Modeling and Imaging Elastic Waves in Heterogeneous Media
Kuang He
University of Connecticut, 2010
Abstract
This thesis presents five studies that apply geophysical tools to image and interpret the heterogeneity
of physical properties of the Earth at a range of spatial scales. Applications of the studies include
resource exploration, environmental remediation, earthquake hazard assessment, and the evolution of
Earth's magnetic field from a process of compositional convection. The first study presents seismic
waveform tomography, a technique that uses whole waveforms to image seismic velocity variations in
two-dimensions. The second study applies wave-equation re-datuming through downward and
upward continuation of surface recordings of ground penetrating radar (GPR) to image
two-dimensional electromagnetic properties of the soil and weathered rock region near the surface.
The third and fourth studies use a pseudospectral modeling technique of the seismic wavefield to
simulate the effects of receiver-side heterogeneities on wavefronts at different angles and to study the
strong ground motion in the Beijing area caused by a hypothetical magnitude 8.0 earthquake. The
fifth study uses body wave data from arrays of seismic stations and Monte-Carlo simulations based on
radiative transfer theory to study the solidification texture of the uppermost inner core.
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Modeling and Imaging Elastic Waves in Heterogeneous Media
Kuang He
B.S., University of Science and Technology of China, Hefei, Anhui, China, 2005
M.S., University of Connecticut, Storrs, CT, U.S.A., 2009
A Dissertation
Submitted in Partial Fulfillment of the
Requirements for the Degree of
Doctor of Philosophy
at the
University of Connecticut
2010
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APPROVAL PAGE
Doctor of Philosophy Dissertation
Modeling and Imaging Elastic Waves in Heterogeneous Media
Presented by
Kuang He, B.S., M.S.
Major Advisor ________________________________________
Vernon F. Cormier
Associate Advisor _____________________________________
Lanbo Liu
Associate Advisor ______________________________________
Winthrop W. Smith
University of Connecticut
2010
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To Vernon and Lanbo
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Acknowledgments
This thesis was made possible by the support and encouragement of many people. Among those
people, I would like to thank Vernon Cormier and Lanbo Liu, to whom this thesis is dedicated, with
the greatest debts of gratitude. Vernon, with all his brilliance, experience and most importantly
patience, walked me into the intriguing and amazing geophysical world. His mentorship was
paramount in providing a well-rounded experience consistent with my long-term career goals. He
encouraged me to pursue different ideas and interests during the fives years of studies and research
work at University of Connecticut, a process from which I will benefit for the rest of my life. Lanbo,
one of the kindest advisors I have ever come to know, worked with me both as a teacher and as a
friend. His passion both in research and in life always urged me to work harder to reach my goals.
Faculty, staff, and students at UConn were always helpful, enthusiastic, and informative. I would
especially like to thank Michael Rozman, who not only always maintained a fine computing
environment, but also mentored and enlightened me on many computer and computing related issues.
Richard Jones also enlightened me many times on parallel computing issues. Anastasia Stroujkova
provided me with valuable seismic data she has collected through the years and guided me step by
step to process data from seismic array stations. I would also like to thank members of my doctoral
committee for their input and valuable discussions. I am grateful to Carol Artacho Guerra, Kim Giard,
Nicole Hryvniak, Dawn Rawlinson, and Lorraine Smurra for their logistic support.
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I have received a great deal of friendship and help from many former and current UConn students
who have created a friendly and stimulating research environment at UConn. They are Januka
Attanayake, Betty Chau, Hui Chen, Li Chen, Yongping Chen, Zhe Chen, Hong Cong, Hao Dong,
Liang Dong, Yu Fan, Kun Fang, Li Fang, Michele Fitzpatrick, Marko Gacesa, Xugang He, Yi Li,
Zhandi Liao, Qi Lu, Shaozhen Ma, Yuefeng Nie, Qing Peng, Yu Shi, Hongzhi Sun, Tun Wang, Hua
Yang, Yongkun Yang, Xiaohu Yu, Lili Zhang, Lu Zhao, Yuan Zhao, Zhao Zhao, Kai Zhou, Ran Zhou,
Rong Zhou, and Lieyuan Zhu.
I took an internship with Petroleum Geo-Services during the summer of 2007 and another one with
GeoTomo during the summer of 2009. During these times, I am indebted to Peter Anthony, Jing Chen,
Chongyang Dong, David Lowrey, Shansong Jiang, Chuntao Liang, Zhaojun Liu, Yan Mei, Boyi Ou,
Weizhong Wang, Jie Zhang, and Chaoguang Zhou for their help and supervision.
Finally, I thank my family and friends from afar who cared a great deal about me. Bin Chen, Bingbing
Dai, Shan Dou, Junwei Du, Lili Gao, Xiaojie Hao, Ting He, Yang He, Hao Hu, Shan Huang, Hao Li,
Qingsong Li, Xiao Li, Sicheng Liao, Hao Lu, Dong Sheng, Ningyu Shi, Zheng Sun, Yun Wang,
Zhong Wang, Hongfeng Yang, Yang Zhang, Yingjuan Zhang, Yang Zhao, Huifeng Zhu, Jian Zhu,
Zuihong Zou, among others, all offered tremendous encouragement. My grandparents, aunts, uncles,
cousins, and in-laws supported me even when it wasn't clear to them what I was working on anyway.
My parents have always supported my choices, listened when I needed a sounding board, and
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welcomed me home when I needed a break. And my wife Wanjun Sun has been supportive, generous,
kind, funny and wonderful in every way possible.
This thesis was funded by grants from the National Science Foundation and teaching assistantships
from Department of Physics, University of Connecticut.
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Table of Contents
Chapter 1 Introduction ........................................................................................................ 1
Chapter 2 Multi-Scale Waveform Tomography ................................................................ 5
2.1
Introduction ......................................................................................................................... 5
2.2
Frequency-Domain Waveform Tomography ....................................................................... 9
2.2.1
Frequency-Domain Finite-Difference Forward Modeling ....................................... 9
2.2.2
Frequency-Domain Waveform Inversion ............................................................... 10
2.3
Multi-Scale Inversion ........................................................................................................ 13
2.4
Numerical Results ............................................................................................................. 19
2.5
2.6
2.4.1
The Foothills Model ............................................................................................... 19
2.4.2
Cross-Well Waveform Tomography ....................................................................... 23
Limitations of Waveform Tomography ............................................................................. 26
2.5.1
Sensitivity to the Initial Model............................................................................... 26
2.5.2
Application to Field Data ....................................................................................... 28
Conclusions ....................................................................................................................... 28
Chapter 3 Image Enhancement With Wave-Equation Redatuming: Application to
GPR Data Collected at Public Transportation Sites ................................................... 31
3.1
Introduction ....................................................................................................................... 32
3.2
Wave-equation Redatuming Applied to Synthetic Models................................................ 35
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3.3
3.2.1
Reproduction of the Original Berryhill Model with Electromagnetic Waves ........ 35
3.2.2
Wave-Equation Redatuming Applied to a Piece of Synthetic Data ....................... 36
Wave-Equation Redatuming Applied to Real Field Data .................................................. 39
3.3.1
A Highway Bridge Refurbish Project in Connecticut, USA .................................. 39
3.3.2
Locating a Drain Pipe at the Edge of an Airport Runway in Houston, USA ......... 41
3.3.3
Assessment of Mortar Consolidation Quality Behind Tunnel Wall in Xiangyin
Tunnel, Shanghai, China....................................................................................................... 43
3.4
3.3.4
Dielectric Permittivity of Sediments and Mortars.................................................. 44
3.3.5
Field Acquisition of GPR Data in Xiangyin Tunnel .............................................. 46
3.3.6
Wave-Equation Redatuming for the Profile of Line 21 ......................................... 48
Conclusions ....................................................................................................................... 49
Chapter 4 Simulation of the Effects of Receiver-Side Heterogeneities on the
Wavefronts at Different Angles: an Application of the Pseudospectral Modeling
Technique ........................................................................................................................ 51
4.1
Introduction ....................................................................................................................... 51
4.2
Two-Dimensional Pseudospectral Modeling Technique ................................................... 54
4.3
Simulation of the Effects of Receiver-side Heterogeneities on the Wavefronts at Different
Angles ........................................................................................................................................... 58
4.4
Conclusions ....................................................................................................................... 64
Chapter 5 Study of the Strong Ground Motion Scenario in the Beijing Area Caused by
the 1679 M8 Sanhe-Pinggu Earthquake ...................................................................... 67
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5.1
Abstract ............................................................................................................................. 67
5.2
Introduction ....................................................................................................................... 68
5.3
Summary of the Numerical Model .................................................................................... 69
5.3.1
Modeled Area ......................................................................................................... 69
5.3.2
Model Implementation ........................................................................................... 70
5.4
The Source Parameters ...................................................................................................... 72
5.5
Discussions ........................................................................................................................ 75
5.6
5.5.1
Surface Peak Ground Velocity (PGV) and Peak Ground Acceleration (PGV) ...... 75
5.5.2
Synthetic Seismograms at Specific Sites ............................................................... 78
Conclusions ....................................................................................................................... 79
Chapter 6 Texture of the Uppermost Inner Core from Seismic Coda Waves .............. 81
6.1
Introduction ....................................................................................................................... 81
6.2
Radiative Transfer Theory ................................................................................................. 85
6.3
6.4
6.2.1
Elastic Radiative Transfer Equations ..................................................................... 87
6.2.2
Random Medium.................................................................................................... 89
6.2.3
Scattering Coefficients ........................................................................................... 92
6.2.4
Monte-Carlo Solution of Radiative Transfer Equations......................................... 94
Data Analysis ..................................................................................................................... 98
6.3.1
Data ........................................................................................................................ 98
6.3.2
Analysis Procedure .............................................................................................. 100
Forward Modeling of the PKiKP Coda Envelope ........................................................... 106
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6.5
Conclusions ..................................................................................................................... 117
Bibliography ........................................................................................................................ 121
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Chapter 1 Introduction
There are all types of heterogeneities at all depths of our planet: those in the crust due to tectonic
activity, volcanoes, etc., those in the mantle due to mantle convection, those near the core-mantle
boundary (CMB) due to the interactions between the mantle and the liquid outer core, and those in
the uppermost inner core due to the solidification process of the inner core from the liquid outer
core. These heterogeneities at different depths may have very different, and possibly anisotropic,
scale lengths. Studying heterogeneities at different depths of the Earth will help us understand more
about the planet we live and depend upon. For example, studying the heterogeneities at the top
inner core will lead us to better understanding of the solidification process of the inner core and
possibly the convection at the bottom of the outer core as well. This in turn will further assist us in
understanding the geodynamo process better in the outer core, which we think is the reason why the
Earth has a magnetic field.
Since the Earth is not transparent, we usually cannot take a direct good look at its deeper parts. So
far, the deepest depth human beings can drill into the Earth is about 12 km. Compared to the radius
of the Earth, which is 6371 km, the part of the Earth directly accessible to human beings, even
without any consideration of costs, is tiny. Aside from the direct methods, which are not always
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available due to technical and/or cost issues, most of time we have to reply on indirect methods
such as those often employed in seismological studies, including modeling seismic wave
propagation, inverting seismic records, etc., to image the Earth.
This thesis presents five studies that apply geophysical tools to image and interpret the
heterogeneity of physical properties of the Earth at a range of spatial scales. Applications of the
studies include resource exploration, environmental remediation, earthquake hazard assessment,
and the evolution of Earth's magnetic field from a process of compositional convection. First, two
different imaging techniques, multi-scale waveform tomography and wave-equation redatuming,
are introduced to give us better images of the Earth's subsurface. Second, a forward modeling
technique called the pseudospectral method is used to simulate the effects of receiver-side
heterogeneities on wavefronts at different angles and to study the strong ground motion in the
Beijing area caused by a magnitude 8.0 scenario Earthquake. Third, body wave data from arrays of
seismic stations and Monte-Carlo simulations based on radiative transfer theory are used to study
the solidification texture of the uppermost inner core. While these studies each employ very
different techniques, together they build a useful arsenal with which we can better understand our
heterogeneous planet.
In Chapter 2, an inversion technique named waveform tomography is introduced. Based on the
solution of full wave equations, this technique avoids the high-frequency limitation of travel-time
tomography due to its use of a ray-tracing based forward modeling technique. At the price of being
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more computationally intensive, waveform tomography tries to minimize the difference between
the observed data and the synthetic data by fitting the whole waveform. The implementation and
numerical results of frequency-domain waveform tomography are shown to prove the effectiveness
of this method. A multi-scale approach, which exploits the redundancy of wavenumber coverage in
the data, is discussed to make the algorithm more efficient.
Chapter 3 presents the application of wave-equation redatuming through downward and upward
continuation to several cases in public transportation engineering projects. This approach is an
effective and economical way to suppress and eliminate the strong reflections and diffractions from
rebars in the surface concrete slab to highlight events in later times of a ground penetrating radar
(GPR) profile. By applying downward and upward continuations to both synthetic and field GPR
data, we have demonstrated the effectiveness of this method in suppressing the diffractions and
interferences caused by not only the geometric heterogeneity, but also the material property
heterogeneity in the surface layer.
In Chapter 4, simulations are carried out using a 2-D pseudospectral modeling technique to model
the effects on the incident seismic wavefield of increased heterogeneity beneath the seismograph
stations. We have demonstrated that the interaction of the wavefield with near-receiver
heterogeneities is an important additional source of amplitude fluctuations across arrays of stations
and used this to explain the general strong negative correlation in the detection of PcP and PKiKP
waves in a particular dataset.
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In Chapter 5, 3-D simulations are carried out using a staggered grid pseudospectral method to
simulate the 1679 M8 Sanhe-Pinggu earthquake in the Beijing area, with a crust model having
high-resolution sediment thickness obtained from microtremor measurements in this area.
Three-component synthetic seismograms of four important sites including the 2008 Summer
Olympic Stadium are shown. From the surface peak ground velocity (PGV) and peak ground
acceleration (PGA) studies, together with H/V calculations, we have observed ground motion
amplification effects possibly associated with local site effects in the north part and the southeast
corner of the Beijing area, which have also been confirmed by other studies. Our result
demonstrates the effectiveness of numerical modeling in seismic hazard assessment and provides
valuable information for mitigating losses for possible earthquakes in this area.
In Chapter 6, we have used high frequency coda of the P wave reflected from the inner core
boundary (PKiKP) to study the small-scale heterogeneity in the uppermost inner core. This region
has been revealed by recent studies to have strong lateral variations in elastic structure, anisotropy,
attenuation, and scattering. The detailed texture of the uppermost inner core is important for
understanding how the inner core is solidifying from the liquid outer core, possibly providing a
mechanism for compositional convection that can drive the geodynamo. Monte-Carlo simulations
of high frequency P wave coda based on radiative transfer theory have also been carried out to
compare with seismic array data in order to put more constraints on the spatial distribution of the
small-scale heterogeneity in the inner core.
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Chapter 2 Multi-Scale Waveform Tomography
2.1 Introduction
Traveltime tomography (Luo and Shuster, 1991; Nemeth et al., 1997; Pratt and Goulty, 1991;
Schuster and Quintus-Bosz, 1993; Zhu and McMechan, 1989; Aki and Richards, 2002) has been
extensively used to estimate the subsurface velocity structure. The velocity model is updated by
iteratively back-projecting the traveltime residuals along rays. This method is fast, cost-effective,
and has long been a standard for delineating the Earth's velocity distribution. However, it generally
uses a ray theory based forward modeling technique. Ray theory is an approximation that becomes
exact at infinite frequency. The data are collected at finite frequency, where ray theory becomes an
approximation. Moreover, traveltime tomography only takes into account the traveltime
information of the seismic traces while totally ignoring all the amplitude information.
To overcome this high-frequency limitation, much effort has been focused on fat-ray or
Fresnel-volume tomography (Cerveny and Soares, 1992; Schuster and Quintus-Bosz, 1993; Vasco
et al., 1995; Montelli et al., 2004; Zhou et al., 2006) and waveform tomography (Lailly, 1984;
Tarantola, 1984, 1986, 1988; Mora, 1987, 1988, Pica et al., 1990), in which multiple paths and
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multiple arrivals can also be taken into account. Waveform tomography, sometimes called
full-waveform tomography or waveform inversion, is an inversion technique that uses the whole
waveform to invert for velocity profiles. Since the whole information contained in the recorded
wavefields is being used in the inversion, this technique has higher accuracy and resolution than
traveltime tomography does. By using the solution of full wave equations, this technique avoids the
high-frequency assumption required by traveltime tomography, at the price of being much more
computationally intensive. Waveform tomography, like traveltime tomography, is usually
formulated as an iterative descent method, in which the minimization of residuals is achieved. The
gradient calculated at each iteration provides the direction of minimization of the object function,
usually the L2 norm of the data residuals, possibly combined with some form of regularization.
Waveform tomography can be implemented in either the frequency domain (Pratt, 1990; Pratt et al.,
1996, 1998; Pratt and Worthington, 1990; Liao and McMechan, 1996) or the time domain
(Tarantola, 1984, 1986; Mora, 1987; Bunks et al., 1995; Zhou et al., 1995, 1997; Shipp and Singh,
2002). Implementations for both acoustic and elastic wave equations exist. The frequency-domain
approach is equivalent to the time-domain approach when all frequencies are inverted
simultaneously (Pratt et al., 1998).
Waveform tomography, however, does have its limitations. The misfit function is often strongly
non-linear, making the inversion difficult to reliably converge. The presence of numerous local
minima in the object function impedes iterative techniques from finding the global minimum unless
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the initial model for the velocity field is already in the neighborhood of the global minimum. To
overcome this problem, several approaches have been proposed. One of them is called early arrival
waveform tomography (EWT), proposed by Pratt and Worthington (1988) and Sheng et al. (2006).
EWT applies a time window to the early arrivals (events that arrive within a few periods of the first
arrivals). By excluding the rest of the wavefield, thus fitting fewer events, the misfit function
becomes more linear with respect to velocity, therefore more reliable convergence can be obtained.
Alternatively, a multi-scale approach could be used. Since the low-frequency data are more linear
than high-frequency data, inverting the low-frequency data first would lead to better chances of
converging to the vicinity of the global minimum of the whole-bandwidth data. The inversion will
proceed sequentially from low to high frequencies, gradually resolving structures of finer scales. In
the time domain, Bunks et al. (1995) proposed that the seismic traces would be filtered using a
finite-impulse response (FIR) Hamming windowed low-pass filter (Oppenheim and Schafer, 1975;
Rabiner and Gold, 1975) so that different frequency bands of the data following a low to high
scheme would be inverted sequentially. In the frequency domain, the recorded seismic traces need
to be discretized first into different frequency components before the inversion; thus it is
straightforward to carry out the waveform inversion adopting the multi-scale strategy -- we just
need to properly select a few frequencies (from low to high) to be inverted. Sirgue and Pratt (2004)
have developed a frequency selection strategy so that a few properly selected frequencies could
yield a result that is comparable to the time domain inversion. This works because in a surface
seismic survey, there is usually redundancy in the wavenumber coverage of the target we want to
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resolve, and the redundancy usually is proportional to the range of offsets available in the surface
surveys. This means that even a single frequency has a finite band of wavenumber coverage. If we
choose a few individual frequencies to be inverted optimally, we could get the whole wavenumber
coverage using only a few discrete frequencies instead of the whole frequency spectrum.
When applied to 2D wide-aperture data, the frequency-domain approach has been shown to be
more efficient than the time-domain approach for three major reasons (Sourbier et al., 2008a). First,
only a few discrete frequencies are necessary to develop a reliable image of the medium by
decimating the wavenumber redundancy provided by multi-aperture geometries (Sirgue and Pratt,
2004; Pratt and Worthington, 1990; Pratt, 1999). This makes the frequency-domain implementation
less computationally intensive than the time-domain approach, which is equivalent to inverting all
the frequencies together in the frequency domain. Second, the multi-scale approach could be
naturally implemented in frequency-domain waveform tomography, thus further reducing the
amount of computing needed. Third, the forward modeling problem in the frequency-domain
reduces to the resolution of a large sparse system of linear equations per frequency whose
right-hand side (RHS) term is the source and the solution is the monochromatic wavefield, so the
few frequencies involved in the inverse problem can be efficiently modeled in the frequency
domain for a large number of sources with the use of a direct solver (Soubier et al., 2008a).
Therefore, in this chapter, only the frequency-domain approach will be discussed in detail. The
frequency selection strategy developed by Sirgue and Pratt (2004) will be reviewed and numerical
results with different acquisition geometries will be presented to show the effectiveness of
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multi-scale waveform tomography in delineating the velocity profile of the subsurface. The
sensitivity of waveform tomography to the initial model will be shown, and the possibility of
applying waveform tomography to field data will also be discussed.
2.2 Frequency-Domain Waveform Tomography
2.2.1 Frequency-Domain Finite-Difference Forward Modeling
The finite-difference method is widely used to numerically solve differential equations. It can be
implemented in either the time domain (Virieux, 1984, 1986) or the frequency domain (Pratt and
Worthington, 1990; Jo et al., 1996). Below is a brief review of the frequency-domain approach.
The 2D visco-acoustic wave equation in the frequency domain can be written as
2
 ( x, z )
p( x, z,  ) 

1 p( x, z,  )

1 p( x, z,  )
(
) (
)  s( x, z, ) (2.1)
x  ( x, z )
x
z  ( x, z )
z
where  (x, z) is the density,  (x, z) the complex bulk modulus,  the frequency, p(x, z,  )
the pressure field and s (x, z,  ) the source. Attenuation effects can be implemented with complex
elastic moduli.
Since the pressure wavefield varies linearly with the source in equation (2.1), this equation can be
written in a matrix form as
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Ap  s
(2.2)
where the complex-valued matrix A depends on the frequency and medium properties. The 2D
pressure fields p and sources s at each frequency  are matrices of dimension nx * nz , where nx
and nz represent the dimension of the finite-difference grid. Typically, the pressure fields are solved
from equation (2.2) using a direct solver after matrix A is factorized using an LU decomposition
scheme. Once the usually quite computationally intensive factorization is performed, solutions for
multiple sources can be achieved by forward and backward substitutions, which are relatively less
computationally intensive.
Equation (2.1) is discretized using a parsimonious mixed-grid method (Hustedt et al., 2004), which
could lead to an accurate and spatially compact finite-difference stencil for the frequency-domain
forward modeling based on a direct solver. Absorbing boundary conditions are implemented using
a combination of a 45˚ paraxial condition (Clayton and Engquist, 1977) and perfectly matched
layer (PML) conditions (Berenger, 1994). The solution of equation (2.2) is computed using a
distributed-memory parallel library called MUMPS (Amestoy et al., 2006).
2.2.2 Frequency-Domain Waveform Inversion
One of the most important advantages of a frequency-domain inversion is the ability to provide a
unaliased image using a very limited number of frequencies, as observed by Pratt and Worthington
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(1988), Liao and McMechan (1996), and Forgues et al. (1998). The theory and formulae of the
frequency-domain waveform inversion described below have been developed by Pratt et al. (1998)
and Operto et al. (2006).
The weighted least-squares object function is written as
C (m)  d *Wd d
(2.3)
where d is the vector of residuals (i.e., the difference between observed data and synthetic data
calculated using model m ). The superscript * indicates the adjoint (i.e., the transpose conjugate).
Wd  Wd (Osr )  exp( g log Osr ) , where g is a constant and Osr represents the source-receiver
offset, is a weighting operator applied to the data in order to scale the relative contribution of each
component in vector d in the inversion. This weighting is used to suppress the relative
contribution of data with small offsets, which generally carry little information on the deep
structure.
In the steepest-descent method, a model perturbation m is searched in the vicinity of the
starting model m0 in the direction that is opposite to the gradient of the object function.
m   mC   Re( J tWd d * )
(2.4)
where m is the model perturbation (the updated model m  m0  m ),  mC is the
gradient of the object function,  is a step length controlling the amplitude of the perturbations,
J t is the transpose of the Jacobian matrix (i.e., the Frechet derivative matrix), d * is the
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conjugate of data residuals and Re denotes the real part of a complex number. The step length
 is usually computed using a linear estimate (Tarantola, 1984; Mora, 1987).
The Jacobian matrix J is give by
J   A 1
A
p
m
(2.5)
where p denotes the forward modeled pressure wavefields. The matrix A1 is constructed using
Green’s functions for sources located at each grid node of the velocity model. From equations
(2.4) and (2.5), and also A1  A1 , we can write the ith model perturbation as
t
  At
mi   Re  p t 
  mi

 1
*
 A Wd d 


(2.6)
In order to obtain reliable perturbation models, we have applied some scaling and
regularization to the gradient in equation (2.6) so that
mi    diagH a   I 
1
 t  At  1

*
Gm Re  p 
 A Wd d 
  mi 

(2.7)
where diagH a denotes the diagonal elements of H a , the weighted approximate Hessian,
and Gm is a spatial smoothing operator implemented as a 2D Gaussian spatial filter (Ravaut
et al., 2004). The diagonal of the approximate Hessian provides a preconditioner of the
gradient that properly scales the perturbation model (Shin et al., 2001).
We stop the inversion iterations when at least one of the following two criteria is met: (1) the
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iteration count reaches a maximum number of iterations we specify; (2) the difference
between
object
functions
in
two
adjacent
iterations
is
not
large
enough,
i.e., C ( j -1)  C ( j )  thres  C ( j  1) .
2.3 Multi-Scale Inversion
Conventional time-domain waveform tomography using a single frequency band of the data can
lead to a local minima problem. By using several frequency bands of the data, the multi-scale
method proposed by Bunks et al. (1995) and Sirgue and Pratt (2004) in the time and frequency
domains, respectively, was very successful in inverting for the complex synthetic models such as
the 2D Marmousi model (Versteeg, 1994). We review here, the frequency selection strategy
proposed by Sirgue and Pratt (2004) for the frequency domain, which depends on the maximum
effective offset present in the surface seismic survey: the larger the range of offsets, the fewer
frequencies are required.
In the frequency domain, the gradient of the object function equation (2.4) can be written as
m   2  Re G0* ( x, s )G0* ( x, r )d (r , s )  ,
s
(2.8)
r
where G0 ( x, s ) and G0 ( x, r ) are the Green’s functions for an excitation at the source and
receiver positions in the reference medium, respectively; and d is the data residual. To proceed
to a frequency selection strategy, Sirgue and Pratt (2004) have introduced some assumptions with
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respect to the Green’s functions:
(1)
Amplitude effects may be ignored;
(2) The reference medium is homogeneous, with velocity c0 ;
(3) We are in the far field, so that we may replace the Green’s functions with plane-wave
approximations.
Under these three assumptions, both Green’s functions G0 ( x, s ) and G0 ( x, r ) may be
approximated by incident and scattered plane waves:
G0 ( x, s )  exp(ik0 s  x)
G0 ( x, r )  exp(ik0 r  x),
(2.9)
where k0   / c0 is the wavenumber of the incident and scattered waves in the homogeneous
reference medium, and s and r are, respectively, unit vectors in the incident propagation
direction (source to scatterer) and the inverse scattering direction (receiver to scatter). Inserting
equation (2.9) into equation (2.8) yields
m   2  Re exp(ik0 s  x) exp(ik0 r  x)d (r , s ) 


s
r
 
2
 Re exp(ik (s  r )  x)d (r , s) .
(2.10)
0
s
r
Equation (2.10) shows that the contribution to the gradient image of a single source-receiver
pair has only a single wavenumber component, given by the vector k0 ( s  r ) . If the model is
1D, then all reflection points in the subsurface are midpoint reflection points. A basic
- 14 -
configuration is illustrated in Figure 2-1.
Figure 2-1: The 1D basic scattering experiment with wavenumber illumination. The incident plane
wave is reflected from a 1D thin layer at the midpoint between source and receiver. A single
frequency component of a single source-receiver pair generates a single wavenumber vector of the
gradient image. The illuminated wave vector is given by the sum of the source-to-scatterer and
scatterer-to-receiver wave vectors. For the homogeneous 1D case, the incident and reflected angles,
 and  are equal (after Sirgue and Pratt, 2004).
For a 1D earth, the incident and scattering angles are symmetric, so that
k0 s  (k0 sin  , k0 cos  )
k0 r  (k0 sin  , k0 cos  ).
(2.11)
From Figure 2-1, we can see that
cos  
sin  
z
h  z2
h
2
h2  z 2
(2.12)
,
- 15 -
where h is the half offset and z is the depth of the scattering layer. Plugging equation (2.12) into
equation (2.11), we find that the components of the vector k0 ( s  r ) are
kx  0
(2.13)
k z  2k0 ,
with
  cos  
z
h z
2
2

1
1  R2
2
,
(2.14)
where R  h / z is the half offset-to-depth ratio.
Using equations (2.13) and (2.14) for a range of offsets, we find that for a giving surface
seismic acquisition geometry characterized by an offset range
0,
2hmax  , the vertical
wavenumber coverage k z of a 1D thin layer for a giving frequency is limited to the range
kz min , kz max  , where
k z min  2k0 min
k z max  2k0 ,
with  min 
1
12  Rmax 2
(2.15)
, where Rmax  hmax / z is the half offset-to-depth ratio obtained at the
maximum half offset hmax , and z is the depth of the target layer. The minimum wavenumber
k z min and maximum wavenumber k z max are produced by the farthest and nearest offsets,
respectively. Expressing equation (2.15) in terms of frequency, we have
- 16 -
k z min  4 f  min / c0
k z max  4 f / c0 ,
(2.16)
where f is the frequency and c0 is the background velocity. Clearly,
k z max
1

 1  Rmax 2 .
k z min  min
(2.17)
From equation (2.16), we may define the wavenumber coverage of a multi-offset acquisition as
k z max  k z max  k z min
 4 (1   min ) f / c0 .
(2.18)
The strategy for choosing frequencies by Sirgue and Pratt (2004) is as follows. Since each
frequency has a limited, finite-band contribution to the wavenumber spectrum of the gradient image,
in order to recover the target accurately over a broad range of wavenumbers, the continuity of the
object in the wavenumber domain must be preserved as the imaging frequencies are selected. They
choose
k z min ( f n1 )  k z max ( f n ),
(2.19)
where f n 1 is the next frequency to be chosen following the frequency f n . The strategy
(illustrated in Figure 2-2) is that the maximum wavenumber of the smaller frequency must be equal
to the minimum wavenumber of the next, larger frequency. This strategy thus relies on using the
full range of offsets available. Using the condition defined in equation (2.19) and substituting
equation (2.16), we arrive at the formula for choosing frequencies proposed by Sirgue and Pratt
- 17 -
(2004):
f n 1  f n 1  f n
 1   min 

 fn

 min 
 1   min  f n 1.
(2.20)
Equation (2.20) shows that the optimum frequency increment is not constant but increases linearly
with frequency. The selection scheme defines a much sparser set of frequencies than does the
frequency sampling theorem.
Figure 2-2: Illustration of the frequency selection strategy for the frequency-domain waveform
tomography. A range of vertical wavenumbers can be recovered from a single frequency component
- 18 -
of the data by using a range of source-receiver pairs. A continuous coverage in the wavenumber
domain is the key for choosing the next frequency (after Sirgue and Pratt, 2004).
2.4 Numerical Results
In this section, numerical results from applying multi-scale waveform tomography to two 2D
models are presented.
2.4.1 The Foothills Model
The foothills model developed by Gray and Marfurt (1995) is a synthetic model widely used in
seismic exploration. It represents a geologic cross-section consisting of a number of faulted and
folded layers typical of mountainous thrust regions. The model has a very irregular topography and
the top layer is air. A slightly modified version of the model has been used in this study, with the
velocities below the air layer ranging from 3600 m/s to 5700 m/s (see Figure 2-3). This model
covers an area of 25 km x 6 km and is discretized with 30 m grid cells, representing a uniform grid
of 832 x 200 nodes. The model is then augmented with 6 km-thick PML layers along the four
edges.
- 19 -
Figure 2-3: The slightly modified version of the Foothills model developed by Gray and Marfurt
(1995). This model covers an area of 25 km x 6 km and is discretized with 30 m grid cells,
representing a uniform grid of 832 x 200 nodes. The velocities in this model excluding the air layer
range from 3600 m/s to 5700 m/s.
We have used a frequency-domain finite difference method to generate the synthetic seismograms,
using 250 frequencies within the range of 0.06-250 Hz. The synthetic data we invert are generated
using a Ricker wavelet centered on 7.5 Hz. In the forward modeling, 278 shots with a spacing of 90
m have been used along the surface of the model. For each shot, receivers with a maximum offset
of 3.6 km and a spacing of 15 m have been used to record the traces. As a result, 123,600 traces in
total have been collected. An example shot gather from the forward modeling result using the
Foothills model is shown in Figure 2-4. These seismic traces will be converted to frequency domain
using the Fourier transform before being inverted.
- 20 -
Figure 2-4: An example shot gather from the forward modeling result using the Foothills model.
Due to the non-linearity of the inversion problem, the initial model used in the waveform
tomography must be in the vicinity of the true model; otherwise, the inversion is likely to get stuck
in one of the local minima (this will be discussed again in section 2.5.1). In this study, for simplicity,
a highly smoothed version of the true model has been used as the initial model for the inversion
(see Figure 2-5).
- 21 -
Figure 2-5: The initial model used in the waveform inversion is a highly smoothed version of the
true Foothills model.
Using the strategy developed by Sirgue and Pratt (2004), we have inverted 11 discrete frequencies
between 2 and 9.31 Hz (the complete list is: 2, 2.33, 2.72, 3.17, 3.70, 4.31, 5.03, 5.87, 6.84, 7.98,
and 9.31 Hz), one by one. In the inversion process, larger structures will be delineated first
followed by finer structures. Waveform inversion results from after accumulating contributions of
progressively higher frequencies are shown in Figure 2-6. It is clear that as iterations progress to
higher frequencies, the waveform inversion result becomes closer and closer to the true model
(Figure 2-3). Comparison between the final waveform tomography model (Figure 2-6d) and the
true model (Figure 2-3) confirms the effectiveness of the tomographic reconstruction and that the
final waveform tomography model is a low-pass version of the true velocity model due to the
limited bandwidth of the source (Sourbier et al., 2008b). Except at the very edges of the model
where the coverage is incomplete, the full complexity of the 2D structure is recovered.
- 22 -
Figure 2-6: Waveform inversion results from after accumulating contributions of progressively
higher frequencies for the Foothills model.
2.4.2 Cross-Well Waveform Tomography
Cross-well seismic tomography is one of the most promising tools that helps increase the yield of
existing oil and gas reservoirs. This technique implies seismic sources in one well and data
recording in an adjacent well. Data analysis through tomography will result in detailed knowledge
of lithology and fluid pathways, which can be crucial to the successful execution of enhanced oil
recovery operations. Previous studies by Zhou et al. (1995, 1997) have shown that waveform
tomograms provide a sharper interface image than delineated in the traveltime tomograms.
- 23 -
In this study, we have made up a 2D model (Figure 2-7a) to test the effectiveness of the waveform
tomography in a cross-well setting. This model covers an area of 5 km x 5 km and is discretized
with 25 m grid cells, representing a uniform grid of 201 x 201 nodes. The velocities in this model
range from 2000 m/s to 4500 m/s. 194 shots are laid along the left edge of the model at a spacing of
25 m and for each shot, there are 194 receivers laid along the right edge of the model with the same
spacing. The synthetic data we invert are generated using a delta wavelet, which has a white
spectrum in frequency. The initial model (Figure 2-7b) used for inversion is a highly smoothed
version of the true model.
Figure 2-7: (a) The true model used in the cross-well study. This model covers an area of 5 km x 5
km and is discretized with 25 m grid cells, representing a uniform grid of 201 x 201 nodes. The
velocities in this model range from 2000 m/s to 4500 m/s. (b) The initial model used for inversion
is a highly smoothed version of the true model.
Since the frequency selection strategy developed by Sirgue and Pratt (2004) only applies to cases
- 24 -
with surface acquisition geometries, it is not applicable to this case. Instead, we have simply
inverted 19 frequencies from 2 to 20 Hz at an interval of 1 Hz. Some of the waveform inversion
results from after accumulating contributions of progressively higher frequencies are shown in
Figure 2-8. The final result after inverting 20 Hz waves (Figure 2-8d) clearly shows that waveform
tomography can resolve all the major features in this cross-well case.
Figure 2-8: Waveform inversion results from after accumulating contributions from progressively
higher frequencies for the cross-well model.
- 25 -
2.5 Limitations of Waveform Tomography
2.5.1 Sensitivity to the Initial Model
Due to the highly non-linear nature of the misfit function in waveform tomography, the inversion
typically converges to a local minimum if the starting model is not in the vicinity of the global
minimum. We show the severity of the problem by an example using a dip section (Fig. 6 in
Sourbier et al., 2008b) of the 3D SEG/EAGE overthrust model (Figure 2-9a), originally designed
by the oil exploration community to test 3D seismic imaging methods (Aminzadeh et al., 1995).
This model covers an area of 20 km x 4.7 km and is discretized with 25 m grid cells, representing a
uniform grid of 801 x 187 nodes. The velocities in this model range from 2360 m/s to 6000 m/s.
The synthetic data we invert are generated using a Ricker wavelet centered on 9.5 Hz.
For one experiment, the starting model we use (Figure 2-9b) is a smoothed version of the true
model. True velocities in the first hundred meters have been used to remove the instabilities during
the waveform inversion (Sourbier et al., 2008b). After sequentially inverting 7 discrete frequencies
between 3.5 and 21 Hz following the frequency selection strategy developed by Sirgue and Pratt
(2004), we get a very reasonable result (Figure 2-9c). For the other experiment, we invert the same
data but starting from a homogeneous model with velocity 3500 m/s. The same set of discrete
frequencies has been inverted and the result is shown in Figure 2-9d.
Comparing the results in Figure 2-9c and Figure 2-9d, we can clearly see that the inversion result is
- 26 -
very sensitive to the starting model we use. In Figure 2-9d, besides a small part in the shallow
region and some large scale structures, the result is not at all a reasonable representation of the true
model. Specifically, the structure of the overthrust is nowhere to be seen after the inversion.
Figure 2-9: (a) The true model used in the test is a dip section (Fig 6 in Sourbier et al., 2008b) of
the 3D SEG/EAGE overthrust model (Aminzadeh et al., 1995). This model covers an area of 20 km
x 4.7 km and is discretized with 25 m grid cells, representing a uniform grid of 801 x 187 nodes.
The velocities in this model range from 2360 m/s to 6000 m/s. (b) The initial made from smoothing
the true model in (a). True velocities in the first hundred meters have been used to remove the
instabilities during the waveform inversion (Sourbier et al., 2008b). This initial model can lead to a
reasonable inversion result. (c) The reasonable inversion result from using the initial model in (b).
(d) The inversion result from using a homogeneous model with velocity 3500 m/s.
- 27 -
2.5.2 Application to Field Data
Multi-scale waveform tomography has been successfully applied in velocity estimation in a few
complex, synthetic models (see, for example, Sourbier et al., 2008; Boonyasiriwat et al., 2009). As
we can see from the examples we have shown, both near-surface and deep, complex structures can
be accurately recovered, as long as we start the inversion with a good enough initial model. These
promising results indicate that waveform tomography can be a good option in complex
environments. However, with its non-linear nature and sensitivity to initial models, it is very
challenging to apply this technique to process field data. In practice, traveltime tomography is
typically used to get the shallow velocity structure which will in turn be used as the starting model
for the waveform tomography. When working with field data, however, other factors including
random noise, surface waves, non-uniform source radiation, attenuation, anisotropy, etc. will
further make using waveform tomography more difficult.
2.6 Conclusions
Waveform tomography is an inversion technique that has been gaining a lot of popularity recently.
It is based on the solution of full wave equations; therefore it avoids the high-frequency limitation
of traveltime tomography due to its use of ray theory based forward modeling technique, which is
an approximation that becomes exact at infinite frequency. At the price of being more
computationally intensive, waveform tomography tries to minimize the difference between
- 28 -
observed data and the synthetic data by fitting the whole waveform.
Waveform tomography implementations exist in both the frequency domain and the time domain.
The frequency domain implementation has been discussed in detail in this chapter. Due to the
highly non-linear nature of the misfit function in waveform tomography, different approaches have
been developed in order to get more reliable convergences. Among them, the multi-scale approach
exploits the redundancy of wavenumber coverage in the data so that only a few properly selected
discrete frequencies need to be inverted in order to get an optimal inversion result.
Several numerical results with different acquisition geometries have been presented to demonstrate
the effectiveness of the multi-scale waveform tomography in delineating the velocity profile of the
subsurface. Waveform tomography has been shown to be very effective with both the surface
acquisition geometry and the cross-well geometry. With wide-aperture data, this technique can be
used to image very complicated structures, with higher accuracy and resolution than traveltime
tomography can provide.
However, waveform tomography is also very sensitive to the initial model used in the inversion. If
the initial model deviates too much from the true model, this technique will not be able to reach a
good convergence. This makes it very challenging to apply this technique to field data, where the
exact true model is unknown, and other factors including random noise, surface waves,
non-uniform source radiation, attenuation, anisotropy, etc. could also interfere with the waveform
- 29 -
fitting.
- 30 -
Chapter 3 Image
Enhancement
With
Wave-Equation Redatuming: Application to
GPR Data Collected at Public Transportation
Sites1
Abstract: In this paper, we described the application of wave-equation redatuming through
downward and upward continuation of surface ground penetrating radar (GPR) data to image the
target beneath the surface layer with geometric or material property heterogeneities. We first tested
this technique with the use of a synthetic radar profile generated by the finite difference time
domain method to show its effectiveness. We then applied it to the surface GPR data collected in
several urban public transportation infrastructure sites including a project to use GPR to assess the
quality of mortar consolidation behind tunnel wall in the Xiangyin highway tunnel, Shanghai,
China. The purpose of this redatuming processing is to enhance the target images in later times of a
GPR profile, by eliminating the contaminations caused by the strong diffractive scattering from the
steel rebars in the surface concrete slab or tunnel wall. The application results for synthetic and
1
Previously published as Liu, L., K. He, X. Xie, and J. Du (2007), Image enhancement with wave-equation redatuming:
application to GPR data collected at public transportation sites, J. Geophys. Eng., 4 (2), 139-147, doi:
10.1088/1742-2132/4/2/003.
- 31 -
field GPR data showed that the wave-equation redatuming technique is an effective way to
eliminate the unwanted diffraction signatures and ease GPR image interpretation for data collected
at sites with a strong heterogeneous surface layer.
3.1 Introduction
As a non-destructive testing technique, ground penetrating radar (GPR) is widely used at public
transportation infrastructure construction and maintenance sites in urban environments for quality
control and structure condition assessment purposes. Typical GPR application includes pavement
condition assessment at both project and network levels for public highways, deterioration of
bridge decks (Annan et al., 2002), and consolidation condition assessment behind tunnel lining (Xie
et al., 2007). It is typical that many sites are covered by a surface layer with strong heterogeneities.
This kind of surface layer has either an irregularly shaped interface contacting with the deeper
materials, a very heterogeneous material composition, or a combination of both. Road pavement
over a rough ground surface is a perfect example of the first kind heterogeneity, i.e., a surface layer
of pavement with non-uniform thickness. The composition of rebars and cemented aggregates in
the prefabricated reinforced concrete is a good example of the second kind, i.e., the heterogeneous
material property. Furthermore, the reinforced concrete with an uneven interface is the example of
the surface layer with a combination of geometric and physical heterogeneities.
One way to eliminate the adverse effects of the heterogeneous surface layer on image
- 32 -
reconstruction of targets in deeper formations (or later times in a GPR profile) is shifting the
reference datum from the original one (the surface on which the GPR data were taken) to an
imaginary interface at a greater depth (or later time in a GPR profile) to exclude the heterogeneous
surface layer. This process is called the downward continuation (Claerbout, 2001). Once the effect
of the surface layer can be modeled and separated, a more uniform layer can be virtually put back
and the datum can be shifted back to the original one. This procedure is called the upward
continuation, and the whole process is called redatuming through downward and upward
continuations (Clearbout, 2001). In this paper we report the approach of applying the redatuming
scheme through downward and upward continuations based on the full wave equation (Berryhill,
1979, 1984, 1986) to GPR data to suppress the surface layer effect and enhance the target images in
later times of a GPR profile.
The approach originated in seismic reflection for petroleum exploration and has been proven to be
a very effective way to carry out the static correction for exploration sites with a rough topography
or an undulating interface between the sediments and the bedrock (Berryhill, 1979, 1984, 1986).
This kind of application deals with the geometric heterogeneities, as characterized above. In this
paper, we used the wave-equation redatuming method (Berryhill, 1979), which uses the Kirchhoff
integral formulation of the wave equation. The implementation of the wave-equation redatuming is
essentially a precise and efficient computerized form of the Huygens principle. Unlike redatuming
with static shifts, wave-equation redatuming removes the distortions caused by material interface
irregularity in a manner consistent with wavefield propagation. This ensures that subsequent
- 33 -
processing steps that assume the hyperbolic form, or even more complicated trajectories consistent
with wave propagation, can be accurately applied (Bevc, 1997). The application of this approach to
material property heterogeneities, however, has not been widely tested, and is the theme of this
paper.
The paper is organized as follows. First, we present the principle of wave-equation redatuming
through two numerical examples: the model for the first example has a geometrically
heterogeneous surface layer, similar to the one originally proposed by Berryhill (1979); the model
for the second example has material heterogeneities in a flat surface layer. We aim to validate the
approach using models with a priori information of the subsurface.
Second, we apply the synthetic data validated approach to real-world field GPR data obtained from
several urban public transportation sites. The first one is from a highway bridge in Connecticut,
USA. The second one is from an airport runway in Houston, Texas, USA; and the last one is from
the Xiangyin tunnel, a highway transportation tunnel crossing the Huangpu River in Shanghai,
China.
For all field examples the purpose of the process is to suppress the influence of the heterogeneity
caused by the steel-bar in the surface layer (the reinforced concrete pavement for the first two cases
and the tunnel wall segments in the second case) to obtain a better imaging result of deeper targets
in later times of a GPR profile.
- 34 -
Finally, we conclude that the wave-equation based redatuming approach through downward and
upward continuations is an effective way to eliminate the interference caused by the geometric and
material property heterogeneity in the surface layer (e.g., a concrete slab). This process is capable
of enhancing target recognition in deeper sub-grades in an engineering infrastructure.
3.2 Wave-equation Redatuming Applied to Synthetic Models
3.2.1 Reproduction of the Original Berryhill Model with Electromagnetic
Waves
The best way to validate the wave-equation redatuming algorithm is to reproduce the original
example. First we generated the model using the full wave simulation technique of FDTD (Liu and
Arcone, 2003, 2005) for the original Berryhill (1979) example. However, this model is for the
electromagnetic wave case, but we keep the velocity ratio the same as the elastic wave case. The
velocity model is shown as Figure 3-1a. The model contains a surface layer with an irregular
bottom interface, and two linear reflectors in the sub-stratum: one horizontal and one with
16-degree dip to the right (Figure 3-1a). The original synthetic record is shown in Figure 3-1b. It
corresponds to the post-stacking case of the seismic reflection surveys, and resembles Berryhill’s
original example. After the procedure of downward (Figure 3-1c) and the complete
downward-upward continuation (Figure 3-1d), it is clear the effect of the irregular interface has
- 35 -
been greatly eliminated, and the two linear reflectors were fundamentally recovered. This
reproduction validated the algorithm that we will use to treat GPR data.
(c) w/ Downward Continu.
(a) Berryhill Model
=1.96
 =1.0
=4.0
200
400
600
20
Horizontal Grid Number
(b) Original Record
200
400
400
20
40
60
80
Trace Number
100
60
80
100
(d) w/ Down/upward Continu.
200
600
40
Trace Number
600
20
40
60
80
100
Trace Number
Figure 3-1: The original model (a) and the results of wave-equation redatuming (b-d) for the
original Berryhill model (1979). The model contains a surface layer (in light blue color) with an
EM velocity 1.4 times slower than the sub-stratum (dark red). The two linear reflectors have a
velocity half of that in the sub-stratum. After redatuming the two straight linear reflectors recovers
two linear lines (d).
3.2.2 Wave-Equation Redatuming Applied to a Piece of Synthetic Data
Penetrating Radar is often used to detect gaps and cracks between engineered surface layer such as
- 36 -
road pavement, tunnel lining, and retaining walls, etc. Metal-fabric reinforced concrete is a popular
practice in constructing such a surface layer. This section deals with the application of the
wave-equation redatuming through downward-upward continuation to a set of synthetic GPR data
generated on such a model as shown in Figure 3-2a. The concrete has a dielectric constant of 6 and
an electric conductivity of 0.005 S/m. Below the concrete surface layer is the bedrock whose
dielectric constant is 5 and the electric conductivity is 0.0001 S/m. Two gaps exist between the
surface concrete layer and the bedrock as shown in Figure 3-2a. The one on the left is air-filled; and
the one on the right is water-filled. The central frequency of the impulse source is 900 MHz, and
the source-receiver offset is 10 cm. Constant-offset reflection mode records are collected at 2 cm
trace spacing. Strong reflections and diffractions from the rebars result in severe interference with
the reflection scattering signatures from the two gaps, as seen in the original profile shown in
Figure 3-2b. The reflection from the two gaps can hardly be identified due to the strong interference
with the diffractions caused by the existence of the metal rebars in the concrete layer. Figure 3-2c
shows exactly the same geometric setting, the only exception being that there are no rebars in the
concrete slab. This time, the reflection from the gaps at about 4 ns can be easily identified with
strong amplitude standout. We can also see a multiple for the left gap at around 6 ns, and two
multiples for right gap at around 6 and 9 ns. The model and simulation techniques are discussed in
detail by Liu et al. (2007b).
- 37 -
Figure 3-2: The results of wave-equation redatuming for the synthetic 900-MHz GPR data from
numerical modeling result (Liu et al., 2007b) and the results after wave-equation redatuming (d and
e).
Figure 3-2d shows the synthetic profile after the downward continuation and Figure 3-2e shows the
final redatuming result by completing the downward and upward continuation cycle. By comparing
the GPR profile after redatuming (Figure 3-2e) with the one without metal rebars in the concrete
slab (Figure 3-2c), it becomes obvious that the redatuming process makes the identification of the
- 38 -
gaps and the multiple reflections from the GPR profile a much easier task. In the redatuming
processed profile, the reflections from the gaps possess the same polarity as shown in the profile
without rebars. Moreover, it has relatively less edge diffraction hyperbolas originated from the tips
and edges of the gaps.
3.3 Wave-Equation Redatuming Applied to Real Field Data
3.3.1 A Highway Bridge Refurbish Project in Connecticut, USA
We now present the application of wave-equation redatuming to real GPR data sets. The first
example deals with rebar diffraction suppression for GPR data collected using the 1 GHz antenna
on a rebar reinforced bridge deck in Connecticut. Since the bridge deck is essentially a simple
layered structure, no target in the sub-base of the concrete slab is expected. This first example is
only for testing the algorithm on the real data to see the effectiveness. Figure 3-3a is a photo picture
taken at the bridge refurbish project was conducted. The thickness of the concrete slab is 30.5 cm.
There are two layers of steel rebars in the concrete bridge deck, one layer is about 8.9 cm below the
surface, and the other is about 25.4 cm. Figure 3-3b shows the raw GPR profile with the DC
component removed. Severe interference characterized by the obliquely conjugated ‘meshes’ can
be seen in the entire time window. Figure 3-3c shows the data after downward continuation, and
Figure 3-3d shows the same profile after the complete redatuming process. Clearly, the redatuming
process essentially suppressed all the interference caused by the rebars in the surface layer, only
- 39 -
horizontal strips are left and these are strong reflection multiples which can be suppressed by other
data process techniques not to be intended to discuss here in this paper. This example gives us more
confidence to use this process to deal with more complicated engineering structures as shown in the
following field application examples.
Figure 3-3: The photo picture of the reinforced bridge deck (a) and the results of wave-equation
redatuming (b-d) for the 1-GHz GPR survey over a highway bridge in Connecticut.
- 40 -
3.3.2 Locating a Drain Pipe at the Edge of an Airport Runway in Houston,
USA
The purpose of this GPR survey was to locate the position of a 15.2 cm PVC drain pipe along the
edge of a runway of an airport. The runway edge has a 38.1 cm thick slab of concrete, the outer
91.4 cm of which has rebar reinforcement that is 11.4 cm deep with 30.5 cm x 30.5 cm spacing
between rebars. Twenty one GPR survey lines were collected on 15.2 cm line spacing. Along the
survey line, data was collected every 1.3 cm with 512 samples spread over a 57 ns time window.
Figure 3-4a shows the engineering sketch of portion of the cross-section of the runway edge. The
15.2 cm PVC drain pipe lies within the deepest potion of the drainage ditch. The drainage layer
beneath the runway consists of coarse, unconsolidated aggregates (symbol 2 in Figure 3-4b). Figure
3-4b shows one original GPR profile of those 21 GPR survey lines collected at this site. Four
hyperbolas resulting from diffraction of the rebars in the concrete layer are clearly seen at about 2
ns two-way travel time in depth from the surface (about 12 cm in depth). The data also shows the
bottom of the concrete layer being slightly dipping towards right. Figure 3-4c shows the profile
after downward continuation, and Figure 3-4d shows the profile after the complete cycle of
redatuming with downward and upward continuations. With wave-equation redatuming process to
suppress the effect of the surface concrete slab, a few events in later time of the GPR profile are
shown up more pronounced. First, at the horizontal distance of 1.7 m, the sudden interruption of the
continuing coherent phase corresponding to the bottom of the concrete slab (about 16 ns) is clearly
- 41 -
more visible. Second, the reflections from the 15.2 cm diameter PVC pipe at ~26 ns become more
pronounced. The slant interface between the drainage ditch and host formation on the right makes
the events more complicated. Reflection events below 30 ns are mostly multiples of shallower
events.
Figure 3-4: The engineering sketch (a) and the results of wave-equation redatuming (b-d) for the
GPR survey for searching a 15.2 cm PVC pine beneath the edge of the runway of an airport in
Houston, Texas. The numbered symbol in (a) stands for the following layers, respectively. 1: 38.1
cm PCC pavement; 2: 15.2 cm drainage layer base course; 3: geotextile fabric; 4: 20.3 cm
compacted subbase; 5: 30.5 cm selected fill.
- 42 -
3.3.3 Assessment of Mortar Consolidation Quality Behind Tunnel Wall in
Xiangyin Tunnel, Shanghai, China
Shanghai City has been under a rapid urban development in the last decade. One major
improvement in the metropolitan public transportation system is the construction of a number of
cross-river underwater highways tunnels to link the East Huangpujiang River District which is
experiencing the highest economical growing rate to city center located west of the river. The
Xiangyin tunnel is one of the major cross-river transportation projects. The tunnel is constructed by
a geotechnical construction technique called shield tunneling. Shield tunneling permits fast,
relatively safe construction work. It neither interferes with traffic above ground nor adversely
affects nearby underground structures.
The shield tunneling technique employs a large, cylindrical-shaped tunneling machine. The shield
tunneling machine excavates by rotation of an excavation face shield (steel shell) and a circular
cutter head equipped with hard metal cutter bits (Koyama, 1997). Pressurized slurry is injected into
the excavation face to form a suitably viscous mixture of slurry. The cutter chamber, once filled
with this mixture, enables stabilization of the excavation face. Simultaneous to excavation, segment
concrete blocks prefabricated elsewhere are assembled automatically inside the shield behind the
work face to finish the tunnel.
- 43 -
3.3.4 Dielectric Permittivity of Sediments and Mortars
In the shield tunneling construction process, after the tunnel wall segments are placed, backfill
grouting mortar is injected to the gaps between the segment concrete tunnel wall and the sediments
through holed cast in tunnel wall blocks. After proper consolidation, the grouting mortar provides
further stability of the tunnel, and seals the tunnel from water seepage. This is especially true for
tunnels placed underneath an open water body like rivers, lakes, and sea straits. The uniformity in
coverage and quality of consolidation of the injected grouting mortar behind the prefabricated
tunnel wall is a main concern for tunnel safety, and can only be tested non-destructively. As one of
the non-destructive testing (NDT) methods, ground penetrating radar (GPR) has been used in
tunnel construction projects and other concrete structures and road pavement for quality assurance
(QA) purposes (Annan et al., 2002; Liu and Guo, 2003; Korhonen et al., 1997; Robert, 1998; Yelf
2004).
In order to design effective GPR surveys in Shanghai highway tunnels, the dielectric permittivities
of the sedimentary formations and grouting mortar were measured in laboratory. Figure 3-5a shows
the frequency dependence of the dielectric permittivity for the grouting mortar during the period of
curing and Figure 3-5b shows some typical results for the sediment at the project site in Shanghai.
In general, it is noteworthy to point out that the permittivity of any earth material or engineered
material is heavily dependent on water content at the time of data collection (Bungey et al., 1997;
Liu and Guo, 2003).
- 44 -
Figure 3-5: The measured frequency dependence of the dielectric permittivity of the grouting
mortar tested at the time of curing (a); and the measured frequency dependence of the dielectric
permittivity of the sediment samples acquired at the site (b) (see Xie et al., 2007).
The dielectric permittivity of the grouting mortar (Figure 3-5a) dropped by 43% in 11 days during
the early stage of the curing. For the grouting mortar (mostly Portland cement) long time after the
curing period, the relative dielectric permittivity can still vary in a wide range from 5.4 (oven-dry
condition) to 12.0 (saturated surface-dry) (e.g., Bungey et al., 1997). Whenever possible, it is
always advised to use the in-situ calibrated value rather than the typical value provided by the
literature for dry concrete. We used the relative dielectric permittivity of 6.25 for the dry concrete,
equivalent to the electromagnetic wave velocity of 0.12 m/ns, which is a value calibrated by the
known thickness of the tunnel wall. The electromagnetic parameters for all materials involved are
listed in Table 3-1.
- 45 -
Table 3-1: Material properties used in redatuming by downward and upward continuations
Material
r
Air
1
0
Concrete
6.25
0.001
Mortar
9
0．005
Sediment
16
0．005
3.3.5 Field Acquisition of GPR Data in Xiangyin Tunnel
A set of GPR data was collected in the Xiangyin Tunnel on July 14, 2005 with the Senor &
Software Noggin Plus system. The nominal central frequency of the antenna for this system is 250
MHz. GPR data were collected longitudinally along the axis of the tunnel, and transversely
perpendicular to the tunnel axis on the floor of the tunnel at a number of locations in the Xiangyin
tunnel.
The wall of the Xiangyin tunnel was formed by the prefabricated circular steel-bar reinforced
concrete segments. Each tunnel segment is a 60-degree arc and 6 of them form a complete wall
segment. The diameter of the Xiangyin Tunnel is 11.58 m. The thickness of the prefabricated tunnel
wall is 0.48 m; and the width of each arc-segment is 1.5 m. The diameter of the major rebars is
16-25 mm, and the diameter of the minor rebars is 10-13 mm. The spacing of the major rebar is
10-20 cm. The distance from the surface of the concrete wall segment to the center of the major
rebar is about 35-45 mm. There is an additional 8 cm thick concrete pavement to make the tunnel
floor flat so that the total thickness varies from 0.48 to 0.56 m from the edge of the floor to the
central axis of the tunnel on the floor. The grouting mortar is used to fill the gap between the
- 46 -
prefabricated tunnel segments and the formation by a dual-fluid injection procedure through the
pre-cast holes on wall segments. The thickness of the mortar is 0.12 m but usually between 0.1 to
0.3 m.
Figure 3-6: (a) A 134-m long longitudinal 250-MHz GPR reflection profile (Line 21) acquired in
the Xiangyin highway tunnel, Shanghai, China. (b) The portion between 40-56m in Line 21.
The Xiangyin tunnel was completed in early 2005 and the grouting mortar injection and
consolidation was carried at the time of completion. Therefore the mortar has cured for a half year
before the GPR survey in July 2005. Figure 3-6a shows a longitudinal GPR reflection profile (Line
21) 134 m long acquired in the Xiangyin tunnel. The total recording time is 48 ns for each trace.
- 47 -
There are 2681 traces, with 1 trace per 0.05 m. The sampling interval is 0.4 ns. Figure 3-6b is an
extraction of a portion (40-56 m) without any suspicious features and events from the long profile
to show the details of the diffraction from the rebars in the concrete tunnel floor (the black dots
inside the concrete tunnel floor layer). From the given thickness of the tunnel wall it is
straightforward to calculate that the radar wave velocity is about 0.12 m/ns, and the two-way travel
time of the reflected wave from the wall-mortar interface is about 9.3 ns. Figure 3-6a barely shows
that the horizontal phase continuity is broken into two major segments: one from 60 to 78 m, and
the other from 122 to 130 m. The reason is unknown; one possibility is an incomplete injection of
the mortar between the tunnel wall and the host. The reflection multiples in later time are also
interrupted, featured by much smaller amplitudes (Figure 3-6a).
3.3.6 Wave-Equation Redatuming for the Profile of Line 21
Results of applying wave-equation redatuming to GPR profile Line 21 are shown in Figure 3-7.
The reference surface was shifted 0.56 m to the bottom of the tunnel wall by downward
continuation (Figure 3-7b). It is equivalent to 9.3 ns in time axis. Next, by applying upward
continuation back to the original tunnel surface, the effect of the tunnel wall was essentially
eliminated, and the events in later times become more pronounced. The mortar layer (16-17 ns)
appears to be relatively uniform. The two possible segments of imperfect mortar layer
consolidation at the 2 locations (60-78 m and 122-130 m) on Line 21 were emphasized by this
treatment (Figure 3-7c).
- 48 -
(a)
0
10
20
30
(b)
0
10
20
30
(c)
0
10
concrete tunnel floor
20
30
Figure 3-7: The results of redatuming for GPR profile Line 21 acquired inside the Xiangyin tunnel.
From top to bottom: the original profile (a); the profile after the downward continuation (b); and the
profile after the downward-upward continuation (c).
3.4 Conclusions
In this paper we presented the application of wave-equation redatuming through downward and
upward continuations to several cases in public transportation engineering projects. The purpose is
to enhance the targets in later times of a GPR profile for a more confident target characterization.
Wave-equation redatuming by downward and upward continuations were applied to both synthetic
and field GPR data to test the method’s effectiveness. The examples of wave-equation redatuming
- 49 -
presented in this paper demonstrate that this method is capable of suppressing the diffractions and
interferences caused by not only the geometric heterogeneity, but also the material property
heterogeneity in the surface layer. This approach is an effective and economical way to suppress
and eliminate the strong reflections and diffractions from rebars in the surface concrete slab to
highlight events in the later times of a GPR profile.
- 50 -
Chapter 4 Simulation
of
the
Effects
of
Receiver-Side Heterogeneities on the Wavefronts
at Different Angles: an Application of the
Pseudospectral Modeling Technique2
4.1 Introduction
Seismic waves reflected from the inner-core boundary (PKiKP waves) have been typically difficult
to observe at subcritical distances and have been argued to arrive in rare cases, and with enough
energy to get barely recorded above noise, thus only providing estimates of the upper bound of the
ICB density jump (e.g., Souriau and Souriau, 1989; Shearer and Masters, 1990). Souriau and
Souriau (1989) stated that PKiKP emerges only slightly out of the noise, which may be due to the
focusing by some heterogeneity. Shearer and Masters (1990) made similar remarks, identifying
only one convincing simultaneous arrival of PKiKP and PcP waves (seismic waves reflected from
the core-mantle boundary) on the global scale. Attempts to collect PcP waves globally sometimes
reveal very clean reflections of the CMB, which at other times are completely buried in noise or
2
Part of this chapter was previously published in Tkalcic, H., V.F. Cormier, B.L.N. Kennett, and K. He (2010), Steep
reflections from the earth's core reveal small-scale heterogeneity in the upper mantle, Physics of the Earth and Planetary
Interiors, 178 (1-2), pp. 80-91, doi:10.1016/j.pepi.2009.08.004.
- 51 -
hardly visible (Tkalcic et al., 2002; Tkalcic and Romanowicz, 2002). The existence of high-quality
PcP arrivals at one station does not guarantee the existence of PcP arrivals on another nearby
station of a similar quality. Although this could be attributed to the topography at the CMB and the
variability in source radiation pattern, this “patchy” pattern of PcP observations is still not well
understood.
Tkalcic et al. (2010) have investigated arrivals of PcP and PKiKP waves at subcritical angles using
arrays of stations, with the goal of measuring their amplitude ratios. It is important to estimate and
understand all the amplitude effects so that the amplitude ratio of PKiKP/PcP can better constrain
the density jump at the inner-core boundary (ICB). This density jump is important for the
mechanism of compositional convection in the outer core that drives the geodynamo. The bigger
the density jump, the bigger the compositional difference is between the inner core and the outer
core, and the more energy is available in the form of gravitational energy and latent heat of
crystallization to provide the heat source that provides the energy to drive convection in the outer
core.
This study by Tkalcic et al. (2010) presented high-quality observations of a large number of PcP
and PKiKP phases at multiple stations from an earthquake and a nuclear explosion, both with
favorable focal mechanisms. Interestingly, there are only a limited number of detections of both
types of waves on the same seismogram, while more frequently, either one or another of the two
phases is detected. Therefore, for those cases where at least one phase is above a detectable level, a
- 52 -
highly significant negative correlation (anti-correlation) of their appearances on seismograms is
observed. The fact that similar anti-correlation is observed for both the explosive and tectonic
sources makes less likely the possibility that source effects or a specific near source structure is
responsible for this phenomenon. Although laterally varying structure near the core-mantle
boundary (CMB) can account for the magnitude of observed fluctuations in the amplitude ratio of
PKiKP to PcP, the Fresnel volumes surrounding their ray paths are well separated at the CMB at the
frequencies of interest. This separation excludes the possibility that complex structure at or near the
CMB is the dominant effect responsible for the observed anti-correlation.
Either volumetric heterogeneity or topography of a discontinuity near the source can produce large
changes in ray direction and potentially affect geometric spreading, but ray theory cannot be used
to accurately predict both the effects of wavefront healing and near-field source effects. Instead,
either a fully numerical or a hybrid numerical/approximate technique must be used to model the
wavefield throughout the whole earth. From elastic reciprocity, the effects of heterogeneity near
either the source or receiver will be similar for body waves observed at either arrays of sources or
arrays of receivers. To avoid problems associated with the description of the source, we have
chosen to simply model the effects of increased heterogeneity beneath the receivers.
In this chapter, results from numerical synthesis of the wavefield associated with either a PKiKP or
PcP plane wave incident at an array of receivers using a 2-D pseudospectral modeling technique
(Cormier, 2000, 2007a) will be shown to demonstrate the effects of receiver-side heterogeneities on
- 53 -
the wavefronts at different angles. Three types of 2-D heterogeneities having either an isotropic or
anisotropic distribution of scale lengths have been considered. We demonstrate that the interaction
of the wavefield with near-receiver heterogeneities is an important additional source of amplitude
fluctuations across arrays of stations, and the likely cause of the anti-correlation pattern.
4.2 Two-Dimensional Pseudospectral Modeling Technique
The pseudospectral method (Kosloff and Kessler, 1990; Fornberg, 1998; Comier, 2007) is a widely
used forward modeling technique which solves a "strong" (differential) form of the wave equations.
This technique is distinguished by its use of Fourier transforms to estimate the spatial derivatives in
the elastic equations of motion, although it has been shown that the pseudospectral approach is also
the high-order limit of the finite difference method (Fornberg, 1998). This method is
computationally more expensive than higher-order finite differences but achieves much higher
accuracy at long ranges, typically having little grid dispersion in applications to ranges of 10,000
wavelengths or higher. For this reason, the pseudospectral method can be an ideal choice for
teleseismic applications, where high accuracy is desirable at both regional and teleseismic range.
Compared to the amount of literature that exists in the application of finite difference and
pseudospectral solutions for local-scale problems up to 100 km (e.g., Harzell et al., 1999; Olsen,
2000) and the amount of published work existing for applications at regional distances (100-2000
km), a much smaller amount exists for teleseismic propagation (e.g., Furumura et al., 1998; Igel,
1999; Cormier, 2000).
- 54 -
The velocity-stress equations (Virieux, 1984, 1986; Van Ark, 2007) are summarized here, with all
variables used in this section defined in Table 4-1:
vr 1

t

1  r  
1 

 r r  r rr   r   r  f r 


(4.1)
v 1  1  2
1  

  2
r  r 
 f 
t
  r r
r 

(4.2)
  P
  vr 1 v vr 
  S
  1 v vr  
rrr
1 

  rrr     1 

   2    1 
 
t   
 
  r r  r 
 
  r  r  
(4.3)
 P
  v 1 v vr 
 S
  v  
r 
1 
  r      1  r 
   2      1  r  
t
  
 
  r r  r 
 
  r  
(4.4)
 S
  v 1 vr v  
rr
1 
  rr       1   
 
t   
 
  r r  r  
(4.5)
 P  vr 1 v vr 
 S  1 v vr 
 rr




 2

r
t
   r r  r 
   r  r  rr
(4.6)


 
 P  v 1 v vr
   r 

t
   r r  r
 r
S
  
t

 S vr



2

r

  r 

 1 vr v v 
 r   r  r   rr


(4.7)
(4.8)
Table 4-1: Pseudospectral method notation
- 55 -
First order velocity-stress equations are solved on a staggered grid (Virieux, 1984, 1986; Witte,
1989; Witte and Richards, 1990) in order to increase stability and to accommodate the half-grid
step Fourier shift required to maintain real first-order spectral derivatives. When calculating spatial
derivatives, we apply the Fourier shift theorem and evaluate the derivative halfway between the
sampled points, i.e., on a staggered grid. The spatial derivatives of the stress field used to step the
velocity components forward in time can be written as:
- 56 -
1
1

   ix  , iy, it  
  
1
2
2

 ix, iy, it   
 
2

1
1

 rr  ix  , iy, it  
 rr 
1
1
1
2
2

 ix  , iy  , it   
r 
2
2
2
r
1
1

  r  ix, iy  , it  
  r 
1
2
2

 ix, iy, it   
r 
2
r
1
1

  r  ix, iy  , it  
  r 
1
1
1
2
2

 ix  , iy  , it   
 
2
2
2

(4.9)
(4.10)
(4.11)
(4.12)
And the spatial derivatives of the velocity components used to step the stress tensor components
forward in time can be written as:
1
1 

vr  ix  , iy  , it 
vr 
1
2
2 


 ix  , iy, it  
r 
2
r

1
1 

vr  ix  , iy  , it 
vr 
1 
2
2 

 ix, iy  , it  
 
2 

v

1

 v  ix, iy, it 
 ix  , iy, it  
2



(4.13)
(4.14)
(4.15)
- 57 -
v
r
1  v  ix, iy, it 

 ix, iy  , it  
2 
r

(4.16)
The numerical solution scheme presented here uses a simple first order finite difference algorithm
in time. Attenuation is optionally included through the use of stress and strain dependent memory
functions (Robertsson et al., 1994). Boundary conditions on both sides of the model are
implemented using cancellation of periodic and anti-periodic wrap-around wavefields (Furumura
and Takenaka, 1995). The computational implementation of this scheme uses the message passing
interface (MPI) parallel instruction set (Gropp et al., 1998, 1999) to take advantage of parallel
computers and computer clusters.
4.3 Simulation of the Effects of Receiver-side Heterogeneities on
the Wavefronts at Different Angles
We have used a 2-D pseudospectral method (Cormier, 2000) to numerically synthesize the
wavefield associated with either a PKiKP or PcP plane wave incident at an array of receivers. The
effects of 2-D heterogeneity specified by three different spectra of scale lengths were considered
(Figure 4-1), having either an isotropic or anisotropic distribution of scale lengths specified by
Gaussian autocorrelations of P wave velocity having 5 per cent RMS fluctuation. We took spatial
and temporal gridding to be x = z = 0.5 km and t = 0.0135 sec; scaling of P velocity, S velocity,
and density variations to be VS/VS = 2VP/VP, and  = 0. Point sources were distributed at
- 58 -
neighboring grid points to simulate PKiKP (126 points) and PcP (136) plane waves arriving at a
receiver array whose center is 20o from the source. A Gaussian far-field displacement source time
function was assumed having a half width of 0.18 sec to simulate the typical frequency content of
the observations of band-passed particle velocity. In Figure 4-2, several snapshots of the wavefield
have been shown to illustrate the wave propagation simulated by the pseudospectral method.
Seismograms were synthesized in the homogeneous background model to check for stability of
amplitudes and any waveform distortion induced by numerical inaccuracy. Displacement to particle
velocity and 2-D line to 3-D point source corrections were applied. This test found that amplitude
fluctuations at the receiver array were less than several percent in the homogeneous background
medium and waveforms visually agreed with those predicted by ray theory. Figure 4-3 to Figure
4-5 summarize the effects of peak-to-peak amplitude fluctuations at the receiver array measured
from synthetics for PKiKP and PcP incident plane waves for the three types of heterogeneity
spectra or textures shown in Figure 4-1. The PKiKP/PcP ratios shown in these figures show only
the effect of heterogeneity beneath the receiver array and are not corrected for the effects of
transmission, reflection, and geometric spreading beneath the zone of heterogeneity, which would
significantly reduce the value of the ratio by the larger absolute amplitude of PcP compared to
PKiKP predicted by ray theory in the distance range around 20o. In cases where one phase is
detected and the other is not these simulations illustrate that the spatial coherence of detections or
non-detections at dense arrays of receivers of these weak, partially reflected, outer and inner core
boundary waves may be simply explained by heterogeneity beneath the receiver.
- 59 -
Figure 4-1: Wavefronts of PcP (blue) and PKiKP (red) in the range 16 to 25 incident on an earth
model perturbed by statistically described heterogeneity in the crust and upper mantle beneath a
receiver array. A 5 per cent RMS perturbation in P velocity is assumed in each case (a-c) with an
exponential autocorrelation. (a) isotropic distribution of scale lengths with a cutoff at 2 km; (b)
anisotropic scale distribution of scale lengths with a horizontal cutoff at 100 km and vertical cutoff
at 2 km; (c) anisotropic distribution of scale lengths with a vertical cutoff at 100 km and horizontal
cutoff at 2 km.
- 60 -
Figure 4-2: Several snapshots of the wavefield illustrating the wave propagation simulated by the
pseudospectral method. Point sources were distributed at neighboring grid points to simulate plane
waves. Note in Figure 4-2d that the second, weaker plane wave arrivals are an undesirable
reflection from the bottom of the model.
Figure 4-3: Predicted amplitude fluctuations of PKiKP and PcP at an array of surface receivers
- 61 -
measured from bandpassed seismograms of particle velocity synthesized by a numerical
pseudospectral method in the isotropic texture of crust and upper mantle heterogeneity shown in
Figure 4-1a. The PKiKP/PcP ratio includes only the effects of heterogeneity beneath the receiver
array; the effects of differing geometric spreading, viscoelastic attenuation, and
reflection/transmission coefficients in deep structure are not included in the PKiKP/PcP ratio.
Figure 4-4: Same as Figure 4-3 but for the anisotropic texture of crust and upper mantle
heterogeneity shown in Figure 4-1b (horizontal scale lengths longer than vertical scale lengths).
- 62 -
Figure 4-5: Same as Figure 4-3 but for the anisotropic texture of crust and upper mantle
heterogeneity shown in Figure 4-1c (vertical scale lengths longer than horizontal scale lengths).
The simulations in Figure 4-3 to Figure 4-5 are in qualitative agreement with recent calculations
using the phase screen method by Zheng and Wu (2008). They calculated the lateral coherence
between the logarithms of amplitudes of two different, near vertically incident, plane waves
transmitted through a heterogeneous upper mantle. They found that for plane waves separated by as
little as 5 degrees in vertical incidence the coherence of log amplitudes decreases by a factor of 2 at
zero lag. This result was predicted at receiver arrays underlain by a 300 km zone of heterogeneity
having a Gaussian correlation function with a scale length of 10 km and 1 per cent RMS
perturbation in P velocity. Scale lengths of this order have long been posed as an explanation of
amplitude fluctuations of body waves observed at short-period arrays (e.g. Aki, 1973).
- 63 -
Since the signal to noise ratio of the PcP and PKiKP observations in the cases where one phase is
observed and the other is not is often a factor of 2 or less, very similar types of heterogeneity
models can easily explain an observed negative correlation between the PKiKP and PcP detections
(Tkalcic et al., 2010). In the case of an isotropic distribution of scale lengths (Figure 4-3), focusing
and defocusing of PKiKP is about equally likely to be correlated or uncorrelated with focusing and
defocusing of PcP. In the case of horizontally stretched heterogeneity (Figure 4-4), focusing and
defocusing of PKiKP seems to be generally more anti-correlated with the focusing and defocusing
of PcP. In the case of vertically stretched heterogeneity (Figure 4-5), PcP is more attenuated by
energy lost to backscattering compared to PKiKP. This type of heterogeneity, i.e., one in which
quasi-vertically stretched heterogeneity is at a larger angle with respect to the ray normal of the
incident PcP wavefront than to the ray normal of the incident PKiKP wavefront, may explain the
more exceptional observations in which PKiKP is detected in the 20o to 40o range but PcP is not
(Tkalcic et al., 2010).
4.4 Conclusions
From abundant records of PcP and PKiKP travel times and amplitudes observed at numerous
seismic stations from two events (a single earthquake and a nuclear explosion), Tkalcic et al. (2010)
have found a general strong negative correlation in the detection of PcP and PKiKP waves, which
has not been reported before. The possibility that the negative correlation pattern stems from the
- 64 -
source radiation pattern has been excluded. Although core-mantle boundary structure can
significantly contribute to the scatter in ratio of PKiKP/PcP, it cannot explain a more systematic
negative correlation in PKiKP and PcP detections. We carried out simulations using a 2-D
pseudospectral modeling technique to model the effects of increased heterogeneity beneath the
receivers, and demonstrated that the interaction of the wavefield with near-receiver heterogeneities
is an important additional source of amplitude fluctuations across arrays of stations. We conclude
that the volumetric heterogeneity in the crust and upper mantle on the receiver side must be the
cause of the observed spatial coherence in anti-correlation of PKiKP and PcP detections.
- 65 -
Chapter 5 Study of the Strong Ground Motion
Scenario in the Beijing Area Caused by the 1679
M8 Sanhe-Pinggu Earthquake3
5.1 Abstract
The great Sanhe-Pinggu earthquake (M~8, with a maximum intensity of XI at the epicenter and
VIII in Beijing city) occurred in September, 1679 is the largest historic event in the past 500 years
within a 100-km radius from the center of Beijing. A detailed understanding of the ground motion
scenario caused by this earthquake will provide valuable information for seismic hazard reduction
in the metropolitan Beijing area. Numerical simulation of seismic wave propagation and strong
ground motion using high performance computational methods is a proven tool in seismic hazard
zonation and local site effect assessment based on scenario earthquakes. We studied the effect of
the 1679 Sanhe-Pinggu M8 earthquake in Beijing area with the staggered grid pseudospectral time
domain (PSTD) method, with stretched coordinate perfectly matched layer (PML) absorption
3
This chapter is a revised version of what was previously published as He, K., L. Liu, W. Wang, Y. Tan, and Q. Chen
(2008), Study of the strong ground motion scenario in the Beijing area caused by the 1679 M8 Sanhe-Pinggu earthquake,
Proceedings of the 14th World Conference on Earthquake Engineering (14WCEE), Beijing, China, October 12-17, Paper
No. 14-0135.
- 67 -
boundary conditions. The epicenter of this great event was about 60 km ENE of the center of
Beijing city. The simulated seismic source is the distributed 5-m right-lateral, strike-slip rupture
along the ~150 km long, NE oriented near-vertical Xiadian Fault. The seismic wave propagation is
simulated for up to 1 Hz. The grid size is about 0.18 km. High resolution near-surface velocity
structure obtained by the newly acquired microtremor data was incorporated in the crust model.
The result suggests that the numerical modeling approach is effective in seismic hazard assessment
and provides valuable information for mitigating losses for possible earthquakes in the future. This
project is supported by the Ministry of Science and Technology of China with Project No.
2006DFA21650 and the Institute of Earthquake Science (Project No.0207690229).
5.2 Introduction
It has long been recognized that the structure and material properties of the uppermost sediments are
important factors to determine the intensity of strong ground motion caused by earthquake shaking
(Anderson, 1996). Therefore, study of local site effects is a critical part in seismic hazard mitigation
effort.
Historic earthquake records indicate that moderate to strong earthquakes have been frequently striking
the greater Beijing area. During the past 500 years (the Ming and Qing Dynasties), there have been at
least 11 earthquakes with a maximum intensity of VI or greater that occurred within a 100-km radius
centered at the Tiananmen Square, the center of Beijing City. The Sanhe-Pinggu (M~8) earthquake,
- 68 -
the largest historic event, occurred 65 km ENE of Beijing, and severely damaged the city on
September 2, 1679 (Institute of Geophysics, 1990).
To quantitatively assess the seismic risks of the Beijing area, especially at a few critical sites
associated with the 2008 Summer Olympic Games, we used numerical simulations to study the strong
ground motion scenarios based on historic earthquake records. To improve the modeling accuracy,
high-resolution near-surface sediment layer thickness information obtained by a microtremor
measurement campaign conducted in summer 2007 (Chen et al., 2008) was incorporated into the
crustal model. With more constraints on near-surface geological information, we have the opportunity
to better estimate the local site effects on strong ground motion in the greater Beijing area generated
by potential pending earthquakes (scenario events) similar to the 1679 Sanhe-Pinggu M8 event.
5.3 Summary of the Numerical Model
5.3.1 Modeled Area
Figure 5-1 shows a map view of the modeled area, with the 2nd to 5th Beijing beltways and main roads
highlighted for reference. We have also highlighted the epicenter of the Sanhe-Pinggu event and four
other locations we are concerned about: the Tiananmen Square, an often used national symbol which
also marks the center of Beijing city; the 2008 Summer Olympics Stadium (also known for its
nickname “Bird Nest”), where the 2008 Summer Olympics Games is being hosted when we write this
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paper; the Wukesong Culture and Sports Center, an indoor arena for the 2008 Summer Olympics
basketball preliminaries and finals; and the Beijing University of Technology (BUT), whose stadium
is hosting the Olympics badminton matches. Detailed shear wave velocity profiling experiments have
been done at all the 3 Olympics game-related sites (e.g., Wang et al., 2008). The thickness of the
Quaternary sediment layer is also shown in Figure 5-1, with data taken from Jia et al. (2005), and
Chen et al. (2008).
5.3.2 Model Implementation
We have used a staggered grid pseudospectral time domain (PSTD) method (e.g., Witte, 1989; Liu
1997; Cormier, 2000; Liu and Arcone, 2005) to simulate the seismic wave propagation generated by a
scenario earthquake based on the 1679 Sanhe-Pinggu M8 event. Our 3D model consists of a system
of 512x256x64 grids, with dx=dy=dx=0.18 km, which translates into a volume of size 92.16 km x
46.08 km x 11.52 km. The model consists of three layers, similar to what is used in Ding et al. (2004):
the Quaternary sediment layer (vp = 1.56 km/s, vs = 0.9 km/s), the Tertiary sediment layer (vp = 4.5
km/s, vs = 2.6 km/s), and the bedrock (vp = 6 km/s, vs = 3.46 km/s). The modeled surface area has a
rectangular shape, with the southwest vertex at (39.8000˚N, 116.1000˚E) and the northeast vertex at
(40.2149˚N, 117.1823˚E). For simplicity, we have assumed a uniform density 2,500 kg/m3 throughout
our model domain.
PML absorption boundary conditions (Chew and Weedon, 1994; Liu and Tao, 1997) have been
- 70 -
used in our modeling, and its thickness is 10 grid points, which is a good trade-off for this case. The
time step used is 7.5 ms, and we have calculated 11,466 time steps, equivalent to 85.995 seconds in
time, which we originally thought would sufficient for all the wave phases of interest to us to
propagate through the entire model domain. Upon close scrutiny, 85.995 seconds might be way too
short to simulate possible damage from such a big rupture. Although the rupture itself might be
over in about 70 seconds or less, the basins can continue to "ring" with surface waves bouncing
back and forth between boundaries of thickly sedimented regions.
Figure 5-1 Thickness of the Quaternary layer of the model, with Beijing city beltways, main roads,
and some specific places we are particularly interested in highlighted. Also shown on the map is the
fault that we modeled.
- 71 -
5.4 The Source Parameters
The parameters used for modeling the scenario seismic source are listed in Table 5-1. Most of these
parameters are directly obtained, indirectly derived, or inferred based on previous studies (Xu et al.,
2002; Shen et al., 2004; Biasi and Weldon, 2006; Liu et al., 2007a). The length and width of the fault
are 140 km and 30 km, respectively. The length of the fault is in agreement with the magnitude and
rupture length relation (Biasi and Weldon, 2006). However, due to the limited size of our model, we
have only been able to construct a fault of 42.48 km in length and 11.52 km in width, with the source
time window 30.20 second long. For implementation simplicity, we did not use the values of strike,
dip and rake in Table 5-1 when simulating the fault. Instead, we constructed a strike-slip fault along
the North-South direction (see Figure 5-1). The rupture is assumed to initiate at a depth of 10 km and
we also assumed that the source mechanism has a rupture velocity 0.8 km/s. In actuality, since we
have assumed the rupture initiates in the bedrock layer, the real rupture velocity would be much
bigger than the values we have used here. The study will be repeated for publication with a more
correct and probable value. We used particle velocity on the fault plane as the source, which is
distributed over the modeled fault plane, and which decay in accordance with a sinusoidal relation to
the surface and the lateral border of the model (Figure 5-2). The source time function that has been
used is a Ricker wavelet with a central frequency of 1 Hz.
- 72 -
Figure 5-2: Source intensity in terms of particle velocity (m/s) of the point sources on the fault
plane.
The largest particle velocity used is 66.67 m/s, which is justified by the argument shown as follows.
To estimate the magnitude of the particle velocity caused by an earthquake rupturing, we assume that
a substantial portion of the elastic potential energy Ep has been converted to the kinetic energy Ek:
Ek ~ Ep
(5.1)
As we all know, the kinetic energy of a volume element can be expressed as
1
mv 2
2
(5.2)
1
V  ij ij
2
(5.3)
Ek 
And the elastic potential energy is
Ep 
where m and V are the mass and volume of a volume element on the fault plane, respectively;
 ij and  ij are stress and strain tensor components; and v is the average particle velocity over the
whole fault plane. To model the strike-slip faulting, only the shear motion is involved so that we have
the simplified Hooke’s Law
- 73 -
 ij   ij
(5.4)
where  is the shear modules, and only the shear components of stress and strain get involved.
Moreover, the relation between seismic moment and stress tensor can be characterized as:
 ij  M 0
M ij
V
(5.5)
where M0 is the scalar moment carrying the magnitude and dimension of the seismic moment caused
by an earthquake rupturing, and Mij is the unit moment tensor carrying the relativity of different
motion with a maximum of unity. Then, from equations (5.1)-(5.5), the average particle velocity on
the fault plane can be estimated as:
v
M0
V
1

(5.6)
From the M0-M relation (Hanks and Kanamori, 1979; Deichmann, 2006):
M
2
log M 0  6
3
(5.7)
we can estimate M0, hence the average velocity v , from equation (5.6) with a given density  and
shear modulus .
- 74 -
Table 5-1 Source parameters of the Sanhe-Pinggu earthquake
Parameters
Value
Parameters
Value
seismic moment
stress drop
length of the fault
vertical extension
epicenter location
1.26x1028x10-7 N·m
20 MPa
140 km
30 km
40˚N / 117˚E
source depth
dip
strike
rake
10 km
70˚
50˚
18˚
5.5 Discussions
5.5.1 Surface Peak Ground Velocity (PGV) and Peak Ground
Acceleration (PGV)
PGV and PGA plots are very useful for earthquake engineering purposes. Surface PGV and PGA
plots in the Beijing city area are shown in Figure 5-3, with the top row showing the three-component
PGV, and the bottom row showing the three-component PGA. Beltways and main roads are also
shown for reference. We have noticed that in the region shown in Figure 5-3, the most prominent
PGV and PGA is in the east part. That is understandable, since the epicenter is ENE of the center of
Beijing city.
- 75 -
Figure 5-3 Top row: three component surface PGV (from left to right: East-West component,
North-South component, and vertical component) in m/s. Bottom row: three component surface
PGA (from left to right: East-West component, North-South component, and vertical component)
in m/s2.
In order to count for geometric spreading and to show more clearly seismic wave amplitude
amplification effects which are possibly due to the sediment layers, we performed a horizontal
versus vertical (H/V) PGV calculation. The result is shown in Figure 5-4, and we can see clearly
from the rightmost plot in Figure 5-4 that there are strong amplitude amplification effects to the
north of the city and in the southeastern corner of the Beijing city. It is worthy to point out that
during the 1976 Tangshan Earthquake, abnormally high amplitude ground motion were reported in
- 76 -
the Haidian District (the NW corner between the 4th and the 5th Beltway), and the southeast of
Beijing area (Ding et al., 2004), which has also been confirmed by observations in a recent
microtremor study (Figure 5-5).
Figure 5-4 (Left) Horizontal PGV; (Middle) Vertical PGV; (Right) Horizontal PGV divided by
vertical PGV (the H/V ratio).
Figure 5-5 (a) The amplification factor of the horizontal to vertical spectral ratio at the predominant
resonance frequency for Beijing area from microtremor measurements (Chen et al., 2008); (b) the
- 77 -
horizontal to vertical ratio of PGA generated from the synthetic simulation of the 1679 M8
Sanhe-Pinggu scenario earthquake.
5.5.2 Synthetic Seismograms at Specific Sites
Shown in Figure 5-6 are the three-component synthetic seismograms recorded at four sites: the 2008
Summer Olympic Stadium, Beijing Wukesong Culture and Sports Center, Tiananmen Square, and
Beijing University of Technology (BUT). The sampling rate of these seismograms is 75 ms.
Figure 5-6 Three-component synthetic seismograms recorded at four sites (the 2008 Summer
Olympic Stadium, Beijing Wukesong Culture and Sports Center, Tiananmen Square, and BUT),
with a sampling rate of 75 ms.
- 78 -
5.6 Conclusions
We have used a staggered grid PSTD method to simulate the 1679 M8 Sanhe-Pinggu earthquake in
the Beijing area, using a crust model with high-resolution sediment thickness information obtained
from microtremor measurements in this area. Three-component synthetic seismograms of four
important sites including the 2008 Summer Olympic Stadium are shown. From the surface PGV
and PGA studies, together with H/V calculations, we have observed ground motion amplification
effects possibly associated with local site effects in the north part and the southeast corner of
Beijing area, which have also been confirmed by other studies. Our result suggests that a numerical
modeling approach is effective in seismic hazard assessment and provides valuable information for
mitigating losses for possible pending earthquakes in the future.
- 79 -
Chapter 6 Texture of the Uppermost Inner Core
from Seismic Coda Waves
6.1 Introduction
Seismic wavefields interacting with the uppermost few hundred kilometers of the inner core,
although limited in uniformity of geographic sampling, reveal this region to have strong lateral
variations in elastic structure, anisotropy, attenuation, and scattering.
Vidale and Earle (2000) have discovered the existence of a scattering fabric or texture (spatial
distribution) in the uppermost inner core from the coda of back-scattered PKiKP waves. Using
classical, single-scattering theory, Leyton and Koper (2007a) examined the possibility of PKiKP
coda wave generation from heterogeneities in different depths of the Earth: the lower mantle on the
source side, the core-mantle boundary on the source side, the inner core boundary, and within the
inner core. They find that PKiKP coda is best explained by scattering from volumetric
heterogeneity in the inner core rather than by topography on the inner core boundary. The inferred
heterogeneities with a length scale of 1-10 kilometers in the uppermost inner core could be related
to misalignments of crystals (Bergman et al., 2002, 2003, 2005), small-scale variations in
- 81 -
orientation of anisotropy and attenuation (Cormier et al., 1998; Cormier and Li, 2002), presence of
partial melt (Singh et al., 2000), and impurities (Jephcoat and Olson, 1987; Lin et al., 2002).
The detailed texture of the uppermost inner core is important for understanding how the inner core
is solidifying from the liquid outer core, possibly providing a mechanism for compositional
convection that can drive the geodynamo (Gubbins et al., 1979, 2004). The origin of lateral
variations in the uppermost inner core may be related to lateral differences in solidification or
viscous flow and recrystallization, which are closely coupled to variations in fluid flow at the
bottom of the liquid outer core. Laboratory experiments examining the complex textures of
crystallized ices and hcp metals in convecting and rotating melts have led Bergman et al. (2002,
2003, 2005) to speculate that “the convective pattern at the base of the outer core is recorded in the
texture of the inner core”. If true, lateral variations in outer core flow would be preserved in lateral
variations in texture of the inner core, and maps of the lateral variation in the anisotropy of
heterogeneity scale lengths can separate regions of growth by active new crystallization
perpendicular to the inner core boundary from regions of viscous flow and recrystallization parallel
to the inner core boundary. Recently, studies have found that the anisotropy in the inner core may
be related to the thermal evolution of the core mantle boundary region, the structure of the
magnetic field (Aubert et al. 2008) and locations of the magnetic flux patches (Deuss et al., 2010).
Seismic wave traveltime tomography has been very successful in imaging lateral variations from an
initial layered Earth model, mostly for P and S wave velocities (e.g., Romanowicz, 2003); however,
- 82 -
its success is limited by the requirement that the scale length of the heterogeneity must be much
larger than the wavelength of the relevant seismic waves. In global mantle models, this leads to
resolution of no better than hundreds of kilometers, even in the best sampled regions. To examine
smaller-scale heterogeneities in the uppermost inner core, it is necessary to use scattered seismic
energy, as found for example in the extended codas of seismic waves. Following the pioneering
work of Aki (1969), several authors have devoted their work to the characterization of coda,
primarily to deduce the properties of crustal heterogeneities located near a seismograph station (e.g.,
Aki, 1973; Berteussen et al., 1975; Bannister et al., 1990; Revenaugh, 1995; Nishigami, 1997;
Snieder et al., 2002). There are a smaller number of studies focusing on the characterization of coda
created in the deep interior of the Earth (e.g. Vidale and Earle, 2000; Vidale et al., 2000; Koper et
al., 2004; Poupinet and Kennett, 2004).
In this study, we have used high frequency coda of the P wave reflected from the inner core
boundary (PKiKP) to study the small-scale heterogeneities in the uppermost inner core. The data
coverage of body waves is sparse, since we are limited by both the availability in location of
seismic stations at the surface of the Earth and the big, deep earthquakes where clean PKiKP data
could be generated. Since PKiKP is a very weak phase, it is often hard to be observed in the data
unless standard array techniques (Rost and Thomas, 2002) are used to boost the sign-to-noise ratio.
The radiative transfer theory (RTT), sometimes also called radiative transport theory, describes
energy transport through a random heterogeneous medium neglecting phase information and has
- 83 -
been frequently used to simulate observed mean square envelopes of high-frequency waves. The
radiative transfer equations can be numerically solved by Monte Carlo simulations. With the effect
of multiple scattering in mind, we have carried out Monte-Carlo simulations to model the high
frequency coda of the PKiKP to infer the spatial distribution of 1 to 100 km scale lengths of
heterogeneity in the inner core. As has been shown by Leyton and Koper (2007a), volumetric
scattering has a strong trade-off between the model used to describe the medium (autocorrelation
function), the characteristic wavelength, total volume, and P wave impedance contrast of the
scatterers, precluding the determination of unique properties of the heterogeneities inside the inner
core.
Fundamentally different sensitivities of forward- versus back-scattered body waves from regions of
heterogeneity enable constraints to be placed on the anisotropy of heterogeneity scale lengths. By
examining both the coda of PKiKP, for back-scattering effects, and the pulse broadening and
amplitude reduction of PKIKP, for forward-scattering effects (Figure 6-1), we can put more
constraints to the spatial distribution of the small-scale heterogeneity in the inner core.
- 84 -
Figure 6-1: Ray paths of PKiKP and short range PKIKP. Large-scale lateral variation in an
elastically isotropic uppermost inner core is shown in a polar cross section (after Cormier, 2007b).
6.2 Radiative Transfer Theory
Short period wave propagation through the lithosphere results in complex wave trains that are
mainly composed of waves multiply scattered at small-scale heterogeneities of the Earth medium.
Seismograms at distant receivers exhibit long-lasting and highly variable wave trains usually called
coda waves (Aki, 1969; Aki and Chouet, 1975). The phase of coda waves is more or less random,
but their envelopes show a much simpler behavior with smooth variations depending on distance
- 85 -
and frequency. Therefore it has become common practice to use envelopes instead of complete
waveforms to gain an understanding of the heterogeneous structure and the statistical
characteristics of the propagation medium. Amplitude attenuation, peak delay, envelope broadening,
and coda decay rate are some of the parameters that can be deduced from observed envelope shapes
(e.g., Scherbaum and Sato, 1991; Gusev and Abubakirov, 1999).
There have been a number of approaches to model envelope shapes in media with random
fluctuations of the elastic parameters. Among them are single-scattering Born theory (Sato, 1977),
mean wave theory (Muller and Shapiro, 2001), energy flux model (Frankel and Wennerberg, 1987;
Korn, 1993), diffusion theory (Dainty and Toksoz, 1977; Margerin et al., 1998; Tregoures and van
Tiggelen, 2002; Wegler, 2004), and more recently, the Markov approximation (Williamson, 1972;
Sato and Fehler, 1998; Saito et al., 2002; Korn and Sato, 2005).
RTT (Wu, 1985; Gusev and Abubakirov, 1987; Hoshiba, 1991; Zeng et al., 1991; Apresyan and
Kravtsov, 1996), on the other hand, describes energy transport through a scattering medium
neglecting phase information. It can be strictly derived from the elastic wave equation (Rytov et al.,
1987; Weaver, 1990; Ryzhik et al., 1996) and is, in principle, capable of modeling both short and
long lapse time coda. Thus it is more general than Markov theory which is only valid for short lapse
times and diffusion theory which is valid for long lapse times and strong scattering.
Originally, RTT was used in atmospheric sciences to describe the scattering of light in the
- 86 -
atmosphere (Chandrasekar, 1960). Only later was it introduced phenomenologically into
seismology. Aki & Chouet (1975) proposed a single backscattering model to explain the generation
of the seismic coda by seismic wave scattering at the heterogeneous structure of the Earth. Sato
(1977) developed an isotropic single scattering model for arbitrary source–receiver configurations.
Later, Sato (1984) developed three-component envelope synthesis, based on the single-scattering
approximation with Born scattering coefficients in random elastic media. Wu (1985) presented a
radiative transfer method to separate scattering attenuation from intrinsic attenuation. Multiple
scattering models were developed, for example, by Zeng et al. (1991), Hoshiba (1991) and Gusev
& Abubakirov (1996). A strict derivation of radiative transfer from the linear elastic wave equation
in random media could also be achieved (Rytov et al., 1987; Weaver, 1990; Ryzhik et al., 1996).
The basic assumptions for the validity of RTT can be expressed as follows: (1) scattering is weak,
(2) wavelength and scale length of the heterogeneities are of comparable size, and (3) phases of
waves from different scatterers are independent of each other, i.e., the energy of scattered wave
packets can be stacked.
6.2.1 Elastic Radiative Transfer Equations
For a detailed derivation of radiative transfer equations from wave theory in random media, we
refer to Ryzhik et al. (1996). The key quantity in these equations is I  x, k , t  of a wave at point x,
time t, moving in direction k. Here we introduce I p  x, k , t  and I s  x, k , t  for P and S waves,
- 87 -
respectively. Following Ryzhik et al. (1996), the elastic radiative transfer equations in 3-D can be
written as
P
1 I  x, k , t 
 k  gradI P  x, k , t 
0
t
1
g pp  k , k   I P  x, k , t  dk   g pp 0 I P  x, k , t 
4 
1

g sp  k , k   I S  x, k , t  dk   g ps 0 I P  x, k , t   Q P  x, k , t 
4 
S
1 I  x, k , t 
 k  gradI S  x, k , t 
0
t

1
g ss  k , k   I s  x, k , t  dk   g ss 0 I S  x, k , t 

4
1

g ps  k , k   I P  x, k , t  dk   g sp 0 I S  x, k , t   Q S  x, k , t  .

4

(6.1)
In equation (6.1), unit wave numbers k  and k denote incidence and scattered wave directions.
Mean velocities are  0 and  0 for P and S waves. The left-hand sides of the coupled equations
describe the intensity transport of P and S modes and represent the total time derivative of
intensities. The right-hand sides contain the intensity loss from the direction of propagation into all
other directions k  through the total scattering coefficients gij 0 (i, j = P or S), and the intensity
gain from all directions into the propagation direction k through the integral over g ij . Coupling
between both equations is given by conversion scattering coefficients g ps and g sp . Q P  x, k , t 
and QS  x, k , t  represent sources of P and S waves, respectively. The radiative transfer equations
can be derived from the Bethe-Salpeter equation using the so called ladder approximation with the
- 88 -
validity range (ka )2
1 (Rytov et al., 1987, p.151). Energy dissipation and velocity dispersion
are neglected in equation (6.1).
6.2.2 Random Medium
A random medium can be described by velocity and density fluctuations around a background
mean value. We write the velocity as
V ( x)  V0   V ( x)  V0 (1   ( x)),
(6.2)
where  ( x) is the fluctuation of wave velocity and
V0  V ( x) ,  ( x)  0.
(6.3)
The autocorrelation function (ACF) is defined as ensemble mean value
R( y)   ( x) ( x  y) ,
(6.4)
 2  R(0)   ( x) 2 .
(6.5)
with variance
The ACF is a statistical measure of the spatial scale and the magnitude of irregularity in the
medium.
Exponential ACF, which is a special case for von Karman ACF (see Sato and Fehler, 1998, pp.
15-16), and Gaussian ACF are typically used to represent different types of media. Exponential
- 89 -
ACF in 3-D is given by
RE ( x)  RE (r )   2 e  r / a ,
(6.6)
where a is the correlation distance, and Gaussian ACF in 3-D is given by
RG ( x)  RG (r )   2e r
2
/ a2
.
(6.7)
The Fourier transform of the ACF in 3-D space is the power spectral density function (PSDF)
where m is the wavenumber vector.
PE (m) 
8 2 a3
1  a m 
2
2
(6.8)
2
PG (m)   2  3 a 3e  m a
2 2
/4
(6.9)
Figure 6-2 gives a visualization of the comparison between random media described by Guassian
ACF and exponential ACF, with the later obviously containing heterogeneities of a wider spectrum
of scale lengths.
- 90 -
Figure 6-2: Comparison between random media described by Guassian ACF and exponential ACF
shows that the random medium described by exponential ACF contains of heterogeneities of a
wider spectrum of scale lengths.
To reduce the number of independent medium parameters, the following correlation between
velocity and density fluctuations from Birch’s law (Birch, 1961) is often assumed:


 


,
0
 0 0
(6.10)
where  0 and  0 are P and S wave mean velocities,  is density, and   0.8 according to
Sato and Fehler (1998). Larger values of  will generally increase the amount of backward
scattering. We think the value   0.8 is a bit too high and prefer to use   0.2 instead. So
equation (6.10) becomes:
0.2

 
 0.2

.
0
 0 0
(6.11)
- 91 -
6.2.3 Scattering Coefficients
Assuming a random media model, the scattered power per unit volume is given by the scattering
coefficients ( g ij in equation (6.1)) for the various types of scattering (P to P, P to S, etc.; Sato and
Fehler, 1998, pp. 104-105):
2
 2l
l4

g ( ,  ;  ) 
X rPP ( ,  ) P  sin 
4
2
 0
PP
g PS ( ,  ;  ) 
2
 l

1 l4
XPS ( ,  ) P 
1   02  2 0 cos 
 0 4
 0

2
 l

l4
g SP ( ,  ;  )   0
X rSP ( ,  ) P 
1   02  2 0 cos 
4
 0

4
2
2
l


g SS ( ,  ;  ) 
XSS ( ,  )  X SS ( ,  ) P  2l sin  ,
4
2


(6.12)

where l   / 0 is the S wavenumber for angular frequency  , P is the PSDF for the random
media model, and X the basic scattering patterns, which are given by (Sato and Fehler, 1998, p.
102):
- 92 -
X rPP ( ,  ) 


1  
2
4
v 1  cos  2 sin 2    2  2 sin 2  
2  
0  
0
0


 

 4
2
XPS ( ,  )   sin v 1  cos   cos 
 0
  0

 

 4
1
2
X rSP ( ,  )  2 sin cos  v 1  cos   cos 
0
 0
  0

(6.13)
XSS ( ,  )  cos  v  cos  cos 2   2 cos 2 
X SS ( ,  )  sin  v  cos  1  2 cos  ,
where X rPP is the radial component of P-to-P scattering, XPS is the  component of P-to-S
scattering, etc. The angles  and  are defined in the ray-centered coordinate system (Figure
6-3) and the velocity ratio  0   0 / 0 .
Figure 6-3: The ray-centered coordinate system used in the scattering equations. The incident ray is
in the x3 direction. For S waves, the initial polarization is in the x1 direction. The scattered ray
direction is defined by the angles ψ and ζ . The scattered ray polarization is defined by X r, Xψ and
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Xζ .
The average of gij over the solid angle gives the total scattering coefficients gij 0 . The mean free
path l for a ray between scattering events is given by the reciprocals of these coefficients:
lP 
g
PP
0
1
 g 0PS
1
l  SP
.
g 0  g 0SS
(6.14)
S
Despite the apparent complexity of the scattering equations, there are only three free parameters
used to describe this model: the root mean square (RMS) fractional fluctuation  , the correlation
distance a , and the velocity density scaling factor  . Of course, a more general PSDF than the
exponential model would require more parameters.
6.2.4 Monte-Carlo Solution of Radiative Transfer Equations
For some special cases, approximate analytical solutions of the radiative transfer equations exist. A
good example would be scalar waves and isotropic scattering (Paaschens, 1997), which have been
applied to the interpretation of S-wave coda from local events (Sens-Schonfelder and Wegler, 2006;
Padhy et al., 2007). For long lapse times and/or strong multiple scattering, radiative transfer
equations approach the diffusion equation. In more general cases, however, numerical solutions of
the radiative transfer equations have to be considered. They are usually based on Monte Carlo
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schemes where a random walk of energy particles through the heterogeneous medium is realized.
Each particle is moved along ballistic ray paths between individual scattering events. Gusev and
Abubakirov (1987) and Hoshiba (1991) were among the first to use this method in seismology.
Yoshimoto (2000) simulated seismogram envelopes for isotropic scattering and scalar waves in a
background medium with a velocity gradient with depth, which can be implemented by moving the
particles along curved ray trajectories between scattering events. Wegler et al. (2006) compared the
performance of isotropic and non-isotropic scattering approximations for scalar waves and found
that isotropic scattering yields considerable deviations in the early parts of the coda. Margerin et al.
(2000), for the first time, developed a Monte Carlo scheme for the full elastic vector wave case in a
medium with discrete scatterers, where conversions between P- and S-wave modes are taken into
account, and the S-wave polarization is taken care of with the help of the Stokes vector. Elastic
RTT has been applied to deep mantle scattering (Margerin and Nolet, 2003; Shearer and Earle,
2004).
Monte Carlo methods have been used in physics since the 1950s to model radiation transport by
using a computer to simulate the random scattering of large numbers of individual particles (see
Dupree and Fraley, 2002, for a recent introduction to many of these techniques). The Monte Carlo
approach uses computer-generated random numbers to sample the different possible variables in a
problem. For example, neutron scattering can be simulated by tracking the behavior of individual
neutrons, radiated in random directions from a source and scattered in random directions during
their propagation, thus in effect simulating the results of an actual experiment. In general, the
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accuracy of the solution grows with the number of particle trajectories that are computed and thus
Monte Carlo methods have become increasingly useful as faster computers have become available.
Typically, the algorithms converge such that the variance of the results decreases as 1/ n ,
where n is the number of particles.
The concept of seismic "particles" may not seem useful upon initial consideration because there is
no wave-particle duality for seismic waves, as exists for electromagnetic waves. However, if one is
willing to consider energy transport alone and discard phase information in seismic records, then a
particle-based, Monte Carlo approach can be very valuable.
Wegler et al. (2006), Przybilla et al. (2006) and Przybilla and Korn (2008) performed a series of
tests of Monte Carlo simulations based on radiative transfer theory, for both acoustic and fully
elastic P and S scattering, and compared their results to those predicted by various analytical
solutions in 3-D and finite difference solutions to the full wave equation in both 2-D and 3-D. In
general, they found good agreement between the Monte Carlo approach and other methods, except
in the case of extreme velocity perturbations.
Simply put, the Monto Carlo simulation requires three steps, which are schematically illustrated in
Figure 6-4. First, the random walk of "particles" representing seismic wave packets is simulated.
The path length distribution is given by an exponential probability law. Second, at each scattering
event the choice of new mode, propagation direction, and path length is determined. Finally, the
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energy contribution of the particle is calculated at the receiver.
Figure 6-4: Schematic illustration of the Monte Carlo simulation. The seismic “particle” starts at a
point source and makes a random walk in the medium. Upon each scattering, the choice of new
mode, propagation direction, and path length is determined. The energy contribution of the particle
is also calculated at the receiver (after Margerin et al., 2000).
The Monte Carlo program we have used is based on Shearer and Earle (2004, 2008). We have
parallelized the code using MPI so that it can run on both multi-core PCs and computer clusters. We
have added to the program the ability to handle heterogeneity of anisotropic scale lengths. This is
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done mainly through changing subroutines related to calculating mean free paths and radiation
coefficients. For more detailed information such as how to calculate particle trajectories, how to
handle interfaces with velocity discontinuity, how to determine scattering angles and mode
conversion during scattering events using random number generators, and how to incorporate
intrinsic attenuation, please refer to Shearer and Earle (2008).
6.3 Data Analysis
6.3.1 Data
Our dataset comes from the archives maintained by the Incorporated Research Institutes for
Seismology (IRIS), previously downloaded by Anastasia Stroujkova. These data have been
recorded by small-aperture arrays of short-period, vertical component seismometers located
throughout the world (yellow triangles in Figure 6-5). This enables us to use standard array
techniques, a key point in the observations made here. The selected events come from a dataset of
166 event-station pairs, in which most of the events have a moment magnitude over 5.5 and a
source depth deeper than 200 km. Table 6-1 shows some properties of the array stations used in this
study. In Figure 6-5 we show the location of the stations, events and their corresponding turning
points in the inner core (red dots). Note that our sampling is mostly restricted to the northern
hemisphere due to locations of the array stations we used.
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Figure 6-5: Location of IMS arrays (yellow triangles), events (black stars), and ray paths (grey lines)
used in this study. The bounce points at the inner core boundary are shown with red dots. All of
arrays are short-period, vertical component seismometers.
Table 6-1: Properties of the IMS stations used
Array Station
Latitude (˚N)
Longitude (˚E)
Number of Elements
ASAR
-23.66
133.95
19
ILAR
64.77
-146.89
19
NVAR
38.43
-118.30
11
PDAR
42.78
-109.58
13
TXAR
29.33
-103.67
9
- 99 -
6.3.2 Analysis Procedure
We use a modified version of the Array Processing Tool (APT) software package (Stroujkova,
personal communication) to analyze the data. First, we filter our data using a three-pole
Butterworth band-pass filter, centered on 1, 2, 3, 4, or 5 Hz, respectively, similar to the what
Tsujiura (1978) has done in the Qc study. The corresponding corner frequencies are 0.5-1.5, 1.5-2.5,
2.5-3.5, 3.5-4.5, and 4.5-5.5 Hz, respectively. Next, we use standard array technique (e.g. Rost and
Thomas, 2002) to get the beam power by stacking the traces using the correct slowness vector. Beam
power is defined as the RMS amplitude over the selected time window, with amplitude proportional
to the ground velocity. An example of the array data processing is illustrated in Figure 6-6.
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Figure 6-6: Illustration of array data processing using the APT software package for an mb=6.4
earthquake in Fiji Islands dated 12/18/2000, with an epicenter depth of 600 km. The PKiKP turning
point is at (22.12˚N, 169.30˚W). (a) The original traces filtered around 2Hz, in a time window
containing the PKiKP phase. (b) Beam power after stacking the traces in (a). (c) The PKiKP coda
envelope calculated using the Hilbert transform. (d) Same as (c), but drawn using a semi-logarithm
scale.
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Figure 6-7: Illustration of the Multiple Signal Classification (MUSIC) estimator in the APT
software package for the same mb=6.4 earthquake in Fiji Islands dated 12/18/2000. The red dot in
the lowermost panel indicates the apparent velocity (VP) and back azimuth as a result of the
analysis, which could in turn be used to make sure we are looking at the correct phase
We then construct the PKiKP coda envelope using the Hilbert transform. The coda envelopes
shown in Figure 6-6 have been smoothened using a running window average of 3 seconds. Extra
analysis such as f-k processing based on the Multiple Signal Classification (MUSIC) algorithm can
be performed to further confirm the existence of the PKiKP phase (Figure 6-7). At each frequency
pass band, we categorize our observations of PKiKP coda into three groups: clear coda, no coda
- 102 -
and “hard to tell”. The following figures (from Figure 6-9 through Figure 6-12) help us visualize
the places of where we can or cannot observe clean PKiKP coda. Each clean coda is indicated in
these maps using a green vector at the PKiKP turning point, whereas red vectors indicating places
where no coda has been observed. For this dataset, it is very obvious that the 2 Hz band is the most
coherent, in the sense of having both the most PKiKP coda observations as well as large regions
that have consistent observations and non-observations. Possible reasons for this are: (1) body wave
energy around 3 Hz or higher suffers scattering by small scale heterogeneity as well as strong
intrinsic attenuation both in the inner core and as well in the mantle that will act as a low pass filter;
(2) coda at 1 Hz or less is less amenable to a statistical description of the heterogeneity that results
in a simple envelope shape because it can be dominated by scattering from larger scale structures,
one or two of which will result in isolated peaks in the coda rather than an exponential decaying
envelope for many single scattering events or the spindle shape for many multiple scattering events.
- 103 -
Figure 6-8: Map showing where clear PKiKP coda around 1Hz could be observed. Events (black
stars) and stations (yellow triangles) together with raypaths are shown. The bounce points at the
inner core boundary are shown with small vectors (red: no coda is observed; green: clear coda is
observed; black: the coda we observe is indefinite).
Figure 6-9: Same as Figure 6-8, but at 2Hz.
- 104 -
Figure 6-10: Same as Figure 6-8, but at 3Hz.
Figure 6-11: Same as Figure 6-8, but at 4Hz.
Figure 6-12: Same as Figure 6-8, but at 5Hz.
- 105 -
6.4 Forward Modeling of the PKiKP Coda Envelope
Since PKiKP coda observations in the 2 Hz band are the most coherent, we have carried out
Monte-Carlo simulations based on RTT (also called the “phonon” method according to Shearer and
Earle, 2004) to simulate PKiKP coda envelope at 2 Hz. This approach generates synthetic codas for
radial Earth models in which heterogeneity is parameterized by an exponential autocorrelation
function that depends on three parameters: a RMS velocity perturbation  , a RMS density
perturbation, and a correlation length a . These parameters can vary from layer to layer within the
Earth, so non-uniform models are easily evaluated. The P- and S-velocity perturbations are
assumed to be equal, and vary linearly with density perturbation (see equation (6.11)). Travel times
and amplitudes are computed using classical ray theory, and anelastic effects are also included.
One of our modeling results, as illustrated in Figure 6-13, clearly shows that volumetric
heterogeneity in the uppermost several hundred kilometers of the inner core can result in
spindle-shaped PKiKP coda (Leyton and Koper, 2007a, 2007b). Without the volumetric
heterogeneity in the uppermost inner core, velocity contrast at the inner core boundary alone will
only cause a very small PKiKP signal and no long-lasting coda.
- 106 -
Figure 6-13: PKiKP coda envelopes modeled using the phonon method based on radiative transfer
modeling. The vertical axis (amplitude) is in arbitrary units. Note the obvious difference between
the result from volumetric heterogeneity in the uppermost 300 km of the inner core (red) and that
from no volumetric heterogeneity in the uppermost inner core (green). The dashed line indicates the
theoretical PKiKP travel time.
There are trade-offs between different modeling parameters in generating PKiKP coda waves, such
as the RMS velocity perturbation in the uppermost inner core (  ) , the corresponding scale length
of the heterogeneities ( a ), the thickness of the volumetric heterogeneous zone in the uppermost
inner core, etc. Further, anisotropy could also have effects on the modeling results. Evaluation of
candidate Earth models is computationally intensive because only a relatively small number of
phonons contribute to the PKiKP coda, and the smoothness of simulated codas scales as the square
root of the number of phonons. We find that at least 400 million phonons are required to generate
robust PKiKP codas, corresponding to several hours of computing time on an 8-core computer with
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Intel Xeon 1.86 GHz processors.
In this study, we will try to find a possible Earth model that could result in the PKiKP coda
envelope shown in Figure 6-6 through a trial-and-error approach. In each of the following
experiments, we maintain a basic Earth model while changing only one modeling parameter at a
time. This gives us the ability to isolate the effects of different parameters one by one. Our
background model contains 2% RMS velocity perturbations at 4 km scale length for the depth
range of 0 to 600 km, 0.5% RMS velocity perturbations at 8 km scale length for the remainder of
the mantle, and 2% RMS velocity perturbations at 10 km scale length for the uppermost 300 km of
the inner core. Density perturbations are calculated as 0.2 of the velocity perturbations (see
equation (6.10)). The intrinsic attenuation Q model is 227 for 0 to 220 km, 1383 for 220 km to
the CMB, and 360 for the inner core. This model is very similar to the mantle scattering and
intrinsic attenuation model of Shearer and Earle (2004) derived from modeling teleseismic P wave
coda. In each experiment, we have used 600 million phonons to generate the coda waves at 2 Hz
and a distance of 89 great circle degrees (same as the event-station distance shown in Figure 6-6).
In our first experiment, we have synthesized PKiKP coda waves at 2 Hz with varying thickness
(100 km vs. 300 km) of volumetric heterogeneity in the uppermost inner core. The result is shown
in Figure 6-14. The main peak of the PKiKP wave is not terribly sensitive to this parameter change.
This could possibly be explained by the fact that our event-station distance (89˚) is very close to the
critical distance (110˚~115˚) beyond which seismic waves start to get fully reflected by the inner
- 108 -
core boundary and therefore the energy won’t have much chance to travel too far into the inner core.
Still, a thicker heterogeneity layer in the uppermost inner core (black curve in Figure 6-14) causes
stronger and more slowly decaying coda.
Figure 6-14: PKiKP coda waves synthesized using radiative transfer modeling at 2 Hz and a
distance of 89 great circle degrees with volumetric heterogeneity in the uppermost 100 km (red) vs.
300 km (black) of the inner core. Dashed line indicates the theoretical traveltime of PKiKP. The
vertical axis (amplitude) is in arbitrary units.
In our second experiment, we have varied the isotropic scale length of heterogeneity in the
uppermost 300 km of the inner core. Results from heterogeneity of scale length 1km, 5km and
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10km are shown in Figure 6-15. Sato and Fehler (1998) has pointed out that backscattering will be
the strongest when ka 
2 a

1 , where k is the wave number, a is the scale length of the
heterogeneity, and  is the wavelength. The wavelength of the P wave at the uppermost inner
core is about 11 km, so a

 1.75km when the PKiKP coda would be the strongest. This
2
well agrees with our observation that heterogeneity of scale length 1km generates the most
prominent PKiKP coda (red curve in Figure 6-15).
Figure 6-15: PKiKP coda waves synthesized using radiative transfer modeling at 2 Hz and a
distance of 89 great circle degrees with heterogeneity of different scale lengths in the uppermost
300 km of the inner core: (red) 1 km isotropic scale length (black) 5 km isotropic scale length
(green) 10 km isotropic scale length. Dashed line indicates the theoretical traveltime of PKiKP. The
- 110 -
vertical axis (amplitude) is in arbitrary units.
In our third experiment, we have synthesized PKiKP coda with different intrinsic Q values (100 vs.
360) in the inner core (Figure 6-16). Cleary, a small intrinsic Q value at the topmost inner core has
knocked down the amplitude of PKiKP coda wave quite a bit and has made the coda much shorter.
This is understandable since a smaller intrinsic Q value means more energy loss into heat through
friction, viscosity, and thermal relaxation processes.
Figure 6-16: PKiKP coda waves synthesized using radiative transfer modeling at 2 Hz and a
distance of 89 great circle degrees with different intrinsic Q in the inner core: (black) intrinsic Qic =
360 (red) intrinsic Qic = 100 for the uppermost 300 km of the inner core, and 360 for the rest of the
- 111 -
inner core. Dashed line indicates the theoretical traveltime of PKiKP. The vertical axis (amplitude)
is in arbitrary units.
In our fourth experiment, we have synthesized PKiKP coda with different amounts of RMS
velocity perturbations in the uppermost 300 km of the inner core: 2%, 5% and 10% (Figure 6-17).
As we can see from the figure, the PKiKP coda shape is not very sensitive to the amount of velocity
perturbations at this distance range. This insensitivity could also be explained by our observation
distance being very close to the critical distance of PKiKP waves. Do note that much bigger (10%)
RMS velocity perturbations cause the PKiKP wave to peak in less time.
- 112 -
Figure 6-17: PKiKP coda waves synthesized using radiative transfer modeling at 2 Hz and a
distance of 89 great circle degrees with different amounts of RMS velocity perturbations in the
uppermost 300 km of the inner core: (red) 2% (black) 5% (green) 10%. Dashed line indicates the
theoretical traveltime of PKiKP. The vertical axis (amplitude) is in arbitrary units.
The above experiments have all used heterogeneity of isotropic scale lengths. Next, let us consider
the effects of anisotropy on the modeled PKiKP coda shape. Using waveform and travel time
observations, Cormier (2007b) proposed several simple textures with anisotropic scale lengths that
could exist in the uppermost inner core (Figure 6-18). To characterize these textures, we need to
replace the single scale length a used in the above isotropic cases with three values a , b , and c ,
which represent scale lengths in the x, y, and z directions, respectively. For example, the texture in
Figure 6-18a may be described with a  b  10km and c  200km ; whereas the texture in
Figure 6-18c may be described with a  b  200km and c  10km . Our modeling code can not
handle azimuthal difference in the textures right now, so we can not model the effect of the texture
in Figure 6-18b yet.
- 113 -
Figure 6-18: (a) Possible texture in the equatorial eastern hemisphere to explain isotropic velocities
and high isotropic attenuation of transmitted PKIKP (120˚ to 140˚) and lack of backscattered coda
of PKiKP. (b) Possible dominant texture in the equatorial western hemisphere to explain more
pronounced equatorial versus polar anisotropy in velocity and attenuation of transmitted PKIKP
(120˚ to 140˚) and weaker backscattered coda of PKiKP. (c) Possible texture in certain regions of
the western hemisphere to explain isotropic velocities, weak isotropic attenuation of transmitted
PKIKP (120˚ to 140˚), and high amplitudes of backscattered coda of PKiKP (after Cormier, 2007b).
We have synthesized PKiKP coda waves using the phonon method at 2 Hz and a distance of 89
great circle degrees with three different textures, each having either isotropic or anisotropic scale
lengths. As illustrated in Figure 6-19, the horizontally stretched heterogeneity (green) has made the
strongest PKiKP coda waves whereas the vertically stretched heterogeneity (black) and
heterogeneity with isotropic scale length (red) have made approximately the same level of less
- 114 -
strong coda. This observation agrees with results from the pseudospectral forward modeling given
by Cormier (2007b).
Figure 6-19: PKiKP coda waves synthesized using radiative transfer modeling at 2 Hz and a
distance of 89 great circle degrees with three different textures: (red) heterogeneity with isotropic
scale length of 10km (black) vertically stretched heterogeneity (green) horizontally stretched
heterogeneity. Dashed line indicates the theoretical traveltime of PKiKP. The vertical axis
(amplitude) is in arbitrary units.
From the above experiments, we have found that the duration of the PKiKP coda seems to be
sensitive only to the intrinsic Q value in the uppermost inner core. Since the duration of the
example PKiKP coda shown in Figure 6-6 is much shorter than most of those shown in the above
- 115 -
figures, in order to fit that example PKiKP coda, we have tried lowering Q values even more, with
a combination of other parameter changes. Figure 6-20 shows an attempt we have made to fit the
example coda. In this figure, the pink curve shows the example PKiKP coda, and the best fit we get
is the green curve, through reducing both the intrinsic Q value to 50 and RMS velocity
perturbations to 0.5% in the top inner core. The PKiKP turning point in this example data is at
(22.12˚N, 169.30˚W). A recent study by Yu and Wen (2006) has shown that along equatorial paths
in the western hemisphere, the intrinsic Q value is between 450 and 700, higher than the global
average Qp  200 in the uppermost 200 km. The discrepancy between our low Q value and their
high Q value in the same region may be due to the fact that we have only compared the normalized
PKiKP coda envelope in Figure 6-20. The curves in the 950 to 1000 second window of Figure 6-20
indicates that the green curve is too high compared to the pink data curve, probably because PKiKP
is attenuated too much. When we then scale on PKiKP, the preceding coda will increase in
amplitude. In order to work around this problem, we need to somehow include the effect of both the
preceding and the following coda. Possible solutions are getting additional constraints like
amplitude ratios of a reference phase (e.g. P, PP or PcP) to PKiKP, frequency content and
waveform of the transmitted phase, PKIKP, etc.
- 116 -
Figure 6-20: PKiKP coda waves synthesized using radiative transfer modeling at 2 Hz and a
distance of 89 great circle degrees with different scattering layers in the uppermost inner core: (red)
volumetric heterogeneity of 100 km thick with 1km scale length, 2% RMS velocity perturbations,
and intrinsic Q= 100; (black) volumetric heterogeneity of 300 km thick with 1km scale length, 2%
RMS velocity perturbations, and intrinsic Q=50; (green) volumetric heterogeneity of 300 km thick
with 1km scale length, 0.5% RMS velocity perturbations, and intrinsic Q=50. Dashed line indicates
the theoretical traveltime of PKiKP.
6.5 Conclusions
Recent studies have confirmed the existence of scattering by a fabric of small-scale heterogeneities
in the uppermost inner core. Seismic waves interacting with the uppermost few hundred kilometers
of the inner core reveal this region to have strong lateral variations in elastic structure, anisotropy,
- 117 -
attenuation, and scattering. The detailed spatial distribution (or texture) of the uppermost inner core
is important for understanding how the inner core is solidifying from the liquid outer core, since
lateral variations in this texture may record variations in the solidification process of the inner core
and fluid flow of the outer core at the inner core boundary.
Traditional seismic imaging techniques such as traveltime tomography do not have resolution fine
enough to study these small-scale heterogeneities in the uppermost inner core, so in this study, we
have used high frequency coda of the P wave reflected from the inner core boundary (PKiKP) to
study them. We have analyzed seismic array data of 166 event-station pairs from short-period,
vertical component seismometers and found that PKiKP coda waves are more consistently
observed around 2 Hz.
RTT, a theory first introduced in astrophysics to study energy transport of light through the
atmosphere, has been introduced into seismology in order to study coda waves. The radiative
transfer equations are usually solved numerically using Monte-Carlo simulations. PKiKP coda
waves have been synthesized using the phonon method in order to search for an Earth model that is
able to generate a specific coda shape. Trade-offs between different modeling parameters, such as
the RMS velocity perturbation in the uppermost inner core, the corresponding scale length of the
heterogeneities, the thickness of the volumetric heterogeneous zone in the uppermost inner core, as
well as anisotropy, have been discussed. We have found that at 2 Hz and a distance of 89 great
circle degrees, factors such as a thicker scattering layer, a smaller intrinsic Q value, heterogeneity
- 118 -
of scale lengths closer to 1.75 km, and horizontally stretched heterogeneity in the uppermost inner
core, all contribute to stronger PKiKP coda. However, different values of RMS velocity
perturbations do not change the amplitude of the coda much, probably because our event-station
distance (89˚) is very close to the critical distance (110˚~115˚). We have also found that the
duration of the PKiKP coda is not sensitive to anything but the intrinsic Q value in the top inner
core.
- 119 -
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