97]. The 4D
images produced with the LoFS data provide a structural framework for identifying
undrained areas, managing existing wells, and analyzing geohazard potentials [97, 76].
Integrated with reservoir modeling, LoFS system reduces the uncertainties in reservoir
performance predictions [98]. We expect the constraints on the reservoir model to
be improved and enhanced by extracting quantitative 4D changes from the LoFS
data with time-lapse FWI [10]. Since FWI includes information on both structure
and properties from all the data in the surveys, individual analyses on overburden
changes, reservoir compaction and reservoir property changes are naturally integrated
in time-lapse FWI.
3.4.1
Acquisition, Repeatability and Preprocessing
As shown in Figure 3-4, an area of 15 km x 8 km is densely covered by 50,000 shots
(white points) on a 50 x 50-m grid. The missing shots in the middle of the acquisition
are due to the center platform. Around 2500 receivers (blue dots show 1/4 of the
receivers covering the same area) are placed a meter into the sea floor comprising 45
km2 of coverage. The distance between receivers along the cable is 50 in, and the
distance between cables is 300 meters. The seismic experiment is repeated roughly
every six months. The datasets used in this study are LoFS 10 and LoFS 12 that are
12 months apart.
Minimum preprocessing applied to the raw shot gathers before input to FWI was
90
limited to proper denoising and low-pass filtering up to 7 Hz. No cross-matching was
applied between surveys. The positions of the receivers are not changed between surveys except several cables were offline in LoFS 12. Shot positioning is very accurate.
Compared to the pre-designed shot network, i.e., a regular 50 x 50 m spacing grid
system, the mean of the shot positioning error in LoFS 10 is close to zero, with a
standard deviation of less than 5 meters [97]. The positioning accuracy is improved in
LoFS 12, in which the standard deviation of the error is less than 2 meters (Figure 35). Regarding the position discrepancies of the paired shots between LoFS 10 and
LoFS 12, 50% of the shot pairs have distances less than 1 m, and 95% have distances
less than 10 m. Because the data residual needs to be injected on regular grids in
finite difference modeling, we adopt the method in [45] to interpolate and resample
both LoFS 10 and LoFS 12 data to the same regular grids.
To demonstrate the excellent survey repeatability, we show example trace pairs
in Figure 3-6.
Both pairs are from the same common receiver gather. Traces in
Figure 3-6a are from the same near-offset shot. Not only do the early arrivals fit each
other well, but the coda waves are also very similar. Traces in Figure 3-6b are from
the same far-offset shot. Despite having traveled for more than 10 kin, the diving
waves and the direct waves are still very close in both phase and amplitude.
3.4.2
Inversion Setup
In this study, FWI is implemented in the time-domain. As a result, CPU runtime is
linearly dependent on the number of sources simulated in each iteration. Therefore,
reciprocity is applied to generate common-receiver gathers as FWI data input instead
of shot gathers. To further reduce the computation, we downsample the receivers
along cables to a spacing of 200 m; in the end 380 receivers are used in FWI.
A few assumptions are made in the process.
First, only the pressure data are
used, and so the acoustic wave equation is solved to simulate the wavefield. Second,
only the isotropic P-wave velocity is inverted for. The density model is derived from
the Gardner Equation [33] with the updated velocity model in each iteration. Third,
attenuation is not included in the modeling. Instead, a trace by trace energy scaling
91
strategy is used to mitigate amplitude differences [53].
We extracted the source wavelet from a raw near offset trace. As it is recorded on
the sea floor, the first event is a mixture of source side ghosts, direct waves, and free
surface multiples. An effective wavelet is derived after the removal of multiples and
ghosts and the application of a low-pass filter. Its quality is confirmed by carefully
comparing a synthetic shot gather with the recorded data before FWI [53].
3.4.3
Initial Velocity Model
It is difficult in practice to use only FWI to invert for a good quality model starting
from a poor initial guess.
Several studies about FWI applications on Valhall use
tomographic models as initial models [70, 71, 82, 54]. We use a smoothed version
of the Valhall velocity model presented in [53] as shown in Figure 3-7a, to avoid the
elaborate process of initial model building, since this study focuses on the time-lapse
application. The details about how we handle the initial model building and obtain
the model in Figure 3-7a can be found in [53, 54].
3.4.4
Baseline Inversion Result
We run acoustic FWI for the baseline survey data (LoFS 10) starting from the model
in Figure(3-7a). After 200 iterations, the baseline inversion is considered converged;
the resulting model is shown in Figure(3-7b). The final model is of higher resolution,
and shows a lot more detail about the geological structures. The image of the gas
cloud (marked by the black arrow in y-z slice in Figure 3-7b) is markedly improved.
The thin layer under the gas cloud that is not visible in the starting model is resolved
remarkably well. The differences between the field data and the synthetics before
and after the inversion are shown in Figure 3-8 for one common receiver gather. The
traces are aligned according to the order of shots. The magnitude of the data residual
is significantly reduced by FWI. The residuals of both the long offset diving waves
(white circle) and the near offset reflections (black circle) are greatly minimized.
92
3.4.5
Time-lapse Inversion Result
As in the synthetic examples, three schemes are applied to the time-lapse dataset
(LoFS 12).
For Scheme I, we start from the smooth model in Figure 3-7a, and
run the same number of iterations to invert LoFS 12 data for the time-lapse model.
The P-wave velocity model difference is shown in Figure 3-9a. In the shallow part,
the differences are relatively weak, whereas the differences in the deeper part are
stronger and also spread out. For Scheme II, the model in Figure 3-7b is used as the
starting model. Figure 3-9b shows the model difference. Compared to Scheme I, the
magnitude of the difference is generally stronger. Both in the shallow part (around
the gas cloud) and in the deep part (below the gas cloud) we find distinct velocity
changes. For Scheme III (DDWI), starting from the model in Figure 3-7b, we invert
the data differences (LoFS 12 minus LoFS 10) for the velocity differences. As shown
in Figure 3-9c, the velocity changes found by DDWI are much more localized than
the results from Scheme I and II.
To better visualize and compare the results, we plot the 2D slices in Figures 3-10
and 3-11. Depth slices at the location of the maximum time-lapse velocity changes
are shown in Figure 3-10. The three black squares mark the holes in the survey
(Figure 3-4). Although there are some common features among the three images in
Figure 3-10, the changes from Scheme I and II cover a much bigger area than the
changes from DDWI. In particular, the changes in Scheme II would be even broader
if shown on the same color-scale as those for Scheme I and III.
In the cross-sectional views in Figure 3-11, the same velocity change volume is
shown in the X-Z axis. The model changes have completely different patterns. In
both Scheme I and II, the velocity changes spread horizontally over most of the area
in the deeper part of the model. Some strong changes are also found in the shallow
parts. By contrast, in the DDWI case, the dominant change is localized in the center
of the model beneath the gas cloud. The changes in other parts are much weaker,
and no evident changes are found in the shallow part of the model.
93
3.5
Discussion
The synthetic examples and the Valhall data results exhibit similar behaviors. The
nonlinearity of the inversion makes Scheme I generate spurious model differences. For
real data, it is more difficult to control the convergence for velocities at all positions.
Because deeper reflections have lower signal to noise ratio, velocities at greater depths
are more likely less constrained and so differ more between independent inversions,
which explains why the magnitude of changes increases with depth.
The model differences in Scheme II are strongly contaminated by the extra updates
to the background model (i.e., model parts without time-lapse changes) because we
try to reduce the data residuals with velocity perturbations that are not related to
time-lapse changes. The residuals left after the baseline inversion are much stronger
than the time-lapse signals for the real data case, which explains the significant model
differences in Figure 3-11b. In addition, the deeper part is less resolved than the
shallow part in the baseline inversion. Consequently, we observe more updates to
the deeper part in the time-lapse inversion in Scheme II. One might argue that the
situation would be improved by running the same number of FWI iterations in extra
on the baseline data (LoFS 10) as those run on the monitor data, and then subtracting
the two models. In other words, if we run N iterations to get the baseline model,
and another N to go from baseline to monitor, then baseline should have another N
iterations to equally resolve unchanging structures. In fact, it reduces to Scheme I
with a better starting model. We conducted this practice, however, no remarkable
improvements were achieved.
DDWI gives localized results in both the synthetic and real data case studied
here. Because only the velocity perturbations that can explain the data differences
are used to update the model, it is not difficult to understand why the synthetic
noise-free DDWI result in Figure 3-3c is so clean. One might feel uncomfortable about
subtracting real datasets when there are so many uncertainties between surveys. Nonrepeatability issues like random noise, source wavelet discrepancy, source position
error and overburden changes, can generate significant data differences that may
94
overwhelm the real time-lapse signals. These non-repeatability effects are discussed
and tested in detail in [116], which concludes that DDWI is robust to random noise,
and mild non-repeatabilities. For the LoFS 10 and LoFS 12 surveys, the standard
deviation of the source positioning error is less than 5 m. Source wavelets are well
repeated in the frequency range used in FWI, and the water velocity change does not
have a huge impact because it is a shallow water environment. Overburden changes
are expected to be small since the two surveys are only one year apart. All the issues
are within the range where DDWI is tested to be robust.
If we take the field data results at face-value, DDWI is definitely finding a timelapse velocity change that is cleaner and easier to interpret. But to understand why
this is the case, and thus to increase our confidence in our interpretation, we need to
describe what we are fitting in DDWI and how this contrasts with traditional FWI.
To this end, Figure 3-12 summarizes the various effects that we expect to see in the
time-lapse data, showing those that are suppressed with DDWI as compared with
standard FWI in black and gray. The data can be decomposed into two parts as
shown in Figure 3-12: signal and noise. Within the signal branch, all the information
is related to real changes in Earth properties. The part of the residual signal that
can be modeled (Black in Figure 3-12), but is due to either under-fitting the data or
being caught in a local minimum is what we expect to cancel in DDWI and not in
Scheme I and II. In the noise branch, we classify noise as either coherent or random.
The random component will contribute relatively little to the final image because
of stacking. Coherent noise should lead to changes throughout the model, if it is
constructively interfering and significant enough. Non-repeatibilities can introduce
coherent noise but are less likely to be modeled in the simulation, which is why
DDWI is robust to them [116]. The signal that is not modeled due to incomplete
physics (Gray in Figure 3-12) in the model equations are considered as noise, and has
a second-order effect on the velocity change. For example, the common background
anisotropy and attenuation effects are subtracted out in DDWI, and those induced by
reservoir changes are relatively weak and localized. Because the model change in the
DDWI example is clean and localized, it is credible that the recovered velocity change
95
is actually the reservoir change rather than simply the movement into a different local
minimum of the objective function, or simply the change one might expect if the
inversion were to be continued to additional iterations.
3.6
Conclusion
Advanced acquisition technologies like ocean bottom cables provide the opportunity
to use high resolution imaging methods to monitor subsurface changes. We applied
double-difference waveform inversion on two time-lapse datasets from the Valhall field,
and resolved cleaner and more interpretable time-lapse velocity changes compared to
those from independent inversion schemes. The results are supported by previous
studies and the synthetic tests included in this work. The non-repeatabilities of the
two surveys are mild and allow double-difference waveform inversion to invert for
credible time-lapse P-wave velocity changes.
3.7
Acknowledgments
This work was supported by the MIT Earth Resources Laboratory Founding Members
Consortium and Hess Corporation. The authors would like to especially thank Hess
Corporation for the permission on publishing this work, and BP for providing the
datasets. We also want to thank our collaborators Faqi Liu and Scott Morton from
Hess for their contributions.
96
Velocity (km/s)
0
5
E
4
N
3
2
0
1
2
3
4
5
6
7
8
9
X (km)
(a)
Velocity (km/s)
0.2
0
E
0.1
1
0
N
-0.1
0
1
2
3
4
5
6
7
8
9
-0.2
X (km)
(b)
Figure 3-1: (a) True P-wave velocity baseline model. The reservoir is located in
the anticline below the salt layers (white wedges) that have the highest velocities.
Five stars mark the source locations that are used in both baseline and monitor
acquisitions. (b) True time-lapse P-wave velocity changes. The layer is located in the
reservoir, and has an uniform velocity increase of 200 m/s, simulating a hardening
effect when the reservoir is compacting.
97
Velocity (km/s)
0
E
5
4
1
3
2
2
0
1
2
3
4
5
6
7
8
X (km)
9
(a)
Velocity (km/s)
0
5
P
N
1
4
3
2
2
0
1
2
3
4
5
X (km)
6
7
8
9
(b)
Figure 3-2: (a) The starting velocity model for FWI. The model is obtained by
smoothing the true velocity model with a Gaussian window. (b) The velocity model
obtained after 90 iterations of FWI. Details of the layers are significantly improved.
The color-scales in both figures are the same.
98
Velocity (km/s)
0
0.2
0.1
E1
0
N
-0.1
0
1
2
3
4
5
6
7
X (km)
8
9
0.2
(a)
Velocity (km/s)
0
0.2
0.1
1
0
2
-0.1
0
1
2
3
4
5
X (km)
6
7
8
9
1-0.2
(b)
Velocity (km/s)
0
0.2
0.1
E 1
0
N
-0.1
0
1
2
3
4
5
6
7
8
9
-. 2
X (km)
(c)
Figure 3-3: Time-lapse velocity changes recovered by Scheme I (a), Scheme II (b)
and Scheme III (c). The differences are obtained by subtracting the final baseline
inversion models from the final time-lapse inversion models for each scheme. The
final baseline inversion models are the same model that is recovered by the baseline
inversion. Both (a) and (b) contain strong artifacts, while (c) is clean and localized.
99
Figure 3-4: Layout of the LoFS survey. White points denote the positions of common
shots used in the acquisitions in LoFS10 and LoFS12. Blue dots denote the common
receiver positions. The missing shot lines are those with low quality in either survey.
The holes in the shot map are the locations of platforms.
100
x14
I-0- LoFS
2.5
-
10 Data
LoFS 12 Datal
C,)
0
2
L1.5
E
:
z
1
0.5
0
-15
-10
-5
0
5
Shot Positioning Error (m)
10
15
Figure 3-5: The shot positioning error distributions of survey LoFS 10 (circled line)
and LoFS 12 (solid line). The error is between the designed positions and the actual
positions measured by GPS. Both distributions have mean values close to zero. LoFS
12 acquisition is improved with a much smaller standard deviation of less than 2
meters.
101
(a)
(b)
Figure 3-6: Traces from LoFS 10 (white line) and LoFS 12 (yellow line) are plotted
together to show their similarity. All traces are from the same common-receiver
gather. The pair from a near offset shot is plotted in (a), and the pair from a far
offset shot is plotted in (b). The strong phases like the diving waves and direct waves,
and the coda waves match well between surveys.
102
(a)
(b)
Figure 3-7: (a) Initial model for baseline FWI obtained by smoothing the model built
by [53] using a combination of FWI and tomography. (b) Baseline model obtained
after 200 iterations starting from (a). The shallow structures are improved with
higher resolution. The black arrow points to the gas cloud area. The low velocity
layer beneath the gas cloud that is not visible in the starting model is recovered.
103
(a)
(b)
Figure 3-8: Data residuals of one common receiver gather (a) before the baseline
inversion and (b) after the baseline inversion are compared to show the convergence
of FWI. The traces are ordered by the shot index. Residuals in far offset diving waves
(marked by the white dashed circles) and near offset reflected waves (marked by the
black dashed circles) are both reduced significantly.
104
(a)
(b)
(c)
Figure 3-9: 3D view of time-lapse P-wave velocity changes resolved by Scheme I (a),
II (b) and III (c). The slices are at the same coordinates as those in Figure 3-7.
105
(a)
(b)
(c)
Figure 3-10: X-Y slice at the depth where the maximum time-lapse changes occur.
Time-lapse P-wave velocity changes resolved by Scheme I (a), II (b) and III (c) are
compared. Note that the color-scale in (b) is larger than those in (a) and (c) meaning
stronger magnitudes. Black squares show the locations of platforms. Note the better
focusing of time-lapse changes with Scheme III.
106
(a)
(b)
(c)
Figure 3-11: Y-Z slice at the location where maximum time-lapse changes occur along
the X-axis. Time-lapse P-wave velocity changes resolved by Scheme I (a), II (b) and
III (c) are compared. The Scheme I result (a) shows changes of similar magnitude at
both shallow and deep locations. The Scheme II result (b) has fewer shallow changes
but contains strong and broad changes in the deeper part. The Scheme III result (c)
shows localized changes in the layer underneath the gas cloud. The gas cloud region
is marked with a black dashed circle.
107
Monitor Data
Yes
Can we model it?
No
Signal
Time-lapse
Noise
Baseline Signal
Signal
Yes
Is it explained by the
Random Noise
No
Yes
Coherent Noise
Is it physical?
No
baseline model?
Baseline
Modeling
Nonrepeatability
Figure 3-12: The decomposition of the monitor dataset. The monitor data can be
separated into two branches by the modeling capability. The parts that can be simulated by the modeling engine are considered as signal, while the rest is treated as
noise. In the signal branch, part of the baseline signal can not be explained by the
current baseline model due to the imperfection of the baseline inversion. This part
would generate artificial time-lapse changes in Scheme I and II, but will be canceled
in DDWI. In the noise branch, these non-repeatable components will remain in all
schemes, but the repeatable components will be canceled in DDWI.
108
Chapter 4
Alternating Time-lapse Full
Waveform Inversion with Different
Survey Geometries
Summary
The repeatability of acquisitions is a key factor determining the success of conventional time-lapse analysis because the background data need to be cross-equalized
to highlight the time-lapse signals. To discriminate unchanged and changed model
parameters, FWI approaches with differenced data also require the surveys to be wellrepeated (e.g., double-difference waveform inversion in Chapter 2 and Chapter 3).
In this chapter, we present a different way to achieve this discrimination. Instead of focusing on acquisition quality and data processing, we regularize the model parameters
during inversion to emphasize the time-lapse changes. To obtain the regularization
parameters, a confidence map of time-lapse changes is constructed from the convergence curves of model parameters that are obtained by fitting baseline and monitor
datasets alternately. Synthetic examples are used to demonstrate the robustness of
the method to different time-lapse survey geometries.
109
4.1
Introduction
To differentiate the real time-lapse changes from the background model (i.e., the
baseline model), inversions with differential datasets are being developed by others [107, 119, 112], and in Chapter 2 and Chapter 3. However, these methods
require highly repeatable surveys because the datasets are subtracted to highlight
the time-lapse signals before inversion. The data differences caused by survey nonrepeatability can be much stronger than the real differences induced by subsurface
changes. Therefore, careful survey design and cautious data co-processing [74] are important to the success of both conventional time-lapse analyses and inversions with
differenced data.
However, in some situations, non-repeatability can be severe enough to break down
all the methods mentioned above. As discussed in [116] and Chapter 2, among nonrepeatability issues, random noise is the most pervasive, but the least concerning
factor in double-difference waveform inversion (DDWI), which is one kind of timelapse FWIs that uses differenced data. Source wavelet discrepancies can be mitigated
by spectrum shape filtering. These two issues can be successfully mitigated by data
co-processing [74]. Survey geometry differences can be minimized by interpolation
when the source and receiver spatial samplings are dense enough, and the survey
coverages overlap approximately.
When surveys are not designed to be repeated
(e.g., a legacy streamer baseline dataset and an ocean bottom cable (OBC) monitor
dataset), it is difficult to regularize the datasets and preserve the real time-lapse
responses. When receivers are sparse like in ocean bottom node (OBN) acquisitions,
interpolating the wavefields is unreliable. Overburden changes technically are not nonrepeatability issues but real time-lapse changes. Nonetheless, as shown in Chapter 2,
when overburden changes are significant, DDWI introduces artificial model changes.
To mitigate the problems mentioned above, significant resources are devoted to
repeating seismic surveys, especially to repeating the survey geometries. Time-lapse
data processing to further improve the repeatability is also costly. In addition, a
perfectly repeated survey loses the opportunity to explore additional coverage of the
110
subsurface. The time-lapse surveys could be designed to cover the area of potential
changes and to maximize the marginal illumination. If we look at the time-lapse problem from the perspective of model changes, the surveys do not have to be repeated.
Instead of trying to highlight the time-lapse signals, the ultimate target should be
highlighting the model changes, both inside and outside of the reservoirs. This could
be achieved directly in the model space, without regularizing the data.
In this work, we present a framework for time-lapse waveform inversion which
jointly inverts baseline and time-lapse datasets that are collected with different survey
geometries.
In this framework the background model is improved by exploiting
information from both datasets, and the time-lapse changes are differentiated from
the background. At the same time, the framework provides a confidence level map
to show how reliable the results are. We first describe the theory of the method,
then use the Marmousi model to demonstrate the performance of the method under
various acquisition conditions.
4.2
Theory
For stand alone FWI, the cost function to be minimized is:
E(m) = 1JF(m) - d1 2 7
(4.1)
where F is the forward modeling operator, m is the model to be recovered, and d is
the observed data. The model parameters m can be iteratively updated using the
adjoint method [68] as the synthetic waveforms F(m) fit d better.
In a time-lapse situation, we use the cost function
E(mo, mi) = 1jF(mo) - d0 12 +
IF(mi) - d1 j2 + 1| (MO - in)
12,
(4.2)
to include both datasets. The first two terms in Equation 4.2 are data misfit functions,
like Equation 4.1, in which mo and do are the baseline model and data, and m, and
d, are the time-lapse model and data, respectively. The parameter 3 is a confidence
111
map of time-lapse changes. If 3 is uniform everywhere, it means we do not have any
knowledge about where the most probable changes are, and the term
(MO-m) 2
will
equally penalize all the possible changes. A better / has different weights at different
locations. At places where we are more certain about the existence of time-lapse
changes, / should be bigger. By contrast, / should be very small where we believe
there should not be changes between surveys. In this way, we are able to resolve real
time-lapse changes, and suppress spurious model differences.
The other way to interpret / is to consider it as a prior information. In [119]
who showed that differential inversion results can be markedly improved with the
knowledge of the location of time-lapse changes.
The a prior information can be
derived from non-seismic data (e.g. well logs, reservoir simulations) when the area
of interest is a local target. However, the purpose of monitoring is not only to verify
the changes within targeted regions, but also to find unexpected changes that help
prevent accidents. A priori information could also be derived from the seismic data
themselves, which does not limit the scope of searching for changes.
Therefore, we would like to introduce a seismic data driven approach to obtain /.
We break the joint inversion in Equation 4.2 into a process which minimizes the cost
function Eo(mo) = 1)F(mo) - do1 2 and E1 (mi) = 1IF(mi) - d 1j 2 in an alternating
manner: (1) for a given starting model mi, we minimize EO for n iterations, and get
an updated model mi+1 ; (2) then we use mi+ 1 as the starting model, and minimize
El for n iterations and get an updated model mi+2 , and repeat step (1) and (2) for
a number of rounds. We refer to this process as alternating FWI (AFWI).
Figure 4-la illustrates a time-lapse acquisition geometry where the source location
is different from that in the baseline. The receivers, shown as the dashed line, are
assumed to cover the same area. This mimics an OBC acquisition where the cables
are not moved, but the sources are. With the process above, we would obtain a
series of baseline and time-lapse models. If the background structure and the timelapse change (in Figure 4-la) are illuminated by both surveys, the parameters in
the background structure should converge monotonically to the true values, while
the parameters in the time-lapse change area should exhibit strong oscillations. If
112
we plot the convergence curves of the two types of parameters, they would have the
patterns as shown in Figure 4-la. We can derive
One example formula for
/
3 from
the convergence patterns.
is:
1
jEjsgn(mj+j - mi)V
One may choose other formulas to calculate
/,
(4.3)
as long as an oscillating pattern leads
to a large number and a monotonic pattern leads to a small number. In the presence of
noise, we need a more robust way to build the confidence map because non-repeatable
noise will also cause oscillations in the convergence curve. Our value for
/
would be
more reliable if we include the amplitude of the oscillation in the calculation. One
example is:
/3 = E(1 -
sgn[(mj-1 - mi)(mi+1 - Mi)) - mi+1 - mil
(4.4)
With 3, we can optimize Equation 4.2 to better recover the time-lapse changes.
4.3
Synthetic Examples with Marmousi Model
In this section, the Marmousi model is used to demonstrate the advantages of our
framework over traditional inversion schemes. Figure 4-2a shows the true baseline
P-wave velocity model. To simplify the investigation, we use acoustic modeling and
inversions for P-wave velocities.
Constant density is assumed.
We consider two
scenarios representing two types of non-repeated time-lapse acquisitions. The first
one mimics the situation where the ocean bottom cable is fixed but sources are shot
differently in the monitor survey.
The illuminations of the baseline and monitor
surveys are similar. In the second scenario, the survey illuminations are very different,
but have an area of overlap. This mimics a situation in which the monitor survey is
planned for both exploration and monitoring purposes. Using the second scenario,
we also discuss the influence of random noise on the performance of AFWI.
113
4.3.1
Surveys with Shifted Sources
Figure 4-2b shows the true velocity changes in the reservoir in the time-lapse model.
In the baseline survey, 19 sources are evenly placed on the water surface (480-meter
spacing), and 400 receivers are placed on the surface covering the entire model. In
the time-lapse survey, all the sources are shifted 240 meters to the left from the
baseline survey. We generate the synthetic baseline and time-lapse surveys using a
Ricker wavelet centered at 6-Hz. The goal of the inversion process is to retrieve the
time-lapse changes as precisely as possible.
We use a Gaussian-blurred version of the true model as the starting model (Figure 4-3a). With FWI, we can successfully recover the baseline velocity model as shown
in Figure 4-3b. Here we choose to run the time-lapse inversion from the smooth model.
The inversion starting from the baseline result produces similar results.
To do the time-lapse inversion, we can conduct an independent FWI for the timelapse data starting from either the same smooth model in Figure 4-3a, or the inverted
baseline model in Figure 4-3b. Figure 4-4a shows the model difference between the
inverted time-lapse and baseline models. The velocity changes are resolved, but there
is also a footprint of the background structure in the image. We also observe strong
amplitude differences at locations other than those of true changes, which makes
interpretation ambiguous. For example, for the two spots marked by the white star
and the red star in Figure 4-4a, it is difficult to tell which is a real time-lapse change
because they are both strong in amplitude, and consistent with geology.
Following the algorithm presented in the method section, the convergence curves
at these two locations (red and white stars) are obtained with alternative baseline and
time-lapse inversions. As shown in Figure 4-5b, the white star curve asymptotically
converges to the true value. The red star curve has a distinctive zig-zag shape which
indicates that the two datasets provide conflicting information about this location.
With Equation 4.4, we can calculate 3 as shown in Figure 4-5a. It is clear that in
the region around the white star, the likelihood of time-lapse changes is very low. In
the region around the red star, brighter color indicates higher confidence about the
114
existence of time-lapse changes.
We can use this 3 and Equation 4.2 to invert the baseline and time-lapse datasets
jointly. Since we have already built up a very good background model with the previous alternating inversions, the final joint inversion with 3 converges very quickly. The
result is shown in Figure 4-4b. Compared to the changes inverted from a traditional
scheme, the joint inversion result has almost no spurious changes, and the velocity
anomaly is recovered well in both amplitude and shape.
4.3.2
Surveys with Different Illuminations
In this scenario, a different time-lapse model is used. Figure 4-6 shows the true timelapse P-wave velocity changes. The stars mark the locations for which convergence
curves will be displayed later. The sources and receivers in both surveys are placed on
the surface. In the baseline survey, 5 sources are evenly spaced from 0.33 to 6.1 km
(black stars in Figure 4-7a), and 270 receivers are placed from 0 to 6.4 km (blue
triangles in Figure 4-7a). In the monitor survey, 5 sources are evenly spaced from
3.1 to 8.8 km (black stars in Figure 4-7b), and 270 receivers are placed from 2.7 to
9.2 km (blue triangles in Figure 4-7b).
The sensitivity of Equation 4.1 to parameter perturbations (i.e., gradient) is a
good indicator of the survey illumination. In Figure 4-7, we show the normalized
gradients for both surveys to compare their illuminations. Black lines outline the
locations of the time-lapse anomalies. Due to the limited acquisition apertures, only
the anomaly in the center is illuminated by both surveys. The baseline survey has no
energy on the anomaly on the right side, and the monitor survey has no energy on
the left side. The inversion with the baseline dataset alone can only resolve the part
of the model highlighted in Figure 4-8a. The right side of the model is improved by
the inversion with the monitor dataset as shown in Figure 4-8b.
With AFWI, we obtain the confidence map for time-lapse changes (Figure 49a).
The only distinct high confidence area is the anomaly in the center.
Both
of the anomalies on the sides are of low confidence in the map because they are
not illuminated by both surveys. Figure 4-9b shows the convergence curves of the
115
sampled parameters marked in Figure 4-6. As expected, the curve of the center
anomaly (yellow star) exhibits strong oscillations. For the other two positions, only
one inversion (baseline or monitor) has significant contributions. As a result, the
curves (white and red stars) show a strong update and a weak update alternatively
like staircases, leading to low confidence levels.
4.3.3
Surveys with Strong Random Noise
As discussed in the theory section, the success of AFWI is based on the assumption
that oscillations in the convergence curves are caused by time-lapse changes. However,
this assumption could be violated in noisy data. In this test, we use the same survey
geometries as in Figure 4-7, and add random noise to the data. Figure 4-10a shows
one shot gather in the baseline survey with noise. The noise is generated with the
same power spectrum of the source wavelet and random phases. Spatial correlation
(cross-trace) is generated with a uniform power spectrum in the wavenumber domain.
The black dashed line marks the sample trace shown in Figure 4-10b. The trace with
noise is compared to the clean trace to show that the amplitude of the noise is as
strong as the reflections.
With the noisy datasets, AFWI finds the confidence map shown in Figure 4-11a.
The overall quality of the map is similar to that of Figure 4-9a. The area of the center
anomaly exhibits high confidence, while the confidence in the other two locations of
time-lapse changes is low. However, due to the presence of noise, the confidence
outside the center anomaly but within the common illumination of both surveys is
increased. In particular, a second high confidence area is detected and marked by the
black star in Figure 4-9a where no time-lapse changes exist, although it is close to a
region that does have changes.
In Figure 4-11b, convergence curves are plotted for the sampled parameters (stars
in Figure 4-6) and the mis-detected high confidence area. The curves in real timelapse changes areas have similar patterns to those in Figure 4-9b, except that the
white-star curve is slightly perturbed. The curve in the mis-detected area (black star)
shows strong oscillations which build up the confidence. This is because random noise
116
generates spurious and different model perturbations between baseline and monitor
inversions. The strong perturbations would likely be detected as high confidence areas
of changes. We use strong noise in this test. The mis-detection would be mitigated if
the noise level is lower or the correlation between noise is weaker. Using more sources
would also improve the situation by suppressing the spurious perturbations.
4.4
Discussion
How to discriminate effects of noise and real time-lapse changes is a philosophical
question. Noise can be viewed as a type of time-lapse signal because it varies between
surveys. If we understand the noise very well, we could use the characteristics of the
noise to mitigate its effects.
In most cases, noise is not fully-understood, and is
difficult to separate from signals. AFWI detects the model changes that are most
probable, but is not designed to identify whether the changes are induced by the
signal or noise.
Although AFWI is an ad hoc methodology, it detects time-lapse changes effectively
with small additional computations compared to conventional FWI. It provides a
new perspective on 4D acquisition designs. Instead of repeating the surveys, new
acquisitions can cover the targeted area that needs monitoring, and explore new
neighboring areas. For locations where acquisition conditions change, AFWI is not
restricted as long as the targeted area is covered. Nonetheless, illumination analysis
is necessary to the success of AFWI.
4.5
Conclusion
We proposed a joint inversion strategy for arbitrary time-lapse acquisition geometries.
Instead of only looking for the changes, the information in both the baseline survey
and time-lapse surveys is utilized to build the baseline model. Our numerical example
shows that from the patterns of model convergence curves, the confidence map can be
computed to provide guidance to the interpreter about how much one can trust the
117
time-lapse results. With the confidence map, time-lapse changes are better resolved
with less contamination compared to traditional methods. The framework can be
extended to inversions with multiple time-lapse datasets. The extension to multiparameter inversion requires future research.
4.6
Acknowledgments
This work was supported by the MIT Earth Resources Laboratory Founding Members
Consortium.
118
Time-lapse Shot
Baseline Shot
(:
Time-l4ape,
hange
Background
Structure
(a)
(b)
Time-lapse
Baseline
(b)
Figure 4-1: (a) Illustration of the inversion with different source locations in baseline
(yellow) and time-lapse (red) surveys. (b) Cartoon of the convergence curves of the
model parameters inside (blue) and outside (red) of the time-lapse change region.
119
True Baseline Model
Velocity (m/s)
0
5000
4000
3000
S2
2000
2
4
6
Distance (km)
8
(a)
True Time-lapse Change
Velocity (m/s)
200
0
100
0
-100
2
4
6
Distance (km)
8
-200
(b)
Figure 4-2: (a) The true baseline P-wave velocity model. (b) The true time-lapse
P-wave velocity changes.
120
Starting Model
Velocity (m/s)
0
5000
4000
3000
2000
2
4
6
Distance (kin)
8
(a)
Inverted Baseline Model
Velocity (m/s)
0
5000
4000
4-
3000
2
2000
2
4
6
Distance (km)
8
(b)
Figure 4-3: (a) Starting P-wave velocity model. (b) The baseline P-wave velocity
model inverted by FWI.
121
Inverted Changes: Direct Subtraction
Velocity (m/s)
200
-
0
100
0
-100
2
4
6
Distance (kin)
8
-200
(a)
Joint Inversion Result
Velocity (m/s)
200
0
100
0
-100
2
4
6
Distance (kin)
8
-200
(b)
Figure 4-4: (a) Inverted time-lapse changes by subtracting two independent inversions. (b) The time-lapse changes recovered by the joint inversion.
122
Confidence Map of Changes
0
I
I
02
2
4
6
Distance (kin)
8
1
0.5
0
(a)
Convergence Curves
0)
0ai)
400200
-
E
OF
It'
I0,000-
14
16
-200
.-
0)
-400
2
4
6
8
10
12
Iterations
(b)
Figure 4-5: (a) The confidence map / obtained by AFWI. (b) The convergence curves
of the parameters marked as stars in (a). To better compare the curves of different
parameters, we subtract a reference value from the parameter estimates for each
curve. The curve of the parameter within the time-lapse changes (red star) exhibits
strong oscillations. The curve of the parameter outside the time-lapse changes (white
star) is monotonic.
123
Time-lapse Changes
Velocity (m/s)
200
0
100
E
0
S2
0
-100
2
4
6
Distance (km)
8
-200
Figure 4-6: The true time-lapse changes with three anomalies. The stars mark the
positions in each anomaly for which convergence comparisons are shown later.
124
Baseline Survey Sensitivity
I
)2
0
2
6
4
Distance (kin)
8
I
0.5
0
-0.5
(a)
Monitor Survey Sensitivity
0
0.5
1
0
E
0
2
4
6
Distance (kn)
8
-0.5
(b)
Figure 4-7: (a) The gradient of the baseline cost function. The right side of the model
is not illuminated. The black stars show the locations of the baseline sources, and the
blue triangles mark the width of the receiver array. (b) The gradient of the monitor
cost function. The left side of the model is not illuminated. The black stars show
the locations of the monitor sources, and the blue triangles mark the width of the
receiver array. Black lines outline the time-lapse anomalies. Only the center anomaly
is illuminated by both surveys.
125
Baseline Survey Inversion
Velocity (m/s)
5000
4000
3000
0
2000
0
2
4
6
Distance (kin)
8
(a)
Total Inversion
Velocity (m/s)
0
5000
E
4000
3000
(D 2
2000
0
2
4
6
Distance (kin)
8
(b)
Figure 4-8: (a) The recovered model with the baseline dataset. The smooth model in
Figure 4-3a is used as the starting model. Due to the limited illumination, the right
side of the model is not inverted. (b) The recovered model with both the baseline
and monitor datasets. The whole model is resolved.
126
Confidence Map of Changes
0
I
E
I
2
0
2
4
6
Distance (kin)
8
1
0.5
0
(a)
Convergence Curves
400
E
C>
200
0
-2001
a)
-400
II
0
5
I
I
I
10
15
20
Iterations
25
(b)
Figure 4-9: (a) The confidence map # obtained with AFWI. The white lines outline
the locations of the time-lapse changes. Only the area of the center anomaly exhibits
high confidence. (b) Convergence curves of the parameters marked by the corresponding colored stars in Figure 4-6. Only the curve of the center anomaly (yellow stars)
shows strong oscillations.
127
Baseline Shot Gather
E
FiO f e
--(k
Offset
(km
(a)
Trace Sample
2
-
--
1
clean trace
noisy trace
E
0
N
CO
L0
z
A. A
AAAAAAA
1
0 aA
A
'n
0
1
2
Time (s)
3
4
5
(b)
Figure 4-10: (a) One baseline shot gather with random noise. The black dashed line
marks the location of the trace shown in (b). In (b), the noisy trace is compared to
the clean trace. The amplitude of the noise is as strong as the reflections.
128
Confidence Map of Changes
0
I
1
n-
I
(D2
0
2
4
6
Distance (kin)
8
1
0.5
0
(a)
Convergence Curves
400
E
a)
0
200F
OF*
-200 F
a)
-400'
0
I
5
a
10
Iterations
a
15
I
20
25
(b)
Figure 4-11: (a) The confidence map 3 obtained by applying AFWI on the noisy
datasets. The white lines outline the locations of the time-lapse changes. The area of
the center anomaly exhibits high confidence. Another area marked by the black star
shows relatively high confidence, but is not an area of time-lapse changes. (b) Convergence curves of the parameters marked by the corresponding colored stars in (a)
and Figure 4-6. The curve of the center anomaly (yellow stars) shows strong oscillations. The black star curve is also oscillatory, because different noise between surveys
causes conflicting parameter estimates. The red star curve decreases monotonically.
The white star curve shows very weak oscillations due to the noise.
129
130
Chapter 5
Time-Lapse Walkaway VSP
Monitoring for CO 2 Injection at
the SACROC EOR Field: A Case
Study
Summary
Geological carbon storage involves large-scale injections of carbon dioxide into underground geologic formations. Changes in reservoir properties resulting from CO
2
injection and migration can be characterized using monitoring methods with timelapse seismic data. To achieve economical monitoring, Vertical Seismic Profile (VSP)
data are often acquired to survey the local injection area. In this study, we investigate the capability of walkaway VSP monitoring for CO 2 injection into an enhanced
oil recovery (EOR) field at SACROC, West Texas. VSP datasets were acquired in
2008 and 2009, while CO 2 injection took place after the first data acquisition. Since
the receivers were located above the injection zone, only reflection data contain the
information from the reservoir. Qualitative comparison between reverse-time migration (RTM) images at different times shows vertical shifts of the reflectors' centers,
131
indicating the presence of velocity changes. We examine two methods to quantify
the changes in velocity: standard full-waveform inversion (FWI) and image-domain
wavefield tomography (IDWT). FWI directly inverts seismic waveforms for velocity
models. IDWT inverts for the time-lapse velocity changes by matching the baseline
and time-lapse migration images. We find that, for the constrained geometry of VSP
surveys, the IDWT result is significantly more consistent with a localized change in
velocity, as would be expected from a few months of CO 2 injection. A synthetic
example is used to verify the result from the field data. By contrast, FWI fails to
provide quantitative information about the volumetric velocity changes because of
the survey geometry and data frequency content.
5.1
Introduction
Public acceptance of geological carbon storage as an effective and environmentally
friendly solution to the mitigation of green house gas emission is a major prerequisite for the method to be widely implemented on the scale necessary to reduce the
atmospheric CO 2 concentration. The injected CO 2 needs to be monitored over time
to demonstrate that the fluid is contained within the targeted formation. It is also
crucial to detect fluid migration in the subsurface and potential leakage to ensure safe
and reliable storage [14]. CO 2 is usually injected into reservoirs like saline aquifers and
depleted oil and gas fields, which axe predominately water-saturated formations [13].
The displacement of water by CO 2 tends to reduce the bulk modulus and density of
the rock-pore fluid system [72]. These properties determine wave speeds changes that
can be observed using seismic methods.
Time-lapse seismic monitoring is widely used in reservoir management in the oil
industry to obtain information about reservoir changes caused by fluid injection and
subsequent production of fluids from heterogeneous reservoirs. It helps identify bypassed oil to be targeted for infill drilling, and extends the economic life of a field [57].
It is also capable of monitoring the progress of fluid fronts providing information for
injection optimization in enhanced oil recovery and long-term CO 2 sequestration.
132
Generally, one baseline survey and subsequent monitoring surveys are acquired over
time. Qualitative analysis of time-lapse seismic data gives information about the
temporal reservoir changes with amplitude maps and time-shifts at certain horizons.
Impedance contrasts and seismic response changes such as amplitude changes and
tuning effects have been used to characterize CO 2 accumulations in thin layers, and
velocity push-down effects that are caused by slower propagation of seismic waves
through the CO 2 saturated area have been identified [3]. Quantitative methods have
also been proposed to directly deliver reservoir property changes like pore pressure
and fluid saturation by linking the rock-physics modeling, reservoir simulation and
4D seismic response simulation [50, 96]. However these methods are conducted with
post-stack data or even 1D wave propagation which focus on a local region and lose
general information during the stacking process. For example, the time-lapse changes
illuminated by the seismic waves from a certain angle could be indistinct in the
stacked data. The amplitude changes axe also not well preserved after stacking without an updated velocity model. [55] proposed a high-resolution quantitative method
to estimate the volume of CO 2 underground by combining 4D seismic,EM, gravity
and inSAR satellite data, however, time-lapse seismic is used to provide qualitative
information in this process.
Full-waveform inversion (FWI) has the potential to estimate subsurface density
and elasticity parameters quantitatively [93, 101], and it is becoming more feasible
with increasing computing power. However, FWI often requires large-offset surveys
and low frequency data to resolve low wavenumber velocities. Large monitoring networks on land or on the sea floor have been successfully deployed for CO 2 monitoring [8, 2]. For small pilot carbon sequestration projects, economic considerations
mean that limited acquisition is generally employed to monitor CO 2 injection. Vertical Seismic Profile (VSP) data have been acquired in a few carbon-sequestration
demonstration projects [22]. The vertical resolution of VSP data is typically higher
than that of surface seismic data because VSP data contain higher frequencies than
surface seismic data. Unfortunately, the VSP survey geometry reduces the ability of
FWI to resolve volumetric velocity changes. In this geometry, FWI tends to produce
133
a reflectivity model like that obtained using least-squares migration. Between baseline and time-lapse surveys, amplitude changes can be transformed into reflectivity
differences between images. Kinematic information is indicated by changes in the
apparent depths of reflectors instead of direct measures of velocity changes. If a constraint that forces the locations of reflectors can be used in FWI, the velocity-depth
bias would be removed. However, we did not find an efficient way to implement such
constraints.
As an alternative to FWI, we apply an image domain wavefield tomography
method (IDWT) [113] to time-lapse VSP data. Based on the assumption that the
geology has not changed dramatically over time, both the baseline and time-lapse
seismic data should be able to image the same area in the subsurface. If the correct velocity models are provided for both datasets, the reflectors should be at the
same location assuming that reservoir compaction is negligible compared to the seismic wavelength. In an inverse problem setting, given a baseline velocity model, the
time-lapse velocity anomaly can be resolved by matching the reflector locations in
time-lapse images with those in the baseline image. The amplitude differences between images, which could be caused by reflectivity changes, are not sensitive to the
smooth (low wavenumber) velocity perturbations in the inversion.
The goal of this chapter is to investigate the practical capability of VSP reflection
data for monitoring CO 2 injection. We first introduce the theory of the imaging
and inversion methods that we apply to the time-lapse VSP data, including reversetime migration, full-waveform inversion, and image-domain wavefield tomography. In
the following sections, we describe the geologic background of the SACROC EOR
site, the injection history, and the seismic data acquisition and processing. Images
and models obtained from different methods are compared to demonstrate how they
provide different types of information about changes in the reservoir. Preliminary
interpretation is given about the mechanism of reservoir response to CO 2 injection,
and the CO 2 fluid migration at the SACROC EOR field.
134
5.2
Methodology
In this section, we briefly introduce the methods used in this study: reverse-time
migration, full-waveform inversion and image-domain wavefield tomography.
5.2.1
Reverse-Time Migration
A reverse-time migration algorithm consists of three steps: (1) forward propagation
of the source wavefield; (2) backward propagation of the receiver wavefield; (3) application of the imaging condition. The wavefield extrapolation is conducted by solving
the wave equation
1
a2
c (f) Bu(t, y)
2
-
V 2 u(t, Z) = S(t, g),
(5.1)
where u(t, 9) is the wavefield at a spatial location i and time t, c(Y) is the P-wave
velocity in the medium, and S(t, 9) is the source function. The image is constructed
by the zero-lag cross-correlation of the source wavefield u,(t, 9) and receiver wavefield
Ur(t, 9) at the image point as follows:
T
U'S(t, 9)U,(T - t, 9)dt.
1(9) =
(5.2)
0
us(t, 9) is calculated by solving Equation 5.1 with an estimated source signature
S(t, 9). Ur(t, 9) is computed by solving Equation 5.1 with the data, reversed in time,
as the boundary condition. More details about RTM can be found in [11], and [61].
5.2.2
Full-Waveform Inversion
Full-waveform inversion minimizes an objective function formed from the difference
between modeled data and field data:
E(m)
=
-|Iu - d| 2 = -6uT j,
2
2
135
(5.3)
where u and d are the waveform measurements from forward modeling, and the field
experiment, respectively, and 3u = u - d. The superscript T denotes the transpose,
and m is the P-wave velocity model to be updated. The gradient of the objective
function is derived by taking its derivative with respect to m, giving by
VmE =
u.(5.4)
The gradient can be calculated efficiently by cross-correlating the forward propagating
wavefields from the sources with the back propagating residual wavefields from the
receivers [93]. The objective function can be minimized via e.g. the Gauss-Newton
or conjugate gradient methods. Because of the computation and memory cost of
calculating the Hessian matrix [85], we use the nonlinear conjugate gradient method
as it does not require the Hessian matrix, and has a better convergence rate than the
steepest descent method [75]. The model parameters are updated in each iteration
according to
mi+1 = mi - aGi+1
(5.5)
where Gi+ 1 is the search direction defined by the gradient of the current step ViE and
the search direction of the previous step Gi [75]. The parameter a is the step length
obtained from a line search algorithm to reach the minimum cost for each iteration.
We first apply FWI to the baseline data. The model that best approximates the wave
events in the baseline data is used as the initial model in the inversion for the timelapse dataset. The differences between the inverted baseline and time-lapse models
are then used as an estimate of the effect of CO 2 injections.
5.2.3
Image-Domain Wavefield Tomography
[91] introduced the principle that if the background velocity is correct, the migrated
images with neighboring shot gathers should show the reflectors at the same depth.
In the time-lapse situation, if the subsurface interfaces have not changed in space or
the changes are much smaller compared to the seismic wavelength (e.g. weak com136
pactions), we can introduce a similar principle that if the time-lapse velocity model
is correct, the time-lapse migration images should show the same structures at the
same locations as the baseline migration images do. Hence we apply the image-domain
wavefield tomography (IDWT) to resolve volumetric time-lapse velocity changes. The
cost function is very similar to Equation 5.3, but in the image domain:
J(m) = 1I1Ibaseline(9) - Itimelapse()11 2,
(5.6)
where Ixaseline is the migration image produced with the baseline data and velocity
model, and Itmelapse is the migration image produced with the time-lapse data and
the velocity model that is being updated iteratively.
The gradient of the objective function can be efficiently calculated by two crosscorrelations:
VmJ(m)
= -
f
T
&
2
(2A,(t+) a2 A,(T - t,
Z)
*U(t,
) +
t2
*U(T - t,
g))d,
(5.7)
0
where u.,(t, 9) and u,(t, Y) are source and receiver wavefields used to form the timelapse migration image Itimelapse( ) as in Equation 5.2. A,(t, 9) and Ar(t, Y) are adjoint
wavefields computed by solving Equation 5.1 with adjoint sources. The adjoint sources
are the multiplication of the image residual Isa,eline(9) - Itimelapse (Y) and the wavefields
Ur (t, Y)
and u, (t, Y) [113]. Similar derivations can be found in [68]. We use the same
nonlinear conjugate gradient-method as used for FWI described above to update the
model. The velocity difference is resolved as the image difference is minimized.
With RTM, we transform the traveltime changes in the two datasets into depth
changes of the reflectors in the images. FWI inverts the amplitude and phase differences between the two datasets to obtain differences in reflectivity. Through IDWT,
we transform the depth changes of the reflectors between the baseline and monitoring
images into velocity differences between the two models. The amplitude differences
between images are not sensitive to the low-wavenumber velocity perturbations, which
makes IDWT focus on the kinematics. In the data application, we show the advan137
tages and limitations of these methods for time-lapse walkaway VSP surveys.
5.3
5.3.1
Site Background of SACROC
Geology and Injection History
The SACROC EOR field is located in the southeastern segment of the Horseshoe
Atoll within the Midland basin of west Texas. It is composed of several layers of
limestone and thin shale beds representing the Strawn, Canyon and Cisco Groups of
the Pennsylvanian. The Wolfcamp Shale Formation of the lower Permian provides a
low permeability caprock above the Pennsylvanian Cisco and Canyon Groups in the
SACROC Unit [40]. The limestone is mostly calcite with minor ankerite, quartz and
thin clay lenses.
Hydrocarbons have been produced from the SACROC field using the solution gas
drive mechanism since 1948 when the unit was discovered. To maintain the subsurface
pressure level and also improve the fluidity of the oil within the reservoir, the field
has been flooded with water since 1954 [25]. Although the water injection facilitated
oil recovery, tremendous reserves still remained in the field by the end of the waterflooding period. CO 2 injections were considered as the best tertiary recovery plan and
they were initiated in 1972. Over 175 million metric tons of CO 2 have been injected
into the SACROC field, and about half of that amount is assumed to be sequestrated
between depths of 1829 to 2134 feet below the surface [72]. As a part of the Phase II
project of the Southwest Regional Partnership for Carbon Sequestration, time-lapse
walkaway VSP data were collected before and after the first CO 2 injection in this
region that began in October 2008, in collaboration with Kinder Morgan, Inc. The
purpose of this project is to study the combined EOR and CO 2 storage.
Figure 5-1 shows a map of the well distribution within the area of study. Red
dots mark the wells from which we have logging data. The distribution of the wells
with logs sample roughly a northeast-southwest trend. Green squares denote the two
injection wells (56-4 and.56-6). The monitoring well (59-2) at the blue star is to the
138
north of the injection wells. The black circle encloses an area of one kilometer in
radius. The injection well 56-6 is 350 meters away from the monitoring well 59-2.
In between the two VSP surveys, CO 2 was injected in wells 56-4 and 56-6 at two
intervals (centered at depths of approximately 1980 and 2040 meters) [20].
5.3.2
Well Logs and Reservoir Properties
Well logs such as gamma ray, resistivity and sonic velocity can be combined to verify
the formation qualities within the range of interest. The gamma ray logs from well
37-11, 59-2a and 56-23 are shown in Figure 5-2. The interval with relatively high
gamma ray values (green blocks in Figure 5-2) is interpreted to be a Wolfcamp shale
formation, which is the cap-rock for the CO 2 sequestration process.
The wells in
Figure 5-2 are along an approximate NE-SW trend through the injection wells. The
shale formation gets slightly thicker and deeper towards the southwest.
Figure 5-3 shows resistivity, porosity and sonic velocity logs in well 59-2a which
is close to the monitoring well. Combining all the readings in Figure 5-3, and the
gamma ray log of 59-2a in Figure 5-2, we can estimate the thickness and depth
of the shale formation at the well location.
As indicated by the green blocks in
Figure 5-3, the lower bound of the shale formation is at 1900 meters, from where
the reservoir formation starts. Low resistivity values indicate that very little organic
matter remains in the reservoir and shale formations. At the interface between the
shale and the limestone, there is a thin layer of high resistivity, which is interpreted
to be residual gas. It is also a proof of the good quality of the shale formation as a
cap-rock with very low permeability. The relatively high porosity (10% to 15%) of the
limestone makes it a good candidate for CO 2 storage. Since the field has previously
been flooded with water, the overall geology of the injection zone is comparable to
a large class of potential brine storage reservoirs. The two injection intervals are
located within the reservoir layer.
139
5.4
5.4.1
Seismic Imaging and Inversions
Data Acquisition and Processing
The schematic configuration of the surveys is shown in Figure 5-4. The walkaway
VSP source line is oriented along the north-south direction marked by a blue dashline in Figure 5-1. It intersects the monitoring well location. The injection well 56-6
is slightly off the survey line. Two walkaway VSP datasets were acquired using the
same well (59-2) in July 2008 and April 2009. The baseline data were acquired before
the CO 2 injection that started in September, 2008. Each survey consists of one zerooffset VSP, two far-offset VSPs (with offsets 1143 meters and 848 meters), and one
walkaway VSP. Vibrators were used as sources, and were spaced at an interval of 37
meters, with a total of 100 shot points. The data were collected in the monitoring
well (59-2) using 13 receivers at depths ranging from 1555 to 1735 meters, spaced at
an interval of 15 meters. Between the two surveys, CO 2 was injected through two
injection wells (56-4 and 56-6 in Figure 5-1). We use the best-quality data from 97
shot points in this study.
The raw datasets were carefully processed by Cambridge Geosciences, Ltd. As
illustrated in Figure 5-4, the downgoing waves do not contain any information from
the reservoir because all the receivers are located above the reservoir layer.
The
downgoing waves and upgoing waves of the VSP data are separated using median
filters [20]. We use the traveltimes of the downgoing waves to constrain the upper
part of the velocity model. Static corrections are applied to compensate for the lateral
heterogeneities of the weathered zone.
The amplitudes of upgoing waves in the 2009 dataset are different from those in the
2008 dataset. Based on the assumption that the geologic structures and physical properties have not changed above the reservoir (e.g. no earthquakes and compactions),
the first reflection that is from the top of the shale formation should be identical in
both datasets. [102] conducted amplitude balancing on the common-receiver gathers
of upgoing waves using the spectral ratios of the first wavelets (the first reflection).
After the amplitude balancing, we align the first arrivals in the time-lapse dataset
140
with those in the baseline dataset to eliminate traveltime inconsistencies.
Figure 5-5 shows the processed common-receiver gathers collected by the receiver
at a depth of 1585 meters. As expected, the first reflection signals from both data
sets have the same amplitude and traveltime. The small time-shifts of later events
between two datasets are the time-lapse signal that we want to invert for velocity
changes between the two surveys. There is no clear observation of new scattered
waves in the time-lapse data. The dominant time-lapse effect is manifested by the
small phase shifts.
We conduct our imaging and inversions in two-dimensional space. The amplitudes
of the data are compensated for the difference between 3-D and 2-D geometric spreading by applying a T-gain (multiply the data by Vt where t is time). In addition, the
waveforms from 2-D propagation contain a ir/4 phase shift, so we adjust the phase of
the data to ensure that there is no phase shift when comparing the synthetics to the
data.
5.4.2
Initial Velocity Model
Since the shear-waves are weak in the vertical components of the VSP data, all the
methods we applied in this study use the acoustic assumption and so only the Pwave velocity model is used to propagate the data. We use a layered P-wave velocity
model obtained from the zero-offset VSP data and sonic logging data to build the
initial model. The 1-D layered model and the sonic log at well 59-2 are shown in
Figure 5-6. From the surface to the maximum depth of the zero-offset VSP receivers,
(which is 1737 meters), we build a blocky layered velocity model using the direct wave
travel-time at each zero-offset VSP receiver. For structures deeper than 1737 meters,
we use the sonic logging in well 59-2 to build a smooth velocity model by applying a
moving average window as follows:
z+w/2
V(z) =
V_'
141
,
(5.8)
where w is the width of the moving average window, v(i) is the sonic velocity value at
depth of i meters from the well log and V(z) is the averaged velocity at depth z. Here
w is 110 meters which is the P-wave wavelength at the data center frequency 45 Hz
with 5000 m/s velocity. The logging profile only reaches to a depth of 2134 meters,
which is still shallower than the depth where the strong reflections (around 0.8 second
shown in Figure 5-5) occurred from below the reservoir. We linearly extrapolate the
model beyond 2134 meters. The density model is directly built from the density
profile in the logging data.
5.4.3
Reverse-Time Migration
Figure 5-7a and Figure 5-7b show the RTM images produced with the 2008 and 2009
datasets using the initial model. Two major features are clearly distinguished in both
images: the interface between the shale formation and the reservoir, and a deeper
reflector. The location of the first reflector is at 1900 meters, which is in agreement
with the depth of the top of the reservoir inferred by the logging in Figure 5-3. The
length of the first reflector in both images is about 200 meters. The length of the
second reflector at about 2300 meters is about 600 meters. There are several factors
that contribute to this difference. First, the geometry of the survey results in the
recording of reflections of a wider aperture from a deeper reflector. Second, we use
only the vertical component of the 3C VSP data in this study. For a flat interface, it
is clear that for P-waves, larger reflection angles lead to weaker vertical signals at the
receiver. Third, some of the wave energy is converted to S-waves at the reflectors.
The shallower reflector causes more conversion because of the larger reflection angle
at the edges. All three effects combined give rise to weaker signals from the shallower
reflector compared to those from the deeper reflector as offset increases as shown in
Figure 5-5. RTM as a linear stacking process shows weaker reflectivity with weaker
signals. That partially explains the significant difference between the lengths of the
reflectors in our images, however, other amplitudes factors like attenuation, and the
reflectivities as a function of angle may also contribute to the observed differences.
The location of the lower reflector in Figure 5-7b is shifted slightly downwards
142
compared its position in the image in Figure 5-7a. To further investigate the shift, we
plot a few columns of the images as traces in Figure 5-8, in which blue lines are from
Figure 5-7a and red lines are from Figure 5-7b. The tops of the first reflectors are
matched showing almost no shifts. The magnitude of the shift accumulates as depth
increases, and plateaus below around 2100 meters. This implies a velocity decrease
below 1900 meters during the time between the two VSP surveys.
Figure 5-9 shows a direct subtraction of the 2008 and 2009 migration images. The
differences at the deeper reflector dominate the image. However, as demonstrated in
Figure 5-8, the differences are caused primarily by the slight shifts between the two
images. The bigger amplitudes of the deeper reflector lead to the bigger amplitudes
in the image difference. It is misleading to interpret the subsurface changes directly
from the image subtraction, because the location of true subsurface changes is not
directly linked to the location of the image differences. The differences below the
reservoir due to the misalignment are because the time-lapse velocity model is not
updated. In the following sections, We update the model using FWI and IDWT.
5.4.4
Full-Waveform Inversion
Before applying FWI to the data, we need to make a few assumptions. First, without
a good estimation of the S-wave velocity model, we invert for only the P-wave velocity
model. Second, the available data are measurements of the vertical components of
the particle velocity measurements, so in the cost function (Equation 5.3), we only
minimize the differences between the vertical components of the'synthetics and field
data.
Starting from the model in Figure 5-6, we invert for the baseline model with the
data from 2008. The result is shown in Figure 5-10a. Structures that are similar to
those in Figure 5-7a are resolved. The lengths of the reflectors are extended compared
to those in the migration images. The ratio of lengths between the shallower reflector
and the deeper reflector is increased, which is more reasonable for the migration
image of a layered model. Unlike the linear stacking in migration, FWI compensates
for the weak vertical components of the signals by taking into account the effects of
143
the survey geometry, and decreasing amplitudes with increasing reflection angle. In
other words, the effect of the incomplete data (only vertical components) is mitigated
in FWI.
One additional reflector at around 2100 meters, which is interpreted to be the
bottom of the reservoir formation, is clearly resolved. In the corresponding migrated
image (Figure 5-7a), this reflector is visible but very weak in amplitude.
This is
because the reflectivities are not correctly balanced in RTM. By contrast, in FWI,
the reflectivities are closer to true amplitude.
Although the image of the subsurface is markedly improved, FWI is not successful
in resolving the smooth (low wavenumber) velocity changes. Figure 5-10b shows the
P-wave velocity model found using FWI on the data in 2009 and starting from the
model obtained from the 2008 data. Similar to the RTM results, the 2009 model is a
slightly downward shifted version of the 2008 model. Figure 5-11 shows differences in
the models obtained by subtracting the 2008 model from the 2009 model. Compared
to the image difference in Figure 5-9, the differences in the interval of 1900-2000
meters and 2300-2400 meters are comparable in amplitude. However the differences
are oscillating rather than smooth. With the walkaway VSP survey geometry, and
only reflected waves used, FWI reduces to a least-squares migration that gives only a
reflectivity model based on the background kinematics from the initial model [68]. The
traveltime delay in the data is mapped to a depth shift rather than a velocity change
in the inversion. With such small offsets and high-frequency data, the ambiguity
between interval velocity and reflector depth is difficult to eliminate.
5.4.5
Image-Domain Wavefield Tomography
From the migrated images, we estimate that the maximum depth shift is about 3
meters. It is very unlikely that the reservoir would have compacted this much in
between the two surveys (i.e. in 10 months) when oil production and CO 2 injection
occurred simultaneously. The injection of CO 2 increases the pore pressure, preventing
significant collapse of the reservoir rock. Moreover, if the reservoir top remains at the
same location, and the lower reflectors sink, like we observed in the migrated images,
144
it actually means the reservoir layer is stretched by 3 meters, which is even more
unlikely. Since the physical displacements of the interfaces are not expected to be
this large, the traveltime delay is more likely caused by a P-wave velocity decrease.
To invert the traveltime change for the amount of velocity change, we apply IDWT
to the time-lapse walkaway VSP data from the SACROC EOR field.
Figure 5-12 shows the velocity changes resolved by IDWT. The most prominent
feature is the low velocity zone below 1900 meters. It indicates that the CO 2 has
probably migrated from the injection well toward the monitoring well. The top of
the velocity changes is right beneath the caprock. The initial injection was between
depths of 1980 meters and 2040 meters. It is possible that the CO 2 migrated upwards
because of buoyancy and accumulated at the bottom of the caprock, resulting in
local velocity changes. The length of the velocity anomaly is about the same length
as the reflectors in the RTM images (Figure 5-7a and 5-7b).
As described in the
methodology section, IDWT inverts for velocity changes by matching the time-lapse
reflection image with the baseline image. The extent of the recovered velocity anomaly
is constrained by the extent of the images. The area imaged in this survey is only
around the monitoring well, which is about 350 meters away from the injection well
56-6. If what we resolve in Figure 5-12 is real, the velocity changes are spreading
over the area between the two wells (59-2 and 56-6) because the reservoir formation
is permeable and connected.
To verify the result in Figure 5-12, we use a synthetic example as a benchmark.
The 1-D layered model is used as the initial model to build the upper part of the
synthetic baseline model (from 0 to 1500 meters). From 1500 meters to 3000 meters,
we construct the layers according to the image in Figure 5-7a. Figure 5-13 shows the
lower part of the model in which the low velocity layer represents the shale formation.
For the time-lapse model, we assume that a low velocity anomaly is caused by the
injected CO 2 flooding from the right side of the model to the area adjacent to the
monitoring well as shown in Figure 5-14. Synthetic data are generated using finitedifference wave-equation modeling with the shot-receiver geometry exactly the same
as that of the time-lapse walkaway VSP surveys at the SACROC EOR field. Figure 5145
15 shows the RTM result with baseline data. Figure 5-16 shows the velocity anomaly
reconstructed using IDWT. It is clear that the velocity changes are well bounded by
the length of the reflectors and their vertical spacing. The velocity change to the
right of the image is not recovered at all because of the acquisition geometry.
When the velocity is corrected using IDWT, the spatial shifts at the deeper reflector are eliminated. As a result, there are no anomalies present below the reservoir
in Figure 5-16. In the SACROC case, the velocity changes at the deeper reflector are
significantly weaker compared to the strong differences in Figures 5-9 and 5-11. However, the inverted model is not as clear as the synthetic result. Several factors might
contribute to these differences. One issue is the noise in the data. The time-lapse
image is not an exact shifted version of the baseline image. Some differences between
the images caused by noise are also minimized in IDWT, giving rise to the scattered
velocity anomalies. Another issue is that the size of the low velocity anomaly in Figure 5-12 is not big enough (limited width) to correct for all the shifting effects at the
deeper reflector. Because of the acquisition geometry, some of the delay in the image
is thus converted to local velocity updates. As a comparison, the velocity anomaly
in the synthetic case (Figure 5-16) is wide enough to account for most of the deeper
time delays. Hence there are almost no anomalous velocity updates in the deeper
part of the model.
5.5
Discussion
The dominant time-lapse effect we observe in the time-lapse walkaway VSP surveys
at the SACROC EOR field is a traveltime delay. Similar observations have been
reported in several other papers [3, 4, 22]. If the physical displacements of subsurface
structures are relatively small, most of the traveltime delay is presumed to be caused
by seismic velocity changes induced by the injections. There have been laboratory
measurements of the velocity decrease of rock samples with different levels of CO 2
saturation [72]. If we are able to retrieve the velocity changes quantitatively, it will
.
be possible to investigate fluid migration and the mineralization of CO 2
146
To track the movement of the fluids, the time-lapse migration images can give
qualitative information about the location of changes in the horizontal direction. As
we observe from our RTM results, RTM converts a traveltime delay to a subsidence
of the migration images beneath the top of the reservoir. It is inaccurate to use RTM
images or image differences to interpret the changes in depth if the seismic velocity
model is not updated after the injections.
Although FWI is considered an effective method of inverting seismic data for velocity models, in the SACROC walkaway VSP surveys, all the receivers are aligned
in one monitoring well. For the area beneath the receivers, the ambiguity between
depth and velocity is hard to reconcile without additional information. The FWI
results with the SACROC VSP data are reflectivity models suffering from the same
problems as RTM, despite the fact that the quality of the images is improved through
the optimization process. Other downhole surveys like cross-well and transmission
VSP [22] have been successfully utilized to do tomographic inversions for CO 2 monitoring. If more monitoring wells are employed, FWI may be capable of recovering
tomographic velocity changes with VSP reflection data. Fewer receivers in two wells
may be better able to resolve tomographic changes than more receivers in a single
well.
In this study, IDWT successfully resolves the P-wave velocity changes within the
reservoir layer. It is also clear that the quality of the IDWT result depends on the
quality of the migration images. FWI does improve the image quality, however, to
use FWI as an imaging operator in IDWT is too computationally expensive. As we
discuss in the RTM result section, the horizontal components of the particle velocity
measurements can improve the image by compensating for small signal amplitudes
from far offsets. To suppress the noise in the IDWT result, a preconditioning or
filtering of the images might mitigate the influence of amplitude mismatches. Future
research is needed to improve the performance of IDWT.
With the assumption that there is no compaction within the reservoir and the pore
pressure stays approximately the same, we can give a rough estimation of the P-wave
147
Kmin
Kdry
Kbrine
Kco 2
80 GPa
41 GPa
3.4 GPa
0.2 GPa
Pmin
Pbrine
2.75 g/cc 1 g/cc
PC0 2
#
Table 5.1: Rock and fluid properties derived from well logs. Symbols are defined as
in Equations 5.9 and 5.10.
0.85 g/cc
10%
velocity change due to a simple fluid substitution using Gassmann equation [34, 104]:
Ksat = K
(1
+
dry
Kfp
Kdry )2
,
"
Kmnin
(5.9)
Ki
where Ksat, Kmin, Kdry, Kfp are the bulk moduli of the saturated rock, the forming
minerals, the dry rock and the fluid, respectively. 4 is the porosity. The density of
the saturated rock is given by:
Peat = Ppf 4' + Pmin(1 -
4),
(5.10)
where Pat, pfp and Pmin are the densities of the saturated rock, the fluid and the forming minerals. If we assume a simple process of CO 2 replacing brine in the reservoir,
the velocity change can be derived by changing values of Kfp and pf, in Equation 5.9
and 5.10. Based on the well log information, we obtained the parameters in Table 5.1.
Then, the calculated P-wave velocity change is about 250 m/s, which is very close to
our IDWT result.
To further link the velocity changes to quantitative measures of CO 2 content,
production data and a good reservoir simulator should be used to calibrate the seismic
inversion results and to obtain the reservoir parameters like pore pressure and fluid
saturation. Although we have not been able to do this here, it remains an important
topic of future research.
148
5.6
Conclusions
We have applied the image-domain wavefield tomography method to time-lapse walkaway VSP data acquired at the SACROC EOR field for monitoring CO 2 injection.
Our inversion result shows a velocity decrease within a region beneath the top of the
reservoir. This may indicate where the injected CO 2 migrated. For image-domain
wavefield tomography, data processing and balancing must be conducted carefully
to suppress amplitude inconsistencies and preserve time-lapse signals. The high frequency of the data gives high image resolution, but relatively small aperture limits
the monitoring range. Neither reverse-time migration nor full-waveform inversion is
able to quantify the localized velocity changes, which are indicated by depth shifts
of certain reflectors in reverse-time migration images and full-waveform inversion results. Image-domain wavefield tomography can resolve a localized low velocity zone
consistent with the geology and the injection pattern, which is interpreted to be the
most likely change induced by the CO 2 injections.
5.7
Acknowledgment
The authors wish to thank MIT Earth Resources Laboratory Consortium members
for supporting this research. I started this work at Los Alamos National Laboratory
(LANL). LANL's work was supported by the U.S. Department of Energy through
contract DE-AC52-06NA25396 to LANL. LANL's work was part of a research effort
in collaboration with Kinder Morgan, Inc. and the Southwest Regional Partnership
on Carbon Sequestration that was supported by the U.S. Department of Energy
and managed by the National Energy Technology Laboratory. We also want to thank
Lianjie Huang from LANL for his insightful comments and contributions to this work.
149
S37-11
*59-5
59-2A
59-2
56-170
U
56-4
59-2-ST
I
56-16
M|
56-6
eI
56-23
Figure 5-1: Schematic illustration of walkaway VSP surveys and CO 2 injection and
monitoring wells at the SACROC EOR field. The red dots denote the wells with
logging records. The green squares denote the two CO 2 injection wells. The blue star
marks the VSP monitoring well where downhole receivers are installed. The black
circle has a radius of 1 km. The blue dashed line is the walkaway VSP source line.
150
Well 59-2a
Well 37-11
Well 56-23
0
0
01
500
500
500
1000
1000
1000
1500
1500
1500
2000
2000
0
2500'
0
100
200
25001
200
100
0
Gamma Ray (API unit)
25001
0
100
200
Figure 5-2: Gamma ray logs from three wells: 37-11, 59-2a, and 56-23. Green blocks
mark the interval of the Wolfcamp shale formations that have high Gamma ray values.
151
Well 59-2a
0
01
0
500F
1000
1000
1 000-
1500
15001
1500
2000
2000-
2000
1I~
2500'
0
100
200
Resistivity (Ohmm)
-
500
500
2500'
0.5
0
Porosity
25001
4000
6000
Sonic Velocity (m/s)
Figure 5-3: The resistivity, porosity and sonic velocity profiles from the logging record
at well 59-2a. Green blocks mark the interval of the Wolfcamp shale formation. The
carbonate reservoir is beneath the shale formation. It is clear that the interface
between the shale and the carbonate is at 1900 meters.
152
0
Shot Series
I
S
I
500
I
I
I
I
II
Monitqr ng Well
I
I
Dowgolig Wave
C.,
Injection Well
1000E
0-
I
1500
hin Wave
I
I
Receive
'
,
2000F
Reservoir
2500'
C
--
I
3000
2000
1000
Distance from First Shot (meters)
I
4000
Figure 5-4: The schematic configuration of a VSP survey. The injection well is slightly
out of the plane. Black and red dashed lines illustrate the downgoing (black) and
upgoing (red) portions of paths for waves propagating from sources to receivers. The
blue dashed line sketches the reservoir location.
153
Data Recorded at the Depth of 1585 meters
0.5111111
1'
0.6
0.7
0.8
0.9
Co)
E
1
1.1
1.2
1.3
1.4
1.5-
-1500
-1000
-500
0
Offset (in)
500
1000
1500
Figure 5-5: The processed common-receiver gathers of the data in 2008 (blue) and
2009 (red). The receiver is at 1585 meters in well 59-2. The datasets are balanced
in amplitude and traveltime using their first reflections. The traveltime differences in
the later arrivals are the time-lapse-change signals.
154
Logging Velocity Profile and Initial Model
3.5 -
CL
-
-
uta
o i
og
n
25
-0
0.5
1.5
I
2
2.5
Depth (km)
Figure 5-6: Black line: the sonic velocity profile from logging records in well 59-2a.
Red line: the initial model built using the zero-offset VSP and the sonic velocity
profile.
155
Migration Image with Data in 2008
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
-0.5
-0.4
-0.3
-0.2
-0.1
0.1
0
Offset (km)
0.2
0.3
0.4
0.5
0.3
0.4
0.5
(a)
Migration Image with Data in 2009
1.8
z
.
2.1
0S
2.2
2
2.5
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
Offset (km)
0.2
(b)
Figure 5-7: (a) RTM image produced with data from 2008; (b) RTM image produced
with data from 2009. Both images show the local layered structures. The shorter
reflector is at 1900 meters that is the top of the reservoir. For the 2009 image, the
reflector below the reservoir is shifted slightly downwards compared to the baseline
image.
156
Comparison between Migration Images
1.8-
1.9-
2-
2.2-
-
2.3-
2.4-
2.5-
-0.1
-0.05
0
Offset (km)
0.05
0.1
0.15
Figure 5-8: Sample traces from the RTM images of 2008 (blue) and 2009 (red). The
lower reflectors in the 2009 image are shifted downwards compared to the 2008 image.
157
Migration Image Difference
1.8
1.9
2.1
-C
0)
2.5
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
Offset (km)
0.2
0.3
0.4
0.5
Figure 5-9: The image difference by subtracting the RTM image of 2008 from that
of 2009. The changes at the deeper reflector are stronger than those in the reservoir
layer.
158
Velocity (m/s)
FWI Result with Data in 2008
5800
5600
5400
5200
5000
4800
4600
0
4400
4200
4000
3800
3600
0
Offset (kin)
0.3
0.2
0.1
0.4
0.5
(a)
Velocity (m/s)
5800
FWI Result with Data in 2009
5600
5400
5200
5000
4800
2.
4600
4400
4200
4000
3800
3600
2.5
-0.5
-0.4
W-10.3
-0.2t
-0.1
V
Offset (krn)
V.
s.e
(.k
vm v.)
(b)
Figure 5-10: (a) The P-wave velocity model reconstructed using FWI with data from
2008; (b) The P-wave velocity model reconstructed using FWI with data from 2009.
Both models contain similar structures. The 2009 model is shifted slightly downwards
compared to the 2008 model.
159
Velocity Difference
Velocity (m/s)
150
1.8
1.9
S100
2
50
2.1
0
2.2
-50
2.3
.
Aft
impI -100
2.4
-150
5
-0.4
-0.3
-0.2
-0.1
0
Offset (km)
0.1
0.2
0.3
0.4
0.5
Figure 5-11: The P-wave velocity model difference obtained by subtracting the model
of 2008 from that of 2009. The changes in the reservoir layer is comparable in amplitude with those at the deeper reflector. The changes are oscillating rather than
smooth.
160
Velocity (m/s)
300
IDVI Result
1.8
200
1.9
100
2
2.1
0
2.2
-100
2.3
-200
2.4
-0.5
-0.4
-0.3
2.5
-0.2
-0.1
0
Offset (km)
0.1
0.2
0.3
0.4
0.5
-300
Figure 5-12: The P-wave velocity changes reconstructed using IDWT. A smooth lowvelocity zone is resolved within the reservoir. Some scattered velocity changes caused
by image noise are also observed.
161
Velocity (m/s)
Baseline Model for Synthetic Test
5400
1.6
5200
5000
1.8
4800
'0'
4600
2.2
4400
4200
4000
2.6
-0.8
-0.6
-0.4
-0.2
0
Offset (kin)
0.2
0.4
0.6
0.8
3800
Figure 5-13: A synthetic layered model with the same geometry as the SACROC
model. The blue layer is the shale formation, below which is the reservoir layer (red).
162
Velocity (m/s)
Time-lapse Velocity Change for Synthetic Test
200
150
1
inn
1.8
50
0
ci,
-50
2.2
AW -100
2.4
-150
2.61
1
-0.8
-0.6
-0.4
-0.2
0
Offset (km)
0.2
0.4
0.6
0.8
1
-200
Figure 5-14: The synthetic P-wave velocity change caused by a fluid injection into a
borehole located on the right side of the model.
163
Baseline Image for Synthetic Test
0
a
2.2
2.4
2.6
-1
-0.8
-0.6
-0.4
-0.2
0
Offset (km)
0.2
0.4
0.6
0.8
1
Figure 5-15: The baseline RTM image obtained using one common-receiver gather.
164
Velocity (m/s)
200
Inverted Time-lapse Velocity Change
150
inn
1
50
2
0
-50
2.2
-100
-150
2.61
-1
-0.8
-0.6
-0.4
-0.2
0
Offset (km)
0.2
0.4
0.6
0.8
-200
Figure 5-16: The P-wave velocity changes reconstructed using IDWT with the synthetic data. The low-velocity zone is confined within the reservoir and limited in
width by the with of the reflectors in the image.
165
166
Chapter 6
Using Image Warping for
Time-lapse Image Domain
Wavefield Tomography
Summary
Time-lapse full waveform inversion has been proposed as a way to retrieve quantitative estimates of subsurface property changes through seismic waveform fitting as
presented in Chapters 2, 3 and 4. However, for some monitoring systems, the offset range versus depth of interest is not large enough to provide information about
the low wavenumber component of the velocity model. In Chapter 5, we use an
image domain wavefield tomography (IDWT) method to successfully invert for velocity changes with this type of narrow-offset survey. In this chapter, we present an
improved IDWT using the local warping between baseline and monitor images as the
cost function to mitigate the cycle-skipping effects. This cost function is sensitive to
volumetric velocity anomalies, and capable of handling large velocity changes with
very limited acquisition apertures, where traditional full waveform inversion fails. In
this chapter, we first describe the theory and workflow of our method. Layered model
examples are used to investigate the performance of the algorithm, and its robustness
167
to velocity errors and acquisition geometry perturbations. The Marmousi model is
used to simulate a realistic situation in which IDWT successfully recovers time-lapse
velocity changes.
6.1
Introduction
For a time-lapse seismic dataset, information about the changes in model parameters
in the target zone can be categorized into two groups: amplitude changes and time
shifts. Amplitude changes could be induced by new scattering in the target interval, or
differences in reflectivity at the interfaces. Time shifts are the response to a physically
shifted geologic interface (e.g. a compacting reservoir), or a velocity perturbation
along the signal's ray-path. To better link the changes in measured signals to inferred
reservoir responses, it is essential to quantify the changes from different mechanisms.
In some time-lapse seismic analysis, the time shift information is omitted because the
monitor data or images are aligned with the baseline to compare the amplitudes. In
other studies, time shifts picked at certain horizons are used to study the reservoir
velocity changes, or the strain field changes above the reservoir [9, 51]. However these
analyses are conducted on post-stack data, which have already lost some information
during the stacking process. In this study, we focus primarily on time shifts in prestack data, and velocities in model space. We do not consider amplitude changes,
which can be better inverted or interpreted after the inversion for a corrected timelapse velocity model.
To recover the seismic velocities, full waveform inversion (FWI) [93, 101] has been
applied to individual surveys.
The application of FWI to time-lapse data seems
straight-forward, however, in practice it is constrained by the survey design, data
quality and the nonlinear nature of FWI. Inversion strategies tailored for time-lapse
data have addressed issues like repeatability, computation efficiency [111] and local
minima [107, 24, 110, 6]. Traditional FWI requires low-frequency data and large survey offsets to invert for the low-wavenumber component of the velocity model [101].
However, seismic surveys with large offsets are expensive particularly when the region
168
of interest is relatively small. Small-offset reflection data do not provide constraints on
the model from a wide enough range of different angles to allow for the estimation of
low-wavenumber structures. With small offsets, FWI functions more like least-square
migration which only finds reflectivity. Image-domain methods, often involving velocity analysis, have been proposed to obtain the low-wavenumber part of the velocity
model from reflection data [90, 15]. Some image-domain methods are computationally expensive because they require the calculation of angle gathers or offset gathers,
which require many sources and receivers. These methods are more suitable for initial
model building. [86] extended an image-domain tomography method to 4D, however,
the inverted velocity changes can be smeared.
When a seismic reflection is shifted in time, there is an ambiguity as to whether the
reflector has shifted or there is a velocity change above the reflector. However, in many
cases, the changes in the depths of structures are not expected to be as significant as
the depth shifts of the reflectors in images due to velocity changes. For example, the
physical displacement of the reservoir boundaries caused by compaction may be only
a fraction of a sampling interval of the migration image (e.g. half a meter per year in
the North Sea [9]). However, volumetric strain in the overburden due to compaction
may cause changes in its seismic velocities.Velocity in the reservoir itself might also
change due to depletion or fluid substitution. In cases like CO 2 sequestration, large
amounts of fluid are injected into the subsurface, without significant changes in pore
pressure. Compared to physical structure changes, velocity changes are expected to
be the dominant effect on time-lapse images from these settings [3]. In this chapter,
we assume that seismic reflectors do not shift over the period during which time-lapse
surveys are collected. We also assume that the waveforms reflected from interfaces in
the targeted area do not change significantly. Based on this assumption, successive
acquisitions that illuminate similar areas should produce similar images without depth
shifts if correct velocity models are used.
In this chapter, we present an image-domain wavefield tomography (IDWT) method
specialized for time-lapse reservoir monitoring. With a baseline velocity model, migrated images for both baseline and monitor data can be produced with a reverse time
169
migration algorithm. With the assumptions above, depth differences between images
should be primarily caused by time-lapse changes in the velocity and not by physical
changes in reflector position. Dynamic image warping [37] is used to measure the image shifts in a way that is robust to cycle skipping and amplitude differences between
images. By minimizing the warping function (the shifts between baseline and monitor
images), we invert for velocity changes iteratively using the adjoint-state method [68].
The inversion is only sensitive to low-wavenumber velocity perturbations that control
wavefield kinematics. The inverted velocity changes are found to be localized between
reflectors, which aids interpretation of fluid migration like gas leakage. [115] applied
this method to time-lapse datasets from a CO 2 injection field. In this chapter, we
describe the theory and workflow of the IDWT approach. Synthetic examples are
used to demonstrate its capability and limitations. The robustness of the method to
baseline velocity errors and survey geometry non-repeatability is also investigated.
6.2
Theory
Iterative inversion methods like full waveform inversion, are designed to estimate
model parameters by fitting observed data with simulated data. In the time-lapse
IDWT method, the model parameters are seismic velocity changes, and the observed
data are the migrated images that are constructed from baseline and monitor seismic
surveys. We estimate velocity changes by matching monitor migrated images with
baseline migrated images. The cost function here can be written as the L-2 norm
of some measure of dissimilarity between two images. The simplest measure is the
amplitude difference:
Es.ubtract(m) =
2X 8 X
IIi(x, z,x)
-
Io(x, z, x,)1 2 dxdz,
(6.1)
where I is the baseline image, I, is the monitor image, x and z are spatial coordinates,
and x, is the source index. We derive all the equations here in 2D for simplicity, but
the extension to 3D is straight-forward with one additional integral over the third
170
spatial dimension. This cost function has the same drawback as the traditional FWI
cost function. When reflector shifts are too large (> half wavelength, measured normal
to the reflector), cycle skipping makes the cost function insensitive to local velocity
perturbations. The direct subtraction 11 - Io also causes problems when the images
have different amplitudes. These differences could be related to effects other than
velocity perturbations. In these cases, even if the velocity model is correct, the cost
function may not be minimized.
As described by [37], a migration image I based on the incorrect velocity can be
considered a warped version of the true image I based on the correct velocity. In
Equation 6.2, h(x, z) and l(x, z) are warping functions that specify how much the
image point at (x, z) in I is shifted from the same image point in I in horizontal (h)
and vertical (1) directions.
I(x, z) = I(x + h(x, z), z + l(x, z)).
(6.2)
Here we assume that the monitor image based on the baseline velocity model is a
warped version of the baseline image. For images with reflection data, both vertical
and lateral shifts can be measured [21, 38]. In this study, we only measure the vertical
warping l(x, z) for simplicity. The amount of vertical warping can be calculated by
solving an optimization problem. Specifically we compute
w(x, z) = arg min D(l(x, z)),
l(x,z)
(6.3)
where
(Ii(x, z) - Io(x, z + l(x, z))) 2 dxdz.
D(l(x, z)) =
(6.4)
x Z
We use the dynamic warping algorithm [37] to solve the optimization problem above
for the warping function w(x, z).
Since the warping function decreases in magnitude as I, and 1 o become well
171
aligned, we use the L-2 norm of w(x, z) as the cost function:
E(m) = 1
J
(6.5)
w(x,z, x, m)12dxdz,
where m is the squared slowness used for migrating monitor data, and x, is the source
by minimizing E(m) with a gradient-based method.
index. We invert for velocity
To calculate the gradient G, we use an adjoint-state method [68]. In full waveform inversion, the gradient is calculated by cross-correlating the forward propagated
source wavefield and the back propagated residual wavefield (the adjoint wavefield).
In IDWT, the gradient can be similarly written as a correlation between wavefields:
T 2A,(x,
G(x,z)
=
2
Z, t, X,)
A,(x, z, t, x,)
Z
u, , z,t,x)+X))dt
-J( 8(X2
X5 t=0
(6.6)
where u.(x, z, t, x') and Ur(X, z, t, x,) are source and receiver fields from forward and
backward propagation respectively. The associated adjoint wavefields are A,(x, z, t, x,)
and Ar(X, z, t, x,). The adjoint wavefields A are obtained by solving the wave equation:
m
2
A(xz, t)
AA(x, z, t) = d,
(6.7)
where m is the squared wave slowness, and d is the adjoint source. The adjoint sources
for solving for A,(x, z, t, x,) and Ar(X, z, t, x,) are respectively:
d,(x, z, t, x,) = a(x, Z, XS)Ur(x, Z, t, x,)
(6.8)
dr(x, z, t, x,) = a(x, z, x,)u,(x, z, t, x,),
(6.9)
and
in which
a(x, z, x,) = (8IO(X,z+W(XzzXs),Xa))2
Z,)X')
w (x, z,
4921 X,,+W
_ O2 Io(x,z~w(x,z,xa),xa) (I1(x, z, x,) - Io(x, z
172
+ w(x, z, x,), x.))
(6.10)
The derivation is similar to the formula in differential semblance optimization (DSO)
[68]. The details are presented in Appendix A. The wavefield mask a(x, z, x,) is
oscillatory due to the term 0Io(xz+w(x,z,x
),x
8 8 ) in the numerator. The denominator in
ex
a(x, z, x,) acts as an amplitude normalizer; in practice, we add a water-level term
to the denominator to avoid dividing by zero. The warping function w(x, z, x,) tells
us where a should be non-zero, and determines the sign of the adjoint source, which
determines the sign of the velocity update.
The implementation of the inversion
process consists of the following steps:
Given a baseline velocity model mo, and a baseline migration image 10,
(i) for each shot x, migrate the monitor data with the velocity model mo used to
produce Ii(x, z, x,)
(ii) compute the vertical shifts w(x, z, x,) using dynamic warping
(iii) evaluate the cost function E(m) after the summation over shots x, stop (if
small enough) or go to the next step
(iv) for each shot x, compute the adjoint wavefields \,A, and the partial gradient
G(x, z, x,)
(v) sum G(x, z, x,) over all shots to form the gradient G(x, z)
(vi) update the velocity model with G(x, z) to get mi+1
(vii) remigrate the monitor data with the updated model mi+1 , and go to step (ii)
6.3
Examples Using Synthetic Data
In this section, we will use synthetic data to show how the method works, and investigate its performance under different scenarios. First, a simple three-layer model
is used to demonstrate IDWT's ability to recover low-wavenumber velocity changes.
The performance of IDWT with respect to number of shots is tested with the same
model. A model with six layers is used to study the relation between IDWT resolution and the layer spacing. The robustness of IDWT to errors in the baseline
velocity model is tested with two cases in which one large and one small Gaussian
velocity errors are introduced. The robustness of IDWT to source-receiver geometry
173
discrepancies between surveys is investigated for both correct and incorrect baseline
velocity models. Finally, the Marmousi model is used to show how IDWT performs
for a complicated velocity structure.
6.3.1
Three-layer Model
The three-layer model has constant velocity (vp=3000m/s) but different density in
each layer (Figure 6-la). A velocity anomaly is placed in the middle of the timelapse model as shown in Figure 6-1b. The shape of the anomaly is Gaussian with
a maximum velocity increase of 800 m/s. We place 300 receivers (blue triangles in
Figure 6-1a) at an interval of 10 meters, and 5 sources (red stars in Figure 6-1a)
at an interval of 600 meters on the surface. The source is a Ricker wavelet with a
center frequency of 25 Hz. We use a finite difference acoustic wave equation solver to
generate the datasets. In this example, we assume the constant baseline velocity is
known.
Imaging and Warping
Reverse time migration (RTM) [11, 611 is used to produce all the migration images
during the inversion. The baseline and initial monitor images obtained using a single
shot gather (the third shot in Figure 6-1a) are shown in Figure 6-1c and Figure 6-1d,
respectively. The position of the deeper reflector in the monitor image (Fig 6-1d) is
shifted vertically due to the velocity change in Figure 6-1b. We compute w(x, z) using
the dynamic image warping algorithm [37] to describe how much I1 is shifted from 1 o,
as shown in Figure 6-2. The maximum vertical shift is 4 grid points (40 meters). As
in Equation 6.10, w(x, z) is used to calculate a spatial weighting function a(x, z, x,),
to mask the wavefields u, and u, to form adjoint sources (Equations 6.8 and 6.9).
Inversion Results Comparison
Figure 6-3a shows the velocity model change recovered from IDWT with the five
sources shown in Figure 6-la.
The recovered anomaly is centered at the correct
174
location, but it is smeared vertically due to the acquisition geometry. This vertical
smearing is bounded by the two reflectors.
If the inversion attempts to put any
perturbation above the first reflector, the entire image will be shifted. IDWT will
subsequently reduce this shift by reversing that perturbation. Some of the changes
are positioned along the ray-paths due to limited source and receiver coverage. Within
the area of the recovered anomaly, the amplitude is not correctly distributed, and the
maximum velocity increase is only 50% of the true value.
Although the inverted velocity is not perfect, the monitor migrated image based
on it (Fig 6-3b) shows reflectors at the same locations as in the baseline image (Fig 61c). The model from IDWT has the correct background kinematics, and is a good
starting model for FWI. Figure 6-3c shows the velocity change determined with the
application of a standard FWI [93] for the same monitor data using the velocity
model obtained from IDWT as a starting model. Both the amplitude of the anomaly,
and the distribution of the velocity are improved as FWI inverts more phase and
amplitude changes.
For comparison, we compute a standard FWI on the monitor data starting from
the correct baseline velocity and density models. Figure 6-3d shows the result. The
inversion gives poor recovery of the velocity anomaly because of several issues. First,
the velocity change is large enough to cause cycle skipping in the data domain. Second, FWI with this narrow-offset survey geometry reduces to least-squares migration,
so that the volumetric velocity change is barely resolved. Instead, a reflector that does
not exist in the true velocity model is generated to fit the data.
Figure 6-4 shows cost-function curves for IDWT, FWI, and FWI after IDWT.
IDWT converges within 10 iterations, while FWI converges much slower, both after IDWT and for FWI alone. The cost function for FWI alone plateaus after 10
iterations, because the residual is insensitive to velocity perturbations, due to cycleskipping. FWI after IDWT converges with a lower cost than does FWI alone, but
remarkably slower than does IDWT. However IDWT requires four wavefield calculations to obtain the gradient in each iteration, and two wavefield calculations are
required for one migration. Assuming each wave propagation calculation takes time
175
T, and each line search takes 3 migrations, the actual computational cost of IDWT
is 10 times that for computing N wavefields, where N is the number of IDWT iterations. Similarly, because it requires 3 forward models per line search, one FWI
iteration takes 5T. In this example, to get the final model in Figure 6-3c, we used 10
IDWT iterations, and 20 FWI iterations. Thus the total computation time is 200T,
of which 50% is used in IDWT.
6.3.2
Multi-layer Model
As shown in the three-layer model example, the smearing of the time-lapse velocity
change is bounded by the reflectors. We expect that smaller reflector spacing will lead
to a better determined anomaly. To investigate this, we use a multi-layer model to
simulate the case where time-lapse changes span several layers. A constant velocity
(vp = 3000 m/s) is used for the baseline model. The time-lapse velocity model is the
same as that in Figure 6-1b. A six-layer density model as shown in Figure 6-5a is
used to generate reflections. Layer thicknesses in the center of the model are smaller
than the size of the velocity anomaly in Figure 6-1b.
Figure 6-5 shows the velocity changes resolved by IDWT using different numbers of
shots., Only one single shot placed in the center on the surface is used in Figure 6-5b.
Compared with the results in Figure 6-3a, the anomaly is much better constrained
vertically by the second and fourth reflectors in the model.
Correspondingly, the
magnitude of the velocity anomaly is better recovered; 10 and 20 shots are used
evenly spaced at intervals of 265 and 125 meters in Figure 6-5c and 6-5d respectively.
The shape and relative magnitude distribution are improved with additional shots.
6.3.3
Baseline Velocity Errors
For all the previous examples, we assumed that the baseline model was exactly known.
In practice, it is more likely that the baseline velocity model we build is inaccurate. To
study the robustness of IDWT to errors in the baseline velocity, we use the model in
Figure 6-6a, which contains a Gaussian-shaped low velocity zone, as the true baseline
176
velocity model. We assume the anomaly is not resolved by the baseline velocity
model building and so a constant velocity model is used for the baseline migration.
We use the density model in Figure 6-5a with 20 shots evenly spaced at an interval
of 125 meters on the surface to generate synthetic data. The true time-lapse velocity
model (Figure 6-6c) has an additional high velocity Gaussian-shaped anomaly, which
is the net change between baseline and time-lapse models (Figure 6-6b). The peak
magnitude of both anomalies is 200 m/s.
Figure 6-6d shows the IDWT result obtained when using 20 shots. Compared
with the result obtained using the correct baseline model (Figure 6-5d), the resolved
time-lapse anomaly maintains the same quality in both shape and magnitude. More
importantly, there are no negative velocity changes apparent in the result. The baseline velocity model error (the negative Gaussian-shaped anomaly) is not carried over
to the time-lapse inversion. In other words, IDWT detects only the relative changes
in the models. A close scrutiny of Figure 6-5d and Figure 6-6d reveals that the shape
of the resolved change is slightly distorted because of the kinematic error induced
by the unknown Gaussian anomaly in Figure 6-6a. We expect the distortion to get
stronger with bigger errors in the baseline velocity model. We test this with the
model shown in Figure 6-6e, in which we increase the maximum amplitude of the low
velocity error in the baseline model to 800 m/s. The IDWT result with 20 shots,
shown in Figure 6-6f, is severely distorted in shape but the amplitude and position
are still accurately recovered.
6.3.4
Source Geometry Non-repeatability
Seismic survey repeatability is a key factor in achieving successful time-lapse monitoring. One common issue is the discrepancy of source-receiver geometry between
surveys. A small deviation of the source position in the monitor survey from that of
the baseline, can lead to large differences in waveforms, which makes direct comparison between datasets difficult. Time-lapse FWI methods, such as double-difference
waveform tomography, which requires data subtraction [24, 107], must carefully coprocess the baseline and monitor datasets. In IDWT, instead of data, we compare
177
images, which are less sensitive to shot position deviations. With the correct velocity
model, neighboring sources should give very similar images. As a result, when they
are migrated with the same baseline velocity, differences between a monitor image
for shot position x + 6x, and a baseline image for shot position x should still relate
to time-lapse velocity changes. We expect IDWT to be robust to this type of source
geometry difference between surveys.
We employ the baseline velocity models used in previous examples, with the constant velocity, weak Gaussian anomaly (200 m/s), and strong Gaussian anomaly (800
m/s). The maximum value of the time-lapse change is 200 m/s. The density model is
the same as that in Figure 6-5a. For the baseline survey, 15 sources are evenly spaced
at an interval of 170 meters, and 300 receivers are evenly spaced at an interval of 10
meters. For the monitor survey, we only consider source positioning errors. Because
IDWT is conducted with shot gathers, the effects from receiver positioning errors
should be negligible as long as they cover the same area. Two types of source positioning errors are commonly observed in practice: random perturbations (e.g. limited
GPS precision), and systematic perturbations (e.g. feathering effects in acquisition).
For random perturbations, we randomly perturbed each source either one grid
point left or one grid point right from its baseline position. The grid spacing is 10
meters in our tests, which is large compared to position errors observed in some wellrepeated surveys in practice [112]. In addition, position errors in reality would not
be uniformly t10 meters. However, we do not expect this to have a large effect on
the results.
Figure 6-7 shows the IDWT results with different levels of baseline velocity errors
but with the same randomly perturbed source positions. There is no baseline velocity
error in Figure 6-7(a). The baseline velocity models used in Figure 6-7(b) and (c)
have Gaussian-shaped errors of 200 m/s and 800 m/s peak value respectively. The
one-to-one comparison between Figure 6-7(a), (b), (c) and Figure 6-5d, Figure 6-6d,
Figure 6-6f show that the random source position perturbations have little effect on
the performance of IDWT.
To study the effect of systematic perturbations, we move the monitor survey source
178
positions uniformly towards the right. Three levels of shot position error are studied:
6x equals 10 meters, 20 meters and 50 meters. The monitor datasets are generated
and migrated using the perturbed source locations. Figure 6-8(a), (b) and (c) show
the IDWT results with the known constant baseline velocity model. The time-lapse
velocity anomalies are resolved with the same quality in all three cases with increasing
shot positioning error. Artifacts near the sources result from illumination differences
between baseline and monitor surveys. As we discussed for the three-layer model,
when the shot positions are the same in both surveys, the smeared updates near the
sources are diminished by iteratively correcting the image of the shallower reflector.
However, when the shot positions are different, as illustrated in Figure 6-9, parts of the
monitor image have no corresponding parts in the baseline image (dashed circles). As
a result, part of the velocity update cannot be constructed because the unconstrained
parts of the image marked by arrows in Figure 6-9 are insensitive to that velocity
change. At greater depth, this effect is mitigated by stacking shots, but the effect of
stacking is weak near the sources. If the targeted area is deep in the subsurface, these
artifacts will not affect the interpretation. If the monitor image is compared to the
entire image formed by all the baseline shots, this effect will be eliminated because
the shadowed areas in Figure 6-9 will be covered by baseline images of neighboring
shots.
Figure 6-8(d), (e) and (f) show the IDWT results with a weak Gaussian velocity
error (200 m/s) in the baseline model. As the shot positioning error increases, the
error induced by the incorrect baseline velocity model, marked by black circles, gets
stronger. The principle that neighboring shots should give similar images is violated
because the baseline velocity is incorrect. As a result, differences between baseline
and monitor images are caused by both the baseline velocity errors and the time-lapse
velocity changes. The difference caused by baseline velocity error is bigger when two
shots are further apart.Accordingly, velocity error increases as the shot positioning
error increases. In addition, the velocity error is inverted with a reverse sign, because
the monitor image is aligned with the incorrect baseline image. For example, if the
low-velocity region in the baseline model is unknown (i.e., not included in the model
179
for migration), the reflectors imaged by a source that illuminates the anomaly will be
deeper than their true positions. Regardless of the time-lapse changes, IDWT would
assume the baseline image is correct, and perturb the velocity to make monitor image
reflectors deeper, leading to a high velocity update.
Figure 6-8(g), (h) and (i) show the IDWT results with the strong Gaussian velocity
error (800 m/s) in the baseline model. As expected, the larger error induces bigger
false changes (located inside the black circles) in the time-lapse inversions. In Figure 68(i), the false changes already have the same order of magnitude as the time-lapse
changes when the source positioning error is 50 meters. In this case, an interpretation
would likely be affected by the velocity error. However, an 800 m/s velocity error
in the baseline model is significant, and source positioning errors of 50 meters are
excessive in a well-repeated 4D seismic survey. Based on the tests shown in this
section, we conclude that for relatively large errors in the baseline velocity model,
and for both random and systematic source geometry discrepancies between surveys,
IDWT is robust and expected to be capable of delivering useful inversion results.
6.3.5
Marmousi Model
For a more realistic synthetic test, we apply IDWT using the Marmousi model [100].
As shown in Figure 6-10a, only part of the original Marmousi model with complicated
geologic structures is used to better simulate narrow-offset acquisition. Five shots
evenly spaced at an interval of 200 meters (red stars) are used to generate the synthetic
datasets, and 400 receivers are deployed on the surface at an interval of 5 meters.
Figure 6-10b shows the true time-lapse velocity model with a velocity decrease in the
layers at around 1900 meters depth. The actual boundary of the velocity anomaly is
outlined by the black dashed line. The density is constant throughout the model.
We smooth the Marmousi model to generate the baseline model for migration as
shown in Figure 6-11a. Figure 6-11b shows the migrated image with one shot gather
of the baseline datasets. Due to the limited aperture of the acquisition, some of the
structures (marked by arrows in Figure 6-11b) are not illuminated. The layers in
these areas are completely missing in the image. The reflectors above and below
180
the layer containing the time-lapse changes (dashed line in Figure 6-11b) are clearly
imaged.
The IDWT result obtained using these 5 shots is shown in Figure 6-12b. The
resolved anomaly is localized within the area enclosed by the dashed line. Both the
shape and amplitude of the anomaly are well recovered. The true change, as shown
in Figure 6-12a, has small values near the boundary of the anomaly (dashed line).
In contrast, the inverted change appears to be larger in size due to vertical smearing
between reflectors. The arrow in Figure 6-12b points to a location where the inverted
anomaly spreads beyond the boundary of the actual anomaly but is well-constrained
by the reflector below. The smearing occurs because the boundary of the true timelapse change, marked by the arrow in Figurer 6-10b, is in the middle of the layer. As
we observed for the layered-model examples, velocity changes within a single interval
are vertically smeared throughout the layer but bounded by the reflectors. With this
limitation, IDWT is again effective in recovering the local time-lapse velocity change.
6.4
Discussion
From synthetic examples, we see that IDWT is able to robustly recover time-lapse
velocity changes, with acquisition limitations, such as narrow offsets and survey nonrepeatability. As with most tomography methods, IDWT smears velocity changes
along wave-paths. However, the smearing effect is clearly bounded by reflectors above
and below the changes. This effect is important for leakage monitoring when the ambiguity between the smearing and real leakage must be removed. Smaller differences
between the boundary of the changes and the reflector boundaries lead to more reliable estimates of velocity changes. Better estimates of the velocity changes lead to
more reliable interpretations of the changes.
In time-lapse inversions, we are interested in the relative changes between the
surveys at different times. However, the data residuals due to the uncertainty in the
baseline inversion are likely to contaminate the final result of time-lapse FWI. Tailored
FWI schemes have been developed to suppress these sources of noise [24, 110]. In
181
IDWT, errors in the baseline model affect both the baseline and monitor images. As
the monitor images match the baseline ones, any perturbation in the velocity model is
caused by the kinematic difference between monitor and baseline datasets. Even with
large baseline velocity errors, IDWT recovers the correct magnitude and position of
velocity changes.
Another concern for time-lapse monitoring is the repeatability of surveys.
In
practice, shot and receiver locations are not identical between surveys, even for highquality ocean bottom cables [12, 112]. In some cases, after the initial large survey
for exploration, specialized local surveys for monitoring are more economical and
efficient [43]. Deviations between survey geometries cause problems in time-lapse
FWI methods that require data subtractions [24, 107]. In contrast, IDWT depends
only weakly on the survey geometry. With a good baseline model, IDWT delivers
accurate results, as long as the monitor survey illuminates an area of interest that is
also well-imaged with the baseline survey. When large errors (e.g., 800 m/s) exist in
the baseline model, IDWT still produces reasonable results when differences in survey
geometries are considerable (e.g., 50 m).
From a computational point of view, IDWT requires two wavefield extrapolations
for each migration. With the same wave equation solver, it takes twice as much time
as FWI for each iteration. However, it is not necessary to simulate the full wavefield to
form the images. The image warping cost function is sensitive only to misalignments,
and is robust to inaccuracy in simulated waveform amplitudes. In contrast, traditional
FWI needs accurate amplitudes so that differences between waveforms are reliable.
We could potentially use a faster traveltime solver like ray-tracing to speed up IDWT.
Another possible concern is the memory requirement for IDWT. While RTM or FWI
need to store two wavefields for calculating the gradient, IDWT needs to store four
wavefields, which could be too demanding in a 3D application.
[92] presented an
optimal checkpointing method that trades floating point operations for most of the
storage in general adjoint computations. Although the memory requirement is still
going to be twice that of FWI, it should be manageable in practice.
Although the time per iteration is twice that of FWI, IDWT appears to converge
182
more quickly. Therefore, when using IDWT before FWI to resolve velocity anomalies
with high resolution, the actual computation of IDWT does not dominate the cost
of the overall process. As in the first synthetic example in this study, IDWT takes
only 50% of the total cpu runtime of the process.
When large velocity changes
exist, the cycle skipping effect makes the regular FWI cost function insensitive to
velocity updates. IDWT using image warping helps to find a good starting model
with correct large-scale kinematics for FWI. For initial velocity model building, ideas
similar to image-warping can be implemented in the data domain to avoid cycleskipping.
However, with reflection geometries, FWI fails to invert for volumetric
changes in velocity, and the result tends to be like that of a least-squares migration.
[58] have successfully overcome this problem. However, to extend their method to
time-lapse applications requires further study.
Beyond the theory and numerical studies presented here, we have applied IDWT
to field datasets (time-lapse walkaway Vertical Seismic Profiles) that were collected
from a CO 2 sequestration testing site, and successfully recovered P-wave velocity
changes that can not be resolved by full waveform inversion [115]. With very limited
survey apertures and the presence of strong noise in real data, stacking images of
neighboring shots would increase the signal-noise ratio and mitigate imaging artifacts
without losing much angle information if the source distribution is dense. Studies
with more field datasets of different acquisition conditions and different time-lapse
mechanisms (e.g. water flood, gas leakage) are planned for the near future.
6.5
Conclusion
We have proposed a time-lapse wavefield tomography method in the image domain
for reflection data. The warping between baseline and monitor images is used as a
cost function that is sensitive to smooth velocity perturbations, and robust to cycleskipping errors. The method is accurate and wave-equation based, and requires no
linearization or assumptions about the smoothness of the model. It is computationally
efficient with fast convergence, and does not require the computation of angle gathers.
183
Even with limited acquisitions, such as narrow offsets and small numbers of sources,
and for complex subsurface structures, IDWT delivers reliable time-lapse inversion
results. It is also robust with respect to baseline velocity errors and survey geometry
discrepancies between surveys. With IDWT, kinematic effects are distinguished from
other time-lapse effects, thereby providing a good foundation for subsequent analysis
of amplitudes and reservoir characterization.
6.6
Acknowledgments
This work was supported by the MIT Earth Resources Laboratory Founding Members
Consortium. The authors would like to especially thank Yong Ma from Conoco Philips
for in-depth discussions and constructive suggestions. We also thank the associate
editor of Geophysics, Dave Hale, Denis Kiyashchenko, and an anonymous reviewer
for their insightful comments and suggestions to help improve this chapter.
184
i
Dens4
0.4
0.6
0.8
Vdloidtygs)
0
800
202
400
lw0.4
D
1700
1.6
0.6
0.8
0
1.2
-400
-2100
1.4
0.5
.81500
1.5
Goimd OW=in
(Iam)
2
3
2.5
0.5
1.5
2
1
Grouad Diatanc (kum)
(a)
2.5
3
-000
(b)
0.2
0.4
S0.6
0.2
0.4
0.6
} 0.8
0.8
1
1
1.4
1.4
1.6
1.6
0.5
2
1
1.5
Ground Ditance (kim)
2.5
0.
3
(c)
i
Grond Olaaie (kin)
1
2.5
3
(d)
Figure 6-1: (a) The three-layer density model for both baseline and monitor surveys.
Red Stars denote the locations of the shots, and blue triangles denote the receiver
locations. (b) Differences in the P-wave velocities between baseline and monitor
surveys. Maximum velocity change is 800m/s. (c) The baseline image 1 obtained
using one shot gather and the constant velocity model. (d) The monitor image I,
obtained using one shot gather and the constant velocity model. The center part of
the second reflector is vertically shifted due to the absence of the velocity anomaly in
(b).
185
Shift (m)
40
0
0.2
20
0.4
E
S0.6
-0
l.0.8
1
1.2
-20
1.6
0.5
1
1.5
2
Ground Distance (kin)
2.5
3
-40
Figure 6-2: The image warping function w(x, z) calculated from Figure 6-1c and 61d. Units on the color scale are image points. Positive values indicate upwards shifts.
The maximum warping is 4 grid points (i.e. 40 meters).
186
400
0
300
0.2
2200
0.2
-
100
0.4
1.2
-100
14
1.6
-400
0.5
1
1.5
2
Ground Dialun (kml)
2.5
1.6
3
0.5
1
1s
2
Ground Dislance (kill)
(a)
2.5
(b)
elodty(m/s)VeocIty
00
0.4
200
0 '.6
0.
0f
1
-M
m)
150
~400
02
0
0
A50
0.6
0.8
i
1
-50
-150
1
0.5
1
1.5
3
2
2.5
3
0.6
Ground Disuirc (kin)
(c)
1.5
2
Ground Dsance (km)
1
2.5
-2
(d)
Figure 6-3: (a) The velocity changes found by IDWT with 5 sources. The anomaly
is correctly positioned. However, the limited aperture of the acquisition makes the
waves travel primarily in the vertical direction, so the recovered velocity anomaly is
smeared vertically. (b) The monitor migration image obtained using one shot gather
and the velocity model inverted by IDWT. The second reflector is correctly positioned.
(c) The velocity changes refined by FWI after IDWT. The amplitude differences and
subtle phase shifts between data and simulation are minimized to resolve the fine
details in the velocity model. FWI has significantly reduced the vertical smearing observed in Figure 6-3a. (d) The velocity changes obtained with standard FWI applied
to the monitor data, starting from the baseline constant background velocity model.
The Gaussian anomaly is barely visible. An artificial reflector is erroneously created
to account for data differences. This failure is due to the combined effects of cycle
skipping and limited survey geometry.
187
1
--0.8-
IDWT
FWI after IDWT
ra
0.2S0.4
0
0
0
5
10
Iteration Number
15
20
Figure 6-4: Cost function curves for IDWT, FWI after IDWT, and FWI only. The cost
functions are normalized by their values before the 1st iterations. IDWT converged
within 10 iterations. FWI after IDWT converged much slower. The cost function of
FWI starting from the constant velocity plateaued after 10 iterations.
188
1 Source
Veoiy(MIS)
600
0
400
02
02
lawO200
.0.4
O's
0A
17000.
02
1
-200
1
12
12
_4W
.4
im1
1.4
1.6
1.8-
05
1
1.5
Ground Distance
2
(kM)
2.5
3
5
1500
1
1.5
2
Ground Distance (km)
(a)
(b)
10 Sources
20 sourc65
0
400
2.5
3
-6
VelociWrnws)
0400
02
02
2W0
O.4
0.4
200
0.8
S0.8
1I
-2DO
000
-2DO
1.4
-400
1.4
-400
1.6
1.6
05
1
1.5
2
Ground Distance (kM)
2.5
3
0.5
1
1.5
2
Ground Distance (km)
2.5
~
(d)
(c)
Figure 6-5: (a) The six-layer baseline and time-lapse density model. Layers in the
center are smaller in thickness than the size of time-lapse velocity anomaly (white
circle). (b), (c) and (d) show the IDWT results with 1 shot, 10 shots and 20 shots,
respectively. As we include more shots, the amplitude distribution within the anomaly
is corrected. The vertical smearing is well constrained by the reflector. The maximum
velocity change is closer to the true value as the changes are confined to a smaller
area.
189
(/)
Veloc
0
3150
02
3100
0.
0.6
3050
Velochlms)
150
02
100
5OA0
0.A
O.S
0.8
2
50
1
2950
1
290
12
-100
-150
S2850
1.6
1.6
0.5
1
1.5
2
(3rjad Disanc (kin)
2.5
3
05
1
1.5
2
Gmund Distamce (kin)
(a)
2.5
3
(b)
M)
OW
Velocty (mM)
150
3200
3150
100
0.2
310D
0.2
0.4
0
OA
50
0.6
3DOO
0.0
0
0.6
0.2
1
21
.2
2DO0
1A'4
12
-100
2
1.6
1.6
0.5
1
1.5
2
Gmotad Distanc (kin)
2.5
3
0.5
.5
Gaotmd Distanc
2
(kin)
2.5
3
-150
(d)
(c)
vhoc3=)
Vebfcty (M)
150
0
0
3000
100
0.2
20
0
2400
-100
1.6
1.6
0. s 1 1.5
2.5 3 22
0
0.5 1.5 2 2.5 3
Oromd Disance (kin)
-150
&"Wai
(e)
Distnc
(kin)
(f)
Figure 6-6: (a) True baseline velocity model with a Gaussian anomaly with peak
velocity change of 200 m/s. We assume the anomaly is not known, and use a constant
velocity model for the baseline migrations. (b) True time-lapse velocity changes with
peak value of 200 m/s. (c) True time-lapse velocity model I with two Gaussian
anomalies ((a) plus (b)). (d) The time-lapse velocity changes found using IDWT. (e)
True time-lapse velocity model IL We increase the peak amplitude of the Gaussian
anomaly in the baseline velocity model to 800 m/s, and use the same time-lapse
velocity changes as in (b). (f) The time-lapse velocity changes inverted by IDWT.
The shape of the anomaly is distorted because of the large error in the baseline
velocity model, but the basic location and amplitude is preserved.
190
Baseline Velocity Error:0 m/s
0
'E
800 m/s
200 m/s
Velocity (m/s)
0
0.5
100
0.5
0.
1
0
1
1.5
1
2
3
-100
1.5
1.5
1
2
3
Ground Distance (kin)
1
2
3
Figure 6-7: This figure shows robustness tests of IDWT to random source positioning
errors and baseline velocity errors. The sources in the monitor survey are randomly
shifted +10 meters from their baseline positions. The baseline velocity error for each
case has maximum value of 0 (a), 200 (b) and 800 m/s (c). Compared to the case
where there is no mispositioning in Figures 6-5d, 6-6d, and 6-6f, the random source
positioning error has little effect on the performance of IDWT.
191
Position Error
21b.-
20 m
10in
(a)
(b)
(c)
0
0.5
1
1.5
0.5
0
0.5
1
1
1.5
1
2
1
0
-100II
3
1
0
0
-0 0.5
1
0.5
0.5
1
1
1
1.5
1.5
2
3
1
2
3
(0
(D
CD
CD
0
%Z
3
1
(h)
2
3
2
3
(i)
0
0
0.5
0.5
0.5
1
1
1
1.5
1.5
1.5
Velocity (m/s)
2
(1)
0
1
100
2
(e)
(g)
0
1.5
3
(d)
W
50 m
1
2
3
1
2
3
Ground Distance (kn)
Figure 6-8: Robustness tests of IDWT against source positioning error plus baseline
velocity error. In the 3x3 plot, the monitor survey sources are systematically shifted
10, 20 and 50 meters from their correct positions for each column, respectively. The
baseline velocity error for each row has maximum value of 0, 200 and 800 m/s. Black
dotted circles mark the areas where false velocity changes are resolved due to the
baseline velocity error, which is at the same location as shown in Figure 6-6e.
192
Baseline Shot
Time-lapse Shot
Wave Path
Unconstrained
Image
Figure 6-9: Migrated images for one baseline shot and one shifted monitor shot.
Dotted lines show the wave paths along which velocities are updated. Portions of
the monitor migrated image marked as unconstrained image (dashed circles), have
no corresponding image points from the baseline image.
193
VeOdy
Vlocly
(O/")
0.2
0.2
0.4
O.4
0.6
0.6
0.8
0.8
R12
1.2
(m/s)
5500
5000
40
4500
3500
2
3000
2
3Mo
2,2
2.2
0.2
0.4
0.6
08
1
2
1.4
GrowS Obtan0 (lul)
1.6
1.0
P50
02
2
(a)
04
0.6
0.
1
1 .2 14
Gocon D~sWol Qcm)
1.6
1.8
2
(b)
Figure 6-10: (a) The center part of the original Marmousi model is used as the true
baseline velocity model. The maximum source-receiver offset is 2 km. Five shots (red
stars) are used to generated synthetic data. (b) True time-lapse velocity model with
a negative velocity change marked with a black dashed line. The black arrow points
to the area where the boundary of the changes is located in the middle of the layer.
We designed this half-layer velocity change intentionally to show how IDWT would
smear the changes within a layer.
194
V.Iol0
(m's)
5000
0.2
0.2
0.4
0.
4500
0.
048
1.2
1.2
3600
14
1
1e
Is
1.8
3000
1
2.2
0.2
04
06
1
12
1.4
0.8
Growd Owc Qokm)
1.6
1.8
2
602.2
0.2
200
0.4
0.6
&W
.O
1.2
1
mDsar."cm
14
1.6
1.6
2
(b)
(a)
Figure 6-11: (a) A smoothed version of the Marmousi model is used as the baseline
model for migration. (b) Migrated image for one shot (red star). Areas pointed to
by arrows are poorly imaged due to the limited receiver aperture. Dashed lines mark
the boundary of the velocity changes. The interfaces above and below the anomaly
are well-imaged.
195
m/s)
10)
Velocity2
0.2
0.2
150
0,4
0.4
0.A
100
0.8
100
500.s
010.8
1A
14
-50
16
1.
1001
IS
--
-150
02
04
06
0.8
1
1 .2 4
Orool Obtane (On)
1.6
1.8
2
0.2
(a)
OA
0.6
0.
1
1.2
1A
Ocomd DWuiw (km)
1.6
1.8
2
-150
(b)
Figure 6-12: (a) The true time-lapse velocity changes. The anomaly is smooth at its
boundary (dashed lines). (b) The inverted time-lapse changes using IDWT with 5
shots. The black arrow points to the area where the inverted velocity changes diffuse
across the boundary of the true changes (dashed lines), and are both smeared towards
and bounded by the lower interface of this layer.
196
Chapter 7
Image Registration Guided Shear
Wave Velocity Model Building
Summary
Multi-component acquisitions offer the opportunity to form elastic migration images
and to estimate elastic parameters of the subsurface. However, the S-wave velocity
is difficult to obtain because the converted shear waves induce strong nonlinearities
in inversions. In contrast, P-wave velocity inversions are better constrained. In this
study, we propose an image registration guided S-wave velocity inversion method
based on the knowledge of the P-wave velocities. The PS depth migration image is
registered to the PP image with a shift function obtained by dynamic image warping.
In each step, a target image is generated by warping the PS image by a fraction of the
shift function to avoid cycle-skipping. Elastic image domain wavefield tomography
(EIDWT), extended from the IDWT method in Chapter 6, is used to minimize the
image differences between the PS image and the target image to update the S-wave
velocities iteratively.
The method works well with high-frequency reflection data.
Starting from a constant S-wave velocity model, the inversion delivers a high quality
PS image and a smooth velocity model, which serves as a good starting model for full
waveform inversion. We use synthetic examples to demonstrate the efficiency of the
method in simple and complex geologies.
197
7.1
Introduction
Multicomponent data have potential advantages over single component because converted waves are easier to record and identify. With advanced acquisition technologies
like ocean bottom cables or ocean bottom nodes, multicomponent data can be collected in marine environments as well as on land. Imaging with converted-waves
complements compressional wave imaging at locations where higher resolution is required. More importantly, converted-waves provide additional information for lithology estimations and reservoir characterization, which are invaluable in hydrocarbon
explorations [89].
Multicomponent imaging methods have been proposed in the literature in both
time and depth domains. [44] investigated and compared several converted-wave imaging approaches with real data applications, which showed that prestack time migration
provides interpretable results when lateral velocity variations are not significant. As
with any imaging, prestack depth migration is preferred for complex velocity models. [49] first proposed Kirchhoff elastic wave migration based on Kirchhoff-Helmholtz
type integrals. [46] presented multicomponent Kirchhoff migration using the surveysinking concept. Similar to acoustic Kirchhoff migration, these methods would likely
fail when ray-theory breaks down in complex media [35]. One-way migration methods
can also be extended for elastic applications. [105] propose separating wave modes on
the surface before one-way migration. [108] advocate an alternative procedure that
uses the vector wavefields during propagation for reconstructing scalar and vector
potentials and imaging using reverse time migration (RTM).
Although depth migration produces better images, a reliable velocity model is
necessary. A converted-wave migration velocity model is often obtained in the time
domain by tuning the V,/V, ratio [32, 37]. For example, shear wave velocities can
be estimated by registering corresponding PP and PS reflections in time-migrated
sections. Assuming the P wave velocity is correct, the time shifts between PP and
PS events can be transformed into V/V, ratio corrections. However, this method
still suffers from the limitations of time migration, and does not provide a final shear
198
wave velocity model with which to migrate the data. [26] propose a joint migration
velocity analysis in the angle domain for both PP and PS depth images. However,
Kirchoff based migration is used, which is likely to break down in complex structures.
[109] present a wave-equation migration velocity (WEMVA) analysis method for shear
wave velocity inversion based on elastic reverse time migration (ERTM) [108]. It finds
the S-wave velocities and PS depth migration images simultaneously, but this requires
heavy computation, for calculating elastic extended images and angle decomposition,
and loses the constraints from PP images.
[114] proposed an image domain wavefield tomography (IDWT) method for timelapse velocity inversion based on the assumption that corresponding reflectors in
time-lapse images should be at the same locations. A similar matching principle can
be used for shear wave velocity inversion. We assume that reflectors in PS depth
migrated images should be at the same depth as corresponding reflectors in PP depth
migrated images. When the shear wave velocities are incorrect, we can measure and
minimize the depth shifts between PS and PP images to recover the shear wave velocity model. The calculation of depth shifts can be achieved by image registration. [30]
introduce a least-squares optimization method for multicomponent data registration,
but this method requires a good initial guess. The local similarity attribute is used for
registering time-lapse images in [31]. [37] improves a dynamic programming method
developed for speech recognition that computes time shifts in a robust and efficient
manner, and applies it to registering PP and PS time migration images. [7] present a
robust piecewise polynomial dynamic time warping method with low-frequency augmented signals, and successfully combine it with full waveform inversion to mitigate
cycle-skipping effects. All theses methods can potentially be applied for the registration of depth migrated images.
In this study, we propose a methodology for inverting S-wave velocities based on
P-wave velocities by integrating image domain wavefield tomography (IDWT) [114],
dynamic image warping (DIW) [37] and registration guided least-squares (RGLS)
method [7]. The chapter is organized as follows. We first briefly describe the ERTM
and DIW algorithms that are used in this study to form and register PS images,
199
respectively. We then introduce the theory of elastic IDWT, and describe how we
modify it using a RGLS framework. A simple and intuitive three layer model is used
to show how the method works, and a modified elastic Marmousi model is used to
show how robust the method is to complex structures. In the discussion, we cover
the advantages, limitations and practical issues of the method.
7.2
7.2.1
Theory
Elastic Reverse Time Migration
Reverse Time Migration (RTM) [11, 61] is robust for imaging in complex geology. Because RTM images are formed by reconstructing source and receiver wavefields, it is
straightforward to link the algorithm with image domain wavefield tomography [114].
However, the imaging condition for ERTM is more complex than acoustic RTM because the wavefields are vector fields. To form PP and PS images separately, the
wave-modes should be separated during migration [108]. [23] propose separating the
extrapolated wavefield into P and S potentials. We follow this approach. Any vector
field u(x, t) can be written as
u = V<D + V x I,
(7.1)
where <k is the scalar potential, T is the vector potential, and V - I& = 0 [1]. The
potentials can be obtained separately, but indirectly, by applying the divergence and
curl operators to the field u(x, t)
P = V u= V 2 <,
S = V x u =-V
2
Xp.
(7.2)
(7.3)
For isotropic elastic media, the P mode is the compressional component of the wavefield propagating at speed V, and the S mode is the transverse component propagating at speed V,.
200
With the separated potentials, we can form PP, PS, SP and SS images by permuting source and receiver wavemode components [108]. Because the S mode is a
vector field, the imaging conditions for PS and SS images vary in how the vectors are
treated. For example, a PS image can be obtained by applying
T
IPS=
J(VP) - (V x S)dt,
0
or
(7.4)
T
P(Is - S)dt,
IPs =
(7.5)
0
where 13
=
[1 1 1]. Other imaging conditions (e.g., cross-correlating component by
component [108]) can also be applied. The choice of imaging condition does not harm
the generality of our framework. To simplify the discussion, we use Equation 7.5 for
the following sections, and present derivations with both Equation 7.4 and 7.5 in
Appendix B.
7.2.2
Dynamic Image Warping for Elastic Images
As described in [114] and [37], a migration image I made with an incorrect velocity
can be considered a warped version of the true image I made with the correct velocity.
In Equation 7.6, w(x) is a vector warping function that specifies how much the image
point at x in
f(x)
is shifted from the same image point in I(x)
I(x) = I(x + w(x)).
Given I(x) and
f(x),
(7.6)
we can pose the optimization problem to solve for w(x) as:
w(x) = arg min D(l(x)),
(7.7)
1(x)
where
D(l(x))
=
I(x) - f(x + 1(x)) I2 dx.
201
(7.8)
Recent developments in [7] and [37] provide efficient algorithms to solve similar problems for time warping with smooth constraints. The application of these methods
for image warping is straightforward. To be the consistent with our previous work
in [114], we choose to use the dynamic time/image warping in [37] to find the depth
shifts between PS and PP images.
When the S-wave velocity is incorrect, the PS image will be shifted in space relative
to the PP image. However, the differences are more than just mis-alignments. The
noise in PS and PP images is also likely to be different. Wavelengths, amplitudes
and phases of the reflectors may differ due to differences in velocities and reflection
coefficients. [37] presents a field data example in which PP and PS events are well
aligned by registration despite the differences. The robustness of the DIW algorithm
is important to the success of our framework.
We do not further discuss this for
conciseness, but demonstrate it below with synthetic examples.
One issue that does require discussion, however, is that in PS RTM images, the
events with flipped polarity in the PS image will be mis-registered with the PP events.
To mitigate this, we modify Equation 7.8 into
D(l(x))
=
J
E[Ipp(x)] - E[Ips(x + I(x))] I2dx,
(7.9)
x
where E is an operator that fixes the polarity issues of the PS images. There are
several ways to correct the polarities in PS images. One efficient way is to use poynting
vectors [27]. In this case; E is a mask that reverses all the flipped polarities in a PS
image, and has no effect on the PP image.
One can also choose to register the
envelopes or magnitudes of the images for which the corresponding E is the Hilbert
transform or absolute value function. In our experience, DIW functions well with all
three choices. In our synthetic examples, we choose to correct the polarities because
this is also necessary to visualize the final imaging results.
202
7.2.3
Elastic Image Domain Wavefield Tomography
Data domain inversion methods like full waveform inversion [93, 101], are designed to
estimate model parameters by fitting observed data with simulated data. If we assume
an observed image Iob,(x) is available, a similar least-squares fitting cost function can
be written
E(m)
=
I(x,x,, m) - Iob,(x,x8 )j 2 dx,
(7.10)
where I is the image we want to construct, x is the spatial vector, x, is the source index
and m is the velocity model to be recovered. Such methods are not commonly used for
initial model building (e.g., WEMVA) because there are no observed images. Instead,
velocity errors are characterized by the features of the events in image gathers, for
example mis-focusing in time-lag gathers [80, 117] or flatness in angle gathers [79, 108].
In time-lapse situations, the observed image is available: the baseline image. The
time-lapse velocity changes are estimated by fitting baseline images with monitor
images [115]. In the context of S-wave velocity model building, the observed image
is also available, if we have a reasonable P-wave velocity model. However, the PP
image cannot be directly substituted into Equation 7.10 for several reasons. First,
the PP and PS images generally have different wavelengths and amplitudes. Second,
the PS image has zero values at normal incidence locations. Third, the PS and PP
images, from the same shot profile, often have different migration apertures.
We thus require a way of estimating a modeled PS image from the PP image.
This can be achieved by using the reflector locations in the PP image to synthesize a
target PS image IPs. For this to work, we need to be able to match the reflectors in
the two images so that we know which reflectors we need to synthesize the response
from. This can be achieved efficiently by using image registration [37]. Assuming
that we obtain a spatial warping function w(x) from image registration which warps
Ips to Ipp, the cost function for S-wave velocity inversion can be written as:
E()
=
|JIps(x, x,, 1)
203
-
Ips(x, x,)I2 dx,
(7.11)
where Ips(x, x,) = Ips(x + w(x), x,)), and 3 is the S-wave velocity model. As we
minimize the image differences in Equation 7.11, the PS images will have the same
reflector locations as the PP image, and the S-wave velocity will be recovered.
We call this least-square optimization problem in Equation 7.11 elastic image
domain wavefield tomography (EIDWT). Similar to derivations for FWI and acoustic
IDWT [114], the formula for the gradient of E(3) can be derived using the adjointstate method [68]. In full waveform inversion, the gradient is calculated by crosscorrelating the forward propagated source wavefield and the back propagated residual
wavefield (the adjoint wavefield). In EIDWT, the gradient of E(3) with respect to
the Lame parameters can be similarly written as a correlation between wavefields:
8Er
(7.12)
(V - A.)(V - u.) + (V - Ar)(V - Ur)dt
-
0
f{[VA.+(
A
[V7,+ (Vu)T] + [VAr + (VAr)T]
T
[VUr + (VUr)TIdt
0
(7.13)
where - is the dot product and: is the Frobenius inner product. We denote by u, and
u, the forward propagated source wavefield and back propagated receiver wavefield,
respectively; A, and A, are the associated adjoint wavefields. These are obtained by
solving the elastic wave equation
pA = V - (c: VA) +A,
p( 6 jl6 km
+
6smokI)
Cjktm
= AjjkjIm
+
where p is density and the elasticity tensor c can be noted by
(7.14)
for elastic isotropic medium, and A is the adjoint source, which
varies with different elastic imaging conditions. To make the following derivation
concise and consistent with our numerical examples, we consider an ERTM algorithm
that uses a scalar imaging condition:
T
(V - us(t))I 3 - (V X Ur(T - t))dt,
I's =
0
204
(7.15)
where 13
=
[1 1 1]. In this case, the adjoint sources are
As
=
-V(OI
3
(7.16)
. (V X Ur)),
and
Ar = V x (qu13 (V - us)),
for A, and Ar respectively, where Or = Ips -
(7.17)
PS-
Based on the relationship between the shear wave velocity 3 and the Lamd parameters [64], we have
19EE
= 2p
-
-
4po
(7.18)
.
Since we assume the P-wave velocity is known, and the source side wavefield V - Ur
used to form Ips is controlled by only the P-wave velocity, the actual contribution to
O
013
from the source side is negligible. Therefore, a more economical formula for the
gradient can be formed by combining Equation 7.12, 7.13 and 7.18 and dropping all
the source side term resulting in:
[VAr +(VAr)T] : [Vr + (VUr)T]}dt +4p
-p,3J
0
(V - Ar)(V -Ur)dt. (7.19)
0
From this we see that only Ar needs to be calculated with A, (Equation 7.17). Many
shortcuts are made to simplify the above derivation. More details for EIDWT with
different imaging conditions are presented in Appendix B.
7.2.4
Multi-level Optimization
This cost function in Equation 7.11 has the same drawback as the traditional FWI cost
function. When reflector shifts are too large (> half wavelength, measured normal
to the reflector), cycle skipping makes the cost function insensitive to local velocity
perturbations. This is very likely to happen in S-wave velocity model building, especially when little prior information is available. Starting from an empirical V/V
ratio might help, however, the convergence is still not guaranteed.
205
[7] propose an RGLS method that uses a data domain cost function where the
real data are substituted by fractionally warped data to make sure the synthetic
waveforms are less than half a wavelength away from the true ones. Here we borrow
this idea, and use a fractionally warped image If,re to substitute the fully warped
image
fps
in Equation 7.11 to avoid cycle-skipping. The fractionally warped image
is defined as:
frac
(7.20)
= IPs(x + aw(x)),
where 0 < a < 1, and w(x) is the original warping function. A sufficiently small a
should be chosen to ensure that Ifac is close enough to Ips.
Now we can write the registration guided EIDWT (RG-EIDWT) method as the
following multi-level optimization problem. Given the current S-wave velocity model
13 k after iteration
k,
(i) We use DIW to solve Equation 7.7 for the warping function w(x) that registers
the current Ips to Ipp;
(ii) We fractionally warp IPs to Ifrac with aw(x), and use EIDWT to minimize
E(3k 1) = 1 EjjJ IIps(X, X., 3 k 1)
-
Ifrac(X, XG)I
X,
(7.21)
to recover /3 +1 iteratively;
(iii) We go back to step (i), adjust a, and repeat step (ii).
In the process above, the overall image shift between the original Ips and Ipp is
minimized fraction by fraction. The parameter a determines the size of each fraction;
this choice could also be optimized to expedite the overall convergence.
To avoid
cycle-skipping, the safest choice for a should satisfy max(aw(x)) = d/2, where d
is the normal spatial wavelength of the reflectors in Ips. Because d varies with the
S-wave velocity, further optimization can be achieved by using an a(x) that is a
function of space. In this chapter, we use a single-valued a to simply the process.
206
7.3
Synthetic Examples
Synthetic examples are used to show how the steps in our method are executed. A
simple three-layer model is used to demonstrate how RG-IDWT recovers the interval
S-wave velocities when image resolution is high with respect to the layer thickness.
A modified Marmousi model is used to show the robustness of RG-IDWT when subsurface structures are complex. It also shows the potential of the resulting S-wave
velocity model as a starting model for FWI. The derivations of our method in the
theory section and Appendix B are applicable in two or three dimensions. Our
synthetic models are 2D, which is sufficient for the proof of concept. DIW is only
executed in the depth direction which is also sufficient in our tests with reflection
data. For near vertical reflectors, DIW in the horizontal directions will add more
constraints.
7.3.1
Three-layer Model
Figure 7-1 shows the true P- and S-wave velocity models used in this test. Eight
sources are placed on the surface, evenly spaced at an interval of 300 m. We use 300
receivers with a 10 m spacing, also on the surface, to cover the entire model. Synthetic
datasets are generated by an elastic finite difference solver. The source wavelet is a
standard Ricker wavelet centered at 15 Hz. Both x-components and z-components of
the waves are collected.
We assume that through velocity model building, a smooth version of the P-wave
velocity model is available, sufficient for migration. Instead of assigning an empirical
V,/V ratio, we use a constant 1900 m/s S-wave velocity model as the starting model.
We perform ERTM beginning from the smooth P and constant S velocity models. In
Figure 7-2b, we piece the P-P and P-S images together to show the depth mismatch
of the reflectors. The left half is the P-P image, and the right half is the P-S image.
Both are formed by stacking the RTM images of all the shots. Because the constant
S-wave velocity is higher than the true velocity in the first layer (1767 m/s), the first
reflector in the P-S image is shifted downwards by about half wavelength. The S-wave
207
velocity of the second layer is higher than 1900 m/s, so the second reflector is shifted
a little less.
Figure 7-3a shows the PS image produced with one shot gather. The black star
marks the location of the shot. To register with the PP image, the polarity of the
reflectors to the left of the shot is corrected. DIW calculates the warping function in
Figure 7-3b that shows the maximum depth shift of the PS image to be 70 m. To
form If,,,
in Equation 7.20, we need to use the original PS image without polarity
correction as shown in Figure 7-4a. We multiply the warping function by a = 0.5,
and use it to warp Figure 7-4a towards the reflectors in the PP image. Figure 7-4b
presents a zoom of the reflectors (dashed line in Figure 7-4a). The blue wiggles, that
are the warped reflectors, are shifted upwards from the red ones (original image).
The subtraction between the red and blue images generates the 01 in Equation 7.16
and 7.17 to form the adjoint sources.
Figure 7-5a shows the gradient calculated with the adjoint sources. The dominant
energy is on the wave-paths from the adjoint sources to the receivers. This is because
only the shear wavefield from the receivers is used to form the PS image. As a
result, the perturbation of the PS image is only sensitive to the S-wave velocities
along the S-wave propagation paths in the receiver field. Since the P-wave velocity is
correct, there is no energy on the source-side wave-path. By summing the gradients
from all of the shots, we obtain the total gradient in Figure 7-4b used to update the
velocity model. The positive values in the gradient indicate that the current velocity
is generally too high.
After 20 iterations, we obtain the final S-wave velocity model as shown in Figure 76a. Both the low velocity (1757 m/s) in the first layer and the high velocity (2060
m/s) in the second layer are recovered. The third layer still has the starting velocity
because there is no reflection from below. On the edges of the model, the recovery
is poorer particularly for deeper parts, due to illumination limits. We compare the
final PS image with the PP image in Figure 7-6b. It is obvious that the reflectors in
the P-S image (right half) are aligned with those in the P-P image (left half). The
alignment is also not as 'good on the edges due to the same illumination limits. We
208
expect the recovery to be improved with a wider acquisition surface.
7.3.2
Modified Marmousi
The three-layer model example is a good showcase for interval velocity recovery when
the imaging resolution is much higher than the layer thickness. However, the layering
of the real subsurface can be very detailed and complicated, and the resolution of
the seismic images are limited by the frequency content of the data. In this example,
we use a modified Marmousi model to show the behavior of RG-EIDWT in realistic
geology, when the layer thickness is lower than, or comparable to the image resolution.
Figure 7-7 shows the true P-wave and S-wave velocity models. To facilitate numerical experiments, we reduce both velocity ranges, as compared to standard Marmousi,
to allow for a larger grid size and time step in the finite difference modeling. We use
18 sources, evenly distributed with a 480 m spacing on the surface, and 750 receivers
also placed on the surface, covering the entire model, with a spacing of 12 m. We
simulate a land acquisition so the top layer of the model is solid. We collect both xcomponent and z-component data. The source wavelet is a standard Ricker centered
at 20 Hz.
We assume a smooth P-wave velocity model, as shown in Figure 7-8a, is obtained
with initial model building tools.
The S-wave velocity is unknown, so we use a
constant velocity model (V, = 2500 m/s) as the starting model for ERTM. Figure 78b is the PP image produced with all 18 sources using the model in Figure 7-8a. In
ERTM, the wavemodes are separated into P and S potentials, and the propagation
of the P mode is dependent only on the P-wave velocities. As a result, even without
the correct S-wave velocities, the PP image still shows the reflectors at the correct
locations. This image is used as the reference image for registering the PS images.
Figure 7-9a is the PS image generated with one shot gather with the polarities
corrected. The shot location is marked by the black star. Due to the low S-wave
velocities, the entire image is shifted upwards from the PP image. DIW is used to
calculate the warping function as shown in Figure 7-9b, which describes the depth
shift of each image point in the PS image from the corresponding image point in the
209
PP image. As described in [37], DIW is robust to the differences between PS and PP
images. A close scrutiny finds that the amplitudes of the reflectors in the PP and
PS images are different. The PS image also has some imaging artifacts that do not
exist in the PP image. The incorrect S-wave velocities and the limited acquisition
aperture could cause the spurious patterns seen in the PS image. Another source of
error is the separation of the modes since the model clearly violates the homogeneous
P-wave velocity assumption. Nonetheless, DIW is robust to these artifacts because
it looks for a global solution for the entire image volume. The real PS events are
more coherent with those in the PP image and have relatively higher amplitudes.
Mis-registration might happen to individual events, but DIW mitigates these errors
by forcing the smoothness of the warping function along the events. For locations
that are outside of the illumination, for example, the right end in Figure 7-9a, DIW
assigns zeros since no registration can be achieved in these areas shown with white in
Figure 7-9b. The success of image registration is the foundation of the -RG-EIDWT
process.
We warp the original PS image (without polarity correction) by a fraction of the
warping function, and form the adjoint sources to calculate the gradient as shown in
Figure 7-10a. The adjoint sources are distributed sources in the entire image volume.
By comparing Figure 7-10a and Figure 7-9a, we see that the amplitudes in the gradient
are proportional to the amplitudes in the image, because the magnitude of the adjoint
sources are scaled by the magnitude of the image differences (Equations 7.16 and 7.17).
By stacking the gradients from all 18 sources, we obtain the total gradient shown in
Figure 7-10b. It only has negative values, indicating that the current S-wave velocities
are too low.
After 50 iterations, we obtain the final S-wave velocity model in Figure 7-11.
The macro velocity distribution is consistent with the true S-wave velocity model.
However, the interval velocities of each layers are not individually resolved because
the image resolution is not high enough. Here the maximum image resolution is 62.5
m, whereas the typical layer thickness is 60 m. The salt bodies on the sides are also
not recovered because of illumination limitations. The structures beneath the salt
210
layers are not recovered because the converted S-waves are outside the acquisition
aperture due to the reflector dips. We expect this region to improve with a wider
acquisition surface.
Although the model in Figure 7-11 has low resolution, it contains the correct
kinematics to place the PS events at the correct locations. Figure 7-12 shows the PS
image before and after the inversion. We divide the entire image into 7 sections, and
PP and PS image sections are shown in an alternating pattern (i.e., odd sections are
PP images and even sections are PS images). In Figure 7-12a, the mis-match between
the images is obvious, and it gets larger deeper in the model because the kinematic
errors accumulate with depth. Particularly, for the reservoir location outlined by the
dashed circle, the image is severely shifted from the correct location. The coherency
of the reflectors is poor, and the polarity of the strong reflector is wrong. Interpretations would likely be unreliable based on this image. Figure 7-12b shows the same
alternating PP/PS image generated with the inverted S-wave velocity model in Figure 7-11. The PP and'PS events are well-aligned, even for the deep reflectors that are
shifted by more than one wavelength in Figure 7-12a. For the reservoir area (black
dashed circle), the strong reflector is at the correct location, and the polarity is also
corrected. The interfaces that are blurry before the inversion are coherent and well
imaged.
With the corrected background velocities, we can apply FWI to further improve
the resolution of the models. We use the same data but only the reflection part with
a maximum offset of 3.6 km to perform FWI starting from the models in Figure 7-8a
(for P) and Figure 7-11 (for S). Figure 7-13 shows the P- and S-wave reflectivity
models after the inversion. The image resolution is remarkably improved, and the
reflectivities are true physical parameters rather than image points. Except for the
areas with poor illumination, the S-wave reflectivity model is of similar quality as the
P-wave reflectivity model. Without the S-wave velocity inversion with RG-EIDWT,
it is difficult to achieve the image in Figure 7-13b.
211
7.4
Discussion
We have shown that both DIW and RG-EIDWT are robust to imaging artifacts.
However, since the method is based on single shot migration images, random noise in
the image might be a cause for concern. To mitigate this, we could stack images from
adjacent shots, relying on the robustness of both RTM and DIW to random noise to
reduce the artifacts. RG-EIDWT could use the stacked image to do registration and
form adjoint sources.
The two synthetic examples we use to explain the process of RG-EIDWT are also
showcases of two different field data applications we have in mind. The Marmousi
example is meant to highlight the problem encountered in seismic exploration, that
the image resolution is often not high enough the discriminate thin layers in the
subsurface. RG-EIDWT can be used to build a smooth background velocity model
for S-waves, without requiring long-offset and low-frequency acquisitions. FWI or
least-squares migration can improve the resolution of the model based on the RGEIDWT results. With the layered model we mimic the global seismology problem
in which, reflections are often sparse in depth. RG-EIDWT can resolve the interval
S-wave velocities by registering the PP and PS images of deep earth discontinuities,
and provide an average estimate of Poisson's ratio for the layers in the crust and
mantle.
In the derivation and numerical tests in this chapter, we show only the S-wave
velocity inversion with registration between P-P and P-S images. However, within
the same framework, RG-EIDWT can be modified to invert for P-wave velocities in
situations where the S-wave velocity model is known. In addition, SP and SS images
can also be integrated into the method if the source has strong shear components. It
is also a natural extension to apply a similar methodology on 4D inversion, where the
baseline image serves as the target image, and the time-lapse P- and S-wave velocity
changes are inverted by matching monitor images with the baseline images.
212
7.5
Conclusion
We have proposed an image registration guided wavefield tomography method in
the image domain for S-wave velocity model building with the knowledge of P-wave
velocities. The shifts of the PS images with respect to the PP image are minimized
fraction by fraction to recover the S-wave velocities iteratively. The method is waveequation based and has no assumptions about the smoothness of the subsurface.
It works well with high-frequency reflection data, and can start with an arbitrary
constant S-wave velocity model. It is computationally efficient without the calculation
of angle gathers or extended images. The resulting model is smooth and serves as a
good starting model for FWI or least-squares migration.
7.6
Acknowledgment
This work was supported by the MIT Earth Resources Laboratory Founding Members Consortium. We would like to thank Hyoungsu Baek from Saudi Aramco, and
Xuefeng Shang from Shell International for helpful discussions.
213
Velocity (km/s)
0
4
0.2
3.8
0.4
3.6
E0.6
N
0.8
1
3.2
1.2
1.4
0.5
1
1.5
3
2
X (km)
(a)
Velocity (km/s)
.2.15
0
N
0.2
2.1
0.4
2.05
0.6
2
0.8
1.95
1.9
1
1.85
1.2
1.4
0.5
1
1.5
2
U
1.8
X (km)
(b)
Figure 7-1: (a) True P-wave velocity model. (b) True S-wave velocity model. The
S-wave velocities in three layers are 1767, 2060 and 2150 m/s respectively.
214
Velocity (km/s)
2.15
M
0
0.2
2.1
0.4
2.05
2
- 0.6
N
1.95
0.8
1.9
1
1.85
1.2
1.4
1.8
0.5
1
2
1.5
X (km)
(a)
0
0.2
0.4
0.6
N
0.8
1
1.2
1.4
0.5
1
1.5
2
X (km)
(b)
Figure 7-2: (a) Starting model with constant S-wave velocity 1900 m/s. (b) Comparison between PP (left half) and PS (right half) images. Both images are formed with
all 8 sources. The first reflector in the PS image is shifted downwards for about half
of the wavelength.
215
0
0.2
0.4
- 0.6
N 0.8
1
1.2
1.4
0.5
1
1.5
2
X (km)
(a)
Depth Shift (m)
70
0
0.2
1 60
0.4
50
40
-0.6
N
30
0.8
20
1
10
1.2
1.4
0
0.5
1
1.5
2
X (km)
(b)
Figure 7-3: (a) Polarity-corrected PS image formed with one shot gather. Black star
marks the location of the source. (b) The warping function calculated by DIW. It
describes how much the depth shift is for each image point in the PS image in (a).
216
0
0.2
0.4
-0.6
N
0.8
1
1.2
1.4
0.5
1
1.5
2
X (km)
(a)
0.80.91
I
E 1.1
N
1.2
1.3 F
1.4
0.5
1
1.5
2
X (km)
(b)
Figure 7-4: (a) Original PS image without polarity correction. The same source is
used as in Figure 7-3a. (b) The zoom-in view of the reflectors marked by black dashed
line in (a). The original image (red wiggles) and the fractionally warped image (blue
wiggles) are shown together. The differences between them are used to generate the
adjoint sources.
217
1
0.2
0.5
0.4
0.4
0
N0.8
1
-0.5
1.2
1.4
0.5
1
1.5
2
X (km)
(a)
01
0.2
0.40.
E- 0.6
0
N0.8
1
-0.5
1.2
0.5
1
1.5
2
X (km)
(b)
Figure 7-5: (a) The gradient for the source marked by the black star. Dominant energy
is along the receiver wave-path. (b) Total gradient by stacking partial gradients from
each source. Positive value indicates the current velocity is too high.
218
Vel Ocit y (km/s)
0
I
0.2
2.1
2.05
0.4
2
-0.6
N
2.15
1.95
0.8
1.9
1
1.85
1.2
1.4
1.8
0.5
1
2
1.5
X (km)
(a)
0
0.2
0.4
0.6
N
0.8
1
1.2
1.4
0.5
1.5
1
2
X (km)
(b)
Figure 7-6: (a) The recovered S-wave velocity model after 20 iterations. Both the
low and high velocity layers are resolved. The recovery is limited by the illumination
of the survey. (b) The PS image formed with the recovered S-wave velocity model in
(a) is compared with the PP image. Both reflectors are aligned. The alignment is
poor on the edges due to the same illumination limits.
219
Velocity (km/s)
4.5
0
4
E
3.5
N
3
1
2
3
4
5
X (km)
6
7
8
2.5
(a)
Velocity (km/s)
3.5
0
E1
3
N
1
2
3
4
5
X (km)
6
7
8
2.5
(b)
Figure 7-7: The modified elastic Marmousi model. (a) True P-wave velocity model.
(b) True S-wave velocity model. Velocity ranges are modified to be smaller to allow
larger grid size and time step in finite difference. The top layer is solid instead of
water.
220
Velocity (km/s)
4.5
0
4
1
3.5
N
2
3
1
2
3
4
5
X (km)
6
7
8
2.5
(a)
0.5
0
E
N
1
0
2
0
2
4
X (km)
6
8
-0.5
(b)
Figure 7-8: (a) Smooth P-wave velocity model. It is assumed to be obtained with
P-wave velocity model building. (b) The PP RTM image produced with all 18 sources.
221
0
0.5
1
0
N
2
0
2
4
X (km)
6
8
0.5
(a)
Depth Shift (km)
0
-0.2
1b
-0.4
N
2
-0.6
-0.8
0
2
4
X (km)
6
8
(b)
Figure 7-9: (a) The PS image produced with one shot gather. The polarities are
corrected. The black star marks the location of the source. (b) The warping function
that registers the PS image in (a) to the PP image in Figure 7-8b. The shifts are all
negative (upwards) because the current S-wave velocity is lower than all velocities in
the true S-wave velocity model.
222
0
1
0.5
0
N
-0.5
0
2
4
X (km)
6
8
(a)
01
0.5
E0
0
N
-0.5
0
2
4
X (km)
6
8
(b)
Figure 7-10: (a) The normalized gradient calculated with the same source. The black
star marks the source location. (b) The normalized total gradient by summing the
gradients from all 18 sources. Negative values indicate that the current velocities
need to be increased.
223
Velocity (km/s)
-3.5
0
E1
N
3
2
1
2
3
4
5
X (km)
6
7
8
2.5
Figure 7-11: The final S-wave velocity model after 50 iterations. The bottom part
of the model is poorly resolved because the converted S-waves are outside of the
acquisition surface due to the tilted reflectors.
224
0
0.5
1
-15.5--N
2
2.5
1
2
3
4
5
6
ow8
5
6
0?8
X (km)
(a)
0
0.5
1
N
2
-
2.51
2
3
4
X (km)
(b)
Figure 7-12: (a) The comparison between PP and PS images before the inversion.
The image is divided into seven sections in the horizontal direction. Odd sections are
PP images, and even sections are PS images. The mismatch is clear on the section
interfaces. In the circled area, the reflectors are not well imaged due to the incorrect
velocity. (b) The PS image based on the inverted S-wave velocity model is compare
to the PP image in the same setup as in (a). The coherency at the section interfaces
is markedly improved. The reflectors in the circled area are all well resolved and
aligned with those in the PP image.
225
0
1
N
2
0
1
2
3
4
X (km)
5
6
7
8
5
6
7
8
(a)
0
1
N
2
0
1
2
3
4
X (km)
(b)
Figure 7-13: (a) The P-wave reflectivity model resolved by FWI starting from the
smooth model in Figure 7-8a. (b) The S-wave reflectivity model resolved by FWI
starting from the smooth model in Figure 7-11. Except the areas outside of illumination, the recovery of the S-wave model is of a similar quality to that of the P-wave
model in (a). It also proves the success of the S-wave velocity model building with
RG-EIDWT.
226
Chapter 8
Conclusion and Future Directions
8.1
Conclusion
The chapters are organized by method types, but the actual evolution of the work in
the past five years was purely driven by data, and it was not done in the order of the
chapters. The first datasets we obtained are the time-lapse walkaway VSP data from
SACROC. The original idea was to use FWI and DDWI to resolve the velocity changes
caused by the CO2 injections. As shown in Chapter 5, FWI was not successful, but
it took us some effort to understand that the failure was caused by limitations of the
acquisition. Not surprisingly, DDWI cannot improve the situation since the problem
is in the acquisition. The research on the SACROC data was coming to a dead end
until we found IDWT in Chapter 6, to constrain the velocity inversion by restricting
the depths to be unchanged. At the same time, we obtained the Valhall datasets that
are long-offset and low-frequency. The applications of FWI and DDWI to the Valhall
datasets were successful as shown in Chapter 3, but raised more questions about
the credibility of the results. The major concern was that data subtractions between
surveys with realistic non-repeatability issues could lead to erroneous results. The
initial idea that DDWI would be sensitive to non-repeatable signals motivated the
robustness and feasibility study of DDWI in Chapter 2, which actually showed the
opposite. However, if non-repeatability is too severe or the surveys are not designed
to be repeated at all, DDWI would not be applicable. Therefore we invented another
227
joint inversion method in Chapter 4 to handle time-lapse surveys of different source
and receiver geometries. During the DDWI robustness study, we also found shear
wave velocity models very difficult to recover even though the inversion for the Pwave velocity model with the same data was successful. That drove the research in
Chapter 7, where we inverted for S-wave velocities by matching P-P and P-S images
without limitations in acquisition offset and data frequency content.
We would like to conclude this thesis from an application point of view, and summarize how the methods presented here can be integrated to recover the P-wave and
S-wave velocity changes. We assume that P-wave velocity model building is successful
with existing tools like traveltime tomography, MVA and FWI. RG-EIDWT can be
used to build an initial S-wave velocity model, and elastic FWI can refine it to higher
resolution. If subsequent time-lapse surveys are local (small-offset), IDWT can be
used to resolve the low wavenumber P- and S-wave velocity changes. If the surveys
are well-repeated, DDWI can refine the velocity changes to higher resolution, and further invert amplitude changes for other parameters. If the survey geometry difference
is beyond the robustness region of DDWI, alternating FWI can be used instead. If the
subsurface structures are changed significantly (e.g., compacting reservoirs), we need
long-offset acquisitions in monitor datasets to remove the depth-velocity ambiguity,
and alternating FWI can be used to extract the time-lapse changes which include
changes in both the reservoir and the overburden.
8.2
Future Directions
The time-lapse velocity inversion methods presented in this thesis can be improved in
many aspects. In addition, based on our research, more advanced inversion methodologies can be developed to go beyond velocity or elasticity parameters to engineering
measures. Here we list a few possible research directions to extend our thesis work:
Computation Efficiency
The major drawback of full wavefield inversion methods is the computation demand. The most time-consuming part is solving wave equations to extrapolate wave228
fields. All of our methods are developed in the time domain for shot gathers. It would
be beneficial to investigate the effects of parallerization approaches on the time-lapse
inversions. For example, randomized shots and phase-encoding are used to reduce
the computation of FWI by firing shots together [52, 63]. It is known that crosstalk from simultaneous sources causes artifacts in the inverted models. However, we
do not know how much impact the artifacts will have on time-lapse inversions. It
would also be interesting to know how alternating FWI can be integrated with shot
randomization.
For image domain methods, we have the same computational demand.
More
efficient imaging techniques will directly improve the efficiency of image domain inversions. Similar to FWI, better parallerization can speed up RTM, and the impact
of the cross-talk on time-lapse images needs to be examined. In addition, for fields
that have simple structures or weaker velocity contrasts, ray-based imaging methods like Gaussian beam methods are sufficient and much faster. The general idea of
IDWT can be used with propagation using ray theory. We can substitute RTM with
faster imaging methods to speed up the inversion. Both theoretical derivations and
implementations of these adaptations are interesting research topics.
Target Oriented
To monitor the whole subsurface within the illumination of surveys, we should
not arbitrarily assign an area of interest. . For example, in CO 2 sequestration, the
purpose of time-lapse seismic is to track the fluid and monitor any leakage through
all possible paths. However, there are also situations where only local monitoring
is required. For example, in a non-compacting reservoir, overburden structures are
assumed to be unchanged. To monitor the hydrocarbon depletion or fluid injections,
the area of interest is very localized around the reservoir layer. Therefore, a large
portion of the wavefield extrapolations is perfectly repeated and so wasted in the
inversion process. It will make the inversions much more efficient if we can develop a
target oriented time-lapse inversion strategy. The fundamental idea is to propagate
the wavefields to a surface close to the target, and synthesize a subsurface acquisition.
We have done some preliminary work in [111]. It is an interesting research direction
229
for time-lapse inversion methods in both the data and image domains.
Passive Seismic
All the methods in this thesis are based on active seismic acquisitions. Monitoring
with passive seismic acquisitions could also use these ideas with some modifications.
For example, in global seismology, sources are natural earthquakes that cannot be
designed and controlled. With source locations well estimated, similar approaches
to alternating FWI could be used to monitor changes in regional or global scales
like volcano activity and mantle plumes. When the locations of sources are uncertain, EIDWT can be easily modified to invert for P- and S-wave velocity changes by
matching P-S images. In this case, P-S images are formed by P and S potentials,
both from receiver wavefields, which does not require source information [84]. Similar approaches could be applied to smaller scale problems like micro-seismicity to
monitor geothermal reservoirs and unconventional reservoirs.
Anisotropy
All the work in this thesis is based on the assumption that the Earth is isotropic.
Although in the discussion in Chapter 3, we state that anisotropy is a secondary
time-lapse effect, rigorous studies about the importance of anisotropy have not been
conducted. Within the framework we provide in the thesis, the effect of anisotropy
can be included as a natural extension to both time-lapse FWI and IDWT. In some
cases, anisotropy is important for understanding the time-lapse effects in the reservoirs during fracturing for example. For unconventional reservoirs, the operationally
induced anisotropy in velocities could be a potential indicator of reservoir permeability. For compacting reservoirs, compaction induced anisotropy in the overburden
helps to constrain the stress field of the shallow structures, which is very useful for
drilling path design.
Density
Density information is highly valuable in discriminating different time-lapse effects. However, with pressure data, estimates of density with FWI are not reliable
due to their similar radiation patterns as we discussed in Chapter 2. This is also why
we focus on time-shifts in this thesis. With the kinematic changes correctly estimated,
230
amplitude information is isolated and now can be inverted independently for density
changes since the velocity changes are already known. Multi-component seismic data
can also help in density inversion. Our preliminary research related to Chapter 2
has shown the potential, but is not included in this thesis. Future research could also
investigate the robustness of density inversion strategies since amplitude information
is more vulnerable due to noise contamination.
Closing the loop
The ultimate goal of time-lapse seismic is to provide constraints for geologic models and reservoir models. With well estimated elasticity and density changes, the
spatial lithology heterogeneities of the formations are better understood. The uncertainties in porosity, permeability, structure and stratigraphy can be reduced with the
dynamic information inferred from the 4D results. Engineering measures like pore
pressure changes and fluid saturation changes can also be derived from the 4D seismic results using rock physics models. This information can reduce the uncertainty
of the reservoir models substantially because it is not limited to the well locations.
Developing quantitative methods to communicate between geologic, geophysical, and
reservoir models, is a promising research direction.
231
232
Appendix A
Adjoint-state Method for
Time-lapse IDWT
Here we present the mathematical derivation of the adjoint wavefields and the gradient
for IDWT using the associate Lagrangian in the time domain. Following the approach
of [68], the steps of the derivation are: for model parameter m, and cost function J(m),
(i) list all the state equations F = 0;
(ii) build the augmented functional L by associating the independent adjoint state
variables Ai with the state equations F;
(iii) define the adjoint-state equations by
L =
aui
0;
(iiii) compute the gradient by '9L = o.
To make the process less complicated, we derive everything based on a single shot
in a 2-D space. A more general derivation can be easily achieved by summing over
all the shots. The extension to 3-D is straightforward by applying an integral over y.
The least-square functional is:
J(m)
=
w(x, z)| 2 dxdz,
(A.1)
where w(x, z) is the warping function that minimizes
L
D(w(x, z)) = -
Ii(x, z) - Io(x, z + w(x, z))I 2 dxdz,
233
(A.2)
where Io(x, z) is the baseline image that stays invariant throughout the process, and
1 (x, z) is the monitor image based on the slowness model m. The first derivative of
the function with respect to w(x, z) should be close to zero at the minimum point:
9 D(x,
z) = (Ii(x, z) - Io(x, z + w(X, z))) 09I(Xz+ w(xz))
0.
(A.3)
I (x, z) is obtained from the imaging condition:
T
Ii(x, z) = Ju,(x, z, t)u,(x, z, T - t)dt.
(A.4)
0
I)
u, is the source wavefield obtained by solving the following wave equations:
M
- Au,(t) = fS
u,(x, z, 0) = 0
(A.5)
8u,(x,z,0) = 0
.9t
u, is the receiver wavefield obtained by solving the following equations:
ma
-
AUr(t) = d(T - t)
ur(x, z, 0) = 0
au(x,z,) = 0.
(A.6)
For simplicity, the spatial boundary conditions are left unspecified because any condition that guarantees a unique solution is acceptable. In our numerical examples,
we use absorbing boundary conditions.
Using the Lagrangian formulation, we associate the adjoint states AO, SA, t0,
/I
with the initial conditions in Equation A.5 and A.6, respectively. Adjoint states
A,
and Ar are associated with the wave equations in Equation A.5 and A.6. Adjoint
states '/
and 0,q are associated with Equation A.4 and A.3. With the operations
above, the augmented functional is defined by:
L(0,7 0, A,, Ar, 1, /1,
,Ar , A,, Us fi, II, m) =
234
f L
v (x, z) 2 dxdz
T
-
a2
Aii,(t) - fs)x,zdt
-
(A,, ma2i
0
(AS,7 ,s(0))X,Z - (AS 7
t ) ''
T
J (Ar,,m
~ Aflr(t) - d(T - t))x,zdt
( at
(Aiir(0)),,Z
-
-
-2
0
''
T
1 1(x, z)
-(Ni,
J
-
us(x, z, t)Ur(X,
Z,
T - t)dt),,
0
+ (X,
-(2, -((x, z) - io(x, z + i7(x, Z))AI(x z az
(A.7)
Z)) ),,
with (A,, &i)x, = fx fz A,(x, z)ii,(x, z)dxdz the real scalar product in space. By two
integrations over t by parts, we switch the second order time derivative operator from
ii, to
IS:
T
I,
a2 t),zdt =
i
0
T2
S(m
, ii,)x,dt + (,(T), m at T
at2
at )X,z
'' - (A8(0), m
0
aAs (T)
a81 (0)
-(m at,
i,(T))x,z + (m at
,(0)) ,..
T
The same operation is applied to similar terms: f (1, m
(A.8)
2
T
8 i, (t)
&2
0
)x,zdt,
f(As, Af,(t))x,,dt
0
T~
and f (Ar, Aiir(t))x,zdt.
0
With Equation A.7 and A.8, we can compute the derivatives with respect to the
adjoint states, and evaluate them at (A, Ar, U, r, O, 11, 0., w) to obtain the adjointstate equations. With respect to ft, we have equations:
I
m-
- A A(t) = 0(-u,(T
A,(T) =0
aA,(T)
0
at
-
t))
(A.9)
235
With respect to i,, we have equations:
A A,(t) = g(-u.,(T - t))
Ma 2-
(A.10)
Ar(T)
aA, (T)
-0
With respect to 4 and iv- we have equations:
-{1 -
OW( _9IO(xz ~W(XZ)))
w
(-I)=0
-#
=
0
(A.11)
_2O (Z+W (XZ)) (I(x, z) - Io(x, z + w(x, z)))
j = (aIO(Z+w(XZ)))2
By taking the derivative of L with respect to the model parameter m, we have the
gradient of the cost function:
9
J(m)
_
-
T
- -2
0
a A,(x, z, t)
2
+
U,(X , Z , t +
)
aL
236
2
2 z, t)
ur(x,z,t)dt
aAr(X,
8t2
(A.12)
Appendix B
Adjoint-state Method for Elastic
IDWT
Here we present the mathematical derivation of the adjoint wavefields and the gradient
for Elastic IDWT using the associate Lagrangian in the time domain. Following the
approach of [68], the steps of the derivation are: for model parameter m, and cost
function J(m),
(i) list all the state equations F = 0;
(ii) build the augmented functional L by associating the independent adjoint state
variables vi with the state equations F;
(iii) define the adjoint-state equations by 8L = 0;
(iiii) compute the gradient by U =
J.
To simplify the process, we assume a single shot. A more general derivation can
be easily achieved by summing over all shots. The least-square functional is:
J(c)
=
LIps(x)
- IPs(x)| 2 dx,
(B.1)
where the model parameter is c which is the elasticity tensor. Ips(x) is the target
PS image. Ips(x) is the PS image produced with the current elasticity tensor c, with
237
the imaging condition:
T
IPs
= J(V(V -u5 (t))) - (V x (V x ur(T - t)))dt.
(B.2)
0
u. is the source wavefield obtained by solving the following wave equations:
)
pus =f +-V.(c: Vu5
us 1t=O = 0
(B.3)
nslt=o = 0
ur is the receiver wavefield obtained by solving the following equations:
pr = d(T - t) + V - (c: Vur)
Urlt=o =0
nr t=0 =
(B.4)
0
For simplicity, density is assumed to be constant, and the spatial boundary conditions are left unspecified because any condition that guarantees a unique solution
is acceptable. In our numerical examples, we use absorbing boundary conditions.
Using the Lagrangian formulation, we associate the adjoint states
AO,
ii, i,
si
with the initial conditions in Equation B.3 and B.4, respectively. Adjoint states T,
and Flr are associated with the wave equations in Equation B.3 and B.4. Adjoint
state q, is associated with Equation B.2. With the operations above, the augmented
functional is defined by:
L2 IIPs(x)
-
I
S(x)1 2 dx
T
(E;.,I ii - f - V - (c : V5.)).dt
0
T
-
{(Aso, ii5 ) + (As-, 8,,)x}6(t)dt
0
238
T
fJ(v-, u*-r
d(T
-
t)
-
V.- (c : Viir))xdt
-
(B.5)
0
T
iir)x + (Ar~ 1Ur)x}6(t)dt
-f(ArO7
0
T
-
-(q, IPS(x)
IJ(V (V
-f ())-(V x (V
X
fir(T
t)
-
t
(B.6)
with (a, b),
=
f f, fz a - b dxdydz the inner product of vector-valued functions in
space.
By integration by parts, it is easy to prove that both the second-order time derivative operator and the second-order spatial derivative operator are both self-adjoint,
so we have:
T
0
JEr Ifis),xdt=
T
T
+
(is fi)xdt
0
-(us,
0
iis)6(t - T) + (Ps, 6iI)x6(t)}dt,
(B.7)
and
(FJS, V -(C : Vfis))x
+
= (V - (c : VjS), ii)x
{FS - (c : VWi) - (c : VP,) - i'}|X=
0.
For the imaging condition term, we can similarly derive:
(01, (V(V -6i)) - (V
x
(V
X
iir)))x
= (ii,, V((VI) - (V X (V X ir))))x
+{(V - ,is)
- (4IV X (V
X
ir)) - (ii - I3 )(VI) - (V
= (fir, V X ((VJ) x (V(V -6i))))x
239
X
(V
X
Gir))}1x= o,
(B.8)
+( i) - (qII x (V X5r)) +(I x ir) - (V I x V(V - 6i))}|x=IO.,
(B.9)
where 13
=
[1 111.
With Equation B.6, B.7, B.8, and B.9, we can compute the derivatives with
respect to the adjoint states, and evaluate them at (V', Vr, Us, Ur, 11, IpS) to obtain
the adjoint-state equations.
With respect to ii, we have equations:
IV
= V- (c : Vv.) + A.
(B.10)
Vslt=T = 0
sIl t=T - 0
in which
As = -V((VI) - (V x (V
With respect to
fir,
x
ur(T - t)))).
(B.11)
we have equations:
I
Vr
= V. (c: Vvr) + Ar
(B.12)
rlt=T = 0
rlt=T = 0
in which
Ar = -V
X
((VOI) x (V(V -u,(T - t)))).
(B.13)
With respect to Ips, we have equation:
0I = IPS
-
IPS
(B.14)
By taking the derivative of L with respect to the model parameter c, we have the
240
gradient of the cost function:
J(Vv.vus + VvrVur)dt
o
=
=
(B.15)
T
0
- is a fourth-order tensor. If we consider an elastic isotropic medium, the elasticity
tensor c can be noted by Cjklm =
09J
+
Adjkolm
J(V - vs)(V
ip(Sjkm
+
Jjmokt).
Thus, we have:
- u.) + (V - vr)(V - ur)dt
(B.16)
0
+
S-{[Vvs
(Vv.)T] : [Vus + (Vu)T] + [VVr + (Vvr)T] : [VUr + (VUr)T]}dt
(B.17)
where : is the Frobenius inner product. Based on the relationship between the shear
wave velocity 3 and the Lame parameters, we have:
9J
(B.18)
= 2 pOT
Ap
The derivation varies subtly with different kinds of imaging conditions. Suppose
we use a more naive imaging condition like:
T
'PS =
(V X Ur (T - t))dt.
J(V -Uwl
(B.19)
0
Accordingly, for the divergence and curl operators, we can derive:
(01, ((V - fis))I3 - ((V X fir))).
=
=
(figs, V(N1Ih - (V x ir)))x + {Ii
I 3(eV
X ir)}x=oo
(ir,V X (4 1 13(V - fi)))x + {ir - (e4IV - f)}isx= oo.
241
(B.20)
Thus, the adjoint sources are:
A,
-V(4II
3
(B.21)
- (V X Ur)),
and
A,
(B.22)
V X (0113(V - U.)).
This simpler formula makes it easier when we reduce the problem to 2-D, because
the curl term will only have one non-zero component:
V X Ur = (j,(
ax
- rz)j Ok).
(B.23)
ax
Then, the adjoint sources in 2-D are:
arx- -9u)|
[ O8I ax
[Por(Ou
_Z
az
- ex
M )|
k)
(B.24)
and
[
Ar =O(_x,
1
a+
UZ)]
az
242
[&qr(%' +
ax
ax
OU-Z)]
9Z
k)
(B.25)
Bibliography
[1] Aki, K., Richards, P. G., 2002, Quantitative seismology
[2] Aoki, N., Takahashi, A., Xue, Z., 2013, Development of a permanent OBC
system for CCS monitoring in shallow marine environments, Energy Procedia,
37, 4174-4181
[3] Arts, R., Eiken, 0., Chadwick, A., Zweigel, P., van der Meer, B., Kirby, G.,
2004, Seismic monitoring at the Sleipner underground CO 2 storage site (North
Sea), Geological Society, London, Special Publications, 233, 181-191
[4] Arts, R., Eiken, 0., Chadwick, A., Zweigel, P., van der Meer, L., Zinszner,
B., 2004, Monitoring of CO 2 injected at Sleipner using time-lapse seismic data,
Energy, 29, 1383-1392
[5] Arts, R., Chadwick, A., Eiken, 0., Thibeau, S., Nooner, S., 2008, Ten years'
experience of monitoring CO 2 injection in the Utsira Sand at Sleipner, offshore
Norway, First break, 26
[6] Asnaashari, A., Brossier, R., Garambois, S., Audebert, F., Thore, P., Virieux,
J., 2011, Sensitivity analysis of time-lapse images obtained by differential waveform inversion with respect to reference model, SEG Technical Program Expanded Abstracts
[7] Baek, H., Calandra, H., Demanet, L., 2014, Velocity estimation via registrationguided least-squares inversion, Geophysics, 79, R79-R89
[8] Bakulin, A., Jervis, M., Burnstad, R., Smith, R., 2013, Making seismic monitoring work in a desert environment with complex near surface, SEG Technical
Program Expanded Abstracts, pp. 4900-4904
[9] Barkved, 0., Kristiansen, T., 2005, Seismic time-lapse effects and stress
changes: Examples from a compacting reservoir, The Leading Edge, 24, 12441248
[10] Barkved, 0., Heavey, P., Kommedal, J., van Gestel, J., Synnve, R., Pettersen,
H., Kent, C., Albertin, U., 2010, Business impact of full waveform inversion at
Valhall, SEG Technical Program Expanded Abstracts, pp. 925-929
243
[11] Baysal, E., Kosloff, D. D., Sherwood, J. W. C., 1983, Reverse time migration,
Geophysics, 48, 1514-1524
[12] Beasley, C., Chambers, R., Workman, R., Craft, K., Meister, L., 1997, Repeatability of 3-D ocean bottom cable seismic surveys, The Leading Edge, 16,
1281-1286
[13] Benson, S. M., Cole, D. R., 2008, CO 2 sequestration in deep sedimentary formations, Elements, 4, 325-331
[14] Bickle, M., Chadwick, A., Huppert, H., Hallworth, M., Lyle, S., 2007, Modeling
carbon dioxide accumulation at Sleipner: Implications for underground carbon
storage, Earth and Planetary Science Letters, 255, 164-176
[15] Biondi, B., Almomin, A., 2012, Tomographic full waveform inversion (TFWI)
by combining full waveform inversion with wave-equation migration velocity
analysis, SEG Technical Program Expanded Abstracts
[16] Boyd-Gorst, J., Fail, P., Pointing, L., 2001, 4-D time lapse reservoir monitoring
of Nelson field, central North Sea: Successful use of an integrated rock physics
model to predict and track reservoir production, The Leading Edge, 20, 13361350
&
[17] Brechet, E., Jorge, H., Turpin, P., Maultzsch, S., 2010, 4D fast track advanced
interpretation on Dalia field, block 17, Angola, in 72nd EAGE Conference
Exhibition
[18] Campbell, S., Lacombe, C., Brooymans, R., Hoeber, H., White, S., 2011,
Foinaven: 4D processing comes up trumps, The Leading Edge, 30, 1034-1040
[19] Chadwick, A., Williams, G., Delepine, N., Clochard, V., Labat, K., Sturton, S.,
Buddensiek, M.-L., Dillen, M., Nickel, M., Lima, A. L., et al., 2010, Quantitative analysis of time-lapse seismic monitoring data at the Sleipner CO 2 storage
operation, The Leading Edge, 29, 170-177
[20] Cheng, A., Huang, L., Rutledge, J., 2010, Time-lapse VSP data processing for
monitoring CO 2 injection, The Leading Edge, 29, 196-199
[21] Cox, B., Hatchell, P., 2008, Straightening out lateral shifts in time-lapse seismic,
First Break, 26
[22] Daley, T. M., Myer, L. R., Peterson, J. E., Majer, E. L., Hoversten, G. M.,
2008, Time-lapse crosswell seismic and VSP monitoring of injected CO 2 in a
brine aquifer, Environmental Geology, 54, 1657-1665
[23] Dellinger, J., Etgen, J., 1990, Wavefield separation in two dimensional
anisotropic media, Geophysics, 55, 914-919
244
[24] Denli, H., Huang, L., 2009, Double-difference elastic waveform tomography in
the time domain, SEG Technical Program Expanded Abstracts, 28, 2302-2306
[25] Dicharry, R., Perryman, T., Ronquille, J., 1973, Evaluation and design of a CO 2
miscible flood project-SACROC unit, Kelly-Snyder field, Journal of Petroleum
Technology, 25
[26] Du, Q., Li, F., Ba, J., Zhu, Y., Hou, B., 2012, Multicomponent joint migration
velocity analysis in the angle domain for pp-waves and ps-waves, Geophysics,
77, U1-U13
[27] Du, Q., Zhu, Y., Ba, J., 2012, Polarity reversal correction for elastic reverse
time migration, Geophysics, 77, S31-S41
[28] Eastwood, J., Lebel, P., Dilay, A., Blakeslee, S., 1994, Seismic monitoring of
steam-based recovery of bitumen, The Leading Edge, 13, 242-251
[29] Falahat, R., Shams, A., MacBeth, C., 2011, Towards quantitative evaluation
of gas injection using time-lapse seismic data, Geophysical Prospecting, 59,
310-322
[30] Fomel, S., Backus, M., 2003, Multicomponent seismic data registration by least
squares, SEG Technical Program Expanded Abstracts, pp. 781-784
[31] Fomel, S., Jin, L., 2009, Time-lapse image registration using the local similarity
attribute, Geophysics, 74, A7-A11
[32] Fomel, S., Backus, M., Fouad, K., Hardage, B., Winters, G., 2005, A multistep approach to multicomponent seismic image registration with application
to a West Texas carbonate reservoir study, SEG Technical Program Expanded
Abstracts, pp. 1018-1021
[33] Gardner, G., Gardner, L., Gregory, A., 1974, Formation velocity and density
- the diagnostic basics for stratigraphic traps, Geophysics, 39, 770
[34] Gassmann, F., 1951, Elastic waves through a packing of spheres, Geophysics,
16,673-685
[35] Gray, S., Etgen, J., Dellinger, J., Whitmore, D., 2001, Seismic migration problems and solutions, Geophysics, 66, 1622-1640
[36] Gritto, R., Daley, T. M., Myer, L. R., 2004, Joint cross-well and single-well
seismic studies of CO 2 injection in an oil reservoir, Geophysical Prospecting,
52, 323-339
[37] Hale, D., 2013, Dynamic warping of seismic images, Geophysics, 78, S105-S115
[38] Hale, D., Cox, B., Hatchell, P., 2008, Apparent horizontal displacements in
time-lapse seismic images, SEG Technical Program Expanded Abstracts, pp.
3169-3173
245
[39] Hall, S. A., MacBeth, C., Barkved, 0. I., Wild, P., 2005, Cross-matching with
interpreted warping of 3D streamer and 3D ocean-bottom-cable data at Valhall
for time-lapse assessment, Geophysical Prospecting, 53, 283-297
[40] Han, W. S., McPherson, B. J., Lichtner, P. C., Wang, F. P., 2010, Evaluation
of trapping mechanisms in geologic CO 2 sequestration: Case study of SACROC
northern platform, a 35-year CO 2 injection site, American Journal of Science,
310, 282-324
[41] Harris, J. M., Nolen-Hoeksema, R. C., Langan, R. T., Van Schaack, M.,
Lazaratos, S. K., Rector III, J. W., 1995, High-resolution crosswell imaging of
a west Texas carbonate reservoir: Part 1-project summary and interpretation,
Geophysics, 60, 667-681
[42] Hatchell, P., Bourne, S., 2005, Rocks under strain: Strain-induced time-lapse
time shifts are observed for depleting reservoirs, The Leading Edge, 24, 12221225
[43] Hatchell, P., Wang, K., Lopez, J., Stammeijer, J., Davidson, M., 2013, Instantaneous 4D seismic (i4D) for offshore water injection monitoring, SEG Technical
Program Expanded Abstracts, pp. 4885-4889
[44] Herrenschmidt, A., Granger, P.-Y., Audebert, F., Gerea, C., Etienne, G.,
Stopin, A., Alerini, M., Lebegat, S., Lambar, G., Berthet, P., Nebieridze, S.,
Boelle, J.-L., 2001, Comparison of different strategies for velocity model building and imaging of pp and ps real data, The Leading Edge, 20, 984-995
[45] Hicks, G. J., 2002, Arbitrary source and receiver positioning in finite-difference
schemes using kaiser windowed sinc functions, Geophysics, 67, 156-166
[46] Hokstad, K., 2000, Multicomponent kirchhoff migration, Geophysics, 65, 861873
[47] Ivanova, A., Kashubin, A., Juhojuntti, N., Kummerow, J., Henninges, J., Juhlin, C., Liith, S., Ivandic, M., 2012, Monitoring and volumetric estimation of
injected CO 2 using 4D seismic, petrophysical data, core measurements and well
logging: a case study at Ketzin, Germany, Geophysical Prospecting, 60, 957973
[48] Jenkins, S., Waite, M., Bee, M., 1997, Time-lapse monitoring of the Duri steamflood: A pilot and case study, The Leading Edge, 16, 1267-1274
[49] Kuo, J., Dai, T., 1984, Kirchhoff elastic wave migration for the case of noncoincident source and receiver, Geophysics, 49, 1223-1238
[50] Landa, J., Kumar, D., 2011, Joint inversion of 4d seismic and production data,
in SPE Annual Technical Conference and Exhibition
246
[51] Landr0, M., Stammeijer, J., 2004, Quantitative estimation of compaction and
velocity changes using 4D impedance and traveltime changes, Geophysics, 69,
949-957
[52] Li, X., Aravkin, A., van Leeuwen, T., Herrmann, F., 2012, Fast randomized
full-waveform inversion with compressive sensing, Geophysics, 77, A13-A17
[53] Liu, F., Guasch, L., Morton, S. A., Warner, M., Umpleby, A., Meng, Z., Fairhead, S., Checkles, S., 2012, 3-D time-domain full waveform inversion of a
Valhall OBC dataset, SEG Technical Program Expanded Abstracts, pp. 1-5
[54] Liu, F., Morton, S., Ma, X., Checkles, S., 2013, Some key factors for the successful application of full-waveform inversion, The Leading Edge, 32, 1124-1129
[55] Lumley, D., 2010, 4D seismic monitoring of sequestration, The Leading Edge,
29, 150-155
[56] Lumley, D., Behrens, R., 1998, Practical issues of 4D seismic reservoir monitoring: What an engineer needs to know, SPE Reservoir Evaluation & Engineering,
1, 528-538
[57] Lumley, D. E., 2001, Time-lapse seismic reservoir monitoring, Geophysics, 66,
50-53
[58] Ma, Y., Hale, D., 2013, Wave-equation reflection traveltime inversion with dynamic warping and full-waveform inversion, Geophysics, 78, R223-R233
[59] Managi, S., Opaluch, J. J., Jin, D., Grigalunas, T. A., 2005, Technological
change and petroleum exploration in the gulf of mexico, Energy Policy, 33, 619
-632
[60] Mavko, G., Mukerji, T., Dvorkin, J., 2009, The rock physics handbook: Tools
for seismic analysis of porous media, Cambridge University Press
[61] McMechan, G. A., 1983, Migration by extrapolation of time-dependent boundary values, Geophysical Prospecting, 31, 413-420
[62] Medwin, H., 1975, Speed of sound in water: A simple equation for realistic
parameters, Journal of the Acoustic Society of America, 58, 1318-1319
[63] Moghaddam, P., Keers, H., Herrmann, F., Mulder, W., 2013, A new optimization approach for source-encoding full-waveform inversion, Geophysics, 78,
R125-R132
[64] Mora, P., 1987, Nonlinear two-dimensional elastic inversion of multioffset seismic data, Geophysics, 52, 1211-1228
[65] Mora, P., 1989, Inversion= migration + tomography, Parallel Computing 1988,
pp. 78-101
247
[66] Onishi, K., Ueyama, T., Matsuoka, T., Nobuoka, D., Saito, H., Azuma, H., Xue,
Z., 2009, Application of crosswell seismic tomography using difference analysis
with data normalization to monitor CO 2 flooding in an aquifer, International
Journal of Greenhouse Gas Control, 3, 311-321
[67] Osdal, B., Husby, 0., Aronsen, H. A., Chen, N., Alsos, T., 2006, Mapping the
fluid front and pressure buildup using 4D data on Norne field, The Leading
Edge, 25, 1134-1141
[68] Plessix, R.-E., 2006, A review of the adjoint-state method for computing the
gradient of a functional with geophysical applications, Geophysical Journal International, 167, 495-503
[69] Pratt, G., Shin, C., et al., 1998, Gauss-Newton and full Newton methods in
frequency-space seismic waveform inversion, Geophysical Journal International,
133, 341-362
[70] Prieux, V., Brossier, R., Gholami, Y., Operto, S., Virieux, J., Barkved, 0. I.,
Kommedal, J. H., 2011, On the footprint of anisotropy on isotropic full waveform inversion: the Valhall case study, Geophysical Journal International, 187,
1495-1515
[71] Prieux, V., Brossier, R., Operto, S., Virieux, J., 2013, Multiparameter full
waveform inversion of multicomponent ocean-bottom-cable data from the Valhall field. part 1: imaging compressional wave speed, density and attenuation,
Geophysical Journal International
[72] Purcell, C., Mur, A., Soong, Y., McLendon, T. R., Haljasmaa, I. V., Harbert,
W., 2010, Integrating velocity measurements in a reservoir rock sample from
the SACROC unit with an AVO proxy for subsurface supercritical CO 2 , The
Leading Edge, 29, 192-195
[73] Richards, T., 2011, Lessons from 4D seismic monitoring of CO 2 injection at the
Delhi field, First Break, 29
[74] Rickett, J., Lumley, D., 2001, Cross-equalization data processing for time-lapse
seismic reservoir monitoring: A case study from the Gulf of Mexico, Geophysics,
66, 1015-1025
[75] Rodi, W., Mackie, R. L., 2001, Nonlinear conjugate gradients algorithm for 2-D
magnetotelluric inversion, Geophysics, 66, 174-187
[76] Roste, T., Landro, M., Hatchell, P., 2007, Monitoring overburden layer changes
and fault movements from time-lapse seismic data on the Valhall field, Geophysical Journal International, 170, 1100-1118
[77] Rushmere, E., Dyce, M., Campbell, S., Hill, A., 2010, 4D acquisition and processing: a North Sea case study on the relative contributions to improved 4D
248
repeatability, in Geological Society, London, Petroleum Geology Conference series, vol. 7, pp. 517-522, Geological Society of London
[78] Sarkar, S., Gouveia, W. P., Johnston, D. H., et al., 2003, On the inversion of
time-lapse seismic data, SEG Technical Program Expanded Abstracts
[79] Sava, P., Fomel, S., 2003, Angle-domain common-image gathers by wavefield
continuation methods, Geophysics, 68, 1065-1074
[80] Sava, P., Fomel, S., 2006, Time-shift imaging condition in seismic migration,
Geophysics, 71, S209-S217
[81] Sava, P., Vasconcelos, I., 2011, Extended imaging conditions for wave-equation
migration, Geophysical Prospecting, 59, 35-55
[82] Schiemenz, A., Igel, H., 2013, Accelerated 3-D full-waveform inversion using
simultaneously encoded sources in the time domain: application to Valhall
ocean-bottom cable data, Geophysical Journal International, 195, 1970-1988
[83] Sears, T., Barton, P., Singh, S., 2010, Elastic full waveform inversion of multicomponent ocean-bottom cable seismic data: Application to Alba Field, U. K.
North Sea, Geophysics, 75, R109-R119
[84] Shang, X., Hoop, M. V., Hilst, R. D., 2012, Beyond receiver functions: Passive source reverse time migration and inverse scattering of converted waves,
Geophysical Research Letters, 39
[85] Sheen, D.-H., Tuncay, K., Baag, C.-E., Ortoleva, P. J., 2006, Time domain
Gauss-Newton seismic waveform inversion in elastic media, Geophysics Journal
International, 167, 1373-1384
[86] Shragge, J., Yang, T., Sava, P., 2013, Time-lapse image-domain tomography
using adjoint-state methods, Geophysics, 78, A29-A33
[87] Smith, S., Tsvankin, I., 2013, Sensitivity of compaction-induced multicomponent seismic time shifts to variations in reservoir properties, Geophysics, 78,
T151-T163
[88] Staples, R., Cook, A., Braisby, J., Hodgson, B., Mabillard, A., 2006, Integration of 4D seismic data and the dynamic reservoir model reveal new targets in
Gannet C, The Leading Edge, 25, 1126-1133
[89] Stewart, R., Gaiser, J., Brown, R., Lawton, D., 2002, Converted-wave seismic
exploration: Methods, Geophysics, 67, 1348-1363
[90] Sun, D., Symes, W. W., 2012, Waveform inversion via nonlinear differential
semblance optimization, SEG Technical Program Expanded Abstracts
[91] Symes, W., Carazzone, J., 1991, Velocity inversion by differential semblance
optimization, Geophysics, 56, 654-663
249
[92] Symes, W. W., 2007, Reverse time migration with optimal checkpointing, Geophysics, 72, SM213-SM221
[93] Tarantola, A., 1984, Inversion of seismic reflection data in the acoustic approximation, Geophysics, 49, 1259-1266
[94] Tarantola, A., 1986, A strategy for nonlinear elastic inversion of seismic reflection data, Geophysics, 51, 1893-1903
[95] Tatanova, M., Hatchell, P., et al., 2012, Time-lapse AVO on deepwater OBN
seismic at the Mars field, SEG Technical Program Expanded Abstracts
[96] Tolstukhin, E., Lyngnes, B., Sudan, H. H., 2012, Ekofisk 4D seismic-seismic
history matching workflow, in SPE Europec/EAGE Annual Conference
[97] van Gestel, J.-P., Kommedal, J. H., Barkved, 0. I., Mundal, I., Bakke, R., Best,
K. D., 2008, Continuous seismic surveillance of Valhall field, The Leading Edge,
27,1616-1621
[98] van Gestel, J.-P., Best, K. D., Barkved, 0. I., Kommedal, J. H., 2011, Integration of the life of field seismic data with the reservoir model at the Valhall field,
Geophysical Prospecting, 59, 673-681
[99] Vedanti, N., Sen, M. K., 2009, Seismic inversion tracks in situ combustion: A
case study from Balol oil field, India, Geophysics, 74, B103-B112
[100] Versteeg, R., 1994, The Marmousi experience: Velocity model determination
on a synthetic complex data set, The Leading Edge, 13, 927-936
[101] Virieux, J., Operto, S., 2009, An overview of full-waveform inversion in exploration geophysics, Geophysics, 74, WCC1-WCC26
[102] Wang, Y., Huang, L., Zhang, Z., 2011, Reverse time migration of time-lapse
walkaway VSP data for monitoring CO 2 injection at the SACROC EOR field,
SEG Technical Program Expanded Abstracts, pp. 4304-4308
[103] Wang, Z., 2001, Fundamentals of seismic rock physics, Geophysics, 66, 398-412
[104] Wang, Z., Cates, M. E., Langan, R. T., 1998, Seismic monitoring of a CO 2 flood
in a carbonate reservoir: A rock physics study, Geophysics, 63, 1604-1617
[105] Wapenaar, C. P. A., Haime, G. C., 1990, Elastic extrapolation of primary
seismic p- and s-waves, Geophysical Prospecting, 38, 23-60
[106] Warner, M., Ratcliffe, A., Nangoo, T., Morgan, J., Umpleby, A., Shah, N.,
Vinje, V., tekl, I., Guasch, L., Win, C., Conroy, G., Bertrand, A., 2013,
Anisotropic 3D full-waveform inversion, Geophysics, 78, R59-R80
250
[107] Watanabe, T., Shimizu, S., Asakawa, E., Matsuoka, T., 2005, Differential waveform tomography for time-lapse crosswell seismic data with application to gas
hydrate production monitoring, SEG Technical Program Expanded Abstracts
[108] Yan, J., Sava, P., 2008, Isotropic angle-domain elastic reverse-time migration,
Geophysics, 73, S229-S239
[109] Yan, J., Sava, P., 2010, Analysis of converted-wave extended images for migration velocity analysis, SEG Technical Program Expanded Abstracts, pp.
1666-1671
[110] Yang, D., Fehler, M., Malcolm, A., Huang, L., 2011, Carbon sequestration
monitoring with acoustic double-difference waveform inversion: A case study on
SACROC walkaway VSP data, SEG Technical Program Expanded Abstracts,
30, 4273-4277
[111] Yang, D., Zheng, Y., Fehler, M., Malcolm, A., 2012, Target-oriented time-lapse
waveform inversion using virtual survey, SEG Technical Program Expanded
Abstracts, pp. 1-5
[112] Yang, D., Fehler, M., Malcolm, A., Liu, F., Morton, S., 2013, Double-difference
waveform inversion of 4D ocean bottom cable data: Application to Valhall,
North Sea, SEG Technical Program Expanded Abstracts, pp. 4966-4970
[113] Yang, D., Malcolm, A., Fehler, M., 2013, Using image warping for time-lapse
image domain wavefield tomography, SEG Technical Program Expanded Abstracts, pp. 4926-4931
[114] Yang, D., Fehler, M., Malcolm, A., 2014, Using image warping for time-lapse
image domain wavefield tomography, Geophysics, accepted
[115] Yang, D., Malcolm, A., Fehler, M., Huang, L., 2014, Time-lapse walkaway
vertical seismic profile monitoring for CO 2 injection at the SACROC enhanced
oil recovery field: A case study, Geophysics, 79, B51-B61
[116] Yang, D., Meadows, M., Inderweisen, P., Landa, J., Malcolm, A., Fehler, M.,
2014, Double-difference waveform inversion: feasibility and robustness study
with pressure data, Geophysics, under review
[117] Yang, T., Sava, P., 2011, Wave-equation migration velocity analysis with timeshift imaging, Geophysical Prospecting, 59, 635-650
[118] Zadeh, H., Landro, M., Barkved, 0., 2011, Long-offset time-lapse seismic:
Tested on the Valhall LoFS data, Geophysics, 76, 01-013
[119] Zhang, Z., Huang, L., 2013, Double-difference elastic-waveform inversion with
prior information for time-lapse monitoring, Geophysics, 78, R259-R273
251
[120] Zheng, Y., Barton, P., Singh, S., 2011, Strategies for elastic full waveform
inversion of time-lapse ocean bottom cable (OBC) seismic data, SEG Technical
Program Expanded Abstracts, pp. 4195-4200
252