Download Modeling and Inversion of Dispersion Curves of Surface Waves in

University of Nevada, Reno
Dissertation Title
Modeling and Inversion of Dispersion Curves of Surface
Waves in Shallow Site Investigations
A dissertation submitted on partial fulfillment of the
requirement for the degree of Doctor of Philosophy
in Geophysics
By
Donghong Pei
Dr John N. Louie, Dissertation Advisor
August, 2007
Copyright by Donghong Pei 2007
All Rights Reserved
Abstract
The shallow S-wave velocity structure is very important for the seismic
design of engineered structures and facilities, seismic hazard evaluation of a region,
comprehensive earthquake preparedness, development of the national seismic hazard
map, and seismic-resistant design of buildings. The use of surface waves for the
characterization of the shallow subsurface involves three steps: a) acquisition of highfrequency broadband seismic surface wave records generated either by active sources
or passive ambient noise (microtremors or microseisms), b) extraction of phase
dispersion curves from the recorded seismic signals, and c) derivation of S-wave
velocity profiles either using inversion algorithms or manually error and trial forward
modeling. The first two steps have been successfully achieved by several techniques.
However, the third step (inversion) needs more improvements. An accurate and
automatic inversion method is needed to generate shallow S-wave velocity profiles.
With the achievement of a fast forward modeling method, this study focuses
on the inversion of phase velocity dispersion curves of surface waves contained in
ambient seismic noise for a one dimensional, flat-layered S-wave velocity structure.
For the forward modeling, we present a new more efficient algorithm, called
the fast generalized R/T (reflection and transmission) coefficient method, to calculate
the phase velocity of surface waves for a layered earth model. The fast method is
based on but is more efficient than the traditional ones. The improvements by this
study include 1) computation of the generalized reflection and transmission
coefficients without calculation of the modified reflection and transmission
coefficients; 2) presenting an analytic solution for the inverse of the 4X4 layer matrix
E. Compared with traditional R/T methods, the fast generalized R/T coefficient
method, when applied on Rayleigh waves, significantly improves the speed of
computation, cutting the computational time at least by half while keeping the
stability of the traditional R/T method.
On inversion study, the dissertation explored a linear inversion technique, a
non-linear inversion method, and a joint method on the dispersion data of surface
waves. Chapter 3 explores the Occam’s linear inversion technique with a higher-order
Tikhonov regulization. The blind tests on a suite of nine synthetic models and two
field data sets show that the final model is heavily influenced by a) the initial model
(in terms of the number of layers and the initial S-wave velocity of each layer); b) the
minimum and the maximum depth of profiles; c) the number of dispersion picks; d)
the frequency density of dispersion picks; and e) other noise.
To minimize this initial-model-dependence of the Occam’s inversion, the nonlinear simulated annealing (SA) inversion technique is proposed in Chapter 4.
Following previous developments I modified the SA inversion yielding onedimensional shallow S-wave velocity profiles from high frequency fundamentalmode Rayleigh dispersion curves and validated the inversion with blind tests. Unlike
previous applications of SA, this study draws random numbers from a standard
Gaussian distribution. The numbers simultaneously perturb both S-wave velocities
and layer thickness of models. The annealing temperature is gradually decreased
following a polynomial-time cooling schedule. Phase velocities are calculated using
ii
the reflectivity-transmission method. The reliability of the model resulting from our
implementation is evaluated by statistically calculating the expected values of model
parameters and their covariance matrices. Blind tests on the same data sets as these in
Chapter 3 show that the SA implementation works well for S-wave velocity inversion
of dispersion curves from high-frequency fundamental-mode Rayleigh waves. Blind
estimates of layer S-wave velocities fall within one standard deviation of the
velocities of the original synthetic models in 78% of cases. A hybrid method is also
explored in Chapter 4. The hybrid idea is that the models obtained by the SA can
used as input to the Occam’s inversion. Tests show that the hybrid method does not
always provide better results.
Dispersion curves of fundamental mode Rayleigh waves alone do not contain
sufficient information to uniquely determine a model. The velocity-depth trade-off
gives rise to model non-uniqueness. A joint SA inversion method is proposed in
Chapter 5 using the fundamental-mode Love wave dispersion curves to constrain the
Rayleigh wave inversion by the SA optimization. The SA technique described in
Chapter 4 is applied on the dispersion data of both fundamental-mode Love and
Rayleigh waves with equal weighting factor. Three synthetic tests show that Love
wave constraints result in significant improvement of inverted model in terms of
resolution of low velocity zones and high velocity contrasts.
iii
TABLE OF CONTENTS
Abstract ………………………..…..………………...…………………….………… i
Table of Contents ………………………………………………….………………... iv
List of Figures ……………………………………..……….……………….…...…. vii
List of Tables ……………………………………………………….…….………... xii
Acknowledgments ….…..…………………………....…………….………… ...… xiii
Chapter 1 Introduction…………………………………..…………...………..…. 1
1.1 Motivation and research objectives ..………..………………………..…..1
1.2 Ambient seismic noise …………………….…………….………..……... 4
1.3 Surface wave properties …………....………………………..…….…….. 9
1.4 Seismic acquisition techniques used in shallow site investigations ….…. 13
Chapter 2 Forward modeling of surface-wave dispersion ………..…….…....… 27
2.1 Motion-stress vector ..…………………..……………….….….....….…. 29
2.2 Reflection and transmission coefficients ..…………...….....……..….…. 34
2.3 Plane waves in a layered model ..………..……….……….………….…. 35
2.4 Phase velocity of Love waves …………..……………….…………...…. 39
2.5 Phase velocity of Rayleigh waves …………..………….……..…..….…. 42
2.6 Improvements on calculation of phase velocity of Rayleigh waves …..…. 46
2.7 Numerical examples on dispersion calculation of Rayleigh waves …...…. 48
2.8 Improvements on calculation of phase velocity of Love waves .….….…. 51
iv
2.9 Numerical examples on Love waves .……..……….……….…….…..…. 52
2.10 Group velocity calculation of surface waves ….………………....….…. 53
2.11 Published calculation codes .………………….……………….....….…. 54
2.12 RTgen .……………………………………..…………………....…..…. 55
Chapter 3 Linearized inversion of surface-wave dispersion ..……….…..…..… 61
3.1 Linear model estimation …………….…………..……………...…..…… 62
3.2 Solving a linear system ….………………………………………….…… 64
3.3 Regularization ….……………………………………….…….…….…… 66
3.4 Singular value decomposition (SVD) ……………………..…….….…… 68
3.5 Proposed linear inversion algorithm ………………………….…….…… 69
3.6 Model appraisal method ….…………………………………..……..…… 73
3.7 Test data sets and numerical tests ….………………………..…..….…… 75
3.8 Initial model dependence ….…………………………………..…….…… 81
3.9 The effect of minimum and maximum depth ………………..…..….…… 82
3.10 The effect of number of dispersion picks …….……………..….….…… 83
3.11 The effect of frequency density of dispersion picks ………..….….…… 84
3.12 The effect of the weighting matrix ……………………………..….…… 85
Chapter 4 Non-linear inversion of surface-wave dispersion based on simulated
annealing optimization
…..…………..……………….…….….…… 99
4.1 Global searching optimization …………………………..…….........….. 100
v
4.2 Simulated annealing optimization method …………………..……..….. 103
4.3 Model appraisal …….………………………………………….….….... 108
4.4 Inversion results ………………..…………….………….…..….….…... 109
4.5 Comparison with linearized inversion results ………….…..……..…..... 113
4.6 Difference from previous implementation ……………….…..…….…... 114
4.7 A hybrid inversion approach: simulated annealing followed by the
linearized inversion ……………………………………….……….….... 116
Chapter 5 A joint SA inversion using both Rayleigh and Love surface-wave
dispersions ………….……….…………………………...…………. 128
5.1 Equalized cost function …..……………….………………….………… 130
5.2 Synthetic tests .……………….…………………………..……….…..... 132
5.3 Inversion results .……………….………………………………….…..... 132
Chapter 6 Summary and suggestions …….……………………………………. 145
6.1 Summary …..………………………………….………………...……… 145
6.2 Suggestions …….……………….……………………………..….…..... 149
References Cited ………………………………………………………...…….… 152
Appendix A: Matrices for Rayleigh Waves ……….………………….….…...… 163
Appendix B: Matrices for Love waves …………….………………….………… 165
vi
LIST OF FIGURES
Figure 1.1 Three steps involved in utilizing dispersion curves of surface waves for
imaging geologic profiles ……………….………….….…….……..…. 21
Figure 1.2 The acceleration power spectrum of microtremors recorded at 75
permanent seismic observatories throughout the world …………..….. 22
Figure 1.3 Body wave motion ……………………………………….….….…..…. 23
Figure 1.4 Particle motion and amplitude of Rayleigh waves ………….……...…. 23
Figure 1.5 Surface wave dispersion ………………………..……...…..…….……. 24
Figure 1.6 Phase velocities vs. frequencies ………………………..….….……..... 24
Figure 1.7. Modes of surface waves ……..…………………………....…..……..... 25
Figure 1.8 Dispersion curves of higher-mode surface waves ...………….…..…..... 25
Figure 1.9 A typical ReMi field configuration .....…………………….….…..…… 26
Figure 1.10 A typical ReMi analysis …..…………………………....…....…..….... 26
Figure 2.1 Illustration of coefficients of reflection and transmission due to SH
incident down (a) and up (b) to an interface ….…..……...…...…….… 56
Figure 2.2 Illustration of coefficients of reflection and transmission due to SV
incident down (a) and up (b) to an interface and P incident down (c) and
up (d) to an interface …..………………………………………....….. 56
Figure 2.3 Configuration and coordinate system of a multiple-layered half-space
…………………………………………………………..…………...... 57
vii
Figure 2.4 Phase velocity dispersion curves of the fundamental-mode Rayleigh waves
for large scale models 1, 2, and 3 (a) and small scale models 4, 5, 6 (b)
…………………………………………………….…………………… 58
Figure 2.5 The normalized errors between phase velocities calculated by RTgen and
CPS of large scale models 1, 2, and 3 (a) and small scale models 4, 5, 6
(b) ………………..…..………………………………………..….….. 58
Figure 2.6 Phase velocity dispersion curves of Rayleigh waves for the Gutenberg
model ………………………………………………...…...………….. 59
Figure 2.7 Phase velocity dispersion curves of Rayleigh waves for model 4 ….….. 59
Figure 2.8 Computational time against number of layers in models ………………. 60
Figure 2.9 Phase velocity dispersion curves of Love waves for the Gutenberg model.
……..……………………………………………………………...……60
Figure 2.10 Phase velocity dispersion curves of Love waves for model 4 ………... 60
Figure 3.1 The inverse problem viewed as a combination of an estimation problem
plus appraisal problem .……………...…..…..……………………… 86
Figure 3.2 Flow chart showing the Occam’s inversion procedure …....……..…… 87
Figure 3.3 A typical synthetic seismic record with strong Rayleigh waves ..…….. 88
Figure 3.4 Slowness-frequency spectrum (p-f) image with ReMi dispersion picks of a
typical synthetic seismic record …………………...………………..... 88
Figure 3.5 Linearized inverted S-wave velocities against the original synthetic models
for nine synthetic data sets. …………….………….….…….……..…. 89
viii
Figure 3.6 The depth-averaged velocities in m/s against the known values for
Occam’s inverted models.………………………………….……..…… 92
Figure 3.7 Dispersion picks on the slowness-frequency spectrum (p-f) images of
Newhall (left) and Coyote Creek (right) data
…………….…..…..…. 93
Figure 3.8 Linearized inverted profiles of S-wave velocity against the OYO
suspension S-wave logs of Newhall and CCOC data ……………..…. 94
Figure 3.9 Inverted S-wave velocity against the OYO S-wave log of Newhall data,
showing the effect of the number of layers ………..….…….….……. 95
Figure 3.10 Inverted S-wave velocity against the OYO S-wave log of Newhall data,
showing the effect of the layer S-wave velocity ……..….….….…..... 96
Figure 3.11 Inverted S-wave velocity against the OYO S-wave log of Newhall data,
showing the effect of the maximum depth ……………....….….…..... 96
Figure 3.12 Inverted S-wave velocity against the OYO S-wave log of Newhall data,
showing the effect of the number of the picks ………………....…..... 97
Figure 3.13 Inverted S-wave velocity against the OYO S-wave log of Newhall data,
showing the effect of the frequency density of the picks ….…...….… 97
Figure 3.14 Effects of the weighting matrix on the inverted models ....……..….... 98
Figure 4.1 Multimodality of the surface wave dispersion curve inversion problem
……..……..……………………………………………………...…… 119
Figure 4.2 A cartoon showing an annealing process …………...…………....….. 120
Figure 4.3 A flowchart showing the annealing process on inversion of dispersion
curve of surface waves ……………………………….…………...... 120
ix
Figure 4.4 A cartoon showing the role of the conditional acceptance ……...…… 121
Figure 4.5 Inverted profiles with standard deviation of S-wave velocity against the
original synthetic models …………...…..……………………….….. 122
Figure 4.6 The depth-averaged velocities in m/s against the known values for SA
inverted models …………………………………………………….. 123
Figure 4.7 Inverted profiles with standard deviation of layer thickness against the
original synthetic models ……..…………………………...……….. 124
Figure 4.8 Comparison of the OYO suspension S-wave velocity logs and the inverted
models for the Newhall (left) and the Coyote Creek data (right) .…. 125
Figure 4.9 Calculated dispersion curves (lines) of fundamental-mode Rayleigh waves
plotted atop the ReMi dispersion picks (circles) for the Newhall (left) and
the Coyote Creek data (right) ……………………..……….…...……125
Figure 4.10 The flow chart of the hybrid inversion algorithm …………...……... 126
Figure 4.11 Final inverted models for Newhall data using the simulated annealing
method (left) and the hybrid inversion method (right) ……...……… 127
Figure 4.12 Final inverted models for CCOC data using the simulated annealing
method (left) and the hybrid inversion method (right) ...……..…….. 127
Figure 5.1 Two different models with same number of layers (left) and corresponding
dispersion curves (right) …………………………………...……….. 136
Figure 5.2 Two different models with different number of layers (left) and
corresponding dispersion curves (right) …………………………..... 136
x
Figure 5.3 A theoretical distribution of the value of the cost function for the joint
inversion
………………………………...…..………...…..…….… 137
Figure 5.4 SA inversion results of the data N102 …………………………....….. 138
Figure 5.5 Joint inversion results of the data N102 ………………….………...... 138
Figure 5.6 The distribution of the value of cost functions for joint inversion on N102
………………………………………………………..……….……… 139
Figure 5.7 The depth-averaged velocities in m/s against the known values for both SA
and joint inverted models ………..…..………………..….….….….. 140
Figure 5.8 SA inversion results of the data N103 ……………………………….. 141
Figure 5.9 Joint inversion results of the data N103 ……………….....………….. 141
Figure 5.10 The distribution of the value of cost functions of the joint inversion on
N103 ………………………………………………...……...………. 142
Figure 5.11 SA inversion results of the data N104 ……………………….....……143
Figure 5.12 Joint inversion results of the data N104 ……………………..……... 143
Figure 5.13 The distribution of the value of cost functions of joint inversion on N104
………………………...……………...…..…………………...……… 144
xi
LIST OF TABLES
Table 1.1 Summary of characteristics of seismic ambient noise ..….………......… 8
Table 1.2. Seismic acquisition techniques used in shallow site investigations …… 14
Table 2.1 Methods of forward modeling of dispersion curves …………….……… 27
Table 2.2 Definition of elastic constants ………………………………....……….. 31
Table 2.3 Relationship between elastic constants ………………………..……….. 31
Table 2.4 Harmonic wave parameters ……………………………….……….…… 81
Table 2.5 Gutenberg’s layered model of continental structure ..….………....….… 49
Table 2.6 Test models at crustal scale ……………………………………….….… 49
Table 2.7 Test models at local site scale …………………………………..….…… 50
Table 3.1 Linearized inversion methods of surface waves used by major research
groups ………………………………………………………..………….. 70
Table 3.2 Sources of uncertainty in surface wave dispersion measurements …….. 73
Table 3.3. Depth-averaged velocities in m/s for Occam’s inverted models and
percentage difference from known profiles in parentheses ………..…… 79
Table 4.1 SA-inverted depth-averaged velocities in m/s and percentage difference
from known profiles in parentheses ..….…………………………....… 111
Table 4.2 Implementation difference of SA from previous study ………….….… 115
Table 5.1 Joint-inverted depth-averaged velocities in m/s and percentage difference
from known profiles in parentheses…………….……………………… 133
xii
ACKNOWLEDGEMENTS
I thank my advisory committee Drs. John Louie, John Anderson, James Brune,
Satish Pullammanappallil, and Ilya Zaliapin for their thorough reviews that
significantly improved this manuscript. I thank my academic advisor Dr. John Louie
for his financial support at the beginning, critical academic encouragements for
challenges, and positive writing guidance for papers and this dissertation. I am
grateful for the valuable learning experiences I gained during the numerous lab
exercises and fieldworks from him. Without his help, I can’t image how I survive at
UNR. The courses taught by John Anderson and James Brune went a long way in
furthering my understanding of geophysical inversion and earthquake seismology.
This dissertation is financially supported by Optim Inc. I want to give my
thanks to Satish Pullammanappallil and Bill Honjas of Optim Inc. The simulated
annealing methods developed in this dissertation are a direct influence of Dr.
Pullammanappallil’s contribution. He provided his own code and value advice to
keep me on the right track. Most importantly, he continuously funds this study.
Without funding, I would not have been able to pursue the topic that interested me
most.
I spent four years in the Nevada Seismological Lab. I thank Drs. Ken Smith,
David Von Seggern, Glenn Biasi, Rasool Anooshehpoor, and Gary Oppliger for their
knowledge of seismology from which I benefit a lot. During my staying, I was
fortunate to come in contact with students who displayed enthusiasm for research and
xiii
unselfish sharing of scientific ideas. They are Aasha Pancha, James Scott, and
Michelle Heimgartner. I am grateful for the friendship and love shared by all of my
friends at UNR and the department. They help me smoothly settle down and make my
staying in Reno much more enjoyable.
Most of all, I thank my wife, Xin Yu, for her love, encouragement, sacrifice,
and moral support throughout this study; my parents, Pei Yunji and Tao Wanzhi, for
their never-failed-support throughout my education; and my parents-in-law, Fulai Yu
and Jinwen Li, for their caring for my baby Steven Pei during the study.
xiv
Chapter 1 Introduction
1.1 Motivation and research objectives
The factors influencing seismic ground motion were divided into source, path,
and site effects, a distinction that has proven useful for understanding and predicting
seismic shaking (e.g. Aki, 1993). The properties of the geological materials beneath a
site (site condition) have a major impact on the ground motion by modifying the
amplitude, phase, duration, and shape of seismic waves. Historical earthquakes have
taught us that damage is often significantly greater on unconsolidated soil than on rock
sites when the surface structure is more than a few kilometers from the earthquake
source (e.g., the Mw 6.9 1989 Loma Prieta earthquake in California, Borcherdt and
Glassmoyer, 1994). Thus, the characterization of the medium underlying a site is one of
the most important tasks in seismic hazard evaluation of a region, comprehensive
earthquake preparedness, development of the national seismic hazard map, and seismicresistant design of buildings (Field et al., 1992).
The use of surface waves (ground roll) for the characterization of the shallow
subsurface has become of growing interest to geotechnical engineers and geophysicists.
In a vertically heterogeneous medium, the phase velocity of surface waves is a function
of frequency (called dispersion curves). The curve is a function of shear wave (S-wave)
velocity, layer thickness, density, and compressional wave (P-wave) velocity of each
geological layer, listed in a decreasing order of priority according to Xia et al. (1999). If
the dispersion curves are measured experimentally, it is in principle possible to obtain
the mechanical parameters of the medium from the dispersion curves.
1
In fact, the dispersion curve has been employed for imaging geological profiles
in a variety of applications for several reasons. First, it is a robust property that can be
quite easily observed without contamination by other wavefields. Second, various
forward modeling techniques exist to generate the dispersion curves of surface waves
rapidly and accurately for a layered geological structure. Finally, compared to the
inversion of waveforms, the complexity of the inversion of dispersion curves is greatly
reduced.
Three steps are involved in utilizing dispersion curves of surface waves for
imaging geological profiles for seismic hazard assessment (Fig. 1.1):
1) acquire high-frequency (>=1 Hz) broadband ground roll,
2) create efficient and accurate algorithms organized in a basic data processing
sequence designed to extract surface wave dispersion curves from the ground roll, and
3) develop stable and efficient inversion algorithms to obtain shear wave
velocity profiles.
The application of dispersion curves for geotechnical site characterization was
originally proposed during the 1950s (e.g., SPAC method of Aki, 1957). The new
improvements do not appear until the 1980s when the SASW technique (e.g., Nazarian
and Stokoe, 1985) was proposed. The main reason for the slow progress is the lengthy
procedure of data acquisition on site. Since then, FK (Horike, 1985), MSM (Okada,
2003), MASW (Park et al., 1999), DASW (Phillips et al., 2004), and wavefield
transformation (Forbriger, 2003a, 2003b) were developed for surface waves acquisition
in shallow site investigations. A significant simplification of the field acquisition of the
2
surface waves did not appear until Louie published his paper on the ReMi technique in
2001.
Current research is still focused on the first two steps, acquisition of broadband
ground roll and extraction of dispersion curves. They are important to successfully
estimate the geological material properties. However, the third step, inversion, is
essential for obtaining proper geotechnical profiles.
This dissertation focuses on the third step, the inversion of the dispersion curves
of surface waves, with the aim of finding the best procedure to get a more accurate and
reliable estimate of the geological material properties. The inversion actually is
comprised of two sub-steps:
3a) estimate a model employing the theory of surface wave propagation and
mathematical optimization;
3b) appraise the model for its accuracy, either deterministically or statistically.
The dissertation uses surface waves contained in ambient seismic noise, which is
an assemblage of body and surface waves (Toksoz and Lacoss, 1968). The extraction of
dispersion curves of surface waves has been achieved by several techniques. The
refraction microtremor (ReMi) technique (Louie, 2001), licensed as SeisOpt ReMi (©,
Optim Inc.) software, is being used widely for commercial and research purposes (Scott
et al., 2004, 2006; Stephenson et al., 2005; Thelen et al., 2006) to produce reliable
dispersion curves. Thus, the ReMi technique is adopted here to generate dispersion
picks for all test data.
3
1.2 Ambient seismic noise
Ambient seismic noise is defined as the constant vibrations of the Earth’s
surface at seismic frequencies, even without earthquakes (Okada, 2003). They are also
called microtremors or microseisms. The ambient noise is ubiquitous and its amplitude
is generally very small, far below human sensing. With some extreme exceptions, the
displacement amplitudes are on the order of 10-4 to 10-2 mm (Okada, 2003). But they
vary greatly between different sites and different frequencies.
Studying Earth noise has become a part of the science at least since Brune and
Oliver (1959) published curves of high and low seismic background displacement based
on a world-wide survey of station noise. Later development is largely due to the efforts
of Japanese seismologists (Aki, 1957; Horike, 1985; Okada, 2003). Figure 1.2 plots
typical microtremor levels for over 75 permanent seismic observatories from the global
seismic networks (Peterson, 1993). Globally, the microtremor level is high
(microtremor peak) at periods at about 5 to 8 seconds and low at 20 to 200 second
periods. These frequency ranges have very little for engineering seismology. Another
relatively high level appears at periods at about 0.15 to 0.5 seconds with large variation
between stations. Most dispersion acquisition technique (for example ReMi technique)
is sensitive to the noise signals between 0.15 to 0.5 seconds. Thus, the seismic peak
within this range is useful for seismic hazard assessment of sites. The large variation
within the range between sites contains the site-dependent information.
Although the noise is studied for its own intrinsic interest, seismologists have
generally considered it as pure random signal because it hampers observations of small
and/or distant earthquakes at least until emerges of noise cross-correlation technique
4
(Campillo and Paul, 2002). Recent developments in seismology and earthquake
engineering have demonstrated experimentally and theoretically that an estimate of the
Green's function for wave propagation between two seismic stations can be obtained
from the time-derivative of the long-time average cross correlation of ambient noise
between these two stations (Campillo and Paul, 2002; Sabra et al., 2005). Shapiro et al.
(2005) showed that the dispersion characteristics of the estimated Green's functions
provide information about the wave propagation between the stations, hence, about
seismic velocities in the crust and uppermost mantle. Thus, Earth structure can be
gained from analysis of seismic noise.
However, our knowledge of ambient seismic noise is still very incomplete.
Understanding the physical nature and composition of the ambient seismic noise
wavefield, especially in urban areas, requires answering two sets of questions that are
not independent of each other:
1) What is the origin of the ambient vibrations (where and what are the
sources)?
2) What is the nature of the corresponding waves, i.e., body or surface waves?
The second set of questions also includes 2a) what is the ratio of body and
surface waves in the seismic noise wavefield? 2b) within surface waves, what is the
ratio of Rayleigh and Love waves? and 2c) again within surface waves, what is the ratio
of fundamental and higher modes?
While there is a relative consensus on the first question (p.3, Okada, 2003), only
a few and partial answers were proposed for the second set of questions, for which a lot
of experimental and theoretical work still lies ahead.
5
As known and taught for a long time in Japan, sources of ambient vibrations are
usually separated in two main categories, natural and human (Shearer, 1999, P.215).
The ratio of these two sources varies in different frequency bands (particularly within
urban areas).
At low frequencies (f < fn = 1 Hz), the origin is essentially natural, with a
particular emphasis on ocean waves, which emit their maximal energy around 0.2 Hz
(Tanimoto, 2005, 2007). This energy corresponds to the peak at period of 5 to 8 seconds
in Fig. 1.2. They are generally called microseisms by seismologists. Higher frequencies
(around 0.5 Hz) are emitted along coastal areas due to the non-linear interaction
between sea waves and the coast line (Tanimoto, 2007). Some lower frequency waves (f
< 0.1 Hz) are reported (Kobayashi and Nishida, 1998) and often referred to as the
“hum”. The “hum” is associated with atmospheric movements or excitation by oceans
(Rhie and Romanowicz, 2004).
Energetic low frequency sources are often distant (being located at the closest
oceans). The most energy is carried from the source to sites by surface waves guided in
the Earth's crust (Lay and Wallace, 1995). However, locally, these waves may (and
actually often do) interact with the local geological structure (especially deep basins)
(Yamanaka et al., 1993, 1994). Their long wavelength induces a significant penetration
depth, so that the resulting local wavefield contains local geology signature. Subsurface
inhomogeneities, excited by the long period crustal surface waves, may act as
diffraction points and generate local surface waves, and even possibly body waves.
Thus, it is possible to extract the local geologic information by studying microseisms.
Extracting information from microseisms is easier on islands (such as Japan) than in the
6
heart of continental areas because the energy at frequencies between 0.1 and 1.0 Hz
decreases with increasing distance from oceans (SESAME, 2004).
At high frequencies (f > fn = 1 Hz), the origin is predominantly related to human
activities (traffic, machinery) (Shearer, 1999, P.215) and may also be associated with
wind and water flows (Okada, 2003 p.3). These waves are generally called
microtremors by engineers. Their sources are mostly located at the surface of the earth
(except some sources like metros), and often exhibit a strong day/night and
week/weekend variability (Okada, 2003, P.14).
High frequency waves generally have much closer sources, which most of the
time are located very close to the surface. While the wavefield in the immediate vicinity
(less than a few hundred meters) includes both body and surface waves, at longer
distances, surface waves become predominant (Lay and Wallace, 1995).
The 1 Hz limit for fn is only indicative, and may vary from one city to another
(SESAME, 2004). Some specific civil engineering works (highways, dams) involving
large engines and/or trucks may also generate low frequency energy. Locally, this limit
may be found by analyzing the variations of seismic noise amplitude between day and
night, and between work and rest days as well. I do not distinguish between
microseisms and microtremors here. The terms are interchangeable in this dissertation.
Besides this qualitative information, only little information is available on the
quantitative proportions between body and surface waves, and the different kinds of
surface waves that may exist (Rayleigh/Love, fundamental/higher). The few available
results, reviewed in Bonnefoy-Claudet et al. (2006), report that low frequency
microseisms predominantly consist of fundamental mode Rayleigh waves, while there
7
is no real consensus for higher frequencies (> 1 Hz). Different approaches were
followed to reach these results, including analysis of seismic noise amplitude at depth
and array analysis to measure the phase velocity (SESAME, 2004).
The very few investigations on the relative proportion of Rayleigh and Love
waves all agree on more or less comparable amplitudes, with a slight trend toward a
slightly higher energy carried by Love waves (around 60% - 40%) (SESAME, 2004). In
addition, there are a few reports about the presence of higher surface wave modes from
several very different sites (some very shallow, other much thicker, some other with
low velocity zone at depth).
The following table simplifies the above discussion.
Table 1.1 Summary of characteristics of seismic ambient noise
Natural
Human
Name
Microseism
Microtremor
Frequency
0.1 - fnh (0.5 Hz to fnh (0.5 Hz to 1 Hz) - 10 Hz
1 Hz)
Origin
Ocean
Traffic / Industry / Human activity
Incident wavefield Surface waves
Surface + body
Amplitude
Related to oceanic Day / Night, Week / week-end
variability
storms
Rayleigh / Love
Incident wavefield Comparable amplitude – slight
issue
predominantly
indication that Love waves carry a little
Rayleigh
more energy
Fundamental /
Mainly
Possibility of higher modes at high
Higher mode issue Fundamental
frequencies (at least for 2-layer case)
Further Comments Local wavefield
Some monochromatic waves related to
may be different
machines and engines. The proximity of
from incident
sources, as well as the short wavelength,
wavefield
probably limits the quantitative
importance of waves generated by
diffraction at depth
In summary, ambient seismic noise is ubiquitous and its amplitude is small. The
low frequency ambient noise is essentially nature while the high frequencies are related
8
to human activities. The acceleration power spectral of ambient noise shows several
peaks. The peak at periods of about 0.15 – 0.5 seconds is useful for seismic hazard
assessment of sites by providing a potential seismic source for shallow S-wave velocity
investigations. Due to the short wavelength contained in the ambient noise, effective
investigation depth is limited (for example less than 100 m by ReMi technique).
The review on the origin of the ambient seismic noise shows that the seismic
noise wavefield is complex. When extracting dispersion curves of microseisms, one has
therefore to consider the possible contributions to the microseisms from both surface
and body waves, including higher modes of surface waves.
1.3 Surface wave properties
Seismic waves can be categorized by whether they travel through a medium
(body waves) or along the medium’s surface (surface waves). Body waves propagate by
a series of compressions and dilatations of the material or by shearing the material back
and forth. The first type of body wave is variously known as a dilatational, longitudinal,
irrotational, compressional, or P-wave, the latter name being due to the fact that this
type is usually the first (primary) event on an earthquake seismogram. The P-wave
forces particles of the medium to move back and forth parallel to the direction of
propagation (Fig. 1.3). The second type is referred to as the shear, transverse, rotational,
or S-wave (because it is usually the second event observed on an earthquake
seismogram). Under S-waves, the medium is displaced transversely to the direction of
propagation (Fig. 1.3). Moreover, because the rotation varies from point to point at any
given instant, the medium is subjected to varying shearing stresses as the wave moves
9
along. S-wave particle motion is often divided into two components: the motion within
a vertical plane through the propagation vector (SV-wave), and the horizontal motion in
the direction perpendicular the plane (SH-wave)
In an infinite homogeneous isotropic medium, only body waves exist (Aki and
Richards, 2002). However, when the medium does not extend to infinity in all
directions, surface waves (known in seismic exploration as ground roll) can be
generated. The primary type of surface wave is the Rayleigh wave. This wave travels
along the surface of the earth and involves an interference of the P-wave and SV-wave.
The particle motion is confined to the vertical plane that includes the direction of
propagation of the wave. The motion is counter clockwise (retrograde) at the surface,
changing to purely vertical motion at a depth of about one fifth of a wavelength, and
becoming clockwise (prograde) at greater depths (Fig. 1.4). The amplitude of the
Rayleigh wave motion decreases exponentially with depth. Because of the existence of
vertical medium-velocity gradients in the real world, the velocity of the Rayleigh wave
varies with wavelength (called dispersion curves); longer period waves travel faster
because they sense the faster material at greater depth. The second type of surface wave
is the Love wave. Love waves are formed through the constructive interference of high
order SH multiples. The particle motion is horizontal and in the direction of SH waves.
The amplitude of this wave motion decreases exponentially with depth. They exhibit
dispersion as well.
Geometrical spreading for surface waves is proportional to r-0.5, in contrast to the
body wave where the geometrical spreading is proportional to r-1, where r is distance
10
from the source (Anderson, 1991). Rayleigh waves often are dominant events in seismic
records.
The amplitude of surface waves decreases exponentially with depth (Fig. 1.4).
Most of the energy propagates in a shallow zone, roughly equal to one wavelength.
Consequently, the wave propagation is influenced by the properties of this limited, nearsurface portion of the geological or geotechnical profile.
The propagation of surface waves in a vertically heterogeneous medium shows a
dispersive behavior. Dispersion means that different frequencies have different phase
velocities. In a homogeneous medium, the different wavelengths (Rayleigh wave only)
“sample” different depths of the subsoil. Since the material is homogeneous, all the
wavelengths have the same velocity (Fig. 1.5 left). In other words, Rayleigh waves are
non-dispersive and Love waves do not exist in a homogeneous medium. If the medium
is not vertically homogeneous, for instance it is layered, with layers having different
mechanical properties, the different wavelengths “sample” different depths to which
different mechanical properties are associated. Each wavelength propagates at a phase
velocity depending on the mechanical properties of the layers involved in the
propagation (Fig. 1.5 right). So the surface wave does not have a single velocity, but a
phase velocity that is a function of frequency.
This relation between frequency and phase velocity is called a dispersion curve
and depends on the geology underneath. At high frequency, the phase velocity is close
to the S-wave velocity through the uppermost layer. At low frequency, the effect of
deeper layers become important, and the phase velocity tends asymptotically to the Swave velocity of the deepest material, as if it extends infinitely in depth (the half space).
11
The shape of dispersion curves is related to geologic profiles. Longer
wavelengths penetrate deeper than shorter wavelengths for a given mode and are more
sensitive to the elastic properties of the deeper layers. Thus, for a profile where S-wave
velocity increases with depth, a normal dispersion curve (phase velocities decrease with
frequency) will be observed (Fig. 1.6). For a profile where S-wave velocity decreases
with depth, a reverse dispersion curve (phase velocities increase with frequency) will be
observed. In an irregular S-wave velocity profile, the phase velocities show a complex
relation with frequency (Fig. 1.6).
In reality, the S-wave velocities increase with depth in most geological
structures. Most of the observed dispersion curves are normal. A complex shape of
dispersion curves is also observed in our surface wave surveys in Las Vegas. This is due
to the regional distributed cliché layer in Las Vegas basin. The reverse dispersion
curves are observed from our synthetic models. The observation of a reverse dispersion
curve in the surveys might be caused by the limited frequency bandwidth of the
surveys. The observed reverse curves actually is one part of the complex dispersion
curves.
Like a vibrating string, the surface wave propagation in vertically heterogeneous
media is actually a multi-modal phenomenon. For a given geology, at each frequency
different wavelengths can exist (Fig. 1.7). Hence different phase velocities are possible
at each frequency, each corresponding to a mode of propagation. The different modes
can exist simultaneously (Aki and Richards, 2002) (Fig. 1.8).
The different modes, except the first one, exist only above their cut-off
frequency, which is for each mode the lowest limit frequency at which the mode can
12
exist. With a finite number of layers, in a finite frequency range, the number of modes
is limited. At very low frequency, below the cut-off frequency of the first higher mode,
only the fundamental mode exists.
Modes are not just theory or mathematically possible solutions; they are often
observed in experimental data, also in the frequency ranges of interest for engineering
purposes. The energy associated to the different modes depends on many factors, the
geology at first, but also the depth and the kind of source.
The first mode is sometimes dominant over a wide frequency range, but in many
common situations higher modes play important roles and are dominant in energy. So
they cannot be neglected. The different modes have different phase velocities.
Therefore, they are separated at distance from the source. at short distances modes
superimpose on one another, and mode identification can be impossible.
At the engineering scale, the modal superposition is important. The effective
Rayleigh phase velocity deriving from the modal superposition is only an apparent
velocity that depends on the observation layout, source orientation, and position.
1.4 Seismic acquisition techniques used in shallow site investigations
Many seismic methods have been used by seismologists to determine the
velocity structure of the Earth at different scales (Lay and Wallace, 1995). They include
the reflection seismic method used by exploration geophysicists, and the use of bodywave arrival times, surface-wave dispersion, and free-oscillation periods of the Earth.
Those methods are now being successfully adopted in the determination of shallow S-
13
wave structure to help in the specification of design ground motions for engineering
purposes (Horike, 1985, 1988; Nazarian and Stokoe, 1985; Stephenson et al., 2005).
According to Boore (2006), the seismic methods used in shallow site
investigations are categorized according to invasiveness (Table 1.2). The noninvasive
methods are further organized according to number of stations used. The multiplestation group is subdivided into those methods that use active sources, those that use
passive sources, and those that combine active and passive sources.
Table 1.2. Seismic acquisition techniques used in shallow site investigations
Invasive
Noninvasive
Surface
Receiver in borehole
HVSR
source
(Boore &
Receiver in cone
Thompson,
penetrometer
SASW
2007)
Active sources
DASW
Suspension P-S
logger (Nighbor and
MASW
Multiple
Imai, 1994)
F-K
stations
Downhole
Passive sources
SPAC
source
Crosshole
ReMi
(ASTM, 2003)
Combined active and
MASW
passive sources
HVSR: Horizontal/vertical spectral ratio method ( on single station
microtremor data) (Bonnefoy-Claudet et al., 2006)
SASW: Spectral analysis of surface wave (Nazarian and Stokoe, 1984)
DASW: Distance analysis of surface wave (Phillips et al., 2004)
MASW: Multichannel analysis of surface wave (Park et al., 1999)
Acronyms
F-K: Frequency-wavenumber method of processing seismic array data
(Horike, 1985)
SPAC: Spatial autocorrelation method of processing seismic array data
(Okada, 2003)
ReMi: Refraction microtremor method (Louie, 2003)
1.4.1 Invasive methods
These methods require data from seismometers placed beneath the Earth’s
surface. They can be divided into two groups: those using surface sources and those
using down-hole sources.
14
Surface-source methods (Boore & Thompson, 2007) employ the source at the
surface and a sensor either clamped to the edges of a cased borehole at a series of
depths, or mounted near the tip of a special tool (a seismic cone penetrometer) that is
pushed into the ground (seismic cone penetration testing, or SCPT). The surface sources
are activated and the seismic waves are recorded in the sensors. Usually a threecomponent seismometer is used as the sensor, and two types of sources are commonly
used. Either a plank struck with a sledge hammer on the ends or an air-activated slide
hammer (in either case the device is held to the ground by the weight of a truck’s tires)
is used to generate S-wave energy. For P-wave energy, usually a metal plate is struck
with a sledge hammer (e.g. Liu et al., 1988). The first arrivals on the resulting record
section are picked, and then a velocity model is found from these arrivals. In some
cases the velocities are determined from a line fit through adjacent arrivals, thus
providing velocities over various intervals of depths. A model such as this can be used
in correlation with shear-wave velocities and geologic units (e.g., Holzer et al., 2005).
In the past, downhole-source methods usually involved crosshole studies, where
a source in one hole emitted waves that traveled more-or-less horizontally to receivers
in an adjacent hole(s). The crosshole method has several limitations (ASTM, 2003): 1)
it is very expensive in that it requires multiple holes whose spatial orientation needs to
be known precisely; 2) the velocities are measured in the horizontal direction and may
not be appropriate for waves traveling essentially vertically, as are those of most
concern in earthquake engineering; 3) the velocity model may not extend without gaps
from the surface to depth. On the other hand, the method is useful for detecting local
15
variations in soil properties, which might be important for liquefaction potential or for
foundation design.
For most purposes related to earthquake engineering, the crosshole method has
largely been replaced by a method developed by the OYO Corporation (Nighbor and
Imai, 1994). The method is known by several names, the most common being variants
of “Suspension P-S Velocity Logging Method”. Information on this widely-used
method can be found at http://www.geovision.com/PDF/M_PS_Logging.PDF. The
method makes use of a probe lowered into a hole, on which a source near the bottom of
the probe emits acoustic waves that are coupled into P- and S-waves at the edges of the
borehole. These waves travel in the surrounding material and are reconverted into
acoustic waves that are then recorded on two receivers mounted 1 m apart. The wave
velocities are given by the difference in travel times at the two receivers. The method
works best in uncased boreholes and can be used in relatively deep holes. It provides
much finer resolution than the surface-source downhole-receiver methods discussed
earlier. Possible drawbacks are that the method sometimes does not yield accurate
velocities near the surface, and does not formally produce a model extending to the
surface. In addition, it is not possible to interpolate across any zones where data are not
obtained. This is in contrast to the surface-source downhole-receiver method where a
single well-recorded travel time below a depth interval with poor data still provides an
average velocity across the skipped interval.
1.4.2 Noninvasive methods
A major disadvantage of the invasive methods is the need for a borehole and the
cost of drilling. For this reason, many noninvasive methods have been devised for
16
obtaining a subsurface velocity structure. As shown in Table 1.2, these methods are
conveniently divided into those that use active sources, those that use passive sources,
and those that use both. Most of the methods attempt to measure fundamental mode
dispersion curves of Rayleigh waves (Boore, 2006). The velocity models are obtained
by inverting these dispersion curves, using either iterative forward modeling or various
inversion algorithms.
SASW is the popular noninvasive method in earthquake engineering community
(e.g., Nazarian and Stokoe, 1984; Brown et al., 2002). This method uses the phase
difference between two receivers, calculated by cross power spectra of the recorded
signals, and a variety of sources, ranging in size from small hammers for high
frequencies to large vehicles (such as those used in petroleum exploration that emit
vibrations at different frequencies, or a large tractor rocking back and forth) for longer
periods. Given spatial spacing of two receivers, the phase difference gives phase
velocity of Rayleigh waves. DASW (Phillips et al., 2004) is proposed to complement
SASW and to evaluate horizontal homogeneity of a medium by examining the phase of
surface waves with respect to horizontal distance.
The field configuration of MASW is the same as that used in conventional
common midpoint (CMP) body-wave reflection surveys. The generated seismic signals
by various sources are simultaneously recorded by a large number of channels (e.g.,
Park et al., 1999). After a wavefield transformation, 1D Fourier transformation on time
followed by an integral transformation (equation (4) of Park et al., 1998), the recorded
wavefields of a single shot gather give rise to images of dispersion curves. Recent
17
developments of MASW is to use the ambient noise (Park et al., 2004) and both active
and passive sources (Park et al., 2005).
A limitation to the active source methods in general is the difficulty of
generating low frequency waves. The amount of active-source energy to push down the
low frequency end of a dispersion curve often increases by several orders of magnitude,
rendering efforts with an active source impractical and uneconomical (Park et al.,
2004). This limits the depths for which velocity models can be obtained. Passive
sources include microtremors produced by a range of natural phenomena (e.g., ocean
surf and wind) and artificial sources (e.g., traffic, machinery). The frequencies can be
quite low (Earth noise at periods near 8 sec required the development of both long- and
short-period sensors in the first global scale seismographic network) (Peterson, 1993).
Measurements of microtremors are usually made on arrays of instruments placed in
two-dimensional configurations, although one method uses linear arrays (the ReMi
method of Louie, 2001). Extraction of the phase velocities can be done using beamforming or frequency-wavenumber (f-k) methods (e.g., Horike, 1985; Liu et al., 2000),
or by using the SPAC method first proposed by Aki (1957) and now experiencing a
resurgence of interest (e.g., Okada, 2003; Asten, 2005a, 2005b). One limitation in
practice is that 2D instrument arrays are usually not dense enough to resolve nearsurface velocities, and yet these velocities can have an important effect on site
amplifications.
Single-station methods for determining shear-wave velocities have been used
over the years (e.g., Bard, 1998; Scherbaum et al., 2003). There is an excellent project
called Site EffectS assessment using AMbient Excitations (SESAME at http://sesame18
fp5.obs.ujf-grenoble.fr/index.htm). The web site provides excellent reports and
publications for the implementation of the H/V spectral ratio technique on ambient
vibrations (measurements, processing, and interpretation). The methods make use of the
frequency-dependence of Rayleigh-wave ellipticity, which in turn depends on the
subsurface velocities (e.g., Boore and Toksšz, 1969). Contamination by higher modes
can complicate the determination of the velocity structure from the observed ellipticity
(e.g., Arai and Tokimatsu, 2004, 2005). Most methods based on the inversion of
apparent velocities vs. frequency make the assumption that the velocities correspond to
fundamental-mode surface waves. This assumption is not always true, particularly at
longer periods for which the offset between the source and the receivers may not be
sufficient for the body and surface waves to be differentiated in time and in amplitude.
This is one reason that some studies use a combination of active and passive sources,
combining the dispersion curves for the two observation methods (e.g. Yoon and Rix,
2005).
1.4.3 ReMi methods
The Refraction Microtremor (ReMi) method (Louie 2001) transforms the timedomain velocity results of microtremor recordings on a linear array into the frequency
domain by a two-dimensional slant transformation (p-τ) followed by a one-dimensional
Fourier transformation on τ. The method allows for separation of Rayleigh waves from
body waves and other coherent noise, and for easy recognition of dispersive Rayleigh
waves.
ReMi data acquisition consists of setting up a linear array of geophones and
recording ambient seismic noise with no need for a specially cased borehole or any
19
sources (Fig. 1.9). After transformations, a Rayleigh wave dispersion curve is derived
and displayed in a p-f image where dispersion curve picks can be made. These picks are
used to model the subsurface geology and seismic velocities. The effective depth of
investigation is related to the length of the geophone array. Examples of the p-f image,
the dispersion curve fitting, and the shear-wave velocity model are shown in Fig. 1.10.
ReMi surveys provide an effective and efficient means to acquire general, onedimensional, information about large volumes of the subsurface with one setup (Louie,
2001). This method measures ambient seismic noise. It can be conducted in seismically
noisy areas such as construction zones and urban environments. The ReMi method,
licensed as SeisOpt® ReMi™ (©, Optim Inc.) software, is being used widely for
commercial and research purposes (Scott et al., 2004, 2006; Stephenson et al., 2005; Liu
et al., 2005; Thelen et al., 2006) to produce reliable dispersion curves. This dissertation
uses the SeisOpt ReMi software to generate dispersion curve picks, and models, for all
test data.
20
0.0
Trace Sequence
0.0
20.0
Rayleigh waves
Synthetic example
Time, sec
1. Acquisition
4.0
0.0
0.0
5.0
Frequency (Hz)
10.0
15.0
20.0
25.0
2. Extraction
Slowness (s/m)
0.002
0.004
Dispersion
picks
(squares) on p-f image
0.006
0.008
0.01
2.5
ReMi Spectral Ratio
0.0
3. Inversion
0
700
10
600
500
Vs (m/s)
Depth (m)
20
30
40
300
200
50
60
400
100
400
600
Vs (m/s)
800
1D S-wave velocity
profile with uncertainty
0
0
0.2
0.4
Period (s)
0.6
0.8
Observed and calculated
dispersion curves
Figure 1.1 Three steps involved in utilizing dispersion curves of surface waves for
imaging geologic profiles.
21
Figure 1.2 The acceleration power spectrum of microtremors recorded at 75
permanent seismic observatories throughout the world (Peterson, 1993).
22
Figure 1.3 Body wave motion. From http://www.eas.purdue.edu/~braile/edumod/slinky/slinky4.doc
Figure 1.4 Particle motion and amplitude of Rayleigh waves. The motion in a homogenous, isotropic half space is retrograde at the surface, passing through purely verticle
at about lamda/5 then becoming prograde at depth (Cuellar, 1997).
23
Figure 1.5. Surface wave
dispersion. In a homogeneous
half space (left) all the wave
lengths sample the same
material and the phase velocity is constant. When the
properties changes with depth
(right) the phase velocity
depends on the wavelength,
forming a dispersion curve.
z (m)
Normal
c (m/s)
β (m/s)
f (Hz)
z (m)
Inverse
c (m/s)
β (m/s)
f (Hz)
Irregular
c (m/s)
z (m)
β (m/s)
f (Hz)
Figure 1.6. Phase velocities vs frequencies. A normal dispersion curve results from a
profile where S-wave velocity increases with depth. For a profile where S-wave velocity decreases with depth, a reverse dispersion curve will be observed over some range
of frequency. For an irregular S-wave velocity profile, phase velocities show a complex
relation with frequencies.
24
0
5
10
Depth (m)
15
20
25
30
35
40
45
50
Fundamental-mode 1st-higher-mode 2nd-higher-mode
Figure 1.7. Modes of surface waves. For
the same frequency, higher modes
penetrate deeper.
phase velocity
3rd-higher
4th mode
2nd-higher
3rd mode
1st-higher
2nd mode
Fundamental-mode
1st mode
f C1
f C2
f C3
frequency
Figure 1.8. Dispersion curves of higher-mode
surface waves. For the same frequency, higher
modes exist only above their cut-off frequency
and propogate faster than the fundamental mode.
25
noise
a) linear array recording
ambient seismic noise
b) field deployment
Figure 1.9. A typical ReMi field configuration.
a) p-f image with dispersion picks
c) 1D S-wave velocity profile
0
500
1000 1500 2000 2500 3000 3500 4000 4500 5000
0
-10
Vs100' = 1299 ft/s
-20
-30
-50
b) dispersion curve fitting
Rayleigh Wave Phase Velocity,ft/s
-60
Dispersion Curve Showing Picks and Fit
3000
Calculated Dispersion
2500
Depth, ft
-40
-70
Picked Dispersion
2000
-80
1500
1000
-90
500
-100
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Period, s
Shear-Wave Velocity, ft/s
Figure 1.10. A typical ReMi analysis.
26
Chapter 2 Forward modeling of surface-wave dispersion
Many methods have been proposed to calculate dispersion curves of surface
waves. They can be categorized into propagator matrix methods and numerical methods
(Table 2.1).
The matrix methods start from Thomson and Haskell (Thomson 1950; Haskell
1953) who used matrices to solve the eigenvalue problem of the system of differential
equations. The matrix methods construct a dispersion equation (or secular equation),
which is an implicit function of frequency, phase velocity (wave-number), thicknesses,
elastic parameters, and damping of the layers. The dispersion curves are the roots
(eigenvalues) of the dispersion equation for possible modes of propagation at any
particular frequency. Therefore, the solutions are analytic.
Table 2.1 Methods of forward modeling of dispersion curves
Propagator matrix
Transfer matrix method
(Thomson, 1950; Haskell, 1953)
Stiffness matrix method
(Kausel and Roesset, 1981)
Reflection-transmission coefficient
(Kennett, 1983; Luco and Aspel,1983)
Numerical methods
Finite element method
(Lysmer and Drake, 1972)
Finite difference method
(Boore, 1972)
Numerical integral
(Takeuchi and Saito, 1972)
There are basically three matrix methods:
1). Transfer matrix -Most commonly used method, especially in earthquake and
exploration seismology, after Thomson (1950) and Haskell (1953);
2). Stiffness matrix - Complementary method, often favoured by engineers, after
Kausel and Roesset (1981); and
27
3). Reflection-transmission (R/T) coefficient matrix (called R/T method
thereafter) for the entire stack of layers by Kennett (1983) and Luco and Aspel (1983).
The above three categories are collectively known as propagator matrix methods
because these methods allow the propagation of the stress-motion or stress-displacement
field through the layered stack from a known value at a reference depth. They are
analytically exact and all equivalents (Buchen and Ben-Hador, 1996).
The implicit dispersion equation could be solved by numerical methods (Table
2.1). The methods include finite element method, finite difference method, and direct
numerical integration. The fundamental difference is how they approximate the
dispersion equation.
The R/T method is well studied and provides the best numerical technique for
computing the surface waves dispersion curves (Zeng and Anderson, 1995). The method
is stable for high frequencies (Chen 1993; Hisada, 1994, 1995). Phase velocities over 100
Hz for a layered crustal model are calculated (Chen, 1993). Thus, it is suitable for ReMi
phase velocity picks because surface wave phase velocity picks are generally made at
frequencies as low as 2 Hz and as high as 100 Hz (Louie, 2001). However, like other
methods, the R/T method is time-consuming. For example, calculation of phase velocity
dispersion curves for fundamental-mode Rayleigh waves for a twenty-four-layer model
takes about 7 seconds on a 1.33 GHz CPU. This has a negative impact on a non-linear
inversion algorithm which usually requires thousands of forward modeling of dispersion
curves.
Based on the R/T method, this study achieved a new more efficient algorithm,
called the fast generalized R/T coefficient method or the fast R/T method, to calculate the
28
phase velocity of surface waves for a layered earth model. The fast method is based on
but is more efficient than the method of Chen (1993) and Hisada (1994, 1995). Except for
a few modifications, most of the mathematic equations and notations are from Chen
(1993). Specifically this chapter focuses on:
1). A dispersion curve of surface waves is an implicit non-linear function of Swave velocity, thickness, density, and P-wave velocity of each layer, listed in a
decreasing order of priority (Xia et al., 1999). Solving for dispersion curves is an
eigenvalue problem of the system of differential equations. But what is the system of
differential equations?
2). Traditional R/T method is considered one of stable and efficient methods.
How does the traditional R/T method solve the system of differential equations for the
dispersion curves?
3). The fast R/T method is faster than the traditional R/T method while
maintaining the stability. How?
4). The efficiency and stability of the fast R/T method is demonstrated by tests of
six cases at both large and small scales.
2.1 Motion-stress vector
Earth material must behave elastically in order to transmit seismic waves. The
behavior of the material is described by density ρ and elastic constants including shear
modulus (µ), Young’s modulus (E), Bulk Modulus (K), and Poisson’s ratio (σ). Those
constants, along with the two Lame parameters (λ and shear modulus µ) completely
describe the linear stress-strain relation within an isotropic solid. Their definitions and
29
relations are tabulated in Table 2.2 and 2.3. There are numerous, excellent papers on the
elastic behavior and derivation of wave equations. The following summaries are from
Lay and Wallace (1995), Shearer (1999), and Aki and Richards (2002).
The theory of elasticity provides mathematical relationships between stresses and
strains in the medium (for details see Shearer, 1999, Chapter 2) and thus governs the
equation of motion in the medium:
ρ
∂ 2ui
= τ ji , j + fi
∂t 2
(2.1)
where ui the displacement in ith component, ρ the density, τ the stress, f the body force.
Equation (2.1) is known as the equation of motion in the medium. For details, please refer
Shearer (1999, equation (3.6) ) and Aki and Richards (2002, equation (2.13) ).
Table 2.2 Definition of elastic constants
Name
Symbol
Young’s modulus
E
Shear modulus
µ
Poisson’s ratio
σ
Bulk Modulus
K
Definition
longitudinal stress
longitudinal strain
shear stress
shear strain
longitudinal stress
transversal strain
idrostatic stress
volumetric strain
Notes
Free transversal deformation
Free transversal deformation
Starting with the simplest problems, we consider Cartesian coordinates and a
surface wave u propagating in the horizontal direction of increasing x with angular
frequency ω and wavenumber k:
u( x, y, z , t ) = Z( z )ei ( kx −ω t )
(2.2)
30
where z is depth, Z(z) is amplitude exponential decay term for surface waves. Other
harmonic wave parameters are listed on Table 2.4.
Table 2.3 Relationship between elastic constants
µ or σ
λ or µ
λ
λ
µ
µ
K
E
σ
(3λ + 2µ )
3
µ (3λ + 2µ )
E=
λ+µ
K=
σ=
λ=
2 µσ
1 − 2σ
µ
K=
2µ (1 + σ )
3(1 − 2σ )
E = 2µ (1 + σ )
E
σ
σ
λ
2(λ + µ )
K or µ
2
λ=K− µ
3
E or σ
µE
λ=
(1 + σ )(1 − 2σ )
E
µ=
2(1 + σ )
E
K=
3(1 − 2σ )
µ
K
9K µ
3K + µ
3K − 2 µ
σ=
2(3K + µ )
E=
Table 2.4 Harmonic wave parameters
Frequency
f
Hz
Period
T
s
Velocity
c
m/s
Angular frequency
ω
radian/s
Wavelength
λ
m
Wavenumber
k
radian/m
1 ω
c
=
=
T 2π Λ
1 2π Λ
T= =
=
f
c
ω
f =
c= fλ =
λ
=
ω
T k
2π
ω = 2π f =
= ck
T
f
2π
= cT =
c
k
ω 2π 2π f
k= =
=
c
c
λ
λ=
31
Let us now consider surface waves propagating in the x-direction in a vertically
heterogeneous, isotropic, elastic medium occupying a half-space z>0 in which elastic
moduli λ(z), µ(z) and density ρ(z) are arbitrary function of z.
Love waves are SH waves only. Their displacements in three directions (u, v, w)
are in the form of:
u = 0

i ( kx −ω t )
 v = l1 (k , z , ω )e
w = 0

(2.3a)
The stress components associated with the above displacement are:
 τ xx = τ yy = τ zz = τ zx = 0

dl1 i ( kx −ω t )

e
τ yz = µ
dz

τ xy = ik µ l1ei ( kx −ωt )
(2.3b)
where l1 is amplitude exponential decay term in both equations.
Equation of motion (equation (2.1) ) must satisfy the following four boundary
conditions:
BC1 (radiation condition): The displacement in infinite depth is zero u |z →∞ = 0 ;
BC2 (displacement continuity condition): Displacement must be continuous
across any layer boundary ui |z = d = ui +1 |z = d ;
BC3 (traction continuity condition): Traction must be continuous across any
layer boundary τ i |z = d = τ i +1 |z = d ;
BC4 (zero traction at the free surface): Traction must be zero at the free surface
τ |z =0 = 0 .
32
After applying BC1-BC4, equation (2.3) could be rewritten as a set of first-order
ordinary differential equations,
 dl1 ( k , z , ω ) l2 ( k , z , ω )
 dz = µ

 dl2 ( k , z , ω ) = (k 2 µ ( z ) − ω 2 ρ ( z ))l ( k , z , ω )
1

dz
(2.4)
Or in a matrix form

d  l1 ( k , z ,ω )  
0
=
 ( )   2
dz  l2 k , z ,ω   k µ ( z )-ω 2 ρ ( z )
1 

µ (z) 

0 
 l1 
 
 l2 
(2.5)
Equation (2.5) is called the motion-stress vector for Love (SH) waves.
For Rayleigh waves that are resulted from interference between P and SV-waves,
the displacements in three directions (u, v, w) are in the form of:
 u = r1 ( k , z , ω )ei ( kx −ω t )

v = 0
 w = ir ( k , z , ω )ei ( kx −ω t )
2

(2.6a )
The stress components associated with the above displacement are:
 τ yz = τ xy = 0

dr2
i ( kx −ω t )
τ xx = i λ dz + k (λ + 2µ )r1  e

dr
i ( kx −ω t )
τ yy = i λ dz2 + k λ r1  e

dr1
i ( kx −ω t )
= r3ei ( kx −ω t )
 τ zx = µ ( dz − kr2 )e
τ = i (λ + 2µ ) dr2 + k λ r  ei ( kx −ω t ) = ir ei ( kx −ω t )
1
4
dz
 zz

(2.6b)
where r1 and r2 are amplitude exponential decay terms in both equations.
After applying BC1-BC4, the motion-stress vector for Rayleigh waves (r1, r2, r3,
r4)T are obtained as
33
1
k
0
 r1  
µ (z)

 
d  r2   λ ( −z )k+λ2(µz )( z )
0
0
=
dz  r3 
k 2ξ ( z )-ω 2 ρ ( z )
0
0
  

r
 4 
−ω 2 ρ ( z )
0
-k
4 µ ( z )[λ ( z ) + µ ( z )]
where ξ ( z ) =
λ ( z ) + 2µ ( z )


1

λ ( z )+ 2 µ ( z )

kλ ( z )
λ ( z )+2 µ ( z ) 

0

0
 r1 
 
 r2 
 r3 
 
 r4 
(2.7)
Equation (2.7) is a linear differential eigenvalue problem with displacement
eigenfunction r1 and r2 and stress eigenfunction r3 and r4. For a given frequency, ω, nontrivial solutions exist only for special values of the wavenumber, k. These possible values
k1(ω), k2(ω), … kn(ω) are called eigenvalues. The corresponding functions (r1, r2, r3, r4)
are the eigenfunctions.
A general form of the motion-stress vector for both SH and P-SV waves is
df ( z )
= G ( z )f ( z )
dz
(2.8)
2.2 Reflection and transmission coefficients
Waves are scattered between two solid half-space. For SH waves, four possible
scatters occur (Fig. 2.1). The scatter matrix for a SH wave would be
R
 du
 Td
Tu 

Rud 
where R represents reflection coefficient and T represents transmission coefficient. Subindex ‘d’ means down-going waves; ‘u’ up-going waves. For example, Rdu
is the
reflection coefficient of incident down-going SH-wave to reflected up-going SH-wave at
interfaces. Td is the transmission coefficient of incident down-going SH-wave to
34
transmitted down-going SH-wave at interfaces. Other terms have similar physical
meaning.
For P-SV waves, sixteen possible scatters occur (Fig. 2.2). The scatter matrix
would be
 Rdpp
Tu   Rdps
= T

R ud   dpp
 Tdps
 R du
T
 d
Rdsp
Rdss
Tupp
Tups
Tdsp
Tdss
Rupp
Rups


Rusp 

Russ

Tusp
Tuss
Sub-index ‘d’ means down-going waves; ‘u’ up-going waves; ‘p’ P-waves; and ‘s’ SVwaves. Rdps is the reflection coefficient of incident down-going P-wave to reflected SVwave at interfaces. Tdps is the transmission coefficient of incident down-going P-wave to
transmitted down-going SV-wave at interfaces. Other terms have similar physical
meaning.
2.3 Plane waves in a layered model
Let us consider a plane surface wave in a horizontally layered, vertically
heterogeneous, isotropic, elastic medium occupying a half-space z > 0 in which elastic
moduli
µ j λj ρ j
are dependent on depth and are constant within layers (Fig. 2.3). From
equation (2.5) and (2.7), the differential equations for the motion-stress vector are

d  l1j   0
=
  
dz  l2j   k 2 µ j -ω 2 ρ j
1 

 l1j 
 j 
0   l2 
µj
(2.9)
for SH waves and
35
 r1 j   0
 j
j
d  r2   −j k λ j
2
λ
µ
+
=
dz  r3j   k 2ξ j -ω 2 ρ j
 j 

 r4   0
k
1
µj
0
0
0
0
−ω 2 ρ j
-k
 j 
  r1 
1
 rj
λ j +2µ j   2 
j
k λ j   r3 
j
j


λ +2µ
 r j 
4 

0

0
(2.10)
for P-SV waves.
0
1
j
N
N +1
j −1
j
where z < z < z , j = 1, 2,3,", N , N + 1 and 0 = z < z < " < z < " < z < z = ∞
In matrix format, equation (2.9) and (2.10) could be summarized as
df j ( z )
= G j ( z )f j ( z )
dz
(2.11)
j
where f ( z ) is the motion-stress vector for the jth layer and has dimension of 4x1 for P-
SV waves or 2x1 for SH waves. Accordingly, the G matrix (most right matrix in the
right-hand-side of equation (2.9) and (2.10) ) has dimension of 4x4 and 2x2 for P-SV and
SH waves, respectively.
Inside each layer, the analytic solution of the differential equation system
(equation (2.11) ) has the following format (Aki and Richards, 2002):
f j ( z) = A j B j D j
(2.12)
j
j
j
for j = 1, 2,3, ", N , N + 1 . where A and B are know matrices given below. But D are
unknown vectors to be determined. For SH waves we have

A j B j D j = 
1
j
 −µ ν
j
1 

µ jν j 
 e−ν j z

 0
 S j↓
j  j 
eν z   S ↑ 
0
(2.13)
j = 1, 2,3," , N . where ν = ± k 2 − ω
2
2
β
For P-SV waves we have (Aki and Richards, 2002, p.276 equation (7.55) )
36
 α jk
1  α jγ j
j
j
j
ABD =  j j j
ω  −2α µ kγ
 −α j χ j µ j
β jν j
β jν j
β jk
−α j γ
−β jχ jµ j
2α j µ j kγ
−2 β j µ j kν
 −γ j z
e
 0
−β jk
β jχ jµ j 
 0
−2 β j µ j kν j  
 0
α jk
j
j
j
−α j χ j µ j
0
0
j
e−ν z
0
0
j
eγ z
0
0
  P j ↓
  j 
0
  S j ↓ 
0   P ↑
 j 
j   S ↑
eν z 
0
(2.14)
j = 1, 2,3," , N . where γ = ± k 2 − ω
2
2
α
j
BC1-BC4 is to determine the unknown matrix D for each layer. The continuity
condition implies
f j ( z j ) = f j +1 ( z j +1 )
(2.15a)
for j = 1, 2,3," , N .
The radiation condition requires
f N ( z) → 0
(2.15b)
as z → ∞
j
Careful observation reveals that matrix A contains some layer-specified
j
j
constants, like α , β , which are constant within a layer and varies for different layers.
These layer-specified constants could be extremely large to cause over-flow errors during
j
matrix multiplications. The matrix B contains some depth-growth terms, like e
−γ z
, eγ z ,
which are extremely large or small for a deep layer. Multiplication on the matrix causes
the instability of the analytical calculation (Chen, 1993). Excluding of these layerspecified constants and depth-growth terms will significantly increase the stability of the
algorithm. So we can rewrite the equations (2.13) and (2.14) as (Luco and Apsel, 1983;
Chen, 1993; Hisada, 1994)
37
 1

 − µ jν

j
1 

µ jν j 
 e−ν j ( z − z j −1 )

0

  e−ν j z j−1 S j ↓  j j  Csdj  j j j
  ν j z j j  =E Λ  j  =E Λ C
j
j
S ↑ 
e−ν ( z − z )   e
 Csu 
0
(2.16)
j = 1, 2,3, " , N for SH waves .
 k
 γj
 −2 µ j kγ j
 −χ jµ j

νj
k
k
−γ
−χ jµ j
2 µ j kγ
−2 µ j kν
j
 −γ
e

−k
χ jµ j  

−2 µ j kν j  

j
νj
j
j
−χ jµ j
( z−z
j −1
)
0
0
j
j −1
e−ν ( z − z )
0
0
0
j
j
e− γ ( z − z )
0
0
0
0


0

0

j
j

ν
z
z
(
)
−
−
e

0
 j

j j −1
 α e− γ z P j ↓ 
ω


 j
j j −1
β

e−ν z S j ↓ 

 ω


j
j j
α

γ z Pj↑ 

 ω e



 β j ν jz j j
e
S ↑ 

 ω

=E j Λ
 j 
 C pd 


 j 
j  Csd 


C j 
 pu 
 j 
C 
 su 
=E j Λ j C j
(2.17)
j = 1, 2,3, " , N for P-SV waves.
where E is layer matrix, Λ is phase delay matrix, and C is amplitude vector matrix.
j
It is noted that exponential terms in the matrix B in equations (2.13) and (2.14)
γ
has been modified by multiplying the layer-specified constants e
j
z j −1
, eν
j
z j −1
, e −γ
j
zj
, e −ν
j
zj
.
j
Accordingly, the unknown matrix D in equations (2.13) and (2.14) has absorbed these
factors. Absorbing of these constants greatly increase the stability of the algorithm. It
j
should be pointed out that the elements of the matrix D in equations (2.13) and (2.14)
j
represent amplitudes of downgoing and upgoing waves. The matrix C in equations
j
(2.16) and (2.17) is the matrix D that has absorbed the layer-specified constants.
j
Therefore, the elements of the matrix C have lost their original physical meanings. They
are just other layer-specified constants.
38
2.4 Phase velocity of Love waves
Love waves contain only SH waves. The analytic solution to equation (2.9) can be
rewritten as
 l1j   E11j
 j  = Ej
 l2   21
( )

E12j  Λ dj
j 
E22
 0
0
Λuj
 Cdj 
 j
 Cu 
(2.18a )
j = 1, 2,3," , N
 l1N +1   E11N +1Λ dN +1CdN +1 
 N +1  =  N +1 N +1 N +1 
 l2   E21 Λ d Cd 
(2.18b)
j = N + 1.
The equation (2.18b) is obtained by applying the radiation condition that only allows the
decaying solution to exist in the (N+1)th layer (the bottom half-space) so that
N +1
Cu
= 0.
For an arbitrary jth interface, the modified reflection and transmission
coefficients for SH waves are denoted as Rduj , Tdj . They are defined as
Cdj +1 = Tdj Cdj + Rudj Cuj +1
 j
j
j
j
j +1
 Cu = Rdu Cd + Tu Cu
(2.19a)
j = 1, 2,3,", N − 1
CdN +1 = TdN CdN
 N
N
N
 Cu = Rdu Cd
(2.19b)
j=N
Continuity condition requires that
 l1j   l1j +1 
 j  =  j +1 
 l2   l2 
⇒
(
E11j
E12j
E21j
E22j
)(
Λ dj ( z j ) 0
0
1
)
 Cdj 
 j =
 Cu 
(
E11j +1
E12j +1
j +1
E21
E22j +1
)(
1
0
0 Λ uj +1 ( z j )
)
 Cdj+1 
 j+1 
 Cu 
(2.20)
After rearranging equation (2.20) and comparing with equation (2.19), we find
39
(
Tdj
Rudj
Rduj
j
=
Tu
)=(
E11j +1
− E12j
j +1
E21
− E22j
)(
−1
E11j
− E12j +1
E21j
− E22j +1
)(
Λ dj ( z j )
0
0
Λ uj +1 ( z j )
(
)

1
0
2 µ jν j
µ j +1ν j +1 − µ jν j  Λ dj ( z j )

µ jν j + µ j +1ν j +1  µ jν j − µ j +1ν j +1
0
Λ uj +1 ( z j )
2 µ j +1ν j +1 

)
(2.21a )
j = 1, 2,3,", N − 1
 N
 Td 
 RN 
 du 
=
(
N +1
E11
N
− E12
N +1
E21
N
− E22
)
−1
 N N N 
 E11 Λ d ( z ) 
 E N ΛN ( z N )
 21 d

(2.21b)
j=N
ˆj ˆj
The generalized reflection and transmission coefficients ( Rdu , Td ) for SH waves
are defined as following:
Cdj +1 = Tˆdj Cdj
 j
j
j
 Cu = Rˆ du Cd
(2.22a)
j = 1, 2,3,", N − 1
CdN +1 = TˆdN CdN
 N
 Rˆdu = 0
(2.22b)
j=N
Cd1 = Rˆud0 Cu1
(2.22c)
j = 0 for the free surface.
It is noted that TdN = TˆdN
and
RduN = Rˆ duN
(2.23) .
At the free surface the traction-free condition yields
E1 Λ1 (0) µ 1ν 1 −ν 1 ( z1 − z0 ) −ν 1z1
1
1
0 = T 1 (0) = E21
Cd1 + E22
Λ1u (0)Cu1 ⇒ Rˆud0 = 22 u1
= 1 1e
=e
µν
− E21
(2.24)
40
Substituting equation (2.22b) in equation (2.22a), we can obtain the following
recursive formula for computing other generalized R/T coefficients
Tˆdj = (1 − Rudj Rˆ duj +1 ) −1Tdj
 j
j
j
j +1 j
 Rˆ du = Rud + Tu Rˆ du Tˆd
(2.25)
j = 1, 2,3,", N − 1
Inside the top layer the definition of the generalized reflection and transmission
coefficients gives us (equations (2.22))
Cd1 = Rˆud0 Cu1
 1
1
1
Cu = Rˆ du Cd
(2.26)
1
Combining the above two equations leads to (1 − Rˆud0 Rˆ du
)Cd1 = 0
(2.27)
The existence of the non-trivial solution leads to the following implicit dispersion
1
equation for SH waves 1 − Rˆud0 Rˆdu
=0
(2.28)
The phase velocities of SH waves are the non-trivial solution of the implicit
equation (2.28). For a given frequency, only a finite number of roots exist, corresponding
to the phase velocities from fundamental to possible higher modes.
1
According to equation (2.27), Cd (vn ) can take any non-zero value, where vn is
the phase velocity of the nth higher mode. We take unity. Then we have
 Cd1 (vn ) = 1
 1
1
1
Cu (vn ) = Rˆ du Cd (vn )
(2.29)
Applying equation (2.26) into (2.22), we find
j +1
j
j −1
1
 Cd (vn ) = Tˆd (vn )Tˆd (vn )"Tˆd (vn )
 j +1
j +1 j +1
Cu (vn ) = Rˆ du Cd
(2.30a)
41
j = 1, 2,3,", N − 1
CdN +1 (vn ) = TˆdN (vn )TˆdN −1 (vn )"Tˆd1 (vn )
(2.30b)
j=N
Finally, substituting these solved
(l
we obtain the non-trivial solutions
j
1
l2j
(C
j
d
Cuj
) for each mode into equation (2.18),
) which are eigenfunctions of displacement and
traction of SH waves. These eigenfunctions are normalized by the corresponding
maximum value.
2.5 Phase velocity of Rayleigh waves
Rayleigh waves are formed by interference between P and SV waves. The
analytic solution to equation (2.10) can be rewritten as
0  Cdj 
 D j ( z )   E11j E12j   Λ dj ( z )
j
j
j
=

 j =E Λ C


Sj z 
j
j
j
Λ u ( z )   Cu 
 ( )   E21 E22   0
(2.31)
See appendix A for each term.
For an arbitrary jth interface, the modified reflection and transmission
coefficients for Rayleigh waves are denoted as (R duj , R udj , Tdj , Tuj ) and defined by the
following equations:
Cdj +1 = Tdj Cdj + R udj Cuj +1
 j
j
j
j
j +1
 Cu = R du Cd + Tu Cu
(2.32a )
for j = 1, 2,3,", N − 1 and
CdN +1 = TdN CdN
 N
N
N
 Cu = R du Cd
(2.32b)
42
for j = N
 j
 Rdpp
where R duj = 
j
 Rdps
j 
R
dsp 

j 
R 
dss 
 j
 Rupp
, R udj = 
j
 Rups
j 
Rusp


j 
Russ

 j
T
, Tdj =  dpp
j
 Tdps
j 
T
dsp 

j 
T 
dss 
 j
T
, and Tuj =  upp
j
 Tups
j 
Tusp


j 
Tuss

. Sub-index ‘d’
j
means down-going waves; ‘u’ up-going waves; ‘p’ P-waves; and ‘s’ SV-waves. Rdps
is
the reflection coefficient of incident down-going P-wave to reflected SV-wave at
interface j. Tdpsj is the transmission coefficient of incident down-going P-wave to
transmitted down-going SV-wave at interface j. Other terms have similar physical
meaning.
Like SH waves, the modified reflection and transmission coefficients ( R duj , Tdj )
for P-SV waves can be found by applying the continuity condition at an arbitrary jth
interface. Then we have
 D j ( z )   D j +1( z ) 
 j  =  j +1 
 S (z)   S ( z) 
j
j
j +1
j +1 
  C j +1 
 j j   C j 
⇒  E11j E12j   Λd ( z ) 0   dj  =  E11j+1 E12j+1   I j+01 j   dj +1 
 E21 E22   0 I   Cu   E21 E22   0 Λu ( z )   Cu 
After rearranging, we obtain the explicit expressions of the modified R/T matrices as
follows:
 Tdj
 j
 R du
R udj   E11j +1 −E12j 
=

Tuj   E21j +1 −E22j 
−1
 E11j
 j
 E21
−E12j +1   Λ dj ( z j )
0 

j +1  
j +1
−E22   0
Λu ( z j )
(2.33a )
for j = 1, 2,3,", N − 1
 TdN
 N
 R du
N
  E11N +1
R ud
 =  N +1
TuN   E21
−E12N 
N 
−E22

−1
 E11j Λ dj ( z j ) 
 j j j 
 E21Λ d ( z ) 
(2.33b)
for j = N
43
Note that the layer matrix E is composed of elements that are determined by the elastic
parameters of both jth and (j+1)th layers.
For an arbitrary jth interface, the generalized reflection and transmission
ˆ j ,T
ˆ j ) and defined by the following
coefficients for Rayleigh waves are denoted as (R
du
d
equations:
ˆ jC j
Cdj +1 = T
d
d
 j
j
j
ˆ
 Cu = R du Cd
(2.34a)
for j = 1, 2,3,", N − 1
ˆ N CN
 CdN +1 = T
d
d
 N
ˆ
R du = 0
(2.34b)
for j = N
Comparing equations (2.32) and (2.34) we find
ˆ N = TN
T
d
d
and
ˆ N = RN
R
du
du
(2.35) .
Substituting equation (2.34) in equation (2.33), we obtain the recursive formula
for computing other generalized R/T coefficients as
ˆ j = (I − R j R
ˆ j +1 ) −1 T j
T
d
ud
du
d
 j
1
+
j
j
j
ˆ = R +T R
ˆ T
ˆj
 R
du
ud
u
du
d
(2.36)
for j = 1, 2,3,", N − 1
ˆN
Starting from the last interface where R du = 0 , we can use equation (2.36) to find the
ˆj ˆj
generalized reflection and transmission coefficients ( R du , Td ) for Rayleigh waves for all
interfaces above.
44
The Rayleigh dispersion curves can be determined by imposing the traction-free
condition at the free surface (z=0). From equation (2.31) we calculate the traction at the
free surface as
ˆ 1 )C1
S1 ( 0 ) = (E121 + E122 Λ u0 (0)R
du
d
(2.37)
Equation ( 2.3 7) has non-trivial solutions only for some particular phase velocities that
satisfy the following secular equation:
ˆ1 )=0
det(E121 + E122 Λ u0 (0)R
du
(2.38)
Equation ( 2.3 8) is called the secular function for Rayleigh waves. Therefore, the roots of
this equation are the phase velocities for modes that potentially exist.
C1d (vn ) has infinite solutions to satisfy equation (2.37). Let’s
ˆ 0 (v ) R
ˆ 1 (v )
G (vn ) = I − R
ud
n
du
n
(2.39)
We take one of them as our starting point. Thus we find that
−G12
 1
C pd (vn ) =
G112 + G122


G11
C1 (v ) =
pd
n
2

G11 + G122

(2.40)
and
ˆ 1 C1 (v )
C1u (vn ) = R
du d
n
(2.41a)
Correspondingly, we have
ˆ j (v )T
ˆ j −1 (v )" T
ˆ 1 (v )
 Cdj +1 (vn ) = T
d
n
d
n
d
n
 j +1
j +1 j +1
ˆ
Cu (vn ) = R du Cd
(2.41b)
j = 1, 2,3,", N − 1
45
ˆ N (v )T
ˆ N −1 (v )" T
ˆ 1 (v )
CdN +1 (vn ) = T
d
n
d
n
d
n
(2.41c)
j=N
(C
Finally, substituting these solved
yields the non-trivial solutions
(r
j
1
r2j
j
d
Cuj
r3j
) for each mode into equation (2.31)
r4j
)
, which are eigenfunctions of
displacement and traction of P-SV waves.
2.6 Improvements on calculation of phase velocity of Rayleigh waves
One of the most CPU time-consuming parts in the R/T method is to compute the
inverse of the 4x4 layer matrix E in equation (2.33) to obtain the modified R/T
coefficients for P-SV waves. Hisada (1995) presents an analytical solution for the inverse
matrix. However, this analytical solution requires recursive substitution. The layer matrix
E in the equation (2.33) is composed of elements that is determined by the elastic
parameters of both jth and (j+1)th layers. This property prohibits the existence of a
simple analytic solution for the inverse matrix E-1.
Careful observation from equation (2.38) reveals that only the generalized R/T
coefficients are needed to calculate the secular function of Rayleigh waves. In fact, the
generalized R/T coefficients could be directly calculated without knowing the modified
R/T coefficients. Therefore, the modified R/T coefficients are not necessary for the
calculation of the secular function of Rayleigh waves (equation (2.38) ).
The continuity condition at any arbitrary interface j states that
0  Cdj +1 
 E11j E12j   Λ dj ( z j ) 0  Cdj   E11j +1 E12j +1  1
=
 j

  

j +1
j 
j 
1   Cuj   E21j +1 E22j +1  0 Λ u ( z )  Cuj +1 
E21 E22   0
(2.42)
46
The definition of the generalized R/T coefficients implies that
ˆ j Cj
 Cuj = R
du d
 j +1
j
j
ˆ
Cd = Td Cd
 j +1 ˆ j +1 ˆ j j
Cu = R du Td Cd
(2.43)
Directly substituting equation (2.43) into equation (2.42) yields
−1
ˆj

 I   E11j E12j  E11j +1 E12j +1  
T
d
=


ˆ j   j



j
j +1
j +1
ˆ j +1T
ˆ j
 R du   E21 E22  E21 E22   Λ uj +1 ( z j )R
du
d 
(2.44)
ˆN
Starting from the last interface where R du = 0 , equation (2.44) yields the
ˆj ˆj
generalized reflection and transmission coefficients ( R du , Td ) of Rayleigh waves for all
interfaces above.
Thus, we can directly calculate the generalized R/T coefficients without knowing
the modified R/T coefficients. The next step is to derive the inverse matrix of 4X4 E
matrix in equation (2.44). Unlike these of the layer matrix E in equation (2.33), the
elements of the layer matrix E in equation (2.44) are determined by the elastic parameters
of the jth layers only. This characteristic allows many terms to be crossed out during the
derivation of the inverse matrix E-1 and results in a simple analytic solution for the
inverse matrix E-1 (Appendix A). For equation (2.33) it is impossible to derive the similar
analytic form of E-1 as the elements of matrix E are related to the elastic parameters of
both jth and (j+1)th layers. The simplicity of our solution significantly reduces the
computational time. The following test section shows that these improvements cut the
computational time for dispersion curves of Rayleigh waves at least by half.
47
2.7 Numerical examples on dispersion calculation of Rayleigh waves
Three cases are designed for the crustal scale. Model 1 is Gutenberg’s classic
crust and upper mantle model for a continent (Table 2.5), which resembles the velocity
structure of the Earth to the depth of 1000 km. The model consists of a stack of 24
homogeneous and isotropic layers and has been used for many geophysical studies and
provides an excellent reference. Model 2 is an artificial, inverted profile, which has a low
velocity zone for the fourth layer (Table 2.6). Model 3 is another artificial four-layer
profile, which has a high velocity zone for the second layer (Table 2.6). Three other cases
are designed for the local site scale. Model 4 is a regular stack of 4 homogeneous and
isotropic layers (Table 2.7). Model 5 has a low velocity zone for the second layer and
model 6 has a high velocity layer for the second layer (Table 2.7).
In traditional forward calculation (Chen, 1993; Hisada, 1994, 1995), the modified
R/T coefficients (equation (2.33) ) are calculated followed by the generalized R/T
coefficients (equation (2.36) ). The inverse matrix E-1 is computed following Hisada’s
procedure (1995). The bisection root-searching method is employed to find the phase
velocities from the secular equation (2.38). I code the above calculation in a program
called RTmod, emphasizing the fact that the generalized R/T coefficients are based on the
modified R/T coefficients. In the fast R/T method, the generalized R/T coefficients are
computed directly from equation (2.44) without calculations of the modified R/T
coefficients. Keeping other parts identical, I code the fast R/T method in a program called
RTgen. I perform stability and efficiency tests for RTgen at both crustal and local site
scales.
48
Table 2.5 Gutenberg’s layered model of continental structure
Depth to bottom (km)
19
38
50
60
70
80
90
100
125
150
175
200
225
250
300
350
400
450
500
600
700
800
900
1000
Density (g/cm3)
2.74
3.00
3.32
3.34
3.35
3.36
3.37
3.38
3.39
3.41
3.43
3.46
3.48
3.50
3.53
3.58
3.62
3.69
3.82
4.01
4.21
4.40
4.56
4.63
Vp (km/s)
6.14
6.58
8.20
8.17
8.14
8.10
8.07
8.02
7.93
7.85
7.89
7.98
8.10
8.21
8.38
8.62
8.87
9.15
9.45
9.88
10.3
10.71
11.10
11.35
Vs (km/s)
3.55
3.80
4.65
4.62
4.57
4.51
4.46
4.41
4.37
4.35
4.36
4.38
4.42
4.46
4.54
4.68
4.85
5.04
5.21
5.45
5.76
6.03
6.23
6.32
Table 2.6 Test models at crustal scale
Depth to bottom (km)
M2*
M3
18
20
24
25
30
40
50
∞
∞
Density (g/cm3)
M2
M3
2.80
2.8
2.90
3.4
3.50
3.2
3.40
3.4
3.30
Vp (km/s)
Vs (km/s)
M2
M3
M2
M3
6.00
5.6
3.50
2.5
6.30
6.3
3.65
3.2
6.70
6.1
3.90
2.9
6.00
6.3
3.70
3.2
8.20
4.70
*M2 means model 2; the same for M3
49
Table 2.7 Test models at local site scale
Depth to bottom
(m)
M4* M5
M6
12
11
11
24
23
23
36
35
35
∞
130 130
∞
∞
Density (g/cm3)
M4
2.0
2.0
2.0
2.0
M5
2.0
2.0
2.0
2.0
2.0
M6
2.0
2.0
2.0
2.0
2.0
Vp (m/s)
Vs (m/s)
M4
M5
M6
M4 M5
M6
311.8 1558.8 866.0 180 900 500
519.6 866.0 1558.8 300 500 900
866.0 1212.4 1212.4 500 700 700
1212.4 1732.0 1732.0 700 1000 1000
2551.6 2551.6
1300 1300
*M4 means model 4; the same for M5 and M6
The stability tests of the fast generalized R/T coefficients method are done by
comparing the calculated phase velocities for a given model, by RTgen and by CPS.
Herrmann and Ammon (2002) developed a FORTRAN code to calculate the phase
velocity dispersion curves of surface waves, which is a part of a package called Computer
Programs in Seismology (CPS) (http://mnw.eas.slu.edu/People/RBHerrmann/). CPS has
been widely used to compute the dispersion curves of Rayleigh waves (e.g., Stephenson
et al., 2005). I use CPS to calculate dispersion curves of Rayleigh waves for six test
models and compare with those calculated by our code RTgen. Figures 2.4 shows the
fundamental-mode dispersion curves of Rayleigh waves calculated by RTgen plotted atop
those by CPS. The dispersion curves cover a broad frequency range from 0.01 Hz to 100
Hz. Figure 2.5 shows the normalized errors between phase velocities calculated by
RTgen and CPS. Both calculations yield the normalized errors of 0.001% - 0.01% for all
tested models, indicating RTgen is correct and stable. Higher-mode (up to mode 14)
dispersion curves of Rayleigh waves for all six test models by both CPS and RTgen are
50
almost the same up to 5% accuracy. Figures 2.6 and 2.7 show the results of model 1 and
model 4.
An efficiency test of RTgen is done through comparison of computational time
taken by RTgen and RTmod for models with differing numbers of layers. We coded both
RTmod and RTgen in an identical way except in how to calculate the generalized R/T
coefficients. The coefficients are calculated from the modified R/T coefficients (equation
(2.36) ) in RTmod, and directly computed from equation (2.44) in RTgen.
Both codes ran on Linux Pentium machines, Sun workstations, personal Windows
PCs, and OS X Macintosh machines. Figure 2.8 shows the computational times in
seconds for 20 runs on a 1.33 GHz PowerPC G4 Mac notebook. The 24-layer-model used
for efficiency test is Gutenberg’s Earth model. We delete the lower 2 layers of
Gutenberg’s Earth model to make the 22-layer-model; 4 to make the 20-layer-model; and
so on. The figure clearly shows that RTgen saves 55% computational time for the 4layer-model; 57% for the 14-layer-model; 60% for the 24-layer-model. Tests on other
computer platforms also show at least 50% saving on computational time.
2.8 Improvements on calculation of phase velocity of Love waves
The same idea can be applied to Love waves, which consist of only SH waves. In
another words, we directly calculate the generalized R/T coefficients of Love waves
without knowing the modified R/T coefficients.
For Love waves, the continuity condition at any arbitrary interface j states that
(see Appendix B for each term)
51
 E11j
 j
 E21
E12j   Λ dj ( z j ) 0  Cdj   E11j +1

  = 
E22j   0
1  Cuj   E21j +1
0  Cdj +1 
E12j +1  1


j +1
j 
E22j +1  0 Λ u ( z )  Cuj +1 
(2.45)
The definition of the generalized R/T coefficients implies that
 Cuj = Rˆ duj Cdj
 j +1
j
j
Cd = Tˆd Cd
 j +1 ˆ j +1 ˆ j j
Cu = Rdu Td Cd
(2.46)
where Rˆduj and Tˆdj are the generalized reflection and transmission coefficients of incident
down-going SH-wave to reflected SH-wave and transmitted SH-waves at interface j,
respectively. Directly substituting equation (2.46) into equation (2.45) yields
 I   E11j
 ˆj  = j
 Rdu   E21
−1
E12j   E11j +1
 
E22j   E21j +1

Tˆdj
E12j +1  



E22j +1   Λ uj +1 ( z j ) Rˆ duj +1Tˆdj 
(2.47)
Starting from the last interface where Rˆ duN = 0 , equation (2.47) yields the generalized
reflection and transmission coefficients ( Rˆ duj , Tˆdj ) of Love waves for all interfaces above.
Appendix B gives explicit solutions for these coefficients.
2.9 Numerical examples on Love waves
I use CPS to calculate higher-mode dispersion curves of Love waves for six test
models and compare with those calculated by the code RTgen. Figures 2.9 and 2.10 show
that both calculations yield identical results to 1%-5% for models 1 and 4, indicating the
new version is correct and stable. However, efficiency tests show only a 1-5% speed-up.
This is not a surprise due to the fact that inversion of the 2X2 E matrix (equation (2.47) )
is not major CPU time consuming part on SH-wave cases. Tests on other models lead to
the same conclusion.
52
2.10 Group velocity calculation of surface waves
The calculation of group velocity of surface waves is adopted from Aki and
Richards (2002, chapter 7). For a linear elastic body, the Lagrangian density is the kinetic
energy minus the elastic strain energy.
L=
1
1
ρ ui ui − ( λ (ekk ) 2 + µ eij eij )
2
2
(2.48)
In the case of plane Love wave, where
ekk = 0 and
eij eij = 2((
1 ∂v 2 1 ∂v 2
) +(
) ) and v = l1 ( k , z ,ω ) ei ( kx −ω t )
2 ∂x
2 ∂z
.
Then we have
L=
2∫
∞
0
∂l
1 2 2 1
ρ l1 ω − µ (k 2l12 + ( 1 ) 2 ) and
∂z
2
2
L =
∂l
1 2 2 1
ρ l1 ω − µ (k 2l12 + ( 1 )2 )
∂z
4
4
∞
∞
1
1
1 ∞ ∂l
L dz = ω 2 ∫ ρ l12 dz − k 2 ∫ µ l12 dz − ∫ µ ( 1 ) 2 dz = ω 2 I1 − k 2 I 2 − I 3
0
0
2
2
2 0
∂z
(2.49)
(2.50)
where < > means an averaging process. The energy integrals are
I1 =
1 ∞ 2
1 ∞ 2
1 ∞ ∂l1 2
l
dz
,
I
l
dz
,
I
ρ
=
µ
=
µ ( ) dz
1
2
1
3
2 ∫0
2 ∫0
2 ∫0
∂z
(2.51)
After a complicated derivation, the group velocity in terms of integrals is given
as
U=
∆ω kI 2
I
=
= 2
∆k ω I1 cI1
(2.52)
The partial derivative of phase velocity with respect to model S-wave velocity
(fixed frequency and density) and with respect to model density (fixed frequency and
model s-velocity) are given as
53
[
∂c
]ω , ρ =
∂β
dl1 2
) )
c(k 2 dI 2 + dI 3 )
dz
=
∞
β k 2 I2
µ k 2 ∫ µ l12 dz
ρ c β (k 2 µ l12 + µ (
(2.53)
0
[
cl12ω 2
c
c dI 2 2
∂c
]ω , β = −
= − 2 l12ω 2 = − 2
ω
∞
2k I 2
2k I 2 µ
∂ρ
2k 2 µ l 2 dz
∫
0
(2.54)
1
For a Rayleigh wave, the energy integrals are defined as
1 ∞
1 ∞
2
2
ρ
+
=
(
)
((λ + 2 µ ) r12 + µ r22 )dz
r
r
dz
I
1
2
2
∫
∫
0
0
2
2
∞
dr
dr
dr
dr
1 ∞
I 3 = ∫ (λ r1 2 − µ r2 1 ) dz
I 4 = ∫ (λ + 2µ )( 2 ) 2 + µ ( 1 ) 2 )dz
0
0
2
dz
dz
dz
dz
The group velocity in terms of integrals is given as
I1 =
U=
∆ω
=
∆k
I2 +
I3
2k
cI1
(2.55)
(2.56)
The partial derivative of phase velocity with respect to model S-wave velocity (fixed
frequency, density, and model p-wave velocity) and with respect to model density (fixed
frequency, model s-wave velocity, and model p-wave velocity) are given as
[
∂c
1 
1
2k 2 4k 1 
]ω , ρ ,α = 2
(kr2 ) 2 + I 42 −
I −
I

∂β
µ
µ 3 λ 3 
4k UI1 
[
∂c
1
ω2
α 2 k 1 1 1 2kα 2 1
]ω , β ,α = 2 [ I1 −
I − I −
I ]
ρ 1 ρ 4
λ 3
∂ρ
4k UI1 ρ
(2.57)
(2.58)
2.11 Published calculation codes
While many codes to calculate dispersion curves have been developed as part of
research, they are generally not published. Based on the transfer matrix method,
Herrmann and Ammon (2002) developed a FORTRAN code to calculate the phase
54
velocity dispersion curves of surface waves. The code is part of package called Computer
Programs in Seismology (CPS). The CPS also provides modal summation synthetic
seismograms. Yuehua Zeng implemented R/T methods using FORTRAN for calculating
the phase velocity dispersion curves of surface waves. Saito’s code (1988), original in
FORTRAN and later adopted in C by Yuehua Zeng, is based on numerical integration.
Matlab code for isotropic dispersion is listed in Chapman (2003).
2.12 RTgen
Our own code, called RTgen, is based on the R/T method. It is written in C and
has been commercialized through SeisOpt® ReMi™ from Optim Inc. For both Rayleigh
and Love waves, RTgen is able to calculate
1). Phase and group velocities from fundamental mode up to any higher mode
that potentially exists;
2). Partial derivative of phase velocity with respect to model S-wave velocity
(fixed frequency, density, and/or model p-wave velocity) and with respect to model
density (fixed frequency, model s-wave velocity, and/or model p-wave velocity);
3). Coefficients of displacement and traction or normalized eigendisplacements
and eigentractions.
ReMi is using Saito’s code (1988) as its forward engine. The code uses a
numerical integration method for computing dispersion curves of surface waves. It is
adapted from Fortran to C by Yuehua Zeng of Nevada Seismological Laboratory in 1992.
Unlike RTgen, the adapted Saito’s code can only produce the phase velocity dispersion of
the fundamental-mode Rayleigh waves.
55
incident SH
transmitted SH
reflected SH
Interface
incident SH
transmitted SH
(a)
Interface
x
(b)
z
x
reflected SH
z
Figure 2.1. Illustration of coefficients of reflection and transmission due to SH incident
down (a) and up (b) to an interface.
incident SV
transmitted SV
reflected SV
transmitted P
reflected P
Interface
Interface
x
transmitted P
reflected P
incident SV
transmitted SV
(a)
(b)
z
incident P
reflected SV
z
transmitted SV
reflected SV
reflected P
Interface
transmitted P
Interface
x
transmitted P
transmitted SV
(c)
z
x
x
reflected P
incident P
(d)
reflected SV
z
Figure 2.2. Illustration of coefficients of reflection and transmission
due to SV incident down (a) and up (b) to an interface and P incident
down (c) and up (d) to an interface.
56
free surface
x
μ1, λ1, ρ1
μ2, λ2, ρ2
μj-1, λj-1, ρj-1
z (1)
:
≈:
μj, λj, ρj
z (2)
z (j-1)
z (j)
μj+1, λj+1, ρj+1
μN-1, λN-1, ρN-1
z (0)
:
≈:
μN, λN, ρN
z (j+1)
z (N-1)
z (N)
μN+1, λN+1, ρN+1
z
Figure 2.3. Configuration and coordinate system of a multiplelayered half-space.
57
4.2
1200
4
1000
model 1____
3.8
800
____model 2
3.4
Vs (m/s)
Vs (km/s)
3.6
3.2
3
____model 5
600
____model 6
400
____model 4
____model 3
2.8
200
2.6
2.4
0
20
40
60
Period (s)
80
0
100
0
0.2
(a)
0.4
0.6
Period (s)
0.8
1
(b)
Figure 2.4. Phase velocity dispersion curves of the fundamental-mode Rayleigh waves for large scale
models 1, 2, and 3 (a) and small scale models 4, 5, 6 (b). In (a) the crosses are phase velocities calculated by RTgen and the circles by CPS. In (b) the crosses are phase velocities calculated by RTgen and
the circles and diamonds by CPS. Note that models 5 and 6 are closely overlapped.
x 10
−5
12
0.8
10
0.6
8
0.4
6
0.2
4
(Vs1−Vs2)/Vs2
(Vs1−Vs2)/Vs2
1
0
−0.2
0
−2
−0.6
−4
−0.8
−6
0
20
40
60
Period (s)
(a)
80
100
−5
2
−0.4
−1
x 10
−8
0
0.2
0.4
0.6
Period (s)
0.8
(b)
Figure 2.5. The normalized errors between phase velocities calculated by RTgen and CPS of
large scale models 1, 2, and 3 (a) and small scale models 4, 5, 6 (b). Vs1 of Y-axis in both
graphs represents the phase velocities calculated by CPS and Vs2 by RTgen. The normalized
errors fall into 0.001% in (a) and 0.01% in (b).
58
1
6.5
700
6
600
5.5
5
Vs (m/s)
Vs (km/s)
500
4.5
300
4
200
3.5
3
400
0
20
40
60
Period (s)
80
100
Figure 2.6. Phase velocity dispersion curves
of Rayleigh waves for the Gutenberg model.
The crosses are phase velocities from fundamental up to the 14th higher mode calculated by RTgen. The circles are the corresponding mode phase velocities by CPS.
100
0
0.2
0.4
0.6
Period (s)
0.8
1
Figure 2.7. Phase velocity dispersion curves of
Rayleigh waves for model 4. The crosses are
phase velocities from fundamental up to the
6th higher mode calculated by RTgen. The
circles are the corresponding mode phase
velocities by CPS.
59
160
RTgen
RTmod
140
Figure 2.8. Computational time against number of
layers in models. The solid line represents computational time of 20 iterations of phase velocity calculation of fundamental-mode Rayleigh waves taken by
RTgen, and the dash line by RTmod. Clearly, the fast
generalized R/T method cuts the computational time
at least by half.
Computational Time (s)
120
100
80
60
40
20
5
10
15
Number of Layer
20
6.5
700
6
600
5.5
500
Vs (m/s)
Vs (km/s)
0
5
400
4.5
300
4
200
3.5
0
20
40
60
Period (s)
80
100
Figure 2.9. Phase velocity dispersion curves
of Love waves for the Gutenberg model.
The crosses are phase velocities from fundamental up to the 14th higher mode calculated by RTgen. The circles are the corresponding mode phase velocities by CPS.
100
0
0.2
0.4
0.6
Period (s)
0.8
Figure 2.10. Phase velocity dispersion
curves of Love waves for model 4. The
crosses are phase velocities from fundamental up to the 6th higher mode calculated by RTgen. The circles are the corresponding mode phase velocities by CPS.
60
1
Chapter 3 Linearized inversion of surface-wave dispersion
Solutions of geophysical inversion problems are not unique in the sense that there
are many models that explain the data equally well (Tarantola, 1987). There are a number
of reasons for this. Firstly, the theoretical error occurs due to use of an inexact theory in
the prediction of theoretical data. For example, the use of 1D model could cause
theoretical errors since in most cases the earth is truly 3D. Secondly, the observational
errors are ubiquitous. They could be measurement errors, instrumental errors, numerical
truncations. The third reason is the most fundamental. In many inverse problems the
model that one aims to determine is a continuous function of the space variables. This
means that the model has infinitely many degrees of freedom. However, in a realistic
experiment the amount of data that can be used for the determination of the model is
usually finite. A simple count of variables shows that the data cannot carry sufficient
information to determine the model uniquely. In the context of linear inverse problems
this point has been raised by Backus and Gilbert (1967, 1968) and more recently by
Parker (1994). This issue is equally relevant for nonlinear inverse problems.
The model obtained from the inversion of the data is therefore not necessarily
equal to the true model that one seeks (Snieder and Trampert, 1999). This implies that for
realistic problems, inversion really consists of two steps. Let the true model be denoted
by m and the data by d. From the data d one reconstructs an estimated model m̂ , this is
called the estimation problem (Fig. 3.1). Apart from estimating a model m̂ that is
consistent with the data, one also needs to investigate what relation the estimated model
61
m̂ bears to the true model m. In the appraisal problem one determines what properties of
the true model are recovered by the estimated model and what errors are attached to it.
Therefore, inversion = estimation + appraisal. It does not make much sense to make a
physical interpretation of a model without acknowledging the presence of errors and
limited resolution in the model (Trampert, 1998).
There are many excellent textbooks on inversion problems (e.g. Aster et al., 2005;
Tarantola, 2005; Snieder and Trampert, 1999; Park, 1994; Menke, 1989). Before
discussing the case of the surface wave data inversion, some of the basic concepts of the
inversion will be briefly presented to introduce the tools that will be used later.
3.1 Linear model estimation
Suppose that we have a discrete n-point model m and discrete m-point data
vector d that in practice are contaminated with errors e. The recorded data and the model
are related through some fundamental physics by a linear system of equation
d = Gm + e
(3.1)
where G is the data kernel matrix.
From the recorded data one makes an estimate of the model m̂ . There are many
ways for designing an inverse operator that maps the data on the estimated model (e.g.
Menke, 1989; Tarantola, 1987; Parker, 1994). Whatever estimator one may choose, the
most general linear mapping from data to the estimated model can be written as:
ˆ = G−gd
m
(3.2)
The operator G-g is called the generalized inverse of the matrix G. In general, the
number of data is different from the number of model parameters. For this reason G is
62
usually a non-square matrix, and hence its formal inverse does not exist. Later we will
show how the generalized inverse G-g may be chosen, but for the moment G-g does not
need to be specified. The relation between the estimated model m̂ and the true model m
follows by inserting (3.1) in expression (3.2):
ˆ = G − g Gm + G − g e
m
(3.3)
The matrix operator G-gG, called the resolution kernel R, gives
R ≡ G−gG
(3.4)
Equation (3.3) can be interpreted by rewriting it in the following form:
−g
ˆ = m + (G − g G − I )m + G
m
e
ErrorN
Pr opagation
(3.5)
lim ited Re solution
ˆ =m ,
In the ideal case, the estimated model equals the true model vector: m
meaning that our chosen parameters, specified in vector m, may be estimated
independently from each other. The last two terms in equation (3.5) account for
“blurring” and artifacts in the estimated model. The term (G − g G − I )m describes the fact
that components of the estimated model vector are linear combinations of different
components of the true model vector. We only retrieve averages of our parameters and
“blurring” occurs in the model estimation as we are not able to map out the finest details.
In the ideal case this term vanishes; this happens when G-gG is equal to the identity
matrix (the perfect resolution: R ≡ I ). The trace of R ( Tr(R) )is often used as a simple
quantitative measure of the resolution. If Tr(R) is close to the number of layer of models,
then R is relatively close to the identity matrix.
The last term in equation (3.5) describes how the errors e are mapped onto the
estimated model. These errors are not known deterministically (otherwise they could be
63
subtracted from the data). A statistical analysis is needed to describe the errors in the
estimated model due to the errors in the data. When the data dj are uncorrelated and have
standard deviation σdj , the standard deviation σmi in the model estimate m̂ , resulting
from the propagation of data errors only, is given by
σ mi2 = ∑ (Gij− gσ dj2 )2
(3.6)
j
where i = 1, 2, …, n and j = 1, 2, …, m.
Ideally, one would like to obtain both: a perfect resolution and no errors in the
estimated model. Unfortunately this cannot be achieved in practice. The error
propagation is, for instance, completely suppressed by using the generalized inverse G-g =
ˆ = 0 which is indeed not affected by errors.
0. This leads to the (absurd) estimated model m
However, this particular generalized inverse has a resolution matrix given by R=0, which
is far from the ideal resolution matrix. Hence in practice, one has to find an acceptable
trade-off between error-propagation and limitations in the resolution.
3.2 Solving a linear system
Considering a discrete linear problem we begin with a data vector d of m
observations and a model vector m of n model parameters that we wish to determine. The
inversion problem can be written as a linear system of equations
Gm = d
(3.7)
For this simple case, the solutions to equation (3.7) can be obtained with different
approaches from different viewpoints: the length method, the generalized inverse, the
64
maximum likelihood method (Menke, 1989). The follow are the most commonly used
methods.
3.2.1 Least-squares estimation
In some problems the number of unknowns is more than the number of
parameters (over-determined problems). A common way to estimate a model is to seek
the model m̂ that gives the best fit to the data in the sense that the difference, measured
ˆ is made as
by the L2-norm, between the data vector d and the recalculated data Gm
small as possible. This means that the least-squares solution is given by the model that
minimizes the following cost function
S =|| d − Gm ||2
(3.8)
This quantity is minimized by the following model estimate (Aster et al., 2005)
ˆ = (G T G ) −1 G T d
m
(3.9)
3.2.2 Minimum norm estimation
In some problems the number of unknowns is less than the number of parameters
(under-determined problems). The minimum norm solution is defined as the model that
fits the data exactly, Gm = d, and that minimizes || m ||2 . As shown in detail by Menke
(1989) the minimum-norm solution is given by:
ˆ = G T (GG T ) −1 d
m
(3.10)
3.2.3 Mixed determined problems
In the least-squares estimation, we assumed that we had enough information to
evaluate all model parameters, even though contradictions occurred due to measurement
errors. The problem is then purely over-determined and as a consequence GTG is regular,
65
which means its inverse matrix always exists. In the minimum norm solution, we
assumed no contradictions in the available information, but we don't have enough
equations to evaluate all model parameters. This is the case of a purely under-determined
problem and here GGT is regular. The most common case, however, is that we have
contradictory information on some model parameters, while others cannot be assessed
due to a lack of information. Then neither GTG nor GGT can be inverted and the problem
is ill-posed. Even if the inverse matrices formally exist, they are often ill-conditioned
meaning that small changes in the data vector lead to large changes in the model
estimation. This means that errors in the data will be magnified in the model estimation.
Clearly a trick is needed to find a model that is not too sensitive on small changes in the
data. To this effect, Levenberg (1944) introduced a damped least-squares solution
ˆ = (G T G + α I) −1 G T d
m
(3.11)
2
Equation (3.11) minimizes the cost function G (m) − d + α m
2
(3.12)
3.3 Regularization
The initial system of equation that we are solving is equation (3.7) is not quite
correct. We ignore the error e which will always be present. To compensate errors, we
transform the model parameters through matrix L
and the data vector through Q
G
d = Qd
G
m = Lm
(3.13a )
(3.13b)
Assume that L has an inverse. The transformed system of equations can be
derived by substituting equation (3.13) into equation (3.7)
G G
QGL−1m = d
(3.14)
66
Damped least-squares solution for equation (3.14) is then given as
ˆ = (G T QT QG + α LT L) −1 G T QT Qd
m
(3.15)
Let QT Q = Wd and LT L = Wm , equation (3.15) is then given by
ˆ = (G T Wd G + α Wm ) −1 G T Wd d
m
(3.16)
This solution minimizes the following cost function:
S = (d − Gm)T Wd (d − Gm) + α mT Wmm
(3.17)
This expression shows that in general the weight matrices Wd and Wm can be
anything. Written in this way, α may be seen as a trade-off parameter which compromises
between two characteristics of the model: its size and its disagreement with the data. Both
independent properties of the model cannot be arbitrary small simultaneously. Hence
there is a need for a balance. The choice of an optimum, however, is not an easy job.
There is no reason to favor the damped least-squares solution (3.11) over the more
general least-squares solution (3.16) (Snieder and Trampert, 1999). In fact, most inverse
problems are ill-posed (partly underdetermined and partly over-determined) and illconditioned (small errors in the data causes large variations in the model) which goes
hand in hand with large null-spaces and hence non-unique solutions. Regularization is
thus needed, but there is a large ambiguity in its choice (Scales and Snieder, 1997). This
reflects the fundamental difficulty that one faces in solving inverse problems: solving the
system of equations is a minor problem, compared to choosing the regularization.
One approach is to define the misfit function in such a way that it favors models
with given properties (for example the smoothest model) (Parker, 1994). As an example
of the choice of the weighting matrices Wm, the use of Occam's inversion is quite
67
common (Constable et al., 1987) where one seeks the smoothest model that is consistent
with the data. Instead of putting a constraint on the model length, one seeks the square of
its gradient to be as small as possible.
Smoothing model is implemented through the Tikhonov regularization. For
simplification we assume Q = I, the equations (3.16) and (3.17) could be written as
ˆ = (G T G + α LT L) −1 G T d
m
2
S = d − Gm +α 2 Lm
(3.18) and
2
(3.19)
where L is called the roughening (smoothening) matrix and α is the damping factor.
L could take several forms.
L=I
(3.20a ) called the zero-order Tikhonov regulization
0 0

 −1 1




L=
−1 1


"



−1 1 n×n
0 0 0
0 0 0

L = 1 -2 1

"


(3.20b) the first-order Tikhonov regulization






1 −2 1 n×n
(3.20c) the second-order Tikhonov regulization
3.4 Singular value decomposition (SVD)
Conveniently, every matrix has a singular value decomposition (SVD). An m by
n matrix G could be factored into
G m×n = U m×mS m×n VnT×n
(3.21)
68
where U is an m by m orthogonal matrix with columns that are unit basis vectors
spanning the data space Rm. V is an n by n orthogonal matrix with columns that are unit
basis vectors spanning the model space Rn. S is an m by n diagonal matrix with
nonnegative diagonal elements called singular values and S1 ≥ S2 ≥ " ≥ S min ≥ 0 . If only
S
the first p singular values are nonzero, we can partition S as S =  p
0
and G as G =  U p
S
U 0   p
0
0
 Vp
0  
V0 
T
0
0 
(3.22b)
We can simplify the SVD of G into its compact form G = U p S p VpT
The
generalized
G † = Vp S −p1UTp
inverse
of
G,
(3.22a)
called
Moore-Penrose
(3.23)
pseudoinverse,
is
(3.24)
And the pseudoinverse (generalized inverse) solution is
m † = Vp S −p1UTp d
(3.25)
The pseudoinverse solution is a least squares solution. But, unless p=n, the solution is
biased. The covariance of the solution is cov(m ) = σ
†
The model resolution matrix is R m = G †G = Vp VpT
The data resolution matrix is R d = GG † = U p UTp
2
p
V.,iV.,Ti
i =1
si2
∑
(3.26)
(3.27 a )
(3.27b)
3.5 Proposed linear inversion algorithm
This dissertation uses Occam’s inversion technique (Constable et al., 1987) with
a higher-order Tikhonov regulization. The proposed inversion procedure contains two
69
fundamental components. First, an algorithm (forward engine) is required to construct a
theoretical dispersion curve based on the properties of an assumed profile. Second, an
algorithm is required to minimize the objective function which is usually the error
between the theoretical and experimental dispersion curves plus a damping term. Table
3.1 summarized the linear inversion methods of shallow surface waves used by major
research groups. Different groups used different forward engines. But all of them used
the generalized linear inversion algorithm by Wiggins (1972) with different damping
terms, iteration methods, and methods to calculate the partial derivative matrix (Jacobian
matrix). The advantage and disadvantage of these methods will be discussed in the next
sections.
Table 3.1 Linearized inversion methods of surface waves used by major research groups
Reference
Horike
(1985)
Ganji et al.
(1998)
Xia et al.
(1999)
This study
Forward
engine
Numerical
integral
Stiffness
matrix
Tranfer
matrix
R/T
method
Dispersion
method
Inversion method
Partial
derivatives
Data
errors
FK
Newton
Analytic
Neglected
Numerical
Neglected
Numerical
Neglected
Analytic
Modelled
SASW
MASW
Combined Newton
and quasi-Newton
LevenbergMarquardt (LM)
ReMi
Occam
Suppose that we have a discrete n-point model m and discrete m-point data
vector d that are related by a nonlinear system of equation G (m) = d
(3.28)
We can formulate this problem as a damped least squares problem. In another words, we
2
minimize G (m) − d 2 + α 2 Lm
2
2
(3.29)
where L is the roughening matrix and α is the damping factor.
70
As the first step of inversion process, we need linearize the nonlinear equation
(3.28). Given a trial model mk, Taylor’s theorem is applied to obtain the local
approximation
G (m k + ∆m) ≈ G (m k ) + J (m k )∆m = d
(3.30)
where J (m k ) is the Jacobian matrix J (m k ) =


k
∂G1 ( m k ) 
 ∂G1 ( m )
"


∂m1
∂mn




#
%
#


 ∂G ( m k )
k
∂G m ( m ) 
 m
"


∂m1
∂mn


(3.31)
Using the new R/T method described in Chapter 2, the Jacobian matrix is analytically
calculated (equations (2.57) and (2.58) ).
Using equation (3.30), the damped least squares problem (equation (3.29) ) is to
minimize
2
2
2
2
G (m k ) + J (m k )∆m − d + α 2 L(m k + ∆m)
(3.32)
where the variable is ∆m and m k is constant. Reformulating this as a problem in which
the variable is m k +1 = m k + ∆m
(3.33)
and letting d (m k ) = d − G (m k ) + J (m k )m k
(3.34)
2
2
2
2
give J (m k )(m k + ∆m) − (d − G (m k ) + J (m k )m k ) + α 2 L(m k + ∆m)
2
or minimize J (m k )m k +1 − d (m k ) + α 2 Lm k +1
2
2
2
(3.35)
(3.36)
Because J (m k ) and d (m k ) are constant, equation (3.36) is a damped least squares
problem J (m k )m k +1 = d (m k )
(3.37)
The solution is given as m k +1 = (J (m k )T J (m k ) + α 2 LT L) −1 J (m k )T d (m k )
(3.38)
71
The damped least squares solution given by equation (3.38) assumes Q = I
(equation (3.14) to (3.17) ). To incorporate the data standard deviation σ dj into a solution,
I use a weighting matrix Q = diag (1/ σ 1 ,1/ σ 2 ," ,1/ σ m )
(3.39)
The damped least squares solution (3.38) would be in the form of
ˆ k +1 = ((QJ (m k ))T (QJ (m k )) + α 2 LT L) −1 (QJ (m k ))T Qd (m k )
m
(3.40)
Fig. 3.2 shows the flowchart of the Occam’s inversion which implemented as
follows:
1). Set an initial model (S-wave and P-wave velocities, thickness, and density);
2). Calculate the theoretical dispersion curves G(mk) and Jacobian matrix J(mk) of the
current model;
3). Set a range of damping parameters (α0, α1, α2, …, αN), usually logarithmic;
4). For each damping parameters αi
a) calculate and save a new model mk+1(αi) based on equation (3.40);
b) and the corresponding RMS errors RMSk+1(αi);
5). Select the model mk+1(αj) based on the following criterions: If two or more trial
models give RMS errors below a user-specific threshold ε1, the largest α is preferred.
If no such value exists, then pick a value of αj that minimizes the RMSk+1(αj);
6). Check the model mk+1(αj) for convergence, in another words, check that the
RMSk+1(αj) is smaller than a user-specific threshold ε2.
7). If the model does not converge, pass the model to next iteration (Step2) until
maximum iterations reached.
72
3.6 Model appraisal method
In a surface wave survey (like ReMi survey), uncertainty is related to the
dispersion calculation (theory errors), the manual picking of dispersion curves, and the
data measurements (Table 3.2). The first is caused by improper application of dispersion
theory from 1D model into real 3D earth geologies. The second is due to transformation
from time-offset domain to frequency-slowness domain and the human bias during
dispersion curve picking from ReMi p-f image. The human bias may be avoided by
averaging dispersion curve picks from several analysts. The last category (measurement
uncertainties) is due mainly to noise in the recorded signals and to instrument
uncertainties.
Table 3.2 Sources of uncertainty in surface wave dispersion measurements
Theoretical errors
Picking errors
Instruments
Measurement errors
Signal noise
Correlated
Uncorrelated
The instrument uncertainties are related to limited frequency range of geophones,
orientation and spatial spacing of the array, limited recording length, and others. Using
numerical simulations, O’Neill (2003) reports minimal influence from geophone tilt and
coupling, while adding noise in the recorded signals introduce larger uncertainties in the
experimental dispersion. By the data gathered at two sites in Italy, Lai et al. (2005)
experimentally verified that the experimental dispersion data are normally distributed.
Uncertainties in recorded signals are associated with coherent noise and
uncorrelated noise. The latter is externally generated noise (environmental noise) and can
73
be studied via the statistical distribution of the recorded signals if many repetitions of the
test in a given configuration are available. Coherent noise is due to events generated by
the seismic source (e.g., near-field effects) or coherent noise associated with ambient
noise. The proposed approach by this study can be used to estimate the effect of
uncorrelated noise on surface wave measurements and its propagation in inversion.
It can be shown that if an inverse problem is nonlinear, a Gaussian distribution of
the errors in the data will in general be mapped into a distribution of the uncertainty of
the model parameters that is non-Gaussian (Tarantola, 1987; Menke, 1989). However, if
the inverse problem is not too non-linear, especially around the point of maximum
likelihood in the model space (i.e., the probability distribution of model parameters
around the point of maximum likelihood is not too different from a Gaussian function), it
is possible to use a simplified approach to estimate the uncertainty of the model
parameters (Lai et al., 2005).
If in solving the non-linear inverse problem G (m) = d , it is assumed that: a) the
uncertainty of the experimental dispersion curve follows a Gaussian distribution; b) the
inverse problem (equation (3.37) ) is only moderately non-linear around its solution, and
c) the relation G (m) = d is inverted using the method of maximum likelihood, then the
uncertainty associated with the expected shear wave velocity profile can be approximated
by the following formula (Tarantola, 1987; Menke, 1989):
ˆ ) = cov(Hd ) = H cov(d )HT
cov(m
(3.41)
From equation (3.40), we obtain the
H = ((QJ (m k ))T (QJ (m k )) + α 2 LT L) −1 (QJ (m k ))T Q
(3.42)
74
ˆ ).
The standard deviations of each model parameter are the root of diagonal of cov(m
Assuming a normal distribution associated with model parameter mi, the 95% confidence
ˆ ± 1.96 diag (cov(m
ˆ ))
intervals are given by m
(3.43)
The concept of model resolution is an important way to characterize the bias of
the linear inversion. For a linear system (equation (3.7) ), the model resolution matrix R
is defined as equation (3.4). The rows of R are the resolution kernels for each parameter.
The difference from the identity matrix illustrates the degree of non-uniqueness of each
parameter. The R would be the identity matrix if all parameters are perfectly resolved
(uniquely determined). The trace of R is often used as a simple quantitative measure of
the resolution. If the trace of R is close to total number of parameters estimated, R is
relatively close to the identity matrix.
Substituting d (m k ) of equation (3.37) into equation (3.40) yields
ˆ k +1 = ((QJ (m k ))T (QJ (m k )) + α 2 LT L) −1 (QJ (m k ))T QJ (m k )m k +1
m
(3.44)
Therefore, the model resolution matrix of the linearized equation (3.37) is
R = ((QJ (m k ))T (QJ (m k )) + α 2 LT L) −1 (QJ (m k ))T QJ (m k )
(3.45)
3.7 Test data sets and numerical tests
Sheriff (1991) sets out four canonical types of earth resistivity profiles. By
analogy we can generalize the velocity profiles into the four similar types. An A-type
velocity profile is a section where Vs increases with depth at all interfaces; H-type has a
low-velocity layer; K-type has a high-velocity layer; and a Q-type where Vs decreases
with depth at all interfaces. Q-type profiles may be encountered in evaporate settings.
75
However, in our collecting of shallow Vs profiles at hundreds of sites, we have not yet
observed a Q-type profile. Thus we do not test that type here.
The proposed Occam’s linearized inversion method described above was tested
on a suite of nine synthetic models and two field data sets. Nine synthetic models were
designed based on data collected in southern Nevada. Seven models are A-type sections,
one is H-type, and one is K-type. These synthetic models have been used to test the ReMi
technique (Heath et al., 2006). The synthetic trace data were generated using the finite
difference code, E3D (Larsen and Grieger, 1998), and recorded on a virtual 185m-long
linear array with a 15m-spacing and a 200 sample-per-second rate. The E3D is an explicit
2D/3D elastic finite-difference wave propagation code that has been successfully used for
the modeling of seismic waves (Larsen and Grieger, 1998; Liu et al., 2005). Flat-layered
models and a Ricker wavelet source placed at the free surface were used. The wavefield
was simulated on a 2-D grid (dt = 0.0001 s and dh = 0.0005 km) with the Courant
condition and Clayton-Engquist absorbing boundary condition (Clayton and Engquist,
1977, 1980) satisfied. The shortest wavelengths in the calculation cover 25 grid points to
completely avoid grid dispersion. As suggested by Park et al. (1999), near-offsets
(distance between source and first receiver) are given to be greater than half the
maximum desired wavelength to allow the development of surface waves. No noise is
added to the trace data. Figure 3.3 shows an example of a synthetic seismic record.
Clearly Rayleigh waves dominate the record. The ReMi dispersion curves are
independently picked from 2.9 Hz to 20.5 Hz in frequency. Figure 3.4 shows a typical
ReMi dispersion picks on the slowness-frequency (p-f) image.
76
The linearized inversion of the synthetic Rayleigh-wave phase velocity dispersion
curve picks is carried out by inverting for S-wave velocity and fixing density and
Poisson’s ratio at the original values (2.0 g/cm3 and 0.25 of the original synthetic models,
respectively) even though the Poisson’s ratio is low for soils. It is well-accepted that Vs
has the most dominant effect on the Rayleigh wave dispersion followed by layer
thickness. This has been shown by numerical partial derivatives at both crust-mantle
scale (Burkhard and Jackson, 1976) and local site scale (Xia et al., 1999). Xia et al. (1999)
showed that a 25% error in Vp or density of all layers only induces a 10% perturbance of
the fundamental-mode dispersion curves. Therefore constant density and Poisson’s ratio
will not significantly affect results of the inversion.
Due to unavailability of any a prior geological information for the synthetic
simulation, initial models are blindly constructed in the following way for all synthetic
test data. The number of layers (N), Poisson ratio, the maximum depth (Dmax), the
maximum (Vmax) and the minimum (Vmin) S-wave velocities are users-specified
quantities. The velocity gradient is defined as
∇V = (Vmax - Vmin ) / Dmax
(3.46)
The initial model is composed of layers of equal thickness of Dmax/N. The calculated
velocity from the velocity gradient at the middle point of each layer is taken as S-wave
velocity of that layer. P-wave velocity is updated from the corresponding S-wave velocity
based on the given Poisson ratio. Density is fixed at 2.0 g/cm3.
Based on the initial model described above and assuming no data noise (equation
(3.38) ), the inverted S-wave velocity profiles are plotted atop nine original synthetic
models (left column, Fig. 3.5). The 95% confidence intervals of S-wave velocity of each
77
layer are calculated from equation (3.43) and plotted as horizontal error bars. Fig. 3.5 also
shows the resolution matrix and calculated dispersion of the corresponding model against
ReMi dispersion picks. The resolution matrix R describes how the term G − g of equation
(3.5) smears out the original true model, m, into an estimated model, m̂ . In these
synthetic tests, if the trace of R ( Tr(R) ) is close to three (the number of layer of the
synthetic models), then R is relatively close to the identity matrix (Fig. 5.3).
The matches between the inverted models and original synthetics are not bad.
There are some bad-match cases (for example model 5, Fig. 3.5). Almost all of the first
layers are resolved as shown on left column of Fig. 3.5 as well as the resolution matrix
whose first elements are almost units for all nine synthetics. The H-type and K-type
models are generally recovered. High velocity layer of model 5 clearly is an inversion
artifact probably due to bad assignment of layer thickness.
On the contrast, the traces of R ( Tr(R) ) of almost all inverted models are close to
three, indicating a good model resolution. The discrepancy between bad model matches
and good model resolutions is caused by the inversion errors (from equation (3.5) ),
caused by linearization, picking, instruments, and correlated and uncorrelated signal
noise sources (Table 3.2). The last section of this chapter shows that incorporation of the
uncorrelated signal noise into inversion cause a little change in final models, indicating
that errors by linearization, picking, instruments, and correlated signal noise sources
causes the major uncertainties in the final inverted models.
The inverted synthetic models can also be evaluated by how well they recover the
original values of Vs averaged to 30 (V30), 50 (V50), and 100 (V100) m as shown in Fig.
3.6 and tabulated in Table 3.3 in which these values have been rounded to the one
78
decimal place. V30 is a major factor in prediction of earthquake amplification and site
response in sedimentary basins (Field et al., 1992). These values are well recovered.
Average V30 error is 5% over all nine synthetic models. Average V50 and V100 errors are
7% and 6%, respectively, over all nine synthetic models.
Table 3.3. Depth-averaged velocities in m/s for Occam’s inverted
models and percentage difference from known profiles in parentheses
Models
V30
V50
V100
model 1
479 (2%)
539 (1%)
607 (1%)
model 2
557 (4%)
630 (1%)
700 (1%)
model 3
609 (6%)
651 (11%)
838 (1%)
model 4
943 (7%)
1096 (14%)
1340 (13%)
model 5
519 (6%)
599 (7%)
622 (1%)
model 6
471 (4%)
502 (7%)
544 (10%)
model 7
873 (10%)
997 (8%)
1228 (9%)
model 8
449 (6%)
449 (8%)
491 (10%)
model 9
912 (1%)
1070 (6%)
1127 (10%)
Newhall
320 (19%)
365 (4%)
464 (3%)
CCOC
216 (5%)
258 (6%)
336 (11%)
Two field data sets were used to test the linearized inversion. One of them was
obtained at the Los Angeles County Fire Station in Newhall, California by the Resolution
of Site Response Issues from the Northridge Earthquake (ROSRINE) project
(http://geoinfo.usc.edu/rosrine). Another is from the Coyote Creek borehole (CCOC) in
Santa Clara Valley, California (Stephenson et al., 2005). Both data sets include OYO
suspension S-wave velocity logs (see Chapter 1 for OYO technique).
79
The Newhall ReMi experiment was performed using a 200m-long array of
twenty-four 8-Hz vertical geophones centered 4 m from the hole. The geophones
recorded the vertical component of microtremors generated by nearby trains and street
traffic. The picks are independently made from 2.44 Hz to 22.22 Hz. The ReMi survey at
CCOC was conducted by USGS (Wentworth and Tinsley, 2005; Asten et al., 2005c)
using a 220m-long array of forty-five, 4.5 Hz vertical geophones, which is located 200m
southwest of the CCOC borehole. The ReMi picks are independently made from 1.22 Hz
to 24.66 Hz (Fig. 3.7). In both data sets, all acquired records were used in the p-f analysis
unless amplitudes within a given record were clipped.
Inversion starts with an initial model generated in the same way as the synthetic
cases described above. The maximum modeling depths were estimated using the halfmaximum-wavelength discussed by Park et al. (1999). This criterion has been used by
Xia et al. (1999), and Stephenson et al. (2005). Another critical issue in dispersion
inversion is the number of layers. The more layers, the more computation effort, and the
more realistic the inverted models (Dal Moro et al., 2007). Based on synthetic experience,
the number of layers for both tests is set as 6.
Fig. 3.8 displays the inverted models for both the Newhall and CCOC data. The
inverted S-wave velocity for the Newhall data follows the OYO suspension S-wave
velocity log. The significant increase of the S-wave velocity at a depth of about 20 m is
shown on the inverted model. Fits of the calculated dispersion curve against the ReMi
picks is generally good (row 1 column 2 in Fig. 3.8). The traces of R ( Tr(R) ) of the
inverted model is 3.96 out of 6.0. Average Vs errors of the Newhall data are 7% for V30,
4% for V50, and 5% for V100 (Fig. 3.6 and Table 3.3).
80
The maximum depth of the inverted profile for the Coyote Creek data is over two
times deeper than that of the Newhall data. The inverted S-wave velocity of the Coyote
Creek data loosely follows the OYO suspension S-wave velocity log. Fits of the
calculated dispersion curve against the ReMi picks is good at low frequencies and bad at
high frequencies (row 2 column 2 in Fig. 3.8). The traces of R ( Tr(R) ) of the inverted
model is 4.16 out of 6.0. Average Vs errors of the CCOC data are 10% for V30, 5% for
V50, and 1% for V100 (Fig. 3.6 and Table 3.3).
3.8 Initial model dependence
The initial model has an important role on linearized inversions (Tarantola,
1987). A final inverted model determined by linearized inversions inherently depends on
an assumed initial model due to the existence of locally optimal solutions (Yamanaka and
Ishida, 1996). When an appropriate initial model can be generated using a priori
information about subsurface structure, linearized inversions may find an optimal
solution that is the global minimum of a misfit function. If a priori information is either
scant or unavailable, the inversion may find a local optimal solution. Luke et al. (2003)
showed that linear inversion yields excellent dispersion results for simple profiles.
However, for more complex profiles multiple solutions with equally good data fits are
possible.
As far as the initial model is concerned, two aspects have to be considered: the
number of layers and the S-wave velocity of each layer. Using Newhall data, I design two
cases to exam the initial-model-dependence: the effect of number of layers and the effect
of layer S-wave velocities.
81
Fig. 3.9 shows the effect of the number of layers using Newhall dispersion data.
The initial model is composed of equal thickness layers. All layer velocities are set to a
fixed value. Inversion with a few layers (for example 4 layers) cannot fit OYO
suspension S-wave log well. The initial models with the limited number of layers yields
high impedance contrasts between the layers (for example the first and second layer of
the four-layer-model, Fig. 3.9). When the number of layers is increased, the site
characteristics are better identified. The inversion with 10 layers (Fig. 3.9) shows a better
fit. This inverted model shows a quite smooth increase of shear velocity with depth. As
the number of layers is further increased, the uncertainties are generally increased
(horizontal bars is widening). There may be a risk of fitting the noise (for example, large
95% confidence interval for Newhall inversion with layer number more than 16, Fig.
3.9) . When the number of layers becomes high (20 layer, Fig. 3.9), the reliability of the
single shear velocities is decreased (due to larger uncertainty) and a trend between Swave velocity and depth is identified.
The effect of layer S-wave velocity is shown in Fig. 3.10. All initial models have
the same thickness but different S-wave velocities, yielding three different profiles
through linearized inversion. Both effects of the number of the layers and the S-wave
velocity of each layer indicate that the inverted profiles depend on the initial models.
3.9 The effect of minimum and maximum depth
The minimum depth, able to be modeled, depends on the minimum sampled
wavelength. If the acquisition high frequencies are reliably acquired, the minimum
wavelength that can be observed depends on the spatial sampling, and equals the
82
geophone spacing. The shortest propagating wavelength carries the information related to
the shallowest layer, and gives the mean properties of the layers above its investigation
depth. These mean properties can still be influenced by a thinner layer that is not
investigated: in this case it is necessary to incorporate a priori information about this
layer, or to acquire shorter wavelengths.
The maximum depth, able to be modeled, depends on the maximum reliably
estimated wavelength. As a rule of thumb, it is limited to one half of the maximum
wavelength (Park et al., 1999), but sometimes it is possible to go deeper, down to one
wavelength (Herrmann and Al-Eqabi, 1991). Actually the maximum investigated depth
depends on the site. A rigorous approach should give a result with its uncertainties, and
should also compare the results of different inversions with different numbers of layers
and depths.
The maximum depth has an important effect on inversion (Fig. 3.11). Using the
initial model that has equal layer thickness and the same velocities for all layers,
inversion with a shallower maximum depth yields a good match (left, Fig. 3.11). The
maximum penetration depth of the data is deeper than the user-specified value of the
maximum depth. If the user-specified maximum depth is deeper than the maximum
penetration depth of the data, the inverted models might not reflect the subsurface
geology (right, Fig. 3.11).
3.10 The effect of number of dispersion picks
The number of dispersion picks heavily influences the final model. Fig. 3.12
shows the Occam’s inverted results on Newhall data sets with different number of picks.
83
The inversion is performed assuming an exact initial model as one used in row 1 column
2 in Fig. 3.9. We randomly deleted 2 out of 14 total original picks (except the minimum
and maximum frequency picks) to make a 12-pick dispersion data; 6 to make a 8-pick
data; 8 to make a 6-pick data. Among three inverted models, the inversion on 12-pick
data (left, Fig. 3.12) yields the best fitting model in terms of how close the inverted
model follows the OYO S-wave log. It is followed by the inversion on the 8-pick data
(middle, Fig. 3.12) and the inversion on the 6-pick data (right, Fig. 3.12). Each pick of
the data contains the geological information on the subsurface S-wave structure. The
inversion of the less pick data sets yields models that are far away from the true profile
than the inversion of the more pick data sets. This means that more picks provide more
information on the underground geology.
3.11 The effect of frequency density of dispersion picks
In the different frequency bands the information density is different. This time,
we randomly deleted 4 out of 14 total original Newhall dispersion picks (except the
minimum and maximum frequency picks) to make a 10-pick dispersion data (called data
A). Then we delete the 4 lowest frequencies to make another 10-pick dispersion data
(called data B). The last 10-pick dispersion data (called data C) is made by deleting the 4
highest frequencies out of a total of 14 original picks.
Fig. 3.13 shows the Occam’s inverted results on these three data sets. Among
three inverted models, the inversion on data B (middle, Fig. 3.13) yields the worst fitting
model that the inverted S-wave velocities are far below the OYO log while the inversion
on data C produces the best model. This means that in the low frequency range the picks
84
are more important to the inversion. Each of them significantly increases the information
content of the dispersion picks. In the high frequency range, on the contrary, the picks are
less important. They actually replicate the same information. The implication is that more
picks are needed in the low frequency range than in the high frequency range.
3.12 The effect of the weighting matrix
The inverted models of all above tests are calculated following equation (3.38),
assuming an identity weighting matrix. This time we re-run the inversion on nine
synthetic data sets under the same condition as before except that the weighting matrix
are set to 10% of data value (σi=0.1*Vsi). The inverted models for this run are calculated
following equation (3.40). The inverted S-wave velocity profiles are plotted atop of the
nine original synthetic models and the inverted model using the identity weighting matrix
(Fig. 3.14). Clearly adding the weighting matrix of 10% of data value does not cause
much changes of the inverted model.
The less important effect of weighting matrix on the inverted models indicates
that the inverted model is not sensitive to the weighting matrix. In the phase velocity
inversion of surface waves, there are errors caused by in-exact forward modeling and
linearization, by manually picking of phase velocity dispersion curves, and by
measurement errors (Table 3.2). These errors may have more important effect than the
weighting matrix.
85
Forward problem
Data d
True model m
Appraisal problem
Estimation problem
Estimated model m
Figure 3.1 The inverse problem viewed as a combination of an estimation problem plus
appraisal problem.
86
starting m0
Forwarding G(m0)
and partials J(m0)
damping parameter
αi
Solve occam’s
solution
m0 = m1
All α tested?
i=0, 1, ..., N
N
Y
Select the best
damped model m1
N
Converged?
Y
m1 is the final solution.
Done!
Figure 3.2 Flow chart showing the Occam’s inversion procedure. After trialling all α, the best model is chosed based on the
following criterion: If two or more trial models give errors
below the threshold, the largest α is preferred. If no such value
exists, then pick a value of α that minimizes the RMS.
87
0.0
0
5
Trace Sequence
10
15
20
Rayleigh waves
1.0
Time (s)
2.0
3.0
4.0
Figure 3.3 A typical synthetic seismic
record with strong Rayleigh waves.
0.0
0.0
5.0
Frequency (Hz)
10.0
15.0
20.0
25.0
Slowness (s/m)
0.001
0.002
0.003
0.004
0.005
0.0
ReMi Spectral Ratio
2.5
Figure 3.4 Slowness-frequency spectrum (p-f) image
with ReMi dispersion picks of a typical synthetic
seismic record. The small squares indicate analyst’s
picks of phase velocities of the fundamental-mode
Rayleigh waves.
88
Model1
0
Model 1 w/ Tr(R)=2.97
Model1
600
20
30
500
40
2
450
400
50
3
350
60
200
400
600
Vs (m/s)
800
0.05
0.1
0.15
Period (s)
1
600
Vs (m/s)
30
40
550
2
500
450
50
400
600
800
Vs (m/s)
0.05
1000
0.1
Period (s)
0.15
Vs (m/s)
750
700
2
650
600
550
3
500
0.4
0.2
0
3
1
800
60
0.6
2
Model 3 w/ Tr(R)=2.99
850
40
0.8
1
Model3
Model3
20
Depth (m)
3
400
0
1
3
Model 2 w/ Tr(R)=2.96
650
20
80
400
2
700
10
60
1
0.2
Model2
Model2
0
Depth (m)
1
550
Vs (m/s)
Depth (m)
10
600
800 1000 1200 1400
Vs (m/s)
0.05
0.1
0.15
Period (s)
0.2
1
2
3
Figure 3.5 Linearized inverted S-wave velocities against the original synthetic models
for nine synthetic data sets. Left column lists the inverted the S-wave velocities (thick
solid lines) with 95% confidence interval (horizontal bars). The dashed lines represent
the original synthetic models. Middle column shows the original ReMi dispersion picks
(circles) against the calculated dispersion curves (thick solid lines) of the inverted
model listed in left column. Right column contains resolution matrix (scale is at the
bottom). The first seven profiles are A-type sections. The eighth and the ninth profiles
are H-type and K-type sections, respectively.
89
Model4
0
Model4
Model4
1500
1
1400
20
1300
Vs (m/s)
Depth (m)
40
60
1200
2
1100
1000
900
80
3
800
100
1000
1500
Vs (m/s)
2000
0.05
3
Model5
1
550
20
Vs (m/s)
Depth (m)
2
600
10
30
500
2
450
40
400
50
350
60
200 400 600 800 1000 1200 1400
Vs (m/s)
3
0.06 0.08 0.1 0.12 0.14
Period (s)
Model6
0
1
520
Vs (m/s)
30
3
1
500
20
2
Model6
Model6
10
Depth (m)
1
0.15
Model5
Model5
0
480
2
460
40
440
50
420
60
0.1
Period (s)
3
400
400
600
Vs (m/s)
800
0.1
0.15
0.2
Period (s)
0.25
1
2
3
Figure 3.5 Continue.
90
Model7
0
Vs (m/s)
Depth (m)
40
60
80
1000
1500
Vs (m/s)
3
0.05
0.1
0.15
Period (s)
1
0.2
20
30
40
1
440
2
420
400
50
400
600
Vs (m/s)
800
3
0.1
1000
Model9
0.2
Period (s)
0.3
1
2
3
Model9
Model9
1100
1
20
1000
Vs (m/s)
Depth (m)
3
Model8
460
200
2
Model8
Model8
Vs (m/s)
Depth (m)
2
1000
2000
10
0
1200
800
0
60
1
1400
20
100
Model7
Model7
40
60
80
800
2
900
800
1000 1200 1400 1600 1800
Vs (m/s)
3
0.1
0.2
0.3
Period (s)
0.4
1
2
3
Figure 3.5 Continue.
91
1200
Original Vs30, m/s
Inverted Vs30, m/s
1000
800
600
400
200
0
M1
M2
M3
M4
M5
M6
M7
M8
M9
1200
Newhall
CCOC
Original Vs50, m/s
Inverted Vs50, m/s
1000
800
600
400
200
0
M1
M2
M3
M4
M5
M6
M7
M8
M9
Newhall
CCOC
1600
Original Vs100, m/s
1400
Inverted Vs100, m/s
1200
1000
800
600
400
200
0
M1
M2
M3
M4
M5
M6
M7
M8
Figure 3.6 The depth-averaged velocities in m/s against the
known values for Occam’s inverted models.
M9
Newhall
CCOC
92
5.0
Frequency (Hz)
10.0
15.0
20.0
25.0
0.0
0.0
0.002
0.002
0.004
0.004
Slowness (s/m)
Slowness (s/m)
0.0
0.0
0.006
0.008
5.0
Frequency (Hz)
10.0
15.0
20.0
25.0
0.006
0.008
0.01
0.01
0.0
ReMi Spectral Ratio
2.9
0.0
ReMi Spectral Ratio
2.5
Figure 3.7 Dispersion picks on the slowness-frequency spectrum (p-f) images of Newhall
(left) and Coyote Creek (right) data. The small squares on both images indicate analyst’s picks
of phase velocities of the fundamental-mode Rayleigh waves.
93
Newhall
0
1
500
20
40
Vs (m/s)
Depth (m)
Newhall w/ Tr(R)=3.96
Newhall
60
2
400
3
300
4
5
80
200
6
100
200 400 600 800 1000 1200
Vs (m/s)
0.1
Vs (m/s)
Depth (m)
50
100
200
600
1
500
2
400
3
300
4
200
5
100
1000
Vs (m/s)
1500
2
3
4
5
6
CCOC w/ Tr(R)=4.16
150
500
1
0.4
CCOC
CCOC
0
0.2
0.3
Period (s)
6
0
0.2
0.4
0.6
Period (s)
0.8
1
2
3
4
Figure 3.8 Linearized inverted profiles of S-wave velocity against the OYO suspension S-wave logs of Newhall and CCOC data. Left column lists the inverted the S-wave
velocities (thick solid lines) with 95% confidence interval (horizontal bars). The wiggle
lines represent the OYO suspension S-wave velocity log. Middle column shows the
original ReMi dispersion picks (circles) against the calculated dispersion curves (thick
solid lines) of the inverted model listed in left column. Right column contains resolution matrix (scale is at the bottom)
5
6
1
0.8
0.6
0.4
0.2
0
94
Newhall 4 layers
20
40
40
40
60
0
Depth (m)
20
100
60
80
Newhall 10 layers
60
100
200 400 600 800 1000 1200
Vs (m/s)
Newhall 12 layers
0
0
20
20
40
40
40
80
100
0
Depth (m)
20
60
60
80
100
200 400 600 800 1000 1200
Vs (m/s)
Newhall 16 layers
0
0
500
1000
Vs (m/s)
100
1500
Newhall 18 layers
0
40
40
Depth (m)
40
Depth (m)
20
100
60
80
200 400 600 800 1000 1200
Vs (m/s)
100
Newhall 14 layers
80
20
80
200 400 600 800 1000 1200
Vs (m/s)
60
20
60
Newhall 8 layers
80
100
200 400 600 800 1000 1200
Vs (m/s)
Depth (m)
Depth (m)
0
20
80
Depth (m)
Newhall 6 layers
0
Depth (m)
Depth (m)
0
200 400 600 800 1000 1200
Vs (m/s)
Newhall 20 layers
60
80
200 400 600 800 1000 1200
Vs (m/s)
100
200 400 600 800 1000 1200
Vs (m/s)
Figure 3.9 Inverted S-wave velocity against the OYO S-wave log of Newhall data, showing the
effect of the number of layers. The thick solid lines are inverted the S-wave velocities with 95%
confidence interval (horizontal bars). The thin wiggle lines represent the OYO suspension S-wave
velocity log. The dash lines are the initial models.
95
20
20
20
40
40
40
60
80
Depth (m)
0
Depth (m)
0
60
100
200 400 600 800 1000 1200
Vs (m/s)
60
80
80
100
100
200 400 600 800 1000 1200
Vs (m/s)
0
1000
2000
Vs (m/s)
3000
Figure 3.10 Inverted S-wave velocity against the OYO S-wave log of Newhall data, showing the
effect of the layer S-wave velocity. The legends are the same as Figure 3.9.
0
20
20
40
Depth (m)
10
30
40
50
Max depth of 100 m
50
60
400
600 800 1000 1200
Vs (m/s)
100
100
150
80
200
Max depth of 200 m
0
Depth (m)
Max depth of 50 m
0
Depth (m)
Depth (m)
0
200 400 600 800 1000 1200
Vs (m/s)
200
0
500
1000
Vs (m/s)
1500
Figure 3.11 Inverted S-wave velocity against the OYO S-wave log of Newhall data, showing the
effect of the maximum depth. The legends are the same as Figure 3.9.
96
Using 12 picks
Using 8 picks
0
20
40
40
40
60
80
80
100
Depth (m)
20
60
0
500
Vs (m/s)
100
1000
Using 6 picks
0
20
Depth (m)
Depth (m)
0
60
80
0
500
Vs (m/s)
100
1000
0
500
Vs (m/s)
1000
Figure 3.12 Inverted S-wave velocity against the OYO S-wave log of Newhall data, showing
the effect of the number of the picks. The legends are the same as Figure 3.9.
Randomly delete 4 picks
0
Delete 4 highest frequency picks
0
Delete 4 lowest frequency picks
20
20
40
40
40
60
80
100
Depth (m)
20
Depth (m)
Depth (m)
0
60
80
80
0
500
Vs (m/s)
1000
100
60
0
500
Vs (m/s)
1000
100
0
500
Vs (m/s)
1000
Figure 3.13 Inverted S-wave velocity against the OYO S-wave log of Newhall data, showing
the effect of the frequency density of the picks. The legends are the same as Figure 3.9.
97
Model1
10
20
20
30
40
50
400
600
Vs (m/s)
400
60
80
1000
1500
Vs (m/s)
2000
80
400
1000
Model5
10
20
20
30
40
60
200 400 600 800 1000 1200 1400
Vs (m/s)
60
0
10
Depth (m)
40
60
80
400
600
Vs (m/s)
800
Model9
20
20
Depth (m)
20
Model6
40
50
Model8
800 1000 1200 1400
Vs (m/s)
30
50
0
600
0
10
Model7
0
600
800
Vs (m/s)
Depth (m)
40
40
60
0
Depth (m)
Depth (m)
40
60
800
20
Depth (m)
30
Model4
0
100
20
50
60
200
Model3
0
Depth (m)
10
100
Model2
0
Depth (m)
Depth (m)
0
30
40
40
60
50
1000
1500
Vs (m/s)
2000
60
200
400
600
Vs (m/s)
800
1000
80
800
1000 1200 1400 1600 1800
Vs (m/s)
Figure 3.14 Effects of the weighting matrix on the inverted models. Each graph shows the final
S-wave velocities model inverted using an identity weighting matrix (the vertical thick solid line) with
95% confidence interval (the horizontal solid bar), the final model using a weighting matrix of 10%
data value (the vertical thick dash line) with 95% confidence interval (the horizontal dash bar), and the
original synthetic models (the vertical dash line).
98
Chapter 4 Non-linear inversion of surface-wave dispersion
based on simulated annealing optimization
One of primary goals of inversion is to find the minimum of an error function.
Due to the fact that such an error function may have several minima of different heights
(known as multimodal problems), the linearized inversions may become trapped in one
of these local minima. In section 3.8 of Chapter 3, I demonstrated that a final inverted
model determined by the linearized inversions inherently depends on an assumed initial
model. The initial-model-dependence is evaluated in two aspects: the number of layers
and the velocity of each layer. The more layers, the more computation effort, and the
more realistic the inverted models. However, increasing number of layers alone does
not guarantee the escape from these local minimum points.
The multimodal problems of surface waves dispersion curves are highlighted by
Dal Moro et al. (2007) through a simple example. They calculated the synthetic
fundamental mode dispersion curve for a model consisting of six layers. The objective
functions (RMS error, equation (3.8) ) are calculated by varying only thickness and Swave velocity of the second and third layers and by fixing the values of the other
parameters to their known values. In other words, as a surface-wave dispersion curve is
a function of S-and P-wave velocities, density, and thickness, they free only four out of
twenty three possible variables (the bottom layer is semi-infinite). Figure 4.1a shows a
3D plot of objective functions (calculated for the synthetic curve) for 4000 random
models. The abscissas report the ratio between S-wave velocity in the second and third
99
layers (VS ratio) and the ratio between the thickness of the second and third layer (THK
ratio).
The complex result from this extremely simple and constrained example
illustrates that the local minima would cause a trap of searching path in cases where
linearized methods are used to invert the data. To further highlight the problem, Figure
4.1b shows a 2-D plot for objective functions versus the VS ratio, by keeping constant
the layer thickness to the proper values (thickness ratio equal to 1). The distribution of
points (models) characterized by low-value objective functions is large even in the
surrounding of the global-minimum area. This gives evidence of the extreme
nonlinearity and multimodality of the problem.
4.1 Global searching optimization
Several algorithms tackle this problem with the main goal of sampling a wide
search space to detect the global minimum (or maximum) of a given function (e.g. Sen
and Stoffa, 1995). Heuristic optimization schemes can be divided into enumerative or
grid search, random search (uniform distribution of the search space sampling, such as
Monte Carlo methods), and “importance sampling” (the search space is non-uniformly
sampled because some function drives the search, such as Simulated Annealing (SA)
and Evolutionary or Genetic Algorithms (GA) ).
Enumerative or grid search methods of inversion involve the systematic search
through each point in a pre-defined model space to locate the best fit models. For most
geophysical problems, the model space is very large and the forward calculation is
slow. For example, there are twenty three possible variables (the bottom layer is semi100
infinite) for a model consisting of six layers. Assuming 10 discrete grids for each
variable, the model space would be 1023 possible points (models), which is extremely
large. Another drawback of enumerative search is that grid increment may be large so
that the best fit model would be skipped. In the case of 10 grids for S-wave velocity
ranging from 100 m/s to 1000 m/s, the grid increment is 100 m/s, which is too coarse
for surface wave dispersion inversion.
Random search overcomes the discreteness problem. A random number is
drawn from a uniform distribution U[0, 1] and is then mapped into a model parameter.
As an example assume a random number (rn), the new model parameter value can be
given as
minew = mimin + rn × (mimax − mimin )
(4.1)
where mimin ≤ mi ≤ mimax
A new random modal vector can be generated by random perturbation of a
specific number of model parameters in the model vector. Synthetic data are then
generated for the new model and compared with observations. The model is accepted
deterministically based on an acceptance criterion which determines how well the
synthetic data compare with the observations. The generation-acceptance/rejection
process is repeated until a stopping criterion is satisfied. A commonly used stopping
criterion is given by the total number of accepted models.
The random search methods have been successfully used to solve highly
nonlinear inversion problems (Sambridge and Mosegaard, 2002). However, they are
computational expensive. Several different schemes have to be adopted. One approach
is to use prior information to define a prior probability density function (pdf) to sample
101
the models, such that the search space is narrowed. Another is “importance sampling”
such as Simulated Annealing (SA) and Genetic Algorithms (GA).
Genetic algorithms (GAs) are used to invert seismic velocities (Louis et al.,
1999), seismic waveform (Stoffa and Sen, 1991), and shallow elastic parameters
(Rodriguez-Zuniga et al., 1997). In GAs, a series of “genetic operations” (namely
selection, crossover and mutation) acts along various successive steps (generations)
with the aim of working out a solution able to minimize (or maximize) a certain fitness
function that measures how good a certain model is with respect to a desired
characteristic. The final solution is a model that shows the best fitness value. This kind
of procedure, or at least its basic form, does not provide any evaluation of accuracy or
uncertainty of the proposed final solution and the problem is actually seldom considered
in several optimization or inversion schemes. Recently, GA has been applied on surface
wave dispersion inversion (Yamanaka and Ishida, 1996; Dal Moro et al., 2007).
The basic GA is quite robust (Yamanaka and Ishida, 1996). The method is not
sensitive to the starting models. However, there are several problems with the basic
algorithm that are needed to be addressed (Sen and Stoffa, 1995). First, there is no
guarantee that the optimum solution will be found. The algorithm proceeds towards a fit
population. In other words, it converges toward the best model of the population.
Second, the convergence toward the global maximum may be slow because some model
parameters have minor impact on the fitness. Third, the basic algorithm is also
unsatisfying in that once the models have been selected and reproduction occurs, other
previously acquired information is not usually exploited. Recently improvements have
102
been made to overcome these disadvantages (Reeves and Rowe, 2003; Dal Moro et al.,
2007).
This dissertation employs the simulated annealing (SA) method as an alternative
for inversion of phase velocity dispersion curves of the high-frequency fundamentalmode Rayleigh waves contained in microtremors. The implementation is validated by
blind tests against a suite of synthetic data sets. SA is a Monte Carlo process for finding
the global minimum of a non-linear error function and has been successfully used in
many geophysical inverse problems (Pullammanappallil and Louie, 1993, 1994; Sen
and Stoffa, 1995) since the work of Metropolis et al. (1953) and of Kirkpatrick et al.
(1983). Recently, SA has been used to invert Rayleigh wave dispersion data for a Swave velocity profile at both crustal (Martinez et al., 2000) and local-site (Beaty et al.,
2002) scales. Martinez et al. (2000) used random numbers drawn from a uniform
distribution in the interval (-1, 1) to perturb the model during their SA inversion, while
Beaty et al. (2002) used a Cauchy-like distribution. In this implementation, the SA
algorithm draws random numbers from a standard Gaussian distribution. After testing
the implementation on twelve synthetic data sets, I applied it to two shallow dispersion
data sets collected using the ReMi technique. The inversion results from both field data
sets are compared against borehole logs.
4.2 Simulated annealing optimization method
Briefly summarized, simulated annealing is a generalization of a Monte Carlo
method for examining the equations of state and frozen states of n-body systems
(Metropolis et al., 1953). The concept is based on the manner in which liquids freeze or
103
metals re-crystallize in the process of annealing (Fig. 4.2). In an annealing process a
melt, initially at high temperature and disordered, is slowly cooled so that the system at
any time is approximately in thermodynamic equilibrium. As cooling proceeds, the
system becomes more ordered and approaches a "frozen" ground state at T=0. Hence
the process can be thought of as an adiabatic approach to the lowest energy state. If the
initial temperature of the system is too low or cooling is done insufficiently slowly the
system may become quenched forming defects or freezing out in metastable states (i.e.
trapped in a local minimum energy state).
The original Metropolis scheme uses an initial state of a thermodynamic system
that is chosen at energy E and temperature T. Holding T, the initial configuration is
perturbed and the change in energy ∆E is calculated. If the change in energy is zero or
negative, the new configuration is accepted. If the change in energy is positive, the new
configuration may still be accepted with a conditional probability given by the
Boltzmann factor exp(-∆E/T). This process is then repeated sufficient times to give
good sampling statistics for the current temperature. Then the temperature is
decremented and the entire process repeated until a frozen state is achieved at zero
temperature.
By analogy the generalization of the above Metropolis Monte Carlo process to
the inversion of dispersion curves is straightforward (Fig. 4.3). The current state of the
thermodynamic system is analogous to the calculated phase velocities based on the
current model m which involves the shear and compression wave velocities, thickness,
and density of each layer. The energy of the thermodynamic system is analogous to the
104
root mean square (RMS) error E between the calculated and the observed phase
velocities, defined as
E=
1
N
N
obs
cal 2
∑ (Vri
− Vri )
i =1
(4.2)
where N is the total number of the phase velocity picks of the fundamental Rayleigh
waves, Vriobs is the observed phase velocity for picks i, and Vrical is the calculated phase
velocity for picks i given a model vector m. The frozen state is analogous to the global
minimum of the RMS error E. The temperature T is analogous to a free parameter
which controls the whole process.
The conditional acceptance is very important for the simulated annealing. In a
local search method, we start with a reference model. A new model is accepted if and
only if ∆E ≤ 0 , in another words, it always searches in the downhill direction (Fig. 4.4).
However, in SA, every model has a finite probability of acceptance even though
∆E > 0 . Thus local methods can get trapped in a local minimum that may be in the
close neighborhood of the starting model while SA has a finite probability of jumping
out of local minima. As the temperature approaches zero, however, only the moves that
show improvement over the previous trial are likely to be accepted.
The Monte Carlo process at a constant temperature T can be modeled by
generating homogeneous Markov chains of finite length for a finite sequence (Sen and
Stoffa, 1995). A starting model m0 is selected and the corresponding misfit E(m0) is
determined. The shear velocity and the thickness of an arbitrary layer of the initial
model are perturbed by random numbers drawn from the standard Gaussian distribution
N(0, 1) as
105
Vs j = Vs j + aVs j
and
h j = h j + bh j
(4.3)
where a and b are random numbers drawn from the standard Gaussian distribution, Vs j
and h j are the S-wave velocity and the thickness of layer j, respectively. We keep
densities constant and update P-wave velocities (Vp) based on the assumed Poisson’s
ratio which is fixed during the inversion. A new misfit E is calculated for the perturbed
model. If the new misfit has been improved then we accept the perturbed model and
update the starting model m0 into m1. If the misfit has become larger then the perturbed
model is conditionally accepted with an acceptance probability exp(( E1 − E0 ) T ) . The
next iteration is based on the updated model m1. This process is repeated at a constant
temperature until the system arrives at an equilibrium state.
In order to find the global minimum, the homogeneous Markov chain of finite
length must be in an equilibrium state before temperature drops (Aarts and Korst,
1989). Mathematically, it is shown that such an equilibrium state can be obtained at a
constant temperature T after a large number of iterations (Aarts and Korst, 1989;
Geman and Geman, 1984). The equilibrium distribution at the temperature T is given by
the Gibbs’ probability density function
G (m) = exp(
− E (m)
)
T
∑ exp(
M
− E (m)
)
T
(4.4)
where the sum is taken over all models in the model space M.
Once the equilibrium has been achieved, we drop the temperature (annealing)
and repeat the Monte Carlo process at a lower constant temperature. After a large
number of iterations, another new equilibrium state is established for this lower
temperature. Then we repeat the temperature annealing until T is close to zero and the
106
misfit is smaller than a user-specified value ε. The whole iteration is called the
annealing process.
One advantage of the annealing process is that the results are independent of the
starting model (Rothman, 1985; Pullammanappallil and Louie, 1993, 1994). This is
rooted in the limiting probability theory of the Markov chain (Ross, 2003, p.200).
Another advantage is the conditional acceptance probability, which allows a conditional
acceptance of models having higher errors. This will give the annealing process an
opportunity to escape local minima. A disadvantage of the annealing process is that it
can be more expensive computationally than linearized methods.
A set of parameters, called the cooling schedule, govern the convergence of the
annealing process. Careful searching for them can guarantee the achievement of an
equilibrium state for each temperature drop. Different cooling schedules require
different probability distributions from which a random number is drawn to perturb
models. I use a polynomial-time cooling schedule that guarantees the convergence of
the annealing process and leads to a polynomial-time execution of the simulated
annealing (Aarts and Korst, 1989). It consists of four components: an initial temperature
T0, a decrement of the temperature, a stop criterion, and the length of the homogeneous
Markov chains.
The initial temperature T0 is obtained from the requirement that at this value
virtually all proposed models should be accepted. It is measured by the acceptance ratio
which is the ratio of the number of models accepted to the number of models proposed.
In practice this can be achieved by starting off at a small positive value of temperature
107
and multiplying it with a constant factor, larger than 1, until the corresponding
acceptance ratio is close to unity.
The temperature drop should be set carefully. A large drop will lead to the loss
of the equilibrium state. A small drop will cost more computational time. Aarts and Van
Korst (1989) showed
Tk +1 = α Tk
0.8 ≤ α ≤ 0.99
k = 1, 2,"
(4.5)
where k is the step in temperature drop. Our tests show that the value of 0.95 for α is
effective for our dispersion curve inversion.
The stop criterion of annealing process is arbitrary. Reasonable constraints are
that the misfit remains the same for a number of iterations for T close to zero, and that
the misfit is smaller than a user-specified value ε.
The length of the homogeneous Markov chain is based on the requirement that
at each constant temperature, equilibrium is to be restored. It is dynamically determined
during the execution of the annealing process.
4.3 Model appraisal
In linearized inversions, the quality of the inverted results can be investigated by
calculating the resolution matrix (equation (3.45) ). The deviation of the inverted
parameters can be estimated from the deviation of the observed data. However, such
information can not be derived when using the SA inversion. One way to approach the
problem is the Gibbs sampler. If the SA runs for a large number of iterations at the final
temperature, it may be used as a Monte Carlo importance sampling technique, weighted
108
by the problem’s posteriori probability density function, to evaluate quantities such as
mean and variance.
The reliability of the model resulting from the annealing can be evaluated by the
uncertainty estimation process, during which the annealing process is repeated at the
final temperature for at least 500 iterations (Martinez et al., 2000). The expected model
<m> can be expressed as
m = ∫ mG (m)dM
M
(4.6)
and the covariance matrix as
Cm = ∫ mmT G (m)dM - m m
T
M
(4.7)
where G (m) is the Gibbs probability density function for the different models m of the
model space M. In practice the integration in the above equations can be replaced by
finite sums for all accepted models.
Both Vs and thickness of each layer of the models are varied during the
annealing process. However, one of these has to be fixed during the uncertainty
estimation process. Under fixed thicknesses, Equations (4.6) and (4.7) yield a mean
profile with the standard deviations of Vs of each layer; under fixed shear velocities, a
mean profile with the standard deviations of thickness of each layer.
4.4 Inversion results
The SA inversion method described above was tested on a suite of nine
synthetic models and two field data sets. Chapter 3 described the acquisition of seismic
records and extraction of surface wave dispersion.
109
The initial models were generally set to be a uniform half-space. The maximum
modeling depths were estimated using the half-maximum-wavelength discussed by Park
et al. (1999). This criterion has been used by Xia et al. (1999), Beaty et al. (2002), and
Stephenson et al. (2005). Another critical issue in dispersion inversion is the number of
layers. The more layers, the more computation effort, and the more realistic the inverted
models (Dal Moro et al., 2007). From the mathematical point of view, we determine the
number of layers by sign change of the second derivatives of the picked dispersion
curves. All initial S-wave velocities were set to a visually inspected average of the
picked phase velocities. P-wave velocities are updated based on the assumed Poisson’s
ratio. Densities are set at a reasonable value (e.g. 2.0 g/cm3) and kept constant during
the inversion. The maximum modeling depth and number of layers can also be
manually set if known independently. Other a priori information that could be manually
incorporated into the SA optimization process includes Poisson’s ratio, density,
thickness, and S-wave velocity of any layers (at least two layers are free to allow
perturbations).
Inversion on the synthetic ReMi Rayleigh-wave phase velocity dispersion curve
picks is carried out by varying both S-wave velocity and layer thickness and fixing
density and Poisson’s ratio at the original values (2.0 g/cm3 and 0.25 of the original
synthetic models, respectively).
Figure 4.5 shows the mean of the inverted S-wave velocity profiles plotted atop
nine original synthetic models. Under fixed thicknesses, the standard deviations of Swave velocity of each layer are calculated from equation (4.7) and plotted as horizontal
error bars. Among the total twenty-seven layers of nine synthetic models, velocities of
110
twenty-one layers (78% of trial cases) fall within the error bars. Statistics on the
inverted models show that the average errors for Vs are 5.5% for the top layer, 11% for
the second, and 7.9% for the lower half space. The top layer Vs errors for H-type and Ktype models are 14.3% and 0.2%, respectively. The Vs inversions for the second layer
for H-type (0.1% Vs error) and K-type (5.8% Vs error) models are better. Inverted
velocities for the half space of the H-type and K-type models are off 17.9% and 15.3%,
respectively.
The Vs value averaged to 30 (V30), 50 (V50), and 100 (V100) m shown in Fig. 4.6.
Table 4.1 tabulates these values which have been rounded to the one significant figure.
Average V30 error is 3.5% over all nine synthetic models. Only one (Model 4) yields a
large V30 error of 11%. Average V50 and V100 errors are 4.6% and 6.1%, respectively,
over all nine synthetic models.
Table 4.1 SA-inverted depth-averaged velocities in m/s and
percentage difference from known profiles in parentheses
V50
V100
Models
V30
model 1
480 (2%)
540 ( 1%)
610 ( 1%)
model 2
550 ( 3%)
620 ( 0%)
720 ( 1%)
model 3
620 ( 6%)
660 (10%)
800 ( 6%)
model 4
980 (11%)
1070 (11%)
1310 (11%)
model 5
490 ( 1%)
560 ( 0%)
630 ( 2%)
model 6
480 ( 1%)
520 ( 3%)
570 ( 6%)
model 7
920 ( 5%)
1020 ( 6%)
1240 ( 8%)
model 8
430 ( 1%)
460 ( 6%)
490 (11%)
model 9
930 ( 1%)
1040 ( 4%)
1110 ( 9%)
model 7a
920 ( 5%)
1110 ( 3%)
1400 ( 3%)
model 8a
460 ( 9%)
470 ( 4%)
500 (10%)
model 9a
1010 (10%)
1140 (13%)
1230 (20%)
Newhall
300 (11%)
380 ( 7%)
490 ( 4%)
CCOC
240 (14%)
270 (11%)
340 (14%)
111
Under fixed shear velocities, equations (4.6) and (4.7) yield the mean profiles
with the standard deviations of thickness of each layer (Fig. 4.7). The models 7a, 8a,
and 9a were inverted using the same picks as the models 7, 8, and 9 but with a different
uncertainty estimation process. The average Vs values (V30, V50, and V100 ) of models
7a, 8a, and 9a do not differ significantly off from those of models 7, 8, and 9 (Table
4.1).
Inversion of two field data sets (Newhall and Coyote Creek) starts with an initial
model generated in the way described above. Both S-wave velocity and thickness are
allowed to vary for a finer fit. P-wave velocities are updated based on the assumed
Poisson’s ratio of 0.35 (Dal Moro et al., 2007). Densities are set at 2.0 g/cm3 and kept
constant during the inversion.
Figure 4.8 displays the inverted models for both the Newhall and Coyote Creek
data. The mean of the inverted S-wave velocity for the Newhall data follows the OYO
suspension S-wave velocity log well. The significant increase of the S-wave velocity at
a depth of about 26 m is clear on the inverted model. Ninety percent of the OYO
suspension S-wave velocity log values fall within the standard deviations of the inverted
model. The velocity uncertainty increases with depth as expected (resolution of
dispersion curves decreases as depth). The maximum depth of the inverted profile for
the Coyote Creek data is over two times deeper than that of the Newhall data. The mean
of the inverted S-wave velocity of the Coyote Creek data follows the OYO suspension
S-wave velocity log. The rapid increases of Vs at the depths of 20 and 60 m are
effectively inverted. Figure 4.9 compares the calculated dispersion curves and ReMi
dispersion picks. Both match well. Average Vs errors of the Newhall data are 11% for
112
V30, 7% for V50, and 4% for V100 (Table 4.1). Average Vs errors of the Coyote Creek
data are 14% for V30, 11% for V50, and 14% for V100.
4.5 Comparison with linearized inversion results
The same suite of nine synthetic models and two field data sets are used to test
both linearized and SA inversions. Visual inspections on nine synthetic data sets (Fig.
3.5 and 4.5) reveal that the SA inverted models have an equal or better fit than the
linearized inverted models. For example, in the case of model 1, SA-inverted profile
almost perfectly fits the original synthetic model (Fig. 4.5) while the linearized
inversion fit for model 1 is far less perfect. Another example is model 5. The high
velocity layer of the inverted model shows an artifact of linearized inversion (Fig. 3.5)
while SA-inverted final model follows the original model very well (Fig. 4.5). Visual
inspections on Newhall and CCOC data sets also show a significant fitting improvement
that is achieved by the SA inversion. Unlike the linearized inverted model of CCOC
data set (Fig. 3.8), the SA-inverted model follows the OYO log very well and does not
show the low velocity zone. The calculated dispersion curve of CCOC fits the ReMi
picks excellently, especially near period 0.3 second (right, Fig. 4.9) while the curve
misfits the picks near 0.3 second on the linearized inversion (Fig. 3.8).
There are some cases where both linearized inversion and SA inversion yield the
models that fit the data equally well. For example, both inversions on model6 6 and 7
yield almost same final model (Fig. 3.5 and Fig. 4.5).
The difference in the inverted final models by both methods is probably related
to the layer thickness. The S-wave velocities are inverted while the layer thickness is
113
kept the same as these in the initial model during the linearized inversion. But both Swave velocities and layer thickness are varied during SA inversion. A dispersion curve
is a function of S-wave velocity, layer thickness, layer density, and P-wave velocity of
models. Varying layer thickness increases the searching model space, thus, increases the
possibility to find a global minimum solution.
4.6 Difference from previous implementation
The SA begins with Metropolis et al. (1953) and was introduced into geophysics
area by Kirkpatrick (1983) and others. Since then, it has been used for many studies
(Pullammanappallil and Louie, 1993, 1994; Sen and Stoffa, 1995; Martinez et al., 2000;
Beaty et al., 2002). Basic implementation of SA consists in three parts: 1) initial
temperature T0; 2) annealing process; and 3) uncertainty estimation. The annealing
process could be subdivided into a) probability distribution that a random number is
drawn from; b) the length of the homogeneous Markov chains; c) decrement of
temperature ∆T; d) stop criterions. Due to these options, researchers are able to
determine the best combination that fit their problem. Even for the same proplem, there
could be various implementations. For example, this study is for inversion of phase
velocity dispersion curves of the high-frequency fundamental-mode Rayleigh waves
contained in microtremors, which is the similar purpose of Beaty et al. study (2002).
Table 4.2 lists the implementation difference between this study and Beaty’s study.
Beaty et al. (2002) used a fast simulated annealing algorithm (FSA), which is
identical to the Metropolis Monte Carlo process with the exception that model
parameters are drawn from a Cauchy-like distribution (Sen and Stoffa, 1995). Unlike
114
a)
b)
c)
d)
e)
f)
Table 4.2 Implementation difference of SA from previous study
Beaty’s study
This study
Cauchy-like distribution.
A standard Gaussian distribution.
One iteration for each temperature Many iterations until equilibrium state
drop.
is achieved. Then temperature is
dropped.
T(i) = T(0) / i
T(i+1) = αT(i)
Stop the SA when the misfit
1) The misfit remains the same for a
remains the same for a number of
number of iterations for T close to zero,
iterations.
and 2) the misfit is smaller than a userspecified value ε.
Invert for coefficients of
Directly invert for S-wave velocities
Chebyshev polynomials that
represent velocity profiles
The SA inversion employing only The SA (this implementation) works
fundamental-mode dispersion
well for S-wave velocity inversion on
curves does not work well. Higher- dispersion curves of fundamental-mode
mode dispersion data have to be
Rayleigh waves.
used for an eligible SA inversion.
this algorithm that requires the equilibrium state before the temperature drops, the FSA
does not consider whether the current state is in equilibrium. For FSA the temperature is
dropped after each individual iteration. Further, instead of directly inverting for the
velocities of each layer, Beaty et al. (2002) use a sum of Chebyshev polynomials to
represent velocity profiles, and invert the coefficients of these polynomials. Beaty et al.
(2002) stop the SA when the misfit remains the same for a number of iterations. This
stop criterion may be satisfied while the temperature is still high, possibly causing a
high probability of non-global-minimum solutions.
Tests on both field and synthetic Rayleigh dispersion data show that our
implementation of SA works well for S-wave velocity inversion on dispersion curves of
high-frequency fundamental-mode Rayleigh waves. Using the idea of SA with a
115
different implementation, Beaty et al. (2002) suggested that SA inversion employing
only fundamental-mode dispersion curves does not work well. The disparity with our
successful blind tests might be due to their different implementation of SA and the
small number of tests they conducted. Only one synthetic and one field data set were
used for testing in Beaty’s study. In this implementation of SA, temperature is not
dropped until an equilibrium state is achieved through many iterations at the current
temperature. Twelve blind synthetic and two field data sets were used to demonstrate
the validity of the implementation.
4.7 A hybrid inversion approach: simulated
annealing followed by the linearized inversion
The convergence of SA is governed by a cooling schedule. At the final
temperature stage, the annealing process is repeated for hundreds of iterations, sampling
at least a hundred models that have misfits to the dispersion data within a pre-set
tolerance. Then the mean and the standard deviation of either S-wave velocity or depth
can be calculated based on mathematical formula (equations (4.6) and (4.7) ). In most
cases, these model samples are close to each other (standard deviation of the samples is
close to 0) such that they are virtually one model. In other cases the standard deviation
of these samples is significant.
The hybrid algorithm utilizes the output of SA as the input to the Occam’s
linearized inversion that has been described in Chapter three (Fig. 4.10). Although SA
practically produces more than one model samples that fit data equally well, they are
independent of the initial model and closer to the global minimum one than the initial
116
model. If we choose the mean of these model samples as our initial model for the
linearized inversion, the possibility of obtaining the global minimum model would be
higher.
The hybrid inversion method described above was tested on two experimental
data sets, Newhall and CCOC. The acquisition of the seismic records and the extraction
of surface wave dispersion curves at both sites have been described in Chapter 3.
The SA inversion is implemented as described above except that the layer
thickness is fixed. In order to exam the improvement of the hybrid inversion, only Swave velocities are allowed to vary during iteration. P-wave velocities are updated
based on an assumed Poisson’s ratio of 0.35. Densities are set at 2.0 g/cm3 and kept
constant during the SA inversion. These SA-inverted models are taken as the starting
models for the Occam’s inversion. The results from the Occam’s inversion are the final
hybrid-inverted models.
For the Newhall data, the SA-inverted model (left, Fig. 4.11) generally follows
the OYO suspension S-wave velocity log with a RMS error (equation (4.2) ) of 2.4 m/s.
The hybrid inversion yields a model (right, Fig. 4.11) that is the exactly same one as the
SA-inverted model with the same RMS error and the layer S-wave velocities. This
indicates that the further Occam’s inversion does not improve the SA-inverted model
and the SA inversion obtained the global minimum.
The similar conclusion could be drawn from the CCOC data. Both SA-inverted
(left, Fig. 4.12) and the hybrid-inverted models (right, Fig. 4.12) follow the OYO
suspension S-wave velocity log. The hybrid inversion produces a model that has a
smaller RMS error (10.7 m/s) than the SA-inverted model (RMS = 11.8 m/s). However,
117
the decrease of RMS error (7.6%) is insignificant. The further Occam’s inversion does
not significantly improve the SA-inverted model, indicating that the SA-inverted model
is reliable.
For the Occam’s inversion, the inverted model can be evaluated by computing
resolution matrix (equation (3.45) ) and covariance matrix (equation (3.42) ). However,
they can not be computed for some cases of the hybrid inversion algorithm. For the
Occam’s inversion, the direction of descent of the cost function is linear. In another
words, the model in iteration (i) must have a smaller cost function than the model in
iteration (i-1). Otherwise, the model in iteration (i) would be the final model. The
hybrid algorithm utilizes the output of SA as the initial model to the Occam’s linearized
inversion (Fig. 4.10). In some cases, this initial model has a smaller cost function than
models produced by equation (3.40) (for instant the Newhall data set). Then the initial
model is simply taken as the final model. Therefore, there are no resolution and
covariance matrix for this final model is given not calculated.
118
Figure 4.1 Multimodality of the surface wave dispersion curve inversion problem. a) In abscissa: the ratios between the shear-wave velocities (VS) of two overlaying strata and their thicknesses (THK), in ordinate the objective functions for 4000 models. The white arrow indicates
the position of the real model. b) A 2-D plot of 4000 objective functions
versus the VS ratio, by keeping constant the THK ratio at the proper
value. The black arrow indicates global minimum point (From Dal
Moro et al., 2007)
119
Fast drop
Glass
Slow drop
Hot metal block
T=1000 oC
Crystal
Figure 4.2 A cartoon showing an annealing process.
E0
h[0] Vs[0]
h[1] Vs[1]
h[2] Vs[2]
Initial model
∆E ≤ 0
Perturb
E1
Accept model
Pc ≥ Pr
else
else
Accept model
h[1] Vs[1]
Refect model
h[2] Vs[2]
Perturbation
where ∆E = E1 − E0 , Pc = e
−(
h[0] Vs[0]
∆E
)
T
Perturbed model
, and Pr ⊂ U [0, 1]
Figure 4.3 A flowchart showing the annealing process on inversion of dispersion curve
of surface waves.
120
Pc = e
−(
∆E
)
T
Figure 4.4 A cartoon showing the role of the conditional acceptance.
A local search mothod searches the minimum model always in the
downhill direction and can get trapped in a local minimum/valley
(black solid dots) that may be in the close neighborhood of the starting model. In SA, every model has a finite probability of acceptance
(Pc). In other words, it has a finite probabability of jumping out of
local minimum/valley. As temperature gradually decreases, only
moves that show improvement over the previous trial are likely to be
accepted. As temperature closes to zero, the global minimum/valley
(open circle) will be found.
121
Model 1
10
10
20
20
40
10
Depth (m)
30
40
400
600
60
800
400
600
800
80
400
1000
60
80
10
10
20
20
30
40
60
200
2000
400
Model 7
600
800
40
60
1000
400
600
800
Vs (m/s)
Model 8
0
Model 9
0
10
10
20
1200
30
Vs (m/s)
Vs (m/s)
0
1000
50
50
1500
800
Model 6
0
Depth (m)
Depth (m)
40
1000
600
Vs (m/s)
Model 5
0
20
Depth (m)
50
Vs (m/s)
Model 4
0
20
40
60
Depth (m)
Depth (m)
20
Depth (m)
40
70
Vs (m/s)
30
40
30
40
50
60
80
100
30
60
50
50
100
Model 3
0
20
30
60
200
Model 2
0
Depth (m)
Depth (m)
0
50
1000
1500
Vs (m/s)
2000
60
70
200
400
600
Vs (m/s)
800
1000
80
800
1000
1200
1400
1600
Vs (m/s)
Figure 4.5 Inverted profiles with standard deviation of S-wave velocity against the original
synthetic models. The thick solid lines are means of the profile inverted by the simulated annealing algorithm. The horizontal bars are standard deviations associated with the solid lines. The
dashed lines represent the original synthetic models. The first seven profiles are A-type sections.
The eighth and ninth profiles are H-type and K-type sections, respectively.
122
1200
Original Vs30, m/s
Inverted Vs30, m/s
1000
800
600
400
200
C
O
CC
ha
ll
N
ew
9a
M
8a
M
7a
M
9
M
8
M
7
M
6
M
5
M
4
M
3
M
2
M
M
1
0
1200
Original Vs50, m/s
Inverted Vs50, m/s
1000
800
600
400
200
C
O
CC
ha
ll
N
ew
9a
M
8a
M
7a
M
9
M
8
M
7
M
6
M
5
M
4
M
3
M
2
M
M
1
0
1600
Original Vs100, m/s
1400
Inverted Vs100, m/s
1200
1000
800
600
400
200
C
O
CC
ha
ll
N
ew
9a
M
8a
M
7a
M
9
M
8
M
7
M
6
M
5
M
4
M
3
M
2
M
M
1
0
Figure 4.6 The depth-averaged velocities in m/s against the known values for SA inverted models.
123
Model 7a
0
10
20
40
60
Depth (m)
20
Depth (m)
Depth (m)
Model 9a
0
10
20
30
40
30
40
50
60
80
100
Model 8a
0
50
1000
1500
Vs (m/s)
2000
60
70
200
400
600
Vs (m/s)
800
1000
80
800
1000
1200
1400
1600
Vs (m/s)
Figure 4.7 Inverted profiles with standard deviation of layer thickness against the original
synthetic models. The thick solid lines are means of the profile with standard deviation of layer
thickness shown as the vertical bars. The dashed lines represent the original synthetic models.
124
0
0
OYO Vs log
Inverted model
OYO Vs log
Inverted model
10
50
20
30
Depth (m)
Depth (m)
40
50
60
100
150
70
200
80
90
100
250
0
200
400
600
Vs (m/s)
800
1000
0
200
400
600
Vs (m/s)
800
1000
1200
Figure 4.8 Comparison of the OYO suspension S-wave velocity logs and the inverted
models for the Newhall (left) and the Coyote Creek data (right). The thin lines are the
OYO suspension logs and the thick lines are means of the optimized S-wave velocity
profiles with standard deviations shown as horizontal error bars.
700
700
600
600
500
Vs (m/s)
Vs (m/s)
500
400
400
300
300
200
200
100
100
0
0.1
0.15
0.2
0.25
Period (s)
0.3
0.35
0
0.2
0.4
Period (s)
0.6
0.8
Figure 4.9 Calculated dispersion curves (lines) of fundamental-mode Rayleigh waves
plotted atop the ReMi dispersion picks (circles) for the Newhall (left) and the Coyote
Creek data (right).
125
An
arbitrary
initial
model
SA
inversion
Occam’s
linearized
inversion
A final
model
Figure 4.10 The flow chart of the hybrid inversion algorithm
126
0
0
OYO Vs log
75.7 260.3
2.4 295.8
10
20
30
30
40
40
Depth (m)
Depth (m)
20
50
60
50
60
70
70
80
80
90
90
100
200
400
600
800
Vs (m/s)
1000
OYO Vs log
2.4 295.8
10
100
1200
200
400
600
800
Vs (m/s)
1000
1200
Figure 4.11 Final inverted models for Newhall data using the simulated annealing method (left) and the
hybrid inversion method (right). Dash lines are the initial S-wave velocity models and solid lines are
the inverted S-wave velocity models by SA (left) and the hybrid inversion (right). Wiggles are OYO
suspension shear wave velocity log. Numbers in the legend are RMS errors in m/s followed by average
shear wave velocity (m/s) in upper 30 meters.
0
0
OYO Vs log
68.1 206.6
11.8 235.6
20
40
60
60
80
80
Depth (m)
Depth (m)
40
100
120
100
120
140
140
160
160
180
180
200
200
220
200
400
600
800
Vs (m/s)
1000
OYO Vs log
10.7 229.9
20
1200
220
200
400
600
800
Vs (m/s)
1000
1200
Figure 4.12 Final inverted models for CCOC data using the simulated annealing method (left) and the
hybrid inversion method (right). The legends are the same as Figure 4.11.
127
Chapter 5
A joint SA inversion using both Rayleigh and
Love surface-wave dispersions
In Chapter 4 I have shown that in the absence of any a priori information or
other constraints, the SA inverted models, along with the uncertainties, give the analyst
a good idea of the underlying S-wave velocity structure. However, dispersion curves of
fundamental mode Rayleigh waves alone do not contain sufficient information to
uniquely determine a model. The velocity-depth trade-off gives rise to model nonuniqueness.
The effect of the layer thickness on surface wave dispersion curves can be
compensated by altering the layer S-wave velocities. Thin layers with a given S-wave
velocities may have the same effect on surface wave dispersion curves as thick layers
with lower S-wave velocities. This ambiguity tends to increase with depth. Figure 5.1
shows two models consisting of five layers. Changing the thickness of the fourth and
the fifth layers of the model A makes the model B. Properly adjusting the S-wave
velocities of the model B yields almost identical dispersion curves. The model B is
equivalent to the model A in the sense that this is not enough resolution for the finer
distinction between both dispersion curves.
The effect of the trade-off between the layer thickness and the S-wave velocity
also can be seen for models with different number of layers. A complicated stratigraphy
can be simplified to find an equivalent model with fewer layers. Figure 5.2 shows two
models, one with five layers (model C) and the other with eleven layers (model D), and
128
corresponding phase velocity curves (right, Fig. 5.2). Both curves are in practice
identical. Thus, the two models are equivalent.
Using other geophysical data is one way to help resolve this non-uniqueness
problem in estimating subsurface structures using the fundamental-mode Rayleigh wave
dispersion curves. Studies using receiver functions (e.g., Last et al., 1997; Özalaybey et
al., 1997; Du and Foulger, 1999; Julia et al., 2000; Chang et al., 2004), gravity (Hayashi
et al., 2005), and the reflection travel times (Dal Moro and Pipan, 2007) have all shown
that added information results in more robust models.
In this Chapter, I explore the possibility of using the fundamental-mode Love
wave dispersion curves to constrain the Rayleigh wave inversion by simulated
annealing optimization. The SA has been successfully adapted to minimize multiple
cost functions involving disparate parameters (Pullammanappallil and Louie, 1997).
Using Love waves as a constraint presents several advantages. First Love wave, like
Rayleigh waves, are dominant in either seismic or microtremor records. Second, there is
no significant increase in field acquisition effort. Both Rayleigh and Love waves could
be recorded simultaneously using a three component geophone. In the absence of 3components, Love wave can be recorded by simply swapping the vertical geophones
used for Rayleigh waves with horizontal ones. Third, forward modeling of Love waves
is much faster than that of Rayleigh waves. Lastly, Love wave dispersion curves can be
extracted using the same procedure as the standard Rayleigh ReMi data (Louie, 2001).
Both can be done simultaneously.
129
5.1 Equalized cost function
The SA inversion, described in Chapter 4, is performed on extracted
fundamental-mode dispersion picks of both Rayleigh and Love waves. Rayleigh waves
are caused by interference between P and SV waves while Love waves are SH waves
only. Thus, they provide two mutually exclusive sampling of the subsurface properties.
Both data sets are said non-commensurable.
Each data set is characterized by its own forward modeling and number of data
points. For the Rayleigh wave data set consisting of M phase velocity picks of the
fundamental Rayleigh waves, the cost function Er is defined as
Er =
1
M
m
obs
cal 2
∑ (Vri
− Vri )
i =1
(5.1)
where M is the total number of the phase velocity picks of the fundamental-mode
Rayleigh waves, Vriobs is the observed phase velocity for picks i, and Vrical is the
calculated phase velocity for picks i given a model vector m.
For the Love wave data set consisting of N phase velocity picks of the fundamentalmode Love waves, the cost function El is defined as
El =
1
N
n
obs
cal 2
∑ (Vl j − Vl j )
j =1
(5.2)
where N is the total number of the phase velocity picks of the fundamental-mode Love
cal
waves, Vl obs
j is the observed phase velocity for picks j, and Vl j is the calculated phase
velocity for picks j given a model vector m.
The simultaneous inversion of both data sets without equalization may yield
solutions that are dominated by one set. To perform the joint inversion, I define a joint
equalized cost junction E as
130
E = γ Er + (1 − γ ) El
(5.3)
where γ ⊆ [0, 1] is an influence/weighting factor, valued between zero and unit. The
factor is an a priori value that trades off between the relative influences of each data set.
ˆr
The minimization of one data set only would yield an estimate for instance m
(the case of γ = 1.0 in equation (5.3) ), while the minimization of the other data set
ˆ l (the case of γ = 0.0 in equation (5.3) ). It
would yield another estimate for instance m
is very important to realize that two estimates generally are different. For any other
value of γ, the minimization of the equalized cost junction E would yield an estimate for
ˆ r and m
ˆl .
ˆ γ that is between m
instance m
The role of the influence factor γ is illustrated by Figure 5.3 which shows that no
improvement in one cost function can be achieved without producing a simultaneous
degradation in another cost function. Zero γ corresponds to point A in Fig. 5.3 which is
the minimum solution solely based on Love wave dispersion data. Unit γ corresponds to
point B in Fig. 5.3 which is the minimum solution solely based on Rayleigh wave
dispersion data. The influence factor γ should set to a value (for example point C in Fig.
5.3) that corresponds to the point closest to the origin of the Cartesian coordinate
system. If a priori geological information is available or one set of picks is more reliable
than the other, the influence factor γ could set to a value that corresponding to this priori
knowledge. If no such knowledge is available, γ usually is set to 0.5, giving the same
weight to each of data sets.
131
5.2 Synthetic tests
Three synthetic data sets are borrowed from Noise Blind Test Project of
Grenoble, France (http://esg2006.obs.ujf-grenoble.fr/BENCH1/B1-Noise.html). The
noise synthetics are computed using 1D structures. Noise sources were approximated by
surface or subsurface forces with random force orientation and amplitude. Distribution
of sources is random in time. In space, distribution is such that around two-third of total
number of sources is randomly distributed, while one-third is spatially localized. The
source time function is either a delta-like signal or a pseudo-monochromatic signal.
Computation of the associated wave field is performed using wavenumber-based
method of Hisada (1994, 1995) for 1D horizontally layered structures. The same names
(N102, N103, and N104) are used as the Noise Blind Test Project of Grenoble, France
for each synthetic data set.
5.3 Inversion results
The non-linear SA inversion, and joint inversion methods described in Chapter 4
and 5 were tested on three synthetic models. The inverted models, along with the
original synthetics and the inverted depth-averaged velocities, are shown from Fig. 5.4
to 5.13. Table 5.1 tabulates these values which have been rounded to the one significant
decimal.
The initial models for all tests were generally set to be a 12-layer uniform halfspace with equal layer thickness. The maximum modeling depths were user-specified.
The S-wave velocities of the initial models are set from depth assuming a velocitydepth slope.
132
Table 5.1 Joint-inverted depth-averaged velocities in m/s and
percentage difference from known profiles in parentheses
V50
V100
Models*
V30
N102R
290 (34)
299 (10)
443 (19)
N102L
219 (1)
238 (13)
298 (20)
N102J
310 (43)
226 (17)
308 (17)
N103R
349 (0)
349 (0)
349 (0)
N103L
353 (1)
353 (1)
353 (1)
N103J
357 (2)
357 (2)
357 (2)
N104R
320 (5)
446 (2)
651 (15)
N104L
312 (7)
397 (9)
511 (10)
N104J
315 (6)
403 (8)
515 (9)
* Last letter of the model name R represents SA inversion on Rayleigh data, L SA
inversion on Love data, and J joint inversion on both
The SA inversion on the synthetic dispersion curve picks is carried out by
varying both S-wave velocity and layer thickness and fixing density and Poisson’s ratio
at the reasonable values (2.0 g/cm3 and 0.25, respectively). For comparison, we fix the
number of layer at 12 for all tests. The inverted models are plotted without uncertainty.
This is a blind test. No any prior geological information is available. Thus, the
simultaneous SA inversion on both Rayleigh and Love wave dispersion data is preferred
by setting the influence factor γ to 0.5 for the joint inversion.
The original synthetic model N102 is a complex shallow structure with strong
impedance contrast and complex layering including low velocity zones. This model is
chosen for verifying the capacity of inversion methods to resolve fine layering. Both
features of strong impedance contrast and low velocity zones appears on models
inverted by SA and the joint inversion methods (Fig. 5.4 and Fig. 5.5), The high
velocity layer below 150 m is better inverted by the joint method (Fig. 5.5).
133
Unfortunately the fine layers in the shallow depth are poorly resolved by either of the
inversion method. Joint inverted model shows a larger RMS error (21.3 m/s). The larger
error is due to the incorrect low velocity zone mapped between 150 m and 190 m (left,
Fig. 5.5) due to lack of sufficient picks at low frequency (right, Fig. 5.5).
The ideal shape of the cost functions should be like one in Figure 5.3. The
closer to that shape, the more reliable the model to the true profile. The evolution of the
cost functions Er (equation (5.1) ) and El (equation (5.2) ) of N102 are shown in Fig.
5.6. The long distance of the distribution envelope from the origin shows that the
control parameters of the joint inversion should be adjusted. These parameters include
the cooling schedule of the SA process, the maximum depth, the number of layers,
density, and Passion’s ratio.
The Vs value averaged to 30 (V30), 50 (V50), and 100 (V100) m for all tests are
shown in Fig. 5.7 and tabulated in Table 5.1. Compared with the next two tests, the
inverted models of N102 have the large errors of V30, V50, and V100.
The N103 is a deep model with strong impedance contrast in order to check the
ability of inversion methods to resolve deep layers. SA inversion on Rayleigh wave
dispersion data produces low velocity zones which is an inversion artifact (left, Fig.
5.8). The final model from the joint inversion matches the original synthetic model best
in both shallow and deep parts (left, Fig. 5.9). The larger RMS (12.1 m/s) of the joint
inverted model is due to bad picks of Love dispersion data above 10 Hz (right, Fig. 5.9).
Very small errors of V30, V50, and V100 (Fig. 5.7) and the relatively shorter distance of
the distribution envelope from the origin (Fig. 5.10) show that the joint inversion of
N103 is better than N102.
134
The N104 is a simple model involving shallow and deep layers. The shallow
layer is clearly inverted by the SA method. However, the deep layer is poorly defined
(left, Fig. 5.11). Layers in both shallow and deep depth are clearly resolved by the joint
inversion except a relatively low S-wave velocity near 320 m (left, Fig. 5.12). The
shape of the cost functions (Fig. 2.13) is close to that in Fig. 5.3 showing that the joint
inversion on N104 is good.
135
0
1000
10
900
20
800
700
40
Vs (m/s)
Depth (m)
30
50
60
500
400
70
300
80
Model A
Model B
90
100
600
200
400
From model A
From model B
200
600
800
Vs (m/s)
1000
0
1200
0.1
0.2
0.3
Period (s)
0.4
Figure 5.1 Two different models with same number of layers (left) and corresponding
dispersion curves (right). They are equivalent in the sense that there is not enough resolution for the finer distinction.
0
10
900
20
800
700
40
Vs (m/s)
Depth (m)
30
50
60
500
400
70
300
80
Model C
Model D
90
100
600
200
400
From model C
From model D
200
600
800
Vs (m/s)
1000
1200
0
0.1
0.2
Period (s)
0.3
0.4
Figure 5.2 Two different models with different number of layers (left) and corresponding dispersion curves (right). Both models yield almost identical dispersion curves.
136
El
C
(0, 0)
Er
Figure 5.3 A theoretical distribution of the value of the cost
function for the joint inversion. Point A and B are the minimum
points of the cost function El and Er, respectively. Point C represents the minimum value of the joint cost function E. Point C
would be closer to A for γ<0.5 and to B for γ>0.5
137
0
1500
Picked Love
Calculated Love
Picked Rayleigh
Calculated Rayleigh
Synthetic N102
Love RMS=19.4
Rayleigh RMS=9.2
50
100
150
Vs (m/s)
Depth (m)
1000
200
500
250
300
350
0
500
1000
1500
Vs (m/s)
2000
2500
0
3000
0
5
10
15
Frequency (Hz)
20
Figure 5.4 SA inversion results of the data N102. Inverted models based on Rayleigh and
Love dispersion data are plotted atop of the original true synthetic model (solid thin line) as
the dash thick line and the solid thick line on the left graph, respectively. The calculated Rayleigh dispersion curve is based on the model in dash thick line on the left graph; the calculated Love dispersion curve is based on the model in solid thick line on the left graph. Both
calculated curves are plotted atop of the observed dispersion data on the right graph.
0
1500
Picked Love
Calculated Love
Picked Rayleigh
Calculated Rayleigh
Synthetic N102
Final RMS=21.3
50
100
150
Vs (m/s)
Depth (m)
1000
200
500
250
300
350
0
500
1000
1500
Vs (m/s)
2000
2500
3000
0
0
5
10
15
Frequency (Hz)
20
Figure 5.5 Joint inversion results of the data N102. The inverted model and the original true
synthetic model are plotted as the solid thick line and the solid thin line on the left graph,
respectively. The calculated dispersion curves of Rayleigh and Love waves based on the joint
inverted model (solid thick line on the left) are plotted atop of the observed dispersion data
on the right graph.
138
30
25
El (m/s)
20
15
10
5
0
0
10
20
30
Er (m/s)
40
50
60
Figure 5.6 The distribution of the value of cost functions for joint inversion on N102. X-axis is the cost
function Er calculated from equation (5.1) in the text
on Rayleigh wave dispersion picks. Y-axis represents
the cost function El calculated from equation (5.2) on
Love wave dispersion picks.
139
400
Original Vs30, m/s
Inverted Vs30, m/s
350
300
250
200
150
100
50
0
N102R
N102L
N102J
N103R
N103L
N103J
N104R
N104L
N104J
Original Vs50, m/s
500
Inverted Vs50, m/s
450
400
350
300
250
200
150
100
50
0
N102R
N102L
N102J
N103R
N103L
N103J
N104R
N104L
700
N104J
Original Vs100, m/s
Inverted Vs100, m/s
600
500
400
300
200
100
0
N102R
N102L
N102J
N103R
N103L
N103J
N104R
N104L
Figure 5.7 The depth-averaged velocities in m/s against the known
values for both SA and joint inverted models.
N104J
140
0
1400
Synthetic N103
Love RMS=10.7
Rayleigh RMS=8.3
Picked Love
Calculated Love
Picked Rayleigh
Calculated Rayleigh
1200
1000
Vs (m/s)
Depth (m)
500
800
600
1000
400
200
1500
0
1000
2000
Vs (m/s)
3000
4000
0
5
10
15
Frequency (Hz)
Figure 5.8 SA inversion results of the data N103. The legends are the same as figure 5.4.
0
1400
Synthetic N103
Final RMS=12.1
Picked Love
Calculated Love
Picked Rayleigh
Calculated Rayleigh
1200
1000
Vs (m/s)
Depth (m)
500
800
600
1000
400
200
1500
0
1000
2000
Vs (m/s)
3000
4000
0
5
10
15
Frequency (Hz)
Figure 5.9 Joint inversion results of the data N103. The legends are the same as figure 5.5.
141
25
20
El (m/s)
15
10
5
0
0
10
20
30
40
50
Er (m/s)
Figure 5.10 The distribution of the value
of cost functions of the joint inversion
on N103. The legends are the same as
figure 5.6.
142
0
Synthetic N104
Love RMS=2.8
Rayleigh RMS=10.2
50
Picked Love
Calculated Love
Picked Rayleigh
Calculated Rayleigh
1600
1400
1200
150
Vs (m/s)
Depth (m)
100
200
1000
800
600
250
400
300
350
200
0
1000
2000
Vs (m/s)
3000
0
4000
0
5
10
15
Frequency (Hz)
20
Figure 5.11 SA inversion results of the data N104. The legends are the same as figure 5.4.
0
Synthetic N104
Final RMS=18.8
50
Picked Love
Calculated Love
Picked Rayleigh
Calculated Rayleigh
1600
1400
1200
150
Vs (m/s)
Depth (m)
100
200
1000
800
600
250
400
300
350
200
0
1000
2000
Vs (m/s)
3000
4000
0
0
5
10
15
Frequency (Hz)
20
Figure 5.12 Joint inversion results of the data N104. The legends are the same as figure 5.5.
143
160
140
El (m/s)
120
100
80
60
40
20
0
0
20
40
60
Er (m/s)
80
100
Figure 5.13 The distribution of the
value of cost functions of joint inversion on N104. The legends are the same
as figure 5.6.
144
Chapter 6 Summary and suggestions
6.1 Summary
The dispersion curves of surface waves have been successfully used for the
characterization of the shallow subsurface for decades. Three steps are involved in
utilizing dispersion curves of surface waves for imaging geological profiles:
1) acquire high-frequency broadband ground roll,
2) create efficient and accurate algorithms organized in a basic data processing
sequence designed to extract surface wave dispersion curves from the ground roll, and
3) develop stable and efficient inversion algorithms to obtain shear wave
velocity profiles.
This dissertation focuses on the third step, the inversion of the dispersion curves
of surface waves, with the aim of searching the best procedure to get a more accurate
and reliable estimate of the geological material properties. The inversion actually is
comprised of two sub-steps:
3a) estimate a model employing the theory of surface wave propagation and
mathematical optimization;
3b) appraise the model for its accuracy, either deterministically or statistically.
This study uses surface waves contained in ambient seismic noise. The
extraction of dispersion curves of surface waves is achieved by the refraction
microtremor (ReMi) technique (Louie, 2001), licensed as SeisOpt ReMi (©, Optim Inc.)
software.
145
Forward modeling is very important for the accuracy of inversion (Chapter 2).
Based on the R/T method, this study achieved a faster algorithm (called RTgen) to
compute dispersion curves of surface waves while keeping the stability of the R/T
method. The improvements by this study include 1) computation of the generalized
reflection and transmission coefficients without calculation of the modified reflection
and transmission coefficients; 2) presenting an analytic solution for the inverse of the
4X4 layer matrix E. Compared with traditional R/T methods, RTgen, when applied on
Rayleigh waves, significantly improves the speed of computation, cutting the
computational time at least by half while keeping the stability of the traditional R/T
method.
One of major goals of this study is to find the shallow S-wave velocity structure
that explains the observed dispersion curves of surface waves. This is achieved by
geophysical inversion that involves the minimization of the cost/objective function that
characterizes the differences between observed and calculated dispersion data.
However, due to discrete nature of geophysical inversion problems, the model obtained
from the inversion of the data is therefore not necessarily equal to the true model that
one seeks. This implies that for realistic problems, inversion really consists of model
estimation followed by model appraisal.
General speaking, there are three catalogs of inversion techniques based on the
internal physical principle of the geophysical problems: linear inversions, non-linear
inversions, and trial and error methods. In this dissertation, I employed a linear
inversion technique, a non-linear inversion method, and a joint method on the
dispersion data of surface waves.
146
Chapter 3 explores the Occam’s linear inversion technique with a higher-order
Tikhonov regulization. The proposed inversion procedure contains two fundamental
components. First, an algorithm (forward engine) is required to construct a theoretical
dispersion curve based on the properties of an assumed profile. Second, an algorithm is
required to minimize the objective function which is usually the error between the
theoretical and experimental dispersion curves plus a damping term. The advantages of
the application of the Occam’s inversion are 1) the analytic form of the Jacobian matrix
given by RTgen improves the accuracy of the inversion comparing with other linear
method implementation during which the Jacobian matrix is numerically given; 2) the
model in the current iteration directly yields the model in the next iteration without
model update; 3) data noise is incorporated in the inversion process. The blind tests on a
suite of nine synthetic models and two field data sets show that the final model is
heavily influenced by a) the initial model (in terms of the number of layers and the
initial S-wave velocity of each layer); b) the minimum and the maximum depth of
profiles; c) the number of dispersion picks; d) the frequency density of dispersion picks;
and e) other noise. When these effect factors are appropriately set either known or
generated using a priori information about subsurface structure, the Occam’s inversion
finds an optimal solution that is the global minimum of a misfit function. If a priori
information is either scant or unavailable, the inversion may find a local optimal
solution.
One way to help resolve this initial-model-dependence of the Occam’s inversion
is non-linear inversion techniques due to the non-linear nature of the surface wave
dispersion phenomenon. The simulated annealing (SA) inversion technique is explored
147
in Chapter 4. Following previous developments I modified the SA inversion yielding
one-dimensional shallow S-wave velocity profiles from high frequency fundamentalmode Rayleigh dispersion curves and validated the inversion with blind tests. Unlike
previous applications of SA, this study draws random numbers from a standard
Gaussian distribution. The numbers simultaneously perturb both S-wave velocities and
layer thickness of models. The annealing temperature is gradually decreased following a
polynomial-time cooling schedule. Phase velocities are calculated using the reflectivitytransmission method. The reliability of the model resulting from our implementation is
evaluated by statistically calculating the expected values of model parameters and their
covariance matrices. Blind tests on the same data sets as these in Chapter 3 show that
the SA implementation works well for S-wave velocity inversion of dispersion curves
from high-frequency fundamental-mode Rayleigh waves. Blind estimates of layer Swave velocities fall within one standard deviation of the velocities of the original
synthetic models in 78% of cases. A hybrid method is also tried in Chapter 4. The
hybrid idea is that the models obtained by the SA can used as input to the Occam’s
inversion. Tests show that the hybrid method does not always provide better results.
In the absence of any a priori information or other constraints, the SA inverted
models, along with the uncertainties, give the analyst a good idea of the underlying Swave velocity structure. However, dispersion curves of fundamental mode Rayleigh
waves alone do not contain sufficient information to uniquely determine a model. The
velocity-depth trade-off gives rise to model non-uniqueness.
To reduce the problem of the model non-uniqueness, in Chapter 5 I explore the
possibility of using the fundamental-mode Love wave dispersion curves to constrain the
148
Rayleigh wave inversion by the SA optimization. The SA technique described in
Chapter 4 is applied on the dispersion data of both fundamental-mode Love and
Rayleigh waves with equal weighting factor. Three synthetic tests show that Love wave
constraints result in significant improvement of inverted model in terms of resolution of
low velocity zones and high velocity contrasts.
6.2 Suggestions
A natural extension of the joint inversion method will be to add both
fundamental Love wave and higher-mode Rayleigh wave as constraints on Rayleigh
wave inversion. Higher modes of surface wave dispersion curves are independent from
the fundamental-mode phase velocities. Using higher modes as constraints presents
several advantages. First, higher modes have faster phase velocities, thus can penetrate
a greater depth. They exist under a specific frequency condition (Aki and Richards,
2002). It has been reported that the generation of higher modes has been associated with
presence of a velocity reversal (Stokoe et al., 1994) and that higher mode surface
waves, when trapped in a layer, are much more sensitive to the fine structure of the Swave velocity field (Kovach, 1965). Second, in some situations higher modes take more
energy than the fundamental mode does in a higher frequency range, which means the
fundamental mode data may not be available in the higher frequency range and higher
modes are the only choice (Xia et al., 2003). Third, reliable dispersion curves of higher
modes have been observed with ReMi techniques (personal communication with
Pullammanappallil, 2007). Fourth, there is no significant increase in field acquisition
149
effort with ReMi surveys. Finally, current forward code (RTgen) is able to calculate the
dispersion curves of higher modes.
Beaty et al. (2002) used a different implementation of SA to invert fundamental
and higher-mode Rayleigh wave dispersion curves for an S-wave velocity profile. Xia
et al. (2003) used a linear inversion method to invert high frequency surface waves with
fundamental and higher modes of Rayleigh waves. Their tests showed that higher mode
data stabilize the inversion procedure and increase the resolution of the inverted S-wave
velocities. However, errors of the inverted S-wave velocities are large (Xia et al., 2003).
Using both higher modes of Rayleigh waves and fundamental-mode Love waves may
solve the problem.
Two dimensional S-wave velocity structures are desirable for seismic-resistant
design of crucial buildings. This presents more challenges on data acquisition,
dispersion curve extraction, and 2D model inversion.
3D acquisition of microtremors is usually made on arrays of instruments placed
in two-dimensional configurations such as triangle and circle. Extraction of the phase
velocities can be done using beam-forming or frequency-wavenumber (f-k) methods
(e.g., Horike, 1985; Liu et al., 2000), or by using the SPAC method first proposed by
Aki (1957) and now experiencing a resurgence of interest (e.g., Okada, 2003; Asten,
2005a, 2005b).
Optim acquires 3D ambient noise records by successively deploying
independent ReMi linear arrays in two-dimensional configurations such as cross and
star shape. Assuming a consistent noise wavefield, successively recording of surface
waves on each linear line rather than simultaneously on the two-dimensional field
150
configurations for analysis is sufficient. The phase velocities are independently
extracted for each line using ReMi processing routine and independently inverted. The
final 2D model is obtained by extrapolating these independently inverted 1D profiles.
However, these independently inverted profiles are significantly different even for an
area where the lateral S-wave velocity heterogeneity is small.
One possible way to reduce the problem is 3D τ-p transformation from time
domain (x, y, t), to τ-p domain (px, py, τ) followed by 1D Fourier transformation to f-p
domain (px, py, ω). This technique has been tested on synthetic data (Donati and Matin,
1995). The new 3D τ-p transformation provides opportunity for simultaneous inversion
of a 2D S-wave velocity model.
It is possible to implement the SA in a parallel computer. Under a fixed
temperature, a parent model is perturbed on a master node to generate a serial of child
models which is then distributed to nodes for forward calculation of dispersion curves.
These calculated dispersion curves are returned to the master node. Only one model,
which is the basis for next iteration, is selected by comparing these curves and the
conditional probabilities at the temperature. The whole process is repeated until the stop
criterion is satisfied. The forward calculations of dispersion curves, which is the most
CPU-consuming part of SA, are performed simultaneously on each node. The
implementation of SA in a parallel computer will significantly speed up the inversion.
151
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162
APPENDIX A: Matrices for Rayleigh waves
Rayleigh waves consist of P and SV-waves. We give the explicit expression of
the layer matrices, the inverse layer matrices, and downgoing- and upgoing-waves
matrices (equation (2.31) ) for Rayleigh waves here. The expression is derived by Luco
and Apsel (1983) and Hisada (1994). The explicit expression clearly shows that equation
(2.31) is completely free from the exponential growth term with depth.
The motion-stress vectors of the jth layer are expressed by the downgoing and
upgoing P and SV waves in the following matrix form (the same as equation (2.31) ):
0  Cdj 
 D j ( z )   E11j E12j   Λ dj ( z )
j
j
j
=
 j =E Λ C
Sj z   j
j 
j
C
(
)
0
(
)
E
E
Λ
z

  21
u
22  
 u
( A1)
where
 D pj ( z ) 
D ( z ) =  j  ( A2)
 Ds ( z ) 
j
are the motion/displacement vectors for P- and SV-waves at depth z.
S j ( z)
S j ( z ) =  pj  ( A3)
 Ss ( z)
are the stress/traction vectors for P- and SV-waves at depth z.
E
Ej = 
E
j
11
j
21
 −1

j
E12   −γ 1 j
=
E22j   2µ jγ j
 j j
 χ 1 µ
ν1j
1
− χ1 j µ j
−2µ jν j


−1 
( A4)
χ1 j µ j 

−2µ jν j 
ν1j
−1
γ 1j
−2 µ j γ j
χ1 j µ j
is the jth layer matrix, representing the effects of the elastic properties and the vertical
wavenumber in the jth layer.
 e −γ ( z − z
Λ ( z) = 

0
j
j
d
j −1
)


j
j −1
e −ν ( z − z ) 
0
 e −γ ( z
Λ ( z) = 
 0
j
j
u
j
−z)
0
e−ν
j
(z
j


−z)

( A5)
163
are the phase delay matrices for downgoing and upgoing P and SV-waves.
Cdpj 
C = j 
 Cds 
j
d
Cupj 
C = j 
 Cus 
j
u
( A6)
are the amplitude vector matrices for downgoing and upgoing P and SV-waves.
ν j = k 2 − (ω β j ) 2 with Re {ν j } ≥ 0
ν1j = ν j k
γ j = k 2 − (ω α j )2 with Re {γ
γ 1j = γ
j
j
}≥0
k
χ1 j = 2k − (ω β j )2 k
( A7)
where k is wavenumber, ω is frequency, βj is shear velocity of the jth layer, αj is
compressional velocity of the jth layer.
Analytical solution of the 4X4 layer matrix E in equation A4 is
 E11j
 j
 E21

j
 −2k µ

 µ j χ1 j
−
1
−
E12j 
1
ν1j

=

µ j (4k − 2 χ1 j ) 
E22j 
j
 −2k µ

 µ j χ1 j
−
 ν1j
µ j χ1 j
γ 1j
1
γ 1j
2k µ j
1
µ j χ1 j
−
γ 1j
1
− j
γ1
−2 k µ j
−1

−1 

1 
− j
ν1 

−1 

1 
− j
ν1 
( A8)
164
APPENDIX B: Matrices for Love waves
Love waves consist of SH-waves only. The motion-stress vectors of the jth layer
are expressed by the downgoing and upgoing SH waves in the following matrix form (the
same as equation (2.45) ):
 D j ( z )   E11j
S j z  =  j
 ( )   E21
0  Cdj 
E12j   Λ dj ( z )
j
j
j
 j = E Λ C
j 
j
Λ u ( z )  Cu 
E22   0
( B1)
Where D j ( z ) , S j ( z ) , Cdj and Cuj have the same physical meaning as these in Appendix
A but only for SH-waves.
 E11j
 j
 E21
E12j   1
=
E22j   − µ jν j
1 
( B 2)
µ jν j 
and
 Λ dj ( z )
0  e −ν ( z − z

=
Λ uj ( z )  
0
 0
j
j −1
)


j
j
e−ν ( z − z ) 
0
( B3)
The explicit solutions for generalized reflection/transmission coefficients are
given as
Tˆdj =
Λ dj ( z j )
a + bΛ uj +1 ( z j ) Rˆ duj +1
Rˆ duj = (b + aΛ uj +1 ( z j ) Rˆ duj +1 )Tˆdj
( B 4a )
( B 4b)
where
a = 1 2 + µ j +1ν j +1 2 µ jν j
( B5a )
b = 1 2 − µ j +1ν j +1 2 µ jν j
( B5b)
165