csim.kaust.edu.sa4 abstract novel surface wave imaging methods zhaolun liu i develop four novel...

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

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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

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©September, 2019

Zhaolun Liu

All Rights Reserved

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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

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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.

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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.

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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


4.3.1 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


5.4.1 Natural Migration of Synthetic Data . . . . . . . . . . . . . . 154

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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

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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


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


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


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


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


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



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



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


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

[


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


∫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


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


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


3.41.8 2.6

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Z (

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)0

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)

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km

)

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(d) S-velocity Tomogram

3.41.8 2.6

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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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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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)

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(c) Slice of Tomogram at y = 433 m

3.41.8

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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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)

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Y (

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)

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3

0

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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

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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

)

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0

10

20

0

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20

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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

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F1

2.2

0.6

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s

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m)

0

10

20

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0 20 40 60 80 100

X (m)

F1

3

1.2

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F2

F2

F2

F3

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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


(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

[


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:


∫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


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.


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


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.


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


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


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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