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 high-

frequency 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

ii

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 Occams 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 Occams inversion, the non-

linear simulated annealing (SA) inversion technique is proposed in Chapter 4.

Following previous developments I modified the SA inversion yielding one-

dimensional shallow S-wave velocity profiles from high frequency fundamental-

mode 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

iii

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 Occams 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.

iv

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

v

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

vi

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

vii

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

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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 Occams 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

ix

Figure 3.6 The depth-averaged velocities in m/s against the known values for

Occams 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

x

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

xi

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

xii

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 Gutenbergs 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 Occams 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

xiii

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 cant 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.

Pullammanappallils 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

xiv

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.

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 seismic-

resistant 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 Earths

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

1 Hz) fnh (0.5 Hz to 1 Hz) - 10 Hz

Origin Ocean Traffic / Industry / Human activity Incident wavefield Surface waves Surface + body Amplitude variability

Related to oceanic storms

Day / Night, Week / week-end

Rayleigh / Love issue

Incident wavefield predominantly Rayleigh

Comparable amplitude slight indication that Love waves carry a little more energy

Fundamental / Higher mode issue

Mainly Fundamental

Possibility of higher modes at high frequencies (at least for 2-layer case)

Further Comments Local wavefield may be different from incident wavefield

Some monochromatic waves related to machines and engines. The proximity of sources, as well as the short wavelength, 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 mediums 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, near-

surface 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 S-

wave 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 clich 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 body-

wave 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 multiple-

station 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 Receiver in borehole HVSR Surface source

(Boore & Thompson,

2007)

Receiver in cone penetrometer SASW

DASW Active sources

MASW Suspension P-S

logger (Nighbor and Imai, 1994) F-K

SPAC Passive sources ReMi

Downhole source Crosshole

(ASTM, 2003)

Multiple stations

Combined active and passive sources MASW

Acronyms

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) 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 Earths

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 three-

component 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 trucks 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 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). One limitation in

practice is that 2D instrument arrays are usually not dense enough to resolve near-

surface 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://sesame-

18

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 Toksz, 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 time-

domain velocity results of microtremor recordings on a linear array into the frequency

domain by a two-dimensional slant transformation (p-) followed by a one-dimensional

Fourier transformation on . 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, one-

dimensional, 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

4.0

Tim

e, s

ec

0.0 20.0Trace Sequence

Rayleigh waves

Figure 1.1 Three steps involved in utilizing dispersion curves of surface waves for imaging geologic profiles.

Slow

ness

(s/m

)

Frequency (Hz) 25.0

0.01

0.0 0.0

ReMi Spectral Ratio 2.5 0.0

5.0 10.0 15.0 20.0

0.002

0.004

0.006

0.008

400 600 800

0

10

20

30

40

50

60

Vs (m/s)

Dept

h (m

)

0 0.2 0.4 0.6 0.80

100

200

300

400

500

600

700

Period (s)

Vs (m

/s)

1. Acquisition

2. Extraction

3. Inversion

Synthetic example

Dispersion picks (squares) on p-f image

1D S-wave velocity profile with uncertainty

Observed and calculated dispersion curves

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 homog-enous, 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 veloc-ity is constant. When the properties changes with depth (right) the phase velocity depends on the wavelength, forming a dispersion curve.

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 veloc-ity 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.

Normal

Inverse

Irregular

f (Hz)

f (Hz)

f (Hz)

c (m

/s)

c (m

/s)

c (m

/s)

(m/s)

(m/s)

(m/s)

z (m

)z

(m)

z (m

)

24

Figure 1.7. Modes of surface waves. For the same frequency, higher modes penetrate deeper.

0

5

10

15

20

25

30

35

40

45

50Fundamental-mode

Dept

h (m

)

1st-higher-mode 2nd-higher-mode

frequency

phas

e ve

loci

ty

1st mode

2nd mode3rd mode

4th mode

fC1 fC2 fC3 frequency

phas

e ve

loci

ty

f f f

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.

Fundamental-mode

1st-higher 2nd-higher

3rd-higher

25

Figure 1.10. A typical ReMi analysis.

a) linear array recording ambient seismic noise

b) field deployment

b) dispersion curve fitting

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

00 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Shear-Wave Velocity, ft/s

Dep

th, f

t

Vs100' = 1299 ft/s

Dispersion Curve Showing Picks and Fit

0

500

1000

1500

2000

2500

3000

0 0.05 0.1 0.15 0.2 0.25 0.3

Period, s

Ray

leig

h W

ave

Phas

e V

eloc

ity,ft

/s

Calculated Dispersion

Picked Dispersion

Figure 1.9. A typical ReMi field configuration.

noise

c) 1D S-wave velocity profilea) p-f image with dispersion picks

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 Numerical methods Transfer matrix method

(Thomson, 1950; Haskell, 1953) Finite element method

(Lysmer and Drake, 1972) Stiffness matrix method

(Kausel and Roesset, 1981) Finite difference method

(Boore, 1972) Reflection-transmission coefficient

(Kennett, 1983; Luco and Aspel,1983) 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 S-

wave 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 and elastic constants including shear

modulus (), Youngs modulus (E), Bulk Modulus (K), and Poissons ratio (). Those

constants, along with the two Lame parameters ( and shear modulus ) 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:

2

,2 (2.1) i

ji j iu ft

= +

where ui the displacement in ith component, the density, 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 Definition Notes

Youngs modulus E longitudinal stress longitudinal strain Free transversal deformation

Shear modulus shear stress shear strain

Poissons ratio longitudinal stress transversal strain Free transversal deformation

Bulk Modulus K idrostatic stress volumetric strain

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 and wavenumber k:

( )( , , , ) ( ) (2.2) i kx tx y z t z e =u Z

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

or or E or K or 2

1 2 = (1 )(1 2 )

E = + 23

K =

2(1 )E = +

K (3 2 )

3K += 2 (1 )

3(1 2 )K

+= 3(1 2 )EK = K

E (3 2 )E

+= + 2 (1 )E = + E 9

3KEK

= +

2( )

= + 3 2

2(3 )KK

= +

Table 2.4 Harmonic wave parameters

Frequency f Hz 1

2cf

T= = =

Period T s 1 2Tf c

= = =

Velocity c m/s c fT k = = =

Angular frequency radian/s 22 f ckT = = =

Wavelength m 2f cT

c k = = =

Wavenumber k radian/m2 2 fk

c c

= = =

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 (z), (z) and density (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:

( )1

0( , , ) (2.3 )

0

i kx t

uv l k z e aw

= = =

The stress components associated with the above displacement are:

( )1

( )1

0

(2.3 )

xx yy zz zx

i kx tyz

i kx txy

dl e bdz

ik l e

= = = = = =

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 | 0zu = ;

BC2 (displacement continuity condition): Displacement must be continuous

across any layer boundary 1| |i z d i z du u= + == ;

BC3 (traction continuity condition): Traction must be continuous across any

layer boundary 1| |i z d i z d = + == ; BC4 (zero traction at the free surface): Traction must be zero at the free surface

0| 0z = = .

32

After applying BC1-BC4, equation (2.3) could be rewritten as a set of first-order

ordinary differential equations,

( , , ) ( , , )1 2

( , , ) 2 22 ( ) ( ) ( , , )1

(2.4)( )

k z k z

k zz z k z

dl ldz

dl k ldz

= =

Or in a matrix form

( )( )

1, , 01 1( )2 2, , ( )- ( ) 02 2

(2.5) k z

zk z k z z

l ldl ldz

=

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:

( )( , , )1

( )( , , )2

0 (2.6 )

i kx tk z

i kx tk z

u r ev aw ir e

= = =

The stress components associated with the above displacement are:

2

2

1

( )1

( )1

( ) (2 3

0

( 2 )

( )

yz xy

dr i kx txx dz

dr i kx tyy dz

dr i kx t izx dz

i k r e

i k r e

kr e r e

= = = + + = +

= =2

)

( ) ( )1 4

(2.6 )

( 2 )

kx t

dr i kx t i kx tzz dz

b

i k r e ir e

= + + =

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

10 01 1( )( ) 10 02 2( ) 2 ( ) ( ) 2 ( )

( )2 23 3( )- ( ) 0 0( ) 2 ( )

24 40 ( ) - 0

(2.7)

4 ( )[ ( ) ( )]( )( ) 2 ( )

kz

k zz z z z

k zk z zz z

z k

r rr rdr rdzr r

z z zwhere zz z

+ +

+

= += +

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, , non-

trivial solutions exist only for special values of the wavenumber, k. These possible values

k1(), k2(), kn() 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

( ) ( ) ( ) (2.8) d z z zdz

=f G f

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

du u

d ud

R TT R

where R represents reflection coefficient and T represents transmission coefficient. Sub-

index d means down-going waves; u up-going waves. For example, duR is the

reflection coefficient of incident down-going SH-wave to reflected up-going SH-wave at

interfaces. dT 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

R R T Tdpp dsp upp uspR R T Tdu u dps dss ups ussT T R Rdpp dsp upp uspd udT T R Rdps dss ups uss

=

R TT R

Sub-index d means down-going waves; u up-going waves; p P-waves; and s SV-

waves. dpsR is the reflection coefficient of incident down-going P-wave to reflected SV-

wave at interfaces. dpsT 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 0z > in which elastic

moduli j j 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

101 1

2 2- 02 2

(2.9) jj j

j j

j jk

l lddz l l

=

for SH waves and

35

10 01 1

10 02 22 2

2 23 3- 0 02

4 420 - 0

(2.10)

j

j

j j j j

jj j

j j

j

j jk

j jk

j jkk

j j

k

r rr rd

dz r rr r

+ +

+

=

for P-SV waves.

where 1j jz z z < < , 1, 2,3, , , 1j N N= +" and 0 1 10 j N Nz z z z z += < < < < < < = " " In matrix format, equation (2.9) and (2.10) could be summarized as

( ) ( ) ( ) (2.11) j

j jd z z zdz

=f G f

where ( )j zf 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):

( ) (2.12) j j j jz =f A B D

for 1, 2,3, , , 1j N N= +" . where jA and jB are know matrices given below. But jD are unknown vectors to be determined. For SH waves we have

1 1 0

0(2.13)

jj

j j j j j j

Szj j j e

z Se

= A B D

1, 2,3, ,j N= " . where 2

22 k

=

For P-SV waves we have (Aki and Richards, 2002, p.276 equation (7.55) )

36

0 0 0

0 0 0

2 2 0 0 02 2

0 0 0

1 (2.14)

jj j j j j j j

jj j j j j j j

j j j j j j j j j j j j j j

j j j j j j j j j j j j jj

zek k Pzj j j k k e S

k k z Pek k Sze

= A B D

1, 2,3, ,j N= " . where 2

22 k

=

BC1-BC4 is to determine the unknown matrix jD for each layer. The continuity

condition implies

1 1( ) ( ) (2.15 ) j j j jz z a+ +=f f

for 1, 2,3, ,j N= " . The radiation condition requires

( ) 0 (2.15 ) N z bf as z

Careful observation reveals that matrix jA contains some layer-specified

constants, like ,j j , 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

matrix multiplications. The matrix jB contains some depth-growth terms, like ,z ze e ,

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 layer-

specified 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

11( )1 1 0

( )0

(2.16)= = j j j

j j j j j jj j

jj j z Cz z e S sdj j j j jej jzz z Ce Se su

E E C

1, 2,3, ,j N= " for SH waves . 1

1

11

( ) 0 0 0

( )0 0 0

2 2 ( )0 0 02 2 ( )0 0 0

j j jj

j jj j

j j j jj jj j

j j j j j j j j j j j j jj

j j j j j j j jj j

j j j j

ze Pz zek k

ze Sz zk k e

k k z z ze e Pk k z ze ze S

(2.17)

=

=

jCpdjCj j sdjCpujCsu

j j j

E

E C1,2,3, ,j N= " for P-SV waves.

where E is layer matrix, is phase delay matrix, and C is amplitude vector matrix.

It is noted that exponential terms in the matrix jB in equations (2.13) and (2.14)

has been modified by multiplying the layer-specified constants 1 1

, , ,j j j j j j j jz z z ze e e e

.

Accordingly, the unknown matrix jD in equations (2.13) and (2.14) has absorbed these

factors. Absorbing of these constants greatly increase the stability of the algorithm. It

should be pointed out that the elements of the matrix jD in equations (2.13) and (2.14)

represent amplitudes of downgoing and upgoing waves. The matrix jC in equations

(2.16) and (2.17) is the matrix jD that has absorbed the layer-specified constants.

Therefore, the elements of the matrix jC 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

( )11 1221 22

01

02

(2.18 ) j j j

dj j j

u

j jCdE Ej jE E Cu

la

l

=

1, 2,3, ,j N= " 1 1

11

1 121

1 11

1 12

(2.18 ) N N

d

N Nd

N NE CdN NE Cd

lb

l

+ +

+ +

+ ++ +

=

1j N= + . 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 1 0NCu+ = .

For an arbitrary jth interface, the modified reflection and transmission

coefficients for SH waves are denoted as ,j jdu dR T . They are defined as

1 1

1 (2.19 ) j j j j jd d d ud uj j j j ju du d u u

C T C R Ca

C R C T C

+ +

+ = + = +

1, 2,3, , 1j N= " 1

(2.19 ) N N Nd d dN N Nu du d

C T Cb

C R C

+ = =

j N= Continuity condition requires that

( )( ) ( )( ) 11 111 12 11 12 11 1 121 22 21 221 1 01 1 ( ) 01 0 ( )0 12 2 (2.20) = j jj j j jj j d dd j jj j j jj juu uj j C CE E E Ezj j zE E E EC Cl ll l ++ + ++ + ++ + = After rearranging equation (2.20) and comparing with equation (2.19), we find

39

( ) ( ) ( )( )( )

1 111 12 11 12

1 1 121 22 21 22

1 1

1 1 11 1 1 1

1( ) 0

0 ( )

1 ( ) 02

0 ( )2(2.21 )

j j j j j j j jd ud dj j j j jj j j

u udu

j jj j j j j jd

j j j j j jj j j j j ju

T R E E E E z

R T E E E E z

z

za

+ ++ + +

+ ++ + ++ + + +

+

=

=

1, 2,3, , 1j N= "

( )1 1111 1212121 22

1( )

( )(2.21 )

N N N NN Nd dN N N NN Ndu d

T E zE E

R E zE Eb

++

=

j N=

The generalized reflection and transmission coefficients ( ,j jdu dR T ) for SH waves

are defined as following:

1 (2.22 )

j j jd d d

j j ju du d

C T Ca

C R C

+ = =

1, 2,3, , 1j N= " 1

(2.22 ) 0

N N Nd d d

Ndu

C T Cb

R

+ = =

j N= 1 0 1 (2.22 ) d ud uC R C c=

0j = for the free surface.

It is noted that (2.23) N N N Nd d du duT T and R R= = .

At the free surface the traction-free condition yields

1 1 0 1 11 1 1 1

1 1 1 1 1 1 0 ( )2221 22 1 1 1

21

(0)0 (0) (0) (2.24) = z z zud u u udET E C E C R e eE

= = + = =

40

Substituting equation (2.22b) in equation (2.22a), we can obtain the following

recursive formula for computing other generalized R/T coefficients

1 1

1

(1 )(2.25)

j j j jd ud du d

j j j j jdu ud u du d

T R R T

R R T R T

+

+ = = +

1, 2,3, , 1j N= " Inside the top layer the definition of the generalized reflection and transmission

coefficients gives us (equations (2.22))

1 0 1

1 1 1

(2.26)

d ud u

u du d

C R C

C R C

= =

Combining the above two equations leads to 0 1 1 (1 ) 0 (2.27) ud du dR R C =

The existence of the non-trivial solution leads to the following implicit dispersion

equation for SH waves 0 1 1 0 (2.28) ud duR R =

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.

According to equation (2.27), 1 ( )d nC v can take any non-zero value, where nv is

the phase velocity of the nth higher mode. We take unity. Then we have

1

1 1 1

( ) 1(2.29)( ) ( )

d n

u n du d n

C v

C v R C v

= =

Applying equation (2.26) into (2.22), we find

1 1 1

1 1 1

( ) ( ) ( ) ( )(2.30 )

( )

j j jd n d n d n d n

j j ju n du d

C v T v T v T va

C v R C

+

+ + + = =

"

41

1, 2,3, , 1j N= " 1 1 1 ( ) ( ) ( ) ( ) (2.30 ) N N Nd n d n d n d nC v T v T v T v b+ = "

j N=

Finally, substituting these solved ( )j jd uC C for each mode into equation (2.18), we obtain the non-trivial solutions ( )1 2j jl l 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

( )( )

11 12

21 22

( ) 0(2.31)

0 ( )

jj j jj z d j j jdjj j jj z uu

zz

= = CD E E

E CCS E E

See appendix A for each term.

For an arbitrary jth interface, the modified reflection and transmission

coefficients for Rayleigh waves are denoted as ( , , , )j j j jdu ud d uR R T T and defined by the

following equations:

1 1

1 (2.32 ) j j j j jd d d ud uj j j j ju du d u u

a+ +

+ = + = +C T C R CC R C T C

for 1, 2,3, , 1j N= " and 1

(2.32 ) N N Nd d dN N Nu du d

b+ = =

C T CC R C

42

for j N=

where j jR Rj dpp dsp

du j jR Rdps dss

=R , j jR Rj upp uspud j jR Rups uss

=R ,

j jT Tj dpp dspd j jT Tdps dss

=T , and j jT Tj upp uspu j jT Tups uss

=T . Sub-index d

means down-going waves; u up-going waves; p P-waves; and s SV-waves. jdpsR is

the reflection coefficient of incident down-going P-wave to reflected SV-wave at

interface j. jdpsT 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 ( ,j jdu dR T )

for P-SV waves can be found by applying the continuity condition at an arbitrary jth

interface. Then we have

( )( )

( )( )

1 111 12 11 12

1 1 121 22 21 22

1 1( )

11 ( )

0000

j j j jj j

dj j j j j j

u

j j j jz z z d dj jj j zz z u u

+ +

+ + +

+ + ++

= = D D C CE E E E I

E E E EC CS S I

After rearranging, we obtain the explicit expressions of the modified R/T matrices as

follows:

1 111 12 11 12

11 121 22 21 22

1 ( ) 0(2.33 )

0 ( )

j j j jj j j jd ud dj j j jj j j jdu u u

za

z

+ +

++ +

= T R E E E ER T E E E E

for 1, 2,3, , 1j N= " 1

1111 121

2121 22

1 ( )(2.33 )

( )

N N j j jN Nd ud dN N j j jN Ndu u d

zb

z

+

+

= T R E E ER T E E E

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

coefficients for Rayleigh waves are denoted as ( , )j jdu dR T and defined by the following

equations:

1 (2.34 )

j j jd d d

j j ju du d

a+ = =

C T C

C R C

for 1, 2,3, , 1j N= " 1

(2.34 )

N N Nd d d

Ndu

b+ = =

C T C

R 0

for j N= Comparing equations (2.32) and (2.34) we find

(2.35) N N N Nd d du duand= =T T R R .

Substituting equation (2.34) in equation (2.33), we obtain the recursive formula

for computing other generalized R/T coefficients as

1 1

1

( )(2.36)

j j j jd ud du d

j j j j jdu ud u du d

+

+ = = +

T I R R T

R R T R T

for 1, 2,3, , 1j N= "

Starting from the last interface where Ndu =R 0 , we can use equation (2.36) to find the

generalized reflection and transmission coefficients ( ,j jdu dR T ) 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

( )1 1 1 0 1 121 22 0 ( (0) ) (2.37) u du d= +S E E R C Equation ( 2.37) has non-trivial solutions only for some particular phase velocities that

satisfy the following secular equation:

1 1 0 121 22

det( (0) ) 0 (2.38) u du+ =E E R

Equation ( 2.38) is called the secular function for Rayleigh waves. Therefore, the roots of

this equation are the phase velocities for modes that potentially exist.

1 ( )d nvC has infinite solutions to satisfy equation (2.37). Lets

0 1 ( ) ( ) ( ) (2.39) n ud n du nv v v= G I R R

We take one of them as our starting point. Thus we find that

1 122 2

11 12

1 112 2

11 12

( )

(2.40)( )

pd n

pd n

GC vG GGC v

G G

= + = +

and

1 1 1( ) ( ) (2.41 ) u n du d nv v a=C R C

Correspondingly, we have

1 1 1

1 1 1

( ) ( ) ( ) ( )(2.41 )

( )

j j jd n d n d n d n

j j ju n du d

v v v vb

v

+

+ + + = =

C T T T

C R C

"

1, 2,3, , 1j N= "

45

1 1 1 ( ) ( ) ( ) ( ) (2.41 ) N N Nd n d n d n d nv v v v c+ =C T T T"

j N=

Finally, substituting these solved ( )j jd uC C for each mode into equation (2.31) yields the non-trivial solutions ( )1 2 3 4j j j jr r r r , 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 metho