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Novel Surface Wave Imaging Methods
Dissertation by
Zhaolun Liu
In Partial Fulfillment of the Requirements
For the Degree of
Doctor of Philosophy
King Abdullah University of Science and Technology
Thuwal, Kingdom of Saudi Arabia
September, 2019
2
EXAMINATION COMMITTEE PAGE
The dissertation of Zhaolun Liu is approved by the examination committee
Committee Chairperson: Gerard Schuster
Committee Members: Daniel B. Peter, J. Carlos Santamarina, and Ronald L. Bruhn
3
©September, 2019
Zhaolun Liu
All Rights Reserved
4
ABSTRACT
Novel Surface Wave Imaging Methods
Zhaolun Liu
I develop four novel surface-wave inversion and migration methods for reconstruct-
ing the low- and high-wavenumber components of the near-surface S-wave velocity
models.
1. 3D Wave Equation Dispersion Inversion. To invert for the 3D background
S-wave velocity model (low-wavenumber component), I first propose the 3D
wave-equation dispersion inversion (WD) of surface waves. The results from
the synthetic and field data examples show a noticeable improvement in the
accuracy of the 3D tomogram compared to 2D tomographic inversion if there
are significant 3D lateral velocity variations.
2. 3D Wave Equation Dispersion Inversion for Data Recorded on Rough
Topography. Ignoring topography in the 3D WD method can lead to signif-
icant errors in the inverted model. To mitigate these problems, I present a
3D topographic WD (TWD) method that takes into account the topographic
effects in surface-wave propagation modeled by a 3D spectral element solver.
Numerical tests on both synthetic and field data demonstrate that 3D TWD
can accurately invert for the S-velocity model from surface-wave data recorded
on irregular topography.
3. Multiscale and layer-stripping WD. The iterative WD method can suffer
from the local minimum problem when inverting seismic data from complex
Earth models. To mitigate this problem, I develop a multiscale, layer-stripping
5
method to improve the robustness and convergence rate of WD. I verify the
efficacy of our new method using field Rayleigh-wave data.
4. Natural Migration of Surface Waves. The reflectivity images (high-wavenumber
component) of the S-wave velocity model can be calculated by the natural mi-
gration (NM) method. However, its effectiveness is demonstrated only with
ambient noise data. I now explore its application to data generated by con-
trolled sources. Results with synthetic data and field data recorded over known
faults validate the effectiveness of this method. Migrating the surface waves in
recorded 2D and 3D data sets accurately reveals the locations of known faults.
6
ACKNOWLEDGEMENTS
Firstly, I would like to thank my mentor, Prof. Gerard T. Schuster, for his guid-
ance, support and encouragement throughout my Ph.D. study at the King Abdullah
University of Science and Technology. His expertise was invaluable in the formulating
of the research topic and methodology in particular. I am also grateful to the mem-
bers of my dissertation committee: Prof. Daniel Peter, Prof. J. Carlos Santamarina
and Prof. Ronald Bruhn for taking their time, patience and for their insights and
suggestions which tremendously benefited my thesis.
I am grateful to Los Alamos National Laboratory (LANL), the U.S. for offering
me an internship during my Ph.D. I appreciate Dr. Lianjie Huang, who is a great
advisor for me at LANL. I would also like to thank the guidance and help that I
received from Dr. Kai Gao, Dr. Benxi Chi, Dr. Yu Chen, and Dr. Yunsong Huang
during my internship in LANL. I thank Fuchun Gao and Paul Williamson to invite
me to visit TOTAL, where I gain valuable industry experiences.
I also thank all of my colleagues from the Center for Subsurface Imaging and
Modeling (CSIM) for their discussion and assistance. I benefited from my discussions
with Dr. Abdullah AlTheyab, Dr. Gaurav Dutta, Dr. Bowen Guo, Dr. Mrinal Sinha,
Dr. Zongcai Feng, Dr. Jing Li, Dr. Lei Fu, Dr. Han Yu, Dr. Kai Lu, Dr. Yuqing
Chen, and Dr. Shihang Feng. I feel lucky to be a CSIMer, and I appreciate the
friendships that I made in the CSIM family.
Last but not least, I thank my parents and my wife Xiaodan Ge for their uncon-
ditional love and endless support during all these years.
7
TABLE OF CONTENTS
Examination Committee Page 2
Copyright 3
Abstract 4
Acknowledgements 6
List of Figures 10
1 Introduction 21
1.1 Surface-wave Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.2 Surface-wave Migration . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 3D Wave-equation Dispersion Inversion of Rayleigh Waves 28
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.1 Misfit Function . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.2 Connective Function . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.3 Frechet Derivative . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.4 Gradient Update . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3 Workflow and Implementation . . . . . . . . . . . . . . . . . . . . . . 38
2.3.1 3D Dispersion Curves for 3D Data . . . . . . . . . . . . . . . 39
2.3.2 Initial Model for 3D WD . . . . . . . . . . . . . . . . . . . . . 42
2.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4.1 Checkerboard Test . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4.2 Modified Foothills Model . . . . . . . . . . . . . . . . . . . . . 50
2.4.3 Qademah Fault Seismic Data . . . . . . . . . . . . . . . . . . 53
2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.7 Appendix A: Correlation Identity . . . . . . . . . . . . . . . . . . . . 76
2.8 Appendix B: Elastic Gradient . . . . . . . . . . . . . . . . . . . . . . 78
8
3 3D Wave-equation Dispersion Inversion of Surface Waves Recorded
on Irregular Topography 81
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.2.1 Theory of 3D TWD . . . . . . . . . . . . . . . . . . . . . . . . 84
3.2.2 Source-receiver Distance on a 3D Irregular Surface . . . . . . . 86
3.2.3 Workflow of 3D TWD . . . . . . . . . . . . . . . . . . . . . . 87
3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.3.1 Homogeneous Half Space . . . . . . . . . . . . . . . . . . . . . 89
3.3.2 Checkerboard Test . . . . . . . . . . . . . . . . . . . . . . . . 90
3.3.3 3D Foothills Model . . . . . . . . . . . . . . . . . . . . . . . . 95
3.3.4 Washington Fault Seismic Data . . . . . . . . . . . . . . . . . 97
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.7 Appendix A: Calculation of the Geodesic . . . . . . . . . . . . . . . . 116
3.8 Appendix B: Discrete Radon Transform . . . . . . . . . . . . . . . . . 117
4 Multiscale and Layer-Stripping Wave-Equation Dispersion Inversion
of Rayleigh Waves 119
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.2.1 Theory of WD . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.2.2 Workflow of multiscale and layer-stripping WD . . . . . . . . 125
4.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.3.1 Synthetic Model . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.3.2 Surface Seismic Data from the Blue Mountain Geothermal Field 135
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5 Imaging Near-surface Heterogeneities by Natural Migration of Sur-
face Waves: Field Data Test 148
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.2 Theory of natural migration . . . . . . . . . . . . . . . . . . . . . . . 150
5.3 Workflow of natural migration for controlled source data . . . . . . . 151
5.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.4.1 Natural Migration of Synthetic Data . . . . . . . . . . . . . . 154
9
5.4.2 Natural Migration of Aqaba Data . . . . . . . . . . . . . . . . 159
5.4.3 Natural Migration of Qademah Data . . . . . . . . . . . . . . 164
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
5.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6 Conclusions 171
6.1 3D Wave-equation Dispersion Inversion of Rayleigh Waves . . . . . . 171
6.2 3D Wave-equation Dispersion Inversion of Surface Waves Recorded on
Irregular Topography . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.3 Multiscale and Layer-Stripping Wave-Equation Dispersion Inversion of
Rayleigh Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.4 Imaging Near-surface Heterogeneities by Natural Migration of Surface
Waves: Field Data Test . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Papers Published and Submitted 175
References 176
10
LIST OF FIGURES
1.1 (a) True and (b) inverted S-wave velocity models, where full waveform
inversion is used. (after Yuan et al. (2015).) . . . . . . . . . . . . . . 22
1.2 (a) True S-wave velocity model and (b) the reflectivity image. (after
Hyslop and Stewart (2015).) . . . . . . . . . . . . . . . . . . . . . . . 22
2.1 Schematic diagram showing how to calculate the weighted conjugated
data D(g, θ, ω)∗obs, where the red star represents the source, the black
solid square shows the geophone location at g and the red solid squares
represent the geophones along the line C which satisfies (g′−g) ·n = 0.
For the azimuth θ and position g, D(g′, θ, ω)∗obs is integrated along the
dashed line with the weighting term 2πiLei∆κL, where L = g · n. The
blue dot at gc is the stationary point for a homogeneous half-space, and
the line integral in equation 4.4 can be approximated by D(gc, ω)obs
(see Appendix A for the detailed derivation). . . . . . . . . . . . . . 36
2.2 Plan view of the areal acquisition, in which the red star represents the
source, and the grid points at the line crossings represent the locations
of geophones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3 (a) 3D CSG, (b) its spectrum, and (c) frequency slice of the magnitude
spectrum at 50 Hz from (b). (d) Picked dispersion surface according
to the dominant amplitudes of the spectrum. . . . . . . . . . . . . . . 40
2.4 (a) Lines from the source point, located at (30 m, 30 m), to the geo-
phones along the boundary, and (b) R(θ) plotted against the azimuth
angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.5 Workflow for calculating the initial S-velocity model for 3D WD. . . . 43
2.6 (a) True S-velocity model and its (b) depth slice at z = 6 m, (c)
inverted S-velocity tomogram and (d) depth slice at z = 6 m. . . . . . 45
11
2.7 (a) Wavefields of the adjoint source for θ = 90◦ and (b) the gradient at
the depth slice z = 6 m; (c) stacked wavefields of the adjoint sources
for θ from 0◦ to 180◦ and (d) the gradient at the depth slice z = 6 m,
where the maximum source-receiver offset is r1=80 m and the source
is located at s = (60, 0, 0) m. . . . . . . . . . . . . . . . . . . . . . . 47
2.8 Slices of the gradient at z = 6 m for (a) r1 = 40 m and (b) r1 = 120 m. 48
2.9 Observed dispersion curves for sources (a) A, (b) B, (c) C and (d) D
marked in Figure 2.6b, where the black dashed lines, the cyan dash-dot
lines and the red lines represent the contours of the observed, initial
and inverted dispersion curves, respectively. . . . . . . . . . . . . . . 48
2.10 True S-velocity depth slices at (a) z = 15 m and (b) z = 24 m; inverted
S-velocity depth slices at (c) z = 15 m and (d) z = 24 m. . . . . . . . 49
2.11 (a) True S-velocity model, and tomograms inverted by the (b) 1D in-
version, (c) 2D WD, and (d) 3D WD methods. . . . . . . . . . . . . . 50
2.12 Slices of the (a) true, (b) 1D inversion, (c) 2D WD and (d) 3D WD
S-velocity models at y = 120 m, where the black dashed lines indicate
the interfaces with large velocity contrast. . . . . . . . . . . . . . . . 53
2.13 1D inversion results computed with the code SURF96 (Herrmann,
2013): (a) the observed (blue line) and the predicted (red triangles)
dispersion curves for CSG No. 30; (b) the initial (blue dashed line)
and the inverted (red solid line) S-velocity profiles. . . . . . . . . . . 54
2.14 Observed dispersion curves along the azimuth angles of (a) θ = 0◦ and
(b) θ = 180◦ for all the 2D CSGs located at y = 120 m, where the
black dashed lines, the cyan lines and the red dash-dot lines represent
the contours of the observed, initial and inverted dispersion curves,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.15 Observed dispersion curves for sources (a) A, (b) B, (c) C and (d) D
as indicated in Figure 3.10a, where the black dashed lines, the cyan
lines and the red dash-dot lines represent the contours of the observed,
initial and inverted dispersion curves, respectively. . . . . . . . . . . 55
2.16 Depth slices at z = 20 m of (a) the true S-velocity model and the
inverted tomograms computed by the (b) 1D inversion, (c) 2D WD
and (d) 3D WD methods, where the black dashed lines indicate the
large velocity contrast boundaries. . . . . . . . . . . . . . . . . . . . . 56
2.17 RMS error between the inverted S-velocity models by the 1D inversion,
2D WD and 3D WD methods and the true S-velocity model. . . . . . 57
12
2.18 Comparison between the observed (red) and synthetic (blue) traces
at far offsets predicted from the initial model (LHS panels) and 3D
tomogram (RHS panels) for CSG No.1 in (a) and (b), and CSG No.15
in (c) and (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.19 (a) Initial and (b) inverted 2D S-velocity models. The corresponding
true model is shown in Figure 2.12a. Here, the black dashed lines
indicate the large velocity contrast boundaries which are the same as
those in Figure 2.12. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.20 Comparison between the observed (red) and synthetic (blue) traces
predicted from the initial model (LHS panels, Figure 2.19a) and 2D
tomogram (RHS panels, Figure 2.19b) for CSG No.1. . . . . . . . . . 59
2.21 (a) Google map showing the location of the Qademah-fault seismic
experiment (Fu et al., 2018b). (b) Receiver geometry for the Qademah-
fault data, where the red dashed line indicates the location of Qademah
fault. The Green triangles represent the locations of receivers, where
the shots are located at each receiver. The red star represents the
location of source No. 132 and the black stars indicate the locations
of sources A, B, C and D on the surface. θ is the azimuth angle with
respect to the acquisition line of source No. 132. . . . . . . . . . . . . 60
2.22 Seismic traces of CSG No. 12 at the first line (a) before and (b) after
amplitude compensation; and its dispersion images for (c) θ = 0◦ and
(d) θ = 180◦. The two red dashed lines in (b) show the length of the
muting window which masks all other arrivals but the fundamental-
mode Rayleigh waves. The red asterisks in (c) and (d) represent the
maximum value for each frequency, and the blue lines are the picked
observed dispersion curves used for inversion. . . . . . . . . . . . . . . 64
2.23 Observed dispersion curves for (a) θ = 0◦ and (b) θ = 180◦ computed
from the 2D CSGs in the first line, where the black dashed lines, the
cyan lines and the red dash-dot lines represent the contours of the
observed, initial and inverted dispersion curves, respectively. . . . . . 65
2.24 Quality control of the picked dispersion curves by reciprocity, where
the stars represent the sources, and the rectangles represent the re-
ceivers. If the dispersion curves (red) of the CSG are the same as
those (blue) computed from the common receiver gather (CRG) at the
same location, it passes the reciprocity test. Passing the reciprocity
test is a necessary QC test all 3D data must pass prior to inversion. 65
13
2.25 1D dispersion curve inversion results by SURF96 (Herrmann, 2013):
(a) the observed (blue line) and the predicted (red triangles) dispersion
curves for CSG No. 12 (see Figure 2.22c); (b) the initial (blue dashed
line) and the inverted (red solid line) S-velocity profiles. . . . . . . . . 66
2.26 S-velocity tomograms from the 2D CSGs beneath the first line by the
(a) 1D inversion and (b) 2D WD methods. . . . . . . . . . . . . . . . 66
2.27 S-velocity tomograms inverted by the (a) 1D inversion, (b) 2D WD,
and (c) 3D WD methods. The red solid line labeled by “F1” indicates
the location of the conjectured Qademah fault and the dashed red line
labeled by “F2” is conjectured to be a small antithetic fault. The
low-velocity anomaly between faults “F1” and “F2” is the conjectured
colluvial wedge labeled by “CW”. . . . . . . . . . . . . . . . . . . . . 67
2.28 Observed dispersion curves for sources (a) A, (b) B, (c) C and (d) D
indicated in Figure 5.12. The black dashed lines, the cyan lines and
the red dash-dot lines represent the contours of the observed, initial
and inverted dispersion curves, respectively. . . . . . . . . . . . . . . 68
2.29 Comparison between the observed (blue) and synthetic (red) traces at
far source-receiver offsets predicted from the initial model (LHS panels)
and 3D WD tomogram (RHS panels) for CSG No.9 in (a) and (b). The
blue and red matched filters in (c) are calculated from the trace No. 76
(green) in (a) and (b), respectively. Comparison between the observed
(blue) and synthetic (red) traces after applying the matched filters in
(d) and (e). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.30 COGs with the offset of 30 m for the selected lines, where the blue and
red wiggles represent the observed and predicted COGs, respectively.
For each panel, a matched filter is calculated from the green trace and
then applied to the other traces. . . . . . . . . . . . . . . . . . . . . . 70
2.31 Slices of (a) the inverted S-wave velocity model, and (b) natural migra-
tion images (Liu et al., 2017a). The dashed lines indicate the location
of the interpreted Qademah fault. . . . . . . . . . . . . . . . . . . . . 71
2.32 (a) and (b): 2D zoom view of the dashed panels in Figure 2.31, com-
pared with (c) the COGs. . . . . . . . . . . . . . . . . . . . . . . . . 72
3.1 Schematic diagram shows the offset distance l along the (a) flat and
(b) irregular surfaces from the source at s (the red star) to the receiver
at r1, where le is the Euclidean distance. . . . . . . . . . . . . . . . . 87
14
3.2 Schematic diagram shows the offset L and the azimuth θ from the
source at s (red star) to the receiver at r. . . . . . . . . . . . . . . . 87
3.3 (a) Acquisition geometry where the yellow area shows the locations of
the receivers (black asterisks) within the azimuth angle ranged from
277.5◦ to 282.5◦ for the source at A, where the source is represented
by the red star; (b) paths of the geodesics on the topography from
the source at A to the receivers that are marked as the black asterisks
in (a); (c) differences between the geodesic and Euclidean distances,
where the trace number is numbered according to the geodesic distance
in ascending order; (d) CSG for trace No. 1 to 30 from the model with
(red) and without (blue) topography. . . . . . . . . . . . . . . . . . . 91
3.4 Dispersion image calculated by the (a) Euclidean and (b) geodesic
distances for the data recorded in the irregular surface. (c) Dispersion
image calculated for the data recorded in the flat surface. Here, the
green curves are the theoretical phase velocity dispersion curves (c =
919.4 m/s) and the red curves are the picked dispersion curves. . . . . 92
3.5 Dispersion curves for the data from the flat-surface model and their
contours are represented by the black dashed lines. Here, the cyan
lines and the red dash-dot lines represent the contours of the dispersion
curves calculated by the Euclidean and geodesic distances from the
model with the topography, respectively. . . . . . . . . . . . . . . . . 93
3.6 (a) True S-velocity checkerboard model and (b) S-velocity tomogram
by 3D TWD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.7 True S-velocity slices at y = (a) 80 m and (c) 160 m. Inverted S-velocity
slices at y = (b) 80 m and (d) 160 m. . . . . . . . . . . . . . . . . . . 94
3.8 Observed dispersion curves from the CSGs with their sources located
at points (a) A and (b) B (indicated in Figure 3.3a), where the black
dashed lines, the cyan and red dash-dot lines represent the contours of
the observed, initial and inverted dispersion curves, respectively. . . 95
3.9 Topography of the 3D Foothill model, where the red lines are the
geodesic paths for the source marked by the red star. . . . . . . . . . 97
3.10 (a) True S-velocity model, (b) corresponding mesh, (c) initial S-velocity
model and (d) S-velocity tomogram. . . . . . . . . . . . . . . . . . . 98
3.11 Acquisition geometry for the numerical tests with data generated for
the 3D Foothill model, where the red dots and blue circles indicate the
locations of the receivers and sources, respectively. . . . . . . . . . . . 98
15
3.12 Observed dispersion curves for the sources located at (a) A, (b) B, (c)
C and (d) D indicated in Figure 3.11b, where the black dashed lines,
the cyan dash-dot lines and the red lines represent the contours of the
observed, initial and inverted dispersion curves, respectively. . . . . . 99
3.13 Slices of the (a) true, (b) initial, and (c) inverted S-velocity models at
y = 433 m, where the black and white dashed lines indicate the large
velocity contrast boundaries and the boundaries 0.5 km below the free
surface, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.14 Depth slices 300 m below the surface for the (a) true, (b) initial and
(c) inverted Foothill S-velocity models, where the black dashed lines
indicate the large velocity contrast boundaries. . . . . . . . . . . . . . 100
3.15 Comparison between the observed (red) and synthetic (blue) traces
at far offsets predicted from the initial model (LHS panels) and 3D
tomogram (RHS panels) for CSG B in (a) and (b), and CSG C in (c)
and (d). Here, the locations of points B and C and the line numbers
are indicated in Figure 3.11. . . . . . . . . . . . . . . . . . . . . . . . 101
3.16 COGs with the offset of 2.85 km, which are retrieved from the traces
located at the green rectangles in Figure 3.11 of the CSGs with the
sources located at the green stars in Figure 3.11. Here the red and
blue wiggles represent the observed and predicted COGs, respectively. 102
3.17 (a) Map of the Washington fault and the survey site. The location of
the survey site is 5 km south of the Utah-Arizona border. (b) Topo-
graphic map around the seismic survey, where the red and green rect-
angles indicate the locations of the 3D seismic survey and the trench
site, respectively. (After Lund et al. (2015).) . . . . . . . . . . . . . . 102
3.18 Survey geometry for the 3D experiment in the Washington fault zone.
The open red circles denote the locations of sources and the solid blue
dots denote the locations of receivers. The dashed black line denotes
the location of the fault scarp. . . . . . . . . . . . . . . . . . . . . . . 103
3.19 Common shot gather # 87 of Washington fault data. . . . . . . . . . 103
3.20 Traveltime matrices before and after the correction of the acquisition
hardware error for the 2D data set on line #4. . . . . . . . . . . . . . 104
3.21 (a) Observed dispersion curves for the CSGs on Line # 4 along the
azimuthal angles (a) θ = 0◦ and (b) θ = 180◦, where the black dashed
lines, the cyan dash-dot lines and the red lines represent the contours
of the observed, initial and inverted dispersion curves, respectively. . 108
16
3.22 (a) Initial and (b) inverted S-wave velocity models beneath line #4.
(c) P-wave velocity tomogram calculated from the picked traveltimes
in Figure 3.20b. (d) Vp/Vs ratio tomogram beneath line #4. Here
the white lines indicate the boundaries 10 m below the free surface.
The trench is excavated in the locations of the black rectangles. The
lines labeled with “F1” and “F2” are interpreted as the locations of
the main fault and the antithetic fault. The line labeled with “F3” is
the location of another possible fault. “CW” represents the colluvial
wedge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.23 (a) Initial, (b) 2D and (c) 3D S-wave velocity tomograms. Here, the
depth and S-wave velocity of the initial model are calculated by scaling
the wavelength and phase velocity with factors of 0.5 and 1.1, respec-
tively (Liu et al., 2019). . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.24 Comparison between the observed (blue) and synthetic (red) traces
predicted from the (a) initial and (b) inverted S-velocity models for
CSG # 128. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.25 COGs with the offset of 16 m for line # 4 calculated from the (a) initial
and (b) inverted S-velocity models, where the blue and red wiggles
represent the observed and predicted COGs, respectively. . . . . . . 112
3.26 Observed COGs with the offset of 16 m are superposed on the S-
velocity tomogram, where the COGs are adjusted by following the
topography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.27 Zoom views of (a) S-velocity and (b) P-velocity tomograms and (c)
Vp/Vs tomogram in Figure 3.22. (d) Ground truth extracted from a
nearby trench log (Lund et al., 2015; Hanafy et al., 2015). . . . . . . 114
3.28 Schematic diagram of the calculation of the geodesic on a simple surface
mesh by unfolding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.1 (a) Common shot gather d(g, t) and (b) the fundamental dispersion
curve for Rayleigh waves in the kx − ky − ω domain. Here, θ is the
azimuth angle, and κ(θ, ω) is the skeletonized data. . . . . . . . . . . 122
4.2 True (a) and initial (b) S-velocity models together with the S-velocity
tomograms obtained using WD with maximum offsets of (c) R = 8 m
and (d) R = 20 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
17
4.3 Plot of residual vs iteration number for the synthetic examples. The
Y-axis represents the normalized wavenumber residual, and the blue
and red lines represent the WD results with R = 20 m for the data
collected from the model in Figs. 4.2a and 4.6a, respectively. . . . . 129
4.4 Observed dispersion contours for (a) azimuth angle θ = 0◦ with the
maximum offset R = 8 m, (b) θ = 180◦ with R = 8 m, (c) θ = 0◦ with
R = 20 m, and (d) θ = 180◦ with R = 20 m, where the black dashed
lines, the cyan dash-dot lines and the red lines represent the contours of
the observed, initial and inverted dispersion curves, respectively. Here,
the background images are the picked wavenumber for all the common
shot gathers. The shot number is determined to make sure that the
maximum offset is at least 8 m in (a) and (b). For comparison, we
also use the same shot number range in (c) and (d), but the maximum
offset of some of the shots may be less than 20 m. For example, in (c),
only shot no. 1-28 has the maximum offset of 20 m for azimuth 0. . 130
4.5 Vertical-velocity profiles at (a) X = 20 m and (b) X = 38 m for the
true model (blue line), the initial model (black dash-dot line) and the
inverted S-velocity tomograms when R=8 m (magenta line) and R=20
m (red line) shown in Fig. 4.2. . . . . . . . . . . . . . . . . . . . . . . 131
4.6 True (a) and initial (b) S-velocity models together with the S-velocity
tomograms obtained using WD with maximum offsets of (c) R = 8 m
and (d) R = 20 m. The high-velocity anomalies in (a) are 2 m deeper
than the one shown in Fig. 4.2a. . . . . . . . . . . . . . . . . . . . . . 133
4.7 Observed dispersion curves for (a) azimuth angle θ = 0◦ with the
maximum offset R = 8 m, (b) θ = 180◦ with R = 8 m, (c) θ = 0◦ with
R = 20 m, and (d) θ = 180◦ with R = 20 m. The black dashed, cyan
dash-dot and red lines represent the contours of the observed, initial
and inverted dispersion curves, respectively. . . . . . . . . . . . . . . 133
4.8 Vertical-velocity profiles at (a) X = 20 m and (b) X = 38 m for the
true model (blue line), the initial model (black dash-dot line) and the
S-velocity tomograms by setting R=8 m (magenta line) and R=20 m
(red line) shown in Fig. 4.6. . . . . . . . . . . . . . . . . . . . . . . . 134
4.9 Frequency spectrum of the observed data, which are divided into eleven
frequency bands. The frequency bands are plotted as horizontal bars
with their corresponding number tags. . . . . . . . . . . . . . . . . . 136
18
4.10 Depth windows for frequency bands 9 (blue solid line) and 10 (red
dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.11 (a) Initial S-velocity model. (b)-(g) S-velocity tomograms for Steps 1
to 11 with an interval of 2 (Table 4.1). (h) True S-velocity model. . . 137
4.12 Observed dispersion curves (azimuth angle θ = 0◦) for Steps 1 to 11
with an interval of 2 listed in Table 4.1, where the black dashed, cyan
dash-dot and red lines represent the contours of the observed, initial
and inverted dispersion curves, respectively. . . . . . . . . . . . . . . 138
4.13 Vertical-velocity profiles at (a) X = 20 m and (b) X = 38 m for the
true model (blue lines), the initial model (black lines), the inverted
tomograms with (red lines) and without (magenta lines) layer stripping.139
4.14 (a) First CSG, (b) its dispersion image with the maximum offset R=500 m,
and (c) the picked dispersion curve for the fundamental-mode surface
waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.15 Observed dispersion curves for (a) θ = 0◦ and (b) θ = 180◦, where the
black dashed lines, the cyan lines and the red dash-dot lines represent
the contours of the observed dispersion curves, the predicted dispersion
curves obtained without and with layer stripping, respectively. . . . . 142
4.16 (a) initial S-velocity Model; the S-velocity tomograms inverted us-
ing the WD methods (b) without and (c) with multiscale and layer-
stripping strategy; (d) the P-velocity tomogram calculated by travel-
time tomography (Huang et al., 2018). . . . . . . . . . . . . . . . . . 143
4.17 Comparison between the observed (red) and synthetic (blue) traces
from the S-velocity tomogram (a) without and (b) with layer-stripping
methods for CSG No. 30. For each panel, a match filter is calculated
from the black trace and then applied to the other traces. . . . . . . . 144
4.18 Comparison between the observed (blue) and synthetic (red) common-
offset gathers (COGs) with the offset of 335 m from the S-velocity
tomogram without (a) and with (b) layer-stripping method. For each
panel, a match filter is calculated from the black trace and then applied
to the other traces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.1 Natural migration workflow for active-source data. . . . . . . . . . . . 153
5.2 3D S-wave velocity model used for the synthetic tests with a 30-by-15
source and receiver array on the surface. . . . . . . . . . . . . . . . . 154
19
5.3 a) Common shot gather generated from the 3D model. The moveout
velocity of the red dashed lines for the separation of transmitted and
backscattered surface waves is about 500 m/s. The near-source arrivals
are muted along the yellow lines (about 0.1 s). b) Transmitted surface
waves. c) Backscattered surface waves. . . . . . . . . . . . . . . . . . 155
5.4 a) Migration images at z = 0 m computed from the synthetic data
with the narrow-band filters from 1 to 7 (center frequencies change
from 45 Hz to 15 Hz with a 5 Hz interval). The two red dashed lines
are at x = 129 m and 174 m, respectively, and the z axis denotes
pseudodepth calculated from the mapping of frequency to the depth
of 1/3 wavelength. b) Upper portion of the Vs-velocity model and the
red dashed lines are taken from a). . . . . . . . . . . . . . . . . . . . 157
5.5 a) Inline common shot gather for the source at x = 0 m and y = 0 m,
b) its estimated phase velocity dispersion curve, and c) the curve that
plots 1/3 wavelength against frequency. . . . . . . . . . . . . . . . . . 158
5.6 Migration images at z = 0 m computed from the synthetic data with
a finer source and receiver spacing of 6 m, where the two red dashed
lines are at x = 129 m and 174 m, respectively, and the z axis denotes
pseudodepth calculated from the mapping of frequency to the depth of
1/3 wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.7 60th common shot gather from the Aqaba data. . . . . . . . . . . . . 160
5.8 Solid lines denote the amplitude spectra of the nine band-pass filters;
Dashed line denote the amplitude sepctrum of all 120 shot gathers in
the Aqaba data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.9 a) 60th common shot gather filtered by the band-pass filter of 35-45
Hz; b) transmitted surface waves and c) backscattered surface waves
obtained by tapered muting of events above the inclined dashed lines. 162
5.10 a) Migration images for the Aqaba data with nine narrow-band filters,
where the z axis is pseudodepth calculated from 1/3 the wavelength,
b) traveltime tomogram, and c) common offset gather (COG) with
7.5 m offset. The locations denoted by 2-4 are clearly associated with
horizontal velocity anomalies in all three illustrations; the horizontal
velocity anomaly denoted by location 1 is also seen in the traveltime
tomogram. A normal fault breaks the surface at location 2. . . . . . . 163
5.11 a) 1st common shot gather of the Aqaba data, b) phase-velocity disper-
sion curve and c) the curve that plots 1/3 wavelength against frequency.164
20
5.12 Receiver geometry for the Qadema-fault data. Shots are located at
each geophone, and a total of 288 shot gathers are migrated using
equation 5.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.13 a) Common shot gather no. 121 from the Qadema-fault data and b)
the amplitude sepctrum for all 288 shot gathers. . . . . . . . . . . . 165
5.14 ) Common shot gather no. 121 from the Qadema-fault data filtered
by a 20-30 Hz band-pass filter and b) the separated transmitted waves
along the red dip lines (slope = 140 m/s); c) the separated backscat-
tered waves along the horizontal red line (about 0.1 s). . . . . . . . . 166
5.15 a) Migration images of the Qademah-fault data filtered by eight narrow-
band filters, where the center frequencies range from 41 Hz (filter 1)
to 13 Hz (filter 8). b) 3D Rayleigh phase-velocity tomogram (Hanafy,
2015). The location of the Qademah fault indicated by the black lines
in the migration images shown in panel a) correlate with the S-velocity
tomogram shown in b). There is no visible indication of the fault on the
free surface. The dip angle of the fault interpreted from this migration
image is similar to that estimated from the tomogram. . . . . . . . . 167
5.16 a) Common shot gather for traces along the x direction for the first
source shown in Figure 5.12, b) estimated phase-velocity dispersion
curve, and c) wavelength/3 plotted against frequency. . . . . . . . . . 169
21
Chapter 1
Introduction
Determining geological changes in the earth’s subsurface is important for studies
in geological engineering, hydrocarbon exploration, and tectonics. Surface waves are
suitable for imaging near-surface heterogeneities because the recorded seismic data are
usually dominated by surface waves for a wide range of source-receiver offsets within
the recorded time window. Inverting these surface waves can give the background
S-wave velocity model with smooth lateral changes (low-wavenumber of the S-wave
velocity model, e.g., Figure 1.1) and a reflectivity image with sharp lateral reflectivity
variations (high-wavenumber of the S-wave velocity model, e.g., Figure 1.2). The
background S-wave velocity and reflectivity images can be calculated by surface-wave
inversion and migration methods, respectively.
This dissertation develops three novel surface-wave inversion methods in Chapters
2, 3 and 4, which can accurately reconstruct the 3D S-wave velocity model of a
laterally heterogeneous medium. This approach can be used for data recorded on flat
or irregular free surfaces and has much less of a tendency of getting stuck in a local
minimum compared to conventional inversion methods. Then, a novel surface-wave
migration method, natural migration, is developed for active seismic data in Chapter
5, and its advantage over other migration methods is that no velocity model is needed.
22
a
a) True S-wave Velocity Model
b) S-wave Velocity Tomogram
Figure 1.1: (a) True and (b) inverted S-wave velocity models, where full waveforminversion is used. (after Yuan et al. (2015).)
a) True S-wave Velocity Model
b) Re ectivity Image
Figure 1.2: (a) True S-wave velocity model and (b) the reflectivity image. (afterHyslop and Stewart (2015).)
23
1.1 Surface-wave Inversion
Background
There are many methodologies that can calculate the S-wave velocity model from sur-
face waves. The conventional dispersion-inversion method estimates the 1D S-wave
velocity model directly from the surface-wave dispersion curves (Haskell, 1953; Xia
et al., 1999, 2002; Park et al., 1999) by assuming a horizontally layered medium be-
neath the recording data. Unfortunately, this layered-medium assumption is violated
when there are strong lateral gradients in the S-velocity model, such as faults, vugs
or gas channels. To avoid the layered medium approximation, Fang et al. (2015)
developed a surface-wave phase inversion method where the phase is computed along
the surface-wave raypaths computed by ray tracing. The S-wave velocity model is
then adjusted until the predicted phases match those of the recorded surface waves.
This methodology is computationally efficient and robust, but it suffers from the
high-frequency approximation of ray tracing. As an alternative, full-waveform in-
version (FWI) (Groos et al., 2014; Perez Solano et al., 2014; Dou and Ajo-Franklin,
2014; Groos et al., 2017) estimates the S-velocity model that accurately predicts the
surface waves recorded in a heterogeneous S-velocity model. No high-frequency ap-
proximation is required and, consequently, can theoretically achieve λ/2 resolution
in the estimated velocity model. But in practice, FWI can easily get stuck in a
local minimum due to the strongly dispersive nature of surface waves and an inad-
equate initial velocity model. Tape et al. (2010) changed the misfit of FWI to the
frequency-dependent multitaper traveltime differences and the gradient of the mis-
fit function is computed using an adjoint technique. To avoid the assumption of a
layered medium and also mitigate FWI’s sensitivity to getting stuck in a local min-
imum, Li and Schuster (2016) and Li et al. (2017c) proposed a new surface-wave
dispersion inversion method, which is denoted as wave-equation dispersion inversion
24
(WD). Later, Li et al. (2017e, 2019b) developed 2D topographic WD (i.e., topographic
WD, also denoted as TWD) which incorporates the free-surface topography into the
finite-difference solutions of the elastic wave equation.
Problems
• It is expected that the 2D assumptions for the subsurface model cannot fully
approximate wave propagation in the presence of significant 3D variations in
subsurface geology.
• Ignoring topography in 3D surface-wave inversion can lead to significant errors
in the inverted model.
• The iterative WD method can suffer from the local minimum problem when
inverting seismic data from complex Earth models.
Solutions
To solve the problems listed above, I proposed the following solutions:
• In Chapter 2, the 2D wave-equation dispersion inversion method is extended
to 3D wave-equation dispersion inversion of surface waves for the shear-velocity
distribution. The synthetic and field data examples demonstrate that 3D WD
can accurately reconstruct the 3D S-wave velocity model of a laterally hetero-
geneous medium and has much less of a tendency to getting stuck in a local
minimum compared to full waveform inversion. The results from the synthetic
and field data examples show a noticeable improvement in the accuracy of the
3D tomogram compared to 2D tomographic inversion if there are significant
3D lateral velocity variations. The results are written up in a paper which is
published in Geophysics (Liu et al., 2019).
25
• In Chapter 3, we develop a 3D topographic WD (TWD) method that takes
into account the topographic effects modeled by a 3D spectral element solver.
Numerical tests on both synthetic and field data demonstrate that 3D TWD
can accurately invert for the S-velocity model from surface-wave data recorded
on irregular topography. Our results from the field data tests suggest that,
compared to the 3-D P-wave velocity tomogram, the 3D S-wave tomogram
agrees much more closely with the geological model taken from the trench log.
The agreement with the trench log is even better when the Vp/Vs tomogram
is computed, which reveals a sharp change in velocity across the fault. The
localized velocity anomaly in the Vp/Vs tomogram is in very good agreement
with the well log. Our results suggest that integrating the Vp and Vs tomograms
can sometimes give the most accurate estimates of the subsurface geology across
normal faults. The results are written up in a paper which is submitted to
Geophysics.
• In Chapter 4, we develop a multiscale, layer-stripping method to alleviate the
local minimum problem of wave-equation dispersion inversion of Rayleigh waves
and improve the inversion robustness. We use a synthetic model to illustrate
the local minima problem of wave-equation dispersion inversion and how our
multiscale and layer-stripping wave-equation dispersion inversion method can
mitigate the problem. We demonstrate the efficacy of our new method using
field Rayleigh-wave data. The results are written up in a paper which is pub-
lished in Geophys. J. Int. (Liu and Huang, 2019).
26
1.2 Surface-wave Migration
Background
Most of the surface-wave inversion methods invert only the transmitted surface waves
for an S-wave velocity model with smooth lateral changes. However, the backscattered
surface waves can be used to obtain a near-surface image of the S-wave reflectivity
by the surface-wave migration method.
The conventional surface-wave imaging methods are based on the Born approx-
imation of surface waves, which requires an estimation of the background velocity
model and the weak-scattering approximation (Snieder, 1986a; Yu et al., 2014). Al-
Theyab et al. (2015, 2016) introduced the natural migration (NM) method to image
the near-surface heterogeneities, assuming that the scattering bodies are within a
depth of about 1/3 wavelength from the free surface. There are several benefits of
the NM method. First, no Born approximation is used so that strongly scattered
events can be migrated to the surface-projection of their origin. Second, no velocity
model is needed because the Green’s functions in the migration kernels are recorded as
band-limited shot gathers, where the sources and receivers are located on the surface.
Problem
AlTheyab et al. (2016) demonstrated the effectiveness of the NM method with ambi-
ent noise data, but did not show it to be effective for controlled source data.
Solution
In Chapter 5, we have developed a methodology for detecting the presence of near-
surface heterogeneities by naturally migrating backscattered surface waves in controlled-
source data. Results with synthetic data and field data recorded over known faults
validate the effectiveness of this method. Migrating the surface waves in recorded
27
2D and 3D data sets accurately reveals the locations of known faults. This work is
published in Geophysics (Liu et al., 2017a).
28
Chapter 2
3D Wave-equation Dispersion Inversion of Rayleigh Waves1
The 2D wave-equation dispersion inversion (WD) method is extended to 3D wave-
equation dispersion inversion of surface waves for the shear-velocity distribution. The
objective function of 3D WD is the frequency summation of the squared wavenumber
κ(ω) differences along each azimuth angle of the fundamental or higher modes of
Rayleigh waves in each shot gather. The S-wave velocity model is updated by the
weighted zero-lag cross-correlation between the weighted source-side wavefield and the
back-projected receiver-side wavefield for each azimuth angle. A multiscale 3D WD
strategy is provided, which starts from the pseudo 1D S-velocity model, which is then
used to get the 2D WD tomogram, which in turn is used as the starting model for 3D
WD. The synthetic and field data examples demonstrate that 3D WD can accurately
reconstruct the 3D S-wave velocity model of a laterally heterogeneous medium and
has much less of a tendency to getting stuck in a local minimum compared to full
waveform inversion.
2.1 Introduction
Obtaining a reliable S-wave velocity model of the near surface is important for
many groundwater, engineering, scientific and environmental studies (Xia et al., 1999;
Woodhouse and Dziewonski, 1984). In this regard, inversion of the surface-wave dis-
persion curves is one of the most reliable imaging tools for the near-surface S-velocity
1This manuscript was published as: Zhaolun Liu, Jing Li, Sherif M. Hanafy, and Gerard Schuster,(2019), ”3D wave-equation dispersion inversion of Rayleigh waves,” Geophysics 84 (5): R673-R691,doi: https://doi.org/10.1190/geo2018-0543.1
29
distribution. An advantage of surface-wave imaging over body-wave imaging is that
the seismic energy of surface waves spreads out as 1/r from the source, compared to
the 1/r2 geometrical spreading of body waves (Aki and Richards, 2002). Here, r is
the distance along the horizontal propagation path between the source and receiver
on the free surface. Thus, the recorded data are usually dominated by surface waves
for a wide range of source-receiver offsets within the time window of surface-wave ar-
rivals. A practical use of surface waves is that they can be inverted to detect shallow
drilling hazards down to the depth of about the dominant shear wavelength (Ivanov
et al., 2013).
The conventional dispersion-inversion method calculates the S-wave velocity model
directly from the surface-wave dispersion curves (Haskell, 1953; Xia et al., 1999, 2002;
Park et al., 1999) by assuming a 1D velocity profile beneath the recording data. Un-
fortunately, this assumption is violated when there are strong lateral gradients in
the S-velocity model, such as faults, vugs or gas channels. To partially mitigate this
problem, spatial interpolation of 1D velocity models (Pan et al., 2016) and later-
ally constrained inversion (Socco et al., 2009; Bergamo et al., 2012) can be used to
compute an approximation to the 2D S-velocity model.
As an alternative, full-waveform inversion (FWI) (Groos et al., 2014; Perez Solano
et al., 2014; Dou and Ajo-Franklin, 2014; Groos et al., 2017) estimates the S-velocity
model that accurately predicts the surface waves recorded in a heterogeneous S-
velocity model. But in practice, FWI can easily get stuck in a local minimum due
to the strongly dispersive nature of surface waves and an inadequate initial veloc-
ity model. To mitigate this problem, Perez Solano et al. (2014) changed the misfit
function of FWI into the l2 misfit of magnitude spectra of surface waves, and their
synthetic data results showed this to be an effective method for reconstructing the
S-wave velocity model at the near surface. Until now there are few studies to assess
the full benefits and limitations of this method so its effectiveness on a wider variety
30
of data sets is still to be determined.
To combine the inversion of both surface waves with body waves, Yuan et al.
(2015) developed a wavelet multi-scale adjoint method for the joint inversion of both
surface and body waves. The efficacy of this method is validated with synthetic data.
However, further studies are needed to assess its capabilities. To enhance robustness,
layer stripping FWI of surface waves was presented by Masoni et al. (2016) who first
invert the high-frequency and near-offset data for the shallow S-velocity model, and
gradually incorporate lower-frequency data with longer offsets to estimate the deeper
parts of the model. This procedure partly mitigates the local minima problem. All of
these methods, however, are still under development and require more tests to fully
understand their relative benefits and limitations.
To avoid the assumption of a layered medium and also mitigate FWI’s sensitivity
to the local minima, Li and Schuster (2016) and Li et al. (2017c) proposed a new
surface-wave dispersion inversion method, which is denoted as wave-equation dis-
persion inversion (WD). The WD method skeletonizes the complicated surface-wave
arrivals as simpler data, namely the picked dispersion curves in the wavenumber-
angular frequency (k−ω) domain. These curves are obtained by applying a combina-
tion of temporal Fourier and spatial Radon transforms to the Rayleigh waves recorded
by vertical-component geophones. The sum of the squared differences between the
wavenumbers along the predicted and observed dispersion curves is used as the objec-
tive function. The solution to the elastic wave equation and an iterative optimization
method are then used to invert these curves for the S-wave velocity models. Numeri-
cal tests on the 2D synthetic and field data show that WD can accurately reconstruct
the S-wave velocity distributions in laterally heterogeneous media. The WD method
also enjoys robust convergence because the skeletonized data, namely the dispersion
curves, are simpler than traces with many dispersive arrivals. The penalty, however, is
that the inverted S-velocity model has lower resolution than a model that accurately
31
fits both the waveform and phase information. Recently, Fu et al. (2018a) showed
that inverting only the phase information and ignoring the amplitudes of body waves
gives almost the same resolution as obtained by full waveform inversion.
In this paper, we extend the 2D WD method to invert dispersion curves for the
3D S-wave velocity model that accounts for strong velocity variations in all three
dimensions. After the introduction, we describe the theory of 3D WD and its im-
plementation. We also discuss the multiscale procedure for estimating a good initial
model for 3D WD: first use the 1D dispersion-curve inversion method and then use
the 2D WD method. Numerical tests on synthetic and field data are presented in
the third section to validate the theory. The limitations of the proposed method are
discussed in the fourth section and a summary is given in the last section.
2.2 Theory
Let d(g, t) denote a shot gather of vertical particle-velocity traces recorded by the
vertical-component geophone on the surface at g = (xg, yg, 0). The surface waves
are excited by a vertical-component force on the surface at s = (xs, ys, 0), where the
horizontal recording plane is at z = 0. We will assume that the effects of attenuation
on the dispersion curves are insignificant. But, if important, such effects can be
accounted for by using solutions to viscoelastic wave equation (Li et al., 2017a,b).
Assume the data have been filtered so that d(g, t) only contains the fundamental
mode of Rayleigh waves. A 3D Fourier transform is then used to transform d(g, t)
into D(k, ω) in the k− ω domain:
D(k, ω) =
∫ ∞−∞
∫ ∞−∞
∫ ∞−∞
d(g, t)e−i(k·g+ωt)dgdt (2.1)
=
∫ ∞−∞
∫ ∞−∞
D(g, ω)e−ik·gdg
32
where dg = dxgdyg and D(g, ω) represents the data in the space-frequency (x − ω)
domain. Here, the z = 0 notation is silent. The wavenumber vector k = (kx, ky) can
be represented in polar coordinate as (k, θ), where θ = arctan kykx
is the azimuth angle
and k =√k2x + k2
y is the radius. Following this notation, the Fourier transformed
data D(k, ω) are denoted as D(k, θ, ω). We skeletonize the spectrum D(k, θ, ω) as
the dispersion curves associated with the fundamental mode of the Rayleigh waves,
which are the wavenumbers κ(θ, ω) obtained by picking the (k, θ, ω) coordinates of the
fundamental dispersion curve.2 This curve is recognized as the maximum magnitude
spectrum D(k, θ, ω) along the azimuth angle θ and is denoted as κ(θ, ω)obs for the
observed data. In this paper, we assume that the dispersion curves are those for
Rayleigh waves recorded by vertical-component geophones, but this approach is also
valid for Love waves or guided waves at the near surface (Li et al., 2018b).
2.2.1 Misfit Function
The 3D WD method inverts for the S-wave velocity model that minimizes the objec-
tive function J of the dispersion curve for a single shot gather:
J =1
2
∑ω
∑θ
[
residual=∆κ(θ,ω)︷ ︸︸ ︷κ(θ, ω)pre − κ(θ, ω)obs]
2 + penalty term, (2.2)
where the penalty term can be any model-based function that penalizes solutions
far from an a priori model. Here, κ(ω, θ)pre represents the predicted dispersion curve
picked from the simulated spectrum along the azimuth angle θ and κ(ω, θ)obs describes
the observed dispersion curve obtained from the recorded spectrum along the azimuth
θ. For pedagogical clarity, we will ignore the penalty term in further manipulations
of the objective function.
2Higher-order modes can also be picked and inverted.
33
The gradient γ(x) of J with respect to the S-wave velocity vs(x) is given by
γ(x) =∂J
∂vs(x)=∑ω
∑θ
∆κ(θ, ω)∂κ(θ, ω)pre∂vs(x)
, (2.3)
so that the optimal S-wave velocity model vs(x) is obtained from the steepest-descent
formula (Nocedal and Wright, 2006)
vs(x)(k+1) = vs(x)(k) − αγ(x), (2.4)
where α is the step length and the superscript (k) denotes the kth iteration. In
practice, all shot gathers are inverted simultaneously by including a summation over
all shot indices in equation 4.1, and a preconditioned conjugate gradient method is
preferred for faster convergence.
2.2.2 Connective Function
The Frechet derivative ∂κ(θ,ω)pre∂vs(x)
in equation 4.2 is derived by forming a connective
function that relates the dispersion curve κ(θ, ω)pre to the S-wave velocity model
vs(x) (Luo and Schuster, 1991a,b; Li et al., 2017d; Lu et al., 2017; Schuster, 2017).
This connective function Φ(κ, vs(x)) is defined as the cross-correlation between the
predicted D(k, θ, ω) and observed D(k, θ, ω)obs spectra along a specified azimuth θ at
frequency ω in the k− ω domain:
Φ(κ, vs(x)) = R
{∫D(k + κ, θ, ω)∗obsD(k, θ, ω)dk
}, (2.5)
where R denotes the real part and the superscript ∗ stands for the complex conjuga-
tion. Here, κ is an arbitrary wavenumber shift between the predicted and observed
spectra at frequency ω and along the azimuth angle θ. We seek the value of κ that
shifts the predicted spectrum D(k, θ, ω) so that it “best” matches the observed spec-
34
trum D(k, θ, ω)obs. The criterion for “best” match is defined as the wavenumber
residual ∆κ that maximizes the cross-correlation function Φ(κ, vs(x)) in equation 2.5
and the predicted data D is an implicit function of the shear-velocity model vs(x).
In this case, the derivative of the cross-correlation function Φ with respect to the
wavenumber shift κ should be zero at ∆κ:
Φ(κ, vs(x))|κ=∆κ = R
{∫˙D(k + ∆κ, θ, ω)∗obsD(k, θ, ω)dk
}= 0, (2.6)
where ˙D(k + ∆κ, θ, ω)obs = ∂D(k+κ,θ,ω)obs∂κ
|κ=∆κ. Equation 2.6 connects the S-wave
velocity model with the dispersion curve which will be used to derive the Frechet
derivative ∂κ(θ,ω)pre∂vs(x)
.
2.2.3 Frechet Derivative
For equation 2.6, the implicit function theorem (Luo and Schuster, 1991a,b; Li et al.,
2017d; Lu et al., 2017; Schuster, 2017) implies that ∆κ is an implicit function of vs(x)
so that
dΦ =∂Φ
∂vs(x)dvs(x) +
∂Φ
∂∆κd∆κ = 0. (2.7)
Rearranging this equation gives the Frechet derivative
∂∆κ
∂vs(x)=
∂κpre∂vs(x)
= −∂Φ/∂vs(x)
∂Φ/∂∆κ, (2.8)
where the denominator is the normalization term
A =∂Φ
∂∆κ= R
{∫¨D(k + ∆κ, θ, ω)∗obsD(k, θ, ω)dk
}, (2.9)
35
and the numerator is
∂Φ(∆κ, vs(x))
∂vs(x)= R
{∫˙D(k + ∆κ, θ, ω)∗obs
∂D(k, θ, ω)
∂vs(x)dk
}. (2.10)
Inserting equations 2.9 and 2.10 into equation 2.8 gives the Frechet derivative
∂κpre∂vs(x)
= −∂Φ/∂vs(x)
∂Φ/∂∆κ= −
R
{∫˙D(k + ∆κ, θ, ω)∗obs
∂D(k, θ, ω)
∂vs(x)dk
}A
. (2.11)
The integral with respect to the wavenumber k in equation 2.11 can be transformed
into an integral with respect to the receiver location g (see Appendix A), so that
equation 2.11 becomes
∂κpre∂vs(x)
= −R
{∫dg∂D(g, ω)
∂vs(x)D(g, θ, ω)∗obs
}A
, (2.12)
where D(g, ω) is the inverse Fourier transform of D(k, θ, ω), and D(g, θ, ω)∗obs is the
weighted conjugated data function defined in equation 2.21:
D(g, θ, ω)∗obs = 2πig · neig·n∆κ
∫C
D(g′, ω)∗obsdg′. (2.13)
Here, n = (cos θ, sin θ) and C is the line described by (g′ − g) · n = 0. Figure 2.1
depicts the procedure for calculating the weighted conjugated data D(g, θ, ω)∗obs.
For equation 4.3, ∂D(g,ω)∂vs(x)
can be obtained according to the Born approximation
for elastic waves (see Appendix B):
∂D(g, ω)
∂vs(x)= 4vs0(x)ρ0(x)
{G3k,k(g|x)Dj,j(x, ω)− 1
2G3n,k(g|x)
[Dk,n(x, ω) +Dn,k(x, ω)
]},
(2.14)
36
Figure 2.1: Schematic diagram showing how to calculate the weighted conjugateddata D(g, θ, ω)∗obs, where the red star represents the source, the black solid squareshows the geophone location at g and the red solid squares represent the geophonesalong the line C which satisfies (g′ − g) · n = 0. For the azimuth θ and position g,D(g′, θ, ω)∗obs is integrated along the dashed line with the weighting term 2πiLei∆κL,where L = g · n. The blue dot at gc is the stationary point for a homogeneous half-space, and the line integral in equation 4.4 can be approximated by D(gc, ω)obs (seeAppendix A for the detailed derivation).
37
where vs0(x) and ρ0(x) are the reference S-velocity and density models, respectively,
at location x. Di(x, ω) denotes the ith component of the particle velocity recorded
at x due to a vertical-component force. Einstein notation is assumed in equation 4.7
where Di,j = ∂Di
∂xjfor i, j ∈ {1, 2, 3}. The 3D harmonic Green’s tensor G3j(g|x) is
the particle velocity at location g along the jth direction due to a vertical-component
force at x in the reference medium.
2.2.4 Gradient Update
Plugging equations 4.3 and 4.7 into equation 4.2 gives the final expression for the
gradient:
γ(x) =∂J
∂vs(x)= −
∑ω
4vs0(x)ρ0(x)
AR
{backprojected data=Bk,k(x,s,ω)∗︷ ︸︸ ︷∫ ∑
θ
∆κ(θ, ω)D(g, θ, ω)∗obsG3k,k(g|x)dg
source=fj,j(x,s,ω)︷ ︸︸ ︷Dj,j(x, ω)
backprojected data=Bn,k(x,s,ω)∗︷ ︸︸ ︷−1
2
∫ ∑θ
∆κ(θ, ω)D(g, θ, ω)∗obsG3n,k(g|x)dg
source=fn,k(x,s,ω)︷ ︸︸ ︷[Dk,n(x, ω) +Dn,k(x, ω)
]}, (2.15)
where fi,j(x, s, ω) for i and j ∈ {1, 2, 3} is the downgoing source field at x, and
Bi,j(x, s, ω) for i and j ∈ {1, 2, 3} is the backprojected scattered field at x. The above
equation indicates that the gradient is computed by a weighted zero-lag correlation
of the source-side and receiver-side wavefields.
From equation 4.8, we can see that the back-propagated data (adjoint source) for
azimuth angle θ in the time domain are given by
D(θ,g, t) = F−1(∆κ(θ, ω)D(g, θ, ω)obs), (2.16)
38
where F−1 is the inverse Fourier transform operator. Inserting equation 4.8 into equa-
tion 4.9 gives the steepest-descent formula for updating the S-wave velocity model:
vs(x)(k+1) = vs(x)(k)+
α∑ω
4vs(x)ρ(x)
AR
{Bk,k(x, s, ω)∗fj,j(x, s, ω) +Bn,k(x, s, ω)∗fn,k(x, s, ω)
}. (2.17)
2.3 Workflow and Implementation
The workflow for implementing the 3D WD method is summarized in the following
six steps.
1. Remove the first-arrival body waves and higher-order modes of the Rayleigh
waves in the shot gather (Li et al., 2017c).
2. Determine the range of the azimuth angles θ for each shot gather.
3. Apply a 3D Fourier transform or the frequency-sweeping method (Park et al.,
1998) to the predicted and observed common shot gather (CSG) to compute the
dispersion curves κ(θ, ω) and κ(θ, ω)obs along each azimuth angle θ. Calculate
the sum of the squared dispersion residuals in equation 4.1.
4. Calculate the weighted conjugated data D(g, ω)∗obs according to equation 4.4,
which is then used for constructing the backprojected data D(θ,g, t) in equation
2.16. The source-side wavefield fi,j(x, s, ω) in equation 4.8 is also computed by
a finite-difference solution to the 3D elastic wave equation.
5. Calculate and sum the gradients for all the shot gathers. Source illumination is
sometimes needed as a preconditioner (Plessix and Mulder, 2004).
6. Calculate the step length and update the S-wave tomogram using the steepest-
descent or conjugate gradient methods. In practice we use a conjugate gradient
method.
39
2.3.1 3D Dispersion Curves for 3D Data
Figure 2.2: Plan view of the areal acquisition, in which the red star represents thesource, and the grid points at the line crossings represent the locations of geophones.
This subsection describes how to compute the 3D dispersion curves of a shot
gather. Here, we assume a 3D seismic survey (Boiero et al., 2011), where either shots
or receivers are located on a dense areal grid (Figure 2.2). Figures 2.3a and 2.3b
depict a CSG and its spectrum respectively. The frequency slice of the spectrum at
50 Hz is displayed in Figure 2.3c, which shows that the dispersion curves are distinct
only at a range of azimuth angles, denoted as the “dominant azimuth angles”. For
example, the dispersion curve at θ1 is much more pronounced than that at θ2 because
there are more geophones with azimuth θ1. In practice, the dominant angles can be
determined by the following two steps:
• Calculate the distances H(θ) from the source point to all the geophones along
40
(a) 3D CSG (b) Spectrum of 3D CSG
(c) Frequency Slice at 50 Hz
-0.123 0.1260
kx (1/m)
ky (
1/m
)
-0.123
0
0.126
Dominant
azimuths
(d) Picked 3D κ(kx, ky)
0.126
0
-0.123
-0.123
0
0.126
kx (1/m) ky (1/m)
10
30
50
(H
z)
Figure 2.3: (a) 3D CSG, (b) its spectrum, and (c) frequency slice of the magnitudespectrum at 50 Hz from (b). (d) Picked dispersion surface according to the dominantamplitudes of the spectrum.
41
the boundary of the acquisition zone. The maximum distance is denoted as
Hmax. Figure 2.4a shows the lines from the source point to the geophones along
the boundary.
• Define the ratio R(θ) as
R(θ) =H(θ)
Hmax
. (2.18)
The azimuth angle θ is the dominant angle when the value of R exceeds a
threshold value R0. For example, Figure 2.4b shows the dominant azimuth
angles for the source in Figure 2.4a by setting R0 = 0.6. The threshold value
R0 is selected by trial and error.
(a) 3D Array of Geophones
156
01560
Y (
m)
X (m)
(b) R vs θ
Dominant
Azimuths
1
0.6
00 100 200 300
0.2
R
Figure 2.4: (a) Lines from the source point, located at (30 m, 30 m), to the geophonesalong the boundary, and (b) R(θ) plotted against the azimuth angles.
After determining the dominant azimuth angles, we can pick the maximum values
along these angles from the dispersion curves. Figure 2.3c shows the picked dispersion
curves (red circles) for a frequency slice, and the whole dispersion surface is depicted
in Figure 2.3d. An efficient machine learning algorithm can be used to pick the
dispersion curves (Li et al., 2018a).
42
2.3.2 Initial Model for 3D WD
Figure 2.5 shows the workflow for calculating the initial model used with 3D WD.
First, we extract 2D in-line profiles from the 3D data set and retrieve their Rayleigh
wave dispersion curves. Then, a pseudo 1D S-velocity model is obtained from the
dispersion curves and the result is the initial model for the next iteration for inverting
the dispersion curves. The depth z and velocity value vs of the pseudo 1D S-velocity
model are calculated by scaling the wavelength λ and phase velocity c with factors of
0.5 and 1.1, respectively (O’Neill and Matsuoka, 2005). We use SURF96, a dispersion-
curve inversion code developed by Herrmann (2013), to invert the dispersion curves
for the 1D S-velocity models. By interpolating the 1D S-velocity models, a 2D S-
velocity model can be computed, which then serves as the starting model for 2D
WD (Li and Schuster, 2016; Li et al., 2017c). Finally, We interpolate the 2D WD
tomograms to obtain a starting model for 3D WD.
2.4 Numerical Examples
In this section, the 3D WD inversion method is evaluated with synthetic and field
data examples. The data are associated with 1) a simple checkerboard model, 2) the
complex 3D Foothills model, and 3) a surface seismic experiment carried out near the
Qademah area north of KAUST.
In the synthetic examples, the observed and predicted data are generated by an
O(2,8) time-space-domain solution to the first-order 3D elastic wave equations with
a free-surface boundary condition (Graves, 1996). For 3D WD, only the S-wave ve-
locity model is inverted and the true P-wave velocity model is used for modeling the
predicted surface waves. The source wavelet is a Ricker wavelet for the synthetic
data. The density model is homogeneous with ρ =2000 kg/m3 for all synthetic and
field data tests. For the field data, the source wavelet is estimated from the direct ar-
rivals. The fundamental dispersion curves associated with each shot gather are picked
43
Disper.
Curves
Pseudo 1D
S-velocity
Model
1D
S-velocity
Model
Disper. Curve
Inversion
2D WD
2D
S-velocity
Model
3D
S-velocity
Model
3D WD
3D Seismic
Data
Initial
Model
Initial
Model
Initial
Model
Figure 2.5: Workflow for calculating the initial S-velocity model for 3D WD.
44
along the dominant azimuth angles, where the dominant azimuth angle is defined in
equation 2.18 and the dispersion curves are picked for amplitudes above a specified
threshold value R0. For each iteration, source-side illumination compensation is used
as a preconditioner (Plessix and Mulder, 2004; Feng and Schuster, 2017) for the WD
gradient:
γpre(x) =1∑
t,s
√D2
1(t,x, s) +D22(t,x, s) +D2
3(t,x, s)γ(x), (2.19)
where Di(t,x, s) is the ith component of the wavefield at x generated by the source
located at s. No more than 15 iterations were used for the examples, where the
attenuation effects are ignored.
2.4.1 Checkerboard Test
The 3D checkerboard model shown in Figure 2.6a is used to test the 3D WD method.
The size of the model is 20 gridpoints in the z direction and 80 gridpoints in the x and
y directions, where the gridpoint interval is 1.5 m in all three directions. The depth
slice at z = 6 m of the model is shown in Figure 2.6b where the values of the high
and low S-velocities are 690 m/s and 510 m/s, respectively. The initial S-velocity
model is homogeneous with vs = 600 m/s. The P velocity is set to be vp =√
3vs.
100 vertical-component shots are uniformly distributed on a 10 × 10 source grid with
an interval of 12 m along both the x and y directions. Each shot gather is recorded
by a 40 × 40 receiver array with a 3 m spacing. The center frequency of the source
wavelet is 30 Hz.
For the source located at s = (60, 0, 0) m, the adjoint-source wavefield D(θ,g, t)
with θ = 90◦ is shown in Figure 2.7a, where the length r1 of the receiver spread
is 80 m. This adjoint source can be interpreted as a plane-wave source with the
azimuth angle of 90◦. The associated gradient at the depth slice z = 6 m is shown
45
(a) True S-velocity Model(b) Depth Slice at z = 6 m of Model(a)
0 118.5
Y (m)
0
118.5
X (
m)
510
600
690
S-v
elo
cit
y (
m/s
)
AC
B
D
(c) Inverted S-velocity Tomogram(d) Depth Slice at z = 6 m of the (c)Tomogram
0 118.5
Y (m)
0
118.5
X (
m)
510
600
690
S-v
elo
cit
y (
m/s
)
Figure 2.6: (a) True S-velocity model and its (b) depth slice at z = 6 m, (c) invertedS-velocity tomogram and (d) depth slice at z = 6 m.
46
in Figure 2.7b. Figures 2.7c and 2.7d show the accumulated adjoint-source wavefield∑θ D(θ,g, t) for all the azimuth angles from 0◦ to 180◦ with an interval of 5◦ and its
gradient at z = 6 m, respectively.
The choice of the maximum source-receiver offset will affect the inversion results.
For example, if we change the maximum source-receiver offsets r1 to the values 40
m and 120 m, the associated depth slices of the gradients are shown in Figures 2.8a
and 2.8b, respectively. The comparison suggests that the inverting data restricted
to small offset value might give better horizontal resolution than data restricted to
long offset value. However, the data with longer offset values will provide higher
resolution and more accurate picking of dispersion curves in the low frequency range
indicated in Figure 2.3-8 of Yilmaz (2015), which leads to deeper velocity updates
(Perez Solano et al., 2014). As a rule of thumb, surface-wave methods can be sensitive
to S-velocities down to a depth of about one-half of the total aperture of the receiver
array (Foti et al., 2014). To make sure that WD has sufficient depth penetration and
lateral resolution, we usually choose r1 to be about two or three times greater than
the penetration depth of interest.
Next, we reset the length of the receiver spread to be 40 m and repeat the 3D
WD computations. The fundamental dispersion curves for each shot gather are picked
along the dominant azimuths from 0◦ to 360◦ with an interval of 10◦ in the kx−ky−f
domain. For example, Figure 2.9 shows the observed dispersion curves from the CSGs
for the sources located at points A, B, C and D marked in Figure 2.6b, where the
black dashed lines represent the contours of the observed dispersion curves. The cyan
dash-dot lines in Figure 2.9 represent the contours of the initial dispersion curves.
Figure 2.6c displays the inverted S-wave velocity model after ten iterations, and
one of its depth slices at z = 6 m is shown in Figure 2.6d, which agrees well with
the true model. The contours of the predicted dispersion curves for the sources at A,
B, C and D are represented by the red lines in Figure 2.9, which correlate well with
47
the contours of the observed dispersion curves. After ten iterations, the normalized
misfit residual decreases to 2.8% of the starting value at the first iteration.
Figure 2.10 compares the depth slices of the true and the inverted tomograms at
z= 15 m and 24 m. It is evident that the deep part of the velocity model is less
accurate compared to the shallow part. This indicates that the sensitivity of surface
waves to the S-velocity decreases with depth.
(a) Adjoint Source for θ = 90◦
(m) (m)
(s)
(b) Gradient Slice for θ = 90◦
0 118.5
Y (m)
0
118.5X
(m
)-0.05
0
0.05
Gra
die
nt
(c) Adjoint Source for θ from 0◦ to180◦
(m) (m)
(s)
(d) Gradient Slice for θ from 0◦ to 180◦
0 118.5
Y (m)
0
118.5
X (
m)
-0.05
0
0.05
Gra
die
nt
Figure 2.7: (a) Wavefields of the adjoint source for θ = 90◦ and (b) the gradient at thedepth slice z = 6 m; (c) stacked wavefields of the adjoint sources for θ from 0◦ to 180◦
and (d) the gradient at the depth slice z = 6 m, where the maximum source-receiveroffset is r1=80 m and the source is located at s = (60, 0, 0) m.
48
(a) Gradient Slice for θ = 90◦, r1 = 40m
0 118.5
Y (m)
0
118.5
X (
m)
-0.05
0
0.05
Gra
die
nt
(b) Gradient Slice for θ = 90◦, r1 = 120m
0 118.5
Y (m)
0
118.5
X (
m)
-0.05
0
0.05
Gra
die
nt
Figure 2.8: Slices of the gradient at z = 6 m for (a) r1 = 40 m and (b) r1 = 120 m.
(a) Dispersion Curves for A (b) Dispersion Curves for B
(c) Dispersion Curves for C (d) Dispersion Curves for D
Figure 2.9: Observed dispersion curves for sources (a) A, (b) B, (c) C and (d) Dmarked in Figure 2.6b, where the black dashed lines, the cyan dash-dot lines and thered lines represent the contours of the observed, initial and inverted dispersion curves,respectively.
49
(a) True Depth Slice at z = 15 m
0 118.5
Y (m)
0
118.5
X (
m)
510
600
690
S-v
elo
cit
y (
m/s
)
(b) True Depth Slice at z = 24 m
0 118.5
Y (m)
0
118.5X
(m
)510
600
690
S-v
elo
cit
y (
m/s
)
(c) Inverted Depth Slice at z = 15 m
0 118.5
Y (m)
0
118.5
X (
m)
510
600
690
S-v
elo
cit
y (
m/s
)
(d) Inverted Depth Slice at z = 24 m
0 118.5
Y (m)
0
118.5
X (
m)
510
600
690
S-v
elo
cit
y (
m/s
)
Figure 2.10: True S-velocity depth slices at (a) z = 15 m and (b) z = 24 m; invertedS-velocity depth slices at (c) z = 15 m and (d) z = 24 m.
50
(a) True Model
(m)(m)
(b) Inverted Model by 1D Method
(m)(m)
(c) Inverted Model by 2D WD
(m)(m)
(d) Inverted Model by 3D WD
(m)(m)
Figure 2.11: (a) True S-velocity model, and tomograms inverted by the (b) 1D inver-sion, (c) 2D WD, and (d) 3D WD methods.
2.4.2 Modified Foothills Model
The 3D Foothills S-wave velocity model shown in Figure 3.10a is modified from the
2D Foothills model in Figure 2a of Brenders et al. (2008). The P-wave velocity is
defined as vp =√
3vs and the physical size of the velocity model is 1.2 km in the x and
y directions and is 80 m deep in the z direction. The gridpoint interval is 4 m in the
z direction and 6 m in the x and y directions. An array of geophones is distributed
on the surface, where 3600 receivers are arranged in 60 parallel lines along the x
direction and each line has 60 receivers. The inline- and crossline- receiver intervals
are both 20 m. There are 400 vertical-component shots distributed on a 20×10 grid
with source intervals of 60 m and 120 m in the x and y directions, respectively. The
peak frequency of the source is 15 Hz and the observed data are recorded for 0.90
seconds with a 0.3 ms sampling rate.
To construct the initial velocity model, we follow the workflow shown in Figure 2.5.
51
Ten 2D in-line data sets are extracted from the 3D data set. We take the second line
as an example, and all the sources on this line are located at y = 120 m. The vertical
slice of the true S-velocity model at y = 120 m is shown in Figure 2.12a, where the
locations of the first and last sources are indicated by the left-hand-side (LHS) and
right-hand-side (RHS) black stars, respectively. The black horizontal lines near the
stars represent the receiver spreads used to calculate the dispersion curves.
First, an initial model is obtained by applying the 1D inversion method to the
dispersion curves. In Figure 2.13a, the dispersion curve for the first CSG from the
2D line at y = 120 m is shown as the blue solid line. The pseudo 1D S-velocity model
is displayed as the blue dashed line in Figure 2.13b. We use SURF96 (Herrmann,
2013) to invert for the 1D velocity model (see the red solid line in Figure 2.13b). The
predicted dispersion curve is indicated by the red triangles in Figure 2.13a, which
agrees well with the observed one. The inverted 1D depth profile is assumed at the
middle of the receiver spreads, marked by the black solid circles in Figure 2.12. For
comparison with the 3D tomogram by 3D WD, the inverted 1D model is interpolated
as the 3D tomogram shown in Figure 3.10b.
The interpolated 1D profiles (see Figure 2.12b) are used as the starting model for
2D WD. Figure 2.12c shows the inverted 2D model at y = 120 m. Figures 2.14a and
b show the observed dispersion curves for all of the 2D shot gathers at y = 120 m
along the azimuth angles of θ = 0◦ and θ = 180◦, respectively, where the black dashed
lines, the cyan lines and the red dash-dot lines represent the contours of the observed,
initial and inverted dispersion curves, respectively. All 10 inverted 2D tomograms are
interpolated to form a 3D tomogram (Figure 3.10c), which is the starting model for
3D WD.
During the workflow of 3D WD, the fundamental dispersion curves for each shot
gather are picked along the dominant azimuths from 0◦ to 360◦ with an interval of 3◦
in the kx − ky − f domain. For example, Figure 2.15 shows the observed dispersion
52
curves calculated from the CSGs for the sources located at points A, B, C and D
indicated in Figure 3.10a, where the black dashed lines represent the contours of the
observed dispersion curves. The cyan lines represent the contours of initial dispersion
curves.
Figure 3.10d displays the inverted S-wave velocity model, and its 2D slices at
y = 120 m are shown in Figure 2.12d. Figure 2.16 shows the corresponding depth
slices at z = 20 m for the models shown in Figure 3.10. The contours of the predicted
dispersion curves for the sources at A, B, C and D are represented by the red dash-
dot lines in Figure 2.15, which more closely agree with the contours of the observed
dispersion curves.
Figures 2.17 show the root-mean-square deviation between the true model and the
inverted models by the 1D inversion, 2D WD and 3D WD methods. Compared with
the 1D inversion and 2D WD methods, the RMS errors of the 3D WD is 35 percent
and 22 percent lower, respectively.
Figure 2.18 compares the observed (red) and synthetic (blue) traces at far source-
receiver offsets predicted from the initial and inverted models for (a) and (b) with CSG
No.1, and (c) and (d) with CSG No.15. It can be seen that the synthetic waveforms
computed from the 3D WD tomogram more closely agree with the observed ones
compared to those computed from the 2D WD velocity model.
Mitigating Cycle Skipping by WD
We will now demonstrate that WD is less sensitive to cycle skipping compared to FWI
by using the synthetic 2D model shown in Figure 2.12a. Figure 2.19a displays the
initial model, which is far from the true model. The corresponding WD tomogram
is shown in Figure 2.19b. Figure 2.20 compares the observed (red) and synthetic
(blue) seismograms predicted from the initial model (LHS panels) and the inverted
model (RHS panels) for CSG No.1. We can see that the synthetic seismograms from
53
the initial model have a time delay greater than half of the period of wavelet, which
can lead to cycle skipping when using FWI (Virieux and Operto, 2009). However,
the synthetic waveforms computed from the WD tomogram closely agree with the
observed ones, which indicates that WD can sometimes mitigate the cycle skipping
problem of FWI. This seems reasonable because, similar to wave-equation travel time
inversion (WT) (Luo and Schuster, 1991a,b), WD computes a simple dispersion curve
to explain the observed seismograms and ,unlike FWI, does not need to fit all of the
amplitudes and phases of a trace.
(a) Slice of True Model at y = 120 m
(m)
(m)
(b) Inverted 1D Model at y = 120 m
(m)
(m)
(c) Inverted 2D Model at y = 120 m
(m)
(m)
(d) Inverted 3D Model at y = 120 m
(m)
(m)
Figure 2.12: Slices of the (a) true, (b) 1D inversion, (c) 2D WD and (d) 3D WDS-velocity models at y = 120 m, where the black dashed lines indicate the interfaceswith large velocity contrast.
2.4.3 Qademah Fault Seismic Data
A 3D land survey was carried out along the Red Sea coast over the Qademah fault
system, about 30 km north of the KAUST campus and the location is shown on the
Google map in Figure 5.12a (Hanafy, 2015). The survey consisted of 288 receivers
arranged in 12 parallel lines with each line having 24 receivers. The inline receiver
54
(a) Phase Velocity Comparison
20 30 40 502
2.2
2.4
2.6
2.8
3
3.2
3.4
Observed
Inverted
(b) Initial and Inverted S-velocityProfiles
2 2.5 3 3.580
60
40
20
0
Initial
Inverted
Figure 2.13: 1D inversion results computed with the code SURF96 (Herrmann, 2013):(a) the observed (blue line) and the predicted (red triangles) dispersion curves for CSGNo. 30; (b) the initial (blue dashed line) and the inverted (red solid line) S-velocityprofiles.
(a) Dispersion Curves, θ = 0◦ (b) Dispersion Curves, θ = 180◦
Figure 2.14: Observed dispersion curves along the azimuth angles of (a) θ = 0◦ and(b) θ = 180◦ for all the 2D CSGs located at y = 120 m, where the black dashed lines,the cyan lines and the red dash-dot lines represent the contours of the observed, initialand inverted dispersion curves, respectively.
55
(a) Dispersion Curves for Source A (b) Dispersion Curves for Source B
(c) Dispersion Curves for Source C (d) Dispersion Curves for Source D
Figure 2.15: Observed dispersion curves for sources (a) A, (b) B, (c) C and (d) D asindicated in Figure 3.10a, where the black dashed lines, the cyan lines and the reddash-dot lines represent the contours of the observed, initial and inverted dispersioncurves, respectively.
56
(a) True S-velocity Model
(m)
(m)
(b) S-velocity 1D Tomogram
(m)(m
)
(c) S-velocity 2D Tomogram
(m)
(m)
(d) S-velocity 3D Tomogram
(m)
(m)
Figure 2.16: Depth slices at z = 20 m of (a) the true S-velocity model and the invertedtomograms computed by the (b) 1D inversion, (c) 2D WD and (d) 3D WD methods,where the black dashed lines indicate the large velocity contrast boundaries.
57
1D 2D 3D
Inversion Method
130
145
160
175
190
RM
S E
rror
Figure 2.17: RMS error between the inverted S-velocity models by the 1D inversion,2D WD and 3D WD methods and the true S-velocity model.
interval is 5 m and the crossline interval is 10 m, and the source is a 40 kg weight drop
striking a metal plate on the ground next to each geophone position. The receiver
geometry is shown in Figure 5.12b, where one shot is fired at each receiver location
for a total of 288 shot gathers. The observed data were recorded for 0.7 seconds with
a 4 ms sampling rate. The following processing steps are first applied to the data:
• Each trace is normalized to compensate for the effects of attenuation and geo-
metrical spreading. The traces of CSG No. 12 in the first line before and after
amplitude compensation are shown in Figures 2.22a and 2.22b, respectively.
• All other arrivals but the fundamental-mode Rayleigh waves are masked in the
CSG by a muting window. The length of the window is marked by the red
dashed lines in Figure 2.22b.
The dispersion images shown in Figures 2.22c and 2.22d are computed by the
frequency-sweeping method (Park et al., 1998), where the red asterisks represent
the maximum value for each frequency. A false high-mode dispersion curve can be
58
(a) CSG No. 1 from Initial Model
342 370
Trace No.
0.23
0.9
T (
s)
(b) CSG No. 1 from 3D Tomogram
342 370
Trace No.
0.23
0.9 T
(s)
(c) CSG No. 15 from Initial Model
342 370
Trace No.
0.23
0.9
T (
s)
(d) CSG No. 15 from 3D Tomogram
342 370
Trace No.
0.23
0.9
T (
s)
Figure 2.18: Comparison between the observed (red) and synthetic (blue) traces atfar offsets predicted from the initial model (LHS panels) and 3D tomogram (RHSpanels) for CSG No.1 in (a) and (b), and CSG No.15 in (c) and (d).
59
(a) Initial S-velocity Model
(m)
(m)
(b) S-velocity Tomogram
(m)
(m)
Figure 2.19: (a) Initial and (b) inverted 2D S-velocity models. The correspondingtrue model is shown in Figure 2.12a. Here, the black dashed lines indicate the largevelocity contrast boundaries which are the same as those in Figure 2.12.
(a) CSG No. 1 (b) CSG No. 1
Figure 2.20: Comparison between the observed (red) and synthetic (blue) tracespredicted from the initial model (LHS panels, Figure 2.19a) and 2D tomogram (RHSpanels, Figure 2.19b) for CSG No.1.
60
A
B
C
D
Figure 2.21: (a) Google map showing the location of the Qademah-fault seismic ex-periment (Fu et al., 2018b). (b) Receiver geometry for the Qademah-fault data, wherethe red dashed line indicates the location of Qademah fault. The Green triangles rep-resent the locations of receivers, where the shots are located at each receiver. Thered star represents the location of source No. 132 and the black stars indicate thelocations of sources A, B, C and D on the surface. θ is the azimuth angle with respectto the acquisition line of source No. 132.
61
observed clearly on the top-right area, which is caused by the spatial aliasing due
to the large receiver interval. The observed dispersion curves (see the blue lines) are
picked and used for inversion. Figure 2.23 shows the observed dispersion curves for all
of the 2D CSGs at the first line, where the black dashed lines represent the contours
of the observed dispersion curves. At certain frequency ranges, it is difficult to pick
the dispersion curves because of the low signal-to-noise ratio of the data so that some
dispersion curves are missing in Figure 2.23.
Evaluating the accuracy of the picked dispersion curves is important. A reciprocity
test is needed to determine if the dispersion curves of a shot gather are the same
as those of a receiver gather at the same location. The schematic diagram of the
reciprocity test is shown in Figure 2.24.
We first use the 1D inversion method to invert the dispersion curves. For example,
the dispersion curve for the 12th CSG from the first line is shown as the solid blue line
in Figure 2.25a. The pseudo 1D S-velocity model is displayed as the blue dashed line
in Figure 2.25b. SURF96 (Herrmann, 2013) is used to invert for the 1D velocity model
(see the red solid line in Figure 2.25b). The predicted dispersion curve is indicated
by the red triangles in Figure 2.25a. The inverted 1D depth profile is assumed to be
an accurate representative of the velocity model at the middle of the receiver spread.
The 1D velocity profiles are interpolated as the starting model for 2D WD, which is
shown in Figure 2.26a. For comparison with the 3D WD tomogram, we interpolate
the 1D velocity profiles as the 3D model shown in Figure 2.27a.
Then, we apply 2D WD to invert for the 2D velocity model along the 12 lines.
Figures 2.26a and 2.26b show the initial and inverted S-velocity models beneath the
first line. The cyan lines in Figure 2.23 represent the contours of the initial dispersion
curves. The contours of the predicted dispersion curves are represented by the red
dash-dot lines in Figure 2.23, which more closely agree with the contours of the
observed dispersion curves.
62
The 12 inverted 2D S-velocity models are then interpolated to obtain an initial
velocity model (see Figure 2.27b) for 3D WD. For each shot gather, only the receivers
within the distance r1 = 50 m from the source are used to retrieve the dispersion
curves. The frequency range used in WD is from 20 Hz to 60 Hz. Figure 2.28
displays the fundamental dispersion curves calculated from the CSGs for the sources
located at A, B, C and D, which are indicated in Figure 5.12. Here the black dashed
lines represent the contours of the observed dispersion curves. The contours of the
initial dispersion curves are represented by the cyan lines in Figure 2.28.
The S-wave velocity tomogram is shown in Figure 2.27c, where the red line labeled
with “F1” indicates the location of the interpreted Qademah fault3 and the red line
labeled with “F2” refers to a possible small antithetic fault. The low-velocity anomaly
between the two faults is interpreted as a colluvial wedge labeled with “CW”. The
red dash-dot lines in Figure 2.28 show the predicted κ(ω) curves calculated from the
CSGs with sources located at A, B, C and D indicated in Figure 5.12. It is evident
that the WD tomogram has decreased the differences between the initial and observed
dispersion curves.
To further test the accuracy of the 3D tomogram, Figure 2.29 shows the compari-
son between the observed (blue) and synthetic (red) traces at far offsets predicted from
the initial model (LHS panels) and 3D tomogram (RHS panels) for (a) and (b) with
CSG No. 9. Here, two matched filters displayed in Figure 2.29c are calculated from
trace No. 76 in Figures 2.29a and 2.29b, respectively. The matched filters are then
applied to reshape the synthetic waveforms. The CSGs after filtering are displayed
in Figure 2.29d and Figure 2.29e. We can see that the predicted fundamental-mode
surface waves closely match the observed ones. Figure 2.30 shows the common offset
gathers (COGs) with the offset of 30 m for several 2D CSGs, where the blue and red
wiggles represent the observed and predicted COGs, respectively. For each panel in
3The low-velocity zone in this tomogram is interpreted as a fault.
63
Figure 2.30, a matched filter is calculated from the green trace and then applied to
the other traces. The predicted COGs are consistent with the raw data.
The slices of the S-wave velocity tomogram are shown in Figure 2.31a and the
dashed lines indicate the locations of the conjectured Qademah fault. The low-
velocity zone (LVZ) in Figure 2.31a next to the conjectured fault is consistent with
the downthrown-side of an interpreted normal fault. The LVZ is also consistent with
the reflectivities of the migration image (Liu et al., 2016, 2017a) indicated by the blue
zone next to the dashed fault in Figure 2.31b. Figure 2.32 compares the 2D zoom
view of the tomogram, migration image and COG, in which the LVZ of the tomogram
is consistent with the location of the delay in the COG arrivals and the reflectivity
of the migration image. This observation increases our confidence in the accuray of
the S-wave velocity tomogram.
2.5 Discussion
The lateral resolution of the WD tomogram is related to the length of the receiver
spread. Different receiver-spread lengths can lead to different lateral-resolution limits
of the retrieved dispersion curves (Bergamo et al., 2012; Mi et al., 2017). A wide
receiver-spread for a specific azimuth angle can lead to poor lateral resolution along
the azimuth angle of the gradient (Figure 2.8), but can provide a deep penetration
depth (Foti et al., 2014). These resolution limits can be mitigated by a multiscale
strategy (Liu and Huang, 2018): use the long-offset and low-frequency data to update
the deep areas and use the short-offset and high-frequency data to update the shallow
regions. Instead of a multichannel analysis method, the dispersion curves with higher
lateral resolution can possibly be measured by tomographic methods (Krohn and
Routh, 2016, 2017), which might be used in the WD method.
In our work we assumed that the effects of attenuation on dispersion curves are
insignificant by using the isotropic elastic wave equation. However, if the attenu-
64
(a) Traces of CSG No. 12 at the 1stLine
X (m)
T (
s)
0
0.70 115
(b) After Amplitude Compensation
X (m)
T (
s)
0
0.70 115
230 m/s 230 m/s
30 m/s
30 m/s
(c) Dispersion Image for θ = 0◦
0.8
0.110 20 40 60
Spatial
Aliasing
(d) Dispersion Image for θ = 180◦
0.8
0.110 20 40 60
Spatial
Aliasing
Figure 2.22: Seismic traces of CSG No. 12 at the first line (a) before and (b) afteramplitude compensation; and its dispersion images for (c) θ = 0◦ and (d) θ = 180◦.The two red dashed lines in (b) show the length of the muting window which masksall other arrivals but the fundamental-mode Rayleigh waves. The red asterisks in (c)and (d) represent the maximum value for each frequency, and the blue lines are thepicked observed dispersion curves used for inversion.
65
(a) Picked Dispersion Curve for θ = 0◦(b) Picked Dispersion Curve for θ =180◦
Figure 2.23: Observed dispersion curves for (a) θ = 0◦ and (b) θ = 180◦ computedfrom the 2D CSGs in the first line, where the black dashed lines, the cyan lines andthe red dash-dot lines represent the contours of the observed, initial and inverteddispersion curves, respectively.
QC of Picking by Reciprocity
CSG CRG
Figure 2.24: Quality control of the picked dispersion curves by reciprocity, wherethe stars represent the sources, and the rectangles represent the receivers. If thedispersion curves (red) of the CSG are the same as those (blue) computed from thecommon receiver gather (CRG) at the same location, it passes the reciprocity test.Passing the reciprocity test is a necessary QC test all 3D data must pass prior toinversion.
66
(a) Phase Velocity Comparison
30 40 50 600.1
0.2
0.3
0.4
0.5
0.6
Observed
Inverted
(b) Initial and Inverted S-velocityProfiles
0.1 0.2 0.3 0.4 0.5
6
4
2
0
Initial
Inverted
Figure 2.25: 1D dispersion curve inversion results by SURF96 (Herrmann, 2013): (a)the observed (blue line) and the predicted (red triangles) dispersion curves for CSGNo. 12 (see Figure 2.22c); (b) the initial (blue dashed line) and the inverted (redsolid line) S-velocity profiles.
(a) S-velocity Tomogram by 1DMethod
0 40 80 120
X(m)
0
2.5
5
7.5
Z (
m)
100
200
300
400
500
600
S-v
elo
city (
m/s
)
(b) S-velocity Tomogram by 2D WD
0 40 80 120
X (m)
0
2.5
5
7.5
Z (
m)
100
200
300
400
500
600
S-v
elo
city (
m/s
)
Figure 2.26: S-velocity tomograms from the 2D CSGs beneath the first line by the(a) 1D inversion and (b) 2D WD methods.
67
(a) S-velocity Tomogram by 1DMethod (b) S-velocity Tomogram by 2D WD
(c) S-velocity Tomogram by 3D WD
Figure 2.27: S-velocity tomograms inverted by the (a) 1D inversion, (b) 2D WD, and(c) 3D WD methods. The red solid line labeled by “F1” indicates the location of theconjectured Qademah fault and the dashed red line labeled by “F2” is conjectured tobe a small antithetic fault. The low-velocity anomaly between faults “F1” and “F2”is the conjectured colluvial wedge labeled by “CW”.
68
Figure 2.28: Observed dispersion curves for sources (a) A, (b) B, (c) C and (d) Dindicated in Figure 5.12. The black dashed lines, the cyan lines and the red dash-dotlines represent the contours of the observed, initial and inverted dispersion curves,respectively.
69
(a) CSG from Initial Model (b) CSG from 3D Tomogram
(c) Matched Filters from Trace No. 76
(d) CSG in (a) with Blue Filter in(c)
(e) CSG in (b) with Red Filter in(c)
Figure 2.29: Comparison between the observed (blue) and synthetic (red) traces atfar source-receiver offsets predicted from the initial model (LHS panels) and 3D WDtomogram (RHS panels) for CSG No.9 in (a) and (b). The blue and red matchedfilters in (c) are calculated from the trace No. 76 (green) in (a) and (b), respectively.Comparison between the observed (blue) and synthetic (red) traces after applyingthe matched filters in (d) and (e).
70
(a) COG Line 1 (b) COG Line 4
(c) COG Line 6 (d) COG Line 10
Figure 2.30: COGs with the offset of 30 m for the selected lines, where the blue andred wiggles represent the observed and predicted COGs, respectively. For each panel,a matched filter is calculated from the green trace and then applied to the othertraces.
71
750
100
vs
(m/s
)
Y (m)X (m)
Y (m)X (m)
Figure 2.31: Slices of (a) the inverted S-wave velocity model, and (b) natural mi-gration images (Liu et al., 2017a). The dashed lines indicate the location of theinterpreted Qademah fault.
72
Figure 2.32: (a) and (b): 2D zoom view of the dashed panels in Figure 2.31, comparedwith (c) the COGs.
73
ation model is known, the visco-elastic effects can be accounted for by solving the
visco-elastic wave equation to compute the theoretical dispersion curves. Instead of
inverting just for velocity, the WD method can be modified to invert for both the ve-
locity and attenuation models (Li et al., 2017a,b). In this case the visco-elastic wave
equation and its numerical solutions must be computed to estimate the gradients for
the attenuation parameters. However, there is an inherent non-uniqueness problem
when inverting for both velocity and attenuation models, so the dispersion curves and
the data with normalized amplitudes might be preferred as input data.
In the synthetic data test, we use the true P-wave velocity and density models for
the inversion. In practice, there might be errors in the P-wave velocity and density
models, but such errors have a limited effect on the WD results because the Rayleigh
wave dispersion curves are not very sensitive to the P-wave velocity or density models
(Xia et al., 1999). As shown in Xia et al. (1999), the overall average error between
the inverted vs and the true vs is 4.4% for the case where there is no error in the
P-wave velocities or densities and 8% for the other cases which include errors in the
P-wave velocity and density models.
In the field data test, the free surface is horizontal so that we do not need to con-
sider the effect of irregular topography. However, significant topographic variations
can strongly influence the amplitudes and phases of propagating surface waves. Such
effects should be taken into account when there exists significant elevation changes,
otherwise the S-velocity model inverted from the Rayleigh wave or Love wave disper-
sion curves will contain significant inaccuracies (Li et al., 2017e, 2019b).
In our numerical tests, we did not assess the uncertainty of the inverted model
using a covariance matrix (Zhu et al., 2016). Instead, the predicted and observed
common offset gathers are compared to one another and the RMS misfit is used to
determine the degree of error in our solution. In addition, the RMS wavenumber error
is provided. Another popular assessment method is to perform the checkerboard test
74
(Zelt and Barton, 1998; Rawlinson et al., 2014), which is performed for our synthetic
tests.
A limitation of 3D WD is that the fundamental dispersion curves must be picked
for each shot gather. This process can be prone to errors when there is a strong
overlap with higher-order modes (Li et al., 2017c) or there is spatial and temporal
aliasing due to large spatial and temporal sampling intervals. A supervised machine
learning method (Li et al., 2018a) can be used to expedite the picking of dispersion
curves for large data sets. In addition, guided waves that are trapped in near-surface
waveguides can be inverted by 3D WD for the near-surface P-velocity model (Li et al.,
2018b).
2.6 Conclusions
We extend the 2D WD methodology to 3D, where the objective function is the sum of
the squared differences between the wavenumbers along the predicted and observed
dispersion curves for each azimuth angle. The Frechet derivative with respect to
the 3D S-wave velocity model is derived by the implicit function theorem. The WD
gradient is calculated by correlating the back-propagated wavefield with the forward-
propagated source field in the model based on the Born approximation in an isotropic,
elastic reference earth model.
We provide a comprehensive approach to build the initial model for 3D WD,
which starts from the pseudo 1D S-wave velocity model, which is then used to get
the 2D WD tomogram, which in turn is used as the starting model for 3D WD. Our
numerical results from both synthetic and field data show that the 3D WD method can
reconstruct the 3D S-wave velocity tomogram for a laterally heterogeneous medium so
that the predicted surface waves closely match the observed ones for the fundamental
modes. This suggests that the WD tomogram can serve as a good starting model
for surface-wave FWI. The 3D WD method can be easily adapted to also invert the
75
higher-order modes for a more detailed velocity model. In addition, guided waves that
are trapped in near-surface waveguides can be inverted by 3D WD for the near-surface
P-wave velocity model.
The main limitation of 3D WD is its high computational cost, which is more
than an order-of-magnitude greater than that of 2D WD. However, the improvement
in accuracy compared to 2D WD can make this extra cost worthwhile when there
are significant near-surface lateral variations in the S-velocity distribution. If the
attenuation is important, then its effects can be accounted for by solving the visco-
elastic wave equation to compute the theoretical dispersion curves. To expedite the
picking of dispersion curves obtained from large data sets we recommend supervised
machine learning methods that adapt to the data recorded at different sites.
Acknowledgments
The research reported in this publication was supported by the King Abdullah Uni-
versity of Science and Technology (KAUST) in Thuwal, Saudi Arabia. We are grateful
to the sponsors of the Center for Subsurface Imaging and Modeling Consortium for
their financial support. For computer time, this research used the resources of the
Supercomputing Laboratory at KAUST and the IT Research Computing Group. We
thank them for providing the computational resources required for carrying out this
work.
76
2.7 Appendix A: Correlation Identity
The integrand in equation 2.10 can be replaced by its Fourier transform (Li et al.,
2017c)
˙D(k + ∆κ, θ, ω)obs = −∫ ∫
i(x′g cos θ + y′g sin θ)
D(x′g, y′g, ω)obse
−i(k+∆κ)(cos θx′g+sin θy′g)dx′gdy′g,
D(k, θ, ω)pre =
∫ ∫D(xg, yg, ω)pree
−ik(cos θxg+sin θyg)dxgdyg,
to give
∂Φ(∆κ, vs(x))
∂vs(x)= R
{∫ ∫dxgdyg
[ ∫ ∫dx′gdy
′g
[∫eik(cos θ(x′g−xg)+sin θ(y′g−yg))dk
]i(x′g cos θ + y′g sin θ)
D(x′g, y′g, ω)∗obse
i∆κ(cos θx′g+sin θy′g)
]∂D(xg, yg, ω)pre
∂vs(x)
}= R
{∫ ∫dxgdyg
[ ∫ ∫dx′gdy
′g
[2πδ(cos θ(x′g − xg)+
sin θ(y′g − yg))]i(x′g cos θ + y′g sin θ)D(x′g, y
′g, ω)∗obs
ei∆κ(cos θx′g+sin θy′g)
]∂D(xg, yg, ω)pre
∂vs(x)
}= R
{∫ ∫dxgdyg
∂D(xg, yg, ω)pre∂vs(x)
2π
[i(xg cos θ + yg sin θ)
cos θ
ei∆κ(cos θxg+sin θyg)
∫dy′gD(xg − (y′g − yg) tan θ, y′g, ω)∗obs
]}= R
{∫ ∫dxgdyg
∂D(xg, yg, ω)pre∂vs(x)
D(xg, yg, ω)∗obs
}, (2.20)
where the weighted conjugated data function is
D(g, θ, ω)∗obs = 2πig · ncos θ
eig·n∆κ
∫dy′gD(xg − (y′g − yg) tan θ, y′g, ω)∗obs, (2.21)
77
in which g = (xg, yg) and n = (cos θ, sin θ).
To calculate the weighted conjugated data, we first compute the integration of
D(xg − (y′g − yg) tan θ, y′g, ω)∗obs along the line x′g = xg − (y′g − yg) tan θ (see the
schematic diagram in Figure 2.1). The line is passing through the point x = (xg, yg)
and is perpendicular to the direction of n. We can see that the weighted conjugated
data along the line are almost identical, which means it will generate a plane-wave
surface wave for the backprojected wavefield.
Next, we will interpret the weighted conjugated data function by the stationary
phase method. The Green’s function for the fundamental mode of Rayleigh waves
excited by a vertical-component force in the far field can be approximated as (Snieder,
2002b):
G(xg, yg, ω) ' A′e−ik√x2g+y2g+iπ/4√
0.5πk√x2g + y2
g
. (2.22)
where A′ accounts for the source amplitude and radiation patten for a trace at (xg, yg)
by a point source at (0, 0).
Replacing D(xg, yg)obs in the equation 2.21 with the Rayleigh Green’s function in
the equation 2.22 and the source W (ω), we can get
D(g, θ, ω)∗obs = 2πiW (ω)g · ncos θ
eig·n∆κA′eiπ/4∫dy′g
eikf(y′g)√0.5πk
√x2g + y2
g
, (2.23)
where
f(y′g) =√y′g
2/ cos θ − 2(xg + yg tan θ) tan θy′g + (xg + yg tan θ)2. (2.24)
According to the stationary phase approximation, for k � 1, the stationary point
78
is located at y′g = cy so that ∂f(cy)
∂y′g= 0 and cy = (xg cos θ + yg sin θ) sin θ. Because
(x′g, y′g) is located at the line: (g′−g) ·n = 0, the x coordinate of the stationary point
is cx = (xg cos θ + yg sin θ) cos θ. Then equation 2.23 can be approximated as
D(g, θ, ω)∗obs = 2πiW (ω)g · ncos θ
eig·n∆κA′eiπ/4eik(xg cos θ+yg sin θ)√0.5k
√x2g + y2
g/π√2πi(cos2θ(xg cos θ + yg sin θ))
k
= 2πiµg · nW (ω)eig·n∆κA′eiπ/4eikg·n√
0.5k|g|/π
√2πig · n
k,
(2.25)
where µ cos θ = | cos θ|.
2.8 Appendix B: Elastic Gradient
The gradient for the WD method is now derived. For an isotropic heterogeneous
medium, the Born approximation in terms of the 3D elastic Green’s functions for a
harmonic source (Snieder, 2002a) is
δDi(x, ω) = ω2
∫Gij(x|x′)δρ(x′)Di(x
′, ω)dx′3
−∫Gik,k(x|x′)δλ(x′)Dj,j(x
′, ω)dx′3
−∫Gin,k(x|x′)δµ(x′)(Dk,n(x′, ω) +Dn,k(x
′, ω))dx′3, (2.26)
where δDi(x, ω) denotes the ith component of the perturbed particle velocity recorded
at x due to the scattering from the perturbations of density δρ and Lame parameters
δλ and δµ. Einstein notation is assumed in equation 2.26. Di,j = ∂Di
∂xjfor i, j ∈
{1, 2, 3}. Gij is the 3D harmonic Green’s tensor (Snieder, 2002a) for the background
medium with the Lame parameters λ and µ, and density ρ. If we assume density ρ
79
is a constant, equation 2.26 yields the derivative of δDi(x, ω) with respect to δλ and
δµ at x′
δDi(x, ω)
δλ(x′)= −Gik,k(x|x′)Dj,j(x
′, ω),
andδDi(x, ω)
δµ(x′)= −Gin,k(x|x′)(Dk,n(x′, ω) +Dn,k(x
′, ω)). (2.27)
Our interest is confined to the derivative of the vertical component of the particle
velocity at g, so equation 2.27 for i = 3, {1, 2, 3} → {x, y, z} and Di(x, ω)→ D(g, ω)
with respect to λ and µ at x can be written as
∂D(g, ω)
∂λ(x)= −
(∂Gzx(g|x)
∂x+∂Gzy(g|x)
∂y+
∂Gzz(g|x)
∂z
)(∂Dx(x, ω)
∂x+∂Dy(x, ω)
∂y+∂Dz(x, ω)
∂z
), (2.28)
and
∂D(g, ω)
∂µ(x)= −2
(∂Gzx(g|x)
∂x
∂Dx(x, ω)
∂x+
∂Gzy(g|x)
∂y
∂Dy(x, ω)
∂y+∂Gzz(g|x)
∂z
∂Dz(x, ω)
∂z
)−(∂Gzx(g|x)
∂z+∂Gzz(g|x)
∂x
)(∂Dx(x, ω)
∂z+∂Dz(x, ω)
∂x
)−(∂Gzx(g|x)
∂y+∂Gzy(g|x)
∂x
)(∂Dx(x, ω)
∂y+∂Dy(x, ω)
∂x
)−(∂Gzy(g|x)
∂z+∂Gzz(g|x)
∂y
)(∂Dy(x, ω)
∂z+∂Dz(x, ω)
∂y
), (2.29)
where Dx(x, ω), Dy(x, ω) and Dz(x, ω) are finite-difference solutions to the 3D elas-
tic wave equation for the background velocity model. From the definitions vp =√(λ+ 2µ)/ρ and vs =
√µ/ρ, the Frechet derivative of D(g, ω) with respect to vs
80
(Mora, 1987) can be obtained:
∂D(g, ω)
∂vs(x)= −4vs(x)ρ(x)
∂D(g, ω)pre∂λ(x)
+ 2vs(x)ρ(x)∂D(g, ω)pre∂µ(x)
(2.30)
Inserting equations 2.28 and 2.29 into equation 2.30 gives,
∂D(g, ω)
∂vs(x)= 4vs(x)ρ(x)
{(∂Gzy(g|x)
∂y+∂Gzz(g|x)
∂z
)∂Dx(x, ω)
∂x+(∂Gzx(g|x)
∂x+∂Gzz(g|x)
∂z
)∂Dy(x, ω)
∂y+(∂Gzx(g|x)
∂x+∂Gzy(g|x)
∂y+)∂Dz(x, ω)
∂z
− 1
2
(∂Gzx(g|x)
∂z+∂Gzz(g|x)
∂x
)(∂Dx(x, ω)
∂z+∂Dz(x, ω)
∂x
)− 1
2
(∂Gzx(g|x)
∂y+∂Gzy(g|x)
∂x
)(∂Dx(x, ω)
∂y+∂Dy(x, ω)
∂x
)− 1
2
(∂Gzy(g|x)
∂z+∂Gzz(g|x)
∂y
)(∂Dy(x, ω)
∂z+∂Dz(x, ω)
∂y
)}. (2.31)
81
Chapter 3
3D Wave-equation Dispersion Inversion of Surface Waves
Recorded on Irregular Topography 1
Irregular topography can cause strong scattering and defocusing of propagating sur-
face waves, so it is important to account for such effects when inverting surface waves
for the shallow S-velocity structures. We now present a 3D surface-wave disper-
sion inversion method that takes into account the topographic effects modeled by a
3D spectral element solver. The objective function is the frequency summation of
the squared wavenumber differences ∆κ(ω)2 along each azimuthal angle of the fun-
damental mode or higher-order modes of Rayleigh waves in each shot gather. The
wavenumbers κ(ω) associated with the dispersion curves are calculated using the data
recorded along the irregular free surface. Numerical tests on both synthetic and field
data demonstrate that 3D topographic wave equation dispersion inversion (TWD)
can accurately invert for the S-velocity model from surface-wave data recorded on ir-
regular topography. Field data tests for data recorded across an Arizona fault suggest
that, for this example, the 2D TWD can be as accurate as the 3D tomographic model.
This suggests that in some cases the 2D TWD inversion is preferred over 3D TWD
because of its significant reduction in computational costs. Compared to the 3-D
P-wave velocity tomogram, the 3D S-wave tomogram agrees much more closely with
the geological model taken from the trench log. The agreement with the trench log is
even better when the Vp/Vs tomogram is computed, which reveals a sharp change in
1This manuscript was submitted as: Zhaolun Liu, Jing Li, Sherif M. Hanafy, Kai Lu, and GerardSchuster, ”3D Wave-equation Dispersion Inversion of Surface Waves Recorded on Irregular Topog-raphy,” Geophysics.
82
velocity across the fault. The localized velocity anomaly in the Vp/Vs tomogram is
in very good agreement with the well log. Our results suggest that integrating the Vp
and Vs tomograms can sometimes give the most accurate estimates of the subsurface
geology across normal faults.
3.1 Introduction
Irregular topography is known to have a significant impact on the amplitudes and
phases of propagating surface waves (Snieder, 1986b; Fu and Wu, 2001). Ignoring
topography in surface wave inversion can lead to significant errors in the inverted
model. Moreover, it is expected that the 2D assumptions about the subsurface model
cannot fully approximate wave propagation in the presence of significant 3D variations
in topography. In these cases, it is important to employ a 3D surface-wave inversion
method that fully accounts for wave propagation along irregular topography.
Eguiluz and Maradudin (1983) and Mayer et al. (1991) analytically studied the
effect of surface roughness on the dispersion relations of a Rayleigh wave propagating
in an isotropic medium with randomly rough surfaces. For significant topographic
variations on a wavelength scale, they showed that the relief of the free surface induces
attenuation of amplitudes, reduces the phase velocity (Eguiluz and Maradudin, 1983)
and generates both Love waves and higher-order modes of Rayleigh waves (Mayer
et al., 1991). These authors argued that these waves sense the uppermost part of the
model as an upper layer with a reduced effective velocity.
When the wavelength is much smaller than the characteristic length scale of the
topographic relief, the source-receiver distance factor may play a significant role. The
is especially true for the fundamental mode of the Rayleigh waves whose propagation
is strongly influenced by the free surface (Kohler et al., 2012). Kohler et al. (2012)
empirically investigated the effect of topography on the propagation of short-period
Rayleigh waves by elastic simulations with a spectral element code and a 3-D model
83
with significant topographical variations. They showed that topography along a pro-
file could result in an underestimation of the phase velocities associated with the
surface waves.
Accounting for topography is also essential for full waveform inversion (FWI) of
surface waves. Nuber et al. (2016) and Pan et al. (2018) use simulations to demon-
strate that even minor topographic variations of the free surface will have a significant
effect in the accuracy of FWI. They found that neglecting topography with an eleva-
tion fluctuation greater than half the minimum seismic wavelength leads to significant
errors in the inverted image (Nuber et al., 2016).
Li and Schuster (2016) developed a wave equation dispersion inversion (WD)
method for inverting dispersion curves associated with surface waves. Li et al. (2019a)
applied WD to Love waves and Liu et al. (2019) extended it to the 3D case, which
includes the multi-scale and layer-stripping WD proposed by Liu and Huang (2019).
Empirical evidence suggests that WD has the benefit of robust convergence compared
to the tendency of FWI (Groos et al., 2014; Perez Solano et al., 2014; Dou and Ajo-
Franklin, 2014; Yuan et al., 2015; Groos et al., 2017) to getting stuck in a local
minimum. It has the advantage over the traditional inversion of dispersion curves
(Haskell, 1953; Xia et al., 1999, 2002; Park et al., 1999) in that it does not assume
a layered model and is valid for arbitrary 2D or 3D media. Later, Li et al. (2017e,
2019b) developed 2D topographic WD (i.e., topographic WD, also denoted as TWD)
which incorporates the free-surface topography into the finite-difference solutions of
the elastic wave equation. Our new paper now extends 2D TWD to the 3D case.
To account for strong variations in topography, we use the elastic modeling code
SPECFEM3D based on the spectral-element method (SEM) (Komatitsch and Vilotte,
1998; Komatitsch and Tromp, 1999). The inversion algorithm is written in the format
of SeisFlows, an open source Python package that can interface with SPECFEM3D
(Modrak et al., 2018).
84
After the introduction, we describe the theory of 3D TWD and its implementa-
tion. We also discuss how to calculate the source-receiver offset distance along a 3D
irregular surface, which is used to calculate the dispersion curves of the data recorded
on the irregular surface. Numerical tests on synthetic data are presented in the third
section to validate the theory. The field data test is for 3D vertical-component data
recorded over a normal fault located near the Arizona-Utah border. Finally, the
discussion and conclusions are given in the fourth and last sections.
3.2 Theory
We first present the mathematical theory for 3D TWD, following the derivation of Liu
et al. (2019), except it is for a 3D irregular surface. Then, we show how to calculate
the source-receiver distance on a 3D irregular surface. Finally, the workflow of 3D
TWD is given.
3.2.1 Theory of 3D TWD
The basic theory of 3D TWD is the same for 3D WD (Liu et al., 2018, 2019), except
a 3D topographic surface is now included in the formulation. The wave-equation
dispersion inversion method inverts for the S-wave velocity model to minimize the
dispersion objective function
ε =1
2
∑ω
∑θ
[
residual=∆κ(θ,ω)︷ ︸︸ ︷κ(θ, ω)pre − κ(θ, ω)obs]
2, (3.1)
where κ(ω, θ)pre represents the predicted dispersion curve picked from the simulated
spectrum along the azimuth angle θ, and κ(ω, θ)obs describes the observed dispersion
curve obtained from the recorded spectrum along the azimuth θ. In the 2D case, the
azimuthal angles have only two values: 0◦ and 180◦, corresponding to the left and
right directions, respectively.
85
The gradient γ(x) of ε with respect to the S-wave velocity vs(x) is given by Liu
et al. (2018, 2019):
γ(x) =∂ε
∂vs(x)= −
∑ω
4vs0(x)ρ0(x)R
{backprojected data=Bk,k(x,ω)∗︷ ︸︸ ︷∫ ∑
θ
1
A(θ, ω)∆κ(θ, ω)D(g, θ, ω)∗obsG3k,k(g|x)dg
source=fj,j(x,ω)︷ ︸︸ ︷Dj,j(x, ω)
backprojected data=Bn,k(x,ω)∗︷ ︸︸ ︷−1
2
∫ ∑θ
1
A(θ, ω)∆κ(θ, ω)D(g, θ, ω)∗obsG3n,k(g|x)dg
source=fn,k(x,ω)︷ ︸︸ ︷[Dk,n(x, ω) +Dn,k(x, ω)
]},
(3.2)
where vs0(x) and ρ0(x) are the reference S-velocity and density distributions at lo-
cation x, respectively, and A(θ, ω) is given in Liu et al. (2019). Di(x, ω) denotes
the ith component of the particle velocity recorded at x resulting from a vertical-
component force. Einstein notation is assumed in equation 4.8, where Di,j = ∂Di
∂xj
for i, j ∈ {1, 2, 3}. The 3D harmonic Green’s tensor G3j(g|x) is the particle velocity
at location g along the jth direction resulting from a vertical-component source at
x in the reference medium. The term fi,j(x, ω) for i and j ∈ {1, 2, 3} represents
the downgoing source field at x, and Bi,j(x, s, ω) for i and j ∈ {1, 2, 3} denotes the
backprojected scattered field at x. D(g, θ, ω)∗obs represents the weighted conjugated
data defined as
D(g, θ, ω)∗obs = 2πig · neig·n∆κ
∫C
D(g′, ω)∗obsdg′, (3.3)
where n = (cos θ, sin θ) and C is the line (g′ − g) · n = 0. The above equation
indicates that the gradient is computed using a weighted zero-lag correlation between
the source and backward-extrapolated receiver wavefields.
The optimal S-wave velocity model vs(x) is obtained using the steepest-descent
86
formula (Nocedal and Wright, 2006)
vs(x)(k+1) = vs(x)(k) − αγ(x), (3.4)
where α is the step length and the superscript (k) denotes the kth iteration. In practice
a preconditoned conjugate gradient method can be used for faster convergence.
3.2.2 Source-receiver Distance on a 3D Irregular Surface
When the wavelength is smaller than the characteristic wavelength of the topographi-
cal relief, the source-receiver distance factor will play a significant role in the accuracy
of the final tomogram (Kohler et al., 2012). Thus, we should calculate the source-
receiver offset distance along the actual irregular surface instead of assuming it to be
a flat surface.
For the flat free surface shown in Figure 3.1a, the source-receiver offset l along
the surface is the length of the line segment sr1, which is the same as the Euclidean
distance le between the source at s and the receiver at r1. When the surface is irregular
as shown in Figure 3.1b, the source-receiver offset l along the surface is the length of
the segment of a curve on the surface, which is larger than the Euclidean distance le.
The source-receiver offset distance along the irregular surface is called the “geodesic
distance”, which is the shortest route between two points on the surface. Appendix
A introduces the method for calculating the geodesic distance on a triangular mesh
surface.
Figure 3.2 shows the offset L and azimuth θ associated with the source at s to the
receiver at r on an irregular surface. Here, the azimuth is along the direction from
s′ to r′, where s′ and r′ are the perpendicular projections of points s and r on the
plane z = 0, respectively. Once we get the offset and azimuth for the receivers, we
can calculate the dispersion curve of the shot gather by applying to the common shot
87
gather (CSG) the discrete Radon transform in the frequency domain as presented in
Appendix B.
a) Flat Surface b) Irregular Surface
Figure 3.1: Schematic diagram shows the offset distance l along the (a) flat and (b)irregular surfaces from the source at s (the red star) to the receiver at r1, where le isthe Euclidean distance.
Figure 3.2: Schematic diagram shows the offset L and the azimuth θ from the sourceat s (red star) to the receiver at r.
3.2.3 Workflow of 3D TWD
The workflow for implementing 3D TWD is summarized by the following steps.
1. Remove the first-arrival body waves and higher-order modes of the Rayleigh
waves in the shot gathers (Li et al., 2017c).
2. Determine the source-receiver offset along the irregular surface, and the range
of the dominant azimuth angles θ for each shot gather. The dominant azimuth
88
angle is defined in Liu et al. (2019).
3. Apply a discrete Radon transform followed by the temporal Fourier transform
of the predicted and observed common shot gathers to compute the dispersion
curves κ(θ, ω) and κ(θ, ω)obs along each azimuthal angle θ. Calculate the sum
of the squared dispersion residuals in equation 4.1.
4. Calculate the weighted data D(g, ω)∗obs according to equation 4.4. The source-
side and receiver-side wavefields in equation 4.8 are computed by the SEM
solution to the 3D elastic wave equation.
5. Calculate and sum the gradients for all the shot gathers. The source illumination
is sometimes needed as a preconditioner (Plessix and Mulder, 2004).
6. Calculate the step length and update the S-wave tomogram using the steepest-
descent or conjugate gradient methods. In practice, we use a preconditioned
conjugate gradient method.
3.3 Numerical Results
The effect of topography on the calculation of the dispersion curves is first tested for
data computed over a homogeneous half-space model with an irregular free surface.
Then the effectiveness of 3D TWD is evaluated with synthetic and field data examples.
The data are associated with 1) a simple checkerboard model, 2) the complex 3D
Foothills model and a surface seismic experiment carried out in the Washington fault
zone of northern Arizona, U.S..
In the synthetic examples, the observed and predicted data are generated by a
spectral-element solver SPECFEM3D (Komatitsch and Vilotte, 1998; Komatitsch and
Tromp, 1999). The mesh is generated by the software package CUBIT, which is a
software toolkit for robust generation of two- and three-dimensional finite element
89
meshes (grids) and geometry preparation. For 3D TWD, only the S-wave velocity
model is inverted and the true P-wave velocity model is used for modeling the pre-
dicted surface waves. The density model is homogeneous with ρ =2000 kg/m3 for all
synthetic data tests. The source wavelet is a Ricker wavelet.
3.3.1 Homogeneous Half Space
The topography shown in Figure 3.2 is used for testing the effect of topography on
the calculation of the dispersion curves associated with Rayleigh waves. The study
area is 150 m in the x-direction and 220 m in the y-direction. The maximum elevation
difference of the topography is 36 m. We choose a homogeneous medium (vs=1 km/s,
vp =√
3vs, ρ =2300 kg/m3) with a free surface on the top. There are 1024 receivers
represented by the red dots in Figure 3.3a, which are arranged in 32 parallel lines
where each line has 32 receivers. A vertical-component shot is fired at the location A
in Figure 3.3a. The peak frequency of the source wavelet is 30 Hz.
The data recorded by the receivers within the yellow area in Figure 3.3a are chosen
for analysis, and the geodesic paths from the source at A are shown Figure 3.3b.
Figure 3.3c shows the differences between the geodesic and Euclidean distances for
these receivers, which indicates that the source-receiver distance errors introduced by
assuming a flat surface are up to 12 m for the far-offset receivers. Such source-receiver
distance errors will lead to inaccurate estimates of the phase velocity of surface waves,
which can be seen in the following tests. The seismograms recorded from these
receivers are displayed as the red wiggles in Figure 3.3d, where the seismograms
from the flat-surface model (blue) are displayed for comparison.
We apply the discrete Radon transform in the frequency domain to the seismo-
grams in Figure 3.3d to get their dispersion images shown in Figure 3.4. We then
pick the dispersion curves shown as the red curves in Figure 3.4. Here, the Euclidean
and geodesic distances are used in Figures 3.4a and 3.4b, respectively, and the the-
90
oretical dispersion curves are represented by the green curves. The dispersion image
computed from the data recorded in the flat-surface model is shown in Figure 3.4c for
comparison. We can see that the dispersion curves calculated by using the geodesic
distances are more accurate than those calculated by the Euclidean distances.
Figure 3.5 shows the dispersion curves for the azimuths ranged from 0◦ to 360◦
computed from the CSGs recorded in the flat-surface model, where the black dashed
lines represent their contours which are the reference contours. The cyan dash-dot
and red lines in Figure 3.5 represent the contours of the dispersion curves from the
topographic model calculated by the Euclidean and geodesic distances, respectively.
The contour calculated by the geodesic distance is much closer to the reference con-
tour compared to the ones computed by the Euclidean distance, especially for the
frequencies between 35 and 60 Hz.
3.3.2 Checkerboard Test
The 3D checkerboard model is shown in Figure 3.6a, and its vertical slices at y = 80
m and 160 m are shown in Figures 3.7a and 3.7c, respectively. We use the same to-
pography and acquisition geometry as those used in the homogeneous half-space test.
The values of the high and low S-velocities are 1100 m/s and 900 m/s, respectively.
The initial S-velocity model is homogeneous with vs = 1000 m/s and the P velocity
is set to be vp =√
3vs. Eighteen vertical-component shots are distributed on the free
surface which are marked as the red stars in Figure 3.3a. The peak frequency of the
source wavelet is 30 Hz. There are two levels of parallelization, one for the sources
and one for domain decomposition, and the total recording time is 0.32 s with a 0.08
ms time step.
The observed dispersion curve is first picked from the spectrum computed by the
Radon transform in the frequency domain. Each trace of the CSGs is compensated
for attenuation and only the traces with their offsets less than 80 m are used. The
91
(a) Acquisition Geometry (b) Paths of Geodesic
O set
0
50
00
150
220
x (m)z (
m)
y (m)
A
(c) Difference between Geodesic andEuclidean Distances
1 30Trace No.
4
12
Dis
tance E
rror
(d) Data with (Red) and without(Blue) Topography
1 30Trace No.
0.32
0
T (
s)
Figure 3.3: (a) Acquisition geometry where the yellow area shows the locations ofthe receivers (black asterisks) within the azimuth angle ranged from 277.5◦ to 282.5◦
for the source at A, where the source is represented by the red star; (b) paths of thegeodesics on the topography from the source at A to the receivers that are marked asthe black asterisks in (a); (c) differences between the geodesic and Euclidean distances,where the trace number is numbered according to the geodesic distance in ascendingorder; (d) CSG for trace No. 1 to 30 from the model with (red) and without (blue)topography.
92
(a) Euclidean Distance
30 80f (Hz)
0.6
1.1
c (
km
/s)
0.8
1
(b) Geodesic Distance
30 80f (Hz)
0.6
1.1
c (
km
/s)
0.8
1
(c) Flat Free Surface
30 80f (Hz)
0.6
1.1
c (
km
/s)
0.8
1
Figure 3.4: Dispersion image calculated by the (a) Euclidean and (b) geodesic dis-tances for the data recorded in the irregular surface. (c) Dispersion image calculatedfor the data recorded in the flat surface. Here, the green curves are the theoreticalphase velocity dispersion curves (c = 919.4 m/s) and the red curves are the pickeddispersion curves.
93
Dispersion Curves
11 1111 11
21 21 21 21
31 31 31 31
41 41 4141
51 51 5151
61 61 6161
11 11 11 11
21 2121 21
31 3131
31
41 41
41 41
51 5151 51
61 61
61 61
11 11 11 11
21 2121 21
31 3131
31
41 4141
41
51 5151
51
61 6161
61
0 100 200 300
Azimuth (degree)
10
22
34
46
58
Fre
qu
en
cy (
Hz)
0
10
20
30
40
50
60
Pic
ked W
avenum
ber
(1/k
m)
Flat
Euclidean
Geodesic
Figure 3.5: Dispersion curves for the data from the flat-surface model and theircontours are represented by the black dashed lines. Here, the cyan lines and thered dash-dot lines represent the contours of the dispersion curves calculated by theEuclidean and geodesic distances from the model with the topography, respectively.
fundamental dispersion curves for each CSG are picked along the dominant azimuths
from 0◦ to 360◦ with an interval of 5◦. For example, Figure 3.8 shows the observed dis-
persion curves from the CSGs with their sources located at points A and B indicated
in Figure 3.3a, where the black dashed lines represent the contours of the observed
dispersion curves. The cyan dash-dot lines in Figure 3.8 represent the contours of the
initial dispersion curves.
3D TWD is then used to invert the picked dispersion curves for the S-velocity
tomogram. Figure 3.6b displays the inverted S-wave velocity model after 15 iterations,
and its associated vertical slices at y = 80 m and 160 m are shown in Figures 3.7b and
3.7d, respectively; these results agree well with the true model. The contours of the
predicted dispersion curves for the sources located at points A and B are represented
by the red lines in Figure 3.8, which correlate well with the contours of the observed
dispersion curves.
94
(a) True S-velocity Model
y (m)
x (m)
z (
m)
1.10.9 1.0
(b) S-velocity Tomogram
y (m)
x (m)
z (
m)
1.10.9 1.0
Figure 3.6: (a) True S-velocity checkerboard model and (b) S-velocity tomogram by3D TWD.
(a) True S-velocity Slice at y = 80 m
x (m)
z (
m)
(b) Inverted S-velocity Slice at y = 80m
x (m)
z (
m)
(c) True S-velocity Slice at y = 160 m
x (m)
z (
m)
(d) Inverted S-velocity Slice at y = 160m
x (m)
z (
m)
Figure 3.7: True S-velocity slices at y = (a) 80 m and (c) 160 m. Inverted S-velocityslices at y = (b) 80 m and (d) 160 m.
95
(a) Dispersion Curves at Source A
1111
11 11
21
21 21
2121
31
31 31
3131
41
41 41
41 41
51
51
51
51
51
61
61
61
1111
11 11
2121
21
21
21
31 31 31
31
31
41 41 41
41
41
51 5151
51
51
61
61
61
61
1111
11 11
21
21 21
21
21
31
31 31
3131
41
4141
41 41
51
51
51
51
51
61
61
61
0 100 200 300
Angle (degree)
10
18
26
34
42
50
58
Fre
quency (
Hz)
0
10
20
30
40
50
60
Pic
ked W
avenum
ber
(1/k
m)
Observed
Initial
Predicted
(b) Dispersion Curves at Source B
11 11 1111
21
2121 21
21
31
31
31 31 31
41
41
41
41
41
51
51
51
51
51
51
61 61
61
61
61
61
11 1111
21
2121
2121
31
31
3131
31
41
4141
41
41
5151
51 51
51
61
61
6161
61
11 1111
21
2121
2121
31
31
31 3131
41
41
41
41
41
51
51
51
51
51
51
61
61
61
61
61
61
61
61
0 100 200 300
Angle (degree)
10
18
26
34
42
50
58
Fre
quency (
Hz)
0
10
20
30
40
50
60
Pic
ked W
avenum
ber
(1/k
m)
Observed
Initial
Predicted
Figure 3.8: Observed dispersion curves from the CSGs with their sources locatedat points (a) A and (b) B (indicated in Figure 3.3a), where the black dashed lines,the cyan and red dash-dot lines represent the contours of the observed, initial andinverted dispersion curves, respectively.
3.3.3 3D Foothills Model
The topography of the 3D Foothills model shown in Figure 3.9 is extracted from
the 3D SEG Advanced Modeling (SEAM) phase II foothills model (Oristaglio, 2012),
where the red lines are the geodesic paths on the triangular mesh for the source
marked as the red star. The maximum elevation difference of the topography is 1.2
km. The 3D Foothills S-wave velocity model shown in Figure 3.10a is modified from
the 2D Foothills model in Figure 2a of Brenders et al. (2008). The P-wave velocity
is defined as vp =√
3vs and the physical size of the velocity model is 7 km and
3.5 km in the x and y directions, respectively, and is 2 km deep in the z-direction.
The mesh used in the SPECFEM3D is shown in Figure 3.10b. The initial S-velocity
model is shown in Figure 3.10c. Figure 3.11 shows the acquisition geometry for this
experiment, where 2312 geophones are distributed on the surface, which are arranged
in 17 parallel lines along the x-direction, and each line has 136 receivers. The in-
line and cross-line receiver intervals are 50 m and 190 m, respectively. There are 80
vertical-component shots distributed on a 10×8 grid with source intervals of 750 m
and 380 m in the x and y directions, respectively. The peak frequency of the source
96
is 5 Hz and the observed data are recorded for 2.40 seconds with a 0.8 ms sampling
rate.
The fundamental dispersion curves for each CSG are picked for the frequencies
from 2 to 9 Hz along the dominant azimuths from 0◦ to 360◦ with an interval of 5◦.
For example, Figure 3.12 shows the observed dispersion curves calculated from the
CSGs for the sources located at points A, B, C and D indicated in Figure 3.11, where
the black dashed lines represent the contours of the observed dispersion curves. The
cyan lines represent the contours of initial dispersion curves.
3D TWD is then used to invert for the S-velocity tomograms. Figure 3.10d displays
the inverted S-wave velocity model. The vertical slices for the true, initial and inverted
models are shown in Figures 3.13a, 3.13b and 3.13c, respectively, where the black- and
white- dashed lines indicate the large velocity contrast boundaries and the boundaries
0.5 km below the free surface, respectively. The depth slices 300 m below the free
surface for the true, initial and inverted models are shown in Figures 3.14a, 3.14b
and 3.14c, respectively. We can see that the S-velocity model is significantly updated
in the shallow part, where most updates are confined to the region within 0.5 km
from the surface. The overall velocity structure is well recovered, even though some
small-scale features are still missing, which might be caused by the limited frequency
content in the data.
The contours of the predicted dispersion curves for the sources located at points
A, B, C and D in Figure 3.11 are represented by the red dash-dot lines in Figure 3.12,
which agree well with the contours of the observed dispersion curves. Figure 3.15
compares the observed (red) and synthetic (blue) traces at the far source-receiver
offsets predicted from the initial and inverted models for (a) and (b) with the CSG at
B, and (c) and (d) with the CSG at C. Figure 3.16 shows the common offset gathers
(COGs) with offset 2.85 km, which are retrieved from the traces located at the green
rectangles in Figure 3.11 of the CSGs with the sources located at the green stars
97
in Figure 3.11. Here the red and blue wiggles represent the observed and predicted
COGs, respectively. It can be seen that the synthetic waveforms computed from the
3D TWD tomogram closely agree with the observed ones.
0
1.4
00
7.0
3.5
x (km)
z (
km
)
y (km)
1.2 0 Elevation (km)
Figure 3.9: Topography of the 3D Foothill model, where the red lines are the geodesicpaths for the source marked by the red star.
3.3.4 Washington Fault Seismic Data
A 3D seismic survey was conducted across the Washington fault zone of northern
Arizona in 2008 (Figure 3.17a) and then the Utah and Arizona Geological Surveys
(UGS) excavated three trenches over that area (Figure 3.17b). The 3D acquisition
geometry consists of six parallel lines and each line has 80 receivers with a 1 m spacing
near the fault scarp and a 2 m spacing far away from the fault scarp. The length of
each line is 119 m and the cross-line spacing is 1.5 m. The seismic source is a 10-lb
sledgehammer striking a metal plate on the ground. Shots are activated at every
other geophone and the experiment geometry is shown in Figure 3.18. One of the
CSGs (# 87) is shown in Figure 3.19, where the observed data are recorded for 0.5
seconds with a 0.25 ms sampling rate.
The 3D data set was impacted by an unpredictable time delay between the source
98
(a) True S-velocity Model
3.41.8 2.6
X (km)
Y (km)
Z (
km
)0
0
(b) Corresponding Mesh
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Y (km)
Z (
km
)
0
0
-0.6 0.6
(c) Initial S-velocity Model
3.41.8 2.6
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Y (km)
Z (
km
)
0
0
(d) S-velocity Tomogram
3.41.8 2.6
X (km)
Y (km)
Z (
km
)
0
0
Figure 3.10: (a) True S-velocity model, (b) corresponding mesh, (c) initial S-velocitymodel and (d) S-velocity tomogram.
Acquisition Geometry
0 7x (km)0
3.5
y(k
m)
A
B
D
C
1
17
Lin
e N
o.
Figure 3.11: Acquisition geometry for the numerical tests with data generated for the3D Foothill model, where the red dots and blue circles indicate the locations of thereceivers and sources, respectively.
99
(a) Dispersion Curves at Source A
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ke
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Figure 3.12: Observed dispersion curves for the sources located at (a) A, (b) B, (c) Cand (d) D indicated in Figure 3.11b, where the black dashed lines, the cyan dash-dotlines and the red lines represent the contours of the observed, initial and inverteddispersion curves, respectively.
100
(a) Slice of True Model at y = 433 m
3.41.8
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Z (
km
)0 2 4 6
1
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00.5 km
(b) Slice of Initial Model at y = 433 m
3.41.8
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)
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(c) Slice of Tomogram at y = 433 m
3.41.8
X (km)
Z (
km
)
0 2 4 6
1
-0.6
00.5 km
Figure 3.13: Slices of the (a) true, (b) initial, and (c) inverted S-velocity models at y= 433 m, where the black and white dashed lines indicate the large velocity contrastboundaries and the boundaries 0.5 km below the free surface, respectively.
(a) Depth Slice of True Model
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)
0 2 4 6
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0
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)
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1
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3.41.8
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Y (
km
)
0 2 4 6
3
0
2
1
Figure 3.14: Depth slices 300 m below the surface for the (a) true, (b) initial and (c)inverted Foothill S-velocity models, where the black dashed lines indicate the largevelocity contrast boundaries.
101
(a) CSG B Line # 17 from InitialModel
1 129
Trace No.
0.8
2.4
T (
s)
(b) CSG B Line # 17 from 3D Tomo-gram
1 129
Trace No.
0.8
2.4
T (
s)
(c) CSG C Line # 2 from InitialModel
1 129
Trace No.
0.8
2.4
T (
s)
(d) CSG C Line # 2 from 3D Tomo-gram
1 129
Trace No.
0.8
2.4
T (
s)
Figure 3.15: Comparison between the observed (red) and synthetic (blue) traces atfar offsets predicted from the initial model (LHS panels) and 3D tomogram (RHSpanels) for CSG B in (a) and (b), and CSG C in (c) and (d). Here, the locations ofpoints B and C and the line numbers are indicated in Figure 3.11.
102
(a) COG from Initial Model
1 9
Shot No.
0.8
2.4
T (
s)
(b) COG from 3D Tomogram
1 9
Shot No.
0.8
2.4
T (
s)
Figure 3.16: COGs with the offset of 2.85 km, which are retrieved from the traceslocated at the green rectangles in Figure 3.11 of the CSGs with the sources located atthe green stars in Figure 3.11. Here the red and blue wiggles represent the observedand predicted COGs, respectively.
Figure 3.17: (a) Map of the Washington fault and the survey site. The location of thesurvey site is 5 km south of the Utah-Arizona border. (b) Topographic map aroundthe seismic survey, where the red and green rectangles indicate the locations of the3D seismic survey and the trench site, respectively. (After Lund et al. (2015).)
103
0 20 40 60 80 100 120
Inline (m)
-1
0
1
2
3
4
5
6
7
8
Cro
sslin
e (
m)
1 2 40
41 80
201 240
Shot No.
Figure 3.18: Survey geometry for the 3D experiment in the Washington fault zone.The open red circles denote the locations of sources and the solid blue dots denote thelocations of receivers. The dashed black line denotes the location of the fault scarp.
Figure 3.19: Common shot gather # 87 of Washington fault data.
104
initiation time and the onset of the data recording. This issue was identified by non-
zero amplitudes at the zero time for the near-offset trace. To correct this hardware
error, the traces in the shot gather of the 3D data set were advanced by a constant
time value t(s). To correct for this timing error, the timing error t(s) is obtained by
minimizing the summation of the picked traveltime differences,
t(s) = arg mint(s)
∑i
(t(s) + t(gi, s)− t(s,gi))2, (3.5)
where t(gi, s) is the traveltime picked from the trace located at gi of the CSG with the
source located at s. Figure 3.20 shows the picked traveltime matrices of traveltime
picks for all the shot gathers on line #4 before and after correction. After correction,
the picked traveltimes are more continuous in the common receiver gather. Continuity
in the arrival times is more important than absolute times in order to compute the
correct moveout velocity of the surface waves.
T (s
)
0.68
0
Shot No. Shot No.
Receiv
er
No.
a) Traveltime before Correction b) Traveltime after Correction
Figure 3.20: Traveltime matrices before and after the correction of the acquisitionhardware error for the 2D data set on line #4.
The data are processed before the calculation of the dispersion images. Each trace
is normalized to compensate for the effects of attenuation and geometrical spreading.
105
We only include the fundamental-mode Rayleigh waves in the CSG by a muting win-
dow. For each shot gather, only receivers within the distance r1 = 35 m from the
source are used to calculate the dispersion curves. The dominant azimuth angles de-
fined in Liu et al. (2019) for most of the CSGs are approximately 0◦ and 180◦ because
of the narrow acquisition geometry. Thus, the fundamental dispersion values are cal-
culated along the azimuthal angles 0◦ and 180◦. The frequency range in the inversion
is from 20 Hz to 60 Hz. Figures 3.21a and 3.21b show the picked dispersion values
for CSGs along line #4 with the azimuthal angles θ = 0◦ and θ = 180◦, respectively;
here, the black dashed lines denote the contours of the observed dispersion curves.
At certain frequency ranges, it is difficult to pick the dispersion curves because of the
low signal-to-noise ratio of the data so that some dispersion curves are missing.
We first approximate an initial model from the picked dispersion curves, which
is called a “pseudo 1D S-velocity model” in Liu et al. (2019). That is, the depth z
and S-wave velocity vs of the initial model are calculated by scaling the wavelength
λ and phase velocity c with factors of 0.5 and 1.1, respectively. The 1D depth profile
is assumed to be centered at the middle of the receiver spread. The 1D velocity
profiles are interpolated as the starting model for 2D WD. For example, Figure 3.22a
displays the pseudo 1D S-velocity model beneath line #4 which is calculated from
the dispersion curves in Figure 3.21. For comparison with the WD tomogram, we
interpolated the 1D velocity profiles from all six lines as the 3D model shown in
Figure 3.23a.
Then, we apply 2D TWD to invert for the 2D velocity model along the 6 lines.
Figures 3.22a and 3.22b show the initial and inverted S-velocity models beneath the
fourth line, where the white lines indicate the boundaries 10 m below the free surface.
The cyan dash-dot lines in Figure 3.21 represent the contours of the initial dispersion
curves. The contours of the predicted dispersion curves are represented by the red
lines in Figure 3.21, which more closely agree with the contours of the observed
106
dispersion curves, especially for the high frequencies ranging between 45 Hz to 60 Hz.
The 6 inverted 2D S-velocity models are then interpolated to obtain an initial velocity
model (see Figure 3.23b) for 3D TWD. The inverted 3D TWD tomogram is shown
Figure 3.23c. The 3D TWD tomogram is almost the same as the initial S-velocity
model and indicates that 2D TWD is sufficient for this dataset because of the narrow
acquisition geometry.
To further test the accuracy of the TWD tomogram, Figure 3.24 shows the com-
parison between the observed (blue) and synthetic (red) traces predicted from the
(a) initial and (b) inverted S-velocity models for CSGs No. 128. Here, two matched
filters are calculated from trace No. 141 in Figure 3.24, respectively. The matched
filters are then applied to reshape the synthetic waveform. We can see that the pre-
dicted fundamental-mode surface waves closely match the observed ones, especially
at the far offset locations. Figure 3.25 shows the COGs with the offset of 16 m for
CSGs in line No. 4, where the blue and red wiggles represent the observed and pre-
dicted COGs, respectively. The predicted COG is more consistent with the raw data
compared to the initial COG.
For comparison, we calculate the 2D P-velocity tomogram shown in Figure 3.22c
by the raypath traveltime inversion method. The Vp/Vs ratio map is then calculated
and displayed in Figure 3.22d. From the inverted S-velocity tomogram shown in
Figure 3.22b, we can see there is a low-velocity zone (LVZ) between X=40 m and
X=80 m.The LVZ appears in the P-velocity tomogram but the boundaries of the
LVZ are ambiguous compared to those shown in the S-velocity tomogram. The LVZ
is also clearly shown in the Vp/Vs ratio tomogram, which has a high Vp/Vs ratio.
The fault scarp at X=43 m ( see Figure 3.18) is located at the left-hand side of the
LVZ. So, we interpret the line labeled with “F1” as the location of the main fault.
In Figure 3.22, the black lines labeled with “F2” are the locations of the interpreted
antithetic fault and there is also a possible fault labeled with “F3”. The locations of
107
these faults and the S-velocity tomogram are superposed on the COGs in Figure 3.26.
The fault structures appear to be consistent with those seen in the COG image.
The trench was excavated to explore the fault zone by UGS in the spring of 2009
(Lund et al., 2015; Hanafy et al., 2015). The trench is across the fault scarp and
its location is indicated by the black rectangles in Figure 3.22. The zoom view of
the S-velocity and P-velocity tomograms and the Vp/Vs ratio tomogram in the black
rectangles are shown in Figures 3.27a, 3.27b and 3.27c, respectively. The trench
log is displayed in Figure 3.27d and exposes a more complex main fault zone (F1).
The locations of the main fault in S-velocity tomogram and Vp/Vs ratio tomogram
are consistent with those in the trench log. Compared to the 3-D P-wave velocity
tomogram, the 3D S-wave tomogram agrees much more closely with the geological
model taken from the trench log. The agreement with the trench log is even better
when the Vp/Vs tomogram is computed, which reveals a sharp change in velocity
across the fault. The localized velocity anomaly in the Vp/Vs tomogram is in very
good agreement with the well log. Our results suggest that integrating the Vp and Vs
tomograms can sometimes give the most accurate estimates of the subsurface geology
across normal faults.
3.4 Discussion
Irregular topography has a significant impact on surface-wave propagation, which dis-
torts seismic wavefronts by strong scattering and attenuation in a complex manner.
When the wavelength is smaller than the characteristic wavelength of the topographic
relief, the source-receiver distance factor may play a significant role for calculating
the phase velocity of surface waves, which is essential for 3D TWD and discussed
by Li et al. (2019b). Failure to use the actual source-receiver distance in the eval-
uation of the phase velocity can lead to errors in the inverted model for 3D TWD.
Results with the homogeneous model suggest that there will be significant errors in
108
(a) Dispersion Curves for θ = 0◦
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Figure 3.21: (a) Observed dispersion curves for the CSGs on Line # 4 along theazimuthal angles (a) θ = 0◦ and (b) θ = 180◦, where the black dashed lines, the cyandash-dot lines and the red lines represent the contours of the observed, initial andinverted dispersion curves, respectively.
the dispersion curve without consideration of the topography. By considering the
topography properly, we can get a more accurate dispersion curve as shown in the
Foothills examples.
The elevation changes in our 3D Foothill example are between two and four S-wave
wavelengths. Accurate wavefield modeling using a 3D elastic SEM is an important
ingredient for successful 3D TWD inversions in such a challenging geologic setting.
As shown in Figure 3.13, the S-velocity model is significantly updated only in the
shallow part (about 0.5 km deep from the surface). This is reasonable because the
maximum wavelength of the surface waves is 1.0 km approximately and surface waves
are typically most sensitive to the velocity model to a depth of approximately one-half
of their wavelength (Liu et al., 2017a; Hyslop and Stewart, 2015).
Our field data example shows no significant improvement for the 3D TWD results
compared to the 2D results. This is because of the relatively narrow recording geom-
etry of the Arizona survey. Considering the high computational cost for 3D TWD,
3D TWD might be too costly and not significantly beneficial compared to 2D TWD
109
Z (
m)
1.1
0.3
Vs (k
m/s
)
0 20 40 60 80 100
0
10
20
0
10
20
0 20 40 60 80 100
a) Initial S-Velocity Model
b) S-Velocity Tomogram
Z (
m)
F1
F1
Vp (k
m/s
)Z (
m)
0
10
20
c) P-Velocity Tomogram
0 20 40 60 80 100
F1
2.2
0.6
Vp/V
s
Z (
m)
0
10
20
d) Vp/Vs Ratio Map
0 20 40 60 80 100
X (m)
F1
3
1.2
F2
F2
F2
F2
F3
F3
F3
F3
Figure 3.22: (a) Initial and (b) inverted S-wave velocity models beneath line #4. (c)P-wave velocity tomogram calculated from the picked traveltimes in Figure 3.20b. (d)Vp/Vs ratio tomogram beneath line #4. Here the white lines indicate the boundaries10 m below the free surface. The trench is excavated in the locations of the blackrectangles. The lines labeled with “F1” and “F2” are interpreted as the locations ofthe main fault and the antithetic fault. The line labeled with “F3” is the location ofanother possible fault. “CW” represents the colluvial wedge.
110
(a) Initial S-velocity Model
(b) 2D S-velocity Tomogram
(c) 3D S-velocity Tomogram
Figure 3.23: (a) Initial, (b) 2D and (c) 3D S-wave velocity tomograms. Here, thedepth and S-wave velocity of the initial model are calculated by scaling the wavelengthand phase velocity with factors of 0.5 and 1.1, respectively (Liu et al., 2019).
111
(a) CSG from Initial S-velocity Model
0 20 40 60 80 100 120
X (m)
0
0.05
0.1
0.15
0.2
0.25
0.3
T (
s)
(b) CSG from the Inverted S-velocity Model
0 20 40 60 80 100 120
X (m)
0
0.05
0.1
0.15
0.2
0.25
0.3
T (
s)
Figure 3.24: Comparison between the observed (blue) and synthetic (red) tracespredicted from the (a) initial and (b) inverted S-velocity models for CSG # 128.
112
(a) COG from Initial Model
20 40 60 80 100
X (m)
0
0.05
0.1
0.15
0.2
T (
s)
F3 F1 F2
(b) COG from Inverted Model
20 40 60 80 100
X (m)
0
0.05
0.1
0.15
0.2
T (
s)
F3 F1 F2
Figure 3.25: COGs with the offset of 16 m for line # 4 calculated from the (a) initialand (b) inverted S-velocity models, where the blue and red wiggles represent theobserved and predicted COGs, respectively.
113
S-velocity Tomogram with COG
Z (
m)
1.1
0.3
Vs (k
m/s
)
0
10
20
0 20 40 60 80 100
X (m)Figure 3.26: Observed COGs with the offset of 16 m are superposed on the S-velocitytomogram, where the COGs are adjusted by following the topography.
for some field surveys. However, the improvement of 3D WD in accuracy compared
to 2D WD can sometimes make this extra cost worthwhile when there are significant
near-surface lateral variations in the S-velocity distribution (Liu et al., 2019).
High Vp/Vs ratios in the LVZ of the tomograms (Figure 3.22d) might be caused
by the groundwater saturation in the fault zone. Groundwater saturation has a major
effect on P-wave velocities in the near surface, where saturated materials typically
have higher P-wave velocities than unsaturated or partially saturated materials due
to the higher incompressibility (e.g., bulk modulus, K) of the saturated materials.
However, S-wave velocities are much less affected by groundwater saturation because
S waves do not include a bulk modulus term (Catchings et al., 2014). When the
sediment in the LVZ is saturated with groundwater, the P-wave velocity is increased so
that the boundaries of the LVZ are ambiguous in the P-velocity tomogram. However,
the S-wave velocity is much less affected so that the boundaries of the LVZ are more
clearly delineated in the S-velocity tomogram.
114
F1
35 45 5510
8
6
43
Z (
m)
4030 50
X (m)
35 45 5510
8
6
43
Z (
m)
4030 50
35 45 5510
8
6
43
Z (
m)
4030 50
35 45 5510
8
6
43
Z (
m)
4030 50
Scarp colluvium (paleoearthquakes E1 and E2)
Red "old" fan(mixed debris-�ow, stream, and eolian deposits)
Brown "cavy" fan
Coarse debris-�ow deposit
Light "buldgy" fan
1.1
0.3
Vs (k
m/s
)V
p (k
m/s
)
2.2
0.6
F1
F1
F1
Fault scarp
Fault scarp
Fault scarp
a) Zoom View of S-velocity Tomogram
b) Zoom View of P-velocity Tomogram
c) Zoom View of Vp/Vs Tomogram
d) Trench Log
Vp/V
s
3.0
1.2
TRml
Figure 3.27: Zoom views of (a) S-velocity and (b) P-velocity tomograms and (c)Vp/Vs tomogram in Figure 3.22. (d) Ground truth extracted from a nearby trenchlog (Lund et al., 2015; Hanafy et al., 2015).
115
3.5 Conclusions
We extend the 2D TWD methodology to 3D, that accounts for significant 3D vari-
ations in topography by a 3D spectral element solver. The objective function of
3D TWD is the sum of the squared differences between the predicted and observed
dispersion curves. More accurate dispersion curves can be calculated by using the
geodesic distance compared to that using the Euclidean distance, which can lead to
a more accurate inverted model for 3D TWD. The effectiveness of this method is
numerically demonstrated with synthetic and field data recorded on an irregular free
surface. Results with synthetic data suggest that 3D TWD can accurately invert for
the S-velocity model in the Foothills region when there is a huge elevation difference
compared to the S-wave wavelengths. Field data tests suggest that, compared to the
3-D P-wave velocity tomogram, the 3D S-wave tomogram agrees much more closely
with the geological model taken from the trench log. The agreement with the trench
log is even better when the Vp/Vs tomogram is computed, which reveals a sharp
change in velocity across the fault that is in very good agreement with the well log.
Our results suggest that integrating the Vp and Vs tomograms can sometimes give
the most accurate estimates of the subsurface geology across normal faults.
Similar to 3D WD, a limitation of 3D TWD is that the fundamental dispersion
curves must be picked for each shot gather. This process can be prone to errors when
there is a strong overlap with higher-order modes or there is spatial and temporal
aliasing due to large spatial and temporal sampling intervals. This problem might be
mitigated by the machine learning method that automatically picks dispersion curves.
3.6 Acknowledgements
We dedicate this paper to Dimitri Komatitsch and his loving family members, he is
a light that has gone out much too early. We also thank Ron Bruhn for his valuable
116
advice about the geological explanation of our field results. The research reported
in this publication was supported by the King Abdullah University of Science and
Technology (KAUST) in Thuwal, Saudi Arabia. We are grateful to the sponsors
of the Center for Subsurface Imaging and Modeling Consortium for their financial
support. For computer time, this research used the resources of the Supercomputing
Laboratory at KAUST and the IT Research Computing Group. We thank them for
providing the computational resources required for carrying out this work.
3.7 Appendix A: Calculation of the Geodesic
A natural shortest paths problem with many applications is the following: given two
points s and r on the surface of a polyhedron of n vertices, find the shortest path on
the surface from s to r. This type of within-surface shortest path is often called a
geodesic shortest path, in contrast to a Euclidean shortest path (O’Rourke, 1999).
The computation of geodesic paths is a common operation in many computer
graphics applications (Surazhsky et al., 2005) and the computation of seismic travel
times (Sethian and Popovici, 1999). There are many methods for computing the
geodesic distance on the topography represented by the triangle mesh. Most of the
algorithms bear a very close resemblance to the famous Dijkstra algorithm (Dijkstra,
1959) that finds the shortest paths on graphs, for example, the fast marching method
(Sethian, 2001) and the exact geodesic algorithm (Mitchell et al., 1987). The exact
geodesic algorithm uses the continuous Dijkstra method and simulates the continuous
propagation of a wavefront of points equidistant from s across the surface. The
method has O(n2 log n) worst-case time complexity, but in practice can work with
million-node meshes in a reasonable time. An exact geodesic algorithm with worst-
case time complexity of O(n2) was described by Chen and Han (1990).
The most straightforward explanation of the exact geodesic algorithm is unfolding.
If we want to find the shortest path on the surface of a sample surface mesh shown
117
in Figure 3.28, we can find a sequence of edge-adjacent faces f1, f2, · · · , f7 and unfold
face fi+1 onto the plane of fi as shown in Figure 3.28b so that the shortest path is the
straight line. The detailed implementation can be referred to Surazhsky et al. (2005)
and Chen and Han (1990). In this paper, we firstly generate a triangular mesh for
the topography by CUBIT. Then, we compute the geodesic distance using the exact
geodesic algorithm implemented by Surazhsky et al. (2005).
(a) A geodesic on a sim-ple surface mesh
(b) The same geodesic, with its faces unfoldedinto the plane.
Figure 3.28: Schematic diagram of the calculation of the geodesic on a simple surfacemesh by unfolding.
3.8 Appendix B: Discrete Radon Transform
Assume that g = rg(cos θ, sin θ, 0) is the mapping point from the receiver g =
(xg, yg, zg) to plane z = 0, where rg and θ are the geodesic distance and azimuth
angle, respectively. So, the domain of the data d(g, t) is changed to (g, t). The
discrete Radon transform of the shot gather d(g, t) is
m(p, θ, τ) =∑rg
d(rg, θ, t = τ + rgp), (3.6)
118
where p = p(cos θ, sin θ) is the slowness vector, and p is the slowness value along the
azimuth angle θ. Apply a Fourier transform to equation 3.6 gives
m(p, θ, ω) =∑rg
∫ ∞−∞
d(rg, θ, τ + rgp)eiωτdω,
=∑rg
[ ∫ ∞−∞
d(rg, θ, τ)eiωτdω
]e−iωprg ,
=∑rg
d(rg, θ, ω)e−iωprg , (3.7)
where d(rg, θ, ω) is the Fourier spectrum of the data d(rg, θ, t), and m(p, θ, ω) is the
Fourier spectrum of the Radon-transformed data m(p, θ, τ). The fundamental-mode
dispersion curve for the azimuth θ, C(ω, θ), is picked from the magnitude spectrum
of m(p, θ, ω).
119
Chapter 4
Multiscale and Layer-Stripping Wave-Equation Dispersion
Inversion of Rayleigh Waves 1
The iterative wave-equation dispersion inversion can suffer from the local minimum
problem when inverting seismic data from complex Earth models. We develop a
multiscale, layer-stripping method to alleviate the local minimum problem of wave-
equation dispersion inversion of Rayleigh waves and improve the inversion robustness.
We first invert the high-frequency and near-offset data for the shallow S-velocity
model, and gradually incorporate the lower-frequency components of data with longer
offsets to reconstruct the deeper regions of the model. We use a synthetic model to
illustrate the local minima problem of wave-equation dispersion inversion and how
our multiscale and layer-stripping wave-equation dispersion inversion method can
mitigate the problem. We demonstrate the efficacy of our new method using field
Rayleigh-wave data.
4.1 Introduction
Wave-equation dispersion inversion (WD) of Rayleigh waves uses solutions to the 2D
or 3D elastic-wave equation to invert the dispersion curves of surface waves for the S-
velocity model (Li and Schuster, 2016; Li et al., 2017c,a,b,e; Liu et al., 2017b, 2019).
The advantage of WD over the conventional dispersion inversion method (Haskell,
1953; Xia et al., 1999, 2002; Park et al., 1999) is that WD does not assume a 1D
1This manuscript was published as: Zhaolun Liu, and Lianjie Huang, (2019), ”Multiscale andlayer-stripping wave-equation dispersion inversion of Rayleigh waves,” Geophys. J. Int. 218(3):1807-1821, doi: https://doi.org/10.1093/gji/ggz215
120
velocity model and is valid when there are strong lateral gradients in the S-velocity
model. The WD method also enjoys robust convergence because the skeletonized
data, namely the dispersion curves, are simpler than a trace with many dispersive
arrivals. Such traces are used in full waveform inversion (FWI) (Groos et al., 2014;
Perez Solano et al., 2014; Dou and Ajo-Franklin, 2014; Groos et al., 2017).
The iterative WD method can suffer from the local minimum problem when in-
verting seismic data from complex Earth models. One method to tackle this problem
is the multiscale method (Masoni et al., 2016). For Body waves, the low-to-high fre-
quency content of data is first used to update the large-scale velocity structure and
then the more detailed features of the velocity model are reconstructed (Sirgue and
Pratt, 2004; Bunks et al., 1995). However, a high-to-low frequency strategy for surface
waves is needed because the frequency content of surface waves is directly related to
their penetration depth: higher-frequency and shorter-wavelength surface waves sam-
ple the top layers of a medium, while lower-frequency and longer-wavelength surface
waves sample deeper subsurface regions (Masoni et al., 2016).
The WD method needs to determine the source-receiver offset range (denoted as
R) starting from the near offset for retrieving the dispersion curves of the data using
F -K or Radon transforms. A narrow range of offsets corresponding to a small R is not
adequate for accurate retrieval of the low-frequency component of dispersion curves
(indicated in Figs. 2.3-8 of Yilmaz (2015)), but can provide high lateral resolution in
the tomographic image. Conversely, a wide range of offsets is adequate for accurately
retrieving the low-frequency dispersion curves but the penalty is that it only provides
a low-wavenumber estimate of the velocity model. As a rule of thumb, we choose R
to be about three or four times (3.5 is used in this paper) greater than the depth of
interest to make sure that WD has enough penetration depth and lateral resolution
(Liu et al., 2019). However, a fixed value of R would result in a loss of either the
low-frequency information in the dispersion curves or the lateral resolution of the
121
inverted S-velocity model. Thus, an iterative small-to-large offset range strategy is
needed to obtain both high lateral resolution and low-frequency information.
We develop a multiscale, layer-stripping method for wave-equation dispersion in-
version of Rayleigh waves to improve the inversion robustness. We first use the high-
frequency surface-wave data with a small-offset range to update the shallow velocity
model, and then use the low-frequency surface-wave data with a large-offset range
to update the deeper regions of the velocity model. Besides the multi-frequency and
multi-offset strategy, we employ a layer-stripping method (Shi et al., 2015; Masoni
et al., 2016) for WD to reconstruct the velocity model from the shallow to deep re-
gions. The layer-stripping method assumes that all layers above a given layer have
been inverted using the near-offset and high-frequency surface-wave data. We use the
far-offset and low-frequency data to invert for the velocity model of the deep layers.
This procedure is repeated until the entire volume of interest is reconstructed.
After the introduction, we describe the theory of WD and the workflow for the
layer-stripping approach. Numerical tests on synthetic and field surface-wave data
are presented in the third section to demonstrate the improvement of the method,
followed by the discussion and conclusions.
4.2 Theory
We present the formulation for wave-equation dispersion inversion, introduce the
multiscale, layer-stripping strategy for WD, and give the workflow of our multiscale,
layer-stripping WD (MSLSWD) method.
4.2.1 Theory of WD
Let d(g, t) denote a shot gather of vertical particle-velocity traces recorded by a
receiver on the surface at g = (xg, yg, 0), as shown in Fig. 4.1a. The surface waves
are excited by a vertical-component force on the surface at s = (xs, ys, 0), where
122
Fourier
Transform
a) b)
Figure 4.1: (a) Common shot gather d(g, t) and (b) the fundamental dispersion curvefor Rayleigh waves in the kx − ky − ω domain. Here, θ is the azimuth angle, andκ(θ, ω) is the skeletonized data.
the horizontal recording plane is at z = 0. Assuming that data are filtered such that
d(g, t) contains only the fundamental mode of Rayleigh waves, a 3D Fourier transform
of the data transform d(g, t) into D(k, ω) in the k−ω domain, as shown in Fig. 4.1b.
The wavenumber vector k = (kx, ky) can be represented in the polar coordinate
as (k, θ), where θ = arctan(ky/kx) is the azimuth angle and k =√k2x + k2
y is the
radius. Following this notation, the Fourier transformed data D(k, ω) are denoted as
D(k, θ, ω). We skeletonize the spectrum D(k, θ, ω) as the dispersion curves associated
with the fundamental mode of the Rayleigh waves, which are the wavenumbers κ(θ, ω)
obtained by the fundamental dispersion curve in the (k, θ, ω) coordinates.2 This curve
is recognized as the maximum magnitude spectrum D(k, θ, ω) along the azimuth angle
θ and is denoted as κ(θ, ω)obs for the observed data, which is displayed as the red
curves in Fig. 4.1b.
The wave-equation dispersion inversion method inverts for the S-wave velocity
model to minimize the dispersion objective function
ε =1
2
∑ω
∑θ
[
residual=∆κ(θ,ω)︷ ︸︸ ︷κ(θ, ω)pre − κ(θ, ω)obs]
2, (4.1)
2Higher-order modes can also be picked and inverted.
123
where κ(ω, θ)pre represents the predicted dispersion curve picked from the simulated
spectrum D(k, θ, ω) along the azimuth angle θ, and κ(ω, θ)obs describes the observed
dispersion curve obtained from the recorded spectrum D(k, θ, ω)obs along the azimuth
θ. In the 2D case, the azimuth angles have only two values: 0◦ and 180◦, corresponding
to the positive and negative x-directions, respectively.
The gradient γ(x) of ε with respect to the S-wave velocity vs(x) is
γ(x) =∂ε
∂vs(x)=∑ω
∑θ
∆κ(θ, ω)∂κ(θ, ω)pre∂vs(x)
, (4.2)
where the ∂κ(θ,ω)pre∂vs(x)
is (Liu et al., 2019, 2018),
∂κ(θ, ω)pre∂vs(x)
= −R
{∫dg∂D(g, ω)
∂vs(x)D(g, θ, ω)∗obs
}A(θ, ω)
, (4.3)
which is derived by forming a connective function that relates the dispersion curve
κ(θ, ω)pre to the S-wave velocity model vs(x) (Luo and Schuster, 1991a,b; Li et al.,
2017d; Lu et al., 2017; Schuster, 2017). In equation (4.3), R denotes the real part and
the superscript ∗ stands for the complex conjugation. A(θ, ω) is constant and defined
in Liu et al. (2019, 2018). D(g, ω) is the inverse Fourier transform of D(k, θ, ω), and
D(g, θ, ω)∗obs is the weighted conjugated data function:
D(g, θ, ω)∗obs = 2πig · neig·n∆κ
∫C
D(g′, ω)∗obsdg′, (4.4)
where n = (cos θ, sin θ) and C is the line (g′−g) ·n = 0. The line integral in equation
4.4 can be approximated by D(gc, ω)∗obs according to the stationary phase method
(Liu et al., 2019), where gc = (g · n)n is the stationary point, so that equation (4.4)
124
can be written as:
D(g, θ, ω)∗obs = 2πig · neig·n∆κD(gc, ω)∗obs. (4.5)
Inserting equations (4.3) and (4.5) into equation (4.2) gives the gradient:
γ(x) = −∑ω
R
{∫dg∂D(g, ω)
∂vs(x)
adjoint source︷ ︸︸ ︷∑θ
[−2πig · n∆κ(θ, ω)e−ig·n∆κ(θ,ω)D(gc, ω)obs]∗
A(θ, ω)
},
(4.6)
where the adjoint source can be explained from the right to left hand sides: for a
specific azimuth angle, a phase shift e−ig·n∆κ(θ,ω) is first applied to observed data
D(gc, ω)obs, which shifts the phase of the observed data to that of the predicted data,
and then is weighted by −2πig · n∆κ and a constant A(θ, ω). The adjoint source of
WD is different from that of FWI, which is the data residual D(g, ω) − D(g, ω)obs
(Virieux and Operto, 2009).
In equation (4.6), term ∂D(g,ω)∂vs(x)
can be obtained using the Born approximation of
elastic waves (Liu et al., 2019, 2018):
∂D(g, ω)
∂vs(x)= 4vs0(x)ρ0(x)
{G3i,i(g|x)Dj,j(x, ω)− 1
2G3n,i(g|x)
[Di,n(x, ω) +Dn,i(x, ω)
]},
(4.7)
where vs0(x) and ρ0(x) are the reference S-velocity and density models, respectively,
at location x. Di(x, ω) denotes the ith component of the particle velocity recorded at
x from a vertical-component force. The Einstein notation is assumed in equation (4.7)
where Di,j = ∂Di
∂xjfor i, j ∈ {1, 2, 3}. The 3D harmonic Green’s tensor G3j(g|x) is the
particle velocity at location g along the jth direction from a vertical-component force
at x in the reference medium.
Inserting equation (4.7) into equation (4.6) gives the final expression of the gra-
125
dient:
γ(x) =∂ε
∂vs(x)= −
∑ω
4vs0(x)ρ0(x)R
{backprojected data=Bi,i(x,ω)∗︷ ︸︸ ︷∫ ∑
θ
1
A(θ, ω)∆κ(θ, ω)D(g, θ, ω)∗obsG3i,i(g|x)dg
source=fj,j(x,ω)︷ ︸︸ ︷Dj,j(x, ω)
backprojected data=Bn,i(x,ω)∗︷ ︸︸ ︷−1
2
∫ ∑θ
1
A(θ, ω)∆κ(θ, ω)D(g, θ, ω)∗obsG3n,i(g|x)dg
source=fn,i(x,ω)︷ ︸︸ ︷[Di,n(x, ω) +Dn,i(x, ω)
]},
(4.8)
where fi,j(x, ω) for i and j ∈ {1, 2, 3} is the downgoing source field at x, and
Bi,j(x, s, ω) for i and j ∈ {1, 2, 3} is the backprojected scattered field at x. The
above equation indicates that the gradient is computed using a weighted zero-lag
correlation between the source and backward-extrapolated receiver wavefields.
The optimal S-wave velocity model vs(x) is obtained using the steepest-descent
formula (Nocedal and Wright, 2006)
vs(x)(k+1) = vs(x)(k) − αγ(x), (4.9)
where α is the step length and the superscript (k) denotes the kth iteration. We use
a preconditioned conjugate gradient method to update the S-wave velocity model.
4.2.2 Workflow of multiscale and layer-stripping WD
In our multiscale, layer-stripping WD (MSLSWD), we use the high-frequency data
with a small offset R to first update the shallow velocity model. Then we assume this
shallow velocity model is known and use the low-frequency data with a large offset
R to update the deeper regions of the velocity model. For a given frequency band,
126
according to the dispersion curves, we estimate an average wavelength
λ = 1/κ,
where κ is the average wavenumber. The penetration depth z is estimated as half of
the wavelength λ, and the maximum offset R is estimated as three or four wavelengths.
The workflow for implementing our multiscale, layer-stripping WD method is
summarized in the following six steps.
1. Determine the frequency range of observed data. Divide the frequency range
into several frequency bands for each MSLSWD step.
2. Retrieve the dispersion curves from the whole common-shot gather (CSG) and
estimate the range of the average wavenumber k for each frequency band.
3. For a given frequency band, determine the maximum offset R according to the
maximum wavelength λ calculated from the range of k values and estimate the
observed dispersion curves from seismic traces within the maximum offset R
for each CSG.
4. Calculate the gradient according to equation (4.8). Only the region within a
depth window is used to update the S-velocity model. The depth window can
be estimated from half of the wavelength range.
5. Use the updated S-velocity model as the initial model to perform WD for the
next frequency band.
6. Repeat the last three steps for all frequency bands.
4.3 Numerical Results
We first study the effect of the maximum receiver-spread length R on WD using syn-
thetic data, and then we show that the conventional WD method may get stuck to
a local minimum when the model becomes complex. Finally, we verify the effective-
127
ness of our MSLSWD method using synthetic and field data examples. We use the
fundamental dispersion curves from each CSG for inversion along the azimuth angles
of 0◦ (toward the positive x-direction) and 180◦ (toward the negative x-direction).
In the synthetic examples, we generate the observed and predicted data using an
O(2,8) time-space-domain solution to the first-order 2D elastic-wave equation with a
free-surface boundary condition (Graves, 1996). MSLSWD inverts only the S-wave
velocity model. We use the actual P-wave velocity model for modeling predicted
surface waves. In practice, there might be errors in the P-wave velocity and density
models, but such errors have a limited effect on the WD results because the Rayleigh
wave dispersion curves are not very sensitive to the P-wave velocity or density models
(Xia et al., 1999). The source wavelet is a Ricker wavelet with a center frequency
of 40 Hz, which is assumed to be known during inversion. In the field data test,
the P-wave velocity model is obtained from refraction tomography (Huang et al.,
2018). Because the WD method does not match the waveforms themselves, we use a
Ricker wavelet as the source wavelet with a peak frequency of 5 Hz. No more than
15 iterations are used for these examples.
4.3.1 Synthetic Model
We use the S-velocity model in Fig. 4.2a to verify the effectiveness of MSLSWD. We
modify the model from Perez Solano et al. (2014) (Fig. 6d) that was also used by
Masoni et al. (2016) (Fig. 4). The corresponding P-wave velocity model vp is obtained
using the S-wave velocity model vs with the relation vp = 2vs. The synthetic model
has a homogeneous density model of 1000 kg/m3. We generate a total of 40 CSGs
for vertical sources located at z = 0.2 m below the free surface with a spatial interval
of 1.5 m. Each CSG has 150 vertical-component receivers at z = 0.2 m below the
surface with a spatial interval of 0.2 m. The initial S-velocity model used is a model
with a linear gradient in depth (Fig. 4.2b).
128
Influence of the maximum receiver-spread length on WD
We first study the influence of the maximum receiver-spread length R on the penetra-
tion depth and lateral resolution in the conventional WD results. The multi-frequency
and layer-stripping strategy is not used in this test. We perform the first numerical
test by setting R to be 8 m. The resulting inverted S-velocity model is shown in
Fig. 4.2c. We can see that the two high-velocity anomalies are separated clearly.
The observed dispersion curves for all the CSGs along the azimuth angles of 0◦ and
180◦ are shown in Figs. 4.4a and 4.4b respectively, where the black dashed lines, the
cyan dash-dot lines and the red lines represent the contours of the observed, initial
and inverted dispersion curves, respectively. The contours of the inverted dispersion
curves correlate well with those of the observed data.
We carry out the second numerical test by increasing the offset R to 20 m. The
resulting inverted S-velocity model is displayed in Fig. 4.2d. Fig. 4.3 shows the nor-
malized misfit values versus the iteration number. After 12 iterations, the normalized
WD residual (red line) is approximately 0.06. The contours of the observed, initial
and inverted dispersion curves are shown in Figs. 4.4c and 4.4d. The contours of the
inverted dispersion curves correlate well with the observed ones. However, the two
high-velocity anomalies cannot be separated in the inverted model (Fig. 4.2d), indi-
cating that the inverted model with R = 20 m has lower lateral resolution compared
with that obtained using R = 8 m. Figs. 4.5a and 4.5b show the vertical-velocity
profiles at X = 20 m and X = 38 m respectively for the true model (blue line),
the initial model (black dash-dot line) and the inverted S-velocity models by setting
R=8 m (magenta line) and R = 20 m (red line). It can be seen that the inverted
velocity tomogram with the maximum offset R = 20 m is more accurately recon-
structed in the deeper region from z = 2.4 m to z = 6 m than that when R = 8 m.
This suggests that more accurate dispersion curves for the low-frequency part can be
retrieved by longer offsets.
129
d) Inverted S-velocity Tomogram (R=20 m)210
210230
230
230
250
250 250
270 270 270
270
290 290 290
310 310 310
330 330 330
0 10 20 30 40 50
X (m)
0
4
8
Z (
m)
c) Inverted S-velocity Tomogram (R=8 m)210
210230 230
230
250 250 250
250
250
250
270 270 270
270
290 290 290
310 310 310
330 330 330
0 10 20 30 40 50
X (m)
0
4
8
Z (
m)
b) Initial S-velocity Model
210 210 210
230 230 230
250 250 250
270 270 270
290 290 290
310 310 310
330 330 330
0 10 20 30 40 50
0
4
8
Z (
m)
a) True Velocity Model
230
230 230
250 250
250
250
270 270 270
270
270
270
270
270
290 290 290
290 290
290
310 310 310
330 330 330
0 10 20 30 40 50
0
4
8
Z (
m)
200
220
240
260
280
300
320
340
S V
elo
city (
m/s
)
Figure 4.2: True (a) and initial (b) S-velocity models together with the S-velocitytomograms obtained using WD with maximum offsets of (c) R = 8 m and (d) R =20 m.
0 2 4 6 8 10
Iteration No.
0
0.2
0.4
0.6
0.8
1
Norm
alized M
isfit
Data Misfit vs Iteration No. (R=20 m)
Test 1
Test 2
Figure 4.3: Plot of residual vs iteration number for the synthetic examples. TheY-axis represents the normalized wavenumber residual, and the blue and red linesrepresent the WD results with R = 20 m for the data collected from the model inFigs. 4.2a and 4.6a, respectively.
130
71 71 71
141141
141
211211
211
281281
281
351351
351
71 71 71
141 141 141
211 211 211
281 281 281
351 351 351
421 421 421
71 71 71
141 141 141
211211
211
281281
281
351
351
351
1 11 21 31
10
26
42
58
74
Fre
qu
en
cy (
Hz)
71 71 71
141141
141
211211
211
281281
281
351351
351
71 71 71
141 141 141
211 211 211
281 281 281
351 351 351
421 421 421
71 71 71
141 141 141
211211
211
281281
281
351
351
351
8 18 28 38
10
26
42
58
74
Fre
qu
en
cy (
Hz)
Observed
Initial
Predicted
71 71 71
141 141141
211211
211
281
281
281
351
351
71 71 71
141 141 141
211 211 211
281 281 281
351 351 351
71 71 71
141141
141
211211
211
281
281
281
351
351
1 11 21 31
Shot Number
10
26
42
58
74
Fre
qu
en
cy (
Hz)
Dispersion Contour Comparison
71 71 71
141141 141
211211 211
281281
281
351351
71 71 71
141 141 141
211 211 211
281 281 281
351 351 351
71 71 71
141141 141
211
211 211
281
281 281
351
8 18 28 38
Shot Number
10
26
42
58
74
Fre
qu
en
cy (
Hz)
50
100
150
200
250
300
350
400
Pic
ked W
avenum
ber
(1/k
m)
Figure 4.4: Observed dispersion contours for (a) azimuth angle θ = 0◦ with themaximum offset R = 8 m, (b) θ = 180◦ with R = 8 m, (c) θ = 0◦ with R = 20 m, and(d) θ = 180◦ with R = 20 m, where the black dashed lines, the cyan dash-dot lines andthe red lines represent the contours of the observed, initial and inverted dispersioncurves, respectively. Here, the background images are the picked wavenumber for allthe common shot gathers. The shot number is determined to make sure that themaximum offset is at least 8 m in (a) and (b). For comparison, we also use the sameshot number range in (c) and (d), but the maximum offset of some of the shots maybe less than 20 m. For example, in (c), only shot no. 1-28 has the maximum offsetof 20 m for azimuth 0.
131
190 220 250 280 310 340
S Velocity (m/s)
0
4
8
Z (
m)
a) X=20 m
True
Initial
Inverted R=8 m
Inverted R=20 m
190 220 250 280 310 340
S Velocity (m/s)
0
4
8
Z (
m)
b) X=38 m
True
Initial
Inverted R=8 m
Inverted R=20 m
Figure 4.5: Vertical-velocity profiles at (a) X = 20 m and (b) X = 38 m for the truemodel (blue line), the initial model (black dash-dot line) and the inverted S-velocitytomograms when R=8 m (magenta line) and R=20 m (red line) shown in Fig. 4.2.
132
Local minimum of conventional WD
With the same acquisition parameters as above, we conduct two additional numerical
tests on the modified S-velocity model shown in Fig. 4.6a. We add a low-velocity zone
in the shallow region and move the location of the high-velocity anomalies to a deeper
depth. The initial model used in inversion is a linear velocity gradient displayed in
Fig. 4.6b. We first apply single-scale WD to the data without layer-stripping strategy.
Fig. 4.7 shows the contours of the observed, initial and predicted dispersion curves
along the azimuth angles of θ = 0◦ and 180◦ for these two numerical tests. There is
a poor match between the contours of the inverted and observed dispersion curves,
which indicates that the WD is stuck to a local minimum for the modified model.
The inverted S-velocity tomograms with a maximum offset of R = 8 m and R=20
m are shown in Figs. 4.6c and 4.6d, respectively. The high-velocity anomalies are
not detected in the inverted tomograms of these two numerical tests. Figs. 4.8a and
4.8b shows the vertical-velocity profiles at X = 20 m and X = 38 m, respectively, for
the true (blue line), initial (black dash-dot line) and inverted S-velocity models when
using R=8 m (magenta line) and R=20 m (red line). The vertical-velocity profiles
show that WD incorrectly updates the low-velocity zones in the shallow region (z < 1
m). Fig. 4.3 shows the misfit values versus the iteration number. After 10 iterations,
the last normalized WD residual (red line) is approximately 0.43. This suggests that
WD converges to a local minimum because of the shallow low-velocity zone.
Multiscale, layer-stripping WD
We apply our multiscale, layer-stripping WD method to the same data as above to
alleviate the local minimum problem. The frequency spectrum of the data is shown
in Fig. 4.9, where eleven frequency bands are chosen and each of them is plotted as
the horizontal bar with a number tag close to it. We select small frequency windows
for low-frequency bands, because we need to make sure that there is not a great jump
133
d) Inverted S-velocity Tomogram (R=20 m)
184184 184
204
204 204
224 224 224
244 244 244
264 264 264
284 284 284
304 304 304
324 324 324
0 10 20 30 40 50
X (m)
0
4
8
Z (
m)
c) Inverted S-velocity Tomogram (R=8 m)
184 184 184
204204 204
224 224 224
244 244 244
264 264 264
284 284 284
304 304 304
324 324 324
0 10 20 30 40 50
X (m)
0
4
8
Z (
m)
b) Initial S-velocity Model164 164 164
184 184 184
204 204 204
224 224 224
244 244 244
264 264 264
284 284 284
304 304 304
324 324 324
0 10 20 30 40 50
0
4
8
Z (
m)
a) True Velocity Model
184 184
204204 204
224
224 224
224244 244
244
244
264 264 264
264
264
264264
264
284 284 284
284
284
284284
304 304 304324 324 324
0 10 20 30 40 50
0
4
8
Z (
m)
180
200
220
240
260
280
300
320
340
S V
elo
city (
m/s
)
Figure 4.6: True (a) and initial (b) S-velocity models together with the S-velocitytomograms obtained using WD with maximum offsets of (c) R = 8 m and (d) R =20 m. The high-velocity anomalies in (a) are 2 m deeper than the one shown inFig. 4.2a.
91 9191
181 181 181
271 271 271
361 361 361
451 451 451
91 91 91
181 181 181
271 271 271
361 361 361
451 451 451
91 91 91
181 181 181
271 271 271
361 361 361
451 451 451
1 11 21 31
10
26
42
58
74
Fre
qu
en
cy (
Hz)
91 91 91
181 181181
271 271 271
361 361 361
451 451 451
91 91 91
181 181 181
271 271 271
361 361 361
451 451 451
91 91 91
181 181 181
271 271 271
361 361 361
451 451 451
8 18 28 38
10
26
42
58
74
Fre
qu
en
cy (
Hz)
Observed
Initial
Predicted
91 9191
181 181181
271 271 271
361 361361
451 451
91 91 91
181 181 181
271 271 271
361 361 361
451 451 451
91 91 91
181 181181
271 271271
361 361361
451 451451
1 11 21 31
Shot Number
10
26
42
58
74
Fre
qu
en
cy (
Hz)
Dispersion Curve Comparison
91 91 91
181 181 181
271 271 271
361361 361
451451
91 91 91
181 181 181
271 271 271
361 361 361
451 451 451
91 91 91
181 181 181
271 271 271
361361 361
451451 451
8 18 28 38
Shot Number
10
26
42
58
74
Fre
qu
en
cy (
Hz)
50
100
150
200
250
300
350
400
450
Pic
ked W
avenum
ber
(1/k
m)
Figure 4.7: Observed dispersion curves for (a) azimuth angle θ = 0◦ with the maxi-mum offset R = 8 m, (b) θ = 180◦ with R = 8 m, (c) θ = 0◦ with R = 20 m, and (d)θ = 180◦ with R = 20 m. The black dashed, cyan dash-dot and red lines representthe contours of the observed, initial and inverted dispersion curves, respectively.
134
164 194 224 254 284 314
S Velocity (m/s)
0
4
8
Z (
m)
a) X=20 m
True
Initial
Inverted R=8 m
Inverted R=20 m
164 194 224 254 284 314
S Velocity (m/s)
0
4
8
Z (
m)
b) X=38 m
True
Initial
Inverted R=8 m
Inverted R=20 m
Figure 4.8: Vertical-velocity profiles at (a) X = 20 m and (b) X = 38 m for the truemodel (blue line), the initial model (black dash-dot line) and the S-velocity tomogramsby setting R=8 m (magenta line) and R=20 m (red line) shown in Fig. 4.6.
135
in the horizontal resolution for two close frequency bands, where the horizontal reso-
lution is related to the maximum offset R that is related to the maximum wavelength
of the frequency band. The ranges of the eleven frequency bands and the wavelength
range corresponding to each frequency band used in MSLSWD are listed in Table 4.1.
The maximum offset R is calculated using R ≈ 3.5 ∗ λmax (Liu et al., 2019) where
λmax is the maximum wavelength. The misfit-change column shows the normalized
misfit change after 10 iterations. The updated depth window for each frequency band
is determined using half of the wavelength. A taper is used at the top and bottom
boundaries of a depth window as shown in Fig. 4.10. The updated velocity models
for all eleven steps are shown in Figs. 4.11, where the black dashed lines indicate
the location of the high-velocity anomalies. We can see that the deeper region of
the model is gradually updated step by step. The contours of the observed, initial
and inverted dispersion curves for each step are shown in Figs. 4.12. The contours
of the inverted dispersion curves correlate well with the observed ones for each step.
The vertical-velocity profiles at X = 20 m and X = 38 m extracted from the inverted
tomogram with MSLSWD (red lines) are shown in Fig. 4.13. They show better agree-
ment with the true ones (blue lines) than those extracted from the tomogram without
layer stripping (magenta lines). The results demonstrate that MSLSWD can mitigate
the local minimum problem of WD for this model caused by the low-velocity layer in
the shallow region.
4.3.2 Surface Seismic Data from the Blue Mountain Geother-
mal Field
Seven 2D lines of surface seismic data were acquired at the Blue Mountain geothermal
field in Nevada, USA, using dynamite sources. We use one 2D line of data for our
study. The line consists of 121 receivers with an interval of 33.5 m, and 57 dynamite
sources with an interval of 67 m. One of the CSGs is shown in Fig. 4.14a, which
136
Table 4.1: Eleven frequency bands used for MSLSWD, where the wavelength λ isestimated from the dispersion curves; the maximum offset are determined by R =3.5λmax; the depth range is calculated using half of the wavelength range with a taperof 0.2 m at both ends.
No. Freq. Band (Hz) λ Range (m) Max. Offset (m) Depth Range (m)
1 90-110 1.4-1.9 7 0-1.02 70-90 1.9-2.4 9 1-1.43 60-70 2.4-2.9 11 1.4-1.64 50-60 2.9-3.6 13 1.6-2.05 40-50 3.6-4.7 17 2.0-2.66 35-40 4.7-5.45 21 2.6-3.07 30-35 5.45-6.6 24 3.0-3.68 25-30 6.6-8.26 30 3.6-4.29 20-25 8.26-10.87 37 4.2-5.210 15-20 10.87-15.74 46 5.2-7.411 10-15 15.74-33 60 7.4-12
0 20 40 60 80 100 120 140
Frequency (Hz)
0
0.2
0.4
0.6
0.8
1
Am
plit
ud
e
1
2
34
5
6
7
8
910
11
Figure 4.9: Frequency spectrum of the observed data, which are divided into elevenfrequency bands. The frequency bands are plotted as horizontal bars with theircorresponding number tags.
137
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
Freq. Band 9
Freq. Band 10
Figure 4.10: Depth windows for frequency bands 9 (blue solid line) and 10 (red dashedline).
Figure 4.11: (a) Initial S-velocity model. (b)-(g) S-velocity tomograms for Steps 1 to11 with an interval of 2 (Table 4.1). (h) True S-velocity model.
138
526
561
561
596
596
596
631
631
666 666
596 596
631 631
666 666
526
526
561
561
596
596
596
631
631
666 666
1 11 21 31
90
94
98
102
106
110Fre
quency (
Hz)
500
550
600
650
Wa
ve
nu
mb
er
(1/k
m) 361
361
381
381
401 401
361
361
381
381
401
401
421
361
361
381
381
401 401
1 11 21 31
60
64
68340
360
380
400
420
Wa
ve
nu
mb
er
(1/k
m)
221 221
241 241
261 261
221
241 241
261 261
221 221
241241
261 261
1 11 21 31
40
44
48
Fre
quency (
Hz)
220
240
260
Wa
ve
nu
mb
er
(1/k
m)
151151
161
161
171
171
181
151
161
161
171
171
181
151
161
161
171
171
181
1 11 21 31
30
34150
160
170
180
190
Wa
ve
nu
mb
er
(1/k
m)
91
101
101
111
111
121
91
101
101
111
111
121
91
101
101
111
111
121
1 11 21 31
Shot Number
20
24
Fre
quency (
Hz)
90
100
110
120
130
Wa
ve
nu
mb
er
(1/k
m)
31 31
41
41
41
51
51
61
61
31
31
41 41
51
51
61
61
3131
41 41
51
51
61
61
1 11 21 31
Shot Number
10
14 30
40
50
60
70
Wa
ve
nu
mb
er
(1/k
m)
Observed
Initial
Predicted
Figure 4.12: Observed dispersion curves (azimuth angle θ = 0◦) for Steps 1 to 11 withan interval of 2 listed in Table 4.1, where the black dashed, cyan dash-dot and redlines represent the contours of the observed, initial and inverted dispersion curves,respectively.
139
164 194 224 254 284 314
S Velocity (m/s)
0
4
8
Z (
m)
a) X=20 m
True
Initial
No Layer Stripping
Layer Stripping
164 194 224 254 284 314
S Velocity (m/s)
0
4
8
Z (
m)
b) X=38 m
True
Initial
No Layer Stripping
Layer Stripping
Figure 4.13: Vertical-velocity profiles at (a) X = 20 m and (b) X = 38 m for the truemodel (blue lines), the initial model (black lines), the inverted tomograms with (redlines) and without (magenta lines) layer stripping.
140
clearly shows three modes of surface waves. These three modes are also shown in the
dispersion images (Fig. 4.14b) calculated using the frequency-sweeping method (Park
et al., 1998) with the maximum offset R =500 m. We pick only the dispersion curves
of the fundamental-mode surface waves (Fig. 4.14c). The observed dispersion curves
for all CSGs are shown in Fig. 4.15, where the black dashed lines indicate the contours
of the observed dispersion curves. The initial S-velocity model is shown in Fig. 4.16a,
and the S-velocity tomogram obtained using the conventional WD is displayed in
Fig. 4.16b. The contours of the predicted dispersion curves from the conventional
WD tomogram are plotted in Fig. 4.15 using the cyan solid lines. It shows that only
the dispersion contours from the high-frequency components and the CSGs NO. 1-30
match well, which indicates that WD is stuck to a local minimum. To alleviate the
local minima problem, we apply our multiscale and layer-stripping WD method to
the data.
We use three frequency bands for MSLSWD: (a) 7-10 Hz, (b) 5-8 Hz and (c) 2-
6 Hz. The corresponding depth windows are 0-45 m, 45-100 m and 100-250 m. The
comparison of the S-velocity tomograms without and with using the layer stripping
approach are shown in Figs. 4.16b and 4.16c. It can been seen that the deeper regions
are significantly updated using layer-stripping WD.
The predicted dispersion contours obtained with the layer stripping approach are
displayed as the red lines in Fig. 4.15. It can be seen that the dispersion contours
calculated using layer stripping WD for the low-frequency components and the CSGs
No.31-56 correlate better with the observed ones compared to those calculated using
the conventional WD. The S-velocity tomogram is also consistent with the P-wave
tomogram shown in Fig. 4.16d, which is obtained using refraction tomography (Huang
et al., 2018). From the acquisition map shown in Pan and Huang (2019), we can see
that the right-hand side of the line is close to the Blue Mountain where there is a
higher S-velocity value which is consistent with our results.
141
To further test the accuracy of the layer-stripping WD method. Fig.4.17 shows
the comparison between the observed (red) and synthetic (blue) traces from the S-
velocity tomogram without and with layer-stripping methods for CSG No. 30. For
each panel in Fig.4.17, we calculate a match filter using the black trace and then apply
the filter to the other traces to reshape the synthetic waveforms. It is evident that
the predicted surface waves from the S-velocity tomogram with the layer-stripping
method more closely match the observed ones than those without layer-stripping
method. Fig.4.18 show the common offset gathers (COGs) with offsets of 335 m from
the S-velocity tomogram (a) without and (b) with layer-stripping methods. The blue
and red wiggles represent the observed and predicted COGs, respectively. We also
calculate the match filters from the black traces and then apply them to the other
traces. The synthetic COGs computed from the S-velocity tomogram inverted using
the MSLSWD more closely agree with the observed ones compared with to those
inverted with WD without layer stripping.
4.4 Discussion
The lateral resolution of the WD tomogram is related to the length of the receiver
spread. Different receiver-spread lengths lead to different lateral-resolution limits
of the retrieved dispersion curves (Mi et al., 2017; Bergamo et al., 2012). A wide
receiver-spread for a specific azimuth angle can lead to poor lateral resolution along
the azimuth angle of the gradient, but can provide a deep penetration depth (Foti
et al., 2014). Our results of synthetic surface seismic data demonstrate that layer-
stripping wave-equation dispersion inversion can provide better depth penetration
and higher lateral resolution than the conventional wave-equation dispersion inversion
without layer stripping.
In our field data example, the interval of the geophones is so large that only
few geophones are involved when inverting the high-frequency-band data if we set
142
(a) First CSG
0 500 1000 1500 2000 2500 3000 3500 40006
4
2
0
Tim
e (
s)
X (m)
12
3
(b) Dispersion Images
Frequency (Hz)
Ve
locity (
km
/s)
2 4 6 8 10 12 14 16 18
1.4
1.2
1
0.8
0.6
0.4
0.2
12
3
(c) Picked Dispersion Curve
2 3 4 5 6 7 8 9 10
10
20
30
Frequency (Hz)
Pic
ke
d W
ave
nu
mb
er
(1/k
m)
Figure 4.14: (a) First CSG, (b) its dispersion image with the maximum offsetR=500 m, and (c) the picked dispersion curve for the fundamental-mode surfacewaves.
(a) Dispersion Curves for Azimuth 0◦
6
6
6
6
6
6
1212
12
12
12
12
18
18
18
18
18
18
24
24
24
30 30
30
36
36
66
6
6 6
6
12
12
12
12
12
12
1818
18
18
18
24
24
24
30 30
30
36
36
66
6
66
6
12
12
12
12
12
1818
18
18
18
18
2424
24
24
3030
30
36
36
36
1 14 27 40
Shot Number
2
6
10
0
12
24
36
Obser.
LS
No LS
(b) Dispersion Curves for Azimuth180◦
6
6
6
6
6
12
12
12
12
12
18
18
18
18
18
24
24
24
30 30
30
36
36
6 6
6
6 6
6
12
12
12
12
18 18
18
18
24
24
24
30 30
36
36
6
6
6 6
6
12 12
12
12
12
18
18
18
18
18
24
24
24
3030
30
36
36
10 23 36 49
Shot Number
2
6
10
0
12
24
36
Obser.
LS
No LS
Figure 4.15: Observed dispersion curves for (a) θ = 0◦ and (b) θ = 180◦, where theblack dashed lines, the cyan lines and the red dash-dot lines represent the contoursof the observed dispersion curves, the predicted dispersion curves obtained withoutand with layer stripping, respectively.
143
(a) Initial S-velocity Model
0 0.8 1.6 2.4 3.2
X (km)
0
0.1
0.2
Z (
km
)
0.28
1.5
(b) S-velocity Tomogram by Conventional WD
0 0.8 1.6 2.4 3.2
X (km)
0
0.1
0.2
Z (
km
)
0.28
1.5
(c) S-velocity Tomogram by MSLSWD
0 0.8 1.6 2.4 3.2
X (km)
0
0.1
0.2
Z (
km
)
0.28
1.5
(d) P-velocity Tomogram
0 0.8 1.6 2.4 3.2
X (km)
0
0.1
0.2
Z (
km
)
1
3.5
Figure 4.16: (a) initial S-velocity Model; the S-velocity tomograms inverted using theWD methods (b) without and (c) with multiscale and layer-stripping strategy; (d)the P-velocity tomogram calculated by traveltime tomography (Huang et al., 2018).
144
(a) CSG for shot NO. 30 (b) CSG for shot NO. 30
Figure 4.17: Comparison between the observed (red) and synthetic (blue) traces fromthe S-velocity tomogram (a) without and (b) with layer-stripping methods for CSGNo. 30. For each panel, a match filter is calculated from the black trace and thenapplied to the other traces.
the offset according to the maximum wavelength. It is hard to pick the dispersion
curves because of low signal-to-noise ratio when using only a few geophones. Thus,
we use the same receiver-spread length (500 m) for all frequency bands, which is
approximately three times the length of the wavelength of the lowest frequency data
(3 Hz). Here, the data quality below 3 Hz is high only for the first 20 shots and low
for the remaining shots, as shown in Fig. 4.15a. Although there are clear signals at
far offsets (up to 1000 m), we do not set R to 1000 m because it may decrease the
horizontal resolution for the inverted tomogram.
To assess the inverted results, the predicted and observed dispersion curves, com-
mon shot gathers, and common offset gathers are compared to one another to deter-
mine the degree of error in our solution. The vertical-velocity profiles of the inverted
models are also compared to those of the true models in the synthetic test. All the
comparisons show that the MSLSWD results can give a better match compared to
the conventional WD results, which indicates that the improved results are closer to
145
(a) COG with Offset of 335 m
(b) COG with Offset of 335 m
Figure 4.18: Comparison between the observed (blue) and synthetic (red) common-offset gathers (COGs) with the offset of 335 m from the S-velocity tomogram without(a) and with (b) layer-stripping method. For each panel, a match filter is calculatedfrom the black trace and then applied to the other traces.
146
the local minimum. There are other assessment methods, such as the checkerboard
test (Liu et al., 2019) and the covariance matrix method (Zhu et al., 2016).
We assume that the effects of attenuation and topography on dispersion curves
are insignificant in our field data test. However, if the attenuation and topography is
important, the effects can be accounted for by solving the visco-elastic wave equation
with an irregular free surface to compute the theoretical dispersion curves and perform
the inversion (Li et al., 2017a,b,e, 2019b).
We use only the fundamental-mode Rayleigh waves for MSLSWD inversion. Nev-
ertheless, the higher-mode Rayleigh wave data with the same wavelength can have
deeper penetration depth, and higher-mode data can increase the resolution of the
S-velocity tomogram (Xia et al., 2003). Rather than inverting only the fundamental-
mode surface waves, our multiscale and layer-stripping WD method can be extended
to invert both fundamental- and higher-mode surface waves.
The multiscale and layer-stripping strategy can be easily extended to the 3D case.
One challenge for the layer stripping WD is to determine the accurate relationship
between the frequency bands and the depth windows. When there are strong lateral
gradients in the S-velocity model, the penetration depth of different shot gathers
can have a dramatic lateral variation for the same frequency bands. In this case,
it is inappropriate to use the same depth windows for all the shot gathers. The
depth windows can be designed according to the sensitivity kernels, because they can
provide a good estimation of the penetration depth (Masoni et al., 2016).
4.5 Conclusions
We have developed a new multiscale and layer-stripping wave-equation dispersion
inversion method for Rayleigh waves. In this method, the high-frequency and near-
offset data are first used to invert for the shallow S-velocity model, and the lower-
frequency data with longer offsets are gradually incorporated to invert for the deeper
147
regions of the model. Numerical results of both synthetic and field seismic data
demonstrate that the wave-equation dispersion inversion can suffer from the local
minima problem when inverting data from a complex earth model, and our multiscale
and layer-stripping wave-equation dispersion inversion method can mitigate the local
minima problem and enhance convergence to the global minimum.
Acknowledgments
This work was supported by U.S. Department of Energy through contract DE-AC52-
06NA25396 to Los Alamos National Laboratory (LANL). We thank AltaRock Energy,
Inc. and Dr. Trenton Cladouhos for providing surface seismic data from the Blue
Mountain geothermal field. Zhaolun Liu thank King Abdullah University of Sci-
ence and Technology (KAUST) for funding his graduate studies. The computation
was performed using super-computers of LANL’s Institutional Computing Program.
Additional computational resources were made available through the KAUST Super-
computing Laboratory (KSL).
148
Chapter 5
Imaging Near-surface Heterogeneities by Natural Migration
of Surface Waves: Field Data Test1
We have developed a methodology for detecting the presence of near-surface het-
erogeneities by naturally migrating backscattered surface waves in controlled-source
data. The near-surface heterogeneities must be located within a depth of approxi-
mately one-third the dominant wavelength λ of the strong surface-wave arrivals. This
natural migration (NM) method does not require knowledge of the near-surface phase-
velocity distribution because it uses the recorded data to approximate the Green’s
functions for migration. Prior to migration, the backscattered data are separated
from the original records, and the band-passed filtered data are migrated to give an
estimate of the migration image at the depth of approximately one-third λ. Each
band-passed data set gives a migration image at a different depth. Results with syn-
thetic data and field data recorded over known faults validate the effectiveness of this
method. Migrating the surface waves in recorded 2D and 3D data sets accurately
reveals the locations of known faults. The limitation of this method is that it requires
a dense array of receivers with a geophone interval less than approximately one-half
λ.
1This manuscript was published as:Zhaolun Liu, Abdullah AlTheyab, Sherif M. Hanafy, andGerard Schuster, (2017), ”Imaging near-surface heterogeneities by natural migration of backscatteredsurface waves: Field data test,” Geophysics 82(3): S197-S205, doi: https://doi.org/10.1190/geo2016-0253.1
149
5.1 Introduction
The scattered surface wave generated by strong heterogeneities in the shallow sub-
surface is often seen as noise in seismic reflection records (Blonk et al., 1995; Ernst
et al., 2002); however, this noise can also be used as signal if the back-scattered data
are migrated to image the near-surface heterogeneities (Snieder, 1986a; Riyanti, 2005;
Yu et al., 2014; Hyslop and Stewart, 2015; Almuhaidib and Toksoz, 2015).
The conventional surface-wave imaging methods are based on the Born approx-
imation of surface waves, which requires an estimation of the background velocity
model and the weak-scattering approximation. Under the Born approximation, the
backscattered surface wave data d is denoted as d = Lm, where L is the forward
modeling operator for a known background velocity and m is the model perturbation
(Snieder, 1986a; Tanimoto, 1990). To invert for the model perturbation m, Riyanti
(2005) used an iterative optimization method to calculate the solution. In contrast,
Snieder (1986a) and Yu et al. (2014) applied the adjoint of the forward modeling
operator L† to the scattered data to obtain the migration image.
Apart from the methods based on the Born approximation, Hyslop and Stewart
(2015) estimate the surface-wave reflection coefficients at near-surface lateral discon-
tinuities by a processing flow based on a 2D semi-analytic forward modeling method
for surface-wave propagation. They then map the frequency-dependent reflection
coefficients to depth in order to produce a 2D reflectivity map of discontinuities.
Recently, AlTheyab et al. (2015, 2016) introduced the natural migration (NM)
method to image the near-surface heterogeneities, assuming that the scattering bodies
are within a depth of about 1/3 wavelength from the free surface. It also requires a
dense distribution of sources and receivers to avoid aliasing artifacts in the migration
image. There are several benefits of the NM method. First, no Born approximation
is used so that strongly scattered events can be migrated to the surface-projection of
their origin. Second, no velocity model is needed because the Green’s functions in
150
the migration kernels are recorded as band-limited shot gathers, where the sources
and receivers are located on the surface.
AlTheyab et al. (2016) demonstrated the effectiveness of the NM method with
ambient noise data, but did not show it to be effective for controlled source data.
This paper now presents a general procedure for the NM method applied to controlled
source data, and shows the results of applying NM to surface-wave data. Results
show that NM of back-scattered surface waves can detect near-surface heterogeneities,
which can indicate the existence of faults or low velocity zones (LVZ).
5.2 Theory of natural migration
Assuming that the vertical component of the scattered Rayleigh wave u(xs,xr) due
to an impulsive point source in the vertical direction at xs is recorded by the receiver
at xr, the natural migration equation in the frequency domain can be expressed as
(AlTheyab et al., 2016)
m(x) =∑s,r∈B
∫2ω2W (ω)∗G(x|xs)∗G(x|xr)∗u(xs,xr)dω, (5.1)
where m(x) is the perturbation model that represents an arbitrary distribution of
elastic-parameter perturbations at the image point x and ∗ denotes complex conju-
gation. ω is the angular frequency, W (ω) represents the source-wavelet spectrum and
is assumed to be W (ω) = A(ω)e−iwt0 , which is a zero-phase wavelet with the time
delay of t0 and A(ω) is the amplitude spectrum. B is a set of source and receiver
positions at the surface (just below the free surface). x, xs and xr are, respectively,
the migration image, source and receiver positions in the set B. Note that the pos-
sible positions of the trial image point x can only be where the sources or receivers
are located near the surface. The function G(x|xs) is the Green’s function for the
vertical-component harmonic point source at xs and receiver at x, and G(x|xr) is
151
the Green’s function for a vertical-component particle-velocity recording that only
contains the transmitted wavefield without backscattering.
The wavefield u(x|xs) is equal to W (ω)G(x|xs) so that the Green’s function can
be expressed as
G(x|xs) = u(x|xs)W (ω)−1. (5.2)
Substituting equation 5.2 into equation 5.1 gives the natural migration equation for
active-source data
m(x) =∑s,r
∫2ω2
[W (ω)−1u(x|xs)u(x|xr)
]∗u(xs,xr)dω,
=∑s,r
L(xr|x|xs)∗u(xs,xr), (5.3)
where L(xr|x|xs) =∫dω2ω2W (ω)−1u(x|xs)u(x|xr) is the forward modeling operator.
To calculate L(xr|x|xs), the deconvolution filter W (ω)−1 must be estimated. Ignoring
the amplitude term A(ω) of W (ω), we only estimate the time delay t0 from the near-
offset transmitted surface-wave arrivals. The deconvolution filter W (ω)−1 is then
approximated as eiwt0 .
The migration image m(x) for x ∈ B in equation 5.3 can be seen as the projection
of the scatterer at shallow depths onto the surface denoted by the set of points B
(Campman et al., 2005). Moreover, migration images at B can be mapped to different
depths in the medium based on the principle that surface waves at lower frequencies
are more sensitive to the presence of deeper scatterers. So, u in equation 5.3 should
be filtered by a narrow-band filter prior to migration.
5.3 Workflow of natural migration for controlled source data
The workflow for migrating the back-scattered surface waves with equation 5.3 is
shown in Figure 5.1, which is summarized as the next 5 steps. Additional details are
152
given in AlTheyab et al. (2016).
• Find the usable frequency range of the surface waves in the data.
• Determine the center frequencies of overlapping narrow-band filters for data
filtering. The minimum center frequency is selected that provides an accept-
able signal-to-noise ratio in the data. The maximum center frequency has to
be smaller than vmin/(2∆x) to avoid horizontal spatial aliasing of the migra-
tion image, where vmin is the minimum phase velocity, and ∆x is the spatial
spacing of the traces. However, as shown in the following synthetic results,
the NM method can generate usable migration images of near-surface lateral
heterogeneities with aliased data.
• Extract the time delay t0 of the source wavelet W (ω) in equation 5.3 from the
near-offset transmitted surface-wave arrivals. The deconvolution filter W (ω)−1
in equation 5.3 is then approximated as eiwt0 .
• Separate the scattered surface waves from other arrivals, especially the trans-
mitted surface waves. In our examples, the seismic arrivals that arrive earlier
than the transmitted surface waves are muted. An alternative is to use FK
filtering to estimate the backscattered surface waves. The muting window is
computed from the estimated phase velocity of the recorded surface waves. The
near-source wavefields are also muted to avoid the near-field strong artifacts.
• Migrate the processed backscattered data to compute the migration image on
the surface for different frequencies.
5.4 Numerical Results
Results are now shown for natural migration of surface waves for both synthetic data
and field data. The field data are recorded for land surveys near the Gulf of Aqaba
153
Figure 5.1: Natural migration workflow for active-source data.
154
and the Qademah fault system in Saudi Arabia.
Figure 5.2: 3D S-wave velocity model used for the synthetic tests with a 30-by-15source and receiver array on the surface.
5.4.1 Natural Migration of Synthetic Data
Synthetic shot gathers are computed by finite-difference solutions to the 3D elastic
wave equation (Virieux, 1986) with a free-surface boundary condition (Gottschammer
and Olsen, 2001). The source is a Ricker wavelet with a peak frequency of 20 Hz and
a time delay of 0.05 s. The S-wave velocity model for modeling the data is shown
in Figure 5.2, which has a buried fault at the depth of 6 m and a LVZ between 129
m and 174 m. The P-wave velocity is calculated by Vp =√
3Vs and the density is
constant with the value of 2.0 kg/m3. The grid spacing of the model is 3 m in each
direction. An areal acquisition array is distributed just below the free surface, and
the source intervals are 10 m and 20 m along the x and y directions, respectively. The
receivers are at the same position as the sources, and the output data are vertical
particle-velocity displacements. One of the common shot gathers (CSG) is shown in
Figure 5.3a.
155
Figure 5.3: a) Common shot gather generated from the 3D model. The moveoutvelocity of the red dashed lines for the separation of transmitted and backscatteredsurface waves is about 500 m/s. The near-source arrivals are muted along the yellowlines (about 0.1 s). b) Transmitted surface waves. c) Backscattered surface waves.
156
Seven narrow-band filters with the peak frequencies ranging from 15 Hz to 45 Hz
are designed to image the subsurface heterogeneities at different depths. Because the
receiver spacing is 10 m and the minimum phase velocity is approximately 700 m/s
(estimated from the dispersion curve), then 35 Hz is the maximum frequency that
avoids spatial aliasing. The band-pass filters with center frequencies above 35 Hz are
used to assess the aliasing issues. The transmitted surface waves shown in Figure 5.3b
are separated by the arrivals between the traveltimes indicated by the yellow and red
dashed lines in Figure 5.3a. The backscattered surface waves shown in Figure 5.3c
are separated by masking the arrivals earlier than the traveltimes of transmitted
surface-wave arrivals, which are indicated by the red dashed lines in Figure 5.3a.
Each band-passed data set is used according to equation 5.3 and the migration
images are shown in Figure 5.4a, where the two red dashed lines are at x = 129 m and
174 m, respectively. Figure 5.4b shows the upper portion of the Vs-velocity model.
The comparison between the natural migration results by different filters shows that
the migration images of the LVZ become more explicit as the peak frequency de-
creases. This is because the migration image from higher-frequency data delineates
the shallow part of the LVZ, while the deep part of the LVZ is imaged from the
lower-frequency data.
Surface waves are typically most sensitive to the velocity model to a depth of about
1/3 (some references choose 1/2) of their wavelength (Hyslop and Stewart, 2015;
Stokoe and Nazarian, 1985). Therefore, the depth range of each migration image can
be estimated roughly by this relationship. We assign each peak frequency f0 of the
filtered data to the depth of about 1/3 the corresponding wavelength. We should note
that this relationship is a rough approximation; however, similar mappings of direct
surface-wave spectra to depth have been proved useful for near-surface interpretation
(Shtivelman, 2000). The wavelength λ for each frequency can be approximated by λ =
c/f , where c is the average phase velocity that can be obtained from the dispersion
157
Pseudo d
epth
(m
)
5.2
6.0
7.1
8.7
11.0
14.9
19.5
Figure 5.4: a) Migration images at z = 0 m computed from the synthetic data with thenarrow-band filters from 1 to 7 (center frequencies change from 45 Hz to 15 Hz witha 5 Hz interval). The two red dashed lines are at x = 129 m and 174 m, respectively,and the z axis denotes pseudodepth calculated from the mapping of frequency to thedepth of 1/3 wavelength. b) Upper portion of the Vs-velocity model and the reddashed lines are taken from a).
158
curves at selected source positions. An alternative procedure for relating the surface-
wave frequency to depth is by analyzing the sensitivity of the surface-wave phase
velocity to the changes in S-wave velocity at a specified depth (Xia et al., 1999).
Figure 5.5a and b show an inline CSG for the source at x = 0 m and y =
0 m and its estimated phase-velocity dispersion curve. The curve that plots 1/3
wavelength against frequency is shown in Figure 5.5c, where we can estimate the
average wavelength for each frequency in the data. Figure 5.4a shows the pseudodepth
for each migration image in the z axis, where the red dashed lines in Figure 5.4a are
mapped onto Vs-velocity model shown in Figure 5.4b. The migration image provides
a good estimate of the fault boundaries.
b) Dispersion Curve
Frequency (Hz)
10 20 30 40 50 60
Velo
city (
km
/s)
1.3
1.1
0.9
0.7
0.5
a)Shot Gather
x (m)
0 200
t (s
)
0.12
0.24
0.36
0.48
Frequency (Hz)
20 40 60
Wavele
ngth
/3 (
m)
0
10
20
30
40c) Wavelength/3
Figure 5.5: a) Inline common shot gather for the source at x = 0 m and y = 0m, b) its estimated phase velocity dispersion curve, and c) the curve that plots 1/3wavelength against frequency.
As a comparison, the migration images with a finer geophone spacing of 6 m
are shown in Figure 5.6, where the maximum frequency that avoids spatial aliasing
is approximately 58 Hz. The comparison of Figures 5.4a and 5.6 shows that the
spatial aliasing artifacts are more prominent in the coarsely gridded model with center
frequencies from 45 Hz to 25 Hz. However, the migration images with spatial aliasing
still show a blurred boundary at the lateral velocity contrast. A combination of
159
images from different frequencies is helpful for interpreting the geological events.
Pseudo d
epth
(m
)
5.2
6.0
7.1
8.7
11.0
14.9
19.5
Figure 5.6: Migration images at z = 0 m computed from the synthetic data with afiner source and receiver spacing of 6 m, where the two red dashed lines are at x =129 m and 174 m, respectively, and the z axis denotes pseudodepth calculated fromthe mapping of frequency to the depth of 1/3 wavelength.
5.4.2 Natural Migration of Aqaba Data
A 2D land survey was carried out along the Gulf of Aqaba coast in Saudi Arabia
(Hanafy et al., 2014). There were 120 shot gathers recorded, with shot and receiver
intervals of 2.5 m. The source is generated by a 200-lb weight drop striking a metal
plate on the ground, with 10 to 15 stacks at each shot location. A shot gather is
shown in Figure 5.7, and the dashed line in Figure 5.8 shows the amplitude spectrum
of all traces in the CSG.
160
A series of low-pass filters is used to find the maximum usable frequency of the
surface waves in the data. Results show that the maximum frequency of the surface
waves is approximately 45 Hz. Next, we design nine narrow-band filters, and their
center frequencies vary from 15 Hz to 55 Hz with a 5-Hz interval. The amplitude
spectra are shown in Figure 5.8, and Figure 5.9 shows the 60th CSG filtered between
35 to 45 Hz. The time delay of the source wavelet is estimated to be 0.05 s, and
the transmitted and backscattered surface waves are separated along the traveltimes
indicated by the inclined dashed line in Figure 5.9. The traveltimes are calculated
based on the average phase velocity of 300 m/s.
The 60th CSGs
x (m)
0 100 200
Tim
e (
s)
0
0.2
0.4
0.6
0.8
Figure 5.7: 60th common shot gather from the Aqaba data.
Migrating the surface waves after applying nine narrow-band filters to the shot
gathers gives the migration images in Figure 5.10a. For these images, the main fault
is located at x = 150 m on the surface, which is also observed in the field as the
surface expression of a fault (Hanafy et al., 2014).
There are two other lateral velocity anomalies at x = 205 m and x = 280 m (loca-
161
Frequency (Hz)
0 10 20 30 40 50 60 70 80
Am
plit
ude
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4Amplitude Spectra of tha Data and Filters
Figure 5.8: Solid lines denote the amplitude spectra of the nine band-pass filters;Dashed line denote the amplitude sepctrum of all 120 shot gathers in the Aqabadata.
tions 3 and 4 in Figure 5.10b) in Figure 5.10a that are detected in the lower-frequency
migration images, which means that they are deeply buried. This is consistent with
the traveltime tomogram shown in Figure 5.10b, which suggests that there are faults
or LVZ at these locations. The LVZs are clearly seen in the common offset gathers
at x = 200 m and x = 280 m in Figure 5.10c, where abrupt changes in velocity are
accompanied by sharp changes in the arrival times of the surface waves. In fact, a
fault that breaks the surface is observed at the location 2. There also exists velocity
anomalies between 0 and 50 m, which can be seen in the traveltime tomogram.
Figure 5.11b shows the dispersion curve of one CSG in Figure 5.11a. We notice
that there is a discontinuity in the dispersion curve above 55 Hz where the funda-
mental mode is weak. Figure 5.11c shows the wavelength plotted against frequency,
which is calculated from the dispersion curve of the first common shot gather. By
averaging over several source positions, we can estimate the average wavelength for
each frequency in the data set. And then the pseudodepth for each migration image
162
a) The 60th CSG (35-45 Hz)
x (m)
0 100 200
Tim
e (
s)
0
0.2
0.4
0.6
0.8
b) Transmitted Surface Waves
x (m)
0 100 200
Tim
e (
s)
0
0.2
0.4
0.6
0.8
c) Backscattered Surface Waves
x (m)
0 100 200
Tim
e (
s)
0
0.2
0.4
0.6
0.8
Figure 5.9: a) 60th common shot gather filtered by the band-pass filter of 35-45 Hz;b) transmitted surface waves and c) backscattered surface waves obtained by taperedmuting of events above the inclined dashed lines.
163
X (m)
1
0.2
0
0.1
km/s
5
2
3
4
6
7
8
9
2.0
3.4
2.3
2.8
3.0
3.9
4.8
5.9
8.0
Pseudo
depth
(m)
a) Migration Images
Filte
rN
o.
39
3.1
1 2 3 4
b)Traveltime Tomogram
c) Common Offset Gather (offset=7.5m)
Tim
e (
s)
Depth
(m
)
X (m)
X (m)
0 50 100 150 200 250
0 50 100 150 200 250
0
10
3.25 50 100 150 200 250
20
30
1.7
0.3
Figure 5.10: a) Migration images for the Aqaba data with nine narrow-band filters,where the z axis is pseudodepth calculated from 1/3 the wavelength, b) traveltimetomogram, and c) common offset gather (COG) with 7.5 m offset. The locationsdenoted by 2-4 are clearly associated with horizontal velocity anomalies in all threeillustrations; the horizontal velocity anomaly denoted by location 1 is also seen in thetraveltime tomogram. A normal fault breaks the surface at location 2.
164
b) Dispersion Curve
Frequency (Hz)
10 20 30 40 50 60
Velo
city (
km
/s)
0.6
0.4
0.2
a)Shot Gather
x (m)
2.5 52.5
t (s
)
0.15
0.3
0.45
0.6
0.75
Frequency (Hz)
20 40 60
Wavele
ngth
/3 (
m)
0
2
4
6
8
10c) Wavelength/3
Figure 5.11: a) 1st common shot gather of the Aqaba data, b) phase-velocity disper-sion curve and c) the curve that plots 1/3 wavelength against frequency.
is estimated in Figure 5.10a.
This example illustrates that the surface-wave migration image can be interpreted
at the locations of abrupt velocity changes in the tomogram, which can represent the
existence of either a LVZ or a near-surface fault.
5.4.3 Natural Migration of Qademah Data
A 3D land survey was carried out along the Red Sea coast over the Qademah fault
system, about 30 km north of the KAUST campus (Hanafy, 2015). There were 288
receivers arranged in 12 parallel lines, and each line has 24 receivers. The inline
receiver interval is 5 m and the crossline interval is 10 m, which is similar to that of
the 3D survey geometry in Figure 5.2. The receiver geometry is shown in Figure 5.12,
where one shot is fired at each receiver location for a total of 288 shot gathers. The
source is generated by a 200-lb hammer striking a metal plate on the ground, and a
shot gather is shown in Figure 5.13a. Figure 5.13b shows the composite amplitude
spectrum of the all common shot gathers over the frequency range between 15 and
55 Hz.
165
1 24
25 48
288265
2
0 20 40 60 80 100 115
0
20
40
60
80
100
110
Inline (m)
Cro
ssline (
m)
Station No.
Figure 5.12: Receiver geometry for the Qadema-fault data. Shots are located at eachgeophone, and a total of 288 shot gathers are migrated using equation 5.3.
a) The 121th Common Shot Gather
Receiver
1 51 101 151 201 251
Tim
e (
s)
0
0.2
0.4
0.6
b) Amplitude Spectrum for All Shots
Receiver
1 51 101 151 201 251
Fre
quency (
Hz)
0
28
57
85
×108
0
5
10
15
Figure 5.13: a) Common shot gather no. 121 from the Qadema-fault data and b) theamplitude sepctrum for all 288 shot gathers.
166
a) The 121th CSG (20-30 Hz)
Receiver
1 51 101 151 201 251
Tim
e (
s)
0
0.2
0.4
0.6
b) Transmitted Surface Waves
Receiver
1 51 101 151 201 251
Tim
e (
s)
0
0.2
0.4
0.6
c) Backscattered Surface Waves
Receiver
1 51 101 151 201 251
Tim
e (
s)
0
0.2
0.4
0.6
Figure 5.14: ) Common shot gather no. 121 from the Qadema-fault data filtered bya 20-30 Hz band-pass filter and b) the separated transmitted waves along the red diplines (slope = 140 m/s); c) the separated backscattered waves along the horizontalred line (about 0.1 s).
167
Figure 5.15: a) Migration images of the Qademah-fault data filtered by eight narrow-band filters, where the center frequencies range from 41 Hz (filter 1) to 13 Hz (filter8). b) 3D Rayleigh phase-velocity tomogram (Hanafy, 2015). The location of theQademah fault indicated by the black lines in the migration images shown in panela) correlate with the S-velocity tomogram shown in b). There is no visible indicationof the fault on the free surface. The dip angle of the fault interpreted from thismigration image is similar to that estimated from the tomogram.
168
We design eight narrow-band filters with center frequencies ranging from 13 Hz
to 41 Hz to get the depth information of the migration image. CSGs are band-pass
filtered with a center frequency of 25 Hz, and CSG #121 is shown in Figure 5.14a,
in which the time shift of the source wavelet is about 0.1 s. The separated transmit-
ted and backscattered surface waves for the 121 common shot gather are shown in
Figures 5.14b and c.
Applying equation 5.3 to 288 processed (see workflow in Figure 5.1) shot gathers
gives the migration images in Figure 5.15a. The blue areas in Figure 5.15a show
the images of near-surface heterogeneities associated with filters from 4 to 7, and the
positions of these images vary from 45 m to 85 m with increasing frequency in the data,
which mostly agrees with the actual fault location indicated by traveltime tomography
shown in Figure 5.15b (Hanafy, 2015). Figure 5.16a shows the first inline traces of the
first CSG, and Figure 5.16b presents the estimated phase-velocity dispersion curve.
Figure 5.16c plots the wavelength of surface waves for each frequency, based on the
dispersion curve in Figure 5.16b. Averaging over several source positions, the average
wavelength can be estimated for each frequency in the data. The pseudodepth for
each migration image is shown in Figure 5.15a.
5.5 Conclusions
We present the natural migration (NM) method for controlled source data, which can
detect near-surface heterogeneities by naturally migrating the backscattered surface
waves. The assumption is that the near-surface heterogeneities must be within the
depth of about 1/3 the dominant wavelength of the surface waves. A dense-receiver
sampling (half the minimum wavelength of the surface waves) must be used to record
the Green’s functions at the surface to avoid spatial aliasing in the migration images.
The NM method uses the recorded data along the surface to calculate the Green’s
functions instead of computer stimulations that require both the P-velocity and S-
169
b) Dispersion Curve
Frequency (Hz)
14 24 34
Velo
city (
km
/s)
0.6
0.4
0.2
a)Shot Gather
x (m)
0 20 40
t (s
)
0.2
0.4
0.6
Frequency (Hz)
15 25 35
Wavele
ngth
/3 (
m)
0
5
10
15c) Wavelength/3
Figure 5.16: a) Common shot gather for traces along the x direction for the firstsource shown in Figure 5.12, b) estimated phase-velocity dispersion curve, and c)wavelength/3 plotted against frequency.
velocity models. Synthetic and field results demonstrate that lateral near-surface
heterogeneities can be imaged by NM of backscattered surface waves in common shot
gathers. No modeling of the 3D wave equation is needed. The implication is that
more accurate hazard maps can be quickly generated by naturally migrating surface
waves in land surveys in a cost-effective manner. The limitation of this method is
that a dense receiver coverage is needed to get a high-resolution image. However, the
NM method with aliased data can still provide migration images that delineate the
locations of faults. Future research should explore the use of least squares migration
in mitigating these artifacts.
5.6 Acknowledgments
The research reported in this publication was supported by the King Abdullah Uni-
versity of Science and Technology (KAUST) in Thuwal, Saudi Arabia. We thank the
sponsors of the CSIM consortium for their support. We would also like to thank the
high performance computing (HPC) center of KAUST for providing access to super
170
computing facilities. We thank Bowen Guo and Zongcai Feng for editing the paper.
AlTheyab thanks Saudi Aramco for sponsoring his graduate studies. We also thank
the associate editor Joost van der Neut and three anonymous reviewers whose reviews
improved the quality of this manuscript.
171
Chapter 6
Conclusions
In this dissertation, I develop several novel surface-wave imaging methods to improve
the quality of near-surface images, respectively. The main results and conclusions of
my thesis are summarized below:
6.1 3D Wave-equation Dispersion Inversion of Rayleigh Waves
In chapter 2, we extend the 2D WD methodology to 3D, where the objective function
is the sum of the squared differences between the wavenumbers along the predicted
and observed dispersion curves for each azimuth angle. The Frechet derivative with
respect to the 3D S-wave velocity model is derived by the implicit function theorem.
The WD gradient is calculated by correlating the back-propagated wavefield with the
forward-propagated source field in the model based on the Born approximation in an
isotropic, elastic reference earth model.
We provide a comprehensive approach to build the initial model for 3D WD,
which starts from the pseudo 1D S-wave velocity model, which is then used to get
the 2D WD tomogram, which in turn is used as the starting model for 3D WD. Our
numerical results from both synthetic and field data show that the 3D WD method can
reconstruct the 3D S-wave velocity tomogram for a laterally heterogeneous medium so
that the predicted surface waves closely match the observed ones for the fundamental
modes. This suggests that the WD tomogram can serve as a good starting model
for surface-wave FWI. The 3D WD method can be easily adapted to also invert the
172
higher-order modes for a more detailed velocity model. In addition, guided waves that
are trapped in near-surface waveguides can be inverted by 3D WD for the near-surface
P-wave velocity model.
The main limitation of 3D WD is its high computational cost, which is more
than an order-of-magnitude greater than that of 2D WD. However, the improvement
in accuracy compared to 2D WD can make this extra cost worthwhile when there
are significant near-surface lateral variations in the S-velocity distribution. If the
attenuation is important, then its effects can be accounted for by solving the visco-
elastic wave equation to compute the theoretical dispersion curves. To expedite the
picking of dispersion curves obtained from large data sets we recommend supervised
machine learning methods that adapt to the data recorded at different sites.
6.2 3D Wave-equation Dispersion Inversion of Surface Waves
Recorded on Irregular Topography
In chapter 3, we extend the 2D TWD methodology to 3D, that accounts for significant
3D variations in topography by a 3D spectral element solver. The objective function
of 3D TWD is the sum of the squared differences between the predicted and observed
dispersion curves. More accurate dispersion curves can be calculated by using the
geodesic distance compared to that using the Euclidean distance, which can lead to
a more accurate inverted model for 3D TWD. The effectiveness of this method is
numerically demonstrated with synthetic and field data recorded on an irregular free
surface. Results with synthetic data suggest that 3D TWD can accurately invert for
the S-velocity model in the Foothills region when there is a huge elevation difference
compared to the S-wave wavelengths. Field data tests suggest that, compared to the
3-D P-wave velocity tomogram, the 3D S-wave tomogram agrees much more closely
with the geological model taken from the trench log. The agreement with the trench
log is even better when the Vp/Vs tomogram is computed, which reveals a sharp
173
change in velocity across the fault that is in very good agreement with the well log.
Our results suggest that integrating the Vp and Vs tomograms can sometimes give
the most accurate estimates of the subsurface geology across normal faults.
Similar to 3D WD, a limitation of 3D TWD is that the fundamental dispersion
curves must be picked for each shot gather. This process can be prone to errors when
there is a strong overlap with higher-order modes or there is spatial and temporal
aliasing due to large spatial and temporal sampling intervals. This problem might be
mitigated by the machine learning method that automatically picks dispersion curves.
6.3 Multiscale and Layer-Stripping Wave-Equation Disper-
sion Inversion of Rayleigh Waves
In chapter 4, we have developed a new multiscale and layer-stripping wave-equation
dispersion inversion method for Rayleigh waves. In this method, the high-frequency
and near-offset data are first used to invert for the shallow S-velocity model, and the
lower-frequency data with longer offsets are gradually incorporated to invert for the
deeper regions of the model. Numerical results of both synthetic and field seismic
data demonstrate that the wave-equation dispersion inversion can suffer from the local
minima problem when inverting data from a complex earth model, and our multiscale
and layer-stripping wave-equation dispersion inversion method can mitigate the local
minima problem and enhance convergence to the global minimum.
6.4 Imaging Near-surface Heterogeneities by Natural Migra-
tion of Surface Waves: Field Data Test
In chapter 5, we present the natural migration (NM) method for controlled source
data, which can detect near-surface heterogeneities by naturally migrating the backscat-
tered surface waves. The assumption is that the near-surface heterogeneities must
be within the depth of about 1/3 the dominant wavelength of the surface waves. A
174
dense-receiver sampling (half the minimum wavelength of the surface waves) must
be used to record the Green’s functions at the surface to avoid spatial aliasing in
the migration images. The NM method uses the recorded data along the surface to
calculate the Green’s functions instead of computer stimulations that require both
the P-velocity and S-velocity models. Synthetic and field results demonstrate that
lateral near-surface heterogeneities can be imaged by NM of backscattered surface
waves in common shot gathers. No modeling of the 3D wave equation is needed. The
implication is that more accurate hazard maps can be quickly generated by naturally
migrating surface waves in land surveys in a cost-effective manner. The limitation of
this method is that a dense receiver coverage is needed to get a high-resolution im-
age. However, the NM method with aliased data can still provide migration images
that delineate the locations of faults. Future research should explore the use of least
squares migration in mitigating these artifacts.
175
PAPERS PUBLISHED AND SUBMITTED
Journal Papers
• Liu, Z., K. Lu, and G. Schuster, 2019, Convolutional sparse coding for noise atten-
uation of seismic data, Geophysics, in preparation
• Liu, Z., J. Li, S. Hanafy, and G. Schuster, 2019, 3D wave-equation dispersion in-
version for data recorded on irregular topography, Geophysics, under review
• Liu, Z., J. Li, S. Hanafy, and G. Schuster, 2019, 3D wave-equation dispersion in-
version of Rayleigh waves, Geophysics, 84(5), 1-127
• Liu, Z., and L. Huang, 2019, Multiscale and layer-stripping wave-equation disper-
sion inversion of Rayleigh waves, Geophys. J. Int., 218(3), 1807-1821
• Liu, Z., Y. Chen, and G. Schuster, 2019, Multilayer sparse least squares migra-
tion=deep convolutional neural network, arXiv:1904.09321
• Li, J., S. Hanafy, Z. Liu, and G. Schuster, 2019, Wave equation dispersion inversion
of Love waves, Geophysics, 84(5), 1-45
• Li, J., FC Lin, A. Allam, Y. Ben-Zion, Z. Liu and G. Schuster, 2019, Wave equation
dispersion inversion of surface waves recorded on irregular topography, Geophys. J.
Int., 217(1), 346-360
• Fu, L., Z. Liu, and G. Schuster, 2017, Superresolution near-field imaging with sur-
face waves, Geophys. J. Int., 212(2), 1111-1122
• Liu, Z., A. Altheyab, S. Hanafy, and G. Schuster, 2017, Imaging near-surface hetero-
geneities by natural migration of surface waves: field data test, Geophysics, 82(3),
S197-S205
176
Abstracts
• Liu, Z., and G. Schuster, 2019, Multilayer sparse LSM=deep neural network, SEG
Expanded Abstracts
• Liu, Z., J. Li, and G. Schuster, 2019, 3D wave-equation dispersion inversion for
data recorded on irregular topography, SEG Expanded Abstracts
• Liu, Z., and G. Schuster, 2019, Neural network least squares migration, 81st EAGE
Conference and Exhibition
• Liu, Z., and L. Huang, 2019, Multiscale and layer-stripping wave-equation disper-
sion inversion of Rayleigh waves, SEG/DGS Near Surface Modeling & Imaging
Workshop, Bahrain
• Liu, Z., and G. Schuster, 2018, Neural network least squares migration, EAGE/S-
BGF Workshop on Least-Squares Migration, Rio de Janeiro
• Liu, Z., K. Lu, and X. Ge, 2018, Convolutional sparse coding for noise attenuation
of seismic data, SEG Maximizing Asset Value through Artificial Intelligence
and Machine Learning Workshop, Beijing
• Liu, Z., and L. Huang, 2018, Multiscale and layer-stripping wave-equation disper-
sion inversion of Rayleigh waves, SEG Expanded Abstracts
• Liu, Z., S. Hanafy, J. Li, and G. Schuster, 2018, 3D Wave-equation dispersion in-
version of Rayleigh waves, SEG Expanded Abstracts
• Liu, Z., J. Li, and G. Schuster, 2017, 3D wave-equation dispersion inversion of sur-
face waves, SEG 2017 Workshop: Full-waveform Inversion and Beyond, Beijing,
China
• Liu, Z.,A. Altheyab, S. Hanafy, and G. Schuster, 2016, Imaging near-surface hetero-
geneities by natural migration of surface waves, 86th Annual International Meeting,
SEG Expanded Abstracts
177
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