Numerical Modeling, Suppression, and Imaging of
APX4NES
Elastic Wave Scattering by Near-Surface
MASSACHUSETTS INS
s
OF TECHNOLOGY
Heterogeneitiec
I
by
Abdulaziz M. Almuha idib
JUN 10 2014
LIBRARIES
M.S. Geophysics, The University of Texas at Austin, 2008
B.S. Geophysics, The University of Tulsa, 2004
Submitted to the Department of Earth, Atmospheric and Planetary
Sciences
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2014
@ Massachusetts Institute of Technology 2014. All rights reserved.
Signature redacted
A u th o r .................. ............................................
Department of Earth, Atmospheric and Planetary Sciences
Certified by....
Signature redacted-,
May 30, 2014
V
-- ' M. Nafi Toks6z
Robert R. Shrock Professor of Geophysics
Signature redacted
A ccepted by ..
Thesis Supervisor
........................
Robert van der Hilst
Schlumberger Professor of Earth Sciences
Head, Department of Earth, Atmospheric and Planetary Sciences
ijTE
2
Numerical Modeling, Suppression, and Imaging of Elastic
Wave Scattering by Near-Surface Heterogeneities
by
Abdulaziz M. Almuhaidib
Submitted to the Department of Earth, Atmospheric and Planetary Sciences
on June 2, 2014, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
In arid environments, near-surface complexity and surface topography present major
challenges to land seismic data acquisition and processing. These challenges can
severely affect data quality and introduce uncertainty into reservoir imaging and
characterization. The primary objectives of this thesis are to model and study the
contribution of near-surface heterogeneities on seismic wavefield scattering for better
understanding of land seismic data, develop an effective approach to filter out the
scattered noise from the seismic records to enhance the signal-to-noise ratio, and to
accurately image and locate the near-surface heterogeneities.
The first part of this thesis is concerned with simulating the effects of seismic wave
scattering from buried, shallow, subsurface heterogeneities through finite difference
numerical forward modeling. The near-surface scattered wavefield is modeled by
separating the incident (i.e., in the absence of scatterers) from the total wavefield
by means of a perturbation method. Wave propagation is simulated for several earth
models with different near-surface characteristics to isolate and quantify the influence
of scattering on the quality of the seismic signal. We show that the scattered field is
equivalent to the radiation field of an equivalent elastic source excited at the scatterer
locations. Moreover, the scattered waves consist mostly of body waves scattered to
surface waves and are, generally, as large as, or larger than, the reflections. These
scattered waves often obscure weak primary reflections and can severely degrade the
image quality. The results indicate that the scattered energy depends strongly on the
properties of the shallow scatterers and increases with increasing impedance contrast,
increasing sizes of the scatterers, decreasing depth of the scatterers, and increasing
the attenuation factor of the background medium. Also, sources deployed at depth
generate weak surface waves, whereas deep receivers record weak scattered surface
waves. The analysis and quantified results help in the understanding of the scattering
mechanisms and, therefore, can lead to developing new acquisition and processing
techniques to reduce the scattered surface waves and enhance the quality of the seismic
image.
The second part of this thesis develops an approach based on spatially varying
3
local-slope estimation, aiming at alleviating the effects of scattered surface waves
and enhancing the quality of the seismic signal. Understanding the mechanism and
behavior of the simulated scattered surface waves in the first part of this thesis form
the basis for designing the filtering scheme. The algorithm is based on predicting the
spatially varying slope of the noise, using steerable filters, and separating the signal
and noise components by applying a directional non-linear filter oriented toward the
noise direction to predict the noise and then subtract it from the data. The slope
estimation step using steerable filters is very efficient, as it requires only a linear
combination of a set of basis filters at fixed orientation to synthesize an image filtered
at an arbitrary orientation. We apply our filtering approach to simulated data as
well as to seismic data recorded in the field to suppress the scattered surface waves
from reflected body-waves, and demonstrate its superiority over conventional f - k
techniques in signal preservation and noise suppression.
The third part of this thesis presents an approach for prestack elastic reverse
time migration (RTM) to locate and image near-surface heterogeneities using the
near-surface scattered waves (e.g., body to P, S, and surface waves). The approach
back-projects the scattered waves until they are in phase with the incident waves at
the scatterer locations. The P wave components (divergence of the wavefield) are
derived from the spatial derivatives of the measured wavefields. Imaging the nearsurface heterogeneities is important for planning seismic surveys or explaining nearsurface related anomalies in the data. The scattered body-to-surface waves travel
horizontally along the free surface, and, therefore, they provide optimal illumination
of the near-surface compared to scattered body-to-body waves. Additionally, the
elastic RTM scheme preserves the relative amplitude because all wave propagation
losses, including mode conversions, are properly taken into account. We demonstrate,
using synthetic data, that elastic RTM of near-surface scattered waves constructs an
accurate and reliable depth image of near-surface scatterers.
Thesis Supervisor: M. Nafi Toks6z
Title: Robert R. Shrock Professor of Geophysics
4
Acknowledgments
I would like to express my appreciation and gratitude to my advisor, Professor M.
Nafi Toks6z, for his patience, guidance, and support throughout my study at MIT.
His vast knowledge and critical thinking helped me in understanding and answering
various topics and questions related to my research. It has been a privilege and a
pleasure to work with such an outstanding scientist and educator as Professor Toks6z.
I also greatly thank Charlotte Johnson for reading the thesis and making editorial
suggestions.
I am grateful for the effort and time put in by my thesis committee, Professor
Alison Malcolm, Dr. Michael Fehler, Professor John Williams, and Dr. Panos Kelamis, especially for reading my drafts and making excellent suggestions during review
meetings. Their advice and helpful discussions have greatly improved my thesis.
I would also like to thank my professors at MIT, especially Rob van der Hilst, Dale
Morgan, Brian Evans, and Laurent Demanet, for not only the valuable knowledge I
learned from them, but also for their kindness and support. Many thanks also go to
the research staff at ERL, especially Sadi Kuleli, Zhenya Zhu, and Yingcai Zheng, for
many informative technical discussions, and to Dan Burns for his constant interest
and encouragement.
The environment at MIT and ERL provided a friendly atmosphere and stimulating
experience to learn about a wide range of topics. Sincere thanks go to my friends
and fellow students at ERL Yang Zhang, Junlun Li, Fuxian Song, Xin Zhan, Hussam
Busfar, Nasruddin Nazerali, Sudhish Bakku, Yulia Agramakova, Sedar Sahin, Fred
Pearce, Di Yang, Ahmad Zamanian, Saleh Al-Nasser, Chen Gu, Xinding Fang, Alan
Richardson, Andrey Shabelansky, and Xuefeng Shang for all the great times we had
together and for making my life at MIT a pleasant journey. In particular, I am very
grateful to Zeid Alghareeb and Sami Alsaadan, my friends and classmates at the
University of Tulsa and now at MIT for the amazing talks, trips, and fun time we
had together. I am also very thankful to my other friends at MIT and Boston for
making my stay enjoyable and memorable.
5
I would like to thank Saudi Aramco for the financial support during my PhD
studies, and the permission to use and publish the field dataset results in this thesis.
My gratitude goes especially to Samer Al-Ashgar, Manager of the Exploration and
Petroleum Engineering Advanced Research Center (EXPEC ARC), and Panos Kelamis, Chief Technologist of Geophysics at EXPEC ARC, for their endless support
during my graduate studies. I would also like to thank the former managers of EXPEC ARC, Muhammad Al-Saggaf and Nabeel Al-Afaleg for their encouragement to
pursue my PhD studies at MIT.
Last but not least, I am thankful to my family; my parents, brothers, and sisters
for their continued love, support, and encouragement over the years, which have
always helped me to go forward.
6
Contents
1
Introduction
1.1
Background and Motivation
. . . . . . . . .
. . . . . . . . . . . . .
27
1.2
Previous Research and Our Studies . . . . .
. . . . . . . . . . . . .
30
1.2.1
Modeling of Seismic Wave Scattering
. . . . . . . . . . . . .
30
1.2.2
Suppression of Coherent Noise . . . .
. . . . . . . . . . . . .
31
1.2.3
Imaging of Mode-Converted Waves
.
. . . . . . . . . . . . .
33
. . . . . . .
. . . . . . . . . . . . .
34
1.3
2
3
27
Thesis Objective and Overview
Modeling of Elastic Wave Scattering
39
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
2.2
Modeling of Elastic Wave Scattering
. . . . . . . . . . . . . . . . . .
44
2.3
Applications to the Earth Model.
. . . . . . . . . . . . . . . . . . . .
46
2.3.1
Effect of Source Frequency and Source and Receiver Depths
2.3.2
Effects of Scatterers' Depth, Size, Impedance Contrast, and
.
52
A ttenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
2.4
Scattering due to irregular interface scattering . . . . . . . . . . . . .
66
2.5
Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . .
69
Suppression of Scattered Surface Waves
75
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
3.2
Noise Reduction by Spatially Varying Slope Estimation . . . . . . . .
79
3.2.1
Steerable Filters . . . . . . . . . . . . . . . . . . . . . . . . . .
80
3.2.2
Slope Estimation of Local Plane-Waves . . . . . . . . . . . . .
82
7
3.2.3
3.3
3.4
4
5
Signal and Noise Separation . . . . . . . . . . . . . . . . . . .
Synthetic Example
83
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
3.3.1
Finite Difference Modeling . . . . . . . . . . . . . . . . . . . .
85
3.3.2
Application to Synthetic Data . . . . . . . . . . . . . . . . . .
89
Effects of 2D and 3D Heterogeneities on seismic data . . . . . . . . .
95
3.4.1
Irregular Bedrock Interface in 2D and 3D . . . . . . . . . . . .
96
3.4.2
Line and Random Side Scatterers . . . . . . . . . . . . . . . .
96
3.5
Field Data Example
. . . . . . . . . . . . . . . . . . . . . . . . . . .
97
3.6
Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . .
98
Imaging of Scattered Elastic Waves
111
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
111
4.2
Methodology
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
113
4.3
Numerical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
115
4.4
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
117
Discussion and Conclusion
127
5.1
129
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A FD Modeling with an Irregular Free Surface
A.1
131
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
132
A.2 Formulation of Elastic Wave Modeling . . . . . . . . . . . . . . . . . .
133
A.3 Numerical Analysis
134
. . . . . . . . . . . . . . . . . . . . . . . . . . .
A.4 Stability of the ADER Scheme
. . . . . . . . . . . . . . . . . . . . .
137
A.5
Numerical Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . .
137
A.6
Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
138
A.7
Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . .
138
A.8
Implementation of Free Surface Boundary Condition
. . . . . . . . .
138
A.8.1
Free Surface Condition in ID (Flat) . . . . . . . . . . . . . . .
138
A.8.2
Free Surface Condition in 2D (Irregular)
. . . . . . . . . . . .
139
A.9 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . .
140
8
A.9.1
Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . .
140
A.9.2
Planar Boundary . . . . . . . . . . . . . . . . . . . . . . . . .
141
A.9.3
Inclined Straight Boundary
. . . . . . . . . . . . . . . . . . .
141
A.9.4
Gaussian Shaped Hill Topography . . . . . . . . . . . . . . . .
142
A.9.5
Ramp Shape Topography
. . . . . . . . . . . . . . . . . . . .
143
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
144
A.10 Summary
159
B
B.1
Fourth Order Lax-Wendroff-type Scheme . . . . . . . . . . . . . . . .
159
B.2
Free Surface Boundary Condition . . . . . . . . . . . . . . . . . . . .
162
B.3
Fast Marching Level Set Method
. . . . . . . . . . . . . . . . . . . .
163
B.4
Lagrange Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . .
164
B.5
Rotation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
164
C
167
C.1
The Effects of Wavelengths and Scatterer Sizes . . . . . . . . . . . . .
167
C.2 The Effects of Common-Mid-Point (CMP) Stack . . . . . . . . . . . .
167
9
10
List of Figures
2.1
Schematic earth model showing how most of the seismic energy is scattered in the shallow subsurface layers.
2.2
. . . . . . . . . . . . . . . . .
Field data from Saudi Arabia showing upcoming body wave scattering
to surface waves caused by near-surface complexities.
2.3
41
. . . . . . . . .
41
Synthetic earth model. A single layer over half a space with two circular scatterers embedded in the shallow layer. The two scatterers are
located at (x, z) = (360 m, 15 m) and (x, z) = (720 m, 15 m). Each
has a 20 m diameter and an impedance contrast corresponding to 0.36.
The P wave, S wave and density values of the first layer are 1800 m/s,
1000 m/s and 1750 kg/m
3
and for the half-space and scatterers are 3000
m/s, 1500 m/s and 2250 kg/m
3
, respectively. The source is located at
(x, z) = (150 m, 10 m) as indicated by the red star. The receivers are
located on the surface with 50 m near-offset and 5 m space intervals.
2.4
48
Snapshots of the total (u) and scattered (6i) wavefields for the model in
Figure 2.3: (a) the total field at 300 ms, (b) total field at 500 ms, and
(c) the scattered field at 500 ms. The source of scattering is reflected
or refracted body waves. The scatterers excite primary, shear, and also
surface waves due to the proximity to the free surface. The source is
indicated by the black circle. Note that we do not show the scattered
surface-to-surface waves in the scattered wavefield because it is much
larger in amplitude compared to the scattered body-to-surface waves.
11
49
2.5
Finite difference simulations (v,-component) showing the scattering
effects for the model in Figure 2.3; (a) shows the results including
the direct surface wave and (b) with the direct surface wave removed;
(left) incident wavefield simulated using the model without scatterers,
(middle) total wavefield simulated using the model with scatterers,
and (right) scattered wavefield (i.e., the difference between the total
and incident wavefields). An explosive point source with 30 Hz Ricker
wavelet is used. The source is located at 10 m depth and the receivers
are located on the surface. Note the complexity due to scattering of
the reflected arrivals.
2.6
. . . . . . . . . . . . . . . . . . . . . . . . . .
50
Simulated waveforms for the model in Figure 2.3 with vertical source
and receivers on the surface: (a) v,-component, and (b) v,-component.
The incident, total, and scattered wavefields are shown from left to
right, respectively. Note that the direct surface wave is removed. Also
note the strong amplitudes of the shear wave reflection and refraction
as indicated by the yellow circles at mid- and far-offset traces (of the
vz-component) due to the radiation pattern of the vertical source.
2.7
. .
51
Simulated waveforms (v,-component) for the model in Figure 2.3. An
explosive point source is used with (a) 20Hz and (b) 40Hz dominant
frequencies.
The incident, total, and scattered wavefields are shown
from left to right, respectively.
2.8
. . . . . . . . . . . . . . . . . . . . .
53
Finite difference simulations (v,-component) for the scattering model
with different source depths (10 m, 20 m, and 40 m from left to right):
(a) including the direct surface wave, and (b) with the direct surface
wave removed. An explosive point source with 30 Hz Ricker wavelet is
used. The receivers are located on the surface.
12
. . . . . . . . . . . .
54
2.9
Finite difference simulations (v,-component) for the scattering model
with different receiver depths (0 m, 20 m, and 40 m from left to right):
(a) including the direct surface wave, and (b) with the direct surface
wave removed. A vertical source with 30 Hz Ricker wavelet is located at
the surface. Note the strong amplitudes of the shear wave reflection and
refraction as indicated by the yellow circles at mid- and far-offset traces
due to the radiation pattern of the vertical source. At 40 m receiver
depth most of scattered waves appear to be body waves.
Note also,
reflections are not as prominent as they are for surface receivers. At
the surface the amplitude doubles. At depth, upgoing and downgoing
w aves interfere.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
2.10 The effects of source and receiver depths on the SNR due to nearsurface heterogeneities: (a-b) source analysis, and (c-d) receiver analysis. Note that sources at deeper depths generate less surface wave
energy and therefore improve the SNR as shown in (a), but source
depth has no effect on the scattered body-to-surface waves as shown
in (b). Receivers at deeper depths, however, improve the SNR in both
cases: the surface waves (c) and scattered body waves to surface waves
(d ).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
2.11 Simulated waveforms (v,-component) for the scattering model with different impedance contrasts (0.16, 0.27, and 0.36) from left to right. The
source (30 Hz) is located at 10 m depth and the receivers are located
on the surface. (a) The total wavefield simulated using the model with
scatterers, and (b) the scattered wavefield (i.e., the difference between
the total and incident wavefields).
13
. . . . . . . . . . . . . . . . . . .
60
2.12 Simulated waveforms (vz-component) for the scattering model with
different scatterer sizes (10 m, 20 m, and 40 m diameter) from left
to right, with the center of the scatterers at 10 m, 15 m, and 25 m
depth, respectively. Top of the scatterers is 5 m depth below the free
surface. The source (30 Hz) is located at 10 m depth and the receivers
are located on the surface.
(a) The total wavefield simulated using
the model with scatterers, and (b) the scattered wavefield (i.e., the
difference between the total and incident wavefields).
. . . . . . . . .
61
2.13 Simulated waveforms (vz-component) for the scattering model with
different scatterer depths (15 m, 30 m, and 45 m) from left to right. The
source (30 Hz) is located at 10 m depth and the receivers are located
on the surface. (a) The total wavefield simulated using the model with
scatterers, and (b) the scattered wavefield (i.e., the difference between
the total and incident wavefields).
. . . . . . . . . . . . . . . . . . .
62
2.14 Simulated waveforms (v,-component) for the scattering model with
different attenuation factors in the top layer (Q =100, 60, and 30)
from left to right. The source (30 Hz) is located at 10 m depth and the
receivers are located on the surface. (a) The total wavefield simulated
using the model with scatterers, and (b) the scattered wavefield (i.e.,
the difference between the total and incident wavefields).
. . . . . . .
63
2.15 The effects of source depths on the SNR due to characteristics of nearsurface heterogeneities (impedance contrast, depth, size, and quality
factor): (a-d) including the direct surface waves, and (e-h) with the
direct surface waves removed.
. . . . . . . . . . . . . . . . . . . . . .
64
2.16 The effects of receiver depths on the SNR due to characteristics of nearsurface heterogeneities (impedance contrast, depth, size, and quality
factor): (a-d) including the direct surface waves, and (e-h) with the
direct surface waves removed.
. . . . . . . . . . . . . . . . . . . . . .
14
65
2.17 An earth model with near-surface irregular (Gaussian) interface and
deeper flat reflector:
(a) Gaussian surface profile, and (b) the earth
model. Material properties are given in Table 2.3.
. . . . . . . . . .
67
2.18 Finite difference simulations (vz-component) for the irregular (Gaussian) interface at different depths:
(a-c) 15 m, and (d-f) 45 m. The
incident wavefield (a and d) simulated using the model with plane shallow interface; (b and e) total wavefield simulated using the model with
Gaussian shallow interface; and (c and f) scattered wavefield (i.e., the
difference between the total and incident wavefields). Note the strong
dispersive character of the surface wave due to the thin layer (a-c).
Also, note that the amplitudes of scattered (reflected and refracted)
body waves to surface waves decrease rapidly as the interface depth
increases.
Overall, scattering from irregular near-surface interface is
more complex and exhibits more diffusive-type scattering compared to
localized scatterers.
3.1
. . . . . . . . . . . . . . . . . . . . . . . . . . .
68
Schematic earth model showing how most of the seismic energy is
trapped and scattered in the near-surface layers: (a) scattering of direct surface waves and upcoming body-waves to surface waves, and (b)
the ideal model after removing the effects of surface wave scattering.
3.2
77
Derivatives of Gaussian filters: (left) basis filter oriented at 00, (middle)
basis filter oriented at 900, (right) synthesis of the filter oriented at 80'
by linearly combining the basis filters.
3.3
. . . . . . . . . . . . . . . . .
81
The convolution of the input image with different directional filters:
(left) the convolution with the basis filter oriented at 00, (middle) the
convolution with the basis filter oriented at 900, (right) synthesis of the
image filtered at 80' orientation by linearly combining the convolution
of the input image with the basis filters.
15
. . . . . . . . . . . . . . . .
81
3.4
Synthetic earth model. Multiple dipping layers with five circular scatterers (red circles) embedded in the shallow layer. The scatterers are
located at 15 m depth, each is 10 m in diameter and has an impedance
contrast corresponding to 0.36. The source is located at (x,z)=(150 m,
0 m). The receivers are located on the surface with 50 m near-offset
and 5 m space intervals. The color scale (on the right) and associated
numbers refer to material properties given in Table 3.2.
3.5
. . . . . . .
86
Finite difference simulations (vz-component) showing the scattering
effects due to near-surface heterogeneities for the model in Figure 3.4;
(a) shows the results including the direct surface wave and (b) with
the direct surface wave removed; (left) total wavefield simulated using
the model with scattering, (middle) incident wavefield simulated using
the model without scattering, and (right) scattered wavefield (i.e., the
difference between the total and incident wavefields). A vertical source
with 30 Hz Ricker wavelet is used. The source is located at (x,z) =
(150 m, 0 m). The receivers are located on the surface with 5 m space
intervals. Note the complexity due to scattering of the reflected arrivals. 87
3.6
Finite difference simulations (vz-component) showing the scattering
effects due to near-surface heterogeneities for the model in Figure 3.4:
(left) total wavefield, (middle) incident wavefield, and (right) scattered
wavefield.
A vertical source with 30 Hz Ricker wavelet is located at
(x,z) = (850 m, 0 m). The receivers are located on the surface with 5
m space intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7
88
An example of applying the stack-array method with different array
sizes: (left) input image, (middle) stack of five receivers, and (right)
stack of ten receivers. Note that the stack-array method reduces the
scattered surface waves and also the frequency content of the data.
3.8
.
88
Flow diagram of the spatially varying filtering approach to remove
scattered surface waves.
. . . . . . . . . . . . . . . . . . . . . . . . .
16
90
3.9
A schematic diagram showing the frequency-wavenumber domain. The
black lines show the range of wavenumbers constrained by the the
f-
k
filter, and the dashed colored lines (cyan, magenta, and green) show
the frequency-wavenumbers corresponding to different steerable filter
orientations.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
3.10 Histograms of local-slopes calculated using steerable filters (left) for
different receiver patches (right).
The top, middle, and bottom his-
tograms correspond to patch 1, 2, and 3, respectively. The red lines in
the histogram plots correspond to the true orientation of the forward
and backward scattering.
. . . . . . . . . . . . . . . . . . . . . . . .
3.11 Application of the median filter:
92
(left) input data, (middle) filtered
data, and (right) residual (the difference or the removed noise).
3.12 Comparison between the directional filter (left), and the
f
. . .
92
- k filter
(right). Note the edge effects and smearing of reflected signal caused
by the
f
- k filter due to leakage in the transform domain.
. . . . .
93
3.13 Difference between the input data (with noise) and the denoised results
with different methods (Figure 3.12): (left) directional filter and (right)
f
- k filter. Note that the
f
- k filter removed part of the reflected
sign al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
3.14 The frequency-wave number spectrum of: (a) total wavefield (input
image), (b) scattered wavefield (true noise), (c) filtered image (signal),
and (d) residual (removed noise).
. . . . . . . . . . . . . . . . . . . .
94
3.15 Shot gathers simulated using 3D finite difference with the scatterers
in-line with the receivers:
(left) model without scattering, (middle)
with scattering, and (right) the difference.
. . . . . . . . . . . . . . .
102
3.16 Shot gathers simulated using 3D finite difference with the scatterers in
the cross-line direction: (left) model without scattering, (middle) with
scattering, and (right) the difference.
17
. . . . . . . . . . . . . . . . . .
102
3.17 A 3D irregular (Gaussian) bedrock interface model: (a) Gaussian surface profile with 70 m standard deviation, (b) a Gaussian smoothing
operator with 5 m correlation length, (c) a smoothed surface generated
by convolving the Gaussian surface with the smoothing operator, and
(d) an earth model with near-surface irregular bedrock interface at 15
m depth below the free surface and deeper flat reflector at 200 m depth. 103
3.18 Finite difference results for the irregular (Gaussian) bedrock interface
model simulated using: (a) 2D FD, and (b) 3D FD. The total wavefield (left) simulated using the model with Gaussian shallow interface;
the incident wavefield (middle) simulated using the model with plane
shallow interface; and (right) scattered wavefield (i.e., the difference between the total and incident wavefields). The 3D simulations include
scattering phases coming from the cross-line direction.
. . . . . . . .
104
3.19 A 3D earth model with multiple dipping layers and near-surface scatterers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
105
3.20 Modeling and filtering of scattered body-to-surface waves in 3D: (a)
simulated 3D finite difference results, and (b) estimated signal after
application of 3D FK filter.
. . . . . . . . . . . . . . . . . . . . . . . 105
3.21 Application of 3D FK filter to spatially dense sampled 3D simulated
data: (a) the difference between the input and filtered data in Figure
3.20 (filtered noise), and (b) histogram of local-slopes calculated using
3D steerable filters showing the dominant slope for the receiver patch
highlighted in red.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
106
3.22 Application of the steered median filter approach to field data: (left)
input data, (middle) filtered data, and (right) residual (the difference
or noise rem oved).
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
107
3.23 Histograms of local-slopes calculated using steerable filters (left) for
different receiver patches (right). The top, middle, and bottom histograms correspond to patch 1, 2, 3, and 4 respectively. The magenta
and red dashed lines in the histogram plots correspond to 67.5' and
70' orientations.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
108
3.24 Application of the steered median filter approach to field data after
NMO, running average filter, and inverse NMO to enhance the reflections: (left) input data , (middle) filtered data, and (right) residual
(the difference or noise removed). Reflected P waves modeled using
ray-tracing are shown in dashed red lines.
3.25 Application of
f
. . . . . . . . . . . . . . .
109
- k filter to field data after NMO, running average
filter, and inverse NMO to enhance the reflections: (left) input data,
(middle) filtered data, and (right) residual (the difference or noise rem oved).
4.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109
Schematic earth model showing: (a) reflected waves as a source for the
incident or source wavefield, and (b) the receiver wavefield is composed
of near-surface scattered waves.
4.2
. . . . . . . . . . . . . . . . . . . . .
120
Synthetic earth model. Multiple dipping layers with five circular scatterers (red circles near the free surface) embedded in the shallow layer.
The scatterers are located at 15 m depth, each is 10 m in diameter and
has an impedance contrast corresponding to 0.36. Material properties
are given in Table 4.1. The source is located at (x,z)=(150 m, 0 in).
The receivers are located on the surface with 50 m near-offset and 5 m
space intervals. The color scale (on the right) and associated numbers
refer to material properties given in Table 4.1.
19
. . . . . . . . . . . .
120
4.3
Finite difference simulations showing the vz-component (a-c), divergence (d-f), and curl (g-i); (a,d,g) incident wavefield simulated using
the model without scatterers, (b,e,h) total wavefield simulated using
the model with scatterers, and (c,f,i) scattered wavefield (i.e., the difference between the total and incident wavefields). A point source with
30 Hz Ricker wavelet is used. The source is located at 10 m depth and
the receivers are located on the surface. . . . . . . . . . . . . . . . . .
4.4
121
Snapshots of the vz-component (normalized) of the incident (a,c,e) and
scattered (b,d,f) wavefields at 300 ms, 400 ms, and 500 ms from top to
bottom, respectively. The seismic source is located at (x, z) = (150 m,
10 m). The source of scattering is reflected or refracted body waves.
The scatterers excite primary, shear and, also, surface waves due to the
proximity to the free surface. Note that the scattered surface-to-surface
waves are rem oved. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5
122
Snapshots of the divergence (normalized) of the incident (a,c,e) and
scattered (b,d,f) wavefields at 300 ms, 400 ms, and 500 ms from top to
bottom, respectively. The seismic source is located at (x, z) = (150 m,
10 m). The source of scattering is reflected or refracted body waves.
The scatterers excite primary, shear and, also, surface waves due to the
proximity to the free surface. Note that the scattered surface-to-surface
waves are rem oved. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6
123
Snapshots of the curl (normalized) of the incident (a,c,e) and scattered
(b,d,f) wavefields at 300 ms, 400 ms, and 500 ms from top to bottom,
respectively.
The seismic source is located at (x, z) = (150 m, 10
m). The source of scattering is reflected or refracted body waves. The
scatterers excite primary, shear and, also, surface waves due to the
proximity to the free surface. Note that the scattered surface-to-surface
waves are removed. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
124
4.7
Elastic RTM of near-surface scattered waves with receivers placed on
the surface and a free surface boundary condition applied to the upper
boundary. The yellow dashed lines in (a) correspond to the zoomed
area shown in (b).
A.1
ADER
4 th
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
125
order stencil with 25 points. . . . . . . . . . . . . . . . . .
136
A.2 Determination of ghost values required for time-marching at neighboring grid nodes. The blue circles correspond to the ghost point (outside
the domain) and its orthogonal projection on the surface (inside the
domain). Lagrange interpolation (left) and extrapolation (right) in 2D
are used to estimate the point inside the domain that will be then used
to impose the boundary condition at the ghost point. . . . . . . . . .
A.3
148
The computational domain is shown to the left; the source (red) and
receiver (blue) with 1000 m offset. To the right are comparisons of the
recorded v, and v, components. The ADER-CV solution (dashed red)
is plotted against the
4 th
order staggered-grid FD solution (black) at
the selected observation point. . . . . . . . . . . . . . . . . . . . . . .
A.4
149
The computational domain is shown to the left; the source (red) and
receiver (blue), with 1000 m offset and 50 m normal distance from the
free surface. To the right are comparisons of the recorded v, and v,
components. The ADER-CV solution (dashed-red) is plotted against
the
4 th
order staggered-grid FD solution (black) for the flat layer model
at the selected observation point.
A.5
. . . . . . . . . . . . . . . . . . . .
149
The computational domain is shown to the left; the source (red) and
receiver (blue) with 995 m offset and 45 m normal distance from the
(26.50) inclined free surface. To the right, comparisons of the recorded
vx and v, components (i.e., parallel and normal to the inclined surface,
respectively). The ADER-CV solution (dashed-red) is plotted against
the
4 th
order staggered-grid FD solution (black) for the dipping layer
model at the selected observation point.
21
. . . . . . . . . . . . . . . .
150
A.6 The computational domain is shown to the left; the source (red) and
receiver (blue) with 1000 m offset and 50 m normal distance from the
(450) inclined free surface. To the right, comparisons of the recorded
vx and v, components (i.e., parallel and normal to the inclined surface,
respectively). The ADER-CV solution (dashed-red) is plotted against
the
4 t"
order staggered-grid FD solution (black) for the dipping layer
model at the selected observation point.
. . . . . . . . . . . . . . . .
150
A.7 Snapshots of the wavefield at 225 ms and 450 ms for the 45 0 -inclined
free surface model shown in Figure A.6. The left hand side panels
are the horizontal velocity vx, and the right-hand side panels are the
vertical velocity v,.
. . . . . . . . . . . . . . . . . . . . . . . . . . .
A.8 Relative error in terms of the boundary's dip-angle.
151
. . . . . . . . . 151
A.9 The computational domain is shown to the left; source (red) and receivers (blue) with a Gaussian shaped hill free surface (100 m height
and 100 m wide). To the right is the distance function computed using
the fast marching level set method. The color bar indicates the normal
distance in meters to the free surface. . . . . . . . . . . . . . . . . . .
152
A.10 To the left are comparisons of the recorded pressure at the receiver
locations shown in Figure A.9; ADER-CV (dashed red) against the
boundary conformal solution (black). To the right is a comparison of
the ADER-CV solutions with different grid spacings showing convergence of the method as the grid spacing decreases. . . . . . . . . . . .
152
A.11 Snapshots of the wavefield v, component at different instants in time
showing the scattering and multiple reflections caused by the irregular
surface model shown in Figure A.9. . . . . . . . . . . . . . . . . . . .
22
153
A.12 Time series of the velocity components along the free surface of the
ramp shape surface model shown at the top. The middle panel shows
the horizontal velocity component v,
and the bottom panel shows the
vertical velocity v,. The obvious phases are labeled, where P indicates
P wave, R indicates Rayleigh wave, and PRrefl indicates P to Rayleigh
reflection.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
154
A.13 Snapshots of the wavefield at different instants in time showing the
scattering and multiple reflections caused by the ramp shape surface
model with homogeneous velocity. The left hand side panels are the
horizontal velocity v,
velocity v..
and the right-hand side panels are the vertical
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
155
A. 14 Time series of the velocity components along the free surface of the
ramp shape model shown at the top. The middle panel shows the
horizontal velocity component v2, and the bottom panel shows the
vertical velocity v,.
. . . . . . . . . . . . . . . . . . . . . . . . . . .
156
A.15 Snapshots of the wavefield at different instants in time showing the
scattering and multiple reflections caused by the ramp shape surface
model with one layer over half space. The left hand side panels are the
horizontal velocity v2, and the right-hand side panels are the vertical
velocity v..
C.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
157
The scattered wavefields due to different scatterer radiuses are shown
to the left, and their corresponding Frequency-wavenumber domains
are shown to the right: (a) 5 m, (b) 10 m, and (c) 20 m.
C.2
. . . . . . .
169
Finite difference results for the single layer over half a space model in
Figure 2.3: (a-c) without scatterers, and (d-f) with scatterers.
The
gathers are sorted to (a and d) common shot, (b and e) common receiver, and (c and f) common midpoint.
23
. . . . . . . . . . . . . . . .
170
C.3 Common-mid-point gathers: (left) for the model without scatterers,
(middle) with scatterers, and (right) the difference. The results in (b)
have one third the fold of the ones in (a).
. . . . . . . . . . . . . . .
171
C.4 Common-mid-point stacks with the direct surface wave: (a-c) with full
fold, and (d-f) with half the fold, (a and d) stack for the model without
scatterers, (b and e) stack for the model with scatterers, and (c and f)
the difference. Increasing the fold reduced the stacked direct surface
w ave.
C.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
172
Common-mid-point stacks with the direct surface wave removed before
stacking: (a-c) with full fold, and (d-f) with half the fold; (a and d)
stack for the model without scatterers, (b and e) stack for the model
with scatterers, and (c and f) the difference.
The results show that
the scattered noise phases have not been removed by CMP stacking.
Increasing the fold has no effect on the stacked body-to-surface wave
n oise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
173
List of Tables
2.1
Summary of all the cases studied and their corresponding figure numb ers.
2.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
Material properties for models with different contrasts. The impedance
contrasts are calculated for different material properties relative to
layer I.
2.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Material properties (P wave velocity, S wave velocity, and density) of
the model shown in Figure 2.17. . . . . . . . . . . . . . . . . . . . . .
69
. . .
79
3.1
Comparison among different methods for surface wave removal.
3.2
Material properties (P wave velocity, S wave velocity, and density) of
the model shown in Figure 3.4.
4.1
. . . . . . . . . . . . . . . . . . . . .
86
Material properties (P wave velocity, S wave velocity, and density) of
the model shown in Figure 4.2.
A.1
57
. . . . . . . . . . . . . . . . . . . . .
120
Relative misfit of the ADER-CV method compared with the boundary
conformal method for different grid spacings and Gaussian topography
sizes.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
148
26
Chapter 1
Introduction
1.1
Background and Motivation
The ultimate goal in exploration seismology is to obtain an optimum image of the
subsurface to map reservoir boundaries (e.g., structural mapping) and to characterize reservoir properties (e.g., porosity, saturation, permeability, and fracture density
and orientation) for field evaluation and development.
Extracting and monitoring
reservoir properties are essential steps for reservoir modeling, flow simulation studies
and for horizontal drilling to accomplish optimal recovery. These processes can be
achieved by inverting the formation parameters (P-impedance, S-impedance, and density) from the seismic data and by using rock physics information derived from core
and log data to link between the rock properties and seismic parameters. However,
in the real world, many factors affect the quality of seismic data. Surface topography and near-surface heterogeneities can significantly complicate the data with large
static travel time variations, signal attenuation, and wave scattering. These factors
can deteriorate the image quality and increase the uncertainty of subsurface images
and formation properties. To improve our understanding of the near-surface effects
on the quality of the seismic data, this thesis examines the scattering of elastic waves
from scatterers embedded in earth's near-surface, with emphasis on body-to-surface
wave scattering.
In land seismic data, the near-surface complexity and surface topography can
27
severely contaminate the seismic data with complicated wave phenomena that cannot be accounted for by single layered models and conventional processing methods.
Both the direct surface waves and the up-going reflections are scattered by the heterogeneities. The scattering takes place from body waves to surface waves, and from
surface waves to body waves.
Further, the near-surface scattered body-to-surface
waves, which have comparable amplitudes to reflections, can mask the seismic reflections. These difficulties, added to large amplitude direct and back-scattered surface (Rayleigh) waves, create a major reduction in signal-to-noise ratio and degrade
the final subsurface image quality. Also, strong contrast in velocity (and therefore
impedance) between the near-surface and the substrata (e.g., carbonate layers) may
add to the complexity of the energy penetration, as most of the seismic energy is
reflected and trapped near the surface, generating both internal and surface related
multiples. In interpreting noise-contaminated data, the main challenges are caused
by complex topographic and near-surface features such as sand dunes, wadis/large
escarpments, karsted carbonates and dry river beds. In such environments, it is essential to understand the various types of near-surface effects such as direct waves,
mode-conversions, scattering, and attenuation.
The scattered elastic waveforms have strong effects on both the phase and amplitude of the recorded signal and are neglected in most conventional imaging and
interpretation schemes. A commonly used scattering model, the Born approximation, assumes a weak single scattering and an incident wave propagating in a homogeneous or smooth background media. The Born approximation fails in the case
of strong scatterers and multiple scattering waves.
More widely used, the acous-
tic approximation assumes only pressure waves are involved and neglects the effects
of P to S mode-conversion. The acoustic approximation fails when upcoming body
waves scatter from near-surface heterogeneities and generate scattered surface waves.
These assumptions can lead to significant deviations in amplitude and phase of the
scattered wavefield, due to strong impedance and velocity anomalies. Re-datuming
techniques, based on wave-equation migration, common-focus-point technology, and
interferometry, can handle dynamic time shift (i.e., statics) of the P wave reflected
28
energy (acoustic response) due to delays caused by near-surface velocity variations.
However, these techniques cannot handle multiple scattering waves in fully elastic media even when an exact sub-surface velocity model is available. Therefore, the Born
and acoustic assumptions can severely affect the interpretation and imaging techniques that require high quality seismic attribute maps and are sensitive to the level
of signal-to-noise ratio, such as velocity estimation and migration, elastic (e.g., V,
V, p) and anisotropic (e.g., 6, c) parameter estimations for AVO and AVOz studies,
and 4D time-lapse for reservoir monitoring.
The main objective of this thesis is to understand and alleviate the effects of
near-surface complexities on the seismic records to improve the signal-to-noise ratio,
mainly reducing the noise components due to scattered body-to-surface waves. We
address this problem by modeling the noise component in order to explain observations made in the field data. The goal is to understand the characteristics of the
noise component to develop an effective algorithm for its subsequent removal. Thus,
an extensive numerical analysis is conducted to quantitatively study the effects of the
near-surface scatterers (e.g., material contrast, size, and depth) as well as the acquisition setup (e.g., source and receiver depths) on the signal-to-noise ratio. Building
on the analysis of the numerical modeling results, we develop a framework to reduce
scattered body-to-surface waves based on spatially varying slope prediction and separation, using steerable and non-linear median filters. Suppression of these scattered
surface waves can be difficult using conventional filtering methods, such as an
f
- k
filter, without distorting the reflected signal. We show, using synthetic examples as
well as a spatially dense sampled onshore field dataset, the robustness of this method
for noise attenuation. To locate and image the near-surface scatterers, we present
an approach based on elastic RTM that utilizes the scattered body-to-surface waves,
the most sensitive part of the wavefield to near-surface heterogeneities. In this thesis
we mainly emphasize 2D earth models as they are more feasible than 3D in terms
of the computational efficiency, and only show 3D examples when it is essential to
demonstrate the differences between the two modes.
29
1.2
1.2.1
Previous Research and Our Studies
Modeling of Seismic Wave Scattering
Numerical modeling of elastic wave propagation plays a key role in almost every
aspect of seismology as it provides a means of explaining the recorded signal associated with complex earth models. In general, the weathered zone (i.e., heterogeneity near the earth's surface) has great effect on seismic reflection surveys. In such
cases, the recorded seismograms can be severely contaminated by scattering, wave
conversions, and ground rolls.
Therefore, accurate modeling is essential to study
the near-surface effects on seismic wave propagation.
In Chapter 2, we present a
modeling approach for simulating and studying the effects of elastic wave scattering
by near-surface heterogeneities (impedance and velocity anomalies), especially the
scattering of up-coming body waves. Several previous studies have formulated solutions of the forward (Hudson, 1977; Wu and Aki, 1985; Beylkin and Burridge, 1990;
Sato et al., 2012) and inverse (Blonk et al., 1995; Blonk and Herman, 1996; Ernst
et al., 2002) elastic scattering problems for modeling and imaging based on the perturbation method and single-scattering (Born) approximation. These methods have
limitations when dealing with large and high-contrast heterogeneities that violate
the single-scattering (Born) approximation. Even though the FD-injection method
(Robertsson and Chapman, 2000) is more efficient, it cannot handle the interaction
of the scattered wavefield with the free surface and bedrock layers (e.g., second or
high order long range interactions). Campman et al. (2005, 2006) imaged and suppressed near-receiver scattered surface waves assuming that scattering takes place
immediately under the receivers. Other methods, based on solving integral equations
using the method of moments, can take into account multiple scattering and can handle strong contrast and large heterogeneities (Riyanti and Herman, 2005; Campman
and Riyanti, 2007).
However, these methods are limited to laterally homogeneous
embedding consisting of horizontal layers, and are not valid in areas with complex
overburden. Numerical forward modeling of elastic waves, as opposed to analytical
methods, plays a key role in this study. Because we solve the full wave equation
30
based on finite difference, our modeling can handle more complicated background
media, contrasts in both density and Lame parameters, and irregular features, and
can generate synthetic seismograms that are accurate over a wide range of scatterer
to wavelength ratios. Although finite element methods are best in explicitly handling
irregular boundaries to minimize numerical dispersion, finite difference methods are
almost always superior in terms of the computational speed, especially in 3D. We
compute numerical simulations in two-dimensions for simple earth models with nearsurface scatterers. We consider irregular interface and finite scatterers with contrasts
in both density and Lam6 parameters that are embedded in the shallow subsurface
to analyze and assess the effects of near-surface scattering mechanisms on recorded
seismic waveforms. The perturbation method for elastic waves is used to separate
the scattered wavefield from the total wavefield based on a perturbation of the wave
equation with respect to medium parameters. The method decomposes the medium
parameters into background and perturbation parts and allows us to model scattering
from arbitrary shape scatterers and the interaction of the multiply-scattered wavefield with the free surface. In Chapter 2, we carry out extensive calculations to study
the effects of the acquisition geometry (e.g., source and receiver depths), the quality
factor of the background medium, and the elastic properties of shallow subsurface
scatterers (e.g., size, depth, and impedance contrast) on the near-surface scattered
wavefield.
1.2.2
Suppression of Coherent Noise
Surface waves, in general, are characterized by low frequency, linear moveout, large
amplitudes and slower amplitude decay with distance. The direct surface wave is
confined within a fan-shaped window in the time-space domain and is much larger
in amplitude and lower in frequency than body wave reflections.
These character-
istics of direct surface waves make it effective to apply filtering techniques within
this narrow fan-shaped window. However, the scattered body-to-surface waves have
comparable amplitudes to reflections, frequencies dependent on the size of the heterogeneities, and they mask the entire dataset. Several methods have been developed
31
in the geophysical literature to filter and attenuate source-generated noise based on
its characteristics (e.g., frequency, velocity, amplitude, and polarization). Bandpass
frequency filters remove ground rolls based on their low frequency characteristics.
Nevertheless, removing low frequencies from the data may also distort reflections
(due to the overlap of noise and signal in the frequency domain) and may affect subsequent quantitative interpretation and inversion schemes that are mainly dependent
on the low frequency component of the signal. Conventional "global" velocity filtering methods, such as
f
- k and T
-
p can be very effective in attenuating ground
roll or scattered surface wave energy. However, they can distort reflections and introduce ringing due to leakage in the transform domain. Local radial transform requires
window selection to filter the fan-shaped noise (Henley, 2003). Seismic interferometry (i.e., cross-correlation of two receivers due to many sources exhibits a stationary
point) can predict direct and scattered surface waves (Dong et al., 2006; Xue et al.,
2008), but it cannot predict (isolate) scattered body-to-surface waves.
Campman
et al. (2005, 2006) imaged and suppressed near-receiver scattered surface waves, employing model-based inverse scattering schemes, but assumed that scattering takes
place immediately under the receivers. Discrimination between surface waves and
body wave reflections can be achieved using particle motion polarization (Vidale,
1986), but requires multi-component data that is not often available. Stack-array approach during acquisition (Anstey, 1986; Morse and Hildebrandt, 1989; Regone, 1998;
Ozbek, 2000)
can be very effective in reducing scattered noise, but it also reduces the
high frequency components of the signal due to intra-array statics and and therefore
decreases the image resolution (Baeten et al., 2000). Even though high fold acquisition and common-mid-point (CMP) stacking are powerful in reducing random noise,
reducing scattered coherent noise and preserving the relative amplitude of the signal
are essential for amplitude critical processes in the pre-stack domain (Larner et al.,
1983) such as predictive deconvolution, velocity analysis, waveform inversion, migration, and quantitative interpretation studies (e.g., amplitude variation with offset and
azimuth). To overcome the shortfalls of these methods, we develop a multi-stage filtering algorithm in Chapter 3, for the separation of scattered surface waves from body
32
wave reflections based on spatially varying slope estimation and non-linear median
filtering.
1.2.3
Imaging of Mode-Converted Waves
Reverse time migration (RTM) schemes based on the acoustic wave equation have
become a standard tool for imaging complex geological structures due to the low
computational expense compared to elastic RTM. The earth, however, is elastic and
the data recorded in the field contain all complicated wave types, including mode conversions. In recent years, there has been more interest in exploiting all the information
carried by mode-converted seismic data by using elastic RTM. Sun et al. (2006) introduced a modified RTM approach of transmitted PS waves for salt flank imaging.
This separates the wavefield into pure mode (PP) and converted (PS) waves and the
extrapolation is performed using the scalar wave-equation with the corresponding Vp
and Vs velocities. A similar strategy is proposed by Xiao and Leaney (2010) for salt
flank imaging with VSP local elastic RTM but using the vector wave equation to
extrapolate the separated PP and PS waves. Shang et al. (2012) used teleseismic
transmitted P and S waves recorded on the surface to perform passive source RTM
to reconstruct dipping and vertical offset interfaces; this approach is superior to traditional receiver function analysis in complex geological environments. In Chapter 4,
we present a prestack elastic reverse time migration approach for locating and imaging near-surface scatterers. To image near-surface scatterers using elastic RTM, the
scattered body-to-surface waves are separated from the total recorded wavefield and
used for receiver wavefield extrapolation. The P wave components (e.g., divergence
of the wavefield) (Dellinger and Etgen, 1990) are derived after RTM and subjected
to a cross correlation-type imaging condition. The stresses and particle velocities are
migrated simultaneously by solving the first order elastic wave equation. We test the
elastic RTM approach on synthetic data simulated with an elastic finite difference
scheme.
33
1.3
Thesis Objective and Overview
This thesis can be broadly divided into three components: modeling, filtering, and
imaging of near-surface scattered waves. The thesis is organized in such a way that
each chapter corresponds to one paper and is self-contained in its motivation, literature review, methods, and results.
In Chapter 2, we study the near-surface scattering of body-to-surface waves and
demonstrate the effects of source characteristics and near-surface perturbations (e.g.,
volume and interface heterogeneities) on the quality of the recorded waveforms. The
bulk of this chapter has been accepted for publication as:
AlMuhaidib, A. M. and Toks6z, M. N., Numerical modeling of elastic scattering by
near-surface heterogeneities, Geophysics (in press)
In Chapter 3, we develop a framework to separate scattered body-to-surface waves
from the reflected signal based on spatially varying slope prediction and separation,
using steerable and non-linear median filters. One of the most important objectives of
this thesis is applying these techniques to field data. We demonstrate the application
of the filtering algorithm on a spatially dense sampled 2D land field dataset acquired
in an area with substantial near-surface scattering. The bulk of this chapter has been
accepted for publication as:
AlMuhaidib, A. M. and Toksbz, M. N., Suppression of near-surface scattered body-to-surface
waves: A steerable and non-linear filtering approach, Geophysical Prospecting (accepted)
In Chapter 4, we introduce an elastic RTM scheme to image near-surface scatterers
by using the incident and near-surface scattered wavefields as input to the migration
process. The bulk of this chapter is in preparation to be submitted for publication
as:
AlMuhaidib, A. M. and Toks6z, M. N., Imaging of near-surface heterogeneities by
scattered elastic waves (in prep)
The conclusion and future work are discussed in Chapter 5, followed by three
appendices.
Appendix A presents a finite difference scheme based on the characteristic-variable
34
method to model elastic seismic waves in the presence of surface topography. The
solver combines a
4
th- order ADER scheme (Arbitrary high-order accuracy using
DERivatives), which is widely used in aeroacoustics, with the characteristic variable
method at the free surface boundary. The bulk of this appendix is in preparation to
be submitted for publication as:
AlMuhaidib, A. M. and Toks6z, M. N., Finite difference elastic wave modeling with
an irregular free surface using ADER scheme (in prep)
Appendix B describes more details about the implementation of the numerical
scheme and the boundary treatment for the ADER-CV method.
Appendix C demonstrates additional factors affecting elastic wave scattering that
have not been included in the main chapters. Specifically, looking at the effects of
the heterogeneity size on the wavelength of the scattered waves, acquisition fold, and
common-mid-point (CMP) stacking.
Bibliography
Anstey, N. A., 1986, Part 1: Whatever happened to ground roll?: The Leading Edge,
5, 40-45.
Baeten, G., V. Belougne, M. Daly, B. Jeffryes, and J. Martin, 2000, Acquisition and
processing of point source measurements in land seismic: Presented at the 2000
SEG Annual Meeting.
Beylkin, G., and R. Burridge, 1990, Linearized inverse scattering problems in acoustics and elasticity: Wave Motion, 12, 15-52.
Blonk, B., and G. C. Herman, 1996, Removal of scattered surface waves using multicomponent seismic data: Geophysics, 61, 1483-1488.
Blonk, B., G. C. Herman, and G. G. Drijkoningen, 1995, An elastodynamic inverse
scattering method for removing scattered surface waves from field data: Geophysics,
60, 1897-1905.
Campman, X., and C. D. Riyanti, 2007, Non-linear inversion of scattered seismic
surface waves: Geophysical Journal International, 171, 1118-1125.
35
Campman, X. H., G. C. Herman, and E. Muyzert, 2006, Suppressing near-receiver
scattered waves from seismic land data: Geophysics, 71, S121-S128.
Campman, X. H., K. van Wijk, J. A. Scales, and G. C. Herman, 2005, Imaging and
suppressing near-receiver scattered surface waves: Geophysics, 70, V21-V29.
Dellinger, J., and J. Etgen, 1990, Wave-field separation in two-dimensional anisotropic
media: Geophysics, 55, 914-919.
Dong, S., R. He, and G. T. Schuster, 2006, Interferometric predcition and least squares
subtraction of surface waves: Presented at the 2006 SEG Annual Meeting.
Ernst, F. E., G. C. Herman, and A. Ditzel, 2002, Removal of scattered guided waves
from seismic data: Geophysics, 67, 1240-1248.
Henley, D. C., 2003, Coherent noise attenuation in the radial trace domain: Geo-
physics, 68, 1408-1416.
Hudson, J., 1977, Scattered waves in the coda of P: J. Geophys, 43, 359-374.
Larner, K., R. Chambers, M. Yang, W. Lynn, and W. Wai, 1983, Coherent noise in
marine seismic data: Geophysics, 48, 854-886.
Morse, P. F., and G. F. Hildebrandt, 1989, Ground-roll suppression by the stackarray:
Geophysics, 54, 290-301.
Ozbek, A., 2000, Adaptive beamforming with generalized linear constraints: Presented at the 2000 SEG Annual Meeting.
Regone, C. J., 1998, Suppression of coherent noise in 3-D seismology: The Leading
Edge, 17, 1584-1589.
Riyanti, C. D., and G. C. Herman, 2005, Three-dimensional elastic scattering by
near-surface heterogeneities: Geophysical Journal International, 160, 609-620.
Robertsson, J. 0., and C. H. Chapman, 2000, An efficient method for calculating
finite-difference seismograms after model alterations: Geophysics, 65, 907-918.
Sato, H., M. C. Fehler, and T. Maeda, 2012, Seismic wave propagation and scattering
in the heterogeneous earth: Springer.
Shang, X., M. V. Hoop, and R. D. Hilst, 2012, Beyond receiver functions: Passive
source reverse time migration and inverse scattering of converted waves: Geophysical Research Letters, 39.
36
Sun, R., G. A. McMechan, C.-S. Lee, J. Chow, and C.-H. Chen, 2006, Prestack
scalar reverse-time depth migration of 3D elastic seismic data: Geophysics, 71,
S199-S207.
Vidale, J. E., 1986, Complex polarization analysis of particle motion: Bulletin of the
Seismological Society of America, 76, 1393-1405.
Wu, R., and K. Aki, 1985, Scattering characteristics of elastic waves by an elastic
heterogeneity: Geophysics, 50, 582-595.
Xiao, X., and W. S. Leaney, 2010, Local vertical seismic profiling (VSP) elastic
reverse-time migration and migration resolution:
Salt-flank imaging with trans-
mitted P-to-S waves: Geophysics, 75, S35-S49.
Xue, Y., S. Dong, and G. T. Schuster, 2008, Interferometric prediction and subtraction of surface waves with a nonlinear local filter: Geophysics, 74, SI-SI8.
37
38
Chapter 2
Numerical Modeling of Elastic
Wave Scattering by Near-Surface
Heterogeneities*
Abstract
In land seismic data, scattering from surface and near-surface heterogeneities adds
complexity to the recorded signal and masks weak primary reflections. To understand
the effects of near-surface heterogeneities on seismic reflections, we simulate seismic
wave scattering from arbitrary shape, shallow, subsurface heterogeneities through the
use of a perturbation method for elastic waves and finite difference forward modeling.
The near-surface scattered wavefield is modeled by looking at the difference between
the calculated incident (i.e., in the absence of scatterers) and total wavefields. Wave
propagation is simulated for several earth models with different near-surface characteristics to isolate and quantify the influence of scattering on the quality of the
seismic signal. The results show that both the direct surface waves and the up-going
reflections are scattered by the near-surface heterogeneities. The scattering takes
place both from body waves to surface waves, and from surface waves to body waves.
The scattered waves consist mostly of body waves scattered to surface waves and are,
generally, as large as, or larger than, the reflections. They often obscure weak primary reflections and can severely degrade the image quality. The results indicate that
the scattered energy depends strongly on the properties of the shallow scatterers and
increases with increasing impedance contrast, increasing size of the scatterers relative
to the incident wavelength, decreasing depth of the scatterers, and increasing the attenuation factor of the background medium. Also, sources deployed at depth generate
*The bulk of this chapter has been accepted for publication as: AlMuhaidib, A. M. and Toks6z,
M. N., Numerical modeling of elastic scattering by near-surface heterogeneities, Geophysics (in press)
39
weak surface waves, whereas deep receivers record weak surface and scattered bodyto-surface waves. The analysis and quantified results help in the understanding of
the scattering mechanisms and, therefore, can lead to developing new acquisition and
processing techniques to reduce the scattered surface waves and enhance the quality
of the seismic image.
2.1
Introduction
In land seismic data acquisition, most of the seismic energy is scattered in the shallow subsurface layers by near-surface heterogeneities (e.g., wadis, large escarpments,
dry river beds and karst features) that are common in many arid regions such as the
Arabian Peninsula and North Africa (Al-Husseini et al., 1981). When surface irregularities or volume heterogeneities are present (Figure 2.1), the data are contaminated
with scattered surface-to-surface and body-to-surface waves (Levander, 1990), also
known as scattered Rayleigh waves or ground roll. These unwanted coherent noise
features can obscure weak body wave reflections from deep structures. Direct surface
(Rayleigh) wave scattering has been extensively studied in numerous previous studies
(De Bremaecker, 1958; Knopoff and Gangi, 1960; Fuyuki and Matsumoto, 1980; Gelis
et al., 2005), among others. In exploration seismology, however, much less research
has been done on the effects of near-surface heterogeneities on the up-coming reflections (Riyanti and Herman, 2005; Campman et al., 2005, 2006), especially in realistic
cases of more complicated scatterers and background media. Therefore, the emphasis
of this paper is more on the scattering of up-coming body waves.
Among all near-surface challenges, signal-to-noise ratio is most strongly affected
by scattering and requires further investigation to obtain good seismic image quality.
A field data example from Saudi Arabia (Figure 2.2) that was acquired in a desert environment shows the scattering phenomena. The scattered waves have strong effects
on both the phase and amplitude of the recorded signal. They are usually neglected in
most conventional imaging and interpretation schemes under simplified assumptions
of the earth model (e.g., acoustic and single-scattering). They can greatly affect subsequent processes such as migration, full waveform inversion, and amplitude critical
40
VV
Free-surface
VyVVV V V Vyyyy
Direct Surface Waves
Vy
y
Scattered
Reflections
Reflections
Figure 2.1: Schematic earth model showing how most of the seismic energy is scattered
in the shallow subsurface layers.
0
0.05
0.1
0.15
0.2
c/)
E
0.25
0.3
0.35
0.4
0.45
0.5
0
50
100
150
200
250
300
Offset (Meter)
350
400
450
500
Figure 2.2: Field data from Saudi Arabia showing upcoming body wave scattering to
surface waves caused by near-surface complexities.
41
steps like AVO. In order to explore means that could remove or reduce the effects
of near-surface heterogeneities, it is helpful to determine what aspects of the heterogeneities contribute most to degradation of data quality. The scattering mechanisms
can be studied by forward modeling to simulate the interactions between different
wave phenomena caused by near-surface heterogeneities.
In this paper, we present a modeling approach for simulating the effects of elastic
wave scattering by near-surface heterogeneities.
Several previous studies have for-
mulated and examined solutions of the forward (Hudson, 1977; Wu and Aki, 1985;
Beylkin and Burridge, 1990; Sato et al., 2012) and inverse (Blonk et al., 1995; Blonk
and Herman, 1996; Ernst et al., 2002) elastic scattering problems for modeling and
imaging based on the perturbation method and single-scattering (Born) approximation.
These methods have limitations when dealing with large and high-contrast
heterogeneities that violate the single-scattering (Born) approximation. Even though
the FD-injection method (Robertsson and Chapman, 2000) is more efficient, it cannot
handle the interaction of the scattered wavefield with the free surface and bedrock
layers (e.g., second or high order long range interactions).
Herman et al. (2000)
and Campman et al. (2005, 2006) imaged and suppressed near-receiver scattered
surface waves assuming that scattering takes place immediately under the receivers.
Other methods based on solving integral equations using the method of moments
can take into account multiple scattering and can handle strong contrast and large
heterogeneities (Riyanti and Herman, 2005; Campman and Riyanti, 2007). However,
these methods are limited to laterally homogeneous embedding consisting of horizontal layers. These assumptions are not satisfied in areas with complex overburden,
which makes these methods unsuitable for this problem. Numerical forward modeling of elastic waves, as opposed to analytical methods, plays a key role in this study.
Because we solve the full wave equation based on finite difference, our modeling can
handle more complicated background media, large contrasts in both density and Lame
parameters, and irregular features, and can generate synthetic seismograms that are
accurate over a wide range of scatterer to wavelength ratios.
Finite difference schemes have been utilized extensively for elastic wave propaga42
tion (Kelly et al., 1976; Virieux, 1986; Levander, 1988; Graves, 1996).
Treatments
of the irregular free surface boundary condition have also been developed and discussed in the literature (Fornberg, 1988; Tessmer et al., 1992; Hestholm and Ruud,
1994; Robertsson, 1996; Ohminato and Chouet, 1997; Zhang and Chen, 2006; Appel6
and Petersson, 2009; AlMuhaidib et al., 2011).
In this study, we utilize an accu-
rate implementation of the standard staggered-grid (SSG) finite difference scheme
(Virieux, 1986; Levander, 1988; Zhang, 2010) with Convolution Perfectly-MatchedLayer (CPML) absorbing boundary condition (Komatitsch and Martin, 2007; Martin
and Komatitsch, 2009; Zhang and Shen, 2010) to fully model elastic waves in the
presence of heterogeneity. The internal interfaces are represented by the so-called
effective medium parameters (Moczo et al., 2002) to avoid spurious numerical diffractions caused by sharp material discontinuity due to the spatial grid. The density is
calculated by arithmetic average, and the Lam6 parameters are calculated by harmonic average. The SSG scheme is fourth order accurate in space (including the free
surface boundary) and second order accurate in time. The free surface boundary is
treated by adjusting the finite difference approximations to the z-derivative close to
the surface (Kristek et al., 2002), which provides
4 th
order accuracy in space and
minimizes numerical dispersion.
We compute numerical simulations in two-dimensions for simple earth models with
near-surface scatterers.
We consider irregular interface and finite scatterers with
contrasts in both density and Lame' parameters that are embedded in the shallow
subsurface to analyze and assess the effects of near-surface scattering mechanisms
on recorded seismic waveforms. The perturbation method for elastic waves is used
to separate the scattered wavefield from the total wavefield based on a perturbation
of the wave equation with respect to medium parameters. The method decomposes
the medium parameters into background and perturbation parts, and allows us to
model scattering from arbitrary shape scatterers and the interaction of the multiplyscattered wavefield with the free surface.
Modeling elastic seismic data through:
(1) a baseline model (i.e., background) and (2) a monitor model (i.e., background
plus perturbation) is very common in the context of 4D seismic monitoring studies
43
(Greaves and Fulp, 1987; Pullin et al., 1987; Lumley, 1995).
However, the focus of
this study is different as we are looking at elastic wave scattering phenomena due to
near-surface heterogeneities instead of reflected phase and amplitude changes due to
time variant changes in reservoir conditions. In this study, we carry out extensive
calculations to study the effects of the acquisition geometry (e.g., source and receiver
depths), the quality factor of the background medium, and the elastic properties
of shallow subsurface scatterers (e.g., size, depth, and impedance contrast) on the
near-surface scattered wavefield.
2.2
Modeling of Elastic Wave Propagation and Scattering with Near-Surface Heterogeneities
In this section, we present the mathematical approach to explain elastic wave scattering using the perturbation method.
The general wave equation for the elastic
isotropic medium is
pi - (A + 2p)V(V - u) + pV x (V x u) = f,
(2.1)
where u is the displacement vector wavefield, f is the body force term, and the
medium is described by three parameters: the Lame constants A(x) and 11(x), and
density p(x). Seismic P and S wave velocities are cp = V(A + 2p)/p and c, = fy/p.
The perturbation theory decomposes the medium parameters into background and
perturbation parts
pWx)
Po + Sp(x)
A(x) = Ao + 6A(x)
(2.2)
pWx) = po +8P(x).
We denote by 6 and subscript 0 the perturbed and background (reference) medium
parameters, respectively.
The wavefield in the background medium is uo, and it
44
satisfies the elastic wave equation
Podo - (AO + 2po)V(V . uo) + poV x (V x uo) = f.
(2.3)
We consider the total wavefield u in the heterogeneous medium as two parts: the
incident wavefield uO in the background medium, which is the wavefield in the absence
of scatterers; and the scattered wavefield ui, which is the difference between the total
and incident wavefields:
U = U - U0.
(2.4)
The definition of the perturbation quantities leads to the derivation of a wave equation
for the scattered wavefield di. By subtracting equation (2.3) from (2.1) we obtain
poii-(Ao+2po)V(V - i)+poV x (V x d)
-
[6pii - (6A + 26p)V(V - u) + 6pV x (V x u)].
(2.5)
The left-hand side of equation (2.5) describes wavefield scattering in the background
medium (i.e., reference medium parameters) that includes multiple scattering waves.
The right-hand side is equivalent to an elastic source term that depends on the perturbations of the medium parameters, and the Green's function of the heterogeneous
medium.
Solving for the scattered wavefield dJ can be achieved by either solving
equation (2.5) directly based on the perturbation method (Wu, 1989) or by solving
equations (2.1) and (2.3) independently and then subtracting the incident from the
total wavefield. In this paper, we follow the latter approach.
For numerical modeling, we use a 2D Cartesian system with the horizontal positive
x-axis pointing to the right and the positive vertical z-axis pointing down.
The
basic governing equations (i.e., system of first order PDE) that describe elastic wave
propagation in the velocity-stress formulation (Virieux, 1986) are:
p
OV
&axx
&aZz
+
ax
0z'
at
p
at
-
ox
45
+ a-4 .
(2.6)
The constitutive laws for an isotropic medium are:
at
zz
(A+2;)
O+AOz
(A + 2p)
at
+ A
Oz
=0XpOV +
at
az
v
(2.7)
ax
V
Ox)'
where v and vz are the velocity components, og are the stresses, A and /- are the Lame
parameters, and p is density. The system of equations (2.6) and (2.7) is discretized
and solved numerically using finite difference schemes. The finite difference scheme
used in the numerical simulation, can also handle viscoelastic materials by using the
E-K model (Emmerich and Korn, 1987) to include attenuation defined by
We assume
2.3
Q
Q
values.
is constant with frequency.
Applications to the Earth Model with NearSurface Heterogeneities
To study the effects of near-surface heterogeneities on the recorded waveforms, we
consider a simple earth model with a single layer over half a space and two circular
scatterers embedded in the shallow layer (Figure 2.3). The two scatterers are located
at (x, z) = (360 m, 15 m) and (x, z) = (720 m, 15 m), each has a 20 m diameter
and an impedance contrast corresponding to 0.36. The P wave, S wave and density
values of the first layer are 1800 m/s, 1000 m/s and 1750 kg/M 3 and for the half-space
and scatterers are 3000 m/s, 1500 m/s and 2250 kg/m 3 , respectively. The domain
has NX = 1001 and N, = 501 grid points with 1 m grid spacing (i.e., Ax and Az),
that is, 500 m depth (along the z-axis) and 1000 m distance (along the x-axis). The
time step is 0.2 ms. An explosive point source is used with a Ricker wavelet and 30
Hz dominant frequency (~
75 Hz maximum frequency). The source is located at (x,
z) = (150 m, 10 m). The receivers are located on the surface with 50 m near-offset
and 5 m space intervals. In this paper, we consider only the vertical component of
46
the particle velocity field (v,). The scatterers are treated in the numerical scheme as
a density and velocity perturbation. The grid size of the model is small enough to
capture the shape of the scatterers. To avoid spurious numerical diffractions caused
by material discontinuity due to the spatial grid, arithmetic and harmonic averages
(smoothing) (Moczo et al., 2002) are applied to the density and elastic constants at
each grid point.
Snapshots of the total and scattered wavefields that are governed by equations
(2.1) and (2.5), respectively, are shown in Figure 2.4. Note that in this figure we do
not show the scattered surface-to-surface waves in the scattered wavefield because they
are much larger in amplitude compared to the scattered body-to-surface waves. The
removal of the direct surface waves is achieved by first computing the wavefield for a
homogeneous full-space with and without the scatterers. Then, we subtract the direct
surface waves from the incident and total wavefields to look only at scattered body
waves. The upcoming body P and S wave reflections, including multiples, impinge
on the near-surface heterogeneities and scatter to weak P and S waves, acting as
secondary sources. Because the scatterers, which are at 15 m depth, are shallower
than 1/3 of the wavelength (A = 60 m), the body wave reflections (incident wavefield)
scatter to strong surface waves.
These wave features are also shown in the shot
gathers in Figure 2.5. The scattered surface waves are comparable in amplitude to
the reflected signal. A few of these scatterers that are close to the free surface could
mask the primary reflections by the scattered body-to-surface waves.
We also model a vertical source placed at the surface, which represents a more
realistic vibrator-type field acquisition (Figure 2.6).
We observe strong amplitudes
of the shear wave reflection and refraction at mid- and far-offset traces due to the
radiation pattern of the vertical source. In all the cases we study in this paper (except
for the source and receiver depth analysis), we consider an explosive point source at
10 m depth to minimize surface wave energy relative to body wave reflections (see
figure summary in Table 2.1).
47
0
-
0
*
100
E 200
o300
400
500
0
200
400
600
Distance (m)
800
1000
Figure 2.3: Synthetic earth model. A single layer over half a space with two circular
scatterers embedded in the shallow layer. The two scatterers are located at (x, z)
= (360 m, 15 m) and (x, z) = (720 m, 15 m). Each has a 20 m diameter and an
impedance contrast corresponding to 0.36. The P wave, S wave and density values
of the first layer are 1800 m/s, 1000 m/s and 1750 kg/m 3 and for the half-space and
scatterers are 3000 m/s, 1500 m/s and 2250 kg/m 3 , respectively. The source is located
at (x, z) = (150 m, 10 m) as indicated by the red star. The receivers are located on
the surface with 50 m near-offset and 5 m space intervals.
Figure
2.7
2.8
2.9
2.11
2.12
2.13
2.14
2.18
Varying Model Parameter
Source frequency
Source depth
Receiver depth
Contrast of scatterers
Size of scatterers
Depth of scatterers
Quality factor
Interface scattering
Table 2.1: Summary of all the cases studied and their corresponding figure numbers.
48
Time = 300 ms
N
0
100
200
300
400
500
600
700
800
900
1000
X (m)
(a)
Time = 500 ms
500
0
100
200
300
400
500
600
700
800
900
1000
X (in)
(b)
Time = 500 ms
N
500
600
700
800
900
1000
X (m)
(c)
Figure 2.4: Snapshots of the total (u) and scattered (t') wavefields for the model in
Figure 2.3: (a) the total field at 300 ms, (b) total field at 500 ms, and (c) the scattered
field at 500 ms. The source of scattering is reflected or refracted body waves. The
scatterers excite primary, shear, and also surface waves due to the proximity to the
free surface. The source is indicated by the black circle. Note that we do not show
the scattered surface-to-surface waves in the scattered wavefield because it is much
larger in amplitude compared to the scattered body-to-surface waves.
49
with Scattering
No Scattering
The Difference
0.1
0.2
0.A
0.4
0.4
(a
02 0.5
E
E
-
0.6
0.7
0.8
0.9
-4
-2
Offset (m)
Offset (m)
Offset (m)
2
0
4
-4
-2
2
0
4
-4
-2
0
2
4
Amplitude
Amplitude
Amplitude
(a)
with Scattering
No Scattering
The Difference
0.1
0.2
0.3
0.4
0.2
02Ao
0.
E
0.5
0.6
0."
0.7
0.A
0.6
0.9
200
400
600
200
800
Offset (m)
-1.5
-1
-0.5
0
0.5
Amplitude
400
600
800
200
Offset (m)
1
1.5
-1.5
-1
-0.5
0
0.5
Amplitude
400
600
800
Offset (m)
1
1.5
-15
-1
-0.5
0
0.5
1
1.5
Amplitude
(b)
Figure 2.5: Finite difference simulations (v-component) showing the scattering effects
for the model in Figure 2.3; (a) shows the results including the direct surface wave and
(b) with the direct surface wave removed; (left) incident wavefield simulated using
the model without scatterers, (middle) total wavefield simulated using the model with
scatterers, and (right) scattered wavefield (i.e., the difference between the total and
incident wavefields). An explosive point source with 30 Hz Ricker wavelet is used.
The source is located at 10 m depth and the receivers are located on the surface.
Note the complexity due to scattering of the reflected arrivals.
50
No Scattering
with Scattering
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0)0.5
0.5
0.6
0.6
0.7
0.7
0.7
08
08
0.8
09
09
0.9
1
11
C .5
E
F
The Difference
EE
F
0.6
600
400
200
800
200
Offset (M)
-1.5
-1
-0.5
0
400
600
800
200
Offset (m)
0.5
1
1.5
Amplitude
-15
-1
-0.5
0
400
Offset (m)
0.5
1
1.5
-1.5
Amplitude
-1
-0.5
0
600
800
05
1
1.5
Amplitude
(a)
No Scattering
E
with Scattering
The Difference
0.1
01
0.1
0.2
0.2
0.2
0.3
0.3
0.
0.4
0.4
0.4
0.5 0.
E
E
0.6
0.6
0.7
0.7
080.8
08
0.9
0.9
11
200
600
400
800
200
Offset (m)
-1.5
0.5
-1
-0.5
0
0.5
Amplitude
400
1
1.5
-1,5
-1
-0.5
0
1
800
600
Offset
200
(m)
0.5
Amplitude
600
400
800
Offset (m)
1
1.5
-1.5
-1
-0.5
0
0.5
1
1.5
Amplitude
(b)
Figure 2.6: Simulated waveforms for the model in Figure 2.3 with vertical source
and receivers on the surface: (a) v,-component, and (b) v,-component.
The incident,
total, and scattered wavefields are shown from left to right, respectively. Note that
the direct surface wave is removed. Also note the strong amplitudes of the shear wave
reflection and refraction as indicated by the yellow circles at mid- and far-offset traces
(of the v,-component) due to the radiation pattern of the vertical source.
51
2.3.1
Effect of Source Frequency and Source and Receiver
Depths
The seismic source frequency and source and receiver depths have significant impact
on the recorded waveforms, especially on the strength of the surface wave energy.
To examine the effects of these factors, we simulate synthetic seismograms for the
earth model in Figure 2.3 with different source frequencies (20 Hz, 30 Hz, and 40
Hz), source depths (10 m, 20 m, and 40 m), and receiver depths (0 m, 20 m, and 40
m) as shown in Figures 2.7, 2.8, and 2.9, respectively.
The excitation of surface waves depends on the source depth and frequency. Direct
surface wave energy decreases with increasing source depth and increasing frequency
at a given depth. At a depth of 10 m, sources with 20 Hz frequency (Ap = 90 m)
excite stronger surface waves (Figure 2.7) than those with 40 Hz frequency (AP = 45
m).
For sources with 30 Hz dominant frequency (AP = 60 m), the excited surface
wave energy is strong for shallow sources at 10 m depth (Figure 2.8), whereas it is
much weaker for deeper sources at 40 m (> Ap/3).
To quantitatively assess the influence of near-surface heterogeneities, we assume
that scattered waves are noise and calculate the signal-to-noise ratio (SNR) in decibels
(dB)
Z0ji
j2
N 1IM
SNR(dB) = 10log1o
N
$=I
Z[s ZU
Z=1
-O (,
(a (i, )
-
*)
UO (i,
2
(2.8)
2))
where uo (i, J) are the sample values considered to be unaffected by noise (i.e.,
the incident wavefield propagated using the model with homogeneous near-surface
layers), u (i, j) are the data affected by noise (i.e., the total wavefield propagated
using the model with near-surface heterogeneity), and Al and N are the number of
traces and time samples, respectively.
We study the effects of source and receiver depths on the SNR due to near-surface
heterogeneities. The synthetic seismograms as functions of source and receiver depths
are shown in Figures 2.8 and 2.9. The corresponding signal-to-noise ratios are shown
in Figure 2.10. Seismic sources deployed at depth can minimize the amount of propagating direct surface wave energy and, therefore, improve the SNR in the seismic
52
No Scatterina (20Hz)
with Scatterina (20Hz)
The Difference (20Hz)
0.1
0.2
0.3
0.5
0) 0 5
E
0.6
0.6
0.6
0.7
0.7
0.7
0.8
0.8
0.9
Offset (i)
0
-2
-4
Offset (i)
Offset (m)
4
2
4
2
0
-2
-4
-4
0
-2
4
2
Amplitude
Amplitude
Amplitude
(a)
with Scattering (40Hz)
No Scattering (40Hz)
The Difference (40Hz)
0.1
01
0.1
0.2
0.
0"
O.A
E
E,)
0.!
0,
EO
F-
0.
100
-4
200
-2
300
400
Offset (m)
0
Amplitude
500
2
600
100
4
-4
200
400
300
Offset (m)
0
-2
Amplitude
500
2
1UU
OUW
4
-4
zW jW
4W
Offset (m)
-2
UU
0
W V
2
4
Amplitude
(b)
Figure 2.7: Simulated waveforms (v,-component) for the model in Figure 2.3. An
explosive point source is used with (a) 20Hz and (b) 40Hz dominant frequencies. The
incident, total, and scattered wavefields are shown from left to right, respectively.
53
30Hz SRC at 10m
30Hz SRC at 20m
0.1
30Hz SRC at 40m
0.1
0.1
0.2
0.2
0.4
0.4
(D 0.!
E
F- 0.1
E 0.4
01=
0.1
0.7
01
100
200
300
400
Offset (m)
-4
-2
0
500
600
100
2
4
-4
200
300
-2
Amplitude
400
Offset (m)
0
500
600
2
Offset (m)
4
-4
-2
0
Amplitude
2
4
Amplitude
(a)
30Hz SRC at 1Om
30Hz SRC at 20m
30Hz SRC at 40m
0.1
0.1
0.2
0.
0.,
0.4
0.1
0.5
0.1
0.1
0.6
0.1
0.1
0.7
01
0.8
E
E
0.9
Iw
-1.5
-1
zUUw JUU 4Uw
Offset (m)
-0.5
0
0.5
Amplitude
Iw
DW D
1
4Wu
1.5
-1.5
-1
4W
oW ow
Offset (m)
-0.5
0
0.5
Amplitude
]U
aw
_' w
'%W
Offset (m)
1
1.5
-1.5
-1
-0.5
0
0.5
DW CvUUo
1
1.5
Amplitude
(b)
Figure 2.8: Finite difference simulations (v,-component) for the scattering model with
different source depths (10 m, 20 m, and 40 m from left to right): (a) including the
direct surface wave, and (b) with the direct surface wave removed. An explosive point
source with 30 Hz Ricker wavelet is used. The receivers are located on the surface.
54
30Hz RCV at Om
30Hz RCV at 20m
30Hz RCV at 40m
0.1
0.1
0.2
0.2
0.3
0.3
0.4
E
( 0
0.5
0.5
0,6
0.7
0.7
0.
0.8
0.
0.9
IU w
Offset (m)
-4
-2
0
2
4
-4
0uU qw
UV
Offset (m)
-2
0
Amplitude
VUuOffset (m)
ow Quu
2
euu
4
-4
13W
-2
VUu
0
Amplitude
_uvIW
2
4
Amplitude
(a)
30Hz RCV at 20m
30Hz RCV at Om
30Hz RCV at 40m
0.1
01
0.2
0.2
0.3
0.4
E
F-
0,4
0.E
(D o.5
Eo
E
0.6
0.7
0.,
08
08
0.8
0.9
0.9
0.9
100
200
300
400
500
600
100
200
Offset (m)
-1.5
-1
-0.5
0
0.5
Amplitude
400
300
500
600
100
200
Offset (m)
1
1.5
-1.5
-1
-0.5
0
0.5
Amplitude
300
400
500
600
Offset (m)
1
1.5
-1.5
-1
-0.5
0
0.5
1
1.5
Amplitude
(b)
Figure 2.9: Finite difference simulations (vz-component) for the scattering model with
different receiver depths (0 m, 20 m, and 40 m from left to right): (a) including the
direct surface wave, and (b) with the direct surface wave removed. A vertical source
with 30 Hz Ricker wavelet is located at the surface. Note the strong amplitudes of
the shear wave reflection and refraction as indicated by the yellow circles at mid- and
far-offset traces due to the radiation pattern of the vertical source. At 40 m receiver
depth most of scattered waves appear to be body waves. Note also, reflections are not
as prominent as they are for surface receivers. At the surface the amplitude doubles.
At depth, upgoing and downgoing waves interfere.
55
records as the amplitude of surface waves decays exponentially with depth (Figure
2.10a). However, source depths have no effects on the scattered body-to-surface waves,
mainly because scattered waves are excited by the near-surface heterogeneities and
are independent of the seismic source depth (Figure 2.10b). The maximum at ~ 10
m source depth in Figure 2.10b is most likely related to the constructive/destructive
interference between the primary and ghost reflections from the free surface. However, the change in the SNR is very small compared to the other three cases. On
the other hand, deploying receivers at depth (Figure 2.9) can improve the SNR as
they record less of both the direct and scattered surface waves (Figure 2.10c, d). A
recent field data study by Bakulin et al. (2012) agrees with our numerical results and
demonstrates the SNR improvement due to deploying the sources and receivers at
depth. Bakulin et al. (2012) also looked into using dual sensor data (geophones and
hydrophones) to reduce ghost reflections from the free surface to further improve the
stacked section.
2.3.2
Effects of Scatterers' Depth, Size, Impedance Contrast,
and Attenuation
As discussed in the previous section, the seismic source wavelength and source and
receiver depths have great effects on the recorded signal. Nevertheless, the characteristics of near-surface scatterers (e.g., impedance contrast, depth, size, and attenuation
factor of the background medium) have similar, if not even greater, effects. These
characteristics have direct impact on the phase and amplitude of scattered surface
waves and body waves.
Recorded waveforms simulated using models similar to the ones shown in Figure
2.3 with varying scatterer impedances (Table 3.2) corresponding to reflection coefficients (0.16, 0.27, and 0.36), depths (15 m, 30 m, and 45 m), diameters (10 m, 20
m, and 40 m), and attenuation factor
Q
of the background medium (30, 60, 100,
and 200) are shown in Figures 2.11 - 2.14. We show all the figures with the same
amplitude scale for ease of comparison.
The impedance contrast is calculated as
56
Source Analysis without Direct Surface Wave
Source Analysis with Direct Surface Wave
5.5
5.
5
5
4.5
4.5
4
4
3.5
3.5
3
3
z
11*'
2.5
z
CO)
2.5
C')
1.5
2
1.5
1
1
0.5
0.5
0
5
10
15
20
25
30
35
40
45
0
5
10
15
Depth (m)
25
30
35
40
45
(b)
(a)
Receiver Analysis with Direct Surface Wave
Receiver Analysis without Direct Surface Wave
5.5
5.1
5
5
4.5
4.5
4
4
3.5
3.5
3
3
zc:2.5
2
S2.5
CO)
:
20
Depth (m)
eox
1.5
1.5
1
1
0.5
0.5
0
5
10
15
20
25
30
35
40
r
0
45
5
10
Depth (m)
15
20
25
30
35
40
45
Depth (m)
(d)
(c)
Figure 2.10: The effects of source and receiver depths on the SNR due to near-surface
heterogeneities: (a-b) source analysis, and (c-d) receiver analysis. Note that sources
at deeper depths generate less surface wave energy and therefore improve the SNR as
shown in (a), but source depth has no effect on the scattered body-to-surface waves
as shown in (b). Receivers at deeper depths, however, improve the SNR in both cases:
the surface waves (c) and scattered body waves to surface waves (d).
Vp (m/s)
Vs (m/s)
Density (kg/m)
Impedance Contrast
Layer I
Layer II
1800
3000
1000
1500
1750
2250
0.36
Scatterers a
Scatterers b
2400
2700
1200
1350
1800
2025
0.16
0.27
Scatterers c
3000
1500
2250
0.36
Table 2.2: Material properties for models with different contrasts. The impedance
contrasts are calculated for different material properties relative to layer 1.
57
Ro = (Z 2 - Z1 )/(Z2 + Z1 ), where Z1 and Z2 are the impedances (i.e., velocity times
density) of the top layer and the scatterers. In all the cases we study in this and
remaining sections, we use an explosive point source at 10 m depth as the standard
source.
The effects of the scatterer characteristics are demonstrated by showing the total
wavefields, which include incident, multiple scattering, and mode-converted waves.
We quantitatively assess the effects of source and receiver depths and characteristics
of near-surface heterogeneities, such as material properties, depths and sizes, by calculating the signal-to-noise ratios. The aim is to understand when the scattered waves
have significant impact on the quality of the recorded data. As discussed previously,
the scattered energy increases with increasing impedance contrast, increasing size
of the scatterers relative to the source wavelength, decreasing depth, and increasing
attenuation factor of the background medium.
In the first case shown in Figure 2.11, we vary the impedance contrast by changing
both the velocity and density of the scatterers, while keeping the properties of the
embedding layer constant. The simulations demonstrate that the strength of the
scattered energy increases with increasing the impedance contrast of the scatterers.
This is explained mathematically by equation (2.5) in which the right-hand side is
equivalent to an elastic source that depends on the material property perturbations.
The frequency of scattered body-to-surface waves depends on the frequency of
the total wavefield and the perturbations of the medium parameters. The wavefield
scattering amplitude is frequency dependent and, therefore, the size of the scatterers
relative to the wavelength is indeed a controlling factor for the scattered energy. The
dominant wavelength of the incident wavefield is 60 m and the minimum wavelength
is 24 m. We show the simulations for different scatterer sizes in Figure 2.12: 10 m,
20 m, and 40 m diameter. Scattered energy depends on the depth of the scatterers.
When changing the scatterer size, the top edge of the scatterers are kept at 5 m
depth from the free surface and the centers are located at 10 m, 15 m, and 25 m,
respectively. Thus, the frequency of scattered waves is either low or high depending
on whether the size of the scatterers is small or large relative to the wavelength of the
58
incident waves (Figures 2.12). Similar to increasing the impedance contrast, larger
scatterers cause more scattered energy that is low in frequency compared to small
scatterers.
The effects of attenuation are studied by using constant
Q
models and recalculat-
ing some models that were run with no attenuation. We included attenuation only
at the top layer of the model shown in Figure 2.3 with (Q = 100, 60, and 30). The
scatterers and the half-space are assumed to be perfectly elastic (Q = oc) materials.
The results in Figure 2.14 demonstrate that the scattered, and also the reflected,
wave amplitudes decrease due to attenuation.
The results of different simulations with varying properties are summarized in
Figures 2.15 and 2.16 as expressed by SNRs.
The SNR increases with decreasing
impedance contrast, decreasing size of the scatterers relative to the source wavelength,
increasing depth, and decreasing attenuation factor of the background medium. As
discussed in the previous section, deeper receivers improve the SNR as they record
weaker direct and scattered surface waves, whereas deeper sources improve the SNR
only because they excite weaker direct surface waves. The same relationships hold for
different impedance contrast (Figures 2.15a and 2.16a, e), sizes of scatterers (Figures
2.15b and 2.16b, f), and attenuation factors (Figures 2.15d and 2.16d, h).
Note,
however, that deeper source has no effect on the scattered body-to-surface waves as
indicated by the narrow range of SNR values for different source depths in Figure
2.15e-h. These relations hold only when the heterogeneities are shallow (e.g., 15 m
depth) and excite significant scattered surface wave energy. In the case where the
scatterers are close to the free surface, the scattered energy is dominated by bodyto-surface wave scattering (Figure 2.13). When the scatterers are deeper than 1/3 of
the wavelength (e.g., 20 in), weak or no scattered surface waves are generated and,
therefore, there is no SNR improvement due to deploying the receivers below the free
surface.
59
Ro=0.16 (3Hz Source at 10m)
Ro=0.27 (30Hz Source at 1 Om)
Ro=0.36 (30Hz Source at 1 Om)
0.1
0.3
0.3
0.4
0.4
CD o.E
CD0.5
E
E
()
E
0.6
0.7
0.8
0.9
]VUZLU
-1.5
-1
;UU 4LK) 0U
Offset (m)
-0.5
0
UU
100
200
300
400
600
500
100
200
300
Offset (m)
0.5
1
1.5
-1.5
-1
-0.5
Amplitude
0
500
400
600
Offset (m)
0.5
1
1.5
-1.5
-1
-0.5
Amplitude
0
0.5
1
1.5
Amplitude
(a)
Ro=0.16 (30Hz Source at 10m)
Ro=0.27 (30Hz Source at 10m)
Ro=0.36 (30Hz Source at 1 Om)
0.1
0.2
0.3
0.4
0.4
0.5
E
E
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
200
100
300
400
500
600
Cw
'1
Offset (m)
-1.5
-1
-0.5
0
0.5
Amplitude
1
1.5
-1.5
-1
W W
Offset (m)
owu
+u
-0.5
0
0.5
Amplitude
100
L
UU
200
4UU
600
500
Offset (m)
1
1.5
-1.5
-1
-0.5
0
0.5
1
1.5
Amplitude
(b)
Figure 2.11: Simulated waveforms (vz-component) for the scattering model with different impedance contrasts (0.16, 0.27, and 0.36) from left to right. The source (30
Hz) is located at 10 m depth and the receivers are located on the surface. (a) The total
wavefield simulated using the model with scatterers, and (b) the scattered wavefield
(i.e., the difference between the total and incident wavefields).
60
Size=1 Om (3Hz Source at 1 Oml
Size=20m (30Hz Source at 1 Om)
Size=40m (30H-z Source at 1 Om)
0.1
0.1
0."
O.f
Eo0.
EO.
0.E
0.7
0.1
0.0
0.1
-1.5
-1
-0.5
0
0.5
JVwOffset (m)
dVV
Offset (m)
Offset (m)
1
1.5
-1.5
-1
-0.5
0
0.5
1
1.5
-1.5
-1
-0.5
Amplitude
Amplitude
'+V
11W
0
owu
atm
0.5
1
1.5
Amplitude
(a)
Size=20m (30Hz Source at 10m)
Size=10m (30Hz Source at 10m)
0.1
0.1
0.2
0.2
0.7
0.3
Size=40m (30Hz Source at 10m)
0.4
0.5
(o.6
E)
4) 0.5
0.E
0.6
0.7
0.7
0.8
0.8
0.9
0.9
Uttset
-1.5
-1
-0.5
0
100
20
(m)
0.5
Amplitude
1
1.5
-1.5
-1
300
400
Offset (m)
-0.5
0
0.5
Amplitude
600
500
1
100
1.5
-1.5
200
-1
300
400
Offset (m)
-0.5
0
0.5
500
600
1
1,5
Amplitude
(b)
Figure 2.12: Simulated waveforms (v,-component) for the scattering model with different scatterer sizes (10 m, 20 m, and 40 m diameter) from left to right, with the
center of the scatterers at 10 m, 15 m, and 25 m depth, respectively. Top of the
scatterers is 5 m depth below the free surface. The source (30 Hz) is located at 10 m
depth and the receivers are located on the surface. (a) The total wavefield simulated
using the model with scatterers, and (b) the scattered wavefield (i.e., the difference
between the total and incident wavefields).
61
Depth=30m (30Hz Source at 1Om)
Depth=15m (3Hz Source at 10m)
Depth=45m (3OHz Source at 1 Om)
0.:
0.'
0.5
) 0.!
E=
E
1=
100
-1.5
200
-1
300
400
Offset (m)
0
-0.5
500
600
100
200
300
400
600
500
Offset (m)
1
0.5
Amplitude
1.5
-1.5
-1
0
-0.5
Offset (m)
0.5
1
1.5
Amplitude
-1.5
-1
-0.5
0
0.5
1
1.5
Amplitude
(a)
Depth=15m (30Hz Source at 1Gm)
Depth=30m (30Hz Source at 1Gm)
Depth=45m (3GHz Source at 1Gm)
0.1
0.2
0.
0.4
0.
E,
E
0.5
a)-o
0A
100
200
300
400
600
500
100
200
Offset (m)
-1.5
-1
-0.5
0
0.5
Amplitude
1
1.5
-1.5
-1
300
400
Offset (m)
-0.5
0
0.5
Amplitude
500
600
Offset (m)
1
1.5
-1.5
-1
-0.5
0
0.5
1
1.5
Amplitude
(b)
Figure 2.13: Simulated waveforms (v,-component) for the scattering model with different scatterer depths (15 m, 30 m, and 45 m) from left to right. The source (30 Hz)
is located at 10 m depth and the receivers are located on the surface. (a) The total
wavefield simulated using the model with scatterers, and (b) the scattered wavefield
(i.e., the difference between the total and incident wavefields).
62
Q=60 (30Hz Source at
0=100 (30Hz Source at 10m)
Q=30 (30Hz Source at
10m)
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.4
0.5
1Oim)
0.5
E
0.6
0.6
0.7
0.7
0.8
0.9
100
200
300
400
500
600
100
200
Offset (m)
-1.5
-1
-0.5
0
300
400
500
600
Offset (m)
0.5
1
1.5
-1.5
-1
-0.5
Amplitude
0
Offset (m)
0.5
1
1.5
-1.5
-1
-0.5
Amplitude
0
0.5
1
1.5
Amplitude
(a)
Q=100 (30Hz
R
Q=60 (30Hz Source at 10m)
Source at 1Om)
Q=30 (30Hz Source at 10m)
0.1
0.;
0.2
0.
0.2
0.2
0.
E
0.,
0.
E
0)
0)
E
,
0.6
0.6
0.6
0,7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
0.9
100
200
300
400
500
1 100
600
200
Offset (m)
-1.5
-1
-05
0
0.5
Amplitude
300
400
500
1
600
100
200
Offset (m)
1
1.5
-1.5
-1
-0.5
0
0.5
Amplitude
300
400
600
500
Offset (m)
1
1.5
-15
-1
-0.5
0
0.5
1
1.5
Amplitude
(b)
Figure 2.14: Simulated waveforms (vs-component) for the scattering model with different attenuation factors in the top layer (Q =100, 60, and 30) from left to right.
The source (30 Hz) is located at 10 m depth and the receivers are located on the
surface. (a) The total wavefield simulated using the model with scatterers, and (b)
the scattered wavefield (i.e., the difference between the total and incident wavefields).
63
(e)
(a)
1
10
1#
I-
C1
cc
z
cc
z
5
5
I
U)
CO)
0
0
0.2
0.25
0.3
0.2
0.35
Scatterer Impedance
(b)
z
0.3
0.35
10
104
C
0.25
Scatterer Impedance
(f)
5
5
Z
Uz
It
0
0
10
10
40
30
20
20
40
30
Scatterer Size (m)
(g)
Scatterer Size (m)
(c)
25
10 [
20
15
-o 11
cc
10
C,,
0
0
20
30
20
40
30
40
Scatterer Depth (m)
(h)
Scatterer Depth (m)
(d)
10
19
V
5
cc
C,,
0
L
5
0
50
100
150
200
50
Quality Factor
-I-
SR at 10m
0
100
150
200
Quality Factor
A
SR at 20m
SR at 30m
:SR
at 40m I
Figure 2.15: The effects of source depths on the SNR due to characteristics of nearsurface heterogeneities (impedance contrast, depth, size, and quality factor): (a-d)
including the direct surface waves, and (e-h) with the direct surface waves removed.
64
(a)
(e)
101
10
1
t3
5
z
U)
0
0
0.2
0.25
0.3
0.35
0.2
Scatterer Impedance
(b)
0.25
0.3
0.35
Scatterer Impedance
(f)
10
1Qy
M
a:
z
5
U)
C,,
0
10
20
30
10
40
20
Scatterer Size (m)
(c)
10
20
cc
z
40
I
25
M
30
Scatterer Size (m)
(g)
15
V
10
(n)
U)
20
30
40
20
Scatterer Depth (m)
(d)
30
40
Scatterer Depth (m)
(h)
101
10
rV
5
a:
z
U)
5
(n
0
0
50
100
150
200
50
Quality Factor
--
RC at 10m--
100
150
200
Quality Factor
A
RC at 20m
RC at 30m -- *-
RC at 40m
Figure 2.16: The effects of receiver depths on the SNR due to characteristics of nearsurface heterogeneities (impedance contrast, depth, size, and quality factor): (a-d)
including the direct surface waves, and (e-h) with the direct surface waves removed.
65
2.4
Scattering due to Bedrock Topography (Interface Scattering)
In the previous sections we showed the examples of scattering from isolated individual
scatterers. Bedrock topography (e.g., subsurface irregular interface) can also cause
scattering and could have pronounced effects on the quality of recorded waveforms.
The irregular interface not only causes time shifts (as assumed by static corrections)
but also causes complicated scattering. We model a case when the top of the interface
layer is not a plane but irregular.
We consider an earth model with an irregular (Gaussian) surface below a homogeneous surface layer, as shown in Figure 2.17. The irregular interface is modeled
using a set of uncorrelated random numbers drawn from a Gaussian distribution with
zero mean and a standard deviation of 15 m (RMS height). The generated random
numbers (surface) are then correlated by the use of a running average filter with a
Gaussian operator that has 5 m correlation length (Ogilvy and Merklinger, 1991).
The corresponding material properties are given in Table 2.3.
An explosive point
source at 10 m depth is used with a Ricker wavelet and 30 Hz central frequency. The
receivers are located on the surface with 50 m near offset and 5 m space intervals.
Simulated waveforms recorded at surface for an irregular interface at 15 m and 45 m
depths are shown in Figure 2.18.
The influence of the irregular interface is clearly demonstrated, as it acts as a
continuous line of sources that adds to the complexity of the recorded waveforms,
compared to the localized scatterers discussed in the previous section.
Scattering
from irregular interface exhibits more diffusive-type scattering compared to individual
scatterers. The wavelength of Rayleigh waves propagating along the free surface is
(AR
-
930 m s-1
/
30 Hz = 31 m). At 15 m interface depth, we observe a strong
surface wave dispersion due to the thin layer (Figure 2.18a-c). Because the surface
wave amplitude at a depth deeper than one third of the wavelength is very small, both
scattering and dispersion of direct surface waves are very minimal for the interface at
45 m depth (Figure 2.18d-f).
66
15
10
5
-c
0
-5
-10
- .15r
0
100
200
300
400
500
600
Distance (m)
700
800
1000
900
(a)
0[
Laver 1
*
I
100
200
-C
3300
400
500
0
200
600
400
Distance (m)
800
1000
(b)
Figure 2.17: An earth model with near-surface irregular (Gaussian) interface and
deeper flat reflector: (a) Gaussian surface profile, and (b) the earth model. Material
properties are given in Table 2.3.
The irregular interface also causes the up-going reflections and refracted waves to
scatter to P and S waves. Because the irregular interface is shallow, up-going body
waves and refracted waves, which travel along the irregular interface boundary, scatter to surface waves that can mask the data entirely. The energy of scattered surface
waves decreases as the depth of the irregular interface increases, mainly because the
interface irregularities act as a source of scattered waves. The scattered energy is
dominated by body-to-body waves (i.e., relatively small amplitude)'for deep scatterers. However, scattering of reflected and refracted body waves to surface waves (i.e.,
relatively large amplitude waves) is dominated in the case of the shallow irregular
interface.
67
0.1
0.
0.4
0.
0.5
E
0.!
E
0.
0.6
0.7
0.;
Offset (M)
Offset (m)
-5
0
-5
5
Amplitude
(a)
0
Amplitude
Offset (m)
5
-5
0
5
Amplitude
(b)
(c)
0.1
E
F-
0.
E
016
0.7
0.6
0.9
-5
0
Amplitude
(d)
200
Offset (m)
Offset (m)
5
-5
0
Amplitude
(e)
5
400
Offset
-5
600
800
(m)
0
5
Amplitude
(f)
Figure 2.18: Finite difference simulations (v,-component) for the irregular (Gaussian)
interface at different depths: (a-c) 15 m, and (d-f) 45 m. The incident wavefield (a and
d) simulated using the model with plane shallow interface; (b and e) total wavefield
simulated using the model with Gaussian shallow interface; and (c and f) scattered
wavefield (i.e., the difference between the total and incident wavefields). Note the
strong dispersive character of the surface wave due to the thin layer (a-c). Also, note
that the amplitudes of scattered (reflected and refracted) body waves to surface waves
decrease rapidly as the interface depth increases. Overall, scattering from irregular
near-surface interface is more complex and exhibits more diffusive-type scattering
compared to localized scatterers.
68
Layer no.
Vp (m/s)
Vs (m/s)
I
II
III
1800
3000
5000
1000
1500
2250
Density (kg/m
3
)
1750
2250
2750
Table 2.3: Material properties (P wave velocity, S wave velocity, and density) of the
model shown in Figure 2.17.
2.5
Discussion and Conclusion
In this paper, we present a numerical approach based on the perturbation method and
finite-difference forward modeling for simulating the effects of seismic wave scattering
from arbitrary-shaped, shallow, subsurface heterogeneities. The scattered wavefield,
due to the near-surface scatterers only, is modeled by taking the difference between
the incident and total wavefields. We show analytically and numerically that the
scatterers act as secondary sources for the scattered elastic wavefield. The numerical
results show that scattering of upgoing reflections by the heterogeneities to surface
waves can obscure weak primary reflections and contaminate the entire data set. We
carried out extensive numerical experiments to study the effects of scattered surface
waves on SNR.
The results show that the scattered energy depends strongly on the properties of
the shallow scatterers and increases with increasing impedance contrast, increasing
size of the scatterers relative to the incident wavelength, decreasing depth of scatterers, and increasing the attenuation factor of the background medium. Additionally,
sources deployed at depths below one-third of the wavelength excite weak surface
waves and, therefore, improve the SNR due to the reduced surface-wave scattering.
However, source depth does not affect the scattering of reflected body waves. On the
other hand, receivers deployed at depth improve the SNR as they record weak surface
and scattered body-to-surface waves.
In addition to showing the effects of volume scatterers, we also examine the effects
of scattering from a near-surface irregular interface or bedrock topography. Similar to
scattering from near-surface inclusions, the energy of scattered body-to-surface waves
decreases as the depth of the irregular interface increases. The irregular interface acts
69
as a continuous line of sources for scattered (reflected and refracted) body waves to
surface waves, and therefore, the scattered amplitudes decrease as the depth to the
interface increases. Compared to scattering from finite scatterers, scattering from an
irregular interface exhibits more diffusive-type scattering.
The analysis and quantified results help explain the scattering mechanisms and,
therefore, could lead to developing new acquisition and processing techniques to reduce the noise and enhance the quality of the subsurface image. For computational
efficiencies, we consider only 2D models, but the same method can be applied to 3D
modeling. In 3D, however, the computational cost will be much larger, but with current advances in multicore parallel programming and the existence of large clusters,
computations with tens of billions of cells are feasible.
Acknowledgments
We thank Saudi Aramco and ERL founding members for supporting this research. We
also thank Saudi Aramco for granting permission to show the field data example used
in this paper. Xander Campman and three anonymous reviewers provided comments
that helped to improve the manuscript, and we are grateful for their assistance.
70
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74
Chapter 3
Suppression of Near-surface
Scattered Body-to-Surface Waves:
A Steerable and Non-linear
Filtering Approach*
Abstract
We present an approach based on local-slope estimation for the separation of scattered surface waves from reflected body waves. The direct and scattered surface
waves contain a significant amount of seismic energy. They present great challenges
in land seismic data acquisition and processing, especially in arid regions with complex near-surface heterogeneities (e.g., dry river beds, wadis/large escarpments, and
karst features). The near-surface scattered body-to-surface waves, which have comparable amplitudes to reflections, can mask the seismic reflections. These difficulties,
added to large amplitude direct and back-scattered surface (Rayleigh) waves, create a
major reduction in signal-to-noise ratio and degrade the final subsurface image quality. Removal of these waves can be difficult using conventional filtering methods, such
as an f - k filter, without distorting the reflected signal. The filtering algorithm we
present is based on predicting the spatially varying slope of the noise, using steerable
filters, and separating the signal and noise components by applying a directional nonlinear filter oriented toward the noise direction to predict the noise and then subtract
it from the data. The slope estimation step using steerable filters is very efficient.
*The bulk of this chapter has been accepted for publication as: AlMuhaidib, A. M. and Toks6z,
M. N., Suppression of near-surface scattered body-to-surface waves: A steerable and non-linear
filtering approach, Geophysical Prospecting (accepted)
75
It requires only a linear combination of a set of basis filters at fixed orientation to
synthesize an image filtered at an arbitrary orientation. We apply our filtering approach to simulated and field seismic datasets to suppress the scattered surface waves
from reflected body-waves and demonstrate its superiority over conventional f - k
techniques in signal preservation and noise suppression.
3.1
Introduction
Seismic energy recorded in the field represents a true earth response that includes all
complicated wave types (i.e., multiple scattering, elastic mode-conversion, and viscoelastic attenuation). In land seismic data, near-surface complexity not only causes
time shifts (as assumed by static corrections) but also causes phase and amplitude distortions to the recorded signal. A significant fraction of the seismic energy is trapped
and scattered in the near-surface layers (in the form of coherent noise) and masks
the body wave reflections from deeper structures. The ultimate goal is to separate
the scattered surface (Rayleigh) wave energy and remove the effects of near-surface
heterogeneities as demonstrated in Figure 3.1.
In case of a surface seismic source, the main components of surface related noise
include: (1) direct surface waves; (2) forward and back-scattered surface waves; and
(3) body-to-surface scattered waves (Levander, 1990). The upcoming body wave reflections (e.g., P and S waves including primaries, multiples and mode conversions)
impinge on the near-surface heterogeneities and scatter to weak P and S waves, and
also to Rayleigh waves since the heterogeneities act as secondary elastic sources. The
scattered surface-to-body and body-to-body wave components, however, pose less
challenge as they are much weaker in amplitude and attenuate much faster with propagation distance than surface waves. In a previous study (AlMuhaidib and Toks6z,
2014), we demonstrated the scattering from near-surface heterogeneities using the
perturbation theory and finite difference modeling. We showed that the scattered
field is equivalent to the radiation field of an equivalent elastic source excited at the
scatterer locations. In conventional seismic data processing, these scattered body-tosurface waves are not usually removed. In this paper we label these waves as "noise",
76
b)
a)
Free-surfaceV
VVVVVVVV VVVVV
VVV\\
J\N\ N\A
Free-surfaceV
VV VVV VV VVV VVV
AI\\ A\A
Scattered
Reflections 4_-
Direct
Surface Waves
L
Reflections
Reflections
Figure 3.1: Schematic earth model showing how most of the seismic energy is trapped
and scattered in the near-surface layers: (a) scattering of direct surface waves and
upcoming body-waves to surface waves, and (b) the ideal model after removing the
effects of surface wave scattering.
since they cannot be accounted for in standard seismic imaging. However, they carry
information about the near-surface and therefore are considered "signal" in other
applications such as inverse scattering and surface wave inversion for near-surface
shear-wave velocity.
Surface waves, in general, are characterized by low frequency, linear moveout,
large amplitudes and slower amplitude decay with distance. The direct surface wave
is confined within a fan-shaped window in the time-space domain and is much larger in
amplitude and lower in frequency than body wave reflections. These characteristics of
direct surface waves make it effective to apply filtering techniques within this narrow
fan-shaped window. However, the scattered body-to-surface waves have comparable
amplitudes to reflections (especially in the cases of large, high-contrast and shallow
scatterers), frequencies dependent on the size of the heterogeneities, and they could
contaminate the entire dataset.
Several methods have been developed in the geophysical literature to filter and
attenuate source-generated noise based on its characteristics (e.g., frequency, velocity,
amplitude, and polarization). Comparisons between some filtering methods based on
their performance of attenuating different types of near-surface generated noise are
given in Table 3.1. Bandpass frequency filters remove ground rolls based on their low
77
frequency characteristics. However, removing low frequencies from the data may also
distort reflections (due to the overlap of noise and signal in the frequency domain)
and may affect subsequent quantitative interpretation and inversion schemes that
are mainly dependent on the low frequency component of the signal. Conventional
"global" velocity filtering methods, such as f - k and
T
- p can be very effective in
attenuating ground roll or scattered surface wave energy. However, they can distort
reflections and introduce ringing due to leakage in the transform domain. Local radial transform (Henley, 2003) requires window selection to filter the fan-shaped noise.
Directional filters based on local slant-stacking depend on picking maximum values of
stacking semblance (Neidell and Taner, 1971; Chiu and Butler, 1997), which can be
difficult to handle in the case of scattered body-to-surface waves that have comparable
amplitudes to the up-going reflections. Seismic interferometry (i.e., cross-correlation
of two receivers due to many sources exhibits a stationary point) can predict direct and
scattered surface waves (Dong et al., 2006; Xue et al., 2008), but it cannot predict
(isolate) scattered body-to-surface waves.
Methods based on principal component
analysis (PCA/SVD) (Liu, 1999) are computationally expensive, and model-based
inverse scattering schemes (Herman et al., 2000; Campman et al., 2005, 2006) imaged and suppressed near-receiver scattered surface waves assuming that scattering
takes place immediately under the receivers. Discrimination between surface waves
and body wave reflections using particle motion polarization is also used (Vidale,
1986), but requires multi-component data that is not often available.
Stack-array
approach during acquisition (Anstey, 1986; Morse and Hildebrandt, 1989; Regone,
1998; Ozbek, 2000) can be very effective in reducing scattered noise, but it also reduces the high frequency components of the signal due to intra-array statics and and
therefore decreases the image resolution (Baeten et al., 2000). Even though high fold
acquisition and common-mid-point (CMP) stacking are powerful in reducing random
noise, reducing scattered coherent noise and preserving the relative amplitude of the
signal are essential for amplitude critical processes in the pre-stack domain (Larner
et al., 1983) such as predictive deconvolution, velocity analysis, waveform inversion,
migration, and quantitative interpretation studies (e.g., amplitude variation with off-
78
Direct Surface
Waves
Back-scattered
Surface Waves
Scattered Body
to Surface Waves
-
k
Yes
Yes
Yes
T -
p
Yes
Yes
Yes
Radial-Trace
Yes
Yes
No
Interferometry
Yes
Yes
No
PCA/SVD
Yes
Yes
Yes
Inverse
Scattering
Particle
polarization
Yes
Yes
Yes
Stack-array
Yes
Yes
Yes
f
Remarks
Reflection
smearing
Reflection
smearing
Window
selection
Cannot isolate
scattered body
to surface waves
Computationally
expensive
Singlescattering
Multi-comp.
data
Reduce freq.
content
Table 3.1: Comparison among different methods for surface wave removal.
set and azimuth). In the following sections of the paper we first describe a filtering
algorithm for the separation of scattered surface waves from body wave reflections
and then show the results of its application to both simulated and field seismic data.
3.2
Noise Reduction by Spatially Varying Slope
Estimation
To overcome the drawbacks of most conventional methods, we propose a new filtering
framework that exploits the multidimensionality of the seismic data (e.g., temporal,
spatial, directional, and spectral variables) to obtain more reliable results of signal
and noise separation. The approach suppresses locally-linear scattered surface waves
by first estimating the dominant local-slopes as a function of offset using steerable
filters (Freeman and Adelson, 1991), and then applies a local non-linear median filter
oriented toward the noise direction. We assume that the slope of the noise is locally
linear corresponding to a velocity of about 0.9 times the shear velocity. It can vary
79
with offset due to lateral shear-wave velocity variations. The filter is applied to the
positive and negative slope directions to suppress the forward and backward scattered
waves, respectively. This approach does not depend on the strong amplitude contrast
of the coherent noise. The filter is narrow in the
f-
k domain (e.g., tackles a specific
slope instead of a range of slopes), which ensures minimal signal distortions and
obtains a focused subsurface image.
3.2.1
Steerable Filters
Oriented filters are used in many vision and image processing tasks such as edge
detection, texture analysis, segmentation, motion analysis, and image enhancement
and compression.
The steerable filter is one of a class of filters in which a linear
combination of a set of basis filters at fixed orientation is used to synthesize a filter of
an arbitrary orientation (Freeman and Adelson, 1991; Simoncelli and Freeman, 1995).
It is useful to examine filter outputs (images) as a function of phase and orientation,
efficiently, without the need to apply many versions of the same filter rotated at
different angles. The idea of the steerable filter can be simply illustrated using the
partial derivative of a 2D symmetric Gaussian filter
G(x, y) = e-+y).
(3.1)
The first x derivative of a Gaussian is
Go = -2xe-(X2 +y2 )
(3.2)
and the y derivative is rotated 900 as
G90
-2ye7-( 2 2
(3.3)
Therefore, we can synthesize a filter at an arbitrary orientation 0 as shown in
80
Oriented Filter (0=80")
Oriented Filter (0=900)
Oriented Filter (0=00)
2
Figure 3.2: Derivatives of Gaussian filters: (left) basis filter oriented at 00, (middle)
basis filter oriented at 900, (right) synthesis of the filter oriented at 80' by linearly
combining the basis filters.
Filtered Image (0 =
0*)
Filtered Image (0 = 80*)
Filtered Image (0 = 90*)
0.1
0.1
0.2
0.2
C-0.,
CZ 0.4
(0
0.4
a,
E~.
j-
0.5
CA
0.6
0.7
0.7
0.8
0.8
0.9
0.9
200
200
800
600
400
-4
-2
0
4
2
-4
-2
0
4
2
x 10
x 10
400
800
600
Offset (m)
Offset (m)
Offset (m)
-4
-2
0
2
4
x
10
Figure 3.3: The convolution of the input image with different directional filters: (left)
the convolution with the basis filter oriented at 00, (middle) the convolution with the
basis filter oriented at 900, (right) synthesis of the image filtered at 80' orientation
by linearly combining the convolution of the input image with the basis filters.
81
Figure 3.2 by taking a linear combination of x and y derivatives (basis filters):
Go = cos(O)GO + sin(O)G
900
.
(3.4)
We can synthesize a seismic image filtered at an arbitrary orientation R(x, y, 0) as
shown in Figure 3.3 by convolving the input image I(x, y) with the directional filter
Go'
R(x, y, 0) = cos(0)(Goo * I(x, y)) + sin(0)(G9 0 ' * I(x, y)).
3.2.2
(3.5)
Slope Estimation of Local Plane-Waves
The local plane-wave partial differential equation (Fomel, 2002) is
< + s(xt)-
dx
dt
U = 0,
(3.6)
where U denotes the seismic wavefield and s is the slope, which can vary in both
time and space coordinates. The time-space derivative of the plane wave equation is
equivalent to the convolution operator G (0) applied to the data (seismic wavefield)
G(0)U = 0.
The orientation 0 in equation (3.7) is related to the slope s in equation (3.6).
(3.7)
The
convolution operator can be efficiently constructed using steerable filters. When the
steerable filter orientation is perpendicular to the feature orientation in the image,
the output is zero (Freeman and Adelson, 1991). The local-slope (i.e., orientation)
is determined efficiently at each data sample from its neighbors by spanning all possible orientations to find the orientation corresponding to the minimum amplitude
of the steerable filter outputs (e.g., minimizing equation 3.7). The range of orientations corresponds to different apparent velocities of both the signal and noise. An
instantaneous slowness (local-slope) plot is constructed. The Gaussian filter in the
steerable filter definition plays an essential role as a regularizing (smoothing) term.
This can avoid unwanted oscillatory instantaneous slowness estimates in a region
82
where the local-slope is not defined. The smoothing of this operator is determined
by the variance of the Gaussian operator. The table of instantaneous slowness (i.e.,
local-slope) is constructed using steerable filters.
The table is sectioned laterally
across the record as a function of offset. Each section is assigned a single slope that
has maximum probability within each time-offset range
= 0, 1,.. .,90 ,
0,(x) = argmaxP(O,x),
0
(3.8)
where On(x) is the dominant local-slope as a function of offset (i.e., noise orientation),
and P(O, x) is the probability as a function of orientation and offset.
3.2.3
Signal and Noise Separation
The input data d consists of unknown signal d, and noise d, components
d = d, + d,.
(3.9)
The application of the non-linear local median filter (Duncan and Beresford, 1995) enhances (predicts) the data component (i.e., noise) that is aligned with the orientation
On and attenuates all other components (i.e., signal) with different orientations
=
0
f(On)dn =
1
dn,
f(0n)ds
(3.10)
where f(Ori) is the filter operator steered toward the noise orientation, and d, and
dn are the signal and noise components, respectively. The modified table of spatially
varying slopes is used for steering the median filter toward the noise direction. For
each temporal point within a window of receivers (e.g., offsets), we apply a 2Al-point
steered median filter to the data
d[m, n] = mcdian{f [i, j]}, (i,j) E
j
-
A
83
tan(On) < j
j + Al . tan(O,)
(3.11)
where d[m, n] is the data point at the m trace and n time sample, i and
j are the
sample index in the spatial (x) and temporal (t) directions, respectively, and On is
the orientation (i.e., slope) of the noise component to be removed. The center and
median values of the time-space window should be very similar and represent the
amplitude value of the estimated noise component when there is no reflection (signal)
information (f(n)dn= dn). However, if there are reflections, the value of the center
sample will be replaced by the median value of it is neighboring points (f(0)ds=0).
Therefore, the application of this filter on the total data will predict only the noise
component
f(On)d =
f(On)(ds+dn)
=
f(On)dn + f(On)d9
=
dn.
(3.12)
Hence, the signal component is obtained by subtracting the predicted noise (dn) from
the input data (d) to produce the filtered record (d.)
ds
=
=
3.3
d -- dn
d
-
f(0n)d.
1
(3.13)
Synthetic Example
In this section we illustrate, with a numerical example, the application of noise separation based on spatially varying slopes. The modeling and understanding of the
complicated scattered wave features play an essential part in the success of their subsequent removal. Numerical forward modeling based on finite difference methods can
handle complex lateral variations in material properties and produce a complete and
accurate solution to the elastic wave equation, with all direct, converted, scattered
and guided waves.
84
3.3.1
Finite Difference Modeling
To fully model elastic waves in the presence of heterogeneity, we utilize an accurate implementation of the standard staggered-grid (SSG) finite difference scheme
(Virieux, 1986; Levander, 1988; Zhang, 2010), with Convolution Perfectly-MatchedLayer (CPML) absorbing boundary condition (Komatitsch and Martin, 2007; Martin
and Komatitsch, 2009; Zhang and Shen, 2010). The SSG scheme is fourth order accurate in space (including the free surface boundary) and second order accurate in time.
The internal interfaces are represented by the so called effective medium parameters
(Moczo et al., 2002) to avoid spurious numerical diffractions caused by the material
discontinuity due to the spatial grid. The density is calculated by arithmetic average,
and the Lame parameters are calculated by harmonic average.
We consider a two-dimensional earth model with multiple dipping layers and five
scatterers embedded in the uppermost layer (Figure 3.4). The scatterers are located
at 15 m depth below the free surface, and each has a 10 m diameter and an impedance
contrast corresponding to 0.36. The material properties are given in Table 3.2. The
domain has N, = 1001 and N, = 501 grid points with 1 m grid spacing (i.e., Ax and
Az), that is, 500 m depth (along the z-axis) and 1000 m distance (along the x-axis).
The grid size is small enough to capture the shape of the scatterers. The time step is
0.2 ins. A vertical source is used with a Ricker wavelet and 30 Hz central frequency
(- 75 Hz maximum frequency). The source is located at (x,z) = (150 m, 0
in).
The
receivers are located on the surface with 50 m near-offset and 5 m space intervals. We
consider only the vertical component (vZ) of the particle velocity field. The scatterers
are treated in the numerical scheme as a density and velocity perturbation. To avoid
spurious numerical diffractions caused by material discontinuity due to the spatial
grid, arithmetic and harmonic averages (smoothing) are applied to the density and
elastic constants at each grid point.
Calculated waveforms for the scattering model with and without the direct surface
waves are shown in Figure 3.5. The direct and back-scattered surface waves are removed as they are much larger in amplitude than the scattered body-to-surface waves.
85
0
5
100
4
E200
3
3300
2
400
0
200
400
600
800
1000
Distance (m)
Figure 3.4: Synthetic earth model. Multiple dipping layers with five circular scatterers
(red circles) embedded in the shallow layer. The scatterers are located at 15 m depth,
each is 10 m in diameter and has an impedance contrast corresponding to 0.36. The
source is located at (x,z)=(150 m, 0 m). The receivers are located on the surface
with 50 m near-offset and 5 m space intervals. The color scale (on the right) and
associated numbers refer to material properties given in Table 3.2.
Material Index
1 - Dark blue
2 - Blue
3 - Green
4 - Orange
5 - Red
Vp (m/s)
1800
2200
2500
2700
3000
Vs (m/s)
1000
1200
1300
1400
1500
Density (kg/m
1750
1900
2000
2100
2250
3
)
Table 3.2: Material properties (P wave velocity, S wave velocity, and density) of the
model shown in Figure 3.4.
This is achieved by computing the wavefield for a homogeneous full-space, with and
without the near-surface scatterers, and then subtracting the direct surface waves
from the incident and total wavefields, respectively, to isolate the scattered body
waves.
Note the strong amplitudes of the shear wave reflection and refraction at
mid-offsets due to the radiation pattern of the vertical source. The effective medium
properties of the heterogeneities are wavelength dependent. Therefore, the relatively
small near-surface scatterers compared to the incident wavelength produce high
frequency features. In this example, scatterers with a size of 1/6 of the wavelength (10
m) produce significant scattering. Applying the stack-array method, shown in Figure
3.7, suppresses the scattered surface waves but also reduces the frequency content of
the data leading to an image with less resolution.
86
with Scattering
No Scattering
The Difference
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
C6' 0.4
Z50.4
C;) 0.4
E
E
0.5
E 0.5
0.
0.8
0.6
0.
0.7
0.7
0.8
0.8
0.8
0.5
200
0 90.9
600
400
800
200
Offset (m)
-5
0
600
400
0..................
8
200
800
Offset (m)
5
-5
600
400
800
Offset (m)
0
5
x 10,
-5
0
5
x10
x 10
(a)
with Scattering
No Scattering
The Difference
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
CO 0.4
_0.4
0A
0)
(D
W3
E05
E0.5
F-
0.5
0.4
F-05I
0.8
0.8
0.8
0.7
0.7
0.7
0.8
0.8
0.8
0.8
200
400
800
800
0.8
200
Offset (m)
-6
-4
-2
0
2
600
400
800
0.9
200
Offset (m)
4
6
-6
-4
-2
0
2
4
6
x
x10
600
400
800
Offset (m)
-6
0
-4
-2
0
2
4
6
x 10-
(b)
Figure 3.5: Finite difference simulations (v,-component) showing the scattering effects
due to near-surface heterogeneities for the model in Figure 3.4; (a) shows the results
including the direct surface wave and (b) with the direct surface wave removed; (left)
total wavefield simulated using the model with scattering, (middle) incident wavefield
simulated using the model without scattering, and (right) scattered wavefield (i.e.,
the difference between the total and incident wavefields). A vertical source with 30
Hz Ricker wavelet is used. The source is located at (x,z) = (150 m, 0 m). The
receivers are located on the surface with 5 m space intervals. Note the complexity
due to scattering of the reflected arrivals.
87
6
with Scattering
The Difference
0.1
0.1
0.4
0)
E
0.1
0.1
0.2
0.2
0.3
0.3
0.4
E 0.5
0)
0.4
E
0.5
0.6
0.6
0.7
0.7
0.8
0.9
M600
-400
-200
-6
-4
-2
0
0.9
0
Offset (m)
2
-600
4
6
-400
-20D
-6
-4
-2
0
0.9
0
Offset (m)
2
-600
4
6
x 10
-400
-6
-4
-200
0
Offset (m)
-2
0
2
4
6
x10-
x 10
Figure 3.6: Finite difference simulations (v-component) showing the scattering effects
due to near-surface heterogeneities for the model in Figure 3.4: (left) total wavefield,
(middle) incident wavefield, and (right) scattered wavefield. A vertical source with
30 Hz Ricker wavelet is located at (x,z) = (850 m, 0 m). The receivers are located
on the surface with 5 m space intervals.
InDut Data
Stack of 5 Receivers
Stack of 10 Receivers
0.1
P 0.
0 0.!
U) 0.
E
F 0.
0J.
0.7
0.9
200
400
600
600
Offset (m)
-6
-4
-2
0
2
4
6
0.9
200
400
Offset (m)
-6
-4
0
-2
x 10'
2
600
600
200
400
800
600
Offset (m)
4
6
x 10'
-6
-4
-2
0
2
4
6
x
10-,
Figure 3.7: An example of applying the stack-array method with different array sizes:
(left) input image, (middle) stack of five receivers, and (right) stack of ten receivers.
Note that the stack-array method reduces the scattered surface waves and also the
frequency content of the data.
88
3.3.2
Application to Synthetic Data
All upcoming body wave reflections (e.g., primaries , multiples, and mode-conversions)
scatter close to the free surface due to near-surface heterogeneities and, therefore,
excite scattered surface waves.
The scattered surface wave noise covers the whole
time-space domain with locally-linear features (i.e., horizontally propagating planewaves along the free surface) that propagate in both the positive and negative (i.e.,
forward and backward) directions. The scattered surface waves are more sensitive
to the uppermost earth layer. Therefore, their slope can vary due to lateral velocity
variations (i.e., mostly shear-wave velocity) and surface topography. However, the
surface waves exhibit constant local-slope (i.e., slowness) at each receiver location depending on the shear wave velocity near the surface, independent of the source depth
and location (see Figures 3.5 and 3.6).
Because the local-slope of the noise can only vary spatially with different receiver
locations (or patch of receivers), only a small range of slopes is filtered at each offset
unlike a global velocity filter. The workflow for separating scattered body-to-surface
waves from up-going reflections is shown in Figure 3.8. The process consists of four
steps:
1. Dip decomposition: using a velocity filter (e.g.,
f
- k domain) to reduce the
signal-noise interference such that the slope estimation yields a more reliable
result, as illustrated by the solid black lines in Figure 3.9.
2. Steerable filters: compute the directional derivatives (equation 3.7) of the dipdecomposed image for different supplementary orientations, as illustrated by
the dashed color lines in Figure 3.9, and construct an instantaneous slowness
table corresponding to the minimum amplitude of the steerable filter outputs.
3. Prediction: estimate the dominant local-slope of the noise component (0,(x)) as
a function of offset (or patch of offsets), as shown in Figure 3.10, by estimating
the mode of the probability mass function of the instantaneous slowness values
(equation 3.8).
89
4. Separation: apply spatially varying directional non-linear median filter steered
toward the noise directions (i.e., dominant local-slopes predicted in step 3) to
predict the noise and then subtract it from the data (equations 3.11 to 3.13).
Input data
Dip decomposition (f - k)
Compute image orientations using steerable filters
Estimate spatially
varying slopes
Apply spatially varying
directional median filter
Final result
Figure 3.8: Flow diagram of the spatially varying filtering approach to remove scattered surface waves.
Slope estimation is an essential step for separating signal and noise components.
The slope of coherent noise events (e.g., slowness) can be estimated using steerable
filters. The dip decomposition using the frequency-wavenumber fan filter reduces the
signal and noise interference for enhanced slope estimations of the noise component
d,(x, t) = F-2 [Vn(k, W)F
2
[d(x, t)] , n = 1, 2, . .. , N,
where V, corresponds to the fan filter in the spectral domain, and F
2
(3.14)
and .-
2
denote
the forward and inverse two-dimensional Fourier transform, respectively. Histograms
of local-slopes calculated using steerable filters for three patches of offsets from one
shot gather are shown in Figure 3.10. The estimated dominant local-slopes match
the true slopes of the noise components in all cases. The histograms look identical
(because
f
- k values are about the same). The offset patches allow for spatially
90
Wavenumber
Figure 3.9: A schematic diagram showing the frequency-wavenumber domain. The
black lines show the range of wavenumbers constrained by the the f - k filter, and
the dashed colored lines (cyan, magenta, and green) show the frequency-wavenumbers
corresponding to different steerable filter orientations.
varying slope estimation, which can be useful in the case of lateral shear wave velocity variations near the earth's surface. Each orientation represents applying the
directional derivative at the supplementary angles (0) and (1800 - 0). Similarly, the
recorded scattered surface waves propagating at the forward and backward directions
have supplementary angles (i.e., slopes) at each offset location.
The operator of the non-linear median filter steered toward the dominant supplementary slope directions returns the amplitude of the surface wave when there is no
reflection interference, and it replaces the amplitude of the sample with its median
of neighboring samples when reflections are present. The results of separating the
signal and noise components are shown in Figure 3.11. The estimated signal is free
of the scattered body-to-surface wave noise, and only very weak scattered P waves
that have slopes similar to the reflected signal are passed by the filter. We compare
the simulated data after applying our filtering approach and the conventional
f
- k
filter (Figure 3.12). As expected, the f - k filter suffers from edge effects as well as
smearing caused by leakage in the transform domain.
f
- k filtering also removed
part of the signal, as shown in Figure 3.13, which directional filtering has preserved.
The frequency-wavenumber domain of the signal and noise are shown in Figure 3.14.
The spatially varying slope filter has a narrow reject band in the
minimizes signal smearing and distortion.
91
f-
k domain that
lb
Patoh 1
10000-
0
5000-
10
40
20
30
40
50
60
70
80
s0
70
80
70
80
Local-Slope (0)
1500 Patch
2
10000-
0
5000-
0
10
30
20
so
40
15c
0.,
-a)
E*
W
(D
Local-Slope (*)
0
Path 3
10000-
500010
20
30
40
50
Local-Slope
60
(0)
Figure 3.10: Histograms of local-slopes calculated using steerable filters (left) for
different receiver patches (right). The top, middle, and bottom histograms correspond
to patch 1, 2, and 3, respectively. The red lines in the histogram plots correspond to
the true orientation of the forward and backward scattering.
Filtered Image
Input Image
10
The Difference
S0.4
0.'
E)
E)
E
01
0.1
0.7
Ofiset (m)
Offset (m)
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
Offset (m)
4
6
x 10'
x 10'
-6
-4
-2
0
2
4
6
x 10'
Figure 3.11: Application of the median filter: (left) input data, (middle) filtered data,
and (right) residual (the difference or the removed noise).
92
Filtered
Cl,
Imaie
Filtered
Image
(F-K)
( 0.4
(.
0.4
a .
E
0.7
Utnset (m)
-6
-4
-2
unset
0
2
4
6
-6
-4
-2
(m)
0
2
4
6
x 10
x 10
Figure 3.12: Comparison between the directional filter (left), and the f - k filter
(right). Note the edge effects and smearing of reflected signal caused by the f - k
filter due to leakage in the transform domain.
Difference
Difference (F-K)
0.1
0.2
0.2
0.3
0.2
0.4
S0.4
E
0.6
0.6
0.7
0.7
0.8
0.9
0.9
Offset (m)
-6
-4
-2
0
Offset (m)
2
4
6
-6
x
10-7
-4
-2
0
2
4
6
x10
Figure 3.13: Difference between the input data (with noise) and the denoised results
with different methods (Figure 3.12): (left) directional filter and (right) f - k filter.
Note that the f - k filter removed part of the reflected signal.
93
(b) F-K Domain - (True Scattering)
(a) F-K Domain - (input Image)
N
N
C
Cr
Cr
(D
a)
-0.05
0
120
-0.1
U.Ub
C1-
Wavenumber (1/m)
U-1
(c) F-K Domain - (Filtered Image)
a)
U-
-U.Ub
U
U.U
Wavenumber (1/m)
(d) F-K Domain - (Removed Scattering)
N
0
Cr
Cr
U-
U-
O.1
-U.Ub
0
120
-0.1
U.Ub
-U.VO
U
Vubu
Wavenumber (1/m)
Wavenumber (1/m)
Figure 3.14: The frequency-wave number spectrum of: (a) total wavefield (input image), (b) scattered wavefield (true noise), (c) filtered image (signal), and (d) residual
(removed noise).
94
3.4
Effects of 3D Heterogeneities on 2D Data Acquisition and Processing
The earth models discussed so far are based on the two-dimensional assumption,
which is that receivers are in-line with the scatterers. However, the earth is threedimensional in reality, and scatterers can occur in the cross-line direction. We demonstrate the effects of 3D heterogeneities on 2D acquisition and processing by simulating
seismic elastic waves in 3D. The waves in 2D are modeled as cylindrical waves whereas
they are modeled as spherical waves in 3D. Similarly, a circular scatterer and a point
source in 2D are represented by a cylindrical heterogeneity and a line of sources in
3D, respectively. These differences between the 2D and 3D modes not only cause
alterations to the phase, but also can alter the amplitude decay due to geometrical
spreading (i.e., attenuation with propagating distance).
Body waves attenuate as
r-1 in energy or r-1/2 in amplitude in 2D, whereas in 3D they attenuate as r-2 in
energy or r-1 in amplitude. On the other hand, surface waves do not attenuate with
propagation distance in 2D, whereas in 3D they attenuate as r-1 in energy or r-12
in amplitude.
We observe in 3D finite difference simulations that when the scatterers are in-line
with the receivers (Figure 3.15), the results are similar to the 2D cases shown in the
previous sections. However, when the scatterers are in the cross-line direction, the
apext of the scattered surface waves appear as non-linear arrivals (e.g., hyperbolic)
as shown in Figure 3.16, as opposed to the conventional linear direct surface waves.
These features may rise in case of surface and body-to-surface wave scattering from
scatterers in the cross-line direction. These features cannot be easily attenuated using
conventional linear noise removal techniques such as
f
- k filter.
In the case of the direct surface wave generated by sources in the cross-line direction, it is possible to correct for the non-linear move-out given that the source
location and surface wave velocity are known. However, this is not trivial for the case
of scattered body-to-surface waves excited by scatterers in the cross-line direction, as
the locations of the sources, scatterers in this case, are not known.
95
3.4.1
Irregular Bedrock Interface in 2D and 3D
We further study the effects of 2D and 3D wave scattering from an irregular (Gaussian) interface. The Gaussian surface is constructed by generating a random Gaussian
surface that is correlated (smoothed) with a Gaussian operator as shown in Figure
3.17. The irregular interface is placed near the free surface, and a plane interface is
added at 200 m depth. In Figure 3.18, we show the results for the irregular (Gaussian)
bedrock interface model simulated using 2D and 3D finite difference schemes. The
2D velocity model was taken as a slice from the 3D model, at the same location of the
receiver line. The results for the model with a planar shallow interface look similar
for the 2D and 3D cases, except the weak amplitude due to spherical divergence in
the 3D case. On the other hand, the simulated data for the model with Gaussian
bedrock interface are different in 2D than in 3D. The 2D case includes scattering
from only the inline direction, and, therefore, the scattered surface waves are locally
linear. The 3D case, however, include scattering from both the inline and cross line
directions, and, therefore, the recorded scattered surface waves are no longer linear.
3.4.2
Line and Random Side Scatterers
In this section, we look at the cases when multiple scatterers are randomly distributed
in 3D (Figure 3.19). The computed 3D finite difference results are shown in Figure
3.20a, with the scattered waves exhibiting mixed phases between linear arrivals due
to the in-line scatterers and non-linear phases with different curvatures due to the
randomly distributed scatterers in the cross-line direction. These noise features have
frequency content comparable to the size of the scatterers, an amplitude comparable
to the reflected signal and can mask the entire data in the time-space domain. Therefore, filters based on frequency, amplitude contrast, or windowing are not feasible in
this case. However, the scattered noise appears as cones in 3D, and therefore sampling densely in both the in-line and cross-line directions makes the problem similar
to the 2D case, in which it is possible to capture and filter out the cone. We show the
application of a 3D
f
- k filter to a spatially dense sampled dataset in Figure 3.20b.
96
The scattered waves have a cone shape in 3D; showing circular phases in the time
slices (Figure 3.21a), while they appear as non-linear surface waves in the time-offset
domain. Similar to the 2D case, we can divide the receivers into patches, and predict
the dominant local-slopes by computing the 3D steerable filters (Figure 3.21b), to
guide a 3D FK filter.
3.5
Field Data Example
The field data used in this paper (shown in Figure 3.22) have a receiver group interval
of 6.25 meters and a maximum offset of 1000 meters. Each receiver station consists
of twelve (12) bunched geophones arranged randomly within a 25 cm radius from the
center of the trace. This single-sensor type acquisition (i.e., no array stacking applied
in the field) can provide true high resolution spatial sampling (to avoid aliasing) of
both the signal and noise components for optimal design of noise attenuation and
filtering.
The main ground roll mode in the data (dominant direct and back-scattered
ground roll) has a velocity of ~ 1050 m/s and frequencies up to 40Hz.
The min-
imum wavelength is ~ 25m. This ground roll is back scattered, and is visible on the
time-offset domain, mainly at negative wave numbers. The high amplitude slow scattered noise is due to wave propagation encountering the near-surface heterogeneities
(Figure 3.22). We apply our steerable and non-linear filtering approach to the field
data. We first reduce the signal and noise interference using an
f
- k filter and then
compute the steerable filters for the initial noise component. We then estimate the
most probable orientation within four patches of offsets (Figure 3.23). In this example, two orientations have been predicted by the histograms in Figure 3.23: 67.5' for
patch one and two, and 700 for patch three and four, as indicated by the magenta
and red dashed lines, respectively. The change in slope of the direct and scattered
surface waves depends on the shear wave velocity of uppermost layers. This could
be due to either an increase in the shear wave velocity of the rock at far offsets, or
due to attenuation of high frequencies traveling within shallow layer slow velocities
97
and dominance of the low frequency component that travels with faster deeper layer
velocities, assuming velocity increases with depth.
Based on the predicted slopes, we apply a spatially varying directional non-linear
median filter steered toward the noise directions (Figure 3.22). The residual image
shows that only locally linear noise features have been removed by the filter. To
enhance the reflected signal, we apply NMO and running average filter to the results
obtained from our approach (Figure 3.24) and conventional f - k filtering (Figure
3.25). Compared to our approach, the f -k filter not only caused smoothing and edge
effects but also removed part of the signal due to leakage in the transform domain
(Figure 3.25).
The refraction models based on first breaks indicate that both the surface elevation
and the thickness of the low velocity layer are almost constant. Also, the wavelength
of the scattered waves is directly proportional to the size of the scatterers, and, as
demonstrated by the difference plot, the wavelength of the backscattered surface
waves is much shorter than the direct surface waves. This suggests that most of the
near-surface scattering is due to individual or sharp interface scatterers.
3.6
Discussion and Conclusion
In this paper, we presented an approach to estimate dominant local-slopes in the data
as a function of offset (or patch of offsets) using steerable filters and to separate the
signal and noise components using a directional median filter. The slope estimation
step using steerable filters requires only a linear combination of a set of basis filters
at fixed orientation to synthesize an image filtered at an arbitrary orientation. This
makes the process very efficient. The method can handle different slopes, locally-linear
coherent noise that has small amplitude contrast and varying slope with offset. We
successfully implemented this approach on synthetic and field data for the separation
of scattered surface waves from reflected body-waves.
The results show that this
approach is superior to conventional f - k techniques.
Although we only discussed scattered surface waves excited by upcoming body
98
waves impinging on near-surface heterogeneities, our method can also handle surfaceto-surface wave scattering. The method can also be applied to marine data with
ocean-bottom-cable (OBC) acquisition, in which the sea bottom acts in a similar way
to the free surface and, therefore, propagating waves can scatter to Scholte waves by
heterogeneities near the sea bottom.
Acknowledgments
We thank Saudi Aramco and ERL founding members for supporting this research.
We also would like to thank Professor Bill Freeman for the helpful discussion about
steerable filters.
Bibliography
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99
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101
No Scatterinq
with Scatterina
The Difference
0.02
0.02
0.02
0.04
0.04
0.04
0.06
0.06
0.06
0.08
0.08
0.08
0.1
0.1
0.12
0.12
0.14
0.14
0.16
0.16
0.18
0.18
0.2
E
0.2
zUU
-4
400
Offset (m)
-2
0
800
800
2
200
4
x
-4
400
Offset (m)
-2
600
0
200
800
400
600
800
Offset (m)
2
4
10
x
-4
-2
0
2
4
10-
X 10
Figure 3.15: Shot gathers simulated using 3D finite difference with the scatterers
in-line with the receivers: (left) model without scattering, (middle) with scattering,
and (right) the difference.
No Scatterinj
with Scatterina
The Difference
0.02
0.02
0.04
0.0
0.04
0.0E
0.06
0.08
E
0.1
0
0.
E
0.1
0.12
0.12
0.12
0.14
0.14
0.14
0.16
0.18
0.16
0.18
0.18
0.18
0.2
200
400
600
800
0.2
200
Offset (M)
-4
-2
0
2
4
-4
400
600
800
Offset (m)
-2
0
x 10
2
4
x10
0.2
200
-4
-2
400
Offset (m)
0
600
2
800
4
x10
Figure 3.16: Shot gathers simulated using 3D finite difference with the scatterers in
the cross-line direction: (left) model without scattering, (middle) with scattering, and
(right) the difference.
102
300
0.014,
0.012,
200
0.01.
100
0.008 ,
0.006,
0
0.0041
0.002I
-100
1000
-200
40
40
30
500
Distance (m)
50
0
0
Distance
-300
0 0
Y (M)
(m)
20
00
10
(a)
X
(M
(b)
25
20
-20
-200
15
-40
-100.
10
05
-00
5
-20D
-80
0
-300,
-100
.5
-400
-120
-500.
-140
100
200
-10
1000
iU
ROO
1000
-15
-100
500
.20
^01
-10
-25
Distance
(m)
0
0
Distance
(m)
Distance (m)
0
0
Distance (m)
(d)
(c)
Figure 3.17: A 3D irregular (Gaussian) bedrock interface model: (a) Gaussian surface
profile with 70 m standard deviation, (b) a Gaussian smoothing operator with 5 m
correlation length, (c) a smoothed surface generated by convolving the Gaussian surface with the smoothing operator, and (d) an earth model with near-surface irregular
bedrock interface at 15 m depth below the free surface and deeper flat reflector at
200 m depth.
103
with Scattering
No Scattering
The Difference
0.3
0.4
0.5
E
0J
E
0
0.6
0.7
0.7
0.8
0.8
0.9
0.9
200
400
Offset (M)
-.
00
0
00
200
5
400
600
Offset (M)
-6
800
0
X 10'*
Outset (m)
5
-5
0
x 10'
5
x 10 ,
(a)
with Scattering
No Scattering
The Difference
0.1
02
0.3
0.3
0.4
04
0.5
a3
o
(D 0.5
wz
0.6
0.6
0.7
0.7
0.8
Offset (m)
-5
0
Offset (M)
5
-5
Offset (m)
0
X 10,*
5
x 10"
-5
0
5
. 10-6
(b)
Figure 3.18: Finite difference results for the irregular (Gaussian) bedrock interface
model simulated using: (a) 2D FD, and (b) 3D FD. The total wavefield (left) simulated using the model with Gaussian shallow interface; the incident wavefield (middle)
simulated using the model with plane shallow interface; and (right) scattered wavefield
(i.e., the difference between the total and incident wavefields). The 3D simulations
include scattering phases coming from the cross-line direction.
104
0
-100
-200.
-300,
-400w
-
-5004.
200
~
400
>
600
1000
600
800
12
800
400
0
10200
Distance (m)
Distance (m)
Figure 3.19: A 3D earth model with multiple dipping layers and near-surface scatterers.
Filtered Data
Input Data
(b)
Figure 3.20: Modeling and filtering of scattered body-to-surface waves in 3D: (a)
simulated 3D finite difference results, and (b) estimated signal after application of 3D
FK filter.
105
(a)
Filtered Noise
(b)
Figure 3.21: Application of 3D FK filter to spatially dense sampled 3D simulated
data: (a) the difference between the input and filtered data in Figure 3.20 (filtered
noise), and (b) histogram of local-slopes calculated using 3D steerable filters showing
the dominant slope for the receiver patch highlighted in red.
106
Input Image
Filtered Image
The Difference
0.1
0.
0.2
0.:
0.3
0
0.4
E
0.5
E
0.6
0.A
0.7
0.,
0.
0.
0.7
0.8
0.8
0.9
0
200
600
400
800
0
Offset (m)
-1
-0.5
0
200
600
400
800
Offset (m)
0.5
1
-1
0
-0.5
Offset (m)
0.5
-1
-0.5
0
0.5
Figure 3.22: Application of the steered median filter approach to field data: (left) input data, (middle) filtered data, and (right) residual (the difference or noise removed).
107
~Patch
1
I
30000 2000[
.JL
1000I
10
20
30
40
50
60
70
0
J
0.2
80
Local-Slope (0)
500
-D f ,
0.4
4000c
3000-
0.6
0 2000100010
20
30
40
50
60
0.8
80
Local-Slope (0)
-1
'3
500
3000
0
E
1.2
2000
1000
20
10
30
40
50
60
70
1.4
80
Local-Slope (0)
1.6
400040-Patch
C
4
1.8
3000-
0
0
2000
1000
1
2
0
200
400
600
800
Offset (m)
Local-Slope (0)
Figure 3.23: Histograms of local-slopes calculated using steerable filters (left) for
different receiver patches (right). The top, middle, and bottom histograms correspond
to patch 1, 2, 3, and 4 respectively. The magenta and red dashed lines in the histogram
plots correspond to 67.50 and 70' orientations.
108
InDut Imnaae
Filtered
Imane
The Difference
0.2
0.3
0.4
E
E
0
E
0.5
07
0.8
0.8
0.9
0
125
250
375
500
625
750
875
Offset (m)
-0.2
-015
-01
-0.05
0
0.05
Offset (m)
01
0.15
0.2
-0.2
-0.15
-0.1
0
-05
Offset (m)
0.05
01
0.15
0.2
-0.2
-015
-0.1
0
-005
0
.
01
015
0.2
Figure 3.24: Application of the steered median filter approach to field data after
NMO, running average filter, and inverse NMO to enhance the reflections: (left) input
data , (middle) filtered data, and (right) residual (the difference or noise removed).
Reflected P waves modeled using ray-tracing are shown in dashed red lines.
Filtered Imaae
E
The Difference
0.)
0).
E
E
0.7
0.8
0.9
0
125
250
370
500
625
750
875
Offset (m)
-02
-015
-01
-005
0
005
Offset (m)
0.1
015
0.2
-02
-015
-0.1
-0.05
0
0.05
Offset (m)
0.1
015
0.2
-0.2
-0.15
-0.1
-0.05
0
0.0
01
0,15
0.2
Figure 3.25: Application of f - k filter to field data after NMO, running average filter,
and inverse NMO to enhance the reflections: (left) input data, (middle) filtered
data,
and (right) residual (the difference or noise removed).
109
110
Chapter 4
Imaging of Near-Surface
Heterogeneities by Scattered
Elastic Waves*
Abstract
We introduce an elastic reverse time migration (RTM) approach for imaging nearsurface heterogeneities using the near-surface scattered waves (e.g., body to P, S,
and surface waves). Wavefield extrapolation is performed using an elastic staggeredgrid finite difference scheme. The divergence of the wavefield is derived from the
spatial derivatives of the measured wavefields. Imaging and locating the near-surface
heterogeneities are essential for planning seismic surveys or explaining near-surface
related anomalies in the data. The scattered body-to-surface waves provide optimal
illumination of the near-surface as they travel horizontally along the free surface
boundary. We demonstrate the robustness of the elastic RTM approach on synthetic
data calculated with finite difference.
4.1
Introduction
In general, depth migration algorithms are categorized as ray-based (e.g., high frequency asymptotic methods such as Kirchhoff and beam migration) and wave equationbased (e.g., one-way and two-way wave equation-based migration).
The concept
*Th11e bulk of this chapter is in preparation for publication as: AlMuhaidib, A. M. and Toksz,
M. N., Imaging of near-surface heterogeneities by scattered elastic waves (in preparation)
111
based on two-way wave equation migration is known as reverse time migration (RTM)
(Baysal et al., 1983; Loewenthal and Mufti, 1983; McMechan, 1983; Whitmore, 1983).
RTM is more attractive over other imaging algorithms as it can handle both multiarrivals and overturned waves, and it has no restrictions with respect to the complexity
of the velocity model or the dip of the structure.
RTM schemes based on the acoustic wave equation have been more widely used
for imaging complex geological structures due to the low computational expense compared to elastic RTM. The earth, however, is elastic and the data recorded in the field
contain all wave types, including P, S, PS, SV, etc. In recent years, there has been
more interest in exploiting all the information carried by mode-converted seismic
data by using elastic RTM. Sun et al. (2006) introduced a modified RTM approach of
transmitted PS waves for salt flank imaging. Their approach separates the wavefield
into pure mode (PP) and converted (PS) waves, and the extrapolation is performed
using the scalar wave-equation with the corresponding Vp and Vs velocities. A similar strategy is proposed by Xiao and Leaney (2010) for salt flank imaging with VSP,
local elastic RTM, and using the vector wave equation to extrapolate the separated
PP and PS waves. Shang et al. (2012) used teleseismic transmitted P and S waves
recorded on the surface to perform passive source RTM to reconstruct dipping and
vertical offset interfaces, an approach superior to traditional receiver function analysis
in complex geological environments.
To address the problem of imaging near-surface heterogeneities, several studies
have formulated solutions of the inverse scattering problems.
Blonk et al. (1995),
Blonk and Herman (1996), and Ernst et al. (2002) used a perturbation method based
on the Born approximation (single-scattering). These methods have difficulties when
dealing with large and high-contrast heterogeneities that violate the Born approximation. Campman et al. (2005, 2006) used an inverse scattering approach based on an
integral-equation formulation to image the near-surface heterogeneities, but assumed
that scattering takes place immediately under the receivers. Other methods, based on
solving integral equations using the method of moments, can handle strong contrast
and large heterogeneities and can take into account multiple scattering (Riyanti and
112
Herman, 2005; Campman and Riyanti, 2007). However, these methods are restricted
to laterally homogeneous background media consisting of horizontal layers.
These
assumptions are not valid in areas with complex overburden.
In this paper, we present a prestack elastic RTM for locating and imaging nearsurface scatterers. The main idea is to separate the near-surface scattered waves from
the total recorded wavefield and to use the scattered waves for receiver wavefield extrapolation.
An elastic staggered-grid finite difference scheme is used for wavefield
extrapolation (Virieux, 1986; Levander, 1988). For the P wave separation (e.g., divergence of the wavefield), the finite difference scheme can be used to calculate the
spatial derivatives of the measured wavefields (Dellinger and Etgen, 1990).
The P
wave separation is derived after wavefield extrapolation and is subjected to a cross
correlation-type imaging condition (Claerbout, 1971). The stresses and particle velocities are migrated simultaneously by solving the first order elastic wave equation.
To the best of our knowledge, this is the first attempt of imaging the near-surface by
incorporating the body waves and the full scattered wavefield. We test the proposed
elastic RTM approach on data simulated with an elastic finite difference scheme.
4.2
Methodology
The main idea underlying elastic RTM for imaging near-surface scatterers is to backproject the near-surface scattered waves (e.g., body to P, S, and surface waves) until
they are in phase (e.g., time of conversion) with the incident waves at the scatterer
locations (Figure 4.1). In the shot profile domain, the image is constructed by forward
propagating the incident wavefield (modeled data with estimated source wavelet) and
back-propagating the scattered wavefield (recorded multicomponent data) as boundary conditions. Then the extrapolated wavefield is separated into a P wave component
(i.e., divergence of the wavefield) before an imaging condition is applied at each image
location. The incident and scattered wavefields are also called the source and receiver
wavefields, respectively.
The multicomponent source and receiver wavefields are extrapolated in time by
113
solving the seismic elastic wave equation in isotropic elastic media
p2U -(A
2p)V(V u) +pVx(Vxu)=f,
(4.1)
where p is density, A and 1t are Lame parameters, u is displacement vector, and f is
a force term (e.g., source wavelet or receiver wavefield injected as a boundary condition). This equation can be separated into scalar and vector potentials by Helmholtz
decomposition, which applies to the vector field u:
u = V#+ V x
where
#
,
(4.2)
and o are the scalar and vector potentials of the wavefield u, respectively.
Substituting equation 4.2 into equation 4.1 and applying vector identities gives equations for P wave potential
1
v2
92
102
a 2 Ot2
= 0,
(4.3)
0,
(4.4)
and S wave potential
1
2
92
-
2
where a and 3 are the P and S wave velocities, respectively.
For reverse-time continuation, the data (i.e., separated scattered noise) are reversed at the corresponding receiver locations and injected as sources (e.g., v, and v,
components) into the computational domain using an elastic wave-equation solver.
The P wave mode is separated at each wavefield extrapolation step by taking the
divergence of the wavefield (V - u), and an imaging condition is applied to form an
image of the scatterers. The zero-lag cross-correlation imaging condition is defined
as follows
tmax
M (x)
=
S(x, t)R(x, t dt,
shots
(4.5)
o
where m is the value of the migration image at a spatial location x; and S and
R are the P wave mode of the forward and time-reversed wavefields. The imaging
condition provides the correct kinematic, and it is simply the scalar product of the two
114
wavefields at each time step and summation over all time levels and shot locations.
4.3
Numerical Tests
In this section, we demonstrate the application of elastic RTM to synthetic data calculated using elastic finite difference modeling. We consider a two-dimensional earth
model with multiple dipping layers and five scatterers embedded in the uppermost
layer (Figure 4.2). The scatterers are located at 15 m depth below the free surface,
and each has a 10 m diameter and an impedance contrast corresponding to 0.36.
The material properties are given in Table 4.1. A point source is used with a Ricker
wavelet and 30 Hz central frequency (-
75 Hz maximum frequency). The simulations
are carried for 19 sources located at 10 m depth with 50 m space intervals.
The
receivers are at the surface and placed at 1 m intervals.
To image the near-surface scatterers, we assume the reflected body wave arrivals
as the incident wavefield (i.e., source wavefield). This implies that the interfaces (i.e.,
reflectors) act as seismic sources (Figure 4.1). The "receiver wavefield" is obtained by
separating the scattered wavefield (i.e., reflected body waves scattered to body and
surface waves) from the total wavefield. The separation of the scattered wavefield can
be achieved by first modeling and then subtracting the incident wavefield from the
total wavefield (Figure 4.3), as demonstrated in Chapter 2.
In Figure 4.4, snapshots of the v,-component of the incident and scattered waves
are shown. The divergence (P waves) and the curl (S waves) of these wavefields are
shown in Figures 4.5 to 4.6, respectively. The snapshots clearly show the wavefield
decomposition into P and S waves by applying the divergence and curl operators.
However, it is difficult to identify the separated phases in the shot gather domain
in the case of surface receivers. This is due to mode conversion occurring right at
the free surface where the receivers are placed. For example, we can observe weak
S waves in the shot gather domain (curl of the wavefield as shown in Figure 4.3i)
recorded with similar slopes to the P wave arrivals. These wave phases are scattered
P waves converted to S waves right at the free surface boundary. The same is also
115
true in the case of the scattered S wave arrivals converted to P waves at the free
surface boundary and recorded with similar slopes to the S wave phases (as shown
by the divergence of the wavefield in Figure 4.3f).
The near-surface scatterers' image is formed by applying the imaging condition
(equation 4.5) to the P wave components of the extrapolated forward incident and
backward scattered wavefields stored at each time step (Figure 4.7).
The image
is constructed when the near-surface scattered waves are in phase with the incident
waves at the scatterer locations. All the scatterers are imaged and located accurately.
Because the reflections are used as the source wavefield, the subsurface interfaces are
not imaged by RTM. The only exceptions, however, are very weak scattered body
waves reflected from the deep interfaces and recorded on the surface. These recorded
phases can slightly contribute to imaging the deep reflectors. In general, artifacts in
the RTM image can be due to many reasons, including the imaging condition, injection
of the receiver wavefield as a boundary condition for backward extrapolation, and the
one side coverage of the receivers.
Scattered body-to-surface waves travel horizontally along the free surface and
attenuate less with distance than body waves. Therefore, they are recorded by all
receivers on the surface and provide optimal illumination of the near-surface layers.
However, because the amplitude of surface waves decays exponentially with depth,
only near-surface heterogeneities that are close to the free surface (e.g., shallower
than one wavelength) can be illuminated and imaged by the scattered body-to-surface
waves. As a result, the intensity of the imaged scatterers with scattered surface waves
decreases with depth and, therefore, body-to-body wave scattering contributes more
to the image.
In terms of data pre-processing, scattered body-to-surface waves have different
slopes than body wave reflections, which can make them easier to separate (e.g.,
using a velocity filter) for subsurface imaging. On the other hand, body-to-body wave
scattering is more challenging to separate from the total wavefield, as the recorded
phases are much weaker in amplitude than scattered surface waves and exhibit similar
slopes to primary P waves.
116
4.4
Conclusion
In this study we have presented a prestack elastic RTM approach for imaging nearsurface scatterers. The image is constructed by forward propagating the source wavefield (e.g., reflected body waves) and back-projecting the receiver wavefield (e.g., nearsurface scattered body to P, S, and surface waves) before a zero-lag imaging condition
is applied to the P wave components (e.g., divergence of the wavefields). The wavefield
extrapolation is performed using an elastic finite difference scheme. We show, using
synthetic data, that elastic RTM of scattered body-to-surface waves constructs a reliable depth image of the near-surface scatterers. The elastic RTM scheme preserves
the relative amplitude because all wave propagation losses, including mode conversions, are properly taken into account. The scattered body-to-surface waves travel
horizontally along the free surface, and, therefore, they provide optimal illumination
of the near-surface. However, the amplitude of scattered body-to-surface waves decays exponentially with depth, and, therefore, only near-surface heterogeneities that
are close to the free surface can be illuminated and imaged by the scattered surface
waves. The proposed imaging approach can be easily extended to 3D problems.
Acknowledgments
We thank Saudi Aramco and ERL founding members for supporting this research.
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53, 1425-1436.
Loewenthal, D., and I. R. Mufti, 1983, Reversed time migration in spatial frequency
domain: Geophysics, 48, 627-635.
McMechan, G. A., 1983, Migration by extrapolation of time-dependent boundary
values: Geophysical Prospecting, 31, 413-420.
Riyanti, C. D., and G. C. Herman, 2005, Three-dimensional elastic scattering by
near-surface heterogeneities: Geophysical Journal International, 160, 609-620.
Shang, X., M. V. Hoop, and R. D. Hilst, 2012, Beyond receiver functions: Passive
source reverse time migration and inverse scattering of converted waves: Geophysical Research Letters, 39.
Sun, R., G. A. McMechan, C.-S. Lee, J. Chow, and C.-H. Chen, 2006, Prestack
scalar reverse-time depth migration of 3D elastic seismic data: Geophysics, 71,
S199-S207.
Virieux, J., 1986, P-SV wave propagation in heterogeneous media: Velocity-stress
finite-difference method: Geophysics, 51, 889-901.
Whitmore, N., 1983, Iterative depth migration by backward time propagation: Presented at the 1983 SEG Annual Meeting.
118
Xiao, X., and W. S. Leaney, 2010, Local vertical seismic profiling (VSP) elastic
reverse-time migration and migration resolution:
mitted P-to-S waves: Geophysics, 75, S35-S49.
119
Salt-flank imaging with trans-
a)
Free-surfaceV
b)
VVV VVVVVV VV
V
Free-surfaceV
V V V V V V V V VV V V V
AV\
AIV\ NVA
/\A
Scattered
body-waves
Reflections
Figure 4.1: Schematic earth model showing: (a) reflected waves as a source for the
incident or source wavefield, and (b) the receiver wavefield is composed of near-surface
scattered waves.
considering the reflected waves as a source for the incident wavefield, and the near-
surface scattered waves as the reciever wavefield
Material Index
1 - Dark blue
2 - Blue
3 - Green
4 - Orange
5 - Red
Vp (m/s)
1800
2200
2500
2700
3000
Vs (m/s)
1000
1200
1300
1400
1500
Density (kg/m 3 )
1750
1900
2000
2100
2250
Table 4.1: Material properties (P wave velocity, S wave velocity, and density) of the
model shown in Figure 4.2.
0
5
100
4
E 200
3
-CL4D-
3~
2
0
200
400
600
Distance (m)
800
1000
i
Figure 4.2: Synthetic earth model. Multiple dipping layers with five circular scatterers
(red circles near the free surface) embedded in the shallow layer. The scatterers are
located at 15 m depth, each is 10 m in diameter and has an impedance contrast
corresponding to 0.36. Material properties are given in Table 4.1. The source is
located at (x,z)=(150 m, 0 m). The receivers are located on the surface with 50 m
near-offset and 5 m space intervals. The color scale (on the right) and associated
numbers refer to material properties given in Table 4.1.
120
(a)
No Scattering
(b)
(c)
with Scatterina
E
E
unset (m)
(x
(d)
Utset
oa t
(
No Scattering
0
-2
(e)
Offset (m)
(M)
2
4
6xatrn
E
Offset (m)
(g)
N-2
-6(
6a
4
w4-2ith
0S 24
X 10
No Scattering
(h)
0
.
2
2
-6
Th
-
4
6xfrn
0.
Di2f0
10,
(i)
4
x108
The Difference
E
utset (m)
unset (m)
1
0
The Difference
uffset (m)
with Scattering
unset (m)
-1
-2
(f)
unset (m)
0 2
E
-2
-6
with Scatterina
0,
(6
The Difference
-2
-1
0
1
2
Figure 4.3: Finite difference simulations showing the v,-component (a-c), divergence
(d-f), and curl (g-i); (a,d,g) incident wavefield simulated using the model without
scatterers, (b,e,h) total wavefield simulated using the model with scatterers, and (c,f,i)
scattered wavefield (i.e., the difference between the total and incident wavefields). A
point source with 30 Hz Ricker wavelet is used. The source is located at 10 m depth
and the receivers are located on the surface.
121
Time = 300 ms
Time = 300 ms
10(
E
20<
N
N 30 (
40
M0
!)W
0U0
/UU
"M0
900
0
1000
100
200
300
400
X (M)
500
X (M)
(a)
(b)
Time = 400 ms
Time = 400 ms
10(
101
E
201
N 30
N 3W
40
401
5a<
400
500
600
700
800
900
1000
0
X (M)
100
200
300
400
500
600
700
800
900
1000
X (M)
(d)
(c)
Time = 500 ms
Time = 500 ms
10C
10(
20(
E
N
40C
500
3a(
40(
0
100
200
300
400
500
600
700
800
900
1000
X (M)
M0
X (M)
(e)
(f)
Figure 4.4: Snapshots of the v,-component (normalized) of the incident (a,ce) and
scattered (b,d,f) wavefields at 300 ms, 400 ms, and 500 ms from top to bottom,
respectively. The seismic source is located at (x, z) = (150 m, 10 m). The source of
scattering is reflected or refracted body waves. The scatterers excite primary, shear
and, also, surface waves due to the proximity to the free surface. Note that the
scattered surface-to-surface waves are removed.
122
Time = 300 ms
N
Time = 300 ms
200
200
300
N 300
400
40O
500
400
500
600
700
800
900
1000
0
X (M)
100
200
300
400
(a)
Time = 400 ms
100
100
E 200
200
N 300
N 300
400
400
200
300
400
500
600
700
800
900
1000
X (M)
X (M)
(c)
(d)
Time =500 ms
Time = 500 ms
loX
10[
20C
N
600
(b)
Time = 400 ms
So0
500
X (M)
20C
30
N 30C
40C
40C
Soc
500
X (M)
U0
700
800
900
1000
500
0
100
200
300
400
500
600
700
X (M)
(e)
(f)
Figure 4.5: Snapshots of the divergence (normalized) of the incident (a,c,e) and scattered (b,d,f) wavefields at 300 ms, 400 ms, and 500 ms from top to bottom, respectively. The seismic source is located at (x, z) = (150 m, 10 m). The source of
scattering is reflected or refracted body waves. The scatterers excite primary, shear
and, also, surface waves due to the proximity to the free surface. Note that the
scattered surface-to-surface waves are removed.
123
Time = 300 ms
Time = 300 ms
10(
10(
-20(
20(
N2 0
N 30(
40
40(
500
600
700
800
900
500
1000
0
100
200
300
400
500
1000
X (M)
X (M)
(a)
(b)
Time = 400 ms
Time = 400 ms
10(
10(
2N(
N 20
401
401
50(
0
100
200
300
400
500
600
700
800
900
5007 100
1000
200
300
400
500
1000
600
X (M)
X (M)
(d)
(c)
Time = 500 ms
Time = 500 ms
10
20
201
N
N 30
0
40(
500
600
700
800
900
500M
0
1000
100
200
300
500
400
600
700
600
900
1000
X (M)
X (M)
(e)
(f)
Figure 4.6: Snapshots of the curl (normalized) of the incident (a,c,e) and scattered
(b,d,f) wavefields at 300 ms, 400 ms, and 500 ms from top to bottom, respectively.
The seismic source is located at (x, z) = (150 m, 10 m). The source of scattering
is reflected or refracted body waves. The scatterers excite primary, shear and, also,
surface waves due to the proximity to the free surface. Note that the scattered
surface-to-surface waves are removed.
124
0
1
100
0.5
E
200
0
N 300
400
500 -0
-0.5
- - - 200
400
600
800
1
1000
(a)
50
0.5
E
100
0
N 150
200
250
-0.5
300
400
500
600
700
_
X (m)
(b)
Figure 4.7: Elastic RTM of near-surface scattered waves with receivers placed on the
surface and a free surface boundary condition applied to the upper boundary. The
yellow dashed lines in (a) correspond to the zoomed area shown in (b).
125
126
Chapter 5
Discussion and Conclusion
This thesis improves our understanding of complexities due to seismic wave scattering by near-surface heterogeneities and develops a method for filtering the scattered
waves. The thesis makes three major contributions. First, it uses finite difference forward modeling to determine the effects of near-surface heterogeneities on the quality
of the seismic data. Second, it develops a multi-stage filtering approach to suppress
the scattered noise from the signal. Last, it introduces an elastic-based reverse time
migration (RTM) approach to image the near-surface scatterers.
To better understand the near-surface scattering mechanisms, we present a numerical approach based on the perturbation method and finite-difference forward
modeling for simulating the effects of seismic wave scattering from shallow subsurface heterogeneities. The scattered wavefield, due to the near-surface scatterers, is
modeled by taking the difference between a reference model without scatterers and a
model with scatterers. As discussed in Chapter 2, the numerical results show that the
direct surface waves and the upcoming P and S wave reflections, including multiples,
scatter by near-surface heterogeneities and generate strong surface-wave energy. In
particular, scattering of upgoing reflections by the heterogeneities to surface waves is
very significant. This scattering can obscure weak primary reflections and contaminate the entire dataset. We carried out extensive numerical experiments to study
the effects of scattered surface waves on S/N. The results show that the scattered
energy depends strongly on the properties of the shallow scatterers. The scattered
127
energy increases with increasing impedance contrast, increasing size of the scatterers
relative to the incident wavelength, decreasing depth of scatterers, and increasing the
attenuation factor of the background medium. Sources located at depths below onethird of the wavelength excite weaker surface waves. However, source depth does not
affect the scattering of reflected body waves. On the other hand, receivers deployed
at depth improve the SNR as they record the weak part of the direct and scattered
surface waves.
In addition to showing the effects of volume scatterers, we also examine the effects
of scattering from an irregular, near-surface interface or bedrock topography. Similar
to scattering from near-surface inclusions, the energy of scattered body-to-surface
waves decreases as the depth of the irregular interface increases. The irregular interface acts as a continuous line of sources for scattered (reflected and refracted) body
waves to surface waves: therefore, the scattered amplitudes decrease as the depth to
the interface increases. Compared to scattering from finite scatterers, scattering from
an irregular interface exhibits more diffusive-type scattering.
In terms of enhancing the quality of the seismic image and reducing the effects of
the scattered surface waves, we developed a framework based on steerable filters to
estimate the spatially varying slopes in the data and on directional non-linear filter
to separate the signal and noise components. The filtering scheme is designed based
on our understanding of the mechanisms and behaviors of the simulated scattered
surface waves as discussed in Chapter 2, in which the noise slope is constant with
time but can vary with offset. The slope estimation step using steerable filters requires
only a linear combination of a set of basis filters at fixed orientation to synthesize an
image filtered at an arbitrary orientation, which makes the process very efficient. The
method can handle locally-linear coherent noise that has small amplitude contrast and
varying slope with offset. We applied our filtering approach to simulated data as well
as to onshore field data to suppress the scattered surface waves from reflected bodywaves, and we demonstrate its superiority over conventional
f
- k techniques in signal
preservation and noise suppression.
To locate and image the near-surface heterogeneities, we introduced a prestack
128
elastic reverse time migration (RTM) approach based on the near-surface scattered
waves. The image is constructed by forward propagating the source wavefield (e.g.,
reflected body waves) and back-projecting the receiver wavefield (e.g., near-surface
scattered body-to-surface waves) before a zero-lag imaging condition is applied to
the P waves (e.g., divergence of the wavefields). The scattered body-to-surface waves
travel horizontally along the free surface, and, therefore, they provide optimal illumination of the near-surface compared to scattered body-to-body waves. However, the
amplitude of scattered body-to-surface waves decays exponentially with depth, and,
therefore, only near-surface heterogeneities that are close to the free surface can be
illuminated and imaged. The elastic RTM scheme preserves the relative amplitude
because all wave propagation losses, including mode conversions, are properly taken
into account. We demonstrate, using synthetic data, that elastic RTM of scattered
body to P, S, and surface waves constructs an accurate and reliable depth image of
near-surface scatterers.
5.1
Future Work
The ultimate goal in modeling near-surface scattering is to study the effects of more
general cases of surface topography with internal scattering in 3D. It is not trivial
to incorporate surface topography in 3D finite difference computations while keeping
the computational cost within reasonable levels. For the ADER-CV scheme discussed
in Appendix A, computing the space partial derivatives with cross terms in 3D can
dramatically increase the computational cost.
The issue of computational cost in
3D elastic wave modeling still persists even in the simple case of a flat free surface
boundary. We have emphasized 2D earth models in most of our studies, but we plan
to extend our simulations and techniques for modeling and imaging more realistic
3D models. Additionally, we plan to apply the filtering approach based on spatially
varying slopes to field data with constant fold, and to investigate the performance of
the filter after CMP stack. Ultimately, we plan to also apply the algorithm to a 3D
field dataset, mainly using steerable filters to guide a local velocity filter such as an
129
f
-
k filter.
130
Appendix A
Finite Difference Elastic Wave
Modeling with an Irregular Free
Surface Using ADER Scheme*
Abstract
To find fast and reliable methods for modeling elastic wave propagation and simulating the scattering effects caused by irregular surface topography. Surface topography
and the weathered zone (i.e., heterogeneity near the earth's surface) have great effects
on elastic wave propagation. Both surface waves and body waves are contaminated
by scattering and conversions by the irregular surface topographic features.
We developed a 2D numerical solver for the elastic wave equation that combines a
4 - order ADER scheme (Arbitrary high-order accuracy using DERivatives), which
is widely used in aeroacoustics, with the characteristic variable method at the free
surface boundary. The method is based on the velocity-stress formulation. The
ultimate goal was to develop a numerical solver for the elastic wave equation that is
stable, accurate, and computationally efficient. The solver treats smooth arbitraryshaped boundaries as simple plane boundaries. The computational cost added by
treating the topography is negligible compared to flat free surface because only small
number of grid points near the boundary need to be computed. In the presence of
topography, using 10 grid points per shortest S-wavelength, the solver yields accurate
results. Benchmark numerical tests using several complex models that are solved by
our method and other independent accurate methods show an excellent agreement,
confirming the validity of our method for modeling elastic waves with an irregular
*The bulk of this appendix has been submitted for publication as: AlMuhaidib, A. M. and
Toks6z, M. N., Finite difference elastic wave modeling with an irregular free surface using ADER,
scheme, Journal of Seismic Exploration (submitted)
131
free surface.
A.1
Introduction
Numerical modeling of elastic wave propagation plays a key role in almost every aspect of seismology as it provides a means of explaining the recorded signal associated
with complex earth models. Most of the numerical schemes for solving the wave equation are either based on the "strong-formulation" (e.g., finite difference and spectral
methods) or the weak-formulation (e.g., finite elements, spectral finite elements, and
discontinuous Galerkin).
Finite-element methods have an advantage over other numerical methods because
they have the flexibility to model irregular boundaries. However, modeling seismic
wave propagation with FE methods is (i) computationally much more expensive than
finite difference especially in 3D, (ii) requires mesh generation and adaption that can
be labor intensive and not easily automated, and (iii) can impose stability restrictions
due to the need for very small geometrical elements near the boundary and thus
requiring very small time steps compared to finite difference schemes.
The difficulties with finite difference modeling are mainly in representing and
constructing the numerical grid near a topographic surface and in determining how to
accurately satisfy the traction-free boundary conditions on the rough surface. Several
approaches to handle irregular free surfaces in finite difference simulations exist in the
literature, with different drawbacks. (I) The simplest approach is the heterogeneous
formulation, also known as the vacuum method (Kelly et al., 1976; Virieux, 1986;
Muir et al., 1992). This implicit approach is implemented easily by setting the elastic
parameters above the free surface to vanish and using a small density value in the
first velocity layer above the free surface to avoid a division by zero. However, the
accuracy of the vacuum method decreases when the angle between the boundary
and the meshing increases (Graves, 1996; Bohlen and Saenger, 2006). (II) A second
approach is to handle the free surface explicitly using the image method in staggeredgrid schemes, which was first developed to deal with flat surfaces (Levander, 1988),
132
and then extended to irregular topography (Jih et al., 1988; Robertsson, 1996; Graves,
1996; Ohminato and Chouet, 1997).
However, the image method suffers from the
discretization error due to the staircase approximation of the surface topography,
which may have an effect on the physical conversion and scattered waves.
(III) A
third approach to handle the surface topography is by using conformal mapping and
solving the elastic wave equation in curvilinear coordinates (Hestholm and Ruud,
1998; Zhang and Chen, 2006; Appel6 and Petersson, 2009). This approach requires
mesh generation and adaption, and it involves expanding the first order hyperbolic
velocity-stress equations in curvilinear coordinates, which can be very expensive for
large-scale problems.
To avoid most of these drawbacks, we developed a 2D finite difference LaxWendroff-type integration scheme that has arbitrary high-order accuracy in time and
space (Schwartzkopff et al., 2004; L6rcher and Munz, 2007). The solver combines a
4th-order ADER scheme (Arbitrary high-order accuracy using DERivatives) with a
characteristic variable method (Bayliss et al., 1986; Giese, 2009) at the free surface
boundary. The ultimate goal was to develop a numerical solver for elastodynamics,
in a time-domain velocity-stress framework that is stable, accurate, computationally
efficient, and can handle smooth arbitrary-shaped boundaries (i.e., topography). To
validate this method, we carried out several numerical tests and benchmarked the
finite difference solver with an independent accurate staggered-grid finite difference
scheme (Virieux, 1986) that is 2 "d and
4 th
order accurate in time and space, respec-
tively, and the conformal mapping method (Zhang and Chen, 2006).
A.2
Formulation of Elastic Wave Modeling
Instead of using the wave equation that is a second order hyperbolic system, we
followed the formulation of the elastodynamic equations. For a 2D Cartesian system
with a horizontal positive x-axis pointing to the right, and a positive vertical z-axis
pointing down, the basic governing equations that describe elastic wave propagation
133
(in the velocity-stress formulation) are the equations of motion (Virieux, 1986)
av ,
OoXX+
arxz
at
ax
az'
pv
at
=
0z +
ax
,0z
Oz
(A. 1)
and the constitutive laws for an isotropic medium:
atX
at
__
at
=
(A+2p) OX +A az,
Oz
=
(A + 2p)
__xz
Z+ A
az
(Va + avz>
at
az
where vx and vz are the velocity components,
ax
(A.2)
ax
'
-ij are the stresses, A and p are the
Lame parameters, and p is density.
A.3
Numerical Analysis
The Lax-Wendroff scheme (Lax and Wendroff, 1960) that has second order accuracy
in time and space, is based on central differences in space and the first three terms of
a Taylor expansion in time:
u (xi, to) + At ut (xi, tn) + A2!2
u (xi, tn+1)
Zutt
(xi, tn) +I
...
(A.3)
Here, U = [vx, vz, oxx, ozz, O-xz]T. The first step to obtain 4 th order accuracy in time
and space (i.e., arbitrary higher order) is to use polynomial interpolation of spatial
derivatives of the state vector (i.e., the governing elastodynamic equations) at each
mesh point Pj that is of
4 th
order in space on a Cartesian grid (Schwartzkopff et al.,
2004). The basis functions used for the interpolation are defined to be monomials. A
basis in 2D domain is provided by:
PT W
= PT(Xy):=
:1 XYXY X2y 2XM
134
IYM}
(A.4)
Hereby, pT (X) is a set of 2D monomials with order m, and n =
(m
+
1)2
elements.
The interpolating polynomials for the scalar field u with values at all grid points is
defined with this basis as:
n
u (x, xQ)Zpi
(x) ai(xQ)
=PT
(A.5)
(x)a (xQ),
where a, (xQ) is an unknown coefficient for the monomial pi (x) corresponding to the
given point xQ (Liu, 2010). The coefficient a, (xQ) can be determined by enforcing
n nodes in the support domain of point xQ as
equation (A.5) to be satisfied at the
follows:
US = PQa,
(A.6)
where U == {U1 , U2, .. , U'n}T is the vector that collects the values of field variables at
all the n nodes in the support domain, and PQ = [p (XI) , p (x 2 ),... , p (X)]T is called
the moment matrix (Liu, 2010). For 2D case:
2
U1
1 x1
Y2
xly 1
U2
1 x2
Y2
x2y2 x 22
2
2
2
X 2 Y2
y1
Y2
2
x 31
al
2
x23
a2
Xn3
an
X 2 Y2
Xlyl
an
1
Xn
yn
xn yn
Xn2
2
2
XnYn
Yn
(A.7)
Note that the number of nodes in the support domain equals the number of basis
functions, and hence PQ is a square matrix with the dimension of (n x n) .
Assuming the inverse of the moment matrix PQ exists, we can find the unknown
coefficients:
a
(A.8)
PfU .
Substituting into the interpolating polynomial of the scalar field:
n
U (x) = PW P 1 U, -- Z
135
#i ()
-:
=
(x) U",
(A.9)
where <D (x) is a matrix of shape functions
#
(x) defined by
(A.10)
'I)(W = PTS-~1 = [#1 (X) , 02 (X) , 03 (X) , - . - , #n (X)]T .
The derivatives of the shape functions can be obtained easily as all the functions
involved are polynomials. The
1 th
<bA)
derivative of the shape functions are given by
(x) =
(A.11)
[p(l (x)]TP1.
The interpolation stencil is symmetric with respect to the interpolation point for
even order of accuracy. Therefore, we obtain a central scheme that is independent of
the direction of wave propagation. The size of the interpolation stencil (operator) for
the 4 th order central difference is 5 x 5 as shown in Figure A.1. As mentioned earlier,
to obtain
4 th
order accuracy in time, we truncate the Taylor expansion in equation
(A.3) at the fifth term and replace the time derivative with central
4 th
order space
derivatives. More details about the discretization of the 4 th order ADER scheme and
the derivation of the time derivative in terms of the space derivatives are provided in
Appendix B.1.
Figure A.1: ADER 4 th order stencil with 25 points.
136
A.4
Stability of the ADER Scheme
The von Neuman stability condition for the ADER method on a regular Cartesian
grid (with Ax = Az) is:
At < Ax
(A.12)
where V is the maximum P wave velocity (Schwartzkopff et al., 2004). The stability
condition is independent of the S wave velocity as P waves are always faster than
shear waves.
A.5
Numerical Dispersion
To resolve the waves that are generated by the source and to avoid numerical dispersion, it is essential to use a grid size that is sufficiently small.
However, using
unnecessarily small grid size is inefficient, because both the computational demand
and the memory requirements increase with decreasing grid size. The number of grid
points per shortest wavelength, P, is a normalized measure of how well a solution is
resolved on the computational grid. Since the shear waves have the lowest velocities
and a shorter wavelength than the compressional waves, the shortest wavelength
Amin
can be estimated by
Amin =
minV
(A.13)
frna
fmax
where V, is the shear velocity of the material properties and
frequency in the source time function
points per wavelength equals
Amin/AX
(fmax=
fmax
is the maximum
2.5fc). Therefore, the number of grid
, which is given by:
P PzminV
=
.f
AX fmax,
(A.14)
The number of grid points per minimum wavelength required for the ADER
scheme to obtain accurate results is 10 (Lombard et al., 2008). For the simulations
involving the free surface interaction, numerical experiments show that our approach
gives accurate results for P > 10, but the exact number depends on the distance
137
between the source and receiver.
A.6
Initial Conditions
Stresses and velocities are set to zero because it is assumed that the medium is in
equilibrium at t = 0. Therefore, propagating time integrated velocities and stresses
are equivalent to propagating velocities and stresses.
A.7
Boundary Conditions
The internal interfaces are represented by the so called effective medium parameters
(Moczo et al., 2002). The density is calculated by arithmetic average, and the Lame
parameters are calculated by harmonic average.
In principle, we would like to model wave propagation in an infinite domain. In
practice, however, we perform the computations in bounded domains (e.g., with artificial boundaries). Therefore, it is essential to prevent waves from reflecting back
into the computational domain from the boundaries (unacceptable artifacts) to correctly describe the physical behavior in unbounded domain. We applied an absorbing
boundary condition for the three artificial edges of the grid (other than the free surface), so only reflections due to the medium interfaces are recorded and reflections
from the three edges are strongly absorbed.
A.8
Implementation of Free Surface Condition with
Surface Topography
A.8.1
Free Surface Condition in 1D (Flat)
The boundary conditions at the free surface are zero normal and shear stresses oz=
OZz = 0. The interior scheme requires the use of two nodes in every direction from
the point being advanced as shown in Figure A.1. The boundaries are taken into
138
account explicitly by using ghost points for points beyond the free surface to impose
the physical boundary condition. Thus, at each time step, all boundary fluxes are
updated at points outside the computational domain using the characteristic variable
method (Gottlieb et al., 1982; Bayliss et al., 1986). The variables v,
v2 , and cTx, are
calculated by mirroring based on the characteristic variables, which are defined as:
V
9
z
U
where v9, vf ,
(9,
z,
9XX=
J p/I
Uzz
VZ +
V9
+
=
X
-_
(A+2p)p
7Z
zz
0
crz
0
A
(A+2p)
,(A.
15)
and oag are the ghost values outside the computational
domain obtained by mirroring the fluxes from the interior domain (see Appendix B.2
for more details). Combining the free surface boundary condition with the mirrored
fluxes of the characteristic variables makes it possible to obtain all of the dependent
variables.
A.8.2
Free Surface Condition in 2D (Irregular)
The boundary treatment described in this section is based on the idea developed
by Forrer and Jeltsch (1998) for the Euler equations.
The first step to find the
mirror points inside the computational domain is to calculate the normal vector to
the boundary for each ghost grid point. Therefore, we used the fast marching level set
method (Sethian, 1996) to compute the signed distance function of the free surface
(see Appendix B.3).
Then, we used the distance function to find the mirror point
inside the domain for each ghost point outside the domain as shown in Figure A.2.
The computation of the normal vector and the projection needs to be done only once,
and its computational cost is negligible.
The mirror point is interpolated using a
2 nd
order accurate Lagrange interpolation
method in two dimensions (see Appendix B.4). In order to mirror the fluxes of the
139
characteristic variables, we rotated the coordinate frame by an angle a, so the normal
direction is pointing in the positive z-direction (vertical) in the new coordinate frame.
Then the ghost values are determined before rotating back to the original coordinate
frame. The idea can be summarized as follows:
" Transform the stresses and velocities into a local coordinate frame U = RaU
" Obtain the ghost state vector U using the characteristics as shown in (A.15)
" Rotate back to the original coordinate frame Ug = R(_,)U
where Ra is a linear transformation matrix given in Appendix B.5.
A.9
Numerical Experiments
In order to validate the ADER-CV method, we consider ten numerical tests; one to
test the method inside the computational domain; four inclined straight boundaries
with different dipping angles; three Gaussian shaped hill free surface models with
different sizes; and two ramp shape free surfaces models.
A.9.1
Configurations
We simulated the excitation of a point source with a Ricker wavelet by adding a
known value to the stress
g (t)
where
fc
{1 - FD [7fc (t - tc)] 2 } e_[7f(tc)1 2
(A.16)
is the central frequency, and t, is the time delay. We used a 15 Hz source
central frequency and 0.1 s time delay for the planar and inclined boundary, and 10
Hz and 0.3 s time delay for the Gaussian and ramp shape surfaces. The maximum
frequency is defined as fm=
2.5fc. A 2D homogeneous earth consisting of a free
surface over half a space is used as an example. The domain has 501 x 501 grid points
with 5 m grid spacing, that is 2500 m depth (along the z-axis) and 2500 m distance
140
(along the x-axis). All benchmark tests were calculated with a Courant number of
(z"vp
=0.9), which satisfies the stability condition discussed earlier. A benchmark
of the ADER against a reference
4
"h order staggered-grid FD scheme for a source and
receiver located inside the domain with no free surface is shown in Figure A.3.
A.9.2
Planar Boundary
We benchmarked the ADER scheme combined with the characteristic variable boundary method (ADER-CV) in the presence of a free surface with a source located at
(750 m, 50 m), and a receiver located at (1750 m, 50 m) as shown in Figure A.4.
The distance between the source and the free surface is 50m < Ap/3, and therefore large Rayleigh waves are generated. The medium properties are; V = 3000 m/s,
V, = 1730 m/s, and p = 2500 kg/m 3 . The ADER-CV results (with only 9 grid points
per shortest S-wavelength) show excellent agreement with the results obtained by the
4 th
order staggered-grid scheme (with 40 grid points per shortest S-wavelength used
to obtain a very accurate reference solution).
A.9.3
Inclined Straight Boundary
Similar tests were carried for 26.50 and 45 0 -inclined straight free surfaces with 1000 m
offset and 50 m depth as shown in Figure A.5 and Figure A.6, respectively. The
medium properties are: Vp = 3000 m/s, V, = 1730 m/s, and p = 2500 kg/m
3
. The
results here are compared with the results obtained from the staggered-grid scheme
for the flat homogeneous case. The agreement between the results obtained by the
two methods is excellent. Snapshots of the wavefield at different instants in time are
shown in Figure A.7 for the 45 0 -inclined straight boundary. One advantage of the
ADER-CV method compared to other methods (e.g., vacuum method) is that the
relative error is independent of the dip angle as shown in Figure A.8. The error is
defined as
£
_
1Ucaic - Uref
2
I Uref H
141
2
11
where |1 . || is the
12
norm, ucaic is the calculated solution using the ADER-CV
scheme with 9 grid points per shortest S-wavelength, and uef is the reference solution
obtained using the
S-wavelength.
4 t"
order staggered-grid scheme with 40 grid points per shortest
The recorded seismograms were rotated by the same dip angle to
match the flat case, so they can be directly compared. Successful modeling of wave
propagation independent of the slope of the surface implies that the algorithm should
also allow for accurate modeling of an arbitrary-shape free surface.
A.9.4
Gaussian Shaped Hill Topography
More tests were performed for a Gaussian shaped hill free surface. The shape of the
hill is defined by a Gaussian function
2
Elevation = h e-(xx0)
/W2
where h and w are the height and width, respectively. xO is the location of the center of
the Gaussian function. The medium properties are: V = 2500 m/s, V = 1200 m/s,
and p = 2000 kg/rn3 . The number of grid points per shortest S-wavelength is about
10. To verify the effect of the size of the Gaussian shaped surface on the waveform
with a fixed source wavelength, Gaussian shaped surfaces with 50 m, 100 M, and
200 m height and width are considered here.
Gaussian Surface with 100 m Height and 100 m Width
The source was located at the middle of the computational domain at 1000 m depth,
simulating an earthquake type scenario with a Gaussian shaped hill free surface (100 m
height and 100 m width) as shown in Figure A.9a. The distance function was computed using the fast marching level set method to find the location of the mirrored
points inside the domain corresponding to the ghost points as shown in Figure A.9b.
Comparisons of the recorded pressure from the ADER-CV method and the boundary
conformal method (Zhang and Chen, 2006) are shown in Figure A.10a. The agreement between the results calculated by the two methods is excellent.
142
Comparison
of the ADER-CV solutions calculated with different grid spacings shows convergence
of the method as Ax decreases as shown in Figure A.10b. Snapshots of the wavefield (v,-component) at different instants in time showing the scattering and multiple
reflections caused by the irregular surface are shown in Figure A.11.
A.9.5
Ramp Shape Topography
Finally, two tests of models (homogeneous and one layer over a half space) with a
ramp shape free surface are presented, in which the topography has a significant
impact on the seismic response. The slope of the ramp edge tends to infinity, making
it an extreme topographic model.
Homogeneous Model
Time series of the velocity components along the free surface of the ramp shape model
are shown in Figure A.12. The middle panel shows the horizontal velocity component
vx; and the bottom panel shows the vertical velocity v,. Due to the existence of sharp
edges (infinite slope), strong and complex multiple body, shear, and several Rayleigh
wave packages are clearly identified on the synthetic seismograms shown in Figure
A.12. Snapshots of the wavefield (both vx and v, components) at different instants in
time showing the scattering and multiple reflections caused by the irregular surface
model with homogeneous velocity are shown in Figure A.13.
One Layer over Half Space
Time series of the velocity components along the free surface of the ramp shape model
are shown in Figure A.14. The middle panel shows the horizontal velocity component
vx; and the bottom panel shows the vertical velocity o.
Snapshots of the wavefield
at different instants in time showing the scattering and multiple reflections caused by
the irregular surface model with one layer over half space are shown in Figure A.15.
The snapshots and the time series clearly illustrate how a simple one layer over a half
space can generate complex waveforms in the presence of surface topography, largely
143
due to scattering and multiple reflections caused by the trapped energy within the
low velocity layer (e.g. the weathered zone).
A.10
Summary
A 2D ADER time-domain single-grid finite difference method that is
4 t"
order accu-
rate in time and space (with velocity-stress formulation) was implemented to model
wave propagation in linearly elastic, isotropic homogeneous media with an irregular
surface. The characteristic variable method is implemented to account for the free
surface B.C., and extended to handle arbitrary smooth boundaries. The scheme does
not require mesh generation and adaption, and it does not involve expanding the
governing equations in curvilinear coordinates, which can be computationally and
labor intensive as in the conformal mapping method. Because only small number of
grid points near the boundary need to be computed, the computational cost added
by treating the topography is negligible compared to flat free surface . The main limitation of this scheme is handling sharp boundaries where the normal vector cannot
be uniquely defined.
The ADER scheme is very memory efficient as it uses only two time levels to
obtain high order of accuracy in time. On the other hand, all explicit Runge-Kutta
time integration schemes require storing more time levels (i.e., integration stages)
than their order of accuracy, which can be memory inefficient specially for order of
accuracy greater than four. The increase of the order of accuracy allows the simulation
of elastic wave propagation over long distances with small dispersion and dissipation
errors and obtains highly accurate results on coarse meshes. Grid spacing and time
steps were carefully chosen to satisfy the stability condition and to avoid numerical
dispersion. The B.Cs. at the free surface are zero normal and shear stresses. The
solver was benchmarked against a standard
4 "h
scheme, and the boundary conformal method.
144
order staggered-grid finite difference
Acknowledgments
We thank Wei Zhang for providing the benchmark results of the Gaussian-shape
topography. We would like to thank Saudi Aramco and ERL founding members for
supporting this research.
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145
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147
4
Z=()
Z=O
Figure A.2: Determination of ghost values required for time-marching at neighboring
grid nodes. The blue circles correspond to the ghost point (outside the domain) and
its orthogonal projection on the surface (inside the domain). Lagrange interpolation
(left) and extrapolation (right) in 2D are used to estimate the point inside the domain
that will be then used to impose the boundary condition at the ghost point.
Relative Seismogram Misfit
Gaus-100 (Ax = 2.5m)
Gaus-100 (Ax = 5m)
Gaus-100 (Ax 1Oin)
Gaus-FDOO (Ax = 5m)
Gaus-50 (Ax = 5m)
(%)
0.64
1.19
3.18
0.58
1.35
Table A.1: Relative misfit of the ADER-CV method compared with the boundary
conformal method for different grid spacings and Gaussian topography sizes.
148
Staggered FD
ADER-CV
-----
05 500
---
> 0
-0.5 -
1000
0
01
02
03
0.4
05
Time (sec)
0.6
07
0.0
09
0
0.1
02
03
0.4
0.5
Time (sec)
086
07
0.8
09
1500
7
00
2000
Vp-3000 ms
2500
500
Vs-1730mIs
1000
c.2500ckgim
1500
0
-0.5
2000
-1
2500
x (M)
Figure A.3: The computational domain is shown to the left; the source (red) and
receiver (blue) with 1000 m offset. To the right are comparisons of the recorded v,
and v, components. The ADER-CV solution (dashed red) is plotted against the 4 th
order staggered-grid FD solution (black) at the selected observation point.
7
*
0,5
---
Staggered FD
ADER-CV
500
-0.5
1000
-1
0
01
02
03
04
0
01
0.2
0.3
0.4
05
06
Time (sec)
07
06
09
1
07
08
019
1
I
1500
05
S0
2000
Vp-3000mis
Vs-1730mts
c-
2500kgIm
3
-005
2500
0
500
1000
1500
2000
-1
2500
x (M)
0S
0.6
Time (sec)
Figure A.4: The computational domain is shown to the left; the source (red) and
receiver (blue), with 1000 m offset and 50 m normal distance from the free surface.
To the right are comparisons of the recorded v., and v, components. The ADERCV solution (dashed-red) is plotted against the 4 th order staggered-grid FD solution
(black) for the flat layer model at the selected observation point.
149
Staggered FD
0.5
ADER-CV
---
500
>
0
-05
1000
-1
0
0.1
0.2
0.3
04
0.1
0,2
03
0.4
N
1500 -
0.5
06
Time (sec)
07
0.0
0.9
1
0.7
0.6
0,9
1
1
0 5-
2000
0
-
c-2500kgmn
Vp-3000 mis Vs-1730mis
2500 'I1-11
0
500
1000
1500
2000
-05
2500
0
x (M)
11
0.5
086
Time (sec)
Figure A.5: The computational domain is shown to the left; the source (red) and
receiver (blue) with 995 m offset and 45 m normal distance from the (26.50) inclined
free surface. To the right, comparisons of the recorded v. and v, components (i.e.,
parallel and normal to the inclined surface, respectively). The ADER-CV solution
(dashed-red) is plotted against the 4 "h order staggered-grid FD solution (black) for
the dipping layer model at the selected observation point.
01
-0.5
500
Staggered
FD
ADER-CV
0
-0.5
1000
1
1500
0
01
02
0.3
0.4
0
0.1
02
0.3
04
05
0.6
Time (sec)
07
0.6
0.9
1
017
0.6
019
1
1
05
2000
0
Vp-3000mis V-173mis
2500
0
500
1000
c-2500kgnm
1500
2000
-0.5
2500
x (M)
05
06
Time (sec)
Figure A.6: The computational domain is shown to the left; the source (red) and
receiver (blue) with 1000 m offset and 50 m normal distance from the (450) inclined
free surface. To the right, comparisons of the recorded v, and v2 components (i.e.,
parallel and normal to the inclined surface, respectively). The ADER-CV solution
(dashed-red) is plotted against the 4t" order staggered-grid FD solution (black) for
the dipping layer model at the selected observation point.
150
Time
-
Time - 225 ms
225 ms
50(
500
0
1
100(
00
N
N
1500
150C
2000
200(
00
2
500
1000
2000
1500
250(
2500
500
0
1000
1500
2000
2500
2000
2500
X (M)
X (M)
rime - 450 ms
Time
-
450 ms
0
50(
1000
1002
N
91111
1500
'
150(
2000
2500 0
500
1500
1000
2000
500
2500
1000
1500
X (m)
x (M)
Figure A.7: Snapshots of the wavefield at 225 ms and 450 ms for the 45 0-inclined
free surface model shown in Figure A.6. The left hand side panels are the horizontal
velocity v,, and the right-hand side panels are the vertical velocity v,.
Error as
function of dipping angle
0
0
7
0.
0
6
--
cc0
0
0
.1. .
0
5
12
15
25
20
Angle(')
N
....
....
35
40
45
Figure A.8: Relative error in terms of the boundary's dip-angle.
151
000
1000
500
1800
1400
1000
1200
1000
800
100
G00
400
2000
Yp-2500mrrs Vs-12OmWS
,!ZUU
500
1000
00
;-2000kgrm3
100
2000
2500
100
X(m)
200
300
400
500
-200
Figure A.9: The computational domain is shown to the left; source (red) and receivers
(blue) with a Gaussian shaped hill free surface (100 m height and 100 m wide). To
the right is the distance function computed using the fast marching level set method.
The color bar indicates the normal distance in meters to the free surface.
Pressure (Txx+.Tzz)
100
--
1-
Pressure (T>.Tzz)
Conformal Mapping
ADER-CV
--- ADER-CV dx-10m
1400
0
-
06
_
04
-
AOER-CV dx-.Sm
eila 1D .. m
Residual (5m-ir5m)
1300
'5
oo
100
04
0.8
000
9001
0
02
04
0.6
0.0
1
10
03
Time (sec)
0.4
05
0,6
07
Time (sec)
0.
0.9
1
Figure A.10: To the left are comparisons of the recorded pressure at the receiver
locations shown in Figure A.9; ADER-CV (dashed red) against the boundary conformal solution (black). To the right is a comparison of the ADER-CV solutions
with different grid spacings showing convergence of the method as the grid spacing
decreases.
152
Time -
600 ms
Time
500
50(
1000
1000
1500
1500
2000
2000
-
600 ms
N
2500
0
500
1500
1000
2000
0
2500
500
1000
Time
- 700
1500
2000
2500
2000
2500
X (i)
x (M)
ms
Time - 900 ms
0
500
500
1000
1000L
1500
1500
2000
2000
N
2500
0
500
1000
X (m)
1500
2000
2500
20
500
1000
X()
1500
Figure A.11: Snapshots of the wavefield v, component at different instants in time
showing the scattering and multiple reflections caused by the irregular surface model
shown in Figure A.9.
153
500
1000
1500
2000
Vp = 2500 m/s
2500C
0
500
Vs = 1200 m/s
1000
p - 2000 kgIm
1500
3
2000
2500
x (m)
Horizontal Velocity Vx
1800
1600
.
1400
M 1200
E
0
1000
800
0.4
0.5
0.6
Time (sec)
0.7
0.8
0.9
1
Vertical Velocity Vz
1800
1600
7
1400
~
1200
E
1000
800
0
0.1
0.2
0.3
0.4
0.5
0.6
Time (sec)
0.7
0.8
0.9
1
Figure A.12: Time series of the velocity components along the free surface of the
ramp shape surface model shown at the top. The middle panel shows the horizontal
velocity component vx, and the bottom panel shows the vertical velocity v. The
obvious phases are labeled, where P indicates P wave, R indicates Rayleigh wave,
and PRrefl indicates P to Rayleigh reflection.
154
Time
-
400
ms
Time
.- 400 ms
500
1000
N
1500
2000
2500,
0
500
1000
1500
2000
2500
500
2500
1000
Time -
1500
2000
2500
2000
2500
X (m)
x (M)
600ms
Time
-
600 ms
500
1000
150050
10001
q
50
00
1500
10
2000
200
2000
500
1000
X ()
1500
2000
2500
0
500
1500
1000
X (m)
Figure A. 13: Snapshots of the wavefield at different instants in time showing the
scattering and multiple reflections caused by the ramp shape surface model with
homogeneous velocity. The left hand side panels are the horizontal velocity v,, and
the right-hand side panels are the vertical velocity v,.
155
500
1000
Vp = 2500 m/s
Vs = 1200 m/s
p = 2000 kg/m3
Vp = 4200 m/s
Vs = 2000 rn/s
p = 2800 kg/m3
N
1500
2000
0
500
1000
1500
2000
2500
x(m)
Horizontal Velocity Vx
CD
M
0
0.2
0.4
0.6
Time (sec)
0.8
1
Vertical Velocity Vz
E
18
0
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure A.14: Time series of the velocity components along the free surface of the
ramp shape model shown at the top. The middle panel shows the horizontal velocity
component vx, and the bottom panel shows the vertical velocity v,.
156
Time -400
mms
Time - 400 ms
50
500
0
100
1000
0
N
150
200
1500
0
2500
0
t
2000
500
1000
1500
2000
2500
0
500
X (m)
Time
-
1500
2000
2500
2000
2500
X (M)
600 ms
Time
0
50
1000
0
-
600 ms
500
1001C
1000
N
150C
1500
200C
2000
2500
500
1000
1500
2000
2500'
2500
5
X (m)
500
1000
1500
x (M)
Figure A.15: Snapshots of the wavefield at different instants in time showing the
scattering and multiple reflections caused by the ramp shape surface model with one
layer over half space. The left hand side panels are the horizontal velocity v2, and
the right-hand side panels are the vertical velocity v.
157
158
Appendix B
In this appendix, more details about the implementation of the numerical scheme and
the boundary treatment for the ADER-CV method are provided. The discretization
of the
4 t"
order ADER scheme and the derivation of the time derivative in terms of the
space derivatives are described in Appendix B.1. The characteristic variable method
(Gottlieb et al., 1982; Bayliss et al., 1986) to update all boundary fluxes at points
outside the computational domain by mirroring the fluxes from the interior domain are
demonstrated in Appendix B.2. The fast marching level set method (Sethian, 1996)
to compute the signed distance function of the free surface is reviewed in Appendix
B.3. The distance function is used to find the normal vector to the boundary and
the mirror point inside the domain for each ghost point outside the domain. In case
the mirror point does not lay on the computational grid, it is interpolated from its
neighboring points using a
2 nd
order accurate Lagrange interpolation method in two
dimensions as shown in Appendix B.4. The linear transformation matrix to rotate
the coordinate frame is given in Appendix B.5.
B.1
Fourth Order Lax-Wendroff-type Scheme
=oU ±t1
Ut
where U = vX, vZ, xxOzz
At 2
0 2Un
At 3
a3 Ur
At 4
4
2
8t2
6
8t 3
24
0t 4
Xz T and
159
Un
b(C[xx + C'
z)
b(C~xz + CO'zz)
+ ACOi
(A + 2p)C±
at
ACf + (A + 2)Coi
P(CIY + COI )
b[(A + 2p)Ci§ff+ 2pCiC + (A + p)Cz]
b[(A + pI)Cvx + 2(A + 2p)Cj + 2pCYg]
2
b[2(A +p-)CTx+2ACjz+2(A+p)Cxz]
b [2AC2x + 2(A + M)CBiz + 2(A + p)CTxz]
b [
pCyxx + C~zz + 2(C2xz + Ciz)]
b2 [6(A + 2p)CfTx + 2pCTx + 2(A
+ p)CUz
b2 [2(A + pt)CT xx+ 2pC2Tz + 6(A + 21 )Cgz
b [6(A+ 2p) 2 Ci + 2(A 2 +2Ap+ ±2p
__
2
+ 2(2A + 3p)C2TZ + 6pCBZI
+ 2(2A + 3p)CTxz
)C~4 +2(A
2
+44 p+2M
+ 61p
Cfz]
2
1
)Cl
[+6A( A±+2p)C~
6(A + 2/p) 2 C
+ 2(A 2 + 4A/i + 2p 2 )CYx + 2(A 2 + 2Apu + 2p 2 )C2V
=
[
+6A(A + 2,p)COizI
byu [2(2 A + 3pt)(CYt + CY ) + 6p(Cli + Cug)]
246 2 C
+4(A 2
4A ++52 )c2
+ 6(A2 +44
+3p 2)(Ci + C)
+24(A + 2pt)2 Cx
+2
&4 U
b2
[
b 242i+ 4(A2+±4AM +5p
2
2 4 1 2 0 + 2p) Cl
2
x
[24(A + 2p) 2 C x + 4(A 2 +2Ap + 2M2 )Cfzx + 4(A 2 +4AM + 2p 2 )CL]z+
24A(A + 2p)C
3 + A2 +
+2
24(A + 2p) 2 Cjz + 4(A 2 + 2Ap + 2
24A(A+ 2M)C
+ 12(A 2 + 2AM+
b2
+ 4A+32(CVZ+i
( 2 i,1'13
)Cl
a
6(2A + 3p)(C
3 Tx + C(
+8(2A
+ 3 M)CLiz
where b =i/p
,
) + 6 (C
2
2
+ 2p2
+ A2 + p2)CUZ
)CLiX + 4(A 2 + 2AM + 2p 2 )CfCz+
2
+ 12(A 2 +4AM + 3
)C
x
+ C 4Z ) +24p(C
2
)CT
t2z +C&Tz
and C is the space derivative of the scalar field a obtained
by multiplying the lthderivative of the shape function <D (x) by the values of the field
160
]
variables at all the n nodes in the support domain U, as given in (A.9). The subscripts
of C denote the partial derivative of the shape function with respect to x and z.
161
B.2
Free Surface Boundary Condition
The first-order hyperbolic system (Virieux, 1986) !U
= A-
U
0 i/p 0
0
0
0
0 i/p
A + 2p
0
0
0
0
A
0
0
0
0
0 A + 2p
0
P1 0
0
0
P
0
B
U + B-LU
0
0
0
0
0
0 1/p
0
0
A
0
0
0
0
0
0
0
0
0
0
0
i/p
where U = [vx, vZ, 7Xu,-zz, -xz] T, and A and B are the jacobian of the flux of
the elastic wave equation along the x and z axes, respectively. Assuming the normal
vector h is parallel to the z-axis, we find the eigenvalues of the jacobian matrix B
and their corresponding right eigenvectors
K
0
0
±4
-
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
p
ip0
-
0
0
0
0
1
/(A+2p)p
K =
0
7(A+2p)p
0
0
0
A
A+2p
A
0
0
1
1
1
0
0
0
0
0
1
1
0
where A is a diagonal matrix of eigenvalues and K is the matrix of right eigenvectors. Then we find the characteristic variables in 1-D along the z-axis
162
rZ~+
2
O'-2
2pVp
2
2pVp
2/(A+2p)p
W = K-U =
2
+
2 p-i
A
-xxA+2iUZ
azz
A+2p Ozz
In order to impose the physical boundary condition, the normal stresses are set
~zz = OxZz = 0, and only the characteristics (i.e., right eigenvectors) that travel at
positive speeds (i.e., positive eigenvalues) are considered:
vx
V9
+ 'xpvsZ
Uzz
VZ
0
B.3
=
xx
xx (A±2/p)
0-gzz
0
(T9
xz
0
Signed Distance Function and the Fast Marching Level Set Method
In order to find the the point inside the computational domain that corresponds
to each ghost point, the distance from the free surface along the normal direction
needs to be calculated. First, a function yo (x) is defined such that the free surface
topography is the zero contour of o (x).
The level set function is a proper signed
function o (x) that satisfies the Eikonal equation
x {(x)
= 0 on the surface
1V (W) = 1
(X) > 0 inside the domain
O (X) < 0 outside the domain
163
Sethian (1996) has developed a fast marching level set method to solve the Eikonal
equation. The method uses an upwind, viscosity solution, finite difference scheme to
numerically solve this equation in order to obtain the signed distance from the surface.
B.4
Lagrange Interpolation
The Lagrange interpolating polynomial is the polynomial P(x) of degree
f
that passes through the n points (x 1 , Y1
(x 1 )), (x 2 , Y2
f
(x 2 )), (X., yn
; (n - 1)
f
(xn)),
and is given by:
n
P3
(X)
=J y
Xj - Xk
k=1
k#j
for second order interpolation (n
P-
(X - X2)(X - X3)
(x)=Y1+
1 - x 2 ) (x 1 - x 3 )
3)
(X - X1)(X - X3)
x)( X2x
2 -- X
(x
(
3)
(X - X1) (X - X2)
Y2+
(x 3 - x 1) (x 3 - X2)
To estimate a point in two dimensions, the Lagrange interpolation is applied twice,
once along each direction.
B.5
Rotation Matrix
The linear transformation matrix is applied to the state vector to rotate the coordinate
frame by an angle a , thus one can find the characteristic variables along the normal
direction.
164
sin (a)
0
0
0
- sin (a) cos (a)
0
0
0
cos (a)
0
0
cos 2 (a)
sin 2 (a)
2 sin (a) cos (a)
0
0
sin 2 (a)
cos 2 (a)
-2 sin (a) cos (a)
0
0
- sin (a) cos (a) sin (a) cos (a) - sin 2 (a) + cos 2 (a)
165
Bibliography
Bayliss, A., K. Jordan, B. LeMesurier, and E. Turkel, 1986, A fourth-order accurate finite-difference scheme for the computation of elastic waves: Bulletin of the
Seismological Society of America, 76, 1115-1132.
Gottlieb, D., M. Gunzburger, and E. Turkel, 1982, On numerical boundary treatment of hyperbolic systems for finite difference and finite element methods: SIAM
Journal on Numerical Analysis, 19, 671-682.
Sethian, J. A., 1996, A fast marching level set method for monotonically advancing
fronts: Proceedings of the National Academy of Sciences, 93, 1591-1595.
Virieux, J., 1986, P-SV wave propagation in heterogeneous media: Velocity-stress
finite-difference method: Geophysics, 51, 889-901.
166
Appendix C
In this appendix, we demonstrate additional factors affecting elastic wave scattering
that have not been included in Chapter 2. Specifically, looking at the effects of the
heterogeneity size on the wavelength of the scattered waves, acquisition fold, and
common-mid-point (CMP) stacking.
C.1
The Effects of Wavelengths and Scatterer Sizes
We show the effects of near-surface scatterer size on the frequency content of the
recorded scattered wavefield. The dominant wavelength of the scattered wavefield can
be different than the fundamental mode as it depends on the size of the scatterers.
The results in Figure C.1 are based on the earth model shown in Figure 2.3, but
with different scatterer sizes. As shown in Figure C.1, the dominant frequency of the
noise (scattered surface waves) in the
f
- k domain appears to be higher than the
incident waves, mainly because the size of the scatterers is much smaller than the
incident wave wavelength. For different scatterer sizes, the dominant frequency of
the scattered surface waves increases with decreasing size of the scatterers as shown
in the frequency-wave number domain of the recorded wavefield.
C.2
The Effects of Common-Mid-Point (CMP) Stack
We demonstrate the influence of common-mid-point stacking on reducing coherent
scattered body-to-surface wave noise for the vz-component. In Figure C.2, we compare
167
different sorting domains (e.g., common source, common receiver, and common midpoint gathers) for the single layer over half a space model without scatterers (Figure
C.2a-c) and with scatterers (Figure C.2d-f).
For the model without scatterers, the
sorting domains show no difference. On the other hand, the model with scatterers
shows similar results for the common source and receiver gathers, but different phases
for the scattered waves in the common mid point domain.
In Figure C.3, we show CMP gathers after NMO with full and one third of the
fold. CMP stacking for the case with and without the direct surface waves are shown
in Figures C.4 and C.5, respectively. The results show that the scattered noise phases
have not been removed by CMP stacking. Increasing the fold reduced the stacked
direct surface wave (Figure C.4), but has no effects on the stacked body-to-surface
wave noise (C.5).
168
Ren.t.-,i W.o-fi.1r
PFO Mmnin
.40
0.
050
0005
0.0
70
S
100
000
000
000
Offset (i)
000
600
-0.1
-0.00
-0.00
-0.04
-0.02
0
0.00
Wavenumber (1/r)
0.00
0.06
0.06
0.04
0.06
0.08
0.06
0.08
(a)
FK domain
Scattered Wavefield
0.0
60
0.7
70
0.0
80
100
00
000
00
60
.- 0.1
600
-0.08
-0.06 -0.04 -0.02
0
0.02
Wavenumber (1/m)
Offset (m)
(b)
FK domain
Scattered Wavefield
0.1
10
0.2
20
0.9
90
0.3
30
0.4
40
0.5
50
-
0.
60
0.7
70
0.0
80
100
200
300
400
Offset (m)
000
S00
-0.1
-0.08 -0.06 -0.04
-0.02
0
0.02
0.04
Wavenumber (1/m)
(c)
Figure C.1: The scattered wavefelds due to different scatterer radiuses are shown to
the left, and their corresponding Frequency-wavenumber domains are shown to the
right: (a) 5 m, (b) 10 m, and (c) 20 m.
169
CSG - No Scattering
C,,
CRG - No Scattering
- CMP -
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.4
0.4
0.2
0.4
0O.
-~
(D.0.
E
E
No Scattering
(D o.~
E
0.6
0.6
0.7
0.7
0.7
0.8
0.8
0.1
0.9
0.0
0.1
0.E
1
Offset (M)
Offset (m)
CSG - with Scattering
CRG - with Scattering
0
0.1
0.1
0.2
0.2
0.3
0.2
C,
0.4
(D o.!
I0.E
0.6
E
0.7
0.7
0.7
0.8
0.1
0.8
0.9
0.4
0.0
Offset (m)
200
400
0.4
O o.E
E
0.6
0
CMP - with Scattering
0.2
0.5
-200
Offset (m)
0.1
0.4
E
-400
Offset (m)
Offset (m)
Figure C.2: Finite difference results for the single layer over half a space model
in Figure 2.3: (a-c) without scatterers, and (d-f) with scatterers. The gathers are
sorted to (a and d) common shot, (b and e) common receiver, and (c and f) common
midpoint.
170
NMO - No Scattering
NMO - Difference
NMO - with Scattering
0.1
0.2
0.3
-
0.4
E
S0.,
0.,
U)o.
0.5
a) 0.!
0.6
I- 0.1
a) 0.!
E
0.0
0.
0.
0.
E
0.7
0.8
0.9
0.9
1
Offset (m)
unset
(a)
Offset (m)
km)
NMO - with Scatterini
NMO - No Scatterina
0.1
0.1
0.2
0.2
0.3
0.3
NMO - Difference
0.:
0.,
0.4
-0.4
U 0.5
U) 0.5
0.6
0.6
0.7
0.7
0.8
0.8
W) o.!
E
E
E
- 0.
0.9
200
400
600
Offset (m)
800
200
400
600
Offset (m)
800
200
400
600
800
Offset (m)
(b)
Figure C.3: Common-mid-point gathers: (left) for the model without scatterers,
(middle) with scatterers, and (right) the difference. The results in (b) have one third
the fold of the ones in (a).
171
CMP Stack - No Scattarino
r.Mp qtart'- with _Qrntte-i"-
.1.1
0.1
0.2
0.3
0.2
0.2
0.3
o.4
0.3
0.4
0.4
0.5
0.6
0.5
0.6
- o.s
P a.6
0..
0.9
0.:
0.
EE
100
200
300
400
500
Offset
600
(m)
700
0.8
0
800
200
200
400
Offset
CMP Stack - No Scattering
OD
700
800
0
100
0.4-
0
0.5
0.6
0.5
-6
P:.0
0.7
0.8
0.7
0.8
0 .9
0*90.
000
800
Offset (m)
700
00
002
0.3
400
8
700
(m)
CMP Stack - Difference
0.4
300
400
00.
0.2
200
000
.00
Offset
0.3
100
000
.00
(m)
CMP Stack - with Scattering
0.1
-
CMP Stack - Difference
800
g00
0
-.
E
.
.
100
200
300
400
000
Offset
800
(m)
700
800
900
1
00
200
00
400
00
800
Offset (M)
700
800
00
Figure C.4: Common-mid-point stacks with the direct surface wave: (a-c) with full
fold, and (d-f) with half the fold, (a and d) stack for the model
without scatterers,
(b and e) stack for the model with scatterers, and (c and f) the difference. Increasing
the fold reduced the stacked direct surface wave.
172
CMP
Stark
- No
Sratterina
CMP
Stark -
with qratrino
CMP
0.11
02
2
0.3
0.
0.4
0
.
W 0.
a00.5
E
C
E
.
06
0.7
0.8
0.9
E
H
0.
5
o..
0.8
0.9
200
300
400
500
600
700
800 900
ID1O
200
300
40D
500
600
700
800 900
Offset (M)
CMP Stack - No Scattering
CMP Stack - with Scattering
loo0 200 3W0
400
500
600
700
Soo goo
Offset (m)
CMP Stack - Difference
0
0.3
0.2
0,3
0.4
0.4
0.5
0.6
0.5
0.6
0.7
0.8
0.7
0.
03
0.
.
0.
0.
o.8
0.9.
0.90.9
100
.
0.
0.6
Offset (m)
0.2
1
0.5
0.
5
031
a
rDiffArnr
01
02
0.3
1 100
Stark -
200
300
400
500 600
Offset (m)
700
800
900
1 loo
2D0
300
500 Boo 700
Offset (m)
400
800 No0
lo10 200
300
400
500 600
700
800 900
Offset (m)
Figure C.5: Common-mid-point stacks with the direct surface wave removed before
stacking: (a-c) with full fold, and (d-f) with half the fold; (a and d) stack for the model
without scatterers, (b and e) stack for the model with scatterers, and (c and f) the
difference. The results show that the scattered noise phases have not been removed
by CMP stacking. Increasing the fold has no effect on the stacked body-to-surface
wave noise.
173