simultaneous inversion of rayleigh phase velocity and attenuation

Georgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of Technology

Simultaneous Inversion

of Rayleigh Phase

Velocity and Attenuation

for Near-Surface Site

Characterization

Carlo G. Lai, PhD

Glenn J. Rix, PhD

National Science Foundation and

U.S. Geological Survey

July 1998

School of Civil andEnvironmental Engineering

i

ACKNOWLEDGMENTS

This research was supported by the National Science Foundation under Grant No.CMS-9402358 and the U.S. Geological Survey under Award No. 1434-95-G-2634. Anyopinions, findings, and conclusions or recommendations expressed in this material arethose of the authors and do not necessarily reflect the views of the National ScienceFoundation and the U.S. Geological Survey. The authors are grateful to Dr. Clifford J.Astill of the National Science Foundation and Dr. John D. Sims of the U.S. GeologicalSurvey for their support and encouragement.

iii

TABLE OF CONTENTS

ACKNOWLEDGMENTS i

LIST OF TABLES vii

LIST OF ILLUSTRATIONS ix

SUMMARY xvii

CHAPTER

1 INTRODUCTION 11.1 Motivation......................................................................................................... 11.2 Research Objectives......................................................................................... 41.3 Dissertation Outline ........................................................................................ 8

2 DYNAMIC BEHAVIOR OF SOILS ........................................................................92.1 Introduction.......................................................................................................92.2 A Survey on Modeling Soil Behavior.......................................................... 10

2.2.1 Overview............................................................................................ 102.2.2 The Continuum Mechanics Approach ......................................... 102.2.3 The Discrete Mechanics Approach............................................... 13

2.3 Phenomenological Modeling of Soil Behavior.......................................... 152.4 Experimental Observations.......................................................................... 16

2.4.1 Overview............................................................................................ 162.4.2 Threshold Strains ............................................................................. 162.4.3 Stiffness Degradation and Entropy Production.......................... 20

2.5 Constitutive Modeling and Model Parameters.......................................... 262.5.1 Overview............................................................................................ 262.5.2 Linear Viscoelastic Constitutive Models....................................... 262.5.3 Low-Strain Kinematical Properties of Soils (LS-KPS)............... 352.5.4 Experimental Measurements of LS-KPS ..................................... 50

3 RAYLEIGH WAVES IN VERTICALLY HETEROGENEOUS MEDIA... 573.1 Introduction.................................................................................................... 573.2 Rayleigh Eigenvalue Problem in Elastic Media ......................................... 58

3.2.1 Solution Techniques ........................................................................ 633.3 Effective Rayleigh Phase Velocity in Elastic Media.................................. 673.4 Rayleigh Green’s Function in Elastic Media.............................................. 713.5 Rayleigh Variational Principle in Elastic Media......................................... 76

3.5.1 Modal Rayleigh Phase Velocity Partial Derivatives..................... 783.5.2 Effective Rayleigh Phase Velocity Partial Derivatives ................ 833.5.3 Attenuation of Rayleigh Waves in Weakly Dissipative

Media.................................................................................................. 87

iv

3.6 Rayleigh Eigenvalue Problem in Viscoelastic Media.................................903.6.1 A Solution Technique ......................................................................91

3.7 Effective Phase Velocity and Green’s Function in ViscoelasticMedia ................................................................................................................99

3.8 Modal and Effective Partial Derivatives in Viscoelastic Media .............101

4 SOLUTION OF THE RAYLEIGH INVERSE PROBLEM...........................1054.1 Introduction ..................................................................................................1054.2 Ill-Posedness of Inverse Problems ............................................................1064.3 Coupled Versus Uncoupled Analysis ........................................................1074.4 Inversion Strategies ......................................................................................1084.5 Occam’s Algorithm ......................................................................................1104.6 Uncoupled Inversion ...................................................................................118

4.6.1 Overview ..........................................................................................1184.6.2 Uncoupled Fundamental Mode Analysis ....................................1204.6.3 Uncoupled Equivalent Multi-Mode Analysis..............................1214.6.4 Uncoupled Effective Multi-Mode Analysis.................................122

4.7 Coupled Inversion........................................................................................1234.7.1 Overview ..........................................................................................1234.7.2 Coupled Fundamental Mode Analysis.........................................1264.7.3 Coupled Equivalent Multi-Mode Analysis ..................................1264.7.4 Coupled Effective Multi-Mode Analysis .....................................127

5 RAYLEIGH PHASE VELOCITY AND ATTENUATION MEASUREMENTS ..................................................................................................1295.1 Overview........................................................................................................1295.2 Conventional Measurements Techniques.................................................130

5.2.1 Phase Velocity Measurements.......................................................1315.2.2 Attenuation Measurements ...........................................................134

5.3 New Measurements Techniques ................................................................1385.3.1 Uncoupled Measurements .............................................................1385.3.2 Coupled Measurements..................................................................142

5.4 Statistical Considerations.............................................................................1435.4.1 Overview ..........................................................................................1435.4.2 Statistical Aspects of Conventional Measurements...................1455.4.3 Statistical Aspects of New Measurements Techniques.............147

5.4.3.1 Uncoupled Analysis ........................................................1475.4.3.2 Coupled Analysis.............................................................148

5.4.4 Statistical Aspects of Uncoupled Rayleigh Inversion................1525.4.5 Statistical Aspects of Coupled Rayleigh Inversion.....................154

6 VALIDATION OF THE ALGORITHMS..........................................................1576.1 Overview........................................................................................................1576.2 Lamb’s Problem............................................................................................1576.3 Numerical Simulations.................................................................................162

v

6.3.1 Uncoupled Analyses....................................................................... 1716.3.1.1 UFUMA Inversion Algorithms.................................... 1716.3.1.2 UEQMA Inversion Algorithms................................... 178

6.3.2 Coupled Analyses ........................................................................... 1886.3.2.1 CFUMA Inversion Algorithms.................................... 1886.3.2.2 CEQMA Inversion Algorithms ................................... 192

6.3.3 Results and Discussion .................................................................. 199

7 EXPERIMENTAL RESULTS............................................................................... 2077.1 Overview ....................................................................................................... 2077.2 Treasure Island Naval Station Site ............................................................ 2077.3 Uncoupled Inversion................................................................................... 2097.4 Coupled Inversion ....................................................................................... 2167.5 Results and Discussion................................................................................ 217

8 CONCLUSIONS AND RECOMMENDATIONS........................................... 2218.1 Conclusions................................................................................................... 2218.2 Recommendations for Future Research................................................... 227

APPENDIX A - Elliptic Hysteretic Loop in Linear Viscoelastic Materials ................. 229A1 Harmonic Constitutive Relations ......................................................................... 229A2 Energy Dissipated in Harmonic Excitations ...................................................... 230A3 Principal Axes of the Elliptic Hysteretic Loop .................................................. 231

APPENDIX B - Effective Rayleigh Phase Velocity Partial Derivatives ........................ 233

APPENDIX C - Description of Computer Codes ........................................................... 241C1 UFUMA (Uncoupled-Fundamental-Mode-Analysis) ....................................... 241C2 UEQMA (Uncoupled-Equivalent-Multi-Mode-Analysis)................................ 243C3 CFUMA (Coupled-Fundamental-Mode-Analysis)............................................ 244C4 CEQMA (Coupled-Equivalent-Multi-Mode-Analysis) .................................... 245

BIBLIOGRAPHY.................................................................................................................. 247

vii

LIST OF TABLES

Number Page

2.1 Phenomenological Soil Responses to Cyclic Excitation as a Function ofShear Strain Level ............................................................................................................. 19

2.2 Measurement of Low-Strain Dynamic Properties of Soils (LS-DPS)Comparison between In-Situ and Laboratory Techniques........................................ 51

6.1 Medium Properties and Frequencies Used for Validation of the ElasticLamb’s Problem.............................................................................................................. 159

6.2 Medium Properties and Frequencies Used for Validation of theViscoelastic Lamb’s Problem........................................................................................ 161

6.3 Medium Properties Used for the Validation of the Inversion Algorithms(Case 1).............................................................................................................................. 163



6.6 Inversion Algorithms RMS Error Misfit for Case 1 Soil Profile ............................. 200



6.9 Inversion Algorithms Performance in Terms of RMS Error Misfit ...................... 204

ix

LIST OF ILLUSTRATIONS

Number Page

1.1 Seismic Energy Path in Ground Response Analysis (Modified fromEPRI, 1993) ..........................................................................................................................1

1.2 Influence of Gmax on the Acceleration Response Spectrum.........................................2

1.3 Influence of DSmin on the Acceleration Response Spectrum........................................3

2.1 Cause-Effects Relationships in Soil Response to Dynamic Excitations .................. 17

2.2 Dependence of Threshold Shear Strains from Plasticity Index(After Vucetic, 1994) ........................................................................................................ 20

2.3(a) Effect of Mean Effective Confining Stress on Modulus DegradationCurves for Non-Plastic Soils (PI = 0) (After Ishibashi, 1992) .................................. 21

2.3(b) Effect of Mean Effective Confining Stress on Modulus DegradationCurves for Plastic Soils (PI = 50) (After Ishibashi, 1992) .......................................... 22

2.4 Modulus Degradation Curves for Soils of Different Plasticity(After Vucetic and Dobry, 1991) ................................................................................... 23

2.5 Dependence of Energy Dissipated within a Soil Mass on Cyclic ShearStrain for Soils of Different Plasticity (After Vucetic and Dobry, 1991)................. 23

2.6 Frequency Dependence of the Energy Dissipated Within a Soil Mass(After Shibuya et al., 1995) .............................................................................................. 25

2.7 Typical Relaxation and Creep Functions for a Viscoelastic Solid............................. 28

2.8 Graphical Representation of the Components of the ComplexModulus.............................................................................................................................. 39

2.9 Stress-Strain Hysteretic Loop Exhibited by a Linear ViscoelasticModel during a Harmonic Excitation............................................................................ 40

2.10(a) Influence of Frequency on Phase Velocity of Viscoelastic Waves asPredicted by the Dispersion Relation Eq. (2.35) ......................................................... 48

2.10(b) Influence of Damping Ratio on Phase Velocity of Viscoelastic Wavesas Predicted by the Dispersion Relation Eq. (2.35)..................................................... 49

2.11 Ranges of Variability of Cyclic Shear Strain Amplitude in Laboratoryand In-Situ Tests (Modified after Ishihara, 1996)........................................................ 52

2.12 Fixed-Free Resonant Column Apparatus (Modified after Ishihara, 1996) .............. 53

3.1 Rayleigh Waves in Vertically Heterogeneous Media ................................................... 61

3.2 Rayleigh Waves Dispersion Curves in Vertically Heterogeneous Media ................. 66

x

3.3 Rayleigh Displacement Eigenfunctions in Vertically HeterogeneousMedia ...................................................................................................................................66

3.4 Geometric Spreading Function for Different Types of Media..................................75

3.5 Partial Derivatives of Rayleigh Phase Velocity with Respect to VP and VS

for an Homogeneous Medium........................................................................................82

3.6 Rayleigh Waves in Viscoelastic Multi-Layered Media ..................................................91

3.7(a) Roots of Rayleigh Secular Function in the Region C of the wR-Plane......................95

3.7(b) Roots of Rayleigh Secular Function in the Region D of the zR-Plane ....................95

4.1 Algorithms for the Solution of the Rayleigh Inverse Problem................................109

4.2 Flow-Chart of Rayleigh Simultaneous Inversion Using Occam’sAlgorithm..........................................................................................................................117

4.3 Algorithms for the Solution of the Uncoupled Rayleigh Inverse Problem ...........119

4.4 Algorithms for the Solution of the Strongly Coupled Rayleigh InverseProblem.............................................................................................................................125

5.1 Typical Configuration of the Equipment Used in SASW Testing...........................131

5.2 Source-Receivers Configuration in SASW Phase VelocityMeasurements ..................................................................................................................132

5.3(a) SASW Arrangement Using Common Receiver Midpoint Array .............................133

5.3(b) SASW Arrangement Using Common Source Array..................................................133

5.4(a) Attenuation Coefficient Computation at Treasure Island Site.................................136

5.4(b) Attenuation Coefficient Computation at Treasure Island Site.................................137

5.5(a) Geometrical Interpretation of Effective Rayleigh Phase Velocity ..........................140

5.5(b) Geometrical Interpretation of Effective Rayleigh AttenuationCoefficient ........................................................................................................................141

6.1(a) Comparison of Solutions for the Elastic Lamb’s Problem (Case 1)........................159

6.1(b) Comparison of Solutions for the Elastic Lamb’s Problem (Case 2)........................160

6.1(c) Comparison of Solutions for the Elastic Lamb’s Problem (Case 3)........................160

6.2(a) Comparison of Solutions for the Viscoelastic Lamb’s Problem (Case 1) ...............161

6.2(b) Comparison of Solutions for the Viscoelastic Lamb’s Problem (Case 2) ...............161

6.2(c) Comparison of Solutions for the Viscoelastic Lamb’s Problem (Case 3) ...............162

6.3 Rayleigh Dispersion Curves for Case 1 Soil Profile....................................................164

6.4 Rayleigh Effective Dispersion Curve for Case 1 Soil Profile ....................................165

6.5 Rayleigh Attenuation Curves for Case 1 Soil Profile ..................................................166

xi

6.6 Rayleigh Dispersion Curves for Case 2 Soil Profile ................................................... 166

6.7 Rayleigh Effective Dispersion Curve for Case 2 Soil Profile.................................... 168

6.8 Rayleigh Attenuation Curves for Case 2 Soil Profile ................................................. 168

6.9 Rayleigh Dispersion Curves for Case 3 Soil Profile ................................................... 169

6.10 Rayleigh Effective Dispersion Curve for Case 3 Soil Profile.................................... 170

6.11 Rayleigh Attenuation Curves for Case 3 Soil Profile ................................................. 170

6.12 Fundamental Mode Theoretical and Synthetic Dispersion Curvesfor Case 1 Soil Profile ..................................................................................................... 171

6.13 Shear Wave Velocity Profile from UFUMA Inversion Algorithmfor Case 1 Soil Profile ..................................................................................................... 172

6.14 Convergence of UFUMA Inversion Algorithm for Case 1 SoilProfile ............................................................................................................................... 172

6.15 Shear Damping Ratio Profile and Theoretical Attenuation Curvefrom UFUMA Inversion Algorithm for Case 1 Soil Profile..................................... 173

6.16 Attenuation Curves RMS Misfit Error using UFUMA InversionAlgorithm for Case 1 Soil Profile.................................................................................. 173



6.19 Convergence of UFUMA Inversion Algorithm for Case 2 SoilProfile ............................................................................................................................... 175





6.24 Non-Convergence of UFUMA Inversion Algorithm for Case 3 SoilProfile ............................................................................................................................... 178



xii

6.27 Effective Theoretical and Synthetic Dispersion Curves for Case 1Soil Profile ........................................................................................................................180

6.28 Shear Wave Velocity Profile from UEQMA Inversion Algorithmfor Case 1 Soil Profile......................................................................................................180

6.29 Convergence of UEQMA Inversion Algorithm for Case 1 SoilProfile ................................................................................................................................181

6.30 Shear Damping Ratio Profile and Theoretical Attenuation Curvefrom UEQMA Inversion Algorithm for Case 1 Soil Profile ....................................181

6.31 Attenuation Curves RMS Misfit Error using UEQMA InversionAlgorithm for Case 1 Soil Profile ..................................................................................182











6.42 Fundamental Mode Theoretical Dispersion and AttenuationCurves for Case 1 Soil Profile ........................................................................................188

6.43 Shear Wave Velocity and Shear Damping Ratio Profile fromCFUMA Inversion Algorithm for Case 1 Soil Profile................................................189

6.44 Convergence of CFUMA Inversion Algorithm for Case 1 SoilProfile ................................................................................................................................189

xiii

6.45 Fundamental Mode Theoretical Dispersion and AttenuationCurves for Case 2 Soil Profile........................................................................................ 190

6.46 Shear Wave Velocity and Shear Damping Ratio Profile fromCFUMA Inversion Algorithm for Case 2 Soil Profile ............................................... 191

6.47 Convergence of CFUMA Inversion Algorithm for Case 2 SoilProfile ............................................................................................................................... 191

6.48 Fundamental Mode Theoretical Dispersion and AttenuationCurves for Case 3 Soil Profile........................................................................................ 192

6.49 Shear Wave Velocity and Shear Damping Ratio Profile fromCFUMA Inversion Algorithm for Case 3 Soil Profile ............................................... 193

6.50 Convergence of CFUMA Inversion Algorithm for Case 3 SoilProfile ............................................................................................................................... 193

6.51 Effective Theoretical Dispersion and Attenuation Curves for Case 1Soil Profile........................................................................................................................ 194

6.52 Shear Wave Velocity and Shear Damping Ratio Profile fromCEQMA Inversion Algorithm for Case 1 Soil Profile .............................................. 195

6.53 RMS Error Misfit of CEQMA Inversion Algorithm for Case 1 SoilProfile ............................................................................................................ .................. 195



6.56 RMS Error Misfit of CEQMA Inversion Algorithm for Case 2 SoilProfile ............................................................................................................................... 197



6.59 RMS Error Misfit of CEQMA Inversion Algorithm for Case 3 SoilProfile ............................................................................................................................... 198

6.60 Inverted Shear Wave Velocity Profiles for Case 1 Soil Stratigraphy ....................... 199

6.61 Inverted Shear Damping Ratio Profiles for Case 1 Soil Stratigraphy...................... 200





xiv

7.1 Treasure Island National Geotechnical Experimentation Site (AfterSpang, 1995) .....................................................................................................................207

7.2 Soil Profile and Properties at the Treasure Island NGES(After Spang, 1995) .........................................................................................................208

7.3 Fundamental Mode Theoretical and Experimental DispersionCurves at Treasure Island NGES .................................................................................210

7.4 Shear Wave Velocity Profile from UFUMA Inversion Algorithm atTreasure Island NGES ...................................................................................................210

7.5 Convergence of UFUMA Inversion Algorithm at Treasure IslandNGES................................................................................................................................211

7.6 Shear Damping Ratio Profile and Theoretical Attenuation Curvefrom UFUMA Inversion Algorithm at Treasure Island NGES ..............................212

7.7 Attenuation Curves RMS Misfit Error using UFUMA InversionAlgorithm at Treasure Island NGES ...........................................................................212

7.8 Effective Theoretical and Experimental Dispersion Curves atTreasure Island NGES ...................................................................................................213

7.9 Shear Wave Velocity Profile from UEQMA Inversion Algorithm atTreasure Island NGES ...................................................................................................213

7.10 Convergence of UEQMA Inversion Algorithm at Treasure IslandNGES................................................................................................................................214

7.11 Shear Damping Ratio Profile and Theoretical Attenuation Curvefrom UEQMA Inversion Algorithm at Treasure Island NGES .............................214

7.12 Attenuation Curves RMS Misfit Error using UEQMA InversionAlgorithm at Treasure Island NGES ...........................................................................215

7.13 Fundamental Mode Theoretical and Experimental Dispersion andAttenuation Curves at Treasure Island NGES...........................................................215

7.14 Shear Wave Velocity and Shear Damping Ratio Profile fromCFUMA Inversion Algorithm at Treasure Island NGES ........................................216

7.15 Convergence of CFUMA Inversion Algorithm at Treasure IslandNGES................................................................................................................................217

7.16 Comparison at Treasure Island NGES of Shear Wave Velocityfrom Surface Wave Test Results with Other IndependentMeasurements ..................................................................................................................218

7.17 Comparison at Treasure Island NGES of Shear Damping Ratiofrom Surface Wave Test Results with Other IndependentMeasurements ..................................................................................................................219

xv

SUMMARY

Surface wave tests are non-invasive seismic techniques that can be used to determinethe low-strain dynamic properties of a soil deposit. In the conventional interpretation ofthese tests, the experimental dispersion and attenuation curves are inverted separately todetermine the shear wave velocity and shear damping ratio profiles at a site. Furthermore,in the inversion procedure, the experimental dispersion and attenuation curves are matchedwith theoretical curves, which include only the fundamental mode of propagation.

The only approach available in the literature that accounts for multi-mode wavepropagation is based on the use of Green’s functions where the partial derivatives ofRayleigh phase velocity with respect to the medium parameters required for the solution ofthe inverse problem are computed numerically, and therefore very inefficiently.

This study presents a new approach to the interpretation of surface wave testing. Thenew approach is developed around three new ideas. First, the definition of the low-straindynamic properties of soils and the Rayleigh wave eigenproblem are revisited andreformulated within the framework of the linear theory of viscoelasticity. Secondly, anexplicit, analytical expression for the effective Rayleigh phase velocity has been derived.

The effective phase velocity concept forms the basis for the development of a newsurface wave inversion algorithm based on multi-mode rather than modal dispersion andattenuation curves. Closed-form expressions for the partial derivatives of the effectiveRayleigh phase velocity with respect to the medium parameters have also been obtained byemploying the variational principle of Rayleigh waves.

Thirdly, a numerical technique for the solution of the complex-valued Rayleigheigenproblem in viscoelastic media has been implemented. An immediate application ofthis solution is the development of a systematic and efficient procedure for simultaneouslydetermining the shear wave velocity and shear damping ratio profiles of a soil deposit fromthe results of surface wave tests. The simultaneous inversion of surface waves data offerstwo major advantages over the corresponding uncoupled analysis. First, it explicitlyrecognizes the inherent coupling existing between the velocity of propagation of seismicwaves and material damping as a consequence of material dispersion. Secondly, thesimultaneous inversion is a better-posed mathematical problem (in the sense ofHadamard). The new approach to surface wave analysis is illustrated using severalnumerical simulations and experimental data.

1

1 INTRODUCTION

1.1 Motivation

Geotechnical earthquake engineering is a well-established discipline concerned withunderstanding the role-played by soils in the effects induced by earthquakes. An essentialpart of geotechnical earthquake engineering is ground response analysis. The objective ofground response analysis is the prediction of the free-field site response induced by acatastrophic event, which may be an earthquake or an explosion, occurring in the interiorof the earth’s crust. A correct implementation of a ground response analysis requires aproper modeling of several aspects of the problem including the rupture mechanism at thesource and all the phenomena associated with the propagation of seismic waves from thesource to the desired site at the free-surface. The latter includes transmission of seismicenergy within the continental and oceanic structures of the earth, as well as wavepropagation within the soil mass overlaying the bedrock. Figure 1.1 is a schematicrepresentation of the spread of seismic energy once it is released from the source. Groundresponse analysis has important applications in several areas of geotechnical earthquakeengineering and soil dynamics. Some of the most common include local site response

Source

Regional Geology

t&& ( )y tF

Local Site Conditions

t&& ( )y tL

Figure 1.1 Seismic Energy Path in Ground Response Analysis (Modified from EPRI,1993)

2 Introduction

analyses for the development of design ground motions and response spectra, studies ofsoil liquefaction potential, seismic stability analyses of slopes and embankments, and studiesof dynamic soil-structure interaction.

A crucial step in implementing a ground response analysis is the selection of theconstitutive models and their associated parameters used to simulate the dynamic behaviorof the soil. Studies have shown that the strain level(s) induced by an earthquake can rangeanywhere from 10-3% up to 1+% depending on several variables including the magnitude ofthe event, the source mechanism, the distance from the epicenter, and the properties ofthe medium (Kramer, 1996). Therefore, an appropriate constitutive model requires adefinition of the model parameters over a broad range of strain levels, ranging from verysmall strain levels (below the linear cyclic threshold strain), where the response of themedium can be considered linear but not necessarily elastic, to intermediate and large strainlevels where non-linear behavior dominates. In many cases, the very small-strain dynamicproperties of soils are sufficient, since there are often circumstances in seismology and soil-dynamics where the assumption of linearity is an acceptable approximation.

The following example illustrates the crucial role played by the very small-straindynamic properties of a soil deposit in controlling the amplification or de-amplification ofan input motion applied at the bedrock. Figure 1.2 and Figure 1.3 illustrate the results of alocal site response analysis performed using the computer program SHAKE91 (Idriss andSun, 1991). This code solves the initial-boundary value problem associated with the one-

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Period [sec]

Spec

tral

Acc

eler

atio

n [g

] Gmax = 150.8 MPa

Gmax = 67.0 MPa

Gmax = 16.8 MPa

Input Motion

ζ = 5 %

Figure 1.2 Influence of Gmax on the Acceleration Response Spectrum

Introduction 3

dimensional wave propagation in layered viscoelastic media using an equivalent linearanalysis. The input motion used in the numerical simulation was the N90E accelerationrecord of the 18 May 1940 El Centro earthquake scaled to a maximum acceleration of0.15g. This acceleration time history was applied at the base of a homogeneous soil depositoverlaying the bedrock and having a thickness m30H = .

Figure 1.2 shows the influence of the initial tangent shear modulus Gmax (or the shearwave velocity VS ) on the acceleration response spectrum. As expected, the maximumresponse of the spectrum is attained at periods close to the fundamental period of the site,calculated with the well-known expression 4H VS/ .

The influence of the initial shear damping ratio DSmin (value of shear damping ratioassociated with a strain level below the linear cyclic threshold strain) on the accelerationresponse spectrum is shown in Figure 1.3. Low values of damping ratio results in a largeamplification of the input motion at the bedrock, particularly at periods close to the naturalperiod of the site. In both response spectra the structural damping ξ was assumed equal to5%.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Period [sec]

Spec

tral

Acc

eler

atio

n [g

]

Dsmin = 0.5 %

Dsmin = 5.5%

Input Motion

ζ = 5 %

Figure 1.3 Influence of DSmin on the Acceleration Response Spectrum

The above figures illustrate the important role played by the low-strain dynamicproperties of a soil deposit in determining the dynamic response of a single degree offreedom system. The low-strain dynamic properties of soil deposits can be measured with avariety of techniques. They are generally classified into laboratory techniques and in-situ or

4 Introduction

field techniques. At the end of Chapter 2, will be presented a summary with the mostimportant advantages and disadvantages of these techniques.

The main focus of this research effort was on the determination of the very small-strain dynamic properties of a soil deposit from the interpretation of the results of surfacewave tests. The use of surface (Rayleigh) waves for geotechnical site characterization hasseveral advantages over more conventional seismic methods such as cross-hole and down-hole tests. The most attractive feature of surface wave tests is that they are non-invasiveand hence they do not require the use of boreholes, which permits the tests to beperformed more rapidly and at lower cost than most invasive methods.

Furthermore, at sites where subsurface conditions (e.g. gravelly soils) or environmentalconcerns (e.g., solid waste landfills) hinder the use of boreholes and probes, surface wavetests may constitute the only possible choice for an in-situ site investigation. The nextsection describes the most important research objectives pursued during this study.

1.2 Research Objectives

Three primary objectives were envisioned at the beginning of this research effort. Thefirst objective was the development of a systematic and efficient procedure forsimultaneously determining the low-strain values of VS and DS from the results of surfacewave tests. The most common application of surface wave methods is the determinationof the shear wave velocity profile at a site (Nazarian, 1984; Sánchez-Salinero, 1987; Rix,1988; Stokoe et al., 1989). Recently, Rix et al. (1998a) developed a procedure to calculatenear-surface values of material damping ratio from measurements of the spatial attenuationof Rayleigh waves. However, until now the two problems of determining the shear wavevelocity and the shear damping ratio profiles at a site have been considered separately andtherefore uncoupled.

One of the goals of this study was to present a different approach to the problem,where Rayleigh wave phase velocity and attenuation measurements are invertedsimultaneously. The simultaneous inversion of Rayleigh wave phase velocity andattenuation measurements has two major advantages over the corresponding uncoupledanalysis: it is an elegant procedure to account for the coupling existing between phasevelocity of seismic waves and material damping and the simultaneous inversion is a better-posed mathematical problem (in the sense of Hadamard).

The numerical solution of a non-linear inverse problem is obtained in most cases fromthe iterative solution of the corresponding forward problem, which in this case is theboundary value problem of Rayleigh waves in dissipative media. In developing the solutionof the Rayleigh forward problem, extensive use was made of the powerful and elegantmethods of complex variable theory, more precisely of the theory of analytic functions.

Introduction 5

Subsequent chapters of this dissertation will provide a description of the theoretical basisof the simultaneous inversion and will illustrate its applications to some experimental data.

The second objective was, in a sense, motivated by the first objective of thisdissertation. The problem of determining the very small-strain dynamic properties of soilsraises fundamental questions about the meaning of words like “properties of soils”. Implicit tothe definition of such a term is assumptions of material behavior to which ascribe certainbehavioral properties. As a result, different idealizations of material behavior will require thedefinition of different types of material properties. It is unfortunate that often in thegeotechnical literature, it is customary to take for granted certain definitions of materialbehavior without ever questioning the validity or the appropriateness of these definitions.One remarkable example is constituted by the so-called dynamic properties of soils a term usedto collectively denote stiffness and material damping ratio of soils. Chapter 2 of thisdissertation attempts to revisit the definition of these parameters within the framework ofa consistent theory of mechanical behaviour. It is shown that whereas it is not a trivial taskto construct a mathematical model describing the behavior of complex materials such assoils, it is still possible to formulate relatively simple and accurate phenomenologicalmodels by restricting the formulation to the low strain spectrum. These and other issuesrelated with constitutive modeling of soils are addressed in this chapter, from a perspectivethat is relevant to problems of geotechnical earthquake engineering.

Finally, the third objective of this research effort was developing a better understandingof the theoretical aspects associated with the interpretation of surface wave measurements.In the current procedure the shear wave velocity and shear damping ratio profiles aredetermined from the application of an inversion algorithm to an experimental dispersionand attenuation curve. Minimization of the distance (specified by an appropriate definitionof norm) between these curves and those predicted theoretically from an assumed profileof model parameters is the most common criterion used for the solution of the inverseproblem of surface waves. This procedure has an important limitation: the simulated(theoretical) dispersion and attenuation curves are defined as modal response functions, i.e. theyare referred to a specific mode of propagation of Rayleigh waves. Conversely, theexperimental dispersion and attenuation curves reflect, in general, the contributions ofseveral modes of Rayleigh wave propagation and also of body waves in the near field.

There are currently two procedures used to overcome this limitation. The first andmost common one is based on comparing the experimental dispersion and attenuationcurves with those of the fundamental mode obtained theoretically. This method is referredto in the literature as a 2-D analysis of surface waves (Roësset et al., 1991;). The resultsprovided by the 2-D analysis are generally satisfactory for normally dispersive (i.e. regular)shear wave velocity profiles (Gucunski and Woods, 1991; Tokimatsu, 1995). The secondmethod of interpretation of surface wave data referred in the literature as a 3-D analysisconsists of reproducing with a numerical simulation the actual set-up of the experiment.The theoretical phase velocities, for instance, are computed from the phase differences

6 Introduction

between theoretical displacements, and the latter are calculated at locations that emulatereceivers spacings used in the experiment. This method is exact, however it has thedisadvantage of requiring the use of a Green’s function program for computing thedisplacement field, which is difficult and time-consuming if one wants to include the bodywave contributions in the near field. Furthermore, in this approach the partial derivativesrequired for the solution of the non-linear inverse problem of determining an unknownshear wave velocity profile that corresponds to a given experimental dispersion curve, arecomputed numerically. Computation of numerical partial derivatives is notoriously an ill-conditioned problem, and in this case is also computationally expensive (if compared withother methods).

This study attempts a new interpretation of surface wave measurements, whichcombines the simplicity of a 2-D analysis with the robustness of a 3-D analysis. This isachieved by deriving an explicit expression for the effective phase velocity of Rayleighwaves in vertically heterogeneous media (in the literature this quantity is often referred toas the apparent phase velocity). This is the phase velocity measured experimentally insurface wave tests if the contribution of the body wave field is neglected. As expected, theeffective phase velocity is a local quantity in the sense that its value varies continuously withthe distance from the source at a given frequency. The effective phase velocity arises fromthe superposition of several modes of propagation of Rayleigh waves, each traveling at adifferent phase velocity, which is denoted as the modal velocity. In dissipative media, theeffective wave propagation leads naturally to the concept of effective attenuationcoefficient, which is also a local quantity. In light of these results, the commonly usednotions of dispersion and attenuation curves should be more properly replaced by those ofdispersion and attenuation surfaces.

Closed-form analytical expressions for the partial derivatives of the effective phasevelocity with respect to the medium parameters (shear and compression wave velocities)were also obtained by employing the variational principle of Rayleigh waves. These partialderivatives are essential for an efficient and accurate solution of the Rayleigh inverseproblem.

Finally, in this attempt to re-formulate the current interpretation of surface wavemeasurements, a new approach is proposed which is based on the replacement of thedispersion and attenuation curves with a different type of response function: thedisplacement spectra. The motivation for introducing this new procedure was largelymotivated by the observation that in surface wave tests the primitive quantities measuredexperimentally are the displacement phase and amplitudes, and not the Rayleigh phasevelocities and attenuation coefficients. In fact, the effective Rayleigh phase velocity isnothing but the partial derivative, at constant frequency, of the displacement phase withrespect to the source-receiver distance. A similar interpretation holds for the Rayleigh waveattenuation coefficient if the notion of displacement phase is replaced by that ofdisplacement amplitude.

Introduction 7

In their efforts to identify the structure of the Earth, seismologists use time historyrecords combined with digital signal processing techniques to obtain modal dispersion andattenuation curves generated by seismic events. Geotechnical engineers use the dispersiveproperties of surface waves generated by active sources for near-surface sitecharacterization. In attempting to find a solution to their respective problems,seismologists and geotechnical engineers face similar problems and difficulties, therefore itis natural that they often come up with similar solution strategies. However, there are twomajor differences that profoundly distinguish the problems faced by seismologists andgeotechnical engineers.

The first and most important difference is the scale factor. Whereas for seismologiststhe layer thickness of their stratified Earth is on the order of kilometers, geotechnicalengineers deal with layers whose size is two or even three order of magnitude smaller. Alsothe frequencies involved in seismology and geotechnical engineering are very different.Most of the energy contained in a seismic record has a frequency range on the order of 0.1to 10 Hz. Geotechnical engineers analyze surface waves having frequencies up to 200 Hzor more. Furthermore, there is a substantial difference in seismology and geotechnicalengineering, concerning the distances over which surface waves are detected and recordedwith seismometers and geophones.

As a result of different spatial and temporal scales involved in seismology andgeotechnical engineering, the phenomenon of surface wave propagation will assume inthese two disciplines certain unique and distinctive features. In seismology for example, themodes of propagation are in most cases well defined and separated from each other, andseismologists can determine them from the interpretation of time-history records. On thecontrary, in geotechnical engineering surface wave modes generated by harmonicoscillators are mostly superimposed rather than separated to each other. It is thereforenatural to expect based on these observations, different methods of interpretation inseismology and geotechnical engineering.

The second difference between the problems faced by seismologists and geotechnicalengineers is that seismologists do not have control over the source of wave energy:earthquakes occur at times, locations and with characteristics (duration, frequency content,source mechanism, etc.) that to this date are not predictable. Conversely, not only cangeotechnical engineers select the source type, but they can also choose its spatial location.As a result, the task of geotechnical engineers in interpreting surface wave data isenormously simplified if compared with that of seismologists, as long as the former canturn to their advantage their ability of control over the source.

In summary, the objectives of this research effort were to reformulate the conventionalinterpretation of surface wave tests by developing a technique to simultaneously invertRayleigh phase velocity and attenuation data, while accounting for the multi-mode nature

8 Introduction

of Rayleigh wave propagation in vertically heterogeneous media. These goals were achievedby first constructing a consistent model of soil dynamic behavior at very-small strain levels.

1.3 Dissertation Outline

The dissertation is organized into eight chapters and three appendices. Chapter 2 is anintroduction to the fundamental problem of modeling soil behavior at low-strain levelsunder dynamic excitation. Objective of this chapter is to provide experimental evidence forsupporting the assumption of linear viscoelasticity as an appropriate constitutive law formodeling dynamic soil behavior at low-strain levels. After a critical overview of theavailable models of soil behavior, the viscoelastic constitutive model is introduced and thecorresponding model parameters are rigorously defined. Chapter 3 reviews the theory ofRayleigh waves propagation in elastic and viscoelastic vertically heterogeneous media. Afterillustrating well-known results, an explicit analytical expression for the effective phasevelocity is derived. The variational principle of Rayleigh waves is then used to obtainexplicit relationship for the partial derivatives of the effective Rayleigh phase velocity withrespect to the shear and compression wave velocity of the medium. An important resultpresented in this chapter is a new numerical technique for the solution of the complexRayleigh eigenproblem in linear viscoelastic media. The technique is quite general and it canalso be applied to strongly dissipative media. Chapter 4 illustrates the main aspectsassociated with the solution of the Rayleigh inverse problem, and presents the inversionalgorithms developed in this study. Chapter 5 reviews the conventional techniques used insurface wave measurements, and introduces a new methodology aimed to improveconsistency, in surface wave testing, between measurement procedures and interpretationof the results. Some statistical considerations related with surface wave measurements arealso analyzed. Chapter 6 presents the results of a systematic numerical simulation for thevalidation of the algorithms developed in this study. Chapter 7 illustrates an example ofapplication of these algorithms to a real site. Finally, Chapter 8 presents the conclusions ofthis research study and illustrates some recommendations for future research.

9

2 DYNAMIC BEHAVIOR OF SOILS

2.1 Introduction

“ Scientific understanding proceeds by way of constructing and analyzing models of the segments oraspects of reality under study. The purpose of these models is not to give a mirror image of reality, not toinclude all its elements in their exact sizes and proportions, but rather to single out and make available forintensive investigation those elements which are decisive. We abstract from non-essentials, we blot out theunimportant to get an unobstructed view of the important, we magnify in order to improve the range andaccuracy of our observation. A model is, and must be, unrealistic in the sense in which the word is mostcommonly used. Nevertheless, and in a sense, paradoxically, if it is a good model it provides the key tounderstanding reality. “ (From Baran and Sweezy, 1968).

Another feature that adds its contribution to the complexity of soil behavior, is thecoupling effect of soil responses. Thermomechanical coupling is one example of a responseinteraction effect, which is usually negligible in soils. However, soils may exhibit othercoupling effects, which may be more important including piezo-electric and chemico-mechanical coupling (Fam and Santamarina, 1996). Accounting for these responseinteraction phenomena may lead to unexpected consequences such as the reformulation ofthe principle of effective stress of classical soil mechanics. Newer formulations of thisprinciple (Mitchell, 1976) recognize that mechanical effects (i.e. change of the effectivestress) may be obtained not only by variations of the gravitational fields (i.e. total stressand/or hydrostatic pressure), but also by means of electro-chemical perturbations (doublelayer theory).

To date, a comprehensive constitutive model able to account in a unified framework forall these phenomena is not available. Even if such a model existed, its complexity would beformidable, and most likely not suitable for applications to real-world engineering problems.However, despite the complexity of soil behavior, soils do not exhibit all their features withthe same degree of importance. Depending on the nature of the problem underinvestigation, which includes its intrinsic spatial and temporal scales, the strain levelsinvolved, and the dominant external fields, many of the features that characterize soilbehavior may be regarded as secondary. They may be interpreted as second or even higherorder effects, and in most cases they may be neglected without appreciable changes in theresulting analyses.

This is true in many other engineering disciplines and applied sciences, and echoing thepreface of Baran and Sweezy, (1968), it may be said that the art of good engineering oftenidentifies with the ability of transforming a difficult problem into a simpler one byattentively discerning what is important from what is superfluous or unessential.

10 Dynamic Behavior of Soils

2.2 A Survey on Modeling Soil Behavior

2.2.1 Overview

Two approaches or philosophies are currently used to model the mechanical behaviorof soils and of solids in general. The classical approach is that of continuum mechanics,which is based on the identification of a deformable medium, in this case soil, with regionsof the three-dimensional Euclidean space. In this approach the mass distribution as well asall the pertinent field variables (deformation gradient tensor, stress tensor, displacementvector, etc.) are assumed to be continuous function of the coordinates.

An alternative to continuum mechanics which is gaining popularity is discretemechanics, which has its roots in explicitly recognizing the discrete nature of soils (and ofmatter in general) and hence modeling them as an aggregate of interacting rigid ordeformable discrete particles.

2.2.2 The Continuum Mechanics Approach

Despite its limitations, continuum mechanics is a formidable tool in the solution of aninnumerable class of practical problems. Most of the strengths of continuum mechanicscome from the consequences of the continuity assumption such as the availability of thepowerful tools of differential and integral calculus (Malvern, 1969). It is by using theconcepts of calculus that the fundamental concepts of stress and strain “ at a point ” may bedefined.

Classical continuum mechanics was originally conceived to describe the mechanicalbehavior of bodies composed of one constituent which could be solid, liquid or gas as longas the continuity assumption of the field variables holds (within an acceptable accuracy). Butthe assumption of continuity itself does not prevent the possibility of describing themechanics of heterogeneous materials. The extension of one-constituent continuummechanics to bodies composed of more than a single substance leads naturally to the socalled theories of mixtures (Truesdell, 1957). Although the origin of such theories may be datedback at the beginning of the century throughout the work of notable chemists andphysicists working on the kinetic theories of gases, the first systematic attempt to constructa multi-component theory of continuum mechanics is due to Truesdell (1957). Since thenthis theory has been extended to include several other features including chemical reactionsoccurring among the constituents and electromagnetic effects (Eringen, 1976).

However, classical mixture theories are based on the fundamental postulate that amixture is represented by “ a sequence of continuous bodies all of which occupy the sameregions of space simultaneously ” (Truesdell, 1957). This assumption of intermiscibility maybe appropriate to model mixtures of fluid-like components; however there are physicalsituations where this assumption is not appropriate. A few examples include soils, porousrocks, granular materials, and multiphase suspensions where the “ mixture ” consists ofidentifiable solid particles or a matrix surrounded by one or more fluids. These types of

Dynamic Behavior of Soils 11

materials lead to the important distinction between multiphase immiscible mixtures andmiscible mixtures (Goodman and Cowin, 1972). The continuity assumption may still be usedbut an additional continuous field variable must be introduced: the volume fraction whichcorresponds to the proportion of volume occupied by each component of the mixture.This scalar field reflects important microstructural features of the mixture subjected to athermomechanical process.

Applications of the theory developed by Goodman and Cowin (1972) to model thebehavior of particulate materials have produced interesting results. In their formulation thebalance laws are essentially the same as those proposed by Truesdell (1957) with theexception that a new equation of balance is included to account for the role-played byvolume fraction changes. This equation is called the balance of equilibrated forces and itdescribes the distribution of microstructural forces which is effective in a multiphasemixture. In essence, the balance of equilibrated forces states that the internal distribution offorces among the constituents of the mixture is directly related to the changes of theirvolume fractions. It can be viewed as a generalization of the principle of effective stress ofclassical soil mechanics. One of the most attractive features of this theory is its ability tomodel dilatancy, a phenomenon that cannot be modeled with classical continuummechanics. Nevertheless, it should be emphasized that although volume fraction is animportant field variable, it is not sufficient to discriminate between two mixtures withuniform distributions of grains, one with large grains and the other with small grains of thesame material density. In other words, volume fraction alone cannot take into account thegrain size distribution of the constituents (Passman, Nunziato and Walsh, 1984) and in thissense is a scaleless theory. An interesting new approach to the construction of a multi-component theory of immiscible mixtures has been proposed recently by Wilmanski (1996).One of the main features of this theory is the replacement of the equation of balance ofequilibrated forces of Goodman and Cowin with a balance equation for porosity. Theintroduction of this new law of balance is motivated by microscopic considerations of thetime rate of change of the geometry of the solid phase of the mixture with respect to thefluid phase.

The application of the theory of immiscible mixtures to multi-phase media composed ofa solid phase and one or more fluid phases leads naturally to the theories of porous media (Biot,1955; Bowen, 1982). Such theories, which are “ special cases ” of the more general theoriesof mixtures, have been the subject of considerable interest over the last 35-40 years. Thisinterest continues today in the form of different formulations and/or assumptions (DeBoer, 1996). Theories of porous media constitute a possible mathematical framework formodeling the mechanical behavior of complex multi-phase materials such as soils.

A different line of thought for modeling particulate materials within the realm ofcontinuum mechanics was developed during the late 1970’s and early 1980’s by Oda (1978),Rothenburg (1980), Nemat-Nasser (1982), and Satake (1982), just to mention few of theearly investigators. Their approach was to supplement classical continuum mechanics with


concepts derived from studies of micromechanics. An important aspect of this theory isthat it provides the link between macroscopic quantities such as the stress tensor and themicrostructural parameters describing the internal arrangement of the particles (such asparticle orientation, orientation of contacts, distribution of inter-particle contact micro-forces, etc.). This link is obtained by means of appropriate averaging procedures of theabove microstructural variables over a representative elemental volume.

As the studies in this area of micromechanics continued, it became apparent the needfor introducing a new quantity able to describe the spatial arrangements of particulatematerials. This new quantity was introduced with the name of fabric tensor (Oda, 1978), andsince then the use of the fabric tensor as a descriptor of the packing of granular materialshas increased. The fabric tensor is defined as a second rank symmetric tensor and itdescribes a continuous field variable. Its use in the mechanics of granular materials has leadto the important definitions of solid phase and void phase fabric tensors.

These quantities, in particular the void phase fabric tensor, have played a major role inthe applications of the concepts of micromechanics to critical state soil mechanics(Muhunthan and Chameau, 1996), particularly because it has been shown how to determinethem experimentally (Muhunthan, 1991; Frost and Kuo, 1996; Kuo and Frost, 1997). Thework in this area of soil modeling has been very intense in the recent years, and realisticconstitutive equations relating micro-scale variables and the macro-scale variables have beenproposed for both granular materials (Christoffersen et al., 1981) and cohesive soils (Masadet al., 1997). The results obtained thus far are encouraging, but more research is required fora definitive validation of these theories.

This brief survey of the use of a continuum mechanics framework to model soilbehavior is concluded with a short introduction to the so-called polar or generalized theories ofcontinuum mechanics. It is an interesting subject, which is appealing for its inherent capabilitiesof modeling continua having an inner microstructure. The first theory on polar continuawas that of the Cosserat brothers in 1907 who laid down the foundations of what today isknown as Cosserat’s elasticity to be distinguished from the classical theory of elasticity alsocalled Cauchy’s elasticity. Since then, there have been a large number of contributors (Greenand Rivlin, 1964; Eringen and Suhubi, 1964, Mindlin, 1964, Eringen, 1976).

Polar materials are defined as those that admit the existence of couple stresses and bodycouples (Truesdell and Noll, 1992). Such a possibility, which is disregarded in classicalcontinuum mechanics, leads to the construction of an “alternative” continuum mechanicswhere the geometrical points of the continua may possess properties similar to those ofrigid or deformable particles. Thus the geometrical points of classical continuum mechanicswhich possess three degrees of freedom are extended to include additional degrees offreedom which may be the three independent rotations (micropolar continua). In practice thisgeneralization may continue by simply ascribing additional degrees of freedom to thematerial point (Eringen, 1976). Micromorphic continua are defined as media having geometrical


points with a total of twelve degrees of freedom: three translations, three rotations, and sixmicrodeformations. Now the material point not only can translate and rotate like a rigidparticle, but it may behave as if it were a deformable particle.

Obviously, the kinematics of polar continua is inherently much more complicated thanthe kinematics of classical continuum mechanics, particularly in the case of micromorphiccontinua. Non-locality is the peculiar feature of polar continuum mechanics, which essentiallymeans that this theory is able to account implicitly for the scale effects induced by the innermicrostructure of the continua (Granik and Ferrari, 1993). In this sense classical continuummechanics is clearly a scaleless theory.

Polar continuum mechanics is not the only type of non-local continuum theory ofmechanics. Others include the so called materials of grade N (Ferrari et al. 1997) which aredefined as those deformable media whose kinematics are described not only by thedeformation gradient of classical continuum mechanics, but also by higher gradientmeasures (Truesdell and Noll, 1992).

Although generalized theories of continuum mechanics have been advancedconsiderably in recent years, very few applications have been implemented, particularly insoil mechanics. Possible explanations include the fact that, despite their elegance, thesetheories are complex (Ferrari et al. 1997). Furthermore, there are additional difficultiesassociated with the physical interpretation and experimental determination of theirconstitutive parameters.

2.2.3 The Discrete Mechanics Approach

The final part of this section is dedicated to alternatives to continuum mechanics as aframework to model the mechanical behavior of soils. As mentioned at the beginning ofthis section, in recent years the popularity of discrete mechanics has increased among soilmechanicians. The attractive feature of discrete mechanics is the explicit recognition by thistheory of the discrete nature of matter, even though it is clear that the manifestation of thisnature is scale dependent.

By analogy to continuum mechanics, there are several classes of discrete mechanicstheories or techniques. Among them, the two mentioned here are Doublet Mechanics (DM)and the Discrete (or Distinct) Element Method (DEM). The most popular theory, at leastwithin the geotechnical community, is certainly the DEM. Its original formulation datesback to the work of Cundall and Strack in 1979, and since then DEM has been usedextensively in studying the mechanical responses of granular materials (Cundall and Strack,1979; Ting et al., 1989).

Recently, DEM has also been applied to study the constitutive behavior of water-saturated cohesive soils (Anandarajah, 1996). The essential feature of DEM is modeling asoil element as a discrete assemblage of interacting rigid or deformable particles. The


interaction among particles is governed by appropriate constitutive laws, which specify themagnitude and the direction of the contact forces. The overall system is subjected to thelaws of dynamics with forces and moments due to the self-weight of the particles and toparticle-to-particle interaction.

The computational procedure of DEM involves an explicit time-integration scheme ofthe equations of motion. DEM simulations are computationally very expensive and thislimits the size of the problems (number of particles) that can be analyzed. Sometimes forthe purpose of reducing the complexity of the simulations, it has been found convenient toanalyze two-dimensional problems using circular or elliptical disks (Ting et al., 1993). Parallelcomputing seems to be the answer for the future of DEM (Kuraoka and Bosscher, 1996),but more research is needed on modeling the particle-to-particle interactions.

A theory that has been recently proposed and that, in view of the authors (Ferrari et al.,1997; Granik and Ferrari, 1993), should bridge the gap between continuum and discretemechanics (DEM) is Doublet Mechanics (DM). The essential feature of DM is its buildingblock, which is constituted by a pair of geometric points separated at a finite distance (adoublet). This elementary unit replaces the differential volume element of continuummechanics and the discrete particle or grain of DEM. In the kinematics of DM, thegeometrical points or nodes of a doublet have the following degrees of freedom: they canmove relative to each other in both the axial and the normal directions to their commonaxis; moreover, they may rotate about their common axis. DM can be constructed withdifferent degrees of approximation (Ferrari et al., 1997).

In its simplest form it is a scaleless theory which reduces to classical continuummechanics. However, higher order approximations of DM are non-local theories and hencethey may account for the scale effects caused by the discrete nature of the medium.Preliminary results from the application of this theory are promising. It has been shown forinstance (Granik and Ferrari, 1993), that DM may solve the well-known Flamant’s paradox(the Flamant’s problem is the two dimensional equivalent of the Boussinesq’s problem) ofthe classical theory of elasticity. However, additional studies and further applications(supported by experimental data) are required for the ultimate validation of the theory.

This section was not intended to be a comprehensive review of the theories and themethods currently used to model the mechanical behavior of soils or more in general ofsolids. The ones briefly mentioned here are a subset of a much broader class of theories incontinuum and discrete mechanics. However, it is the writer’s belief that some of themodels presented in this section are of a significative interest in the problem of modelingsoil behavior.


2.3 Phenomenological Modeling of Soil Behavior

It is apparent from the previous section that the mechanical behavior of soils may bedescribed with a variety of mathematical idealizations. At the present time none of thetheories that have been cited is able to capture simultaneously all of the features exhibitedby soils, particularly under dynamic excitation and for wide ranges of strain and stress levels.Each proposed model has its own domain of validity, and it may predict reasonable resultsif applied to problems satisfying the assumptions laid at basis of the theory. Often in themechanics of materials, the conditions of applicability of a specific theory are dictated bythe intrinsic spatial and temporal scales of a problem. The spatial scale(s) of a problem maybe defined as a measure of the relationships existing among the size of some of itscharacteristic elements. Each problem has its own spatial (and temporal) scale whichpermits attributing meaning to relative terms such as “ small ” and “ large ”. The temporalscale of a problem provides a quantitative description of the relationships existing amongthe duration of some of its characteristic temporal events. By specifying the temporal scaleassociated to a given problem it becomes possible to attribute a relative meaning to termssuch as fast and slow. All of the problems associated with the mechanical behavior of soilsand other materials are characterized by intrinsic spatial and temporal scales. This statementcan ultimately be justified by the experimental evidence that all natural events are neithercontinuous nor instantaneous.

An appropriate assessment of the spatial scales of a problem may show for instance,that even if there are profound differences between the theories of continuum and discretemechanics, the discrepancies between their predictions may be irrelevant for practicalpurposes. In seismology, where most of the seismic energy propagates within the frequencyrange of about 0.001-100 Hz (Aki and Richards, 1980), the discrete nature of the mediumhas no role to play when compared with the lengths of the propagating seismic waves.Sometimes however, multi-scale phenomena may complicate the analysis of a problem. Forinstance in a composite medium characterized by the presence of randomly distributed localinhomogeneities (scatterers) whose size is comparable with the wavelength of the seismicwaves, a continuous model may be inadequate to represent the scattered wave field.

The time dependent deformation processes exhibited by many rheological materialsincluding soils are examples of phenomena involving intrinsic temporal scales. For instance,the process of energy dissipation occurring when a seismic wave propagates within a soildeposit involves the superposition of several dissipation mechanisms, each characterized byits own temporal scale (Liu et al., 1976). It is the frequency of excitation that will dictate therelative importance of the mechanisms activated during the overall process of energydissipation (a measure of the amount of unrecoverable energy produced during thedeformation of inelastic materials is the internal entropy density).

Throughout this study the mechanical behavior of soils was modeled using thephenomenological approach of classical one-constituent continuum mechanics. Theconstitutive model used to simulate soil response to dynamic excitations at very small strain


levels was linear-isotropic viscoelasticity. One of the purposes of the next section is tojustify this choice by illustrating some experimental results.

2.4 Experimental Observations

2.4.1 Overview

During the last 25-30 years, a considerable amount of research has been performed inan effort to better understand the mechanical response of soils to dynamic excitations.These studies were performed using a variety of laboratory techniques (e.g., resonantcolumn tests, cyclic torsional shear tests, cyclic direct simple shear tests, and cyclic triaxialtests), which allowed researchers to investigate the influence of variables including strainamplitude and frequency of excitation on soil behavior. The results obtained from this workhave helped in identifying the most important variables and factors affecting the dynamicbehavior of soils.

These variables and factors can be broadly divided into two categories according to theirorigin: “external variables“ and “intrinsic variables“. The external variables correspond toexternally applied actions and include the stress/strain path, stress/strain magnitude,stress/strain rate, and stress/strain duration depending on the nature of the applied action(i.e., stress-controlled versus strain-controlled tests). The intrinsic variables correspond tothe inherent characteristics of soil deposits and include the soil type, the size of soilparticles, and the state parameters. The latter include the geostatic effective stress tensor(which is a measure of the current state of in-situ effective stress), some measure of thearrangement of soil particles (e.g., the fabric tensor or at least the void ratio, which howeveris scale-dependent), and some measure of the stress-strain history (e.g., the yield surface orat least the preconsolidation pressure). Figure 2.1 summarizes the relationships betweencauses and effects in the response of soils to dynamic excitations.

As described in the previous section, soil behavior may be studied using either aphenomenological or a micromechanical approach. In the phenomenological approach themain concern is understanding the relationship between causes and effects from amacroscopic point of view, without attempting an explanation of the observed phenomenaat a microscopic level. This microscopic interpretation is the objective of themicromechanical approach, which can be implemented by using the framework of eithercontinuum or discrete mechanics. As already mentioned the approach used in this work tomodel the dynamic behavior of soils is phenomenological, and coincides with that ofclassical, one-constituent continuum mechanics.

2.4.2 Threshold Strains

Experimental evidence shows that among the external variables affecting soil responseto dynamic excitations, the one that plays the most important role is the magnitude of theapplied stress or strain, or more precisely, the magnitude of the deviatoric strain tensor in


strain controlled tests. This quantity is a measure of the level of shear strains that wereinduced in the soil mass during the dynamic excitation. Based on these findings, it was thenpossible to define a shear strain spectrum for simple shear conditions where four distincttypes of soil behaviors were identified (EPRI, 1991; Vucetic, 1994).

The very small strain region is defined for values of shear strain in the range 0 < ≤γ γ tl

where γ tl is the so-called linear cyclic threshold shear strain (Vucetic, 1994). Within this region

soil response to cyclic excitation is linear, but not elastic since energy dissipation occurseven at these very small strain levels (Lo Presti et al., 1997; Kramer, 1996). Although nostiffness reduction is observed in the soil response for γ γ≤ t

l (linear behavior), thehysteretic loop in the stress-strain plane is characterized by a non-null area. Thephenomenon of energy dissipation at very small strain levels is caused by the existence of atime-lag between say, a driving cyclic strain and the driven cyclic stress in a strain-controlledtest (the word hysteresis comes from the ancient Greek and means lag or delay). This timelag between cyclic stress and strain is responsible for energy losses over a finite period oftime, which is typical of a viscoelastic behavior. There is little experimental evidence tosupport the existence of appreciable phenomena of instantaneous energy dissipation forγ γ≤ t

l , which would be typical of an elastoplastic behavior.

Another important feature of soil behavior at very small strain levels is that soilproperties do not degrade as the number of cycles increases, and, as a result, the shape of

ExternalCauses

• Stress/Strain Path

• Stress/Strain Magnitude

• Stress/Strain Rate

• Stress/Strain Duration

• Soil Type

• Size of Soil Particles

• Soil Natural State

- Geostatic Stresses

- Void Ratio/Fabric

- Stress/Strain History

IntrinsicCauses

Soil Response

Phenomenological

Linear ViscoelasticNon-Linear Viscoelastic

Non-Linear Elasto-Visco-Plastic

Micromechanical

Response Functions

• Threshold Strains• Stiffness Degradation• Energy Dissipation

Figure 2.1 Cause-Effect Relationships in Soil Response to Dynamic Excitations


the hysteretic loop does not change with the continued loading (EPRI, 1991; Ishihara,1996). The value of γ t

l varies considerably with the soil type. For example, γ tl for sands is

on the order of 10-3%, whereas for normally consolidated clays with a plasticity index (PI) of50, γ t

l is on the order of 10-2% (Lo Presti, 1987; Lo Presti, 1989).

The small strain region corresponds to shear strain levels in the range γ γ γtl

tv< ≤

where γ tv is the so-called volumetric threshold shear strain (Vucetic, 1994). The name for this

threshold strain is suggested from the experimental observation that soil response to cyclicexcitation for values of γ exceeding γ t

v is characterized by irrecoverable changes involume in drained tests and development of pore-water pressure in undrained tests(Vucetic, 1994). This region of the shear strain spectrum is characterized by a non-linear,inelastic soil response. However, the material properties do not change dramatically withincreasing shear strain, and very little degradation of these properties is observed as thenumber of cycles increases (soil hardening or softening). Values of γ t

v , the upper limit forthis region of behavior, are on the order of 5.10-3% for gravels, 10-2% for sands, and 10-1%for normally consolidated, high plasticity clays (Bellotti et al., 1989; Lo Presti, 1989; Vuceticand Dobry, 1991).

Values of γ γ γtv

tpf< ≤ identify an intermediate strain region where γ t

pf may be called

pre-failure threshold shear strain since values of γ γ> tpf characterize soil behavior at large

deformations preceding the conditions at failure. In the intermediate strain region bothinstantaneous energy dissipation and energy losses over a finite period of time take place asthe number of cycles progresses. This is mostly due to the irrecoverable microstructuralchanges that affect soils once the volumetric cyclic threshold shear strain γ t

v is exceeded(Vucetic, 1994). In this range of deformation the degradation of soil properties with theshear strain is apparent not only within the hysteretic loop but also with the increase of thenumber of cycles (Ishihara, 1996).

Finally, values of γ γ γtpf

tf< ≤ identify the region of large strains (EPRI, 1991; Vucetic,

1994) where soil response to cyclic excitation is highly non-linear and inelastic. This is thestate of soils preceding the condition of failure, which is postulated to occur at the failurethreshold shear strain γ t

f . Table 2.1 shows the shear strain spectrum with the four postulatedtypes of soil response to cyclic excitation.

Among the four threshold shear strains previously defined, namely γ tl , γ t

v , γ tpf , and

γ tf , two of them are particularly meaningful: the linear cyclic threshold shear strain γ t

l and

the volumetric threshold shear strain γ tv . The threshold strain γ t

l is important because itseparates the linear (even though inelastic) response from the non-linear response of soilssubjected to cyclic excitations. The volumetric threshold shear strain γ t

v instead, is a


threshold strain used to distinguish between different types of irrecoverable deformationsoccurring in soils undergoing harmonic oscillations. For γ γ≤ t

v it can be postulated that allthe energy losses taking place in a soil specimen are of a viscoelastic nature, i.e. they onlyoccur over a finite period of time. However, for γ γ> t

v both phenomena of instantaneousand finite-time energy dissipation, which is typical of a visco-plastic soil behavior, areobserved experimentally.

The threshold shear strains γ tl and γ t

v were defined by considering simple shear strainpaths. Soil response to both static and dynamic loadings is strain/stress-path dependent andhence, different values of γ t

l and γ tv (and also of γ t

pf , and γ tf ) would be obtained if

different strain/stress-paths had been used. However the relevance of these concepts andtheir implications in understanding the dynamic behavior of soils would be unchanged.

Another factor that affect the values of the threshold shear strains γ tl and γ t

v is themean effective confining pressure which is a measure of the state of effective stresses.Several studies have shown (Iwasaki et al., 1978; Kokoshu, 1980; Ishibashi and Zhang, 1993;Ishihara, 1996) that both values of γ t

l and γ tv increase with increasing confining pressure,

particularly the linear cyclic threshold shear strain γ tl .

Concerning the influence of the intrinsic properties of soils on the response of thesematerials to harmonic excitations, a gradual change has occurred in recent years on how toapproach the problem. Early works (Seed and Idriss, 1970; Hardin and Drnevich, 1972;Hardin, 1978; Dobry and Vucetic, 1987) treated fine-grained and coarse-grained soils

Table 2.1 Phenomenological Soil Responses to Cyclic Excitation as a Function of ShearStrain Level

Shear StrainMagnitude

0 < ≤γ γ tl

Very SmallStrain

γ γ γtl

tv< ≤

Small Strain

γ γ γtv

tpf< ≤

IntermediateStrain

γ γ γtpf

tf< ≤

Large Strain

Soil ResponseLinear

ViscoelasticNon-LinearViscoelastic

Non-LinearElasto-Visco-

Plastic

Non-LinearElasto-Visco-

Plastic

Type ofNon-Linearity

___ Material Material Material andGeometric


separately, and independent correlations were developed for each of these classes. In recentyears (Dobry and Vucetic, 1987; Vucetic and Dobry, 1991; Jamiolkowski et al., 1991;Vucetic, 1994), it has been recognized that this distinction is unnecessary once variablessuch as soil type and size of soil particles are replaced with the plasticity index (PI). Theability of this index parameter to capture the essential features of soil behavior has beenrecognized since the early days of soil mechanics (Casagrande, 1932; Lambe and Whitman,1969). In the case of the threshold strains, recent research (Vucetic, 1994) has shown thatthe plasticity index (PI) plays an important role in determining the magnitude of γ t

l and

γ tv . Figure 2.2 shows the dependence of γ t

l and γ tv on the plasticity index.

Cyclic Shear Strain Amplitude, γ(%)

Pla

stic

ity I

ndex

, PI Very Small Strains

Small Strains

Intermediate toLarge StrainsA

vera

ge γ t

l Lin

e

Ave

rage

γ tv L

ine

0.0001 0.001 0.01 0.1 1.00

10

20

30

40

50

60

Figure 2.2 Dependence of Threshold Shear Strains from Plasticity Index (AfterVucetic, 1994)

The magnitude of both threshold shear strains γ tl and γ t

v increases with the plasticityindex of the soil. The advantage of using the plasticity index as an independent parameter isapparent. It provides a unified description of soil properties where the explicit distinctionbetween fine-grained and coarse-grained becomes unnecessary. It should be remarkedhowever, that the description of soil properties via the plasticity index is inadequate whenattempting to describe non-plastic soils constituted by large size particles.


2.4.3 Stiffness Degradation and Entropy Production

So far the emphasis has been placed on the importance of the threshold strains γ tl and

γ tv as the primary indicators of the fundamental changes occurring in soil response during

dynamic excitation. From a phenomenological point of view (see Figure 2.1) these changesresult in two observable effects: stiffness reduction and entropy density production (whichis a measure of the amount of energy dissipated within the soil mass during the process ofdeformation). The significance of both effects depends strongly on the magnitude of theshear strains according to the pattern dictated by the threshold strains γ t

l and γ tv .

Furthermore, both stiffness reduction and entropy density production are also affected bythe mean effective confining pressure and by the soil properties. Figures 2.3(a) and Figure2.3(b) show the influence of the mean effective confining pressure on stiffness degradationof soils.

Figure 2.3(a) Effect of Mean Effective Confining Stress on Modulus Reduction Curvesfor Non-Plastic Soils (After Ishibashi, 1992)

It is apparent from these figures that the effect of confining pressure may be significant,particularly in soils of low plasticity. Entropy density production is also affected by theeffective confining pressure (Ishibashi and Zhang, 1993). Experimental evidence shows thatthe energy dissipated in a soil mass decreases as the mean effective confining stressincreases, particularly in low plasticity soils. With regard to the influence of soil type onstiffness reduction and entropy density production, recent studies (Dobry and Vucetic,


1987; Vucetic and Dobry, 1991) have shown that the plasticity index (PI) is a significant soilparameter. This is consistent with the strong influence of the PI on the values of thethreshold strains noted above. Figure 2.4 shows the dependence of shear modulus reductioncurves obtained from cyclic laboratory data (Vucetic and Dobry, 1991) on the cyclic shearstrain amplitude and plasticity index of soils. The effects of these parameters on entropydensity production are illustrated in Figure 2.5. Here the amount of energy dissipated withinthe soil mass was measured using a different parameter called damping ratio. A rigorousdefinition of this quantity will be given later in this chapter.

Other intrinsic soil properties affecting stiffness reduction and entropy densityproduction include some measure of the initial soil fabric and of the stress/strain history ofthe soil deposit. The effect of soil fabric has been quantified only in terms of the initial voidratio and for very small strain levels (Hardin, 1978; Jamiolkowski et al., 1991). It has beenfound that an increase of the void ratio of a soil deposit is accompanied by a decrease of thestiffness and an increase of the entropy density production at very small strain levels.

The results of laboratory experiments indicate also that the stress/strain history, asreflected for instance by the yield surface (as a side note, we remark that the overconsolidationratio OCR, is a very poor measure of the soil stress history since it corresponds only to onepoint of the overall yield surface), affects soil response only at very small strains, where it

Figure 2.3(b) Effect of Mean Effective Confining Stress on Modulus Reduction Curvesfor Plastic Soils (After Ishibashi, 1992)


has a moderate influence on soil stiffness of high plasticity soils (Hardin and Drnevich,1972; Dobry and Vucetic, 1987; Jamiolkowski et al., 1994).

Figure 2.4 Modulus Reduction Curves for Soils of Different Plasticity (After Vuceticand Dobry, 1991)

Figure 2.5 Dependence of Energy Dissipated within a Soil Mass on Cyclic Shear Strainfor Soils of Different Plasticity (After Vucetic and Dobry, 1991)


The stress/strain path and the magnitude of the deviatoric strain tensor induced in thesoil mass are the most important external variables, which affect soil response to dynamicexcitations. Other important factors affecting the dynamic behavior of soils include the timerate of change at which the excitation (stress or strain) is applied and its duration (numberof equivalent cycles). An increasing number of studies (Dobry and Vucetic, 1987; Vuceticand Dobry, 1991; Shibuya, 1995; Lo Presti et al., 1996; Malagnini, 1996) have investigatedthis issue known as the strain-rate effect (in strain-controlled tests). Results from thesestudies indicate that the influence of strain-rate effects on soil response is stronglycontrolled by the strain level. In particular for clayey soils, the volumetric threshold shearstrain γ t

v increases with increasing strain-rate (Lo Presti et al., 1996).

The stiffness at very small strains ( γ γ≤ tl ) does not seem to be affected by the strain-

rate in low-plasticity soils; however, plastic soils show an increase of the stiffness at verysmall strain with increasing strain-rate (Dobry and Vucetic, 1987). At higher strain levels,experimental evidence shows that the stiffness reduction curves are affected by the strain-rate (Dobry and Vucetic, 1987; Lo Presti et al., 1996); soil stiffness generally increases withincreasing strain rate.

Less is known about the influence of the strain-rate effects on the entropy densityproduction. Even though several studies have attempted to clarify this issue (Dobry andVucetic, 1987; Shibuya et al., 1995; Malagnini, 1996; Lo Presti et al., 1997), it is difficult todraw any general, definitive conclusions. A key parameter controlling entropy densityproduction is the frequency of excitation. It is observed that at certain frequency bandwithsthe amount of energy dissipated in a soil mass during cyclic excitation is frequencydependent; experimental evidence also shows the existence of frequency ranges where theentropy density production is frequency or rate independent (Hardin and Drnevich, 1972;Shibuya et al., 1995; Lo Presti et al., 1997).

Within this context it is interesting to note that the seismic bandwith, namely thefrequency range (0.001-100 Hz) where most of the seismic energy released during theearthquakes is concentrated, coincides with the observed zone of frequency independenceof entropy density production (Aki and Richards, 1980; Ben-Menahem and Singh, 1981;Shibuya et al., 1995). Although it might be expected to observe different frequencydependence laws at different strain levels, the currently available data do not support thisexpectation (Shibuya et al., 1995).

Figure 2.6 shows a conceptual diagram where the amount of energy dissipated(measured by the material-damping ratio) is plotted as a function of frequency (Shibuya etal., 1995).

Finally, a factor that may affect soil response to dynamic loads significantly is theduration of the excitation. In cyclic tests this duration corresponds to the number ofloading cycles. In clayey soils and dry sands, experimental results show that at very small


strain levels ( γ γ≤ tl ), the duration effect on stiffness and entropy density production is

negligible (Shibuya et al., 1995; Lo Presti et al., 1997). As the magnitude of shear strainincreases, material degradation phenomena begin to occur, and the importance of loadingduration becomes more apparent (Lo Presti et al., 1997).

In general it is observed that at large values of shear strains ( γ γ> tv ) an increase of the

number of loading cycles yields an acceleration of material deterioration effects such asstiffness degradation and entropy density production. There are differences, however,between drained and undrained responses. In sandy soils, stiffness may even increase withthe increase of the number of cycles in drained conditions, whereas the oppositephenomenon is observed under undrained conditions (Dobry and Vucetic, 1987).

Several other factors may affect stiffness reduction and entropy production duringdynamic excitations. Some of these include cyclic prestraining, creep and relaxation,anisotropy (structural and stress induced), geological age, diagenetic processes (e.g.cementation), degree of saturation, and drainage conditions. For most of these factors it isvery difficult to evaluate their effects on stiffness degradation and energy dissipation of soils.An exception is constituted by anisotropy, where some interesting results can be found inthe literature (Kopperman et al., 1982; Jamiolkowski et al., 1994).

phase A phase B phase C

wave loading seismic loading traffic loading

0.001 0.01 0.1 1 10 100

Frequency , f (Hz)

Dam

ping

rat

io,

D

γ max

inc

reas

es

Figure 2.6 Frequency Dependence of the Energy Dissipated within a Soil Mass(After Shibuya et al., 1995)


2.5 Constitutive Modeling and Model Parameters

2.5.1 Overview

The goal of the previous section was to highlight experimental data concerning thefactors (external and intrinsic) affecting the response of soils to dynamic excitation. Shearstrain magnitude is the most important external variable controlling soil response todynamic loading. In particular it was shown that for strain-paths of simple shear, it ispossible to define a shear strain spectrum where four different regions of soil response canbe recognized. Table 2.1 illustrates the definition of these four regions and of the associatedthreshold shear strains.

Since the primary objective of this research work was the determination of the verysmall-strain dynamic properties of soil deposits from the results of surface wave tests, it isof interest to study the dynamic behavior of soils at strain levels γ γ≤ t

l . Experimentalevidence shows that in this region of the strain spectrum soils subjected to dynamicexcitations have both the ability to store strain energy (elastic behavior), and to dissipatestrain energy over a finite period of time (viscous behavior). The mechanical behaviour ofsuch materials can be accurately described from the phenomenological point of view by thetheory of linear viscoelasticity. The following sections illustrate the main features of linearviscoelastic constitutive models, which include a definition of the general three-dimensionalstress-strain relationships, and of their associated model parameters. The chapter ends witha section dedicated to the experimental determination of these model parameters.

2.5.2 Linear Viscoelastic Constitutive Models

In purely hypoelastic materials the current state of stress is completely determined bythe current state of strain in a one-to-one or injective correspondence. Viscoelastic materialshave the distinctive feature that the current state of stress is a function not only of thecurrent state of strain but also of all past states of strains defining the strain-history of thematerial under study. There are other types of material responses characterized by this kindof memory effect (e.g. plastic behaviour), however they are rate independent and hence thematerial response to any prescribed strain or stress history does not have any time scaledependence (Lubliner, 1990).

In the theory of linear viscoelasticity the current state of stress, which is specified by thestress tensor σ ij t( ) , is considered to be related to the past strain history via the following

linear functional (Christensen, 1971) (the summation convention is implied on repeatedindices):

σ τε τ

ττij ijkl

kltt G t

d

dd( ) ( )

( )= −

−∞∫ (2.1)


where klε is the infinitesimal strain tensor and ijklG is a fourth order tensor-valued function

called the relaxation tensor function of the material. In deriving Eq. 2.1 it was assumed thatthe strain history is continuous, however discontinuous strain histories may be representedas well if the integral appearing in Eq. 2.1 is intended in the Stieltjes sense (Fung, 1965).

Another important assumption required for the derivation of Eq. 2.1 is the timetranslation invariance hypothesis, which states that the material response is independent ofany shift along the time axis. The constitutive relationship described by Eq. 2.1 is also calledBoltzmann’s equation since it can also be derived by applying the Boltzmann’ssuperposition principle. The relaxation tensor function G tijkl ( ) has 81 components;

however, only 21 are independent due to the symmetry of the stress and strain tensors in ageneral anisotropic material. Equation 2.1 can be inverted to yield:

ε τσ τ

ττij ijkl

kltt J t

d

dd( ) ( )

( )= −

−∞∫ (2.2)

where J tijkl ( ) is a fourth order tensor-valued function called the creep tensor function of

the material. For an isotropic, linear, viscoelastic material the creep and relaxation tensorfunctions have only two independent components and they are sufficient to completelydescribe the mechanical response of the material. In this case the constitutive relationships(Eq. 2.1) can be rewritten as:

s t G tde

ddij S

ijt( ) ( )

( )= −

−∞∫ 2 ττ

ττ (2.3a)

σ τε τ

ττkk B

kktt G t

d

dd( ) ( )

( )= −

−∞∫ 3 (2.3b)

where s ij ij ij kk= −σ δ σ1

3 and e ij ij ij kk= −ε δ ε

1

3 are the components of the deviatoric

stress and strain tensors, respectively, and δ ij is the Kronecker symbol. The scalar functions

G tS ( ) and G tB ( ) are the shear and bulk relaxation functions, respectively. As expected,shear and volume deformations of viscoelastic isotropic materials are uncoupled, mimickinga well-known fact of linear isotropic elasticity. From Eq. 2.2 it is possible to obtainrelationships analogous to Eq. 2.3 with the relaxation functions G tS ( ) and G tB ( ) replacedby the creep functions )t(JS and J tB ( ) .

The creep and relaxation functions are material response functions. They are analogousto the elastic constants in linear elasticity. The important difference is that the creep and


relaxation functions are no longer constants but time-dependent functions. They both havean important physical interpretation. The relaxation function G tS ( ) represents the shearstress response of a material subjected to a shear strain (in a strain-controlled test) specifiedas a unit step function (the Heaviside function).

Time

G(t)

Time

J(t)

Figure 2.7 Typical Relaxation and Creep Functions for a Viscoelastic Solid

The creep function )t(JS can be viewed as the shear strain response of a materialsubjected to a shear stress (in a stress-controlled test) specified as a unit step function.Analogous interpretations holds for the bulk relaxation and creep functions G tB ( ) andJ tB ( ) . Figure 2.7 illustrates qualitatively the typical shape of a relaxation and a creepfunction of a viscoelastic solid.

Often the constitutive relationships of viscoelastic materials are given a physicalinterpretation in terms of mechanical models. These are formed by various combinations ofelementary units including linear elastic springs and linear viscous dashpots. The simplest ofthese combinations is the Kelvin-Voigt model in which a linear spring and a dashpot arecombined in parallel and the Maxwell model where the above elements are connected inseries. Whereas the former can be used to model certain features of solid behavior, the latteris suitable to represent fluid behaviour. The relaxation and creep functions of the Kelvin-Voigt model can be written as follows:

)t(c)t(Hk)t(G δ⋅+⋅= χχχ (2.4a)

J tH tk

e

k

ct

χχ

χ

χ( )( )

= ⋅ −

− ⋅

1 (2.4b)


where the index B,S=χ is a subscript denoting the shear and the bulk mode ofdeformation, respectively. The constant χk is the stiffness of the elastic spring whereas the

constant cχ is the coefficient of the viscous dashpot. Finally, H(t) and δ(t) are the Heaviside

and the Dirac generalized functions. The Kelvin-Voigt model offers a very simplistic andtherefore poor representation of material behaviour. Because 0)0(J =+

χ it cannot

represent the instantaneous elastic response that every solid material exhibits when it issubjected to a suddenly applied stress. Furthermore, although the Kelvin-Voigt model isable to describe creep, it cannot describe stress relaxation because of the presence of theDirac delta function in the expression of the relaxation function G tχ ( ) .

Better and more accurate mechanical models can be constructed by variouscombinations of linear springs and dashpots. The standard linear solid is a relatively simplemodel formed by a linear spring and a Kelvin-Voigt unit connected in series. It is able todescribe both phenomena of stress relaxation and instantaneous elastic response in a stress-controlled test. The relaxation function of the standard linear solid can be written asfollows:

G t G G G ee g e

t

χ χ χ χ

τ χ( ) = + −

⋅

−

(2.5)

where G G tk k

k ke χ χ

χ χ

χ χ

= → ∞ =⋅

+( )

1 2

1 2

, and G G t kg χ χ χ= = =+( )0 1 are the limiting

values of G tχ ( ) known as the equilibrium and glassy responses, respectively (Pipkin, 1986);

the terms k1χ and k 2 χ are the spring constants of the standard linear solid. Finally

( )τ χχ

χ χ

=+

c

k k1 2

is the relaxation time and it represents the time required by the stress in

the model to reach the equilibrium state represented by G e χ .

Real materials and in particular soils exhibit more than a single relaxation time, andtherefore the standard linear solid is inadequate to represent their behaviour accurately.More complicated networks of linear springs and dashpots are required to simulate complexbehaviors characterized by a series of relaxation times. A generalized Maxwell model iscomposed of a group of N Maxwell elements in parallel (Malvern, 1969) and the relaxationfunction G tχ ( ) of such a model is given by the following expression:


G t G G eei

N t

i

iχ χ χ

τ χ( ) = + ⋅=

−

∑1

(2.6)

This equation could be used as a basis for a model fitting procedure (Ferry, 1980) where anexperimental stress relaxation curve ( )t(Gχ ) is fitted with the model represented by the

right hand side of Eq. 2.6. The model parameters are the N relaxation times τ χ iand the N

amplitudes Giχ .

Although the use of mechanical models to formulate viscoelastic constitutive modelsmay be instructive, it is not necessary (Christensen, 1971). Far more general constitutiverelationships may be constructed which are not linked to networks of springs and dashpots.For example, Eq. 2.6, which represents a model with a discrete spectrum of relaxationtimes, may be generalized to a Fredholm integral equation of the first kind representing amodel with a continuous spectrum as follows:

( )G t G H e de

t

χ χ χττ τ( ) = + ⋅ ⋅

−∞

+∞ −∫ (2.7)

The function ( )H χ τ is called the relaxation spectrum and it provides important information

about the dissipation mechanisms that may be associated to a spectrum of relaxation times.The mechanical properties of a viscoelastic material may alternatively be specified by therelaxation spectrum rather than by the ordinary creep or relaxation functions. However,solution of the inverse problem represented by Eq. 2.7 may be difficult since the solution ofa Fredholm integral equation of the first kind is a notoriously ill-posed problem (Tikhonovand Arsenin, 1977).

The complete description of a linear viscoelastic constitutive model requiresspecification of two material functions for an isotropic material. These functions may beeither the shear and bulk relaxation functions G tS ( ) and G tB ( ) or the shear and bulk creepfunctions )t(JS and J tB ( ) . Alternatively, two other response functions may be used: the

shear and bulk relaxation spectra ( )HS τ and ( )H B τ . Once the two material functions havebeen specified, a constitutive relationship such as Eq. 2.3 may be used to compute thematerial stress response to any given strain history.

When the prescribed strain or stress history is a harmonic function of time, theconstitutive relationships of the viscoelastic material assume a very simple form. Supposefor instance that the strain history in Eq. 2.1 is specified by ti

kl0kl e)t( ω⋅ε=ε where ε 0 kl is


the amplitude of the strain component, i = − 1 and ω is the angular frequency; then theintegral equation (Eq. 2.1) reduces to the following algebraic equation:

( )σ ω ε ωij ijkl kl

i tt G e( ) *= ⋅ 0 (2.8)

where ( )G ijkl* ω is called the complex tensor modulus and its components are related with

the Fourier sine and cosine transforms of the relaxation tensor function via the followingrelations (Christensen, 1971):

( ) ( ) τ⋅ωττ⋅ω+=ω ∫∞

dsinGGG0 ijklijkl)e(ijkl)1( (2.9a)

( ) ( )G G dijkl ijkl( ) cos2 0

ω ω τ ωτ τ= ⋅ ⋅∞

∫ (2.9b)

where ( )ωijkl)1(G and ( )ω

ijkl)2(G are the real and the imaginary parts of the complex

modulus tensor, respectively, i.e.:

( ) ( ) ( )[ ]ω⋅+ω=ωijkl)2(ijkl)1(

*ijkl GiGG (2.10)

In Eq. 2.9 it is understood that the relaxation tensor function ( )tG ijkl subtracted from

the equilibrium response G G te ijkl ijkl( ) ( )= → ∞ is bounded as ∞→t in such a way that the

improper integral converges.

From Eq. 2.9 it is clear that the real and the imaginary parts of the complex tensormodulus are not independent. Their relationship can be easily found to be (Christensen,1971):

( )( )

( )G GG

dijkl e ijkl

ijkl

( ) ( )

( )

1

22

2 20

2ω

π

τ ω

τ ω ττ= + ⋅

⋅

⋅ −⋅

∞

∫ (2.11)


This equation is known in the literature (Tschoegl, 1989) as one form of the Kramers-Krönig relations. It is important because Eq. 2.11 in essence states that viscoelastic materialsare inherently dispersive. Material dispersion is the phenomenon by which the velocity ofpropagation of mechanical waves in dissipative media is frequency dependent. It can beshown (Tschoegl, 1989) that the Kramers-Krönig relation in the form of Eq. 2.11 simplystates that the real and the imaginary part of the complex tensor modulus are the Hilberttransforms of each other. It can also be proven (Bracewell, 1965) that this result constitutesthe necessary and sufficient condition for the response function ( )ω*

ijklG to satisfy the

fundamental principle of causality.

If the material is isotropic the constitutive relationship (2.8) can be rewritten as follows:

( ) ( )s t G e i tij S ij( ) exp*= ⋅ ⋅2 0ω ω (2.12a)

( ) ( )σ ω ε ωkk B kkt G i t( ) exp*= ⋅ ⋅3 0 (2.12a)

where ij0e and kk0ε are the amplitudes of the deviatoric and volumetric strains, respectively,

and ( )GS* ω and ( )ω*

BG are the complex shear and bulk moduli. One important feature ofthe constitutive relationships of linear viscoelastic materials undergoing a steady stateharmonic motion is that stress and strain states are in general out of phase. The amount bywhich the stress lags behind the strain is measured by the argument of the complexmodulus, which is also a measure, as shown in the following section, of the amount ofenergy dissipated by the material during harmonic excitations. The reciprocal of a complexmodulus is called complex compliance and it is denoted by ( )ω*J .

The complex moduli or complex compliances are the fundamental material propertiesthat need to be specified for the solution of any linear viscoelastic boundary value problemwhere all field variables are harmonic functions of time. However, because of therelationship existing between the complex moduli and the relaxation function (Eq. 2.9 andits inverse), the complex moduli may also be considered, more generally, as alternativedefinitions of mechanical properties of linear viscoelastic materials. Experimentalmeasurements of the complex moduli may be accomplished by direct observation of thephase and amplitude relations existing between stress and strain of a material sampleundergoing a cyclic excitation (Ferry, 1980).

In selecting the frequency spectrum to be analyzed, it is important to consider thatinformation on material behavior over short times may be obtained with high frequencycyclic experiments, and conversely information on material behaviour over a long time


period may be obtained by using low frequency tests. However, it should be recognizedthat, in principle, knowledge of the relaxation function over a limited time period is notequivalent to knowing (according to Eq. 2.9) the complex modulus over a finite frequencyrange (Christensen, 1971).

In a manner similar to the relaxation function, the experimental determination of acomplex modulus (sometimes called the complex stiffness) ( )ω*G requires the selection ofan appropriate constitutive model. This model has to be selected by choosing a satisfactorycompromise between its mathematical complexity and its ability to capture the mostimportant features of material behaviour. In this sense the description of the mechanicalproperties of viscoelastic materials differs profoundly from those of elastic materials. In thelatter case the material properties of an isotropic medium are uniquely specified by twoelastic constants (say the shear and bulk moduli G and B), which may be determinedexperimentally from the slopes of appropriate stress-strain curves.

In linear elasticity the difference in response between two materials is simply that onematerial is stiffer than the other, either in bulk or in shear or in both modes of deformation.On the other hand, the mechanical behaviour of a viscoelastic material is more difficult tocharacterize. Although the material properties of a linear isotropic viscoelastic material areuniquely specified by two response functions (say G tS ( ) and G tB ( ) or ( )GS

* ω and ( )G B* ω ),

they may however be constructed in infinitely many ways. In other words, the theory oflinear viscoelasticity allows the construction of a variety of viscoelastic models, with orwithout using networks of linear springs and dashpots.

Each of these models has its own features, which may or may not be desirable formodeling a specific material. Like in any other problem of constitutive modeling, theselection of a viscoelastic model for fitting the experimental data must be guided by thematerial behaviour that the model tries to represent. For example, it has been shown in theprevious section that the amount of energy dissipated by soils during cyclic excitations isindependent from the frequency of excitation, at least within certain frequency bandwidths.A viscoelastic constitutive model that is intended to simulate soil behaviour under cyclicloading should be able to reproduce this important feature.

Even though a constitutive model should be as simple as possible, but still able toreproduce the essential aspects of a material's behavior. A very common viscoelastic modelused to simulate soil behaviour in geotechnical earthquake engineering and soil dynamics isthe Kelvin-Voigt model (Dobry, 1970; EPRI, 1991; Kramer, 1996). Although very simple,this model fails to reproduce important features of soil or even solid behaviour. It wasshown early in this section that the Kelvin-Voigt model is not able to describe theinstantaneous elastic response displayed by a solid subjected to a suddenly applied stress orthe important phenomenon of stress relaxation.


Furthermore, the amount of energy dissipated in one cycle of harmonic excitationpredicted by this model is directly proportional to the frequency of vibration ( ωπ∝ χc );

however, it was pointed out in the previous section that experimental evidence shows thatat very small strain levels energy dissipation in soils is a frequency independent phenomenonwithin the seismic band. The current procedure adopted for correcting this inconsistency isto rearrange the Kelvin-Voigt model to have a viscous dashpot coefficient specified not as aconstant, but as a function inversely proportional to frequency (EPRI, 1991; Kramer, 1996;Ishihara, 1996). The resulting constitutive model (called a non-viscous or rate-independentKelvin-Voigt model by Ishihara (1996)) violates the time-translation invariance hypothesis,which is one of the postulates of linear viscoelasticity. It can also be shown that when thisrearranged Kelvin-Voigt model is applied to solve the wave equation in linear viscoelasticmaterials, it leads to a violation of the elementary principles of causality (Dobry, 1970).

The selection of an oversimplified constitutive model and “forcing” it to match theexperimental results is rarely a correct strategy to model material behavior. Sometimes thestandard linear solid (with constant dashpot coefficient) is used to replace the Kelvin-Voigtmodel for the description of geologic materials (Liu et al., 1976; Jones, 1986). Although astandard linear solid overcomes the major limitations of the Kelvin-Voigt model, it is stillunable with a single relaxation time to reproduce important features of soil behaviorincluding the observed frequency independence of energy dissipation within the seismicband.

A more reasonable approach to the problem of modeling the dynamic behavior of soilsat very small strains would be to assume a material response function (say the relaxation orthe creep function) that is able to capture the essential aspects of the experimental results.This assumed material response function would in general depend on parameters, which willbe the equivalent of the elastic constants of linear elasticity. These parameters will bedetermined by a model fitting procedure applied to some experimental data in a manneridentical to that used when soils are assumed to behave as elastic materials.

This procedure is currently used successfully in modeling other types of viscoelasticmaterials such as polymers (Ferry, 1980). It has the great advantage of generality, whichmake it suitable to accommodate a large variety of experimental results while remainingconsistent with the fundamental postulates of the theory of linear viscoelasticity.

Examples of application of this approach to geologic materials include the work of Liuet al. (1976) who assumed a hyperbolic distribution of the relaxation spectrum, and that ofKjartansson (1979) who adopted a power law time dependence for the creep responsefunction. Both models are able to predict several features of the behavior of geologicmaterials quite accurately including the frequency independence of energy losses in theseismic band and material dispersion effects.


2.5.3 Low-Strain Kinematical Properties of Soils (LS-KPS)

In the previous section it was shown that a linear viscoelastic constitutive model for anisotropic material is completely defined by two response functions which may be giveneither in the time or frequency domain. In the time domain they are the relaxation functionsG tS ( ) and G tB ( ) or the creep functions J tS ( ) and J tB ( ) . In the frequency domain they

are the complex moduli ( )GS* ω and ( )G B

* ω or the complex compliances ( )JS* ω and ( )J B

* ω .

Alternatively, the mechanical properties of a viscoelastic material may be also specifiedby the relaxation spectra ( )HS τ and ( )H B τ or by the retardation spectra ( )LS τ and ( )LB τ ,which are the equivalent of the complex compliances in the retardation time domain (Ferry,1980).

As in linear elasticity, knowledge of a pair of response functions in one domain enablesone to determine (at least in principle) the corresponding response functions in any otherdomain. Furthermore, as in linear elasticity, the solution of initial-boundary value problemsin linear viscoelasticity requires specification of the material properties only in terms of apair of response functions (for dynamic problems, it is also required the knowledge of themass density).

Even though the three types of response functions are in a sense equivalent, meaningthat they all contain the same information, it is often advantageous to specify themechanical properties of a viscoelastic material by means of the complex moduli. In thefrequency domain, the constitutive relationships of linear viscoelasticity (Eqs. 2.8 and 2.12)become simple and compact algebraic equations, which resemble those of linear elasticity.Moreover, in many geotechnical earthquake engineering problems, the dependent variablesare assumed to have an harmonic time dependence or they can be reduced to this case byusing the Fourier transform.

It is a simple matter to show that from the knowledge of the complex moduli one caneasily determine phase velocity and attenuation of harmonic waves propagating in linearviscoelastic media using the elastic-viscoelastic correspondence principle (Read, 1950; Fung,1965; Christensen, 1971). According to this principle the solution of a harmonic boundaryvalue problem in linear viscoelasticity can be easily obtained from the solution of thecorresponding elastic boundary value problem, by extending the validity of the lattersolution to complex values of the field variables. It should be remarked however, that thevalidity of the correspondence principle is restricted to problems where the boundariesconditions (which are the specified stresses and displacements) are time-invariant(Christensen, 1971).

Application of the elastic-viscoelastic correspondence principle to the equations ofmotion (in absence of body forces) governing the propagation of elastic waves in


unbounded media (Achenbach, 1984), leads to the following pair of complex Helmholtz’sequations:

( ) ( )∇ = − ⋅2

2

2divV

divP

$ $*

u uω

(2.13a)

( ) ( )∇ = − ⋅2

2

2curlV

curlS

$ $*

u uω

(2.13b)

where ∇2 denotes the Laplacian operator, the vector ( )$ $u = u x,ω is the Fourier

transformed displacement vector ( x is the position vector), and ( )VS* ω and ( )VP

* ω are thecomplex S-wave and P-wave velocities, respectively; they govern phase velocity andattenuation of body waves propagating in linear viscoelastic unbounded media and aredefined by:

( )V

G GP

B S** *

( )ωω

ρ=

+ ⋅43

(2.14a)

( )V

GS

S**

( )ωω

ρ= (2.14b)

When the equations of motion (Eq. 2.13) are specialized for the two separate cases ofone-dimensional (harmonic) P-wave and S-wave propagation, their general solution can bewritten in the following compact form:

( )

ω+

ω

γ=t

Vi

*

ex

Atx,u (2.15)

where A is a constant to be determined from the boundary conditions, and γ = P S, is asubscript denoting the irrotational (compression) and the equivoluminal (shear) wave


motion, respectively. In Eq. 2.15 the term ω γV* is the complex wavenumber associated

with the propagation of the γ -wave. It is convenient to rewrite Eq. 2.15 as follows:

( ) ( )txkix eeu ω+⋅⋅α− γγ ⋅= Atx, (2.16)

whereα γ is the attenuation coefficient and is a measure of the spatial amplitude decay of

the γ -wave as it propagates through viscoelastic and hence dissipative media. Theterm k Vγ γω= is the (real) wavenumber associated with the γ -wave propagating with a

(real) phase velocity Vγ . From Eq. 2.14 the attenuation coefficients and the phase velocities

of the irrotational and shear waves are given by the following expressions:

( )VP ωρ

= ℜ+

−

G GB S* *4

3

1

(2.17a)

( )α ω ωρ

P = ⋅ ℑ+

G GB S* *4

3(2.17b)

( )VS ωρ

= ℜ

−

GS*

1

(2.18a)

( )α ω ωρ

S = ⋅ ℑ

G S* (2.18b)

In Eqs. 2.17 and 2.18 the symbols ( )ℜ ⋅ and ( )ℑ ⋅ denote the real and imaginary parts of acomplex number, respectively.

The mechanics of wave propagation in linear viscoelastic media is completely describedby the phase velocities, VP and VS , and attenuation coefficients, α P and αS . Whereas VP

and VS give a measure of the speed at which irrotational and equivoluminal disturbancespropagate in a viscoelastic medium, α P and αS give a description of the spatial attenuation


of these waves as they propagate through a dissipative material. The velocities ofpropagation of P and S waves are directly related to the stiffness of the medium.

On the other hand, the coefficients of attenuation α P and αS are directly related to thephysical mechanisms responsible for the energy dissipation phenomena occurring within thematerial. As shown by the above equations, VP VS , and α P , αS are material properties

and, as such, they are uniquely determined from the complex moduli ( )GS* ω and ( )G B

* ω ,and mass density ρ . The other aspect that is important to emphasize is that in a viscoelasticmedium both phase velocities and attenuation coefficients are, in general, frequencydependent functions. Hence, wave propagation in viscoelastic media gives rise to thephenomenon of material dispersion as described in Section 2.4.2. One importantconsequence of material dispersion is that the shape of the pulse associated with disturbancechanges as it propagates in a viscoelastic medium (Aki and Richards, 1980).

It is instructive to rewrite Eqs. 2.17 and 2.18 in a slightly different form which will makeapparent the close relationship between the coefficients of attenuation and the energydissipation phenomena occurring when a wave propagates through a viscoelastic medium.In particular, by defining the complex constrained modulus ( )G P

* ω as:

( ) ( )G G GP B S* * *ω = + 4

3 (2.19)

and by separating real and imaginary parts of the complex moduli ( )G P* ω and ( )GS

* ω as:

( ) ( ) ( )G G i Gγ*

( ) ( )ω ω ωγ γ

= + ⋅

1 2 (2.20)

where again γ = P S, , it is then possible to rewrite Eqs. 2.17 and 2.18 as follows:

( )( )

( )Vγ

γ γ

γ γ γ

ωρ

=+

⋅ + +

2 21

22

21

22 1

G G

G G G

( ) ( )

( ) ( ) ( )

(2.21a)


( ) ( )α ωω

ωγγ

γ γ γ

γ

= ⋅+ −

V

G G G

G

21

22 1

2

( ) ( ) ( )

( )

(2.21b)

Finally, Eq. 2.21 may be rearranged as:

( )Vγγ

γ

γ

γ

γ

ωρ

= ⋅

+

+ +

G

G

G

G

G

( )

( )

( )

( )

( )

1

22

21

22

21

2 1

1 1

(2.22a)

( ) ( )α ωω

ωγγ

γ

γ

γ

γ

= ⋅

+ −

V

G

G

G

G

1 12

2

21

2

1

( )

( )

( )

( )

(2.22b)

The ratio ( ) ( )[ ]G

GG

( )

( )

*arg tan2

1

γ

γγ γϕ ω= = is called the loss tangent or the loss angle, and

( )G ( )1 γω and ( )G ( )2 γ

ω are often referred to as the storage and loss moduli, respectively

(Pipkin, 1986). Figure 2.8 shows a graphical representation of these parameters.

G ( )1 γ ( )ℜ G γ*

( )ℑ G γ*

G ( )2 γ

( )arg *G γ

G γ*

G γ*

Figure 2.8 Graphical Representation of the Components of the Complex Modulus


The loss modulus ( )G ( )2 γω is so named because this parameter is directly related to the

energy dissipated in a viscoelastic material subjected to cyclic loading. It can be shown (seeAppendix A) that the shape of the stress-strain loop predicted by a linear viscoelastic modelduring harmonic excitation is elliptical. The equation of the ellipse can be written as follows:

ε

ε

σ ε

εγ

γ

γ γ γ

γ γ0

21

2 0

2

1

+

− ⋅

⋅

=

G

G

( )

( )

(2.23)

where the term ε γ 0denotes the amplitude of the harmonic strain ε γ . Equation 2.23 is the

equation of an ellipse rotated by an angle ( )ψ γ ω with respect to the strain axis (see Fig. 2.9).

( )ε t

( )σ t

( )ψ ωγ

( )

( )

ε ω ε

σ ω ε

γ γω

γ γ γ

=

= ⋅

0e

G

i t

*

ℜ ⋅ℜ = ⋅ ⋅∫ ( ) ( ) ( )σ ε π εγ γ γ γd Gl 2

2

l

Figure 2.9 Stress-Strain Hysteretic Loop Exhibited by a Linear Viscoelastic Modelduring a Harmonic Excitation

An expression of the angle ( )ψ γ ω in terms of the real and imaginary parts of the

complex modulus is given in Appendix A. The elliptical shape of the stress-strain loopspredicted by the theory of linear viscoelasticity compares fairly well with experimentalstress-strain loops obtained for soils at very small strains (i.e. γ γ τ≤ l ) (Dobry, 1970).


However experimental evidence shows that stress-strain loops obtained at larger strainamplitudes are cusped (Kjartansson, 1979).

The area enclosed by the ellipse is related to the amount of energy (per unit volume)dissipated by the material during a cycle of harmonic loading. It can be easily shown (seeAppendix A) that this quantity, ∆ γWdissip is equal to:

( )∆W Gdissipγ γ γω π ε= ⋅ ⋅( )2

2

(2.24)

and is therefore directly proportional to the loss modulus ( )G ( )2 γω . From a

thermodynamic point of view ∆ γWdissip is equal to the amount of entropy produced in one

cycle of harmonic loading and due to unrecoverable mechanical work.

At the microscopic level different mechanisms have been proposed (Biot, 1956; Stoll,1974; Johnston et al., 1979; White, 1983; Leurer, 1997) to explain the process of energydissipation occurring at very small strain levels in geological materials subjected to dynamicexcitation. These studies indicate that an interactive combination of several individualmechanisms is responsible for most of the phenomena macroscopically called energydissipation. For coarse-grained soils the two mechanisms that have been postulated toaccount for the internal entropy production are frictional losses between soil particles andfluid flow losses due to the relative movement between the solid and fluid phases. Fine-grained soils exhibit more complex phenomena, which are controlled by electromagneticinteractions between water molecules and microscopic solid particles.

Based on Eq. 2.24, several definitions have been proposed in the literature as measuresof energy dissipation in geological materials (O’Connel and Budiansky, 1978; Aki andRichards, 1980; Ishihara, 1996). Some of them, in particular those used by seismologists,were inspired by the definitions of energy losses used in other disciplines such as electricengineering (Cole and Cole, 1941). Most of them are dimensionless parameters proportionalto the ratio between the energy dissipated ∆ γWdissip , and some measures of the stored

energy per unit volume.

All these definitions of energy dissipation are consistent with each other only when theyare applied to weakly dissipative viscoelastic materials. In soil dynamics and geotechnicalearthquake engineering, the parameter traditionally used, as a measure of energy dissipationduring harmonic excitation is the so-called material damping ratio:


( )( )( )D

W

Wtrad

dissip

γγ

γ

ωω

π ω=

⋅

∆

4 max (2.25)

where ( )Wγ ωmax is the maximum stored energy per unit volume during one cycle of

harmonic excitation.

Although in principle plausible, this definition of material damping ratio is inconvenientto use. The reason is that the maximum stored energy ( )Wγ ωmax per unit volume of an

harmonically excited linear viscoelastic material depends not only on the storage modulus( )G ( )1 γω , but also on the loss modulus ( )G ( )2 γ

ω as well as on their derivatives with respect

to frequency (O’Connel and Budiansky, 1978; Tschoegl, 1989).

This interesting result (which is often ignored in the literature of linear viscoelasticity,see Ferry, 1980; Pipkin, 1986) is caused by the phase lag between the various energy storingmechanisms which govern the mechanical response of linear viscoelastic materials duringharmonic excitation (Tschoegl, 1989). As a consequence, when the definition of materialdamping ratio given by Eq. 2.25 is expressed in terms of the complex modulus ( )G γ ω* , the

ensuing result is very cumbersome. The need for the latter operation is motivated by thenecessity to relate the definition of material-damping ratio to the constitutive parameters oflinear viscoelasticity.

The difficulties associated with the definition of material damping ratio given by Eq.2.25 can be completely overcome by redefining this parameter, now simply denoted by

( )Dγ ω , as:

( )( )( )D

W

W

dissip

aveγγ

γ

ωω

π ω=

⋅

∆

8(2.26)

where the term ( )W aveγ ω is defined as the average stored energy over one cycle of harmonic

oscillation. An analogous, dimensionless definition of energy dissipation is used byseismologists, via a material parameter called the quality factor (O’Connel and Budiansky,1978; Aki and Richards, 1980) and denoted by ( )Qγ ω . The two parameters ( )Dγ ω and

( )Qγ ω are related by ( ) ( )Q Dγ γω ω− =1 2 .


It can be shown (Tschoegl, 1989) that the term ( )W aveγ ω in the denominator of Eq.

2.26 is equal to ( )W Gaveγ γ γω ε= ⋅( )1

2

4 . Considering this result and that given by Eq.

2.24, Eq. 2.26 yields:

( )DG

Gγγ

γ

ω =⋅

( )

( )

2

12(2.27)

It is worth noting that when the energy losses in the material are small, the definition ofmaterial damping ratio given by Eq. 2.25 tends to yield the same results as Eqs. 2.26 and2.27. However, the adoption of the definition given by Eq. 2.26 has the advantage of beingindependent from the magnitude of the energy losses. In light of these considerations, thedefinition of material damping ratio adopted in this study is that given by Eq. 2.26.

Equation 2.27 can be used to rewrite Eq. 2.22 as follows:

( )( )

[ ]Vγγ γ

γ

ωρ

= ⋅+

+ +

G D

D

( )12

2

2 1 4

1 1 4(2.28a)

( ) ( )α ωω

ωγγ

γ

γ

= ⋅+ −

V

D

D

1 4 1

2

2

(2.28b)

In geotechnical earthquake engineering, the term low-strain dynamic properties of soilsis used to denote those geotechnical parameters that characterize the dynamic response ofsoils at very small strain levels. The most important of these parameters are the stiffness andthe material damping ratio. The symbols commonly used to represent them are Gmax and

Bmax for the low-strain shear and bulk stiffnesses, and DSmin and DB

min for the low-strain shearand bulk material damping ratios. The low-strain dynamic properties are essentialparameters in the solution of many geotechnical earthquake engineering problems. Asdescribed in Chapter 1, they control the response of soil deposits to dynamic excitation, andhence they play a crucial role in ground response analyses.

However, it is apparent from the above discussion that the low-strain dynamicproperties can be defined more fundamentally in terms of the complex shear and bulk


moduli ( )GS* ω and ( )G B

* ω , which will be denoted here as the low-strain viscoelasticproperties of soils. Yet, as demonstrated by Equations 2.19, 2.27, and 2.28a, knowledge ofthe low-strain dynamic properties of soils is equivalent to specifying the low-strainviscoelastic properties of soils. In fact these equations provide the means of determining thereal and the imaginary parts of the complex moduli once the material damping ratio and thestiffness (or equivalently the phase velocity) of the soil have been specified.

These equations can obviously be used also to solve the inverse problem, where thelow-strain dynamic properties of soils are determined from the specified low-strainviscoelastic properties of soils. Consequently, although the fundamental model parametersof a linear viscoelastic model in the frequency domain are the low-strain viscoelasticproperties of soils, specification of the low-strain dynamic properties of soils is analternative way to provide exactly the same information.

Often in wave propagation problems, it is convenient to replace the low-strain dynamicproperties of soils with the phase velocities VP and VS , and the attenuation coefficients,α P , and αS . Equations 2.21 and 2.28 provide the relationships between these quantities,denoted as the low-strain kinematical properties of soils, and the low-strain viscoelasticproperties of soils and the low-strain dynamic properties of soils.

As already mentioned, the low-strain viscoelastic properties, low-strain dynamicproperties and the low-strain kinematical properties are equivalent. They are simplyalternative ways to characterize the mechanical properties of linear viscoelastic materials.The choice of which parameters is more appropriate is dictated by the experimentalmeasurements. In this study, the low-strain kinematical properties are the most frequentlyused parameters because in surface wave tests the measured quantities are phase velocitiesand attenuation coefficients.

Equation 2.28 give frequency dependent phase velocities and attenuation coefficients,i.e. the low-strain kinematical properties, for a general linear viscoelastic constitutive model

as a function of material damping ratio ( )Dγ ω and elastic phase velocity V Geγ γ

ρ= ( )1 .

These equations can be rewritten in a simpler form as:

( ) ( )[ ]V V eγ γ γ γω φ ω= ⋅ 1 D (2.29a)

( ) ( ) ( )[ ]α ω ω φ ωγ γ γ γ= ⋅k D2 (2.29b)

where ( ) ( )k Vγ γω ω ω= / and the functions φ γ1 and φ γ 2 are defined by:


( )[ ] ( )[ ]φ ωγ γ

γ

γ

1

2

2

2 1 4

1 1 4D

D

D=

+

+ +(2.30a)

( )[ ]φ ωγ γ

γ

γ2

21 4 1

2D

D

D=

+ −(2.30b)

At very small strain levels material damping in geological materials is small (Aki andRichards, 1980; Ishihara, 1996). Therefore it is reasonable to assume:

sup .Dγ ≤ 01 (2.31)

where the symbol sup( )⋅ denotes the least upper bound of a quantity. Thus it is possible toexpand the functions φ γ1 and φ γ 2 of Eq. 2.30 in a Maclaurin series in the variable Dγ . By

retaining only the first and second order terms, Eq. 2.29 simplifies to:

( )( )

V V eγ γ

γ

γ

ω = ⋅+

+

1 2

1

2

2

D

D(2.32a)

( ) ( ) ( )α ω ω ωγ γ γ= ⋅k D (2.32b)

where the same symbols ( )Vγ ω and ( )α ωγ have been used for the exact and approximated

expressions of these parameters. Equation 2.32 defines the low-strain kinematical propertiesof soils of weakly dissipative (also called low-loss or loss-less) media. The theory of surfacewaves in linear viscoelastic media developed in Chapter 3 is referred to weakly dissipativematerials (in this case soil deposits). Thus, in this study the low-strain kinematical propertiesof soils are defined by Eq. 2.32.

Application of the correspondence principle to the solution of boundary valueproblems in linear viscoelastic media, makes use of a formalism that requires theintroduction of the low-strain kinematical properties of soils as complex-valued phasevelocities ( )Vγ ω* . Therefore, as suggested by Eq. 2.16, it is convenient to rewrite Eq. 2.32 in

the following more suitable form:


( ) ( ) ( ) ( ) ( )[ ]Vk i k iD

γγ γ γ γ

ωω

ω α ωω

ω ω* =

+=

⋅ +1(2.33)

Since ( ) ( )k Vγ γω ω ω= / , Eq. 2.33 can be reduced to:

( )( ) ( )[ ]

( )[ ]VV iD

Dγ

γ γ

γ

ωω ω

ω* =

⋅ −

+

1

1 2(2.34)

For the remainder of this study, Eq. 2.34 will be considered as the formal definition oflow-strain kinematical properties of soils. Equation 2.29a shows that phase velocity Vγ (and

hence stiffness) and material damping ratio Dγ are not independent quantities in linear

viscoelastic media. This fact is also true in weakly dissipative media as shown by Eq. 2.32a.The functional coupling between soil stiffness and material damping ratio is a directconsequence of material dispersion, a phenomenon defined in Section 2.5.2 by means of theKramers-Krönig relation (Eq. 2.11).

An important corollary of the functional dependence of Vγ upon Dγ is that a

fundamentally correct procedure for the experimental measurement of these parametersshould determine them simultaneously. However, the current practice in geotechnicalengineering is to determine Vγ and Dγ separately. The next section will illustrate the

principles of an experimental procedure to be conducted in laboratory with the resonantcolumn test where stiffness (or phase velocity) and material damping ratio are determinedsimultaneously.

Thus far the derivation of the low-strain kinematical properties has been quite general inthe sense that it has not required any assumption about a specific constitutive model. As aresult the actual formulation, and in particular Eqs. 2.32a and 2.34 are valid for any type ofweakly dissipative, linear viscoelastic solids. (If Eq. 2.31 does not hold, these equationsshould be replaced by the first of Eq. 2.33 and Eq. 2.29). It is remarkable to note here thatsince V e

γ is a frequency independent property of a purely elastic material, ( )Dγ ω is the only

viscoelastic parameter that needs to be specified for a complete description of low-strainkinematical properties of soils (i.e. Eq. 2.34).

However, as implied by the Kramers-Krönig relation (Eq. 2.11), the real and imaginaryparts of the complex modulus, ( )G ( )1 γ

ω and ( )G ( )2 γω , are not independent, and hence


from Eq. 2.27 material damping ratio ( )Dγ ω cannot be specified arbitrarily. If for example

( )Dγ ω is assumed to be constant and thus frequency independent for the entire frequency

range ] [ω ∈ − ∞ +∞, , then Eq. 2.32a (and Eq. 2.29a in the general case) would predict thatthe velocity of propagation is frequency independent in linear viscoelastic media.

This conclusion is unacceptable not only because it is unsupported by the experimentaldata summarized in Section 2.3.3, but also because it violates the fundamental principle ofcausality, since no Hilbert transform pair may satisfy the Kramers-Krönig relation with aconstant material damping ratio Dγ (Aki and Richards, 1980). The important result that can

be drawn from this analysis is that even if Eq. 2.34 only requires (apart from V eγ ) the

determination of ( )Dγ ω , the frequency dependence of ( )Dγ ω cannot be prescribed

arbitrarily but it must satisfy the causality constraint imposed by the Kramers-Krönigrelation.

In Section 2.3.3 it was shown that most of the available experimental data indicate thatmaterial damping in soils is a frequency independent phenomenon at very small strain levelswithin the seismic frequency band. In the seismological literature (Aki and Richards, 1980;Kennett, 1983; Keilis-Borok, 1989) it is shown that a nearly constant material damping ratio,namely a function ( )Dγ ω which is frequency independent only over the seismic band,

satisfying the Kramers-Krönig relation yields the following dispersion relation:

( )( )

VV

Dref

refγ

γ

γ

ωω

πωω

=+

1

2ln

(2.35)

where ω ref denotes an angular reference frequency (usually ω π πref Hz= ⋅ =2 1 2( ) ).Equation 2.35 is applicable only for weakly dissipative media within the frequency range~0.001-10 Hz, (Liu et al., 1976) which corresponds approximately to the seismic band. Inthis range of frequencies the material damping ratio Dγ is considered to be a constant and

therefore frequency independent. Frequency or rate independent damping ratio is alsonamed hysteretic damping ratio (EPRI, 1991; Kramer, 1996); the word hysteretic is oftenused in physics and other sciences to denote memory effects processes that are scaleindependent (Visintin, 1994).

Theoretically the dispersion relation (Eq. 2.35) could be obtained by imposing theconstraint due to the Kramers-Krönig relation on Eq. 2.32a. This approach, however, turnsout to be very difficult to implement (Tschoegl, 1989). A more practical method (Liu et al.,


1976) is to assume a creep or a relaxation function which is able to reproduce the nearlyconstant Dγ and then use the Kramers-Krönig relation to deduce the dispersion

relationship. An alternative to this approach (Azimi et al., 1968) would be to assume aHilbert transform pair which satisfies the Kramers-Krönig relation with a nearly constantDγ . Both methods, when applied to weakly dissipative media, yield the dispersion relation

(Eq. 2.35).

It should be apparent from these procedures that postulating the validity of Eq. 2.35 orany other type of dispersion relation is equivalent to assuming a specific constitutive modelfor the material; in fact Eq. 2.35 is an alternative way often used in seismology to specify theconstitutive laws of viscoelastic materials. With the definition of the dispersion relation (Eq.2.35), the low-strain kinematical properties of soils are completely described by Eq. 2.34with ( )D D cons tγ γω = = tan .

0 20 40 60 80 1000.8

1

1.2

1.4

Damping Ratio = 0.01Damping Ratio = 0.04Damping Ratio = 0.08

Frequency [Hz]]

Nor

mal

ized

Pha

se V

eloc

ity

Figure 2.10(a) Influence of Frequency on Phase Velocity of Viscoelastic Waves asPredicted by the Dispersion Relation (2.35)

Figure 2.10 illustrates the material dispersion effects predicted by Eq. 2.35. In particularFig. 2.10(a) shows the variation of phase velocity with frequency at constant damping ratio

γD . In this figure the ratio ( ) ( )V V refγ γω ω is plotted against frequency with ω πref = 2 .

As expected, material dispersion effects increase with increasing Dγ . For Dγ = 8 % the


velocity at which a disturbance propagates in a viscoelastic medium may be up to 30%higher than the corresponding velocity in an elastic medium. Figure 2.10(b) shows ananalogous chart where now phase velocity at constant frequency has been plotted againstdamping ratio Dγ .

The description of the low-strain kinematical properties of soils conducted so far hasassumed one-dimensional wave propagation. An important aspect of one-dimensional wavepropagation is that the direction of propagation coincides with the direction of attenuation.In the case of two or three-dimensional wave propagation, these two directions are notnecessarily the same (Aki and Richards, 1980). By alternatively considering harmonic P andS-waves, the general solution to Eq. 2.13 (note that for an P-wave curl u = 0 whereas foran S-wave div u = 0 ) may be written in the form:

( ) ( )tie ω+⋅γ= xAx*

t, ku (2.36)

0 0.02 0.04 0.06 0.080.8

1

1.2

1.4

5 Hz10 Hz50 Hz100 Hz

Damping Ratio

Nor

mal

ized

Pha

se V

eloc

ity

Figure 2.10(b) Influence of Damping Ratio on Phase Velocity of Viscoelastic Waves asPredicted by the Dispersion Relation (2.35)


where x is the position vector, A is a constant vector to be determined from the boundaryconditions, and k kγ γ γ

* = + iαα is the bivector characterizing both the direction of

propagation (specified by the vector k γ ) and the direction of attenuation (specified by the

vector αα γ ) of the γ -wave. The two vectors k γ and αα γ do not need to be parallel. It is

possible to show (Ben-Menahem and Singh, 1981) that whereas the vector k γ is normal to

planes of constant phase defined by k xγ ⋅ = const. , the vector αα γ is normal to planes of

constant amplitude defined by αα γ ⋅ =x const. The phase velocity of the γ -wave is equal to

Vγ ω= k γ . When the vector k γ is parallel to the vector αα γ the corresponding γ -wave

is called simple (Lockett, 1962).

In a simple γ-wave the direction of propagation is always the same as the direction ofmaximum attenuation. Non simple waves may arise as a result of boundary effects (e.g.reflection and refraction of harmonic waves at a plane interface) combined with viscoelasticmaterials obeying specific constitutive laws (Christensen, 1971). All the propagationphenomena involving viscoelastic waves considered in this study were assumed tocorrespond to simple waves.

2.5.4 Experimental Measurements of Low-Strain Kinematical Properties of Soils

A comprehensive formulation of a model describing material behaviour includes threefundamental steps. The first step is material modeling which consists of constructing amathematical model able to capture the main features of material behaviour (in this casesoil) for a specified range of variation of certain state variables (see preface of Chapter 2). Thesecond step is constitutive modeling and consists in refining the material modeling with aspecific constitutive law, which is assumed to completely describe the mechanical behaviourof the material under study. The third step and last step is calibration modeling. This step isassociated with the definition of an appropriate set of experimental procedures fordetermining the model parameters of the previously constructed constitutive model.

In the context of this study, the outcome of the first step was the selection of thephenomenological theory of linear isotropic viscoelasticity to model the dynamic behaviorof soils at very small strain levels. The second step, which was the subject of Sections 2.5.2and 2.5.3, resulted in the adoption of the dispersion relation (Eq. 2.35) which was largelybased on experimental observations that the material damping ratio for many soils ishysteretic in nature within the seismic frequency band.

With respect to calibration modeling, it was noted in Section 2.5.3 that specification ofthe complex moduli ( )GS

* ω and ( )G B* ω over the entire range of frequencies is sufficient to

completely characterize any isotropic linear viscoelastic constitutive material. Thecorresponding relaxation functions G tS ( ) and G tB ( ) in the time domain may be computedfrom the inverse Fourier transform of the complex moduli as suggested by Eq. 2.9. A direct


measurement of the response functions G tS ( ) and G tB ( ) in the time domain is alsopossible, although it is not common practice in geotechnical engineering. There arealternatives to the use of complex moduli as constitutive parameters defining a viscoelasticmodel. In soil dynamics and wave propagation problems, the low-strain kinematicalproperties often constitute a convenient choice. These are the shear and compression phasevelocities VP and VS , and the shear and compression attenuation coefficients α P , and αS ,or equivalently [see Eq. (2.28b) and (2.32b)] the damping ratios DP and DS .

It was emphasized in the previous section that Vγ and Dγ are not independent, and

hence a correct experimental procedure should determine these parameters simultaneously.Yet, in the current practice of experimental soil dynamics (Kramer, 1996; Ishihara, 1996)Vγ and Dγ are determined independently from each other. This section will illustrate the

principles of an experimental procedure for simultaneously determine the low-strain dynamicproperties of soils Vγ and Dγ . The technique presented has to be conducted in laboratory,

with a widely known test called the resonant column test. In section 5.3.2 it will be shown howthe simultaneous measurement of the LS-DPS can be performed with the in-situ surfacewave testing.

In the current practice of geotechnical testing, the uncoupled measurement of low-straindynamic properties of soils and low-strain kinematical properties of soils can be performedwith a variety of techniques using both in-situ and laboratory tests. In general, laboratorytests provide more accurate measurements compared with in-situ tests, however they arenot exempt from certain limitations. Table 2.2 attempts to summarize the main differencesbetween in-situ and laboratory tests.

Table 2.2 Measurement of Low-Strain Dynamic Properties of Soils Comparisonbetween In-Situ and Laboratory Techniques

Type of Test Main Advantages Main Disadvantages

In-Situ Tests Account forScale Factors

Applicable toany Soil Type

No Alterationof Soil Natural

State

Difficulties ofInterpretation

Little Controlof BoundaryConditions

Lack ofGeneral

Standards

LaboratoryTests

AccurateMeasurements

RepeatableControlledBoundary

Conditions

Very Sensitiveto Sampling

Disturb

Results areScale

Dependent

Difficult inGranular Soils

The field techniques used to measure the low-strain dynamic properties of soils aremainly of geophysical type because of the ability of these tests to operate at strain levelsγ ≤ −10 5 . Some of them like the cross-hole, the down-hole or the seismic cone tests are


invasive, and hence they require the use of boreholes and probes. Some others like theseismic reflection, the seismic refraction or the surface wave tests are non-invasive, and theycan be executed from the ground surface without the need of using boreholes or probes.Concerning the use of laboratory techniques to measure the low-strain dynamic propertiesof soils, very few of them can operate at very small strain levels; among them the mostpopular is certainly the resonant column which allow the tests to be performed at strainlevel γ < −10 6 .

Figure 2.11 illustrates the strain levels mobilized by the most common in-situ andlaboratory tests. Current techniques used to measure the low-strain dynamic properties ofsoils consider soil stiffness and material damping ratio as independent parameters. As a resulteach of these parameters is measured separately. Using the results presented in section 2.5.2and Section 2.5.3 it is possible to show how a laboratory technique such as the resonantcolumn test can effectively be used to simultaneously determine stiffness and damping ratio ofa soil specimen. Since these parameters are determined at specific frequencies of excitation,the proposed method is well suited to investigate the frequency dependence laws of theseimportant soil properties.

In the resonant column test, a solid or hollow circular cylindrical soil specimen issubjected to harmonic excitation by an electromagnetic driving system (Drnevich, 1985).The soil specimen can be excited in either the torsional or the longitudinal modes of vibration.

10-4 10-3 10-2 10-1 10 0 101

Cyclic Shear Strain, γ (%)

In-Situ Seismic Wave Methods

Resonant Column

Cyclic Triaxial

Cyclic Direct Simple Shear

Earthquakes

Figure 2.11 Ranges of Variability of Cyclic Shear Strain Amplitude in Laboratory andIn-Situ Tests (Modified from Ishihara, 1996)


The study presented hereinafter refers to a stress-controlled loading resonant column test inthe torsional mode of oscillation. The frequency and amplitude of the harmonic excitationis controlled by the electromagnetic driving system.

Figure 2.12 shows schematically a fixed-free resonant column apparatus, which is fixedat the base and free to rotate at the top where a driving torque T e i t

0ω is applied. In the

proposed setting of the test, the parameter measured experimentally is the shear complexmodulus ( )GS

* ω .

T e i t0

ω

h

z

ϑr

o

Figure 2.12 Fixed-Free Resonant Column Apparatus (Modified after Ishihara, 1996)

Introducing a system of cylindrical coordinates r z, ,ϑ the equation of motion

governing the vibrations of an elastic cylinder is:

∇ = ⋅22

2

2

1u

V

u

tSϑ

ϑ∂∂

(2.37)


where ∇ = + + +22

2 2

2

2

2

2

1 1∂∂

∂∂

∂∂ϑ

∂∂r r r r z

denotes the Laplacian operator in cylindrical

coordinates, VG

SS=

ρ is the elastic shear wave velocity, G S is the elastic shear modulus,

and ρ is the mass density of the soil specimen; finally ( )u r z tϑ , , is the displacementcomponent in the direction ϑ .

If at z = 0 the cylinder is subjected to a specified harmonic torque T e i t0

ω , the solutionto Eq. 2.37 may be sought in the form:

( ) ( ) ( )u r z t r z e i tϑ

ωξ φ, , = ⋅ ⋅ (2.38)

where ω is the angular frequency of oscillation. If Eq. 2.38 is substituted into Eq. 2.37, thissecond order partial differential equation becomes two ordinary differential equations in theunknown functions ( )ξ r and ( )φ z which can be easily solved. The result is:

( )u r z t Az

hB

z

hr e i t

ϑω, , sin cos=

+

⋅ ⋅

Ω Ω(2.39)

where Ω22 2

=

ρ ωh

G S

and h is the height of the cylinder. A and B are two constants to be

determined from the boundary conditions which are:

( )

( )

u r h t

r t ds T ezS

i t

ϑ

ϑωτ

, ,

, ,

=

=∫

0

0 0

(2.40)

where S R= π 2 is the cross sectional area of the cylinder having a radius R.

Since the only non-zero stress component is τ∂∂ϑ

ϑz SG

u

z= , the twisting moment

( )M z t, is computed by integrating ( )τϑz r z t, , over the cross sectional area to yield:


( )M z tR h

Az

hB

z

he i t, cos sin= ⋅ ⋅

−

⋅

π ρω ω4 2

2 ΩΩ Ω

(2.41)

By applying the boundary conditions in Eq. 2.40, it is found that the ratio between the

twisting moment ( )M z t, and the angle of twist ( )u

rz e i tϑ ωφ= at z = 0 is given by

(Christensen, 1971):

( )( ) ( )

M t

e

T R hi t

0

0 0 20

4 2,cot

φ φπ ρω

ω = = ⋅ ⋅Ω

Ω (2.42)

Application of the elastic-viscoelastic correspondence principle allows one to obtain thecorresponding viscoelastic solution from the elastic solution. The result is:

( )T R h0

4 2

0 2φπ ρω

= ⋅ ⋅Ω

Ω**cot (2.43)

where ( )( )Ω*

*ωρω

ω=

2 2h

G S

and ( )GS* ω is the complex shear modulus. An inspection of Eq.

2.43 suggests that experimental measurement of the angle of twist at the top of thespecimen ( )φ 0 will allow the complex modulus ( )GS

* ω to be determined once the amplitudeof the applied torque T0 and the geometry of the specimen are known.

It should be remarked that since the twisting moment ( )M t0, and the angle of twist

( )φ ω0 e i t will be in general out of phase, ( )φ 0 is a complex number. If this analysis is carried

out over a wide range of frequencies, knowledge of ( )GS* ω will permit the determination of

other types of response functions such as the creep and relaxation functions. Finally, thelow-strain dynamic properties of soils in shear are determined from the knowledge of

( )GS* ω using Eq. 2.18a for shear wave velocity, ( )VS ω , and Eq. 2.27 for material damping

ratio, ( )DS ω .

This analysis can be generalized to other modes of excitation. If for example, the soilspecimen in the resonant column test were excited in the longitudinal direction, the


procedure just presented would lead to the experimental measurement of the complexYoung’s modulus ( )GE

* ω .

57

3 RAYLEIGH WAVES IN VERTICALLY HETEROGENEOUSMEDIA

3.1 Introduction

The purpose of this chapter is to illustrate the most salient aspects of the theory ofRayleigh wave propagation in vertically heterogeneous media. The chapter is organized intwo parts: in the first part the theory is developed for linear elastic media, and in the secondpart the media is assumed linear viscoelastic. In both cases the properties of the medium areassumed to be arbitrary (hence not necessarily continuous) functions of the depth y. Explicitsolutions, however, are presented only for the case of a finite number of homogeneouslayers overlaying a homogeneous half-space (a multi-layered medium).

The first topic to be discussed is the Rayleigh eigenvalue problem from whichfundamental results such as the Rayleigh Green’s function and the effective Rayleigh phasevelocity can be easily derived. The solution of the Rayleigh eigenvalue problem also leads tothe important concept of geometric dispersion, a phenomenon by which, in heterogeneousmedia, the phase velocity of Rayleigh waves is a multi-valued function of the frequency ofexcitation. Geometric dispersion needs to be distinguished from material dispersionintroduced in Chapter 2. Whereas the latter is caused by the causality constraint imposed bythe Kramers-Krönig relation, the former arises from constructive interference phenomenaoccurring in media that are either bounded (e.g. rods, plates, and other types of waveguides)or heterogeneous. Geometric dispersion is responsible for the existence of several modes ofpropagation each traveling at a different phase and group velocity (modal velocities). Laterin this chapter it will be shown that another consequence of geometric dispersion is to alterthe geometric spreading law governing the attenuation of Rayleigh waves in heterogeneouselastic media.

For surface waves generated by harmonic forces applied at the boundary or in theinterior of a vertically heterogeneous half space, the various modes of propagation ofRayleigh waves are superimposed like in a spatial Fourier series. The phase velocity of theresulting waveform can be obtained from an appropriate superposition of modal Rayleighquantities (phase and group velocities, eigenfunctions, etc.). This kinematical quantity isgiven the name effective Rayleigh phase velocity and is shown to be a local quantity in thesense that its magnitude depends on the spatial position where it is measured. It will beshown in Chapters 5 and 6 that the notion of effective Rayleigh phase velocity is particularlyrelevant for surface waves measurements conducted with harmonic sources.

58 Rayleigh Waves in Vertically Heterogeneous Media

The last topic to be discussed in this chapter is the Rayleigh variational principle. Theapplication of this powerful principle to the solution of the Rayleigh eigenvalue problemleads to closed-form solutions for the partial derivatives of Rayleigh phase velocity withrespect to the medium parameters, in particular the medium compression and shear wavevelocities. As shown in Chapter 5, these partial derivatives are very important in the solutionof the inverse problem in which a given set of Rayleigh phase velocities (i.e., a dispersioncurve) are used to obtain an unknown profile of medium parameters.

The efficiency and accuracy of an inversion algorithm is strongly dependent upon thetechnique used to compute the partial derivatives of Rayleigh phase velocity with respect tothe medium parameters. The great advantage offered by the partial derivatives obtained withthe Rayleigh variational principle is that they can be computed using only the unperturbedmedium properties. Numerical partial derivatives, on the other hand, are very inefficient tocompute because they require the solution of the Rayleigh eigenvalue problem for bothperturbed and unperturbed medium parameters. Finally it should be remarked thatnumerical computation of partial derivatives is a notoriously ill-conditioned problem. In thischapter it will be shown how the Rayleigh variational principle can successfully be used tocompute the partial derivatives of both Rayleigh modal and effective phase velocities withrespect to the medium parameters, in a systematic and efficient manner.

Another important result that can be obtained with the use of variational techniques isrelated to the attenuation of Rayleigh waves in weakly dissipative media. Some of the mostcommon procedures used by seismologists to solve surface waves propagation problems ininelastic media are based on the assumption of weak dissipation (Keilis-Borok, 1989;Herrmann, 1994). One important consequence of this assumption is that Rayleighattenuation coefficients ( )α ωR can be easily computed from the solution of the Rayleigheigenproblem in elastic media. Later in the chapter, a new technique for the solution of theRayleigh eigenproblem in linear viscoelastic media is presented. The technique is quitegeneral since it can also be applied to strongly dissipative viscoelastic media.

The solutions presented in this chapter are all obtained in the frequency domain. Thischoice was made for two main reasons. The first reason is simplicity. The mathematics ofwave propagation problems is often fairly involved, and explicit non-integral solutions canrarely be obtained. One of the few exceptions is a boundary value problem where theboundary conditions and the body forces are specified as harmonic functions of time. Thesecond reason for choosing to work in the frequency domain is generality. The availabilityof harmonic solutions is often sufficient for obtaining far more general solutions by usingthe Fourier integral theorem.

3.2 Rayleigh Eigenvalue Problem in Elastic Media

It is a well established result of classical mechanics (Goldstein, 1980) that the dynamicbehavior of a continuous system can be described by a scalar function, called the Lagrangian

Rayleigh Waves in Vertically Heterogeneous Media 59

density and denoted by L, of a certain number N of generalized coordinates q i ( , )i N= 1 andtheir spatial and temporal derivatives q i j, and &q i ( , )j = 1 3 . The equations governing the

motion of such a system can be derived from Hamilton’s principle and are called Lagrange’s

equations. When the system is conservative, the Lagrangian density ( )L q i , & , ,q qi i j is equal

to L = −T U where T and U are the kinetic and the potential energy of the system,respectively. For a conservative system Lagrange’s equations are (Achenbach, 1984):

( )∂∂

∂∂

∂∂

∂

∂

∂∂t q x q qi j i jj i

L L L&

,

+

− ==∑

1

3

0 (3.1)

where x j are the components of the position vector x in Cartesian coordinates. For a

linear elastic material the potential energy identifies with the elastic strain energy, and thusthe Lagrangian density is given by:

( )L & , & &,u i u u ui j i i ij ij= −1

2

1

2ρ σ ε (3.2)

where ( )u ii = 1 3, are the components of the displacement vector, σ ij and ε ij are the

components of stress and strain, respectively, and ρ is the mass density that is assumedconstant with time. It is seen from Eq. 3.2 that in a linear elastic body the generalized

coordinates q i identify with the components of displacement vector ( )u x, t .

Considering Hooke’s law σ λ ε δ εij kk ij ijG= + 2 where ( )ε ij i j j iu u= +1

2 , , , δ ij is the

Kronecker symbol, and λ and G are Lamè’s elastic moduli, Eq. 3.2 can be rewritten as:

( ) ( )L & , & &,u i u u u Gi j i i kk ij ij= − +

1

2

1

22

ρ λ ε ε ε (3.3)

In general, Lamè’s parameters and the mass density are functions of the coordinates, namely

( )λ λ= x j , ( )G G x j= , and ( )ρ ρ= x j . However, in this chapter the elastic medium is

assumed vertically heterogeneous hence ( )λ λ= x 3 , ( )G G x= 3 , and ( )ρ ρ= x 3 . Using thedefinition of Lagrangian density given by Eq. 3.3, Lagrange’s equations of motion (Eq. 3.1)yield the following result in vector notation (Ben-Menahem and Singh, 1981):


( ) ( )G G grad divd

dydiv

dG

dycurl

y ty y∇ + + + + × +

=2

2

22u u e u e uu u

λλ ∂

∂ρ

∂∂

(3.4)

Equations 3.4 are the Navier’s equations of motion for vertically heterogeneous mediain absence of body forces. Obviously Eq. 3.4 is written in Cartesian coordinates specified by

a set of basis vectors e j and a set of coordinate axes x j . For convenience the base

vector e3 has been denoted by e y . Finally, the symbol ( ) ( )⋅ × ⋅ is used to indicate the vector

product.

To find a solution of Eq. 3.4 for harmonic Rayleigh waves, the displacement field

( )u x, t is assumed equal to:

( ) ( ) ( ) ( ) u: , , , , , ,u r y k u u i r y ki t kr i t kr1 1 2 3 20= ⋅ = = ⋅ ⋅− −ω ωω ωe e (3.5)

In elastic media the Rayleigh wave particle motion is elliptical with the minor axis of theellipse parallel to the free surface. The horizontal and the vertical components of thedisplacement field are π 2 radians out of phase. Equation 3.5 represents a two-dimensionalplane strain field ( u 2 0= ). This assumption does not imply any loss of generality in thediscussion since it can be proven (Aki and Richards, 1980) that cylindrical Rayleigh waveshave the same y-dependence indicated by Eq. 3.5. In these equations the term ( )k k= ωdenotes the real wavenumber which, in general, is a multi-valued function of the frequencyof excitation ω . Finally the term r is used to indicate the direction of propagation. Figure3.1 illustrates the sign convention assumed for the coordinate axes.

To represent Rayleigh waves Eq. 3.5 must be supplemented with appropriate boundaryconditions: no stresses at the free surface of the half-space and no stresses anddisplacements at infinity (i.e., the radiation condition):

( )( ) ( )

σσ ,,

σσ

r y ⋅ = =

→ ⋅ → → ∞

n 0

u 0 n 0

at y

r y r y as y

0

, , ,(3.6)

where the symbol ( )σσ r y, denotes Cauchy’s stress tensor and n is a unit normal vector.


( ) ( ) ( )yG,y,yPropertiesElasticMedium λρ

n

r

y

Rayleigh Wave

Figure 3.1 Rayleigh Waves in Vertically Heterogeneous Media

In vertically heterogeneous media where the material properties ( λ ρ, ,G and ) havejump discontinuities, the stress and displacement fields must be continuous at each layerinterface:

( ) ( )

( ) ( )u u

n n

r y r y, ,+ −=

⋅ = ⋅σσ ,, σσ ,,r y r y-+(3.7)

If Eq. 3.5 is substituted into the Navier’s equations (Eq. 3.4) the result, written in matrixform, is (Aki and Richards, 1980):


( ) ( )

( ) ( ) ( ) ( )[ ] ( ) ( )[ ]

( )[ ] ( ) ( ) ( ) ( ) ( )[ ]

( ) ( )

d

dy

r

r

r

r

k G y

k y y G y y G y

k y y k y y G y

y k

r

r

r

r

1

2

3

4

1

1 1

2 21

2

1

2

3

4

0 0

2 0 0 2

0 0 2

0 0

=

− ⋅ + +

− +

− −

⋅

−

− −

−

ω

ω λ λ λ

ω ζ ω ρ ω λ λ

ω ρ ω

( )

(3.8)

where r y k Gdr

dykr3

12( , , )ω = −

, and ( )r y k G

dr

dyk r4

212( , , )ω λ λ= + +

. The function

ζ( )y depends on Lamè’s parameters and is given by ( )

( )ζλ

λ( )y G

G

G=

++

42

.

The motivation for introducing the functions ( )r y k3 , ,ω and ( )r y k4 , ,ω is the followingresult:

( ) ( )

( ) ( )

τ ω

τ ω

ω

ω

yri t kr

yyi t kr

r y k

i r y k

= ⋅

= ⋅ ⋅

−

−

3

4

, ,

, ,

e

e(3.9)

By defining a vector ( ) [ ]f y r r r rT

= 1 2 3 4 , and a matrix ( )A y denoting the 4 4×

array above whose elements are functions of ( ) ( )λ ωy G y k, , , , Eq. 3.8 can be rewrittensimply as:

( ) ( ) ( )d y

dyy y

fA f= ⋅ (3.10)

Equation 3.10 defines a linear differential eigenvalue problem with displacement

eigenfunctions ( )r y k1 , ,ω and ( )r y k2 , ,ω and stress eigenfunctions ( )r y k3 , ,ω and

( )r y k4 , ,ω . The boundary conditions associated with the eigenproblem are easily derivedfrom Eq. 3.6 and Eq. 3.9:


( ) ( )( )

r y,k,3 ω ω

ω

= = =

→ →∞

0 0 04, , ,

, ,

r y k at y

y k as yf 0(3.11)

For a given frequency ω , non-trivial solutions of the linear eigenproblem (Eq. 3.10)subjected to the boundary conditions (Eq. 3.11) exist only for special values of the

wavenumber ( )k kj j= ω , ( )j M= 1, . These particular values of k j are the eigenvalues of

the eigenproblem, and the corresponding solutions ( )r y ki j, ,ω , ( )i = 1 4, are the

eigenfunctions. It can be shown (Keilis-Borok, 1989) that the set of eigenfunctions for agiven frequency ω obey certain orthogonality conditions with appropriate weightingfunctions.

The relation ( )k kj j= ω is only known in the implicit form

( ) ( ) ( )[ ]F R λ ρ ωy G y y k j, , , , = 0 where [ ]F R ⋅ is a complicated function of Lamè’s

parameters, the mass density, the wavenumber and the frequency of excitation. Therelationship [ ]F R ⋅ = 0 is called the Rayleigh dispersion equation. It states that in vertically

heterogeneous media the velocity of propagation of Rayleigh waves is, in general, a multi-

valued function of frequency. Each pair ( ) k r y kj i j, , ,ω defines a mode of propagation

and, in general, there are M normal modes of propagation at any given frequency. Thenumber M can be finite or infinite, depending on the y-dependence of the mediumproperties and on the frequency of excitation. Furthermore, the distribution of the modes,namely the mode spectrum, can be continuous or discrete, and in some cases both (Keilis-Borok, 1989). In a medium composed of a finite number of homogeneous layers overlayinga homogeneous half-space, the total number of Rayleigh modes of propagation is alwaysfinite (Ewing et al., 1957).

From a physical point of view, the existence of different modes of propagation at agiven frequency can be explained by the constructive interference occurring among waves.In continuously varying heterogeneous media the ray paths are curved (as a result ofFermat’s principle), and hence they interfere with each other. In multi-layered media the raypaths are rectilinear, and interference phenomena occur among waves undergoing multiplereflections at the layer interfaces. In either case, the dispersion equation is the mathematicalstatement of this condition of constructive interference (Achenbach, 1984).

3.2.1 Solution Techniques

Several techniques may be used to solve the linear eigenvalue problem with variablecoefficients including numerical integration, finite difference, finite element, boundaryelement, and spectral element methods. However, some of the most common and


frequently used techniques are those belonging to the class of propagator-matrix methods(Kennett, 1983). The Thomson-Haskell algorithm (Thomson, 1950; Haskell, 1953) alsocalled the transfer matrix method is probably the most famous of this class of methodsbecause of its conceptual simplicity and ease of computer implementation. The applicationof this algorithm, however, is limited to vertically heterogeneous media that can berepresented by a stack of homogeneous layers overlying a homogeneous half-space.

In the Thomson-Haskell algorithm the non-trivial solutions of the linear eigenproblemare found from the roots of the Rayleigh dispersion equation

( ) ( ) ( )[ ]F R λ ρ ωy G y y k j, , , , = 0 . The dispersion equation is constructed by a sequence of

matrix multiplications involving terms that are transcendental functions of the materialproperties of the layers in the stratified medium. The roots of the Rayleigh dispersionequation are the wavenumbers corresponding to the modes of propagation of Rayleighwaves at each frequency. Once the roots of the Rayleigh dispersion equation are found, theeigenfunctions for each mode of propagation can be computed by means of simplealgebraic manipulations. The Rayleigh eigenfunctions give the depth-dependence of thestress and displacement.

Haskell (1953) also developed asymptotic expressions for the Rayleigh dispersionequation in the important limiting cases of short and long wavelengths. Because the originalformulation of the Thomson-Haskell algorithm suffers numerical instability problems athigh frequencies (Knopoff, 1964), this method has been modified and improved throughoutthe years by numerous researchers (Schwab and Knopoff, 1970; Abo-Zena, 1979; Harvey,1981).

Kausel and Roësset (1981) derived a finite element formulation from the Thomson-Haskell algorithm, which is called the dynamic stiffness matrix method. The main feature ofthis method is the replacement of the Thomson-Haskell transfer matrices with layerstiffness matrices that are similar to conventional stiffness matrices used in structuralanalysis. The advantage of this formulation is the ability to use standard structural analysistechniques such as condensation and substructuring to solve both the eigenproblem and theinhomogeneous elastodynamic problem of layered media subjected to dynamic loads(Kausel, 1981). The first attempts in using finite element techniques to solve wavepropagation problems in seismology and earthquake engineering date back to the early 70’swith the works of Lysmer and Waas (1972) and Lysmer and Drake (1972).

Another important class of algorithms for solving eigenvalue problems of surface wavesis the method of reflection and transmission coefficients developed by Kennett and his co-workers (Kennett, 1974; Kennett and Kerry, 1979; Kennett, 1983), and modified and/orimproved by others researchers (Luco and Apsel, 1983; Chen, 1993; Hisada, 1994; Hisada,1995). This method, like the Thomson-Haskell algorithm to which it is related, is onlysuitable for applications in multi-layered media. It is based on the use of reflection andtransmission coefficients to construct reflection and transmission matrices for a stratified


media. The result is a very efficient iterative algorithm for establishing the Rayleigh

dispersion equation ( ) ( ) ( )[ ]F R λ ρ ωy G y y k j, , , , = 0 . The method of reflection and

transmission coefficients also offers an interesting physical interpretation because itexplicitly models the constructive interference that leads to formation of the surface wavesmodes (Kennett, 1983). Earlier versions of this algorithm were numerically unstable at highfrequencies because of the presence of certain frequency-dependent terms that have beeneliminated in more recent formulations (Chen, 1993).

Most of the computational efforts spent by the algorithms used for the solution of theRayleigh eigenvalue problem are devoted to the following two tasks: construction of the

Rayleigh dispersion equation ( ) ( ) ( )[ ]F R λ ρ ωy G y y k j, , , , = 0 (which is also called the

Rayleigh secular function), and computation of its roots as a function of frequency. Thelatter are the Rayleigh wavenumbers ( )k kj j= ω , and also the eigenvalues solution of the

Rayleigh eigenproblem (Eqs.3.10 and 3.11).

In the case of an elastic medium the use of complex arithmetic in constructing theRayleigh secular function can completely be avoided (Haskell, 1953; Schwab and Knopoff,1971), and the roots of the dispersion equation are generally obtained by means of root-bracketing techniques combined with bisection (Hisada, 1995). The use of these slowconverging root-finding techniques is suggested by the rapidly oscillating behavior of theRayleigh secular function, particularly at high frequencies, which requires the use of methodsthat cannot fail to find the roots (Press et al., 1992).

Figure 3.2 shows a typical plot of the roots of the Rayleigh dispersion equation( )k kj j= ω where the phase velocity, rather than the wavenumber, has been plotted against

frequency. The algorithm used is that developed by Chen (1993) and Hisada (1995) whichuses the method of reflection and transmission coefficients. Each dispersion curve isassociated with a particular mode of propagation.

In general, there are several modes of propagation at a given frequency with the highermodes characterized by a higher velocity of propagation. Another important feature ofmulti-mode Rayleigh wave propagation is readily apparent from Fig. 3.2: as the frequencyincreases the number of modes associated with that frequency also increases and the modesbecome more closely spaced. As ω → ∞ the modes all tend to a common limit which isthe Rayleigh phase velocity of the thin layer bordering the free-surface of the verticallyheterogeneous half-space.

Once the roots of the Rayleigh secular function have been obtained, computation of theeigenfunctions is a straightforward task. Figure 3.3 illustrates the typical mode shapes ofRayleigh displacement eigenfunctions in a vertically heterogeneous medium.


0 20 40 60 80 100125

130

135

140

145

150

155

160

165

Frequency (Hz)

Ray

leig

h P

hase

Vel

ocity

(m/s

ec)

1

6

2

3

4

5

7

Figure 3.2 Rayleigh Waves Dispersion Curves in Vertically Heterogeneous Media

-0.5 0.5 1.5 2.5

-25

-20

-15

-10

-5

0

Normal ized Par t ic le D isp lacement

Dep

th (

m)

M o d e N u m b e r

1 2 3

Figure 3.3 Rayleigh Displacement Eigenfunctions in Vertically Heterogeneous Media

Vertical

Horizontal


The eigenfunctions have been normalized with respect to the maximum value of thevertical displacement. A common feature of multi-mode Rayleigh wave propagation is thathigher modes have a greater penetration depth than lower modes. This property, which isclearly shown in Fig. 3.3, is very important in the solution of the inverse problem becausethe resolution of deeper layers can be directly related to the presence of higher modes ofpropagation.

As mentioned at the beginning of this section, other numerical techniques can be usedto solve the linear eigenvalue problem including the finite difference method (Boore, 1972),numerical integration (Takeuchi and Saito, 1972), the boundary element method (Manolisand Beskos, 1988), and the spectral element method (Komatitsch and Vilotte, 1998; Faccioliet al., 1996). Although these techniques are less popular than the propagator matrixmethods briefly described in this section, they have several advantages. Numericalintegration and spectral element method, for example, can be used in verticallyheterogeneous media where the medium properties vary continuously with depth, andtherefore they are more general than the propagator matrix methods. Boundary elementmethods are best suited for modeling unbounded or semi-infinite media because theyrequire discretization of only the boundaries. As a result, they reduce the dimension of theproblem by one. Moreover, boundary element methods eliminate the need, required byfinite element based methods, of using fictitious or non-reflecting boundaries to simulatethe radiation condition at infinity.

3.3 Effective Rayleigh Phase Velocity in Elastic Media

In Section 3.2 it was shown that several Rayleigh wave modes may propagate in avertically heterogeneous medium at a given frequency ω . Each mode is specified by the pair

( ) k r y kj i j, , ,ω where ( )k j ω ( )j M= 1, is the wavenumber, and ( )r y ki j, ,ω ( )i = 1 4, is

the set of four eigenfunctions. ( )M M= ω is the total number of modes associated with thefrequency ω . The linear eigenvalue problem corresponds formally to a homogeneousboundary value problem, and its solutions are known as free Rayleigh waves (Ewing et al.,1957). In inhomogeneous problems, Rayleigh waves are generated by sources applied at theboundary or in the interior of the half-space. If these sources are harmonic in time, thedifferent Rayleigh modes of propagation are superimposed on one another. The phasevelocity of the resulting wave train is here named the effective Rayleigh phase velocity. Thissection is devoted to deriving an explicit, analytical expression for this kinematical quantityfrom the solution of the Rayleigh eigenproblem.

In isotropic vertically heterogeneous media, Rayleigh waves generated from pointsources acting in a direction perpendicular to the boundary of the half-space propagatealong cylindrical wave fronts (Ewing et al., 1957). It can be shown (Ben-Menahem andSingh, 1981) that the wave field originated by an harmonic point source located at a position


( )r y yS= =0, can be expanded, in the radial direction, in a series of pth order Hankelfunctions (p is an integer). For large values of r the pth order Hankel functions can beapproximated by their asymptotic expansions involving complex exponential functions. As

a result, the particle displacement ( ) ( ) ( )[ ]u e er y u r y u r yr r y y, , , , , ,ω ω ω= + resulting from

the superposition of M distinct Rayleigh modes, can be written in cylindrical coordinates

r y, ,θ , as follows (Aki and Richards, 1980):

( ) ( )[ ] ( )u β βω ϕω ω βr y A r y e

j

i t k r

j

Mj, , , ,= ⋅ ⋅ − ⋅ +

=∑

1

(3.12)

where β = r y, and ( )[ ]A β ωr yj

, , , ( )k j ω are the Rayleigh displacement amplitudes and

wavenumber, respectively, associated with the jth mode of propagation. Finally, ϕ πβ = − 4

for β = r and ϕ πβ = 4 for β = y . Equation 3.12 shows, as expected, that ( )uβ ωr y, , is

independent of the azimuthal angle θ . The actual particle displacement is obtained bytaking either the real or imaginary part of Eq. 3.12. By choosing the latter, Eq. 3.12becomes:

( )[ ] ( )[ ] ( ) ( ) ( ) ( ) ( )ℑ = ℑ

= −

⋅ − ⋅ +

= =∑ ∑u A C Dβ β

ω ϕβ βω ω ω ωβr y r y e t t

j

i t k r

j

M

j jj

Mj, , , , sin cos

1 1

(3.13)

where ( ) ( ) ( )C Aβ β βϕj j jk r= ⋅ ⋅ +cos and ( ) ( ) ( )D k r

j j jβ β βϕ= ⋅ ⋅ +A sin . Now using simple

trigonometric identities Eq. 3.13 can be re-written as follows:

( )[ ] ( ) ( )[ ]ℑ = ⋅ −uβ β βω ω ω ωr y r y t r y, , , , sin , ,U ψ (3.14)

where:

( ) [ ] [ ] ( )[ ]U β β βω ω ωr y A r y r y r k ki j i j

j

M

i

M

, , ( , , ) ( , , )

.

= ⋅ ⋅ ⋅ −

==

∑∑ A cos11

0 5

(3.15)


( )( )[ ] ( )( )[ ] ( )

ψ ωω ϕ

ω ϕβ

β β

β β

r yr y k r

r y k r

i ii

M

j jj

M, , tan, , sin

, , cos=

⋅ ⋅ +

⋅ ⋅ +

− =

=

∑

∑1 1

1

A

A(3.16)

From Eq. 3.14, the expression:

( )[ ]ω ψ ωβt r y cons t− =, , tan (3.17)

represents the equation of a wave front, since it is the locus of points having constant

phase. Assuming the functions ( )ψ ωβ r y, , to be sufficiently smooth, Eq. 3.17 can be

differentiated with respect to time, to give:

( )ω ωβ

− ⋅ =∂ψ

∂rr,y, 0

dr

dt(3.18)

which finally yields:

( ) ( )[ ]$ , ,

, ,,

Vβ

β

ωω

ψ ωr y

r yr

= (3.19)

where the symbol ( )$ , ,Vβ ωr y has been used to denote the effective Rayleigh phase velocity.

It is apparent from Eq. 3.19 that the effective Rayleigh phase velocity is a local quantity,which means that its value depends on the spatial position where it is evaluated. At a fixed

y y c= , the function ( )$ , ,Vβ ωr y c describes what may be called a dispersion surface, i.e. a two

dimensional surface showing the variation of the effective Rayleigh phase velocity withfrequency and distance from the source. From Eq. 3.19 it is also interesting to observe thatsince the effective Rayleigh phase velocity is a vector quantity, different components of

( )$ , ,Vβ ωr y will, in general, travel at different phase velocities. Furthermore, since

( )( ) ( )

∂

∂

ω ψ

ψ ψ

β β β

β β

$ $,

, ,

V V

tr

r r

= −⋅

⋅ is not, in general, equal to zero, the wave train accelerates as it


propagates along the surface of the half-space. In Eq. 3.19 the term ( )ψ β ,rcould be

interpreted as an effective Rayleigh wavenumber and denoted as ( )$ , ,k r yβ ω . However, a

decomposition of the argument of Eq. 3.14 in a form ( )ω βt k r− ⋅$ , which is common for

harmonic waves, is no longer possible here because the effective wavenumber ( )ψ β ,r being

a local quantity, must be integrated over r to yield the phase ( )ψ ωβ r y, , . Considering the

definition of ( )ψ ωβ r y, , given by Eq. 3.16, it is possible to obtain from Eq. 3.19, an explicit

definition for the effective Rayleigh phase velocity which is given by:

( )( ) ( ) ( )[ ]

( ) ( ) ( ) ( )[ ] $ , ,

cos

cosVβ

β β

β β

ω ωr y

r k k

k k r k k

i j i jj

M

i

M

r s r s r ss

M

r

M= ⋅

⋅ −

+ ⋅ −

==

==

∑∑

∑∑2

11

11

A A

A A(3.20)

For an harmonic point source Fy ⋅ e i tω located at r y yS= =0, , the Rayleigh displacement

amplitudes ( )[ ]A β ωr yj

, , of the individual modes of propagation are related to the

displacement eigenfunctions ( )r y k1 , ,ω and ( )r y k2 , ,ω , and to other modal parameters bythe following expression (Aki and Richards, 1980):

[ ] ( ) ( )

( )A

A

A

β ω

ω

ω

ω

π

ω

ω

( , , )

( , , )

( , , )

, ,, ,

, ,

r y

r y

r y

F r y k

V U I r k

r y k

r y kj

r

y j

y S j

j j j j

j

j

=

=⋅

⋅ ⋅ ⋅ ⋅⋅

2

1

2

4 2(3.21)

where V U and kj j j, , are the phase, group velocity and wavenumber of the Rayleigh jth

mode of propagation ( )j M= 1, , respectively. The term ( )I y kj j, ,ω is the first Rayleigh

energy integral associated with the jth mode of propagation (see Section 3.5 for more details)and is defined by (Aki and Richards, 1980):

( ) ( ) ( )[ ]I y k y r y k r y k dyj j j j( , , ) , , , ,ω ρ ω ω= +∞

∫1

2 012

22 (3.22)


By substituting Eq. 3.21 into Eq. 3.20, the expression for the effective Rayleigh phasevelocity becomes:

( )

( ) ( ) ( ) ( ) ( )[ ]( )( )

( ) ( ) ( ) ( )( ) ( )[ ]( )( )

$ , ,

, , , , cos

, , , , cosVr r y

r y k r y k r y k r y k r k k

V U I V U I k k

r y k r y k r y k r y k k k r k k

V U I V U I k k

i j S i S j i j

i i i j j j i jj

M

i

M

r s S r S s r s r s

r r r s s s r ss

M

r

Mω

ω

=

−

+ −

==

==

∑∑

∑

21 1 2 2

11

1 1 2 2

11∑

(3.23a)

( )

( ) ( ) ( ) ( ) ( )[ ]( )( )

( ) ( ) ( ) ( )( ) ( )[ ]( )( )

$ , ,

, , , , cos

, , , , cosVy r y

r y k r y k r y k r y k r k k

V U I V U I k k

r y k r y k r y k r y k k k r k k

V U I V U I k k

i j S i S j i j

i i i j j j i jj

M

i

M

r s S r S s r s r s

r r r s s s r ss

M

r

Mω

ω

=

−

+ −

==

==

∑∑

∑

22 2 2 2

11

2 2 2 2

11∑

(3.23b)

where ( )$ , ,V r yr ω and ( )$ , ,V r yy ω denote the components of the effective Rayleigh phase

velocity along directions r and y respectively. To reduce the length of the above expressions,

the frequency dependence of the eigenfunctions ( )r y k1 , ,ω and ( )r y k2 , ,ω has beenomitted.

As a final remark of this section, it is noted from Eq. 3.23 that the effective Rayleighphase velocity is completely determined by the solution of the Rayleigh eigenproblem. In

fact, recalling that V kj j= ω and U d dkj j= ω ( )j M= 1, , all of the modal quantities

appearing in Eq. 3.23 can be calculated from the pair ( ) k r y kj i j, , ,ω ( )i = 1 2, .

3.4 Rayleigh Green’s Function in Elastic Media

The term Green’s function is generally used in applied mathematics to denote theresponse of a linear system governed by a set of differential or integral equations andassociated boundary conditions (integral equations contain built-in boundary conditions) toan impulsive unit point source represented by a Dirac- ( )δ ⋅ distribution. It is possible toshow (Logan, 1997) that the response of a system to an arbitrary distribution of sources inspace and time can uniquely be determined by the knowledge of the Green’s function of thesystem via a convolution integral. In engineering, the term Green’s function is frequently


used with a somewhat more general and sometimes different meaning than the one used inapplied mathematics. A common example is constituted by the response of a linear systemto a harmonic or a Heaviside unit point source. This response if often denoted as theGreen’s function associated to the linear system.

In this study the term Green’s function is primarily used to denote the response of alinear elastic or viscoelastic half-space to a harmonic unit point source. Of particular interestin this section is the displacement Green’s function, which is defined as the displacement

( )$ , ,u r y ω induced in a linear elastic medium by a harmonic unit point load 1yi te⋅ ω located

at the position ( )r y yS= =0, . The subscript y in 1y denotes the direction of action of the

unit point load.

The particle displacement field ( )$ , ,u r y ω can be separated into two components

( ) ( ) ( )$ , , $ , , $ , ,u u ur y r y r yB Sω ω ω= + . The first component ( )$ , ,uB r y ω represents the bodywave field and is composed of the superposition of P and S waves. The component

( )$ , ,u S r y ω is the surface wave field and, in general, is composed of a superposition of Loveand Rayleigh waves. The body wave field attenuates with distance from the source at amuch higher rate than the surface wave field. It is possible to show (Ewing et. al, 1957) thatat the free surface of a homogeneous half-space, body waves generated by harmonic pointsources attenuate with a factor proportional to r −2 . For the same setting, the spatialattenuation of surface waves is proportional to r −0 5. . These laws of attenuation are notapplicable, in general, to transient wave-forms because in the latter, the spatial attenuationof the wave results from a combination of both geometrical spreading and spreading of thesignal with time (Keilis-Borok, 1989).

It is evident from these considerations that for harmonic oscillations at large distancesfrom the source, the surface wave field dominates the overall particle motion and

( ) ( )$ , , $ , ,u ur y r ySω ω≈ . The distance from the source where the body wave field is notnegligible is usually called the near field. Numerical studies by Holzlohner (1980), Vrettos(1991) and Tokimatsu (1995) of wave propagation in vertically heterogeneous media haveshown that in normally dispersive media the near-field effects are important up to a distancefrom the source equal to λ 2 (where ( )λ λ ω= is the wavelength of the Rayleigh waves).However, in inversely dispersive media (i.e. media where the material properties varyirregularly with depth) the near-field is larger and may extend up to 2λ . All theoreticalstudies and experimental measurements performed during this research program haveassumed the near-field effects to be negligible. Furthermore, even though surface wavestests are suitable for measurements of both Love and Rayleigh waves, most currentapplications including this study focus exclusively on Rayleigh waves.


With these assumptions the displacement Green’s function ( )$ , ,u r y ω can be easilycomputed from the solution of the Rayleigh eigenproblem using the concept of modesuperposition. In the previous section it was shown how to compute the particle

displacement induced by a harmonic point source Fy ⋅ e i tω located at r y yS= =0, of a

vertically heterogeneous half-space. From Eq. 3.14,

( ) ( ) ( )[ ]$ , , $ , ,, ,

uβ βω ωω ω βr y r y e

i t r y= ⋅ −U ψ(3.24)

where the subscript β = r y, denotes the radial and the vertical directions, respectively.

The expressions for ( )$ , ,U β ωr y and ( )ψ ωβ r y, , are obtained from Eqs. 3.15 and 3.16

with the modal amplitudes ( )[ ]A β ωr yj

, , computed from Eq. 3.21 with Fy = 1; the final

result is:

( )( ) ( ) ( ) ( ) ( )[ ]$ , ,

, , , , cos

( )( )

.

U r

i j i S j S i j

i j i i i j j jj

M

i

M

r yr

r k y r k y r k y r k y r k k

k k V U I V U Iω

π=

⋅

−

==∑∑1

4 2

1 1 2 2

11

0 5

(3.25a)

( )( ) ( ) ( ) ( ) ( )[ ]$ , ,

, , , , cos

( )( )

.

U y

i j i S j S i j

i j i i i j j jj

M

i

M

r yr

r k y r k y r k y r k y r k k

k k V U I V U Iω

π=

⋅

−

==∑∑1

4 2

2 2 2 2

11

0 5

(3.25b)

and

( )

( )ψ ω

ω ω π

ω ω πr

i i S

i i i i

ii

M

j j S

j j j j

jj

Mr y

r k y r k y

k V U Ik r

r k y r k y

k V U Ik r

( , , ) tan

( , , ) ( , , )sin

( , , ) ( , , )cos

=⋅

⋅ ⋅ −

⋅⋅ ⋅ −

− =

=

∑

∑1

1 2

1

1 2

1

4

4

(3.26a)


( )

( )ψ ω

ω ω π

ω ω πy

i i S

i i i i

ii

M

j j S

j j j j

jj

Mr y

r k y r k y

k V U Ik r

r k y r k y

k V U Ik r

( , , ) tan

( , , ) ( , , )sin

( , , ) ( , , )cos

=⋅

⋅ ⋅ +

⋅⋅ ⋅ +

− =

=

∑

∑1

2 2

1

2 2

1

4

4

(3.26b)

Equations 3.24 through 3.26 completely define the displacement Green’s function

( )$ , ,u r y ω . Equation 3.24 is informative because it shows that a multiplicative

decomposition of the displacement Green’s function of the type ( )$ $ arg $u

u

β ββ=U e

i is possible

even for multi-mode wave propagation. However, because the wavenumber ( )$ , ,k r yβ ω is

no longer a constant, the spatial variation of the displacement field is no longer harmoniceven though the temporal variation of the source is harmonic. Equations 3.25 and 3.26

show also the remarkable result that the three main factors in the expression of ( )$ , ,u r y ω ,

namely the source depth ( )yS , the receiver depth ( )y , and the distance from the source

( )r , are uncoupled in the sense that their contribution is independent from each other.

With the definition of Green’s function given in this section, the computation of the

displacement ( )u r y, ,ω induced by a point harmonic source Fy ⋅ e i tω located at

r y yS= =0, becomes a trivial task. In fact:

( ) ( )u ur y F r yy, , $ , ,ω ω= ⋅ (3.27)

It is instructive to re-write Eq. 3.27 as follows:

( ) ( ) ( )[ ]uβ βω ψ ωω ω βr y F r y ey

i t r y, , , ,

, ,= ⋅ ⋅ −G (3.28)

where a new function ( ) ( )Gβ βω ωr y r y, , $ , ,=U called the Rayleigh geometrical spreading function,

has been introduced. This function has the important physical interpretation of modelingthe geometric attenuation in vertically heterogeneous media. As already mentioned Rayleighwaves in homogeneous media attenuate by a factor proportional to r −0 5. as a result of theirgeometrical spreading from a localized source. This simple geometric attenuation law, whichfollows directly from the principle of conservation of energy, does not hold in non-


homogeneous media. Under these conditions the Rayleigh wave displacement field resultsfrom the superposition of several modes of propagation (geometric dispersion). Animportant consequence is that the geometric spreading law of Rayleigh waves is altered.

Figure 3.4 shows a plot of the geometric spreading function ( )Gy r y, ,ω at the free-

surface ( )y = 0 of three different types of elastic media: a homogeneous medium, a three-layer soft-stiff-stiffer system (normally dispersive), and a three-layer stiff-soft-stiff system(inversely dispersive). The numerical simulation was carried out at a frequency of 40 Hz.

0.0E+00

2.0E-09

4.0E-09

6.0E-09

8.0E-09

1.0E-08

0 10 20 30 40 50 60

Distance [m]

Gy (r

,y,ω

) [m

]

Homogeneous

Inversely Dispersive

Normally Dispersive

Figure 3.4 Geometric Spreading Function for Different Types of Media

It is apparent from the figure that geometric attenuation in inversely dispersive media ismost strongly affected by geometric dispersion. This has also been observed in othernumerical studies (Tokimatsu et al., 1992; Gucunski and Woods, 1991).

From Eq. 3.25 it can be easily verified that in homogeneous media where M = 1 ,

k k ki j= = , and U V Vi i= = , ( )Gβ ωr y, , reduces as expected, to the frequency

independent function E rβ where [ ]E r y r y V I kr S= 1 224 2( ) ( ) π and

[ ]E r y r y V I ky S= 2 224 2( ) ( ) π . The importance of the explicit factorization of the


Rayleigh displacement field into a product of source magnitude, geometric spreadingfunction, and phase factor will become apparent in Section 5.3, which is dedicated to theattenuation of Rayleigh waves in weakly attenuating media.

3.5 Variational Principle of Rayleigh Waves in Elastic Media

This section will illustrate some interesting results that can be obtained from theapplication of Hamilton’s principle to the solution of the Rayleigh eigenvalue problem. Themost important of these results will be the derivation of closed-form expressions for thepartial derivatives of Rayleigh phase velocity with respect to the body wave velocities of themedium. As already mentioned in Section 3.1, these partial derivatives are very useful in thesolution of the inverse problem, a topic that will be discussed in Chapter 4. The Rayleighvariational principle can be used to compute both the modal and the effective Rayleighphase velocity partial derivatives with respect to the medium parameters. The Rayleighvariational principle can also be used to obtain important results for Rayleigh wavespropagating in weakly attenuating media. In fact, it will be shown later in this section thatthe solution of the Rayleigh eigenproblem in elastic media can be used as a basis to computethe Rayleigh attenuation coefficient ( )α ωR , which is a parameter characterizing theresponse of dissipative media.

Hamilton’s principle applied to a continuous, conservative system of volume V, statesthat among all the possible paths of motion between two instants in time t1 and t 2 theactual path is such that the integral:

( )I L= ∫∫ qi , & , ,q q dVdti i jVt

t

1

2

(3.29)

where L = −T U has a stationary value (Goldstein, 1980). The function ( )L q i , & , ,q qi i j

( , ; , )i N j= =1 1 3 , already introduced in Section 3.2, is called the Lagrangian density,whereas the variables q i describing the behavior of the system are the generalizedcoordinates. Finally T and U are the kinetic and the potential energy of the system,respectively. It is well known from the calculus of variations (Logan, 1997) that for theintegral of Eq. 3.29 to have a stationary value it is required that:

( ) ( )δ δI L= =∫∫ q i , & , ,q q dVdti i jVt

t

1

2

0 (3.30)


That is, the first variation of the integral I vanishes for arbitrary changes δ q i whichvanish at the boundary of the volume V and at times t1 and t 2 . Implementation of Eq.3.30 yields Lagrange’s equations of motion of the system and the associated naturalboundary conditions. It was shown in Section 3.2 that in a continuous linear elastic material

the Lagrangian density is given by ( ) ( )L & , & &,u i u u u Gi j i i kk ij ij= − +

12

12

2ρ λ ε ε ε where

( )λ λ= y , ( )G G y= , and ( )ρ ρ= y . If the elastic body of volume V is identified with avertically heterogeneous half space with no body forces and surface tractions, and if the

displacement field ( )u x, t is specified according to Eq. 3.5, the expression for theLagrangian density becomes:

( )L = + − +

+ −

+ +

14

14

2212

22

12

2

12

2

212 2

2

ρω λr r krdr

dyG

dr

dykr G k r

dr

dy(3.31)

where the symbol ⋅ denotes the average value of a quantity. The average value has beenused to eliminate the time dependence from the definition of the harmonic Lagrangian

density. Equation 3.31 defines the average Lagrangian density L for the homogeneousboundary value problem of Rayleigh waves. Application of Hamilton’s principle with the

Lagrangian density given by Eq. 3.31 for any perturbation of the eigenfunctions ( )r y k1 , ,ω

and ( )r y k2 , ,ω satisfying the boundary conditions given by Eq. 3.11 yields:

( ) [ ]δ δ δ ωI L = = − − − =∞

∫ dy I k I kI I0

21

22 3 4 0 (3.32)

where I I I and I1 2 3 4, , , are the Rayleigh energy integrals, and are defined as (Aki andRichards, 1980):

( ) ( )[ ]I r r dy I G r Gr dy1 12

22

2 12

22

00

1

2

1

22= + = + +

∞∞

∫∫ ρ λ (3.33a)

( )I rdr

dyGr

dr

dydy I G

dr

dyG

dr

dydy3 1

22

14

2

2

1

2

00

1

22= −

= +

+

∞∞

∫∫ λ λ (3.33b)


In Eq. 3.32 the eigenfunctions ( )r y k1 , ,ω and ( )r y k2 , ,ω are the only quantities beingperturbed, hence:

( ) ( )δ δ ω δ δ δ δI L = = − − − =∞

∫ dy I k I k I I0

21

22 3 4 0 (3.34)

From the Rayleigh equations of motion and associated boundary conditions (Eqs. 3.10and 3.11), it is possible to obtain the following result at the stationary point where

δ L dy0

0∞

∫ = (Aki and Richards, 1980):

( )L dy I k I kI I0

21

22 3 4 0

∞

∫ = − − − =ω (3.35)

Equation 3.35, written in the form ω21

22 3 4I k I kI I= + + , can be interpreted as a

statement of conservation of energy, i.e., the average kinetic energy associated with a givenmode of propagation equals the average elastic strain energy.

The combined results given by Eqs. 3.34 and 3.35 will be referenced hereafter as thevariational principle of Rayleigh waves. This should not be confused with the Rayleighprinciple which asserts that first-order perturbations in the Rayleigh eigenvalue (namely thewavenumber k) will only result in second-order perturbations of the correspondingeigenfunctions (Ben-Menahem and Singh, 1981) and may, in fact, be derived from Eqs. 3.34and 3.35 if desired.

3.5.1 Modal Rayleigh Phase Velocity Partial Derivatives

The results obtained in the previous section, namely, the variational principle ofRayleigh waves, will now be used to obtain closed-form expressions for the partialderivatives of the modal Rayleigh phase velocity ( )VR ω with respect to the body wave

velocities of the medium VP and VS . For this purpose, let the triple ( ) ( ) ( ) λ ρy G y y, ,

characterize the material properties of a linear elastic vertically heterogeneous medium M .

The pair ( ) k r G kj i j, , , , ,λ ρ ω with i j M= =1 4 1, ; , will represent the solution of the

Rayleigh eigenvalue problem associated with this medium. In the functional dependence of

the eigenfunctions, the material properties ( )λ y , ( )G y , and ( )ρ y have been considered as

independent variables. Now let ( ) ( )[ ] ( ) ( )[ ] ( ) λ δ λ δ ρy y G y G y y+ +, , denote the

material properties of a medium ~M whose Lamè’s parameters differ slightly from those of

the medium M . At a given frequency ω , the pair


( ) k k r G G k kj j i j j+ + + +δ λ δλ δ ρ δ ω, , , , , represents the solution of the Rayleigh

eigenproblem associated with this new medium.

Later in this section it will be shown that the problem of determining the partial

derivatives of the modal Rayleigh phase velocity ( )[ ]VR jω with respect to the medium

parameters is essentially reduced to that of computing δk j . The latter task is accomplished

by using the variational principle of Rayleigh waves, namely Eqs. 3.34 and 3.35. For ease of

notation, the modal parameters k j and ( )[ ]VR jω will subsequently be denoted without the

subscript j. Application of Eq. 3.35 to the medium ~M yields:

( ) ( )[ ]

( ) ( ) ( )[ ]( ) ( )( )

( ) ( )( ) ( ) ( )( ) ( )

( ) ( )[ ] ( ) ( )

1

2

1

22

1

22

21 1

2

2 2

2

0

21 1

2

2 2

2

0

1 1 2 2 2 2 1 10

2 2

2

1

ω ρ δ δ

δ λ δλ δ δ δ δ

δ λ δλ δ δ δ δ δ

λ δλ δ δ δ

r r r r dy

k k G G r r G G r r dy

k k r rd

dyr r G G r r

d

dyr r dy

G Gd

dyr r G G

d

dyr

+ + + =

+ + + + + + + + +

+ + + + + − + + +

+

+ + + + +

+ +

∞

∞

∞

∫

∫

∫

( )+

∞

∫ δr dy1

2

0

(3.36)

Application of Eqs. 3.34 and 3.35 to Eq. 3.36 expanded to include first order terms leadsto:


( )[ ]

( )

δ λ λ

δ λ

δ

k rdr

dyGr

dr

dyk G r Gr dy

dr

dykr

dr

dyk r G dy

dr

dyk r

dr

dyr

dr

dyk r G

12

21

12

22

0

2

2

12 2

12

0

1

2

21

12 2

22

2

1

2

1

22

1

22

1

2

− + + +

+

+

+ +

+

+

+

− +

+

∞

∞

∫

∫

=

∞

∫ dy 00

(3.37)

Now from Eq. 3.35:

( )δ ω

ωδω ω δ δ δ δ δ δ

21

22 3 4

21 2

22 3 3 42 2 0

I k I kI I

I I k k k I I k k I I

− − − =

= + − − − − − =(3.38)

which considering Eq. 3.34 yields:

Uk

kI I

I= =

−δωδ ω

2

22 3

1

(3.39)

Equation 3.39 is an important result because it is an alternative procedure to compute thegroup velocity without using numerical differentiation. In view of Eq. 3.39, Eq. 3.37 can bere-written as:

( )δω

δ δ λkUI

krdr

dykr

dr

dyGdy kr

dr

dyG dy= − −

−

+ +

+

∞ ∞

∫ ∫1

44 2

12

1

2

12

01

2

2

0

(3.40)

Since VkR =ω

, it follows that δω

δkk

VR= −2

, and hence from Eq. 3.40:

( )δ δ δ λVk UI

krdr

dykr

dr

dyGdy kr

dr

dyG dyR = −

−

+ +

+

∞ ∞

∫ ∫1

44 22

12

1

2

12

01

2

2

0

(3.41)


Equation 3.41 allows one to compute the change in Rayleigh phase velocity δVR

resulting from a small perturbation of the Lamè’s parameters ( )λ y , and ( )G y . Equation3.41 may also be expressed as:

δ∂∂

δ∂∂

δω ω

VV

GG dy

V

MM dyR

R

M

R

G

=

+

∞ ∞

∫ ∫, ,0 0

(3.42)

where ( )M G= +λ 2 is the constrained modulus and:

∂∂

∂∂

ω

ω

V

G k UIkr

dr

dykr

dr

dy

V

M k UIkr

dr

dy

R

M

R

G

= ⋅ −

−

= ⋅ +

,

,

1

44

1

4

21

21

2

12

21

12

2(3.43)

The subscripts outside the brackets indicate the parameters that are held constant.Equations 3.43 are known in seismology as the partial derivatives of the modal Rayleighphase velocity ( )VR ω with respect to the medium parameters G and M . However, the

term partial derivative must be used with care because ( ) ( ) ( ) V V M y G y y kR R= , , , ,ρ ω

and hence ( )VR ω is a functional rather than a function of the parameters ( )M y and ( )G y .Accordingly, the partial derivatives in Eq. 3.43 are understood to refer to a particular depthy.

One more step is required to compute the partial derivatives of ( )VR ω with respect to

the body wave velocities of the medium VP and VS . Since G VS= ρ 2 and M VP= ρ 2 , usingthe chain-rule of calculus:

( )

( )

∂∂

∂∂

ω∂∂

∂∂

ρ

∂∂

∂∂

ω∂∂

∂∂

ω ω ρ

ω ω ρ

V

V

V

Vy k

V

G

G

V

V

k UIkr

dr

dykr

dr

dy

V

V

V

Vy k

V

M

M

V

R

S V

R

S

R

M S

S

R

P V

R

P

R

G P

P

S

= =

⋅

= ⋅ −

−

= =

⋅

=

, ,

, ,

, ,

, ,

242

12

1

2

12

ρV

k UIkr

dr

dyP

2 21

12

2

⋅ +

(3.44)


Equations 3.44 give a measure of the sensitivity of modal Rayleigh phase velocity( )VR ω to small changes of the medium parameters VP and VS at a specific depth y. The

remarkable feature of Eq. 3.44, which makes it so valuable in the solution of the Rayleighinverse problem, is that these partial derivatives can be computed using Rayleigh waveparameters referred to the original and not the perturbed VP and VS profiles. Conversely,it would be very expensive to compute the above partial derivatives numerically with, say, afour-point central finite difference scheme (Spang, 1995); a single computation of∂ ∂V VR S would require the solution of four Rayleigh eigenproblems instead of just oneeigenproblem using the variational approach.

Examining Eq. 3.44, it is found that the phase velocity of Rayleigh waves is relativelyinsensitive to changes in VP (Lee and Solomon, 1979; Ben-Menahem and Singh, 1981), andthus the partial derivative ∂ ∂V VR P is small compared to ∂ ∂V VR S . Figure 3.5 shows thepartial derivatives given by Eq. 3.44 for the case of a homogeneous medium and a frequencyof 40 Hz.

0.0

2.0

4.0

6.0

8.0

10.0

0.0 0.1 0.2 0.3 0.4 0.5

dVR/dVS , dVR/dVP

Dep

th [

m]

d V R /d V S

d V R /d V P

Frequency 40 Hz

VS = 120 m/s

VP = 400 m/s

dVR/dVS

dVR/dVP

Figure 3.5 Partial Derivatives of Rayleigh Phase Velocity with Respect to VP and VS fora Homogeneous Medium


It is apparent from the figure that the shear wave velocity VS controls the Rayleigh

wave velocity ( )VR ω . The largest values of ∂ ∂V VR S and ∂ ∂V VR P are 0.418 and 0.105respectively. Both maxima occur at the free-surface ( )0y = .

In a stratified medium composed of a finite number of homogeneous layers overlayingan homogeneous half-space it may be of interest to evaluate how Rayleigh phase velocity

( )VR ω is affected by a small change in body wave velocities VP and VS of one layer. Thisquantity can be computed by integrating Eq. 3.44 over the layer thickness. Thus, for a

layered medium composed by NL layers L j ( )j NL= 1, the results is:

∂∂

ρ

∂∂

ρ

V

V

V

k UIkr

dr

dykr

dr

dydy

V

V

V

k UIkr

dr

dydy

R

S L

S

y

y

R

P L

P

y

y

j j

j

j j

j

= ⋅ −

−

= ⋅ +

−

−

∫

∫

24

2

21

21

2

12

21

12

2

1

1

(3.45)

3.5.2 Effective Rayleigh Phase Velocity Partial Derivatives

In Section 3.5.1 closed-form expressions for the partial derivatives of Rayleigh phasevelocity with respect to the medium parameters have been obtained. These partialderivatives are modal quantities, in the sense that they refer to a specific mode ofpropagation of Rayleigh waves. However, it was shown in Section 3.3 that in verticallyheterogeneous media excited by harmonic sources, Rayleigh waves propagate in wave trainsthat result from the superposition of different modes of propagation. In thesecircumstances, the relevant kinematical quantity that describes the velocity of propagation ofthe wave train is the effective Rayleigh phase velocity.

In this section, closed-form solutions for the partial derivatives of the effective Rayleigh

phase velocity ( )$ , ,V r yβ ω with respect to the body wave velocities of the medium VP and

VS will be obtained. From Eq. 3.23 the effective Rayleigh phase velocity may be considered

a function of the following variables ( ) ( )[ ]$ $ , , , , , , , ,V V r y k r y k V U kj j j j jβ β ω ω= 1 2 , where

β = r y, and j M= 1, . The explicit dependence of $Vβ on the independent variables r and ωhas been omitted because it is irrelevant in the next developments. It is convenient to re-write Eq. 3.23 as follows:


( )

( ) ( )[ ]

( ) ( ) ( )[ ]$ , ,

cos

cosVβ

β

β

ω

ω

ωr y

r V V

V V

V V r V V

V V

ij i j

i jj

M

i

M

rs r s r s

r ss

M

r

M

=

⋅−

+ −

==

==

∑∑

∑∑

21 1

1 1 1 1

11

11

Φ

Φ(3.46)

where:

( )( ) ( ) ( ) ( )

( )( )Φ r ij

i j S i S j

i i j j

r y k r y k r y k r y k

U I U I=

1 1 2 2, , , ,(3.47a)

( ) ( ) ( ) ( ) ( )( )( )Φ y

ij

i j S i S j

i i j j

r y k r y k r y k r y k

U I U I=

2 2 2 2, , , ,(3.47b)

Hence from Eq. 3.46:

( ) ( )[ ]$ $ , , , , , , ,V V r y k r y k V Uj j j jβ β ω ω= 1 2 (3.48)

since Vj and k Vj j= ω are not independent, and I j is given by Eq. 3.22.

Mimicking the procedure used in Section 3.5.1, the problem of determining the partialderivatives of $Vβ with respect to the medium parameters is essentially reduced to that of

computing δ β$V for small variations of δVS and δVP . From Eq. 3.48, it is apparent that the

latter task requires the computation of quantities such as ( )δ rj1 , ( )δ r

j2 , δVj , and δU j .

However from Rayleigh principle, first-order perturbations in the wavenumber δk j will induce

variations in the corresponding eigenfunctions ( )δ rj1 and ( )δ r

j2 that are of second order.

Hence, in computing the first variation of δ β$V , the terms ( )δ r

j1 and ( )δ rj2 may be

neglected, and from Eq. 3.48:


δ∂

∂δ

∂

∂δβ

β β$$ $

VV

VV

V

UU

jj

jj= +

(3.49)

where the summation convention is implied over the index j. The first variation of δVj is

given by Eq. 3.42, and δU j can be computed from Eq. 3.39 resulting in:

( )( )δ δ

δ δω

ω

ωδU

I

V IV

I

V I

I

I

V I I

V IIj

j

= −⋅

+⋅

− +⋅ −

⋅

22

1

2

1

3

1

3 2

12 12

2

2(3.50)

In Appendix B it is shown that the first variations of δVj and δU j given by Eq. 3.41

and Eq. 3.50, respectively, can be written as follows:

[ ]δ δ δV P V Q V dyj j S j P= +∞

∫0

(3.51)

[ ]δ δ δU V V dyj j S j P= +∞

∫ Π Ω0

(3.52)

where:

( ) ( )P yV

k UIkr

dr

dykr

dr

dyjS

j j

,ωρ

= ⋅ −

−

2

42

12

1

2

12

(3.53a)

( ) ( )Q yV

k UIkr

dr

dyjP

j j

,ωρ

= ⋅ +

2 21

12

2

(3.53b)

and


( ) ( ) ( )Π j

j

S

jj jy

IV kr r

dr

dyr

dr

dyk I P,ω

ωωρ= ⋅ − −

− ⋅

12

21

22

21

12 2

2 (3.54a)

( ) ( ) ( )Ωj

j

P

jj jy

IV kr r

dr

dyk I Q,ω

ωωρ= ⋅ −

− ⋅

12

1

12

12 2

2 (3.54b)

In view of Eqs. 3.51 and 3.52, Eq. 3.49 can be written as follows:

δ∂

∂

∂

∂δ

∂

∂

∂

∂δβ

β β β β$$ $ $ $

VV

VP

V

UV

V

VQ

V

UV dy

jj

jj S

jj

jj P= +

+ +

∞

∫ Π Ω0

(3.55)

which, by analogy to Eq. 3.42 suggests the following definition:

δ∂

∂δ

∂

∂δβ

β

ω

β

ω

$$ $

, ,

VV

VV dy

V

VV dy

S V

SP V

P

P S

=

+

∞ ∞

∫ ∫0 0

(3.56)

where:

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

β

ω

β β

β

ω

β β

$ $ $

$ $ $

,

,

V

V

V

VP

V

U

V

V

V

VQ

V

U

SV

jj

jj

PV

jj

jj

P

S

= +

= +

Π

Ω

(3.57)

Equation 3.57 provides expressions for computing the partial derivatives of the effective

Rayleigh phase velocity ( )$ , ,V r yβ ω with respect to the medium parameters VP and VS . To


complete the formulation requires explicit expressions for the partial derivatives ∂

∂β

$V

Vj

and

∂

∂β

$V

U j

. Appendix B shows the details of this computation. The final result is:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

∂

∂

∂

∂ω

∂

∂

∂

∂ω

β

ω

ββ β β β

β

ω

ββ β β β

$ $, , ~ ~

$ $, , ~ ~

,

,

V

V

V

Vr y P P T T

V

V

V

Vr y Q Q T T

SV

Sij i ij j ij i ij j

j

M

i

M

PV

Pij i ij j ij i ij j

P

S

= = −

+ +

= = −

+ +

==∑∑ Γ Γ Π Π

Γ Γ Ω Ω

11

==

∑∑j

M

i

M

11

(3.58)

The terms ( ) ( )Γ Γβ βij ij, ~ and ( ) ( )T T

ij ijβ β, ~ are complicated functions of the modal

parameters V Uj j, , ( )r y k j1 , ,ω and ( )r y k j2 , ,ω , the frequency of excitation ω , and the

distance from the source r. Because their expressions are rather lengthy, they have beenreported in Appendix B

The distinctive feature of the effective partial derivatives that distinguishes them fromthe modal partial derivatives is their dependence on r. The local properties of thesequantities have been inherited from the effective Rayleigh phase velocity during the processof differentiating Eq.3.46.

3.5.3 Attenuation of Rayleigh Waves in Weakly Dissipative Media

In the previous sections it was shown how variational methods can be used to obtainuseful expressions regarding Rayleigh wave propagation in elastic media. In particular, thesemethods were used to determine closed-form solutions of the partial derivatives of modaland effective Rayleigh phase velocity with respect to the medium parameters. Anotheruseful result was an alternate expression for computing Rayleigh group velocity without theuse of numerical differentiation.

Variational methods can also be used to obtain important results for Rayleigh wavepropagation in weakly dissipative media. In Chapter 2 weakly dissipative media were definedas those media satisfying Eq. 2.31. It was also shown that the mechanical propertiesgoverning the behavior of such media are defined by Eqs. 2.34 and 2.35. In physical termsthese are the phase velocity ( )Vγ ω and attenuation coefficient ( )α ωγ of compression and

shear waves ( )γ = P S, . In this section variational techniques are employed to obtain phase


velocity ( )VR ω and attenuation coefficient ( )α ωR of Rayleigh waves propagating in weaklydissipative media. Equation 3.42 also implies the following result:

δ∂∂

δ∂∂

δω ω

VV

VV dy

V

VV dyR

R

S V

SR

P V

P

P S

=

+

∞ ∞

∫ ∫, ,0 0

(3.59)

In weakly dissipative media the material damping ratio Dγ is a small quantity, and hence

it is reasonable to assume that in Eq. 2.34 the second order terms in Dγ are in most cases

negligible. With this assumption Eq. 2.34 simplifies to ( ) ( ) [ ]V V iDγ γ γω ω* = ⋅ −1 with

( )Vγ ω given by Eq. 3.35. If V eγ denotes the phase velocity of the γ -wave in a linear elastic

medium, the existence of material damping may be thought (Anderson and Archambeau,

1964; Anderson et al., 1965) to introduce a small change in V eγ given by:

( )[ ]δ γ γ γ γ γV V V iV De* = − − (3.60)

Then, substituting Eq. 3.60 in Eq. 3.59 for δVS and δVP :

( )[ ] ( )[ ]δ∂∂

∂∂

ω ω

VV

VV V iV D dy

V

VV V iV D dyR

R

S V

S Se

S SR

P V

P Pe

P P

P S

*

, ,

=

− − +

− −

∞ ∞

∫ ∫0 0

(3.61)

where ( )[ ]δV V V iV DR R Re

R R* = − − . Taking the real and the imaginary parts of Eq. 3.61:

( )

( )( )

V V VV

V

V

Vdy V

V

V

V

Vdy

DV

V DV

Vdy V D

V

Vdy

R Re

SR

S V

Se

SP

R

P V

Pe

P

RR

S SR

S V

P PR

P V

P S

P S

ω∂∂

∂∂

ωω

∂∂

∂∂

ω ω

ω ω

= +

−

+

−

= ⋅

+

∞ ∞

∞ ∞

∫ ∫

∫ ∫

, ,

, ,

1 1

1

0 0

0 0

(3.62)


From Eq. 2.35, ( )1

2−

=

V

V

De

ref

γ

γ

γ

ω πω

ωln where ( )V Vref

eγ γω = since ω ref is the

reference frequency for material dispersion. Also, ( ) ( )α ωω

ωRR

RVD= from Eq. 2.32. With

these results Eq. 3.62 can be rewritten as follows:

( )

( )( )[ ]

V V V DV

Vdy V D

V

Vdy

VV D

V

Vdy V D

V

Vdy

R Re

refS S

R

S V

P PR

P V

R

R

S SR

S V

P PR

P V

P S

P S

ωπ

ωω

∂∂

∂∂

α ωω

ω

∂∂

∂∂

ω ω

ω ω

= +

⋅

+

= ⋅

+

∞ ∞

∞ ∞

∫ ∫

∫ ∫

2

0 0

20 0

ln, ,

, ,

(3.63)

Equation 3.63 is an important result because it shows that Rayleigh phase velocity( )VR ω and attenuation coefficient ( )α ωR in vertically heterogeneous, weakly dissipative

media can be computed from the solution of the elastic Rayleigh eigenvalue problem. Infact Eq. 3.63 forms the basis of an algorithm for the solution of the uncoupled inverseproblem of Rayleigh waves. The procedure, which will be described in detail in Chapter 4,involves three major steps. The first step is the determination of the experimentaldispersion and attenuation curves, namely ( )VR ω and ( )α ωR , from surface wave

measurements. In the second step the experimental dispersion curve ( )VR ω is inverted to

obtain the elastic shear wave velocity profile ( )V yS . The third and final step involves the

use of Eq. 3.63 as the basis of the inversion of the experimental attenuation curve ( )α ωR

to obtain the material damping ratio profile ( )D yS .

It should be noted that the inversion for VS is non-linear, but the inversion for ( )D yS

is linear. Most of the procedures currently used by seismologists to study Rayleigh waveattenuation are based on Eq. 3.63 and the assumption of weak dissipation (Lee andSolomon, 1979; Aki and Richards, 1980; Keilis-Borok, 1989; Herrmann, 1994). The nextsection will illustrate a new technique that can be used to simultaneously invert both thedispersion and attenuation curves. The technique is quite general and can also be applied tostrongly dissipative media.

Although Eq. 3.63 has been derived with reference to individual Rayleigh wave modes, itis also valid for the case of effective Rayleigh wave propagation with small changes ofnotation. The primary modification is the replacement of the modal partial derivatives (Eq.3.44) with the effective partial derivatives (Eq. 3.58). Then the modal phase velocities andattenuation coefficients ( )VR ω and ( )α ωR must be replaced with the corresponding


effective quantities ( )$ ,V rβ ω and ( )$ ,α ωβ r . However, for the latter operation to be

legitimate, Eq. 2.32b, namely ( ) ( ) ( )α ω ω ωR R Rk D= ⋅ , must be assumed to be true for theeffective (instead of modal) quantities.

3.6 Rayleigh Eigenvalue Problem in Viscoelastic Media

The approach used to define the Rayleigh eigenvalue problem in elastic media was basedon the application of Lagrange’s equations to a vertically heterogeneous elastic medium. Asolution of the resulting Navier’s equations of motion was then sought in a form of aharmonic displacement field satisfying the boundary conditions for surface waves. Thegeneralization of this procedure to viscoelastic media requires the use of certain variationaltheorems of linear viscoelasticity (Christensen, 1971). The equations of motion and theassociated boundary conditions are then established from the stationary condition of anenergy functional.

However, integral transform methods provide a more straightforward approach to theformulation of the Rayleigh eigenproblem in viscoelastic media. For boundary valueproblems with time-invariant boundary conditions, the use of integral transform methodsleads naturally to the application of the elastic-viscoelastic correspondence principle.According to this principle, elastic solutions to steady state harmonic problems can beconverted into viscoelastic solutions for identical boundary conditions by simply replacingthe elastic moduli with the corresponding frequency dependent complex moduli (Read,1950; Christensen, 1971). Application of the correspondence principle to the elasticeigenproblem leads to a complex eigenproblem for viscoelastic media. Formally, Eq. 3.10and the associated boundary conditions (Eq. 3.11) are still adequate to describe this complex

eigenproblem if the vector ( )f y and the matrix ( )A y are complex-valued arrays. Most ofthe features described for the elastic eigenproblem are carried over to the complexeigenproblem with the important difference that non-trivial solutions of the latter arecomplex-valued eigenvalues (i.e the wavenumbers) and eigenfunctions. It should beremarked, however, that certain properties of the real eigenproblem require furtherconsideration before they can be generalized to the complex eigenproblem. One example isthe orthogonality of the eigenfunctions.

The techniques available for the solution of the complex eigenproblem are in essencethe same as those used for the elastic eigenproblem. The main difference is that the use ofcomplex arithmetic can no longer be avoided in solving the viscoelastic eigenproblem, andalgorithms such as root finding techniques must be properly generalized to remainapplicable for complex values of the arguments. This generalization is not always trivial asshown in the next section.

Before concluding this section it is worth noting that the viscoelastic eigenproblemincludes an interesting degenerate case. If the complex Lamè’s parameters are such that the


viscoelastic Poisson’s ratio is a frequency-independent real number, it can be shown that theroots of the Rayleigh dispersion equation as well as the corresponding eigenfunctions arereal (Christensen, 1971). Therefore, in this special circumstance, the solution of the complexeigenproblem could be obtained using the same procedures used for the elasticeigenproblem. The wavenumbers for the viscoelastic eigenproblem will still be complex,however.

3.6.1 A Solution Technique

A new technique has been developed for the solution of the complex eigenvalueproblem. This technique is based on the generalization to viscoelastic media of the methoddeveloped by Chen (1993) and Hisada (1995) for finding the normal modes in multi-layeredelastic half-spaces. In their algorithm the authors solve the elastic eigenproblem using themethod of reflection and transmission coefficients. Figure 3.6 shows the parameterscharacterizing the properties of viscoelastic multi-layered media.

Layer 1

Rayleigh Wave n

r

y

Layer 2

Layer nl

h V V D DP S1 P S11 1 1 1, , , , ,ρ

h V V D DP S P S2 2 2 2 2 2, , , , ,ρ

M

h V V D Dn n P n Sn P n Snl l l l l l, , , , ,ρ

M M M M M M

Figure 3.6 Rayleigh Waves in Viscoelastic Multi-Layered Media

As mentioned in Section 3.2.1, most of the computational effort required for thesolution of the elastic eigenvalue problem are spent in constructing the Rayleigh secularfunction and in finding its roots. This is also true for the viscoelastic eigenproblem with newdifficulties arising in connection with the root computation of the Rayleigh secular function.The latter is now a complex-valued function ( )F R ⋅ of the complex-valued Rayleigh phase


velocity VR* defined by Eq. 2.34: symbolically ( )F R VR

* : C C→ (for simplicity, the use of an

asterisk to denote a complex quantity will hereafter be limited to the essential cases).

In general, the problem of computing the roots of a complex-valued function ( )f ⋅ of a

complex variable z, namely ( )f z : C C→ , is not simple, particularly if the function is highlynon-linear and is known only numerically (Henrici, 1974). No general methods are currentlyavailable, and in most cases the adopted strategy consists in breaking up the complexstructure of the function and transforming it into an equivalent pair of non-linear equationsin two real variables. The root-finding problem is then addressed by conventional methods.When this approach was used in this study to find the roots of the real and imaginary partsof the Rayleigh secular function, it was found to have limited success, and only for verysmall values of material damping ratio.

Most of the difficulties associated with computing the roots of functions ( )f z : C C→are overcome if the function ( )f z satisfies the condition of analyticity within an open setD ⊂ C (see Hille, 1973 for a precise definition of the necessary and sufficient conditionsrequired for a function ( )f z : C C→ to be analytic or holomorphic in D ⊂ C ). In this case the

zeros of the function ( )f z may be determined using a completely new class of algorithmswhich are developed by taking full advantage of the theory of analytic functions. Strictly

speaking, the Rayleigh secular function ( )F R VR* is not analytic with respect to the complex

variable Rayleigh phase velocity VR* , because it is not a single-valued function of VR

* .

However, this fundamental property can be restored (within a set D D: *VR ∈ ⊂ C ) during

the process of constructing ( )F R VR* by choosing the appropriate branches of ( )F R VR

* on

the Riemann surfaces. The practical implementation of this process requires that all of the

branch-cuts and branch-points of ( )F R VR* in D be identified first (Båth, 1968; Schwab and

Knopoff, 1972).

In this study a new technique for computing the zeros of ( )F R VR* was developed. The

technique finds the roots of the Rayleigh secular function without breaking up its complexstructure. Conversely, it takes advantage of the intimate connection existing between the

real and the imaginary part of ( )F R VR* , as a consequence of its analyticity with respect to

*RV in the domain of interest, with the exception of at most a finite number of isolated

pole-type singularities. In this case the term meromorphic function would be more appropriate

than holomorphic function to designate ( )F R VR* . The proposed method is based on the

theory of analytic functions, particularly the well-known Cauchy’s residue theorem:


( )12 1πi

z dz s z jj

M

⋅ ==∑∫ f( ) Re

Γ(3.64)

where the integral sign denotes integration along a positively oriented closed contour Γ ,

( )f z is an analytic function inside and on Γ except at the points z j ( )j M= 1, where it may

have isolated singularities; z ∈C is a point of the complex plane ( )z x iy= + , and the

symbol ( )Res z j denotes the residue of the function ( )f z at the point z j . Finally, M is the

number of isolated singularities of ( )f z located inside Γ . Equation 3.64 forms the basis of

the algorithm proposed by Abd-Elall et al. (1970) for computing the roots of ( )f z . In factCauchy’s residue theorem may also be written in the form:

Gi

zz

dz zN

N

j jN

j

m

= ⋅ =∫ ∑=

12 1π

ρf( )Γ

(3.65)

where ρ j are the residues of 1 f( )z at the points z j that are the zeros of ( )f z , m is the

number of zeros of ( )f z which are located inside Γ . By evaluating the contour integraldefined by Eq. 3.65 for different values of N ( , )N m= −0 2 1 , a sequence of complexnumbers GN G

determine the coefficients of the complex polynomial ( )P zm :

( )P z c c z c z c z zm mm m= + + + + +−

−0 1 2

21

1.... (3.66)

by solving the linear system of equations that can be constructed from the modifiedNewton identities (Abd-Elall et al., 1970):

G c G r mr j j r mj

m

+ +=

−

⋅ + = = −∑ 0 0 10

1

, (3.67)

The zeros of the polynomial (Eq. 3.66) coincide with the zeros of ( )f z inside Γ . Itshould be remarked that solution of the system of equations (Eq. 3.67) does not requireknowledge of the residues ρ j .

If ( )f z is identified with ( )F R VR* , and the contour Γ is the boundary of a region D

where the roots of ( )F R VR* are located, then the roots of ( )P zm can be interpreted as the


modal (complex) Rayleigh phase velocities associated with the solution of the complexeigenproblem. A fundamental step required for the implementation of the algorithm is thecomputation of the complex numbers G N for N m= −0 2 1, . This task is accomplished byevaluating the contour integral of Eq. 3.65 numerically.

A crucial step of this calculation is the definition of the region D , delimited by the

boundary Γ , where the roots of ( )F R VR* are located. To define D , one must determine

lower and upper bounds for the real and imaginary parts of VR* . In the plane

w R : ,V DR R lower and upper bounds for VR may be easily established from the roots of

the Rayleigh dispersion equation in homogeneous media, using min( )VS and max ( )VS ,respectively. Bounds for DR are found from the observation that0 ≤ ≤ <D D DR R Smax ( ) max ( ) . However, the plane of analyticity of the Rayleigh secular

function is not the plane w R : ,V DR R but rather the plane z R : ,x yR R , whose

relations with w R : ,V DR R are given by Eq. 2.34, namely:

( ) ( )

( ) ( )

z R :

,

,

x x V DV

D

y y V DV D

D

R R R RR

R

R R R RR R

R

= =+

= = −⋅

+

1

1

2

2

(3.68)

such that ( )V z x iyR R R R* = = + . In Eq. 3.65, the numerical evaluation of the integral

involving ( )F R VR* along the contour Γ must be performed in the zR-plane.

Equations 3.68 give the relationships between a point ( )w R R RV D: , of the wR-plane and

the corresponding point ( )z R R Rx y: , of the zR-plane. Thus, it is possible to map a region

C of the wR-plane into another region D of the zR-plane. Figures 3.7(a) and (b) show the

two regions C and D where the roots of ( )F R VR* are located. It should be noted from

the figures the opposite orientation of the boundaries γ and Γ of the two regions C and Dof the complex plane.

In Eq. 3.65 the numerical evaluation of the contour integral may be simplified byintroducing an admissible parametrization of the contour Γ so that the numbers GN maybe computed as follows:


0 100 200 300 400 500 600-2

0

2

4

6

8

10

VR (m/s)

DR (%

)

A B

CD

γ

C

roots

wR-plane

Figure 3.7(a) Roots of Rayleigh Secular Function in the Region C of the wR-plane

XR (m/s)

YR

(m/s

)

100 200 300 400 500 600-20

-15

-10

-5

0

5

A B

C

D

roots

ΓD

zR-plane

0

Figure 3.7(b) Roots of Rayleigh Secular Function in the Region D of the zR-plane


( )[ ]( )[ ] ( )G

i

z

zdz

i

z

zz dN

RN

R RR

R

N

R RR= ⋅ = ⋅ ⋅ ′∫ ∫

12

12π π

τ

ττ τ

F F( )Γ Γ (3.69)

where τ ∈R is a parameter. A necessary condition for an admissible parametrization ( )zR τof a curve Γ is the existence of the function ( )′zR τ . This requirement is not satisfied if thecurve Γ is parameterized with the Cartesian representation that follows from Eq. 3.68because of the existence of a pole-singularity at the points A and B of the zR-plane. Using apolar rather than a Cartesian representation of Γ overcomes this difficulty. An admissibleparametrization of the contour Γ is therefore given by following equations:

( ) ( ) ( )

[ ]

( ) ( ) ( )

Γ:

( )

( ) cos sin tan

( )

( ) cos sin tan

min max

max max

maxmax

max

max

min

max

min max

arc AB z V V

arc BC z V i a D

arc CD zD

iDV

D

V

D

arc DA z V i a D

R R

R R

RR

R

R

R

R

R R

τ τ τ

τ τ τ τ

ττ

τ

τ τ τ τ

= ≤ ≤

= ⋅ −

≤ ≤

=+

⋅ − + −+

≤ ≤ −+

= ⋅ +

− ≤ ≤

2

2 2 2

2

1

22 0

11

1 1

1

22 0

(3.70)

where V VR Rmin min( )= , V VR Rmax max ( )= , and D DR Rmax max( )= . In calculating the

integral of Eq. 3.69 care must be used because the two curves γ (wR-plane) and Γ (zR-plane)are oriented in opposite directions.

A substantial improvement in the accuracy of the above numerical integration can beachieved by rescaling the contour of integration Γ (zR-plane) via a conformal linear

transformation of the type ( )zR R R R= ⋅ +α ζ β where ζR ∈C and α βR R, ∈R are twoconstants (this type of mapping is a special case of a Möbius transformation). Under thisconformal mapping the region D of the zR-plane is mapped into a similar region D' of theζR -plane. Depending on the value adopted for α R , the region D' is the magnified orcontracted version of D . The factor βR is responsible for shifting the region D from theorigin. After some experimentation it was found that the optimum values for α R and βR

are ( )α βR R RV and= =max 0 .

The method of substitution in C , applied to the contour integral of Eq. 3.69 yields:


( )[ ]( )[ ] ( ) ( )[ ]

( )[ ] ( )Gi

z

zz d

idN

R

N

R RR

R R R R

N

R R R RR= ⋅ ⋅ ′ = ⋅

+

+⋅ ′∫ ∫

1

2 2π

τ

ττ τ

απ

α ζ τ β

α ζ τ βζ τ τ

F FΓ Γ

'

'' '

'(3.71)

where the contour Γ ' denotes the boundary of the region D' in the ζR -plane. Thenumerical integration in Eq. 3.71 was performed using a classical fifty-point Gauss-Legendrequadrature formula (Press et al., 1992).

Once the numbers GN ( N m= −0 2 1, ) have been calculated, the remaining stepsrequired for the complete implementation of the algorithm are 1) the solution of the linearsystems of equations (Eq. 3.67) for the coefficients c j mj ( , )= −0 1 , and 2) the computation

of the roots of the complex polynomial ( )P zm of Eq. 3.66. The task of solving the linearsystem of equations is accomplished with the complex version of the LU decompositionalgorithm, whereas the roots of ( )P zm are computed using La Guerre’s methodsupplemented with the appropriate deflation and polishing techniques (Press et al., 1992).

At each frequency ω , the determining the number of zeros of ( )F R VR* is the first step

in finding the roots of the Rayleigh secular function. A possible strategy for computing m isbased on the observation that all the matrices G j with j m> are singular (Abd-Elall et al.,

1970). Hence, the value of m can be found by a procedure that evaluates the rank ofsuccessive matrices G j until a value of j is found for which G j is singular. In this study the

test for singularity of the matrices G j was performed using their condition number R cond that

was calculated using the singular value decomposition of G j . A matrix G j is considered

singular if its condition number is so small that the logical expression ⟨ + = ⟩10 10. .R cond istrue to machine precision.

If the number of roots of ( )F R VR* which is equivalent to the number of Rayleigh

modes M (M m)= associated with a frequency ω is large, say mmax greater than 15 or 20,

the computation of the roots of the high-degree polynomial ( )P zm is an ill-conditioned

problem. As a result, it may be difficult to compute the roots ( )VR j

* ( , )j m= 1 of the

Rayleigh secular function with a high degree of accuracy. A strategy indicated by Delves andLyness (1967) to overcome this problem is to subdivide the region D' into smaller

subregions where the number of roots of ( )F R VR* in each subregion is less than mmax . The

problem of finding the roots of the Rayleigh secular function in D' is then reduced to that

of computing the roots of ( )F R VR* in each subregion. The calculation of the zeros of


( )P zm completes the problem of finding the eigenvalues ( ) ( )k VR j R j= ω * ( , )j M= 1

associated with the solution of the complex eigenproblem.

Once the roots of ( )F R VR* have been computed for a given frequency ω , the

remaining task is to determine the corresponding (complex) eigenfunctions ( )r y ki R j

* , ,ω

( , )i = 1 4 . The technique used in this study to compute ( )r y ki R j

* , ,ω

is the same

technique used to solve the elastic eigenproblem, and is based on the algorithm developedby Chen (1993) and Hisada (1995). The only difference in computing the elastic and theviscoelastic eigenfunctions is that, in the latter, all the operations must be performed usingcomplex arithmetic. However, the fact that the Rayleigh eigenfunctions are complex-valuedfor viscoelastic media implies that the phase difference between the horizontal and thevertical components of the displacement field is no longer equal to π 2 as in the elasticcase. As a result, the principal axes of the ellipse describing the trajectory of the Rayleighwave particle motion are rotated forward or backward with respect to the free surface of thehalf-space (Båth, 1968).

Determination of the wavenumbers ( ) ( )k VR j R j= ω * and of the associated

eigenfunctions ( )r y ki R j

* , ,ω

completely solves the complex eigenproblem and thus the

homogeneous boundary value problem of Rayleigh waves in viscoelastic, verticallyheterogeneous media. The corresponding dispersion and attenuation curves can be easilyobtained from the inversion of Eq. 3.68 to give:

( ) ( )

( )w R

R R R R

R R

R

R R R RR

R

V V x yx y

x

D D x yy

x

:

,

,

= =+

= = −

2 2

(3.72)

where ( )V V kR R R j=

,ω , and ( )D D kR R R j

=

,ω with j M= 1, . To properly account

for causal material dispersion, the Rayleigh secular function ( )F R VR* must be constructed

using body wave velocities ( )V V yP P= ,ω and ( )V V yS S= ,ω calculated with Eq. 2.35.


3.7 Effective Phase Velocity and Green’s Function in Viscoelastic Media

The purpose of this section is to generalize the results obtained in Sections 3.3 and 3.4for elastic materials to viscoelastic media. The main tool used to implement thisgeneralization will again be the elastic-viscoelastic correspondence principle. The

displacement field ( )uβ ωr y, , ( )β = r y, induced by a harmonic point source Fy ⋅ e i tω in a

vertically heterogeneous elastic half-space is given by either Eq. 3.28 or Eq. 3.12. The latterrepresentation follows directly as solution of the inhomogeneous boundary value problemof Rayleigh waves in elastic, vertically heterogeneous media (Aki and Richards, 1980).

An efficient approach for determining the solution of the corresponding problem inviscoelastic media involves the use of integral transform methods. The governing equationsobtained with this procedure will be formally identical to those of the elastic problem,except that the transformed viscoelastic field variables replace the elastic field variables, andthe viscoelastic complex moduli replace the elastic moduli (if using the Fourier transform).This association between elastic and viscoelastic solutions is the essence of thecorrespondence principle that, if applied to Eq. 3.12, yields:

( ) ( )[ ] ( )u β βω ϕω ω βr y A r y e

j

i t k r

j

Mj, , , ,*

= ⋅ ⋅ − ⋅ +

=∑ *

1

(3.73)

where ( )[ ]A *β ωr y

j, , and ( )k j

* ω are the (complex) Rayleigh displacement amplitude and

wavenumber, respectively, associated with the jth mode of propagation ( )j M= 1, . Equation

3.73 allows the displacement field induced by an harmonic point source Fy ⋅ e i tω in a

vertically heterogeneous, viscoelastic half-space to be determined. The modal amplitude

( )[ ]A *β ωr y

j, , can be determined from Eq. 3.21 if all of the modal quantities are replaced

with those obtained from the solution of the complex eigenproblem, namely

( ) k r y kj i j* * *, , ,ω ( )i = 1 4, . Equation 3.73 also shows that the viscoelastic solution can be

obtained via an extension of the elastic solution (3.12) to complex values of

( ) k r y kj i j, , ,ω . It should be remarked that this extension is restricted only to those

subsets D D D1 2∪ ∪ ⊂... n C where Eq.3.73 is a well-defined, single-valued andcontinuous function of the independent complex arguments.

As already mentioned, in elastic media Eqs. 3.12 and 3.28 provide equivalent

representations for ( )uβ ωr y, , . Therefore, following the procedure that led to Eq. 3.73, the


viscoelastic Green’s function may be obtained from Eq. 3.24 by extending the validity of theelastic Green’s function to complex valued amplitude and phase to yield:

( ) ( ) ( )[ ]$ , , $ , ,* , ,

uβ βω ψ ωω ω βr y r y e

i t r y= ⋅ −U * (3.74)

where the terms $ , *U β βψ* ∈C can be computed using Eqs. 2.25 and 2.26 with complex-

valued modal quantities. The restrictions about the domains of validity of Eq. 3.73 applyequally well to Eq. 3.74. However, the presence in Eqs. 2.25 and 2.26 of rather complicatedmulti-valued functions of several complex variables can make the task of identifying branchcuts, branch points and singularities associated with these expressions very difficult (Krantz,

1982),. By setting ( )$ $ arg $*

U U Uβ β

β* *= ⋅ ⋅e

i Eq. 3.74 may be more conveniently re-written as:

( ) ( ) ( )[ ]$ , , $ , ,* , ,

uβ βω ωω ω βr y r y ev

i t r yv= ⋅ −U Ψ(3.75)

where the terms $U βv ∈ R , Ψβv* ∈C are defined respectively by ( )$ , , $ *U Uβ βωv r y = and

( ) ( )[ ]Ψβ β βω ψv r y* * *, , arg $= − U .

Finally, from Eq. 3.75 the displacement field ( )uβ ωr y, , induced by a harmonic source

Fy ⋅ e i tω located at r y yS= =0, in a linear viscoelastic vertically heterogeneous medium

can be written in a form:

( ) ( ) ( )[ ]uβ βω ωω ω βr y F r y ey

i t r yv, , , ,* , ,= ⋅ ⋅ −G v

Ψ(3.76)

where ( ) ( )G vβ βω ωr y r y, , $ , ,=U v is the Rayleigh geometric spreading function in viscoelastic

media, which is in general different from ( )Gβ ωr y, , .

From Eq. 3.76, the expression ( )[ ]ω ωβt r y cons tv− =Ψ* , , tan represents the equation of

a (complex) wave front, which is characterized by an effective complex phase velocity givenby:


( ) ( )[ ]$ , ,

, ,*

,

V *β

β

ωω

ωr y

r yv r

=Ψ

(3.77)

where ( )[ ]Ψβ ωv rr y*

,, , can be interpreted as an effective complex wavenumber and denoted

by ( )$ , ,*k r yβ ω .

To obtain quantities of physical significance it is convenient to set

( ) ( ) ( )$ , ,$ , ,

$ , ,*k r yV r y

i r yββ

βωω

ωα ω= + ⋅

so that:

( )( ) ( )[ ]

$ , ,arg $* *

,

V r y

r

β

β β

ωω

ψ=

ℜ − U(3.78a)

( ) ( )[ ]$ , , *

,α ω ψβ βr y

r= ℑ (3.78b)

( )$ , ,V r yβ ω and ( )$ , ,α ωβ r y are the effective phase velocity and the effective attenuation

coefficient, respectively, and characterize the harmonic propagation of multi-mode Rayleighwaves in viscoelastic, vertically layered media. At a fixed y y c= , the two-dimensional plots

( )$ , ,V r y cβ ω and ( )$ , ,α ωβ r y c are defined as the dispersion and the attenuation surface,

respectively. It should be remarked that Eq. 3.78 could have also been obtained from theextension of the elastic solution (i.e. Eq. 3.23) for complex values of the arguments.

3.8 Modal and Effective Partial Derivatives in Viscoelastic Media

The results presented in Section 3.5, particularly the closed-form partial derivatives ofRayleigh phase velocity with respect to medium parameters were obtained using theHamilton variational principle. The applicability of this fundamental principle of dynamics isrestricted only to monogenic mechanical systems, i.e., to those systems for which the internaland external forces are derivable from scalar potentials which are functions of particlecoordinates, particle velocity and time (Lanczos, 1970; Goldstein, 1980). Elastic bodiessubjected to specified distributions of body forces and surface tractions fulfill the conditionsset forth for monogenic systems. In fact, they satisfy the even more restrictive conditions ofconservative systems, for which the potential functions (in this case the strain energy) do nothave an explicit dependence on particle velocity and time. For monogenic continuous


systems, the Lagrangian density is simply given by the difference between kinetic energydensity and potential energy density.

Viscoelastic bodies, on the other hand, belong to the category of dissipative systemsthat are not monogenic because dissipative forces do not admit a representation in terms ofpotential functions. As a result, Hamilton’s variational principle is not applicable inviscoelastic bodies, at least in its classical form. These considerations seem to suggest thatthe important results obtained in Section 3.5 and derived from the application ofHamilton’s principle do not hold in viscoelastic media.

Two approaches may be used to show that, fortunately, this is not the case. The firstapproach is based on introducing a type of formalism that allows treating non-conservativesystems as if they were conservative (Ben-Menahem and Singh, 1981). The basic idea is toconsider a mechanical system made up from the combination of two systems: the “real”systemA , which stores and dissipates strain energy, and a “mirror” system B producingenergy in equal amount to that dissipated byA (i.e. the system B acts as a storage for theenergy dissipated byA ). For the joined systemA BU the total energy is conserved, andhence Hamilton’s principle can be applied. The results of this approach show that thevariational principles of Rayleigh and Love waves along with their implications are also validfor viscoelastic media, and thus Eqs.3.44 and 3.58 remain valid for complex values of theparameters. However, the analysis of Ben-Menahem and Singh (1981) indicates that the

results are correct only to a first order approximation in Dγ ( )γ = P S, , and hence they are

rigorously valid only in weakly dissipative media.

The second approach that may be used to show that the Rayleigh variational principle,can be extended to viscoelastic media is based on the application of certain variationaltheorems of linear viscoelasticity (Gurtin, 1963; Christensen, 1971). As mentioned inSection 3.6, these theorems can also be used as an alternative to integral transform methodsto obtain the equations of motion and the associated boundary conditions of variousboundary value problems in linear viscoelasticity. Most of the viscoelastic variationaltheorems are natural extensions of the results obtained in linear elasticity. They provide therigorous procedure for extending the variational theorems of linear elasticity toviscoelasticity, in particular the Rayleigh variational principle, which is formally expressed byEqs. 3.34 and 3.35.

However, as remarked several times in this chapter, integral transform methods are anideal tool for solving boundary value problems in linear viscoelasticity because of theadvantages offered by the application of the elastic-viscoelastic correspondence principle.Among them is the possibility of reinterpreting the variational theorems of linear elasticityas integral transformed viscoelastic variational theorems (Christensen, 1971). For steadystate harmonic problems, this reinterpretation becomes a trivial exercise simply involvingthe extension of the elastic solution to complex values of the field variables. Once theviscoelastic version of the Rayleigh variational principle has been established, the results


obtained in Sections 3.5.1 and 3.5.2, particularly Eqs. 3.34 and 3.35, can be extended tolinear viscoelasticity on a firm theoretical basis.

The approach based integral transform methods does not have any physicalinterpretation and is solely based on the formal analogies between the field equations oflinear elasticity and the integral transformed field equations of linear viscoelasticity. On theother hand, the approach based viscoelastic variational theorems has both a solid theoreticalbasis and a physical, energy-based interpretation. However, the thermodynamical legitimacyof the procedure based on the application of Hamilton’s principle to the combined real-mirror system (Ben-Menahem and Singh, 1981) is questionable. Furthermore, this approachlead to results that are correct only to first order.

105

4 SOLUTION OF THE RAYLEIGH INVERSE PROBLEM

4.1 Introduction

Given the set of medium parameters ( ) ( ) ( ) ρ y V y V yP S, ,* * defining the material

properties of a site, the problem of determining the dispersion and attenuation curves( )VR ω and ( )α ωR associated with that site is often referred to as the Rayleigh direct or

forward problem. Conversely, if ( )VR ω and ( )α ωR are known, then the problem of

determining the unknown medium parameters ( ) ( ) ( ) ρ y V y V yP S, ,* * defines the Rayleigh

backward or inverse problem.

In more general terms, direct problems are concerned with determining the effectsinduced on a physical system by certain causes, whereas in inverse problems the roles ofcauses and effects are reversed, and the objective is to determine the causes that generatethe observed effects (Engl, 1993). Following this definition, for a vertically heterogeneousviscoelastic medium excited by a harmonic source, several types of Rayleigh direct/inverseproblems may be considered. They differ from each other in the function (i.e., the effect)that is chosen to represent the medium response. Dispersion and attenuation functions areone possible type of response function but other choices are possible as well. In the frequencydomain other response functions include the displacement amplitude, the displacementphase, or the displacement spectra. In the time domain and for transient sources, a valuableresponse function is either a short or long period seismogram. Although in geotechnicalearthquake engineering the most common response functions are experimentallydetermined dispersion and attenuation curves (Stokoe et al., 1989; Tokimatsu, 1995; Rix etal., 1998a), other interpretations of the test using alternate response functions are alsopossible as shown later in this chapter.

Inverse problems concerned with the determination of a model physical parametersfrom the measurements of certain model field variables constitute so-called parameteridentification problems (Engl, 1993). In the solution of parameter identification problems, theability to successfully invert the measured field variables to obtain reliable estimates of themodel parameters depends to a significant degree on the choice of the response function.Factors governing the selection of appropriate response functions include the ability toexperimentally measure the field variables, the ability to solve the corresponding inverseproblem (e.g., the procedure used to compute the partial derivatives of the responsefunction with respect to the model parameters), and the information content associatedwith the selected response function.

106 Solution of the Rayleigh Inverse Problem

4.2 Ill-Posedness of Inverse Problems

Inverse problems are often inherently ill-posed or unstable, particularly non-linearproblems of parameter identification such as the Rayleigh inverse problem. According toHadamard’s definition of well-posedness, a mathematical problem is said to be well-posed orstable if it satisfies the following three conditions (Tikhonov and Arsenin, 1977; Engl, 1993):

a. For all admissible data, a solution exists.

b. For all admissible data, the solution is unique. (4.1)

c. The solution depends continuously on the data.

To be more precise, the definition (4.1) should also specify the functional space inwhich the solution is supposed to exist, and the restrictions that a given set of data mustsatisfy to be considered admissible. It should be remarked that the Hadamard’s postulates ofwell posedness apply to both direct and inverse problems. However, it is only in recentyears through studies in non-linear dynamics that the importance of instability has beenrecognized in the solution of forward problems where small perturbations in the initial dataproduce unpredictable changes in the solution (Parker, 1994).

In inverse problems, conditions b. and c. in (4.1) are often violated. Of particularrelevance in parameter identification problems is the violation of condition b., that is theexistence of more than one solution. For the Rayleigh inverse problem this implies that agiven experimental dispersion curve may correspond to more than one body wave velocityprofile, or analogously, that an experimental attenuation curve may correspond to two ormore different material damping ratio profiles. From a mathematical point of view, non-uniqueness in the solution of an inverse problem is caused by a lack of sufficientinformation to constrain the solution. Alternatively, the available information available maynot be independent.

Two strategies can be used to effect uniqueness in the solution of an inverse problem.The first strategy is to add a priori information about the solution of the problem. For theRayleigh inverse problem, this may be information about the body wave velocity and/ormaterial damping ratio of one or more layers (obtained, for example, from laboratory tests).

Adding constraints to the solution is a second widely used strategy to effect uniquenessin the solution of an inverse problem. In some cases, the strategy of providing moreinformation can be regarded as constraining the solution. An obvious example is to requirethe body wave velocities and/or material damping ratios to be within a specified range (e.g.,non-negative). However, there are constraints of a different nature that enforce features ofglobal behavior such as smoothness and regularity rather than requiring the solution toassume specific numerical values or bounds.

Solution of the Rayleigh Inverse Problem 107

In discussing strategies to address non-uniqueness problems, it should be noted that themethods for effecting uniqueness are relatively simple for ideal error-free observations, butthe situation is more complicated for data containing bias and random errors. The nextsection will illustrate the most important aspects of the constrained algorithm used as a basisof the inversion of surface wave experimental data.

Violation of condition c. in (4.1) is also an important concern. Certain inverse problems,like the solution of the Fredholm integral equation of the first-kind, are very sensitive toperturbations in the data. Their instability is inherent to the problem and does not dependon the algorithm used to solve the inverse problem. For linear inverse problems with bothdiscrete and continuous linear operators, a stability analysis can be carried out by means ofthe singular-value expansion method (Menke, 1989; Engl, 1993). The results of this analysis showthat the smallest singular value controls the amplification of the measurement errors. Therate of decay of the singular values arranged in order of decreasing magnitude is used as ameasure to quantify the degree of instability of a given inverse problem.

For very unstable problems there are mathematical techniques, called regularizationmethods, that approximate the ill-posed problem with a parameter-dependent family ofneighboring well-posed problems (Tikhonov and Arsenin, 1977; Engl, 1993). Because someof these regularization methods admit a variational formulation (e.g. Tikhonovregularization) where the objective is the minimization of appropriate functionals, they canalso be applied to non-linear inverse problems successfully.

4.3 Coupled Versus Uncoupled Analysis

For the Rayleigh inverse problem, the violation of condition c. in (4.1) is not an issue of

great concern, at least for low to moderate frequencies of excitation ( )< 50 Hz . At higherfrequencies an uncoupled analysis, where the dispersion and the attenuation curves are invertedindependently, becomes increasingly sensitive to data perturbations. This problem can beovercome, or at least mitigated, by using a constrained smoothed inversion algorithm thathas the remarkable property of acting as a regularization method while effecting uniquenessin the solution. It is interesting to note, however, that a coupled analysis, where the dispersionand the attenuation curves are inverted simultaneously, is more stable than the uncoupledanalysis, even at high frequencies. The following paragraphs explain this somewhatunexpected result.

An uncoupled inversion of dispersion and attenuation data requires the solution of twoinverse problems for 2 nL unknown model parameters, for example the shear wave velocitiesand the shear damping ratios of a nL-layer soil deposit. The solution of these two inverseproblems is not completely independent because the shear wave velocity profile obtainedfrom the non-linear inversion of the dispersion curve will subsequently be used in the linearinversion of the attenuation curve. Therefore, the amplification of the errors resulting from


inversion of the dispersion curve will carry over to the inversion of the attenuation curve, aprocess that is characterized by its own degree of ill-posedness. In other words, theuncoupled inversion suffers a negative synergetic effect resulting from the solution of twoinverse problems where the input data of one problem comes from the solution of theother.

Conversely, the simultaneous inversion of both the dispersion and attenuation curveseliminates this negative coupling effect because both sets of experimental data are invertedsimultaneously in a single, complex-valued, inversion. Furthermore, the solution of thecoupled inverse problem takes advantage of an internal constraint that is embedded in theformalism of the complex inversion. This internal constraint is given by the Cauchy-Riemann equations that are satisfied by the Rayleigh phase velocity when viewed as ananalytic function of the complex-valued shear wave velocity. The intimate connectionbetween the real and the imaginary parts of the variables involved in the simultaneousinversion adds a salutary built-in constraint that makes the coupled inversion a better-posedproblem.

In summary, application of complex variable theory to the simultaneous inversion ofsurface wave data is not only an elegant procedure that accounts for the coupling betweenelastic moduli and dissipative properties of viscoelastic media (as shown in Chapters 2 and3), but it also improves the well-posedness of the inverse problem. It should also beremarked that the simultaneous inversion, in contrast to the uncoupled analysis, is notrestricted by the assumption of weak dissipation.

4.4 Inversion Strategies

In most cases the solution of an inverse problem belonging to the class of parameteridentification problems can be obtained from the solution of an optimization problem whichinvolves finding the stationary condition of an functional subjected to various constraints(Parker, 1994). The techniques used to solve non-linear optimization problems can bebroadly divided into global-search procedures and local-search procedures. This distinctionis motivated by the fact that a non-linear optimization problem will have, in general, severalstationary points in the solution space.

Local-search procedures are iterative schemes that, starting from an initial guess of thesolution, generate a sequence of improved approximations converging, under suitableconditions, toward a stationary point. Most local-search procedures are calculus basedtechniques whose strategy consists of locally linearizing a non-linear functional at eachiteration. These techniques require the functional to be sufficiently smooth so that itsGateaux derivatives with respect to the model parameters exist and are continuous.Furthermore, even if all the smoothness requirements for the functional are satisfied, thesequence of approximations of the solution is guaranteed to converge only if the initialguess is sufficiently close to the solution. However, the most important limitation inherent to


all the local-search procedures is that even when they succeed in finding a stationary point,there are no simple means to determine whether it is a local or a global stationary point inthe solution space.

This dilemma is addressed by global-search procedures, which are optimizationtechniques where the search for a global stationary point is conducted over the entiresolution space. The strategy adopted in a global search method varies according to differentphilosophies, some of which include genetic algorithms, fractal inversion, neural networkinversion, enumerative methods, and Monte Carlo simulation. Global-search procedures arein general more expensive than local-search procedures, both in terms of time andcomputer resources. However, they are more robust and reliable compared to the latter.

Figure 4.1 illustrates some of the possible approaches to the Rayleigh inverse problem.The shaded boxes indicate the options that were considered in this study.

Global-Search-Methods

UnconstrainedOptimization

Occam's Algorithm

ConstrainedOptimization

Local-Search-Methods

Type of Inversion

Uncoupled Analysis

Coupled Analysis

Type of Analysis

Dispersion andAttenuation Functions

Displacement Functions[Complex Spectra]

Frequency Domain

Short/Long PeriodSeismograms

Time Domain

Type of Response Function

Inverse Problem of Rayleigh Waves

Figure 4.1 Algorithms for the Solution of the Rayleigh Inverse Problem

The Rayleigh inverse problem was solved using a local-search procedure, where thestationary point in the solution space was sought with a constrained optimization techniqueknown as Occam’s algorithm. The implementation of the procedure requires the application ofa two-step iterative scheme. During the first step the Rayleigh forward problem is solved forthe current values of the medium parameters. In the second step, the non-linear inverseproblem is linearized in the neighborhood of the current medium parameters so that theresulting constrained linear inverse problem can be solved. The procedure is repeated for asufficient number of iterations until a properly defined convergence criterion is satisfied.

The practical implementation of Occam’s algorithm relies on the ability to solve twocrucial problems: the Rayleigh forward problem and the computation of the partial


derivatives of the response function with respect to the medium parameters. When theresponse functions are the dispersion and attenuation curves, the solution of both problemshas been described in Chapter 3.

4.5 Occam’s Algorithm

The strategy of solving inverse problems, particularly parameter identification problems,using constrained optimization algorithms is not new (Lawson and Hanson, 1974) and it islargely motivated by the need to effect uniqueness in the solution of problems withuncertain data (i.e., data containing bias and random errors). However, inversionsperformed with conventional least-squares techniques where solution is constrained to havethe minimum norm (e.g. the stochastic damped least-squares algorithm) are ofteninadequate, and may lead to physically unreasonable profiles of model parameters(Constable et al., 1987; Menke, 1989). The inadequacy of this class of algorithms can beattributed to a lack of physical justification for assuming the minimum norm constraint.

A more reasonable approach for constraining the solution of an inverse geophysicalproblem is the so-called Occam’s inversion (Constable et al., 1987; Parker, 1994). The strategyof this algorithm can be summarized as follows: given a set of experimental data and theirassociated uncertainties, find the smoothest profile of model parameters subject to theconstraint of a specified misfit between observed and predicted data. The development ofthis class of algorithms was motivated by the following observations. The solution of aparameter identification problem relies on the ability to synthetically reproduce a set ofexperimental data by means of a mathematical model describing a particular physicalproblem. In discrete inverse theory, the mathematical model is assumed to depend on acertain number of unknown model parameters, whose determination is the objective of theinversion algorithm.

For the Rayleigh inverse problem, the mathematical model is given by a linearviscoelastic, multi-layered medium. The model parameters may be the complex body wave

velocities of the individual layers, namely ( ) ( ) ( )[ ]VP P P P nV V V

L

* * * *, ,...=1 2

and

( ) ( ) ( )[ ]VS S S S nV V V

L

* * * *, ,...=1 2

. Additional sets of model parameters may include the mass

density [ ]ρρ = ρ ρ ρ1 2, ,... nL and the thickness [ ]h = h h h nL1 2, ,... of the layers. Because the

number of layers nL is generally assumed, the inverted profile of model parameters willdepend on the a priori assumption about nL, and it may contains large discontinuities orother features that are not essential for matching the experimental data. By enforcingmaximum smoothness and regularity in the solution, one minimizes its dependence uponthe assumed number of layers, and at the same time rejects solutions that are unnecessarilycomplicated.


The implementation of the smoothed least-squares inversion algorithm requires anadequate definition of smoothness of a profile of model parameters. In a multi-layeredmedium, where the variation of the model parameters is discontinuous, smoothness or itsconverse roughness, is defined in terms of difference rather than differential operators (Constableet al., 1987). For complex-valued model parameters represented by the vectors VP

* and VS* ,

roughness may be defined by either one of the two following expressions:

( ) ( )

( ) ( )

R

R

H

H

1 2

2

2 2

22

2

22 2

= = ⋅

= = = ⋅

∂∂ ∂∂ ∂∂

∂∂ ∂∂ ∂∂ ∂∂ ∂∂

V V V

V V V V

γ γ γ

γ γ γ γ

* * *

* * * *

(4.2)

where γ = P S, , the symbol ⋅ ∈2

R denotes the Euclidean norm of a vector in CN , and

( )⋅H

indicates the Hermetian transpose of a complex-valued matrix. Finally ∂∂ is an n nL L×real-valued matrix representing the two-point central finite difference operator and is given by:

∂∂ =−

−−

0

1 1 0

1 1

0 1 1

...

...(4.3)

It can be easily shown that for a continuously varying medium and for real-valued modelparameters, the two definitions of R 1 and R 2 given by Eq. 4.2 correspond to the integralover depth of the square of the first and the second derivative, respectively, of the model

parameter function ( )Vγ y with respect to depth.

The experimental data are a vector of complex-valued Rayleigh phase velocities

measured at different frequencies ( ) ( ) ( )[ ]VR R R R nV V V

F

* * * *, ,. ..=1 2

. For the time being it will

be assumed the Rayleigh phase velocities refer to a specific mode of propagation (i.e., modalRayleigh phase velocities). However, for simplicity of notation the mode index is omitted inthe expression for VR

* .

From a statistical point of view, the experimental measurements are assumed to beindependent and normally distributed. Therefore each experimental datum is completely


described by a pair of complex numbers: the expected value ( )VR j

* ( )j nF= 1, and its standard

deviation σ j* which represents an estimate of the uncertainty associated with ( )VR j

* . A

methodology for determining ( )VR j

* and σ j* is presented in Chapter 5.

In a linear viscoelastic, multi-layered medium characterized by the model parameters VP*

and VS* , the Rayleigh phase velocities VR

* associated with a specified set of frequencies canbe predicted from the solution of the non-linear Rayleigh forward problem:

( )V V V VR R P S* * * *,= (4.4)

For the solution of the Rayleigh inverse problem, it is convenient to expand Eq. 4.4 in a

Taylor series about an initial guess of model parameters V VP S0 0* *, obtaining:

( ) ( ) ( ) ( )

( ) ( )

V V J V V J V V

V V V V

V V V VR R0 P*

S*

P0*

S0*

P0*

S0*

* *P*

P0*

S*

S0*

P*

P0*

S*

S0*

= + ⋅ − + ⋅ − +

+ − + −

, ,

o2 2

2

(4.5)

where VR 0* is the nF × 1 vector of Rayleigh phase velocities corresponding to the solution

of Eq. 4.4 with medium parameters equal to VP0* and VS0

* . The terms ( )JV VP

*

,* *P S0 0

and

( )JV VS

*

,* *P S0 0

are the n nF L× complex-valued Jacobian matrices whose elements are defined by:

( )[ ] ( )( )

( )[ ] ( )( )

J

J

P jk

R j

P k

S jk

R j

S k

P S

P S

P S

P S

*

,

*

*

,

*

,

*

*

,

* *

* *

* *

* *

V

V

V

V

V VV V

V VV V

0 0

0 0

0 0

0 0

=

=

∂

∂

∂

∂

(4.6)


where j nF=1, and k nL=1, . The subscripts outside the brackets indicate the point in

Cn nL L× at which the Jacobian matrices are evaluated. By neglecting terms higher than thefirst order in Eq. 4.5, this equation reduces to:

( ) ( ) ( ) ( )V V J V V J V VV V V VR R0 P

*S*

P0*

S0*

P0*

S0*

* *P*

P0*

S*

S0*= + ⋅ − + ⋅ −

, ,(4.7)

which is the linearization of the functional relationship ( )V V V VR R P S* * * *,= about the initial

model V VP S0 0* *, .

In Section 3.5.1 it was shown that the Rayleigh phase velocity is relatively insensitive tochanges in P-wave velocity in elastic media. It can be easily proved that this result also holdsin viscoelastic media for complex-valued velocities. Based on this observation and notingthat the ill-posedness of the Rayleigh inverse problem can be reduced by minimizing thenumber of independent model parameters, Eq. 4.7 is inverted only for VS

* . With thisassumption Eq. 4.7 simplifies to:

( ) ( ) ( )[ ]J V J V V VV VS

*S*

R R0S0*

S0*⋅ = ⋅ + −S

*S0* * * (4.8)

By setting V VR R* *= , Eq. 4.8 can be used as a basis for determining the unknown profile

of model parameters VS* that correspond to a set of experimental data VR

* . The value

obtained for VS* , say VS1

* , could then be used as a new starting model for determining the

next approximation VS2* .

The process could then be repeated in an iterative fashion to generate a sequence ofmodel parameters VS0

* , VS1* , VS2

* …. VSn* , which under suitable conditions will converge

towards the solution of the non-linear Rayleigh inverse problem. A necessary condition forthe convergence of this iterative scheme is a starting model VS0

* that is sufficiently close tothe true solution.

For each iteration, the technique used to solve the linear system of equations is Occam’salgorithm whose strategy is as follows: given a set of nF measured Rayleigh phase velocities

( )VR j

* and their associated uncertainties σ j* ( )j nF= 1, , find those values of

( )VS k

* ( )k nL= 1, that minimize the roughness R 1 (or R 2 ) of the resulting complex-valued


shear wave velocity profile while predicting the experimental ( )VR j

* with an acceptable

accuracy.

A measure ε 2 of the misfit between measured and predicted Rayleigh phase velocitiescan be obtained with the weighted least-squares criterion applied to complex-valued data:

( )[ ] ( )[ ]ε2 = − ⋅ −W W W W* * * *V V V V V VR R R R* *

S* * *

S*

H

(4.9)

where W* is a complex-valued diagonal n nL L× matrix defined by:

W* * * */ , / ,..... , /= diag nL1 1 11 2σ σ σ (4.10)

which are the uncertainties associated with the experimental data VR* .

A standard procedure used to solve constrained optimization problems is the method ofLagrange multipliers (Constable et al., 1987; Logan, 1997). In this case, the optimizationproblem consists of minimizing the functional R 1 (or R 2 ) defined by Eq. 4.2 subject to the

condition that the residual error function ε 2 given by Eq. 4.9 be equal to ε *2 , a value

considered acceptable considering the uncertainties associated with VR* . The method of

Lagrange multipliers allows the solution of this constrained minimization problem to bedetermined by finding the minimum of the following unconstrained functional:

( ) ( )[ ]( )[ ] ( )[ ]

FU

* *S* * *

S*

= ⋅ +

+ − ⋅ − −−

∂∂ ∂∂V V

W W W W

S

H

S

H

* *

* * * **µ ε1 2V V V V V VR R R R

(4.11)

where the first term is the roughness of the solution VS* , and the second term is the data

misfit multiplied by the Lagrange multiplier µ −1 . From Eq. 4.11 it can be observed that theparameter µ may be interpreted as a smoothing parameter: If µ is large, the value of the

functional FU is controlled by the roughness of the solution VS* and the data misfit does

not significantly affect the solution. Conversely, if µ is small, most of the contribution toFU is given by the data misfit and the roughness term plays only a minor role.


Equation 4.11 is non-linear in VS* because of the presence of the term ( )V VR S

* * thatrepresents the predicted Rayleigh phase velocity. If this term is replaced by its linear partwhich is given by Eq. 4.7 (without the contribution of VP

* ), Eq. 4.11 becomes:

( ) ( )

( ) ( )

FU = ⋅

+

+ − ⋅ ⋅

⋅ − ⋅ ⋅

−

−

∂∂ ∂∂V V

W d W J V W d W J VV V

S

H

S

S S

H

S SS S

* *

* * * * * * * * * *** *

µ ε10 0

2

0 0

(4.12)

A necessary condition for the existence of a minimum of the unconstrained linearfunctional of Eq. 4.12 is the vanishing of its gradient ∇∇

VSFU* with the respect to VS

* , that is:

( ) ( )[ ]( )[ ] ( )[ ]

∇∇ ∇∇ ∂∂ ∂∂

∇∇

V V

V V V

V V

W d W J V W d W J V

S*

S*

S*

FU = ⋅ +

+ − ⋅ ⋅ ⋅ − ⋅ ⋅ −

=−

S

H

S

S S

H

S SS S

* *

* * * * * * * * * *** *µ ε1

0 02

0 00

(4.13)

which can be expanded to yield:

( ) ( ) ( ) ( )

( ) ( )

∇∇ ∂∂ ∂∂ ∇∇ ∂∂ ∂∂

∇∇

V V

V V V

V V V V

W d W J V W d W J V

S*

S*

S*

S

H

S S

H H

S

S S

H

S SS S

* * * *

* * * * * * * * * ** *

⋅ +

⋅ +

+ − ⋅ ⋅

⋅ − ⋅ ⋅

=−

T T

2 010 0

0 0

µ(4.14)

After considering that ( )[ ] ( )∇∇ ∇∇V V V

J W 0S S S

S* * ** *

0= = , Eq. 4.14 simplifies to:

( ) ( ) ( )∂∂ ∂∂T + ⋅

⋅ ⋅

⋅ = ⋅

⋅− −µ µ1 10

0 0 0

W J W J V W J W dV V V

* * * * * * * * ** * *S

H

S S S

H

S S S

(4.15)

which can finally be solved for VS* to yield:


( ) ( ) ( ) ( )V W J W J W J W dV V VS S

H

S S

H

S S S

* * * * * * * * ** * *

= + ⋅

⋅ ⋅

⋅ ⋅

⋅−

µ ∂∂ ∂∂T

0 0 0

1

0 (4.16)

The smoothing parameter µ in Eq. 4.16 must be determined with the additional

constraint that the specified residual error ε *2 is matched with a vector VS

* composed onlyof negative imaginary parts. This condition will insure that the hysteretic shear-dampingratio obtained from the inversion algorithm will be a positive quantity.

Figure 4.2 shows a flow chart of Occam’s algorithm applied to the solution of thecoupled Rayleigh inverse problem. Most of the computational effort of the algorithm isspent in the solution of the complex Rayleigh eigenproblem.

Thus far, the application of Occam’s algorithm to the solution of the Rayleigh inverseproblem is based on comparing the experimental Rayleigh phase velocity with the predictedRayleigh phase velocity of a specific mode of propagation. However, the procedure willremain valid when the modal quantities are replaced with the corresponding effectivequantities. In this case, since the effective Rayleigh phase velocity depends on twoindependent variables, namely the frequency and distance from the source, the total numberof experimental data is not nF but rather n n nT F P= + , where nP is the number oflocations at which the Rayleigh phase velocity has been measured. With few other changes,most of the formalism developed in this section for the modal Occam’s inversion is applicableto the effective Occam’s inversion. As shown in the following sections, this generalization canalso be extended to other formulations of the Rayleigh inverse problem where differenttypes of analyses (coupled versus uncoupled) and response functions are considered.

In Chapter 2 it was shown that the body wave velocities ( )VP k

* , and ( )VS k

* ( )k nL= 1,

are frequency dependent in a linear viscoelastic medium due to material dispersion. Aprecise definition of the frequency dependence is given by Eqs. 2.34 and 2.35. In thesolution of the Rayleigh inverse problem presented in this section, no specific assumptionwas made about the frequency-dependence law of the complex-valued vector VS

* . Atechnique that is often used when accounting for material is to conduct the inversion (a so-called causal inversion) at a reference frequency ω ref (Lee and Solomon, 1979; Herrmann,

1994). This technique is adopted in this study, and the vector VS* can be denoted by

( )V VS ref S ref( )* *= ω . Occam’s algorithm remains valid; however, the partial derivatives

appearing in Eq. 4.6 must now be computed with respect to ( )VP refk

( )* and ( )VS ref

k( )

* . Using

the chain rule of calculus one obtains:


Find Roots of theDispersionEquation

ComputeEigenfunctions

Compute PartialDerivativesCompute Effective

Velocity & PartialCompute Green’s

Function

INPUT

and

Hisada -LaiFORTRAN

Code

Start

Stop

(Input/Output/Storage)

Main Code

OCCAM’S INVERSION ALGORITHM

MATLABProgramInterface

5

SOLUTION OF THE COMPLEX RAYLEIGH EIGENPROBLEM

STOP

YES

NO

Solution of Linearized Inverse Problem

Select a New Profile

VS0*VR

*

( )( )

∂

∂

V

V

R*

S*

j

k

Derivatives

( )V VS i S i Tol( )*

( )*

+ − <1

( ) ( )

( )

∂∂ ∂∂T +

=

=

−

−

µ

µ

1

10

0 0

0

W J W J V

W J W d

V V

V

* * * * *

* * * *

* *

*

S

H

S S

S

H

S S

S

( )V V VS i S i S i( )*

( )*

( )*

+ = +1 ∆

MATLAB 5

Code

Figure 4.2 Flow-Chart of Rayleigh Simultaneous Inversion Using Occam’s Algorithm


( )( )

( )( )

( )( )

( )( )

( )( )

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

V

V

V

V

V

V

V

V

V

V

*

( )*

,

*

*

,

*

( )*

*

( )*

,

*

*

,

( )*

( )* * *

( )*

( )* *

R j

P refk

R j

P k

P k

P refk

R j

S refk

R j

S k

P ref S ref P S

P ref S ref P

=

⋅

=

V V V V

V V V

0 0 0 0

0 0 0

( )( )

VS

S k

S refk0

*

V

V

*

( )*

⋅

∂

∂

(4.17)

where j nF= 1, . Finally from Eqs. 2.34 and 2.35:

( )( )∂

∂π

ωω

γ

γ γ

V

V ln

*

( )*

k

ref kref

D

=

+

1

12

(4.18)

where γ = P S, .

4.6 Uncoupled Inversion

4.6.1 Overview

This section will describe several algorithms that were developed for the solution of theuncoupled Rayleigh inverse problem. Uncoupled inversion refers to the case where thedispersion and the attenuation curves are inverted independently. For weakly dissipativemedia the uncoupled analysis yields satisfactory results although, as mentioned in Section4.3, the uncoupled inversion is more ill-posed than the corresponding coupled analysis.

Figure 4.3 shows classes of possible algorithms for solving the uncoupled Rayleighinverse problem. The shaded boxes indicate algorithms developed during this study,whereas the boxes with dashed borders briefly outline the procedure used to obtain theexperimental data. Each class of algorithm has two components: one to compute the shear

wave velocity profile ( )VS k ( )k nL= 1, and the other to determine the shear damping ratio

profile ( )DS k. The subscript eff appearing in the algorithms UEQMA and UEFMA is used

to denote effective quantities. The term G in the expression α R SG D= ⋅ denotes the matrixformed by the partial derivatives of Rayleigh phase velocity with respect to the shear andcompression wave velocities of the soil layers (hereafter called the G-matrix). This matrix iscalculated using Eq. 3.63b where the partial derivatives are computed using Eq. 3.45.

Experimentallyobtain VR(ω) from linear regression

arg[w(r,ω)] = a + kR r

Non-Linear InversionVR = VR(Vs)

[Compute VR using first mode only]

Shear Wave VelocityInversion

Experimentally

obtain αR(ω) fromnon-linear regression

|w(r,ω)| = Fy Gy(r,ω)exp(-αR r)

Linear Inversion

αR = GDs[Compute G using first mode only]

Non-Linear Inversion|w(r,ω)| = Fy Gy(r,ω) exp(-GDs r)[Account for αR = αR(r) = GDs]

Shear Damping RatioInversion

Fundamental Mode Analysis[UFUMA]

Experimentallyobtain VR(eff)(ω) from

linear regressionarg[w(r,ω)] = a + kR(eff) r

Non-Linear InversionVR(eff) = VR(eff)(Vs)

[Ignore dependenceVR(eff) =VR(eff)(r)]


Experimentally

obtain αR(eff)(ω) fromnon-linear regression

|w(r,ω)| = Fy Gy(r,ω)exp(-αR(eff) r)

Linear Inversion

αR(eff) = G(eff) Ds[Ignore dependence G(eff) = G(eff)(r)]


Equivalent Multi-Mode Analysis[UEQMA]

Experimentallyobtain VR(eff)(ω, r) fromVR(eff)(ω, r) = ω∆r/∆φ

at each receiver spacing


[Account for VR(eff) = VR(eff)(r)]

Experimentallyobtain arg[w(r,ω)]

at each receiver locationand at various ω

Displacement Phase SpectraNon-Linear Inversionarg[w(r,ω)] = Φ(Vs)


Experimentallyobtain |w(r,ω)|

at each receiver locationand at various ω

Non-Linear Inversion|w(r,ω)| = Fy Gy(r,ω) exp(-G(eff)Ds r)

[Accounts for G(eff) = G(eff)(r)]


Effective Multi-Mode Analysis[UEFMA]

Uncoupled Inversion

Figure 4.3 Algorithms for the Solution of the Uncoupled Rayleigh Inverse Problem


In Eq. 3.63b the compression-damping ratio DP has been replaced by the term K DS⋅where K is a parameter defining the ratio of compression to shear damping ratio (Rix et al.,1998a). Previous studies (Spang, 1995) have shown that the value adopted for K has anegligible influence on the backcalculated shear damping ratio profile. This is due to the factthat the phase velocity of Rayleigh waves is relatively insensitive to changes in VP and thusthe partial derivative ∂ ∂V VR P in Eq. 3.63b is small compared to ∂ ∂V VR S (see Section3.5.1). Furthermore, many studies (Winkler and Nur, 1979; Jongmans, 1990; Malagnini,1996; Leurer, 1997) have shown that 0 1< <K in soil. This fact (which is consistent withthe observation that, in fine grained materials such as soils, most of the energy dissipation isexpected to occur in shear mode) combined with small values of ∂ ∂V VR P makes thesecond term of Eq. 3.63b negligible compared to the first. Based on these considerations, alluncoupled analyses were performed using a value of K = 1.

In an uncoupled analysis of Rayleigh wave data, the inversion of the attenuation curve,that is α R SG D= ⋅ , is a linear problem. In this case it can be shown that the formulation ofOccam’s algorithm could be reversed and would yield the same result. In other words, for alinear inversion the smoothest solution with specified error misfit is the same as the solutionwhere the error misfit is minimized for a specified value of smoothness. The Lagrangemultipliers of the two solutions are reciprocals of each other.

The following sections provide a description of the main tasks performed by each classof algorithms. The inversion algorithms were written in MATLAB; the Rayleigh forwardproblem was written in FORTRAN 77 and linked to the main program via the MATLABAPI (Application Program Interface). A description of the main tasks performed by thealgorithms is reported in Appendix C.

4.6.2 Uncoupled Fundamental Mode Analysis

This is the simplest type of uncoupled inversion analysis and is based on independentlycomparing the experimental dispersion and attenuation curves with the correspondingcurves obtained from the theoretical model for the fundamental mode of Rayleigh wavepropagation. The algorithm is composed of two modules called Dispersion and Attenuation.The Dispersion module performs a non-linear inversion of the experimental dispersion curveusing Occam’s algorithm (Eq. 4.16). In performing this task Dispersion calls a MATLABmex-file called Rayleigh, which is a FORTRAN 77 routine that solves the eigenvalue problemof Rayleigh waves in elastic, vertically heterogeneous media (modified from Hisada, 1995).Rayleigh also computes the partial derivatives of the modal Rayleigh phase velocities withrespect to the shear and compression wave velocities of the soil layers using the variationalformulation. Rayleigh computes the theoretical dispersion curve and the partial derivatives ofRayleigh phase velocity with respect to medium parameters for the fundamental mode ofRayleigh wave propagation. The experimental phase velocities are imported into Dispersionfrom an external ASCII file.


The second module of the UFUMA algorithm is called Attenuation and is designed toimplement the linear inversion of the experimental attenuation curve. This task isaccomplished with a two-step procedure. First, the frequency-dependent Rayleighattenuation coefficients ( )α ωR are computed from the experimental displacementamplitudes at multiple receiver offsets using a non-linear regression based on the expression

w r F r e R r( , ) ( , )exp.

ω ω α= ⋅ ⋅ − ⋅y yG where ( )w r,ω is the vertical particle displacement spectrum,

and Gy ( , )r ω is the vertical geometric spreading function. Once the experimental

attenuation curve is determined, the shear damping ratio profile is determined from thelinear inversion of the relation α R SG D= ⋅ . Both Gy ( , )r ω and G and calculated by

Rayleigh.

4.6.3 Uncoupled Equivalent Multi-Mode Analysis

This class of inversion algorithms explicitly recognizes the multi-mode nature of thequantities measured in surface wave tests. Accordingly, during the inversion process theexperimental dispersion and attenuation curves are matched with the effective theoreticaldispersion and attenuation curves instead of the modal curves. However, since the effective

Rayleigh phase velocities ( )$ ,V rR ω and attenuation coefficients ( )$ ,α ωR r are functions oftwo independent variables (frequency and source-receiver distance), a simplification isrequired to reduce the dimensionality of the problem by one, that is to transform dispersionand attenuation surfaces into “equivalent” curves. The simplification is suggested by theexperimental technique used for the uncoupled phase velocity and attenuation

measurements and involves an averaging process of ( )$ ,V rR ω and ( )$ ,α ωR r over thereceiver spacings used in the measurements.

The UEQMA algorithms UEQMA are composed of two modules that are again calledDispersion and Attenuation. The Dispersion module implements the non-linear inversion of theexperimental dispersion curve using Occam’s algorithm. Like UFUMA, Dispersion calls aMATLAB mex-file named Rayleigh that solves the eigenvalue problem of Rayleigh waves inelastic, vertically heterogeneous media. Rayleigh also computes the effective Rayleigh phase

velocity ( )$ ,V rR ω , and the partial derivatives of ( )$ ,V rR ω with respect to the shear andcompression wave velocities of the soil layers. Dispersion computes the effective theoreticaldispersion curve and the effective partial derivatives that account for all of the Rayleighmodes of propagation according to Eqs. 3.23 and 3.58, respectively. Averaging the effectiveRayleigh phase velocities and partial derivatives over the receiver offsets used in themeasurements eliminates the dependence of these quantities on the spatial coordinate. Theexperimental dispersion curve and receiver offsets are imported into Dispersion from anexternal ASCII file.

Attenuation is the second module of the algorithm UEQMA and is used to obtain theshear damping ratio profile from the experimental attenuation curve. Like UFUMA, the


inversion of the experimental attenuation curve is performed using a two-step procedurethat involves computation of the attenuation coefficients ( )α ωR first, followed by an

inversion of the ( )α ωR yielding the DS profile. The key difference is that the linearinversion α R SG D= ⋅ is performed with the effective rather than the modal G-matrix.

4.6.4 Uncoupled Effective Multi-Mode Analysis

UEFMA denotes a class of inversion algorithms that take the multi-mode nature ofRayleigh wave propagation in vertically heterogeneous media, fully into account in theinterpretation of surface wave tests. Ideally, the experimental measurements would consist

of a series of phase velocities ( )$ ,V rR ω and a series of attenuation coefficients ( )$ ,α ωR reach measured at different frequencies and at different receiver locations. These twoindependent sets of data constitute what may be called the experimental dispersion andattenuation surfaces of a site. The UEFMA algorithm then finds the smoothest shear wavevelocity and shear-damping ratio profiles whose corresponding theoretical dispersion andattenuation surfaces match those obtained from the experiments.

Unfortunately, the practical implementation of this algorithm is difficult because of the

problems associated with the experimental determination of ( )$ ,V rR ω and ( )$ ,α ωR r at apoint. The local (i.e., spatially dependent) nature of these quantities suggests that they bedetermined from displacement phase and amplitude measurements performed over verysmall receiver spacings. However, the uncertainty associated with these measurements willincrease as the distance between the receivers decreases. This problem becomes particularlyacute for displacement amplitude measurements because of the combined effect ofgeometric and material attenuation.

In Chapter 5 it is shown that the dispersion surface can be constructed bydifferentiating the displacement phase with respect to the source-to-receiver distance atconstant frequency. The attenuation surface is similarly determined from the displacementamplitude. In both cases, the procedure used to determine the Rayleigh phase velocity andthe attenuation coefficient at different receiver offsets is numerical differentiation, which is anill-conditioned problem particularly with inaccurate or noisy data.

A possible strategy to overcome the difficulties of determining the experimentaldispersion and attenuation surfaces is to apply a numerical fitting procedure to the set ofdata formed by the experimental phases and displacement amplitudes. This approach willenable obtaining smooth phase and corrected amplitude surfaces over which the operation ofdifferentiation could be performed analytically. This requires the selection of an adequatemodel for the numerical fitting procedure, which may be guided by the analyticalexpressions obtained in Chapter 3 for the displacement phase and amplitude.

The difficulties that are inherent in the determination of the experimental dispersionand attenuation surfaces suggest a new approach to interpreting surface wave data. As


mentioned at the beginning of this chapter, the conventional interpretation of surface wavemeasurements is based on adopting dispersion and attenuation curves as the mediumresponse functions. This choice is not unique, and other types of response functions may beselected that may be better suited to the solution of the Rayleigh inverse problem. In thefrequency domain some of these alternate response functions include displacementamplitude, displacement phase, and displacement spectra (see Fig. 4.1).

In surface wave tests displacement phase and amplitude are directly measured; Rayleighphase velocity and attenuation coefficient are derived from the phase and amplitude. Thus,it may be advantageous to choose the displacement phase and amplitude as the responsefunctions for the solution of the Rayleigh inverse problem because they are more basicquantities. For the uncoupled inversion an example of this approach would be to determinethe shear damping ratio profile directly from the experimental displacement amplitudes

using w r F r e G D rS( , ) ( , )exp.

$ω ω= ⋅ ⋅ − ⋅ ⋅y yG . In essence, the response function is changed from

attenuation coefficients to displacement amplitudes. This alternative, denoted in Fig.4.3 witha dotted box, is attractive because it permits the experimental displacement amplitudes to beused directly and avoids the need to calculate the attenuation coefficients.

A direct inversion of the measured displacement phases ( )ψ ω ,r would be theanalogous procedure for determining the shear wave velocity profile. However, theinversion of the displacement phases is far more involved than the inversion of thedisplacement amplitudes, mainly because the functional relationship between thedisplacement phase with the shear wave velocity profile does not admit a simple, explicitrepresentation. Moreover, the partial derivatives of the displacement phase with respect tothe medium parameters, required for the solution of this non-linear inverse problem, canonly be computed numerically. Section 4.7.4 presents a unifying approach for the solutionof the Rayleigh inverse problem where the displacement phases and amplitudes are replacedby complex displacement spectra.

4.7 Coupled Inversion

4.7.1 Overview

In this study, the term strongly coupled inversion describes a procedure in which thedispersion and attenuation curves are inverted simultaneously. A strongly coupled Rayleighinversion requires the ability to solve the complex eigenproblem for linear viscoelastic mediawhere Rayleigh phase velocity ( )VR ω and attenuation coefficient ( )α ωR both depend on

the body wave velocities, ( )V yP and ( )V yS , and the material damping ratios, ( )D yP and

( )D yS . In other words, a strongly coupled inversion refers to any procedure where an


appropriate complex-valued response function ( )R R* * * *,= V VP S is inverted to obtain the

corresponding medium parameters V VP S* *, and .

A consistent, strongly coupled inversion should not only simultaneously invert the real

and the imaginary parts (or amplitude and phase) of ( )R R* * * *,= V VP S , but it should also

include an experimental procedure where both ( )ℜ R * and ( )ℑ R * are measuredsimultaneously. In other words, in a consistent, coupled inversion the dispersion andattenuation curves should be both measured and inverted simultaneously. Chapter 5 willillustrate the details of an experimental procedure conceived for the simultaneousmeasurement of both ( )VR ω and ( )α ωR .

In the seismological literature the term coupled inversion is often used for a procedurethat accounts for the link between surface wave phase velocity and attenuation and thestiffness and damping properties of the layers in weakly dissipative medium (Aki andRichards, 1980; Keilis-Borok, 1989; Herrmann, 1994). As illustrated in Section 3.5.3, thisweakly coupled inversion requires only the solution of the real-valued eigenproblem in elasticmedia. The implementation of a weakly coupled inversion is based on the iteratedapplication of Eq. 3.63.

Figure 4.4 shows some of the possible algorithms that may be used for the solution ofthe coupled Rayleigh inverse problem. The shaded boxes show the strongly coupledinversion algorithms developed in this study and written in MATLAB; the solution of thecorresponding forward problem for linear viscoelastic media was written in FORTRAN 77.A description of the main tasks performed by these computer codes is reported inAppendix C.

The dashed boxes in Fig. 4.4 briefly outline the principles used to obtain the

experimental data. The symbol ( )T r,ω appearing in the dashed boxes denotes thedisplacement transfer function; a quantity that will be defined in Chapter 5. The term

( ) ( )Ψ ΨR vr r, , ,*ω ωβ= 0 denotes the complex-valued phase angle defined by Eq. 3.76.

Finally, the subscript eff appearing in some of the equations is used to distinguish effectivequantities from modal quantities. The next three sections will provide a description of thetasks performed by each of the algorithms outlined in Fig. 4.4.

Experimentally

obtain VR(ω) and αR(ω)from non-linear regression

T(r,ω) = G(r,ω) exp(-ikR r)

Non-Linear InversionVR = VR(Vs)

[Compute VR using first mode only](Note: VR and VS are complex-valued)

Shear Wave Velocity andShear Damping Ratio

Inversion

Fundamental Mode Analysis[CFUMA]

Experimentally

obtain VR(eff)(ω) and αR(eff)(ω)from non-linear regression

T(r,ω) = G(r,ω) exp(-ikR(eff) r)


[Ignore dependence VR(eff) = VR(eff)(r)](Note: VR and Vs are complex-valued)


Inversion

Equivalent Multi-Mode Analysis[CEQMA]

Experimentally

obtain VR(eff)(r,ω) and αR(eff)(r,ω)iteratively from solving

T(r,ω) = G(r,ω) exp[-iΨR(r,ω)]


[Account for VR(eff) = VR(eff)(r)](Note: VR and Vs are complex-valued)

Experimentallyobtain w(r,ω) from

arg[w(r,ω)] and |w(r,ω)|at each receiver location

Complex Displacement SpectraNon-Linear Inversion

w(r,ω) = w(Vs)(Note: w(r,ω) and Vs are complex valued)


Inversion

Effective Multi-Mode Analysis[CEFMA]

Coupled Inversion

Figure 4.4 Algorithms for the Solution of the Strongly Coupled Rayleigh Inverse Problem


4.7.2 Coupled Fundamental Mode Analysis

This algorithm implements the simultaneous, strongly coupled inversion of theexperimental dispersion and attenuation curves by comparing these curves with thetheoretical curves obtained from the viscoelastic model for the fundamental mode ofpropagation. The algorithm CFUMA performs in a single step a task that is performed bythe algorithm UFUMA in two separate operations with the additional, beneficial internalconstraint provided by the Cauchy-Riemann equations. The program that implementsCFUMA is a MATLAB code called ViscoRay, which performs the simultaneous inversion ofthe experimental dispersion and attenuation curves by applying the complex formalism to aconstrained least squares algorithm that enforces maximum smoothness in the invertedcomplex shear wave velocity profile.

ViscoRay uses a MATLAB mex-file called Rayleigh written in FORTRAN 77 that solvesthe complex eigenvalue problem of Rayleigh waves in linear viscoelastic multi-layered media.Rayleigh also computes the partial derivatives of the complex Rayleigh phase velocity withrespect to the complex shear and compression wave velocities of the soil layers using thevariational formulation illustrated in Section 3.8. These partial derivatives are used toconstruct the complex-valued Jacobian matrix required for the solution of the coupledRayleigh inverse problem.

The theoretical dispersion and attenuation curves computed by Rayleigh include only thefundamental mode of propagation of Rayleigh waves. However, the frequency dependentattenuation coefficients ( )α ωR are calculated using a geometric spreading functionGyv ( , )r ω that accounts for all of the modes of propagation (Rix et al., 1998a). The

experimental Rayleigh phase velocities and attenuation coefficients are imported intoViscoRay from external ASCII files.

4.7.3 Coupled Equivalent Multi-Mode Analysis

This algorithm simultaneously inverts the experimental dispersion and attenuationcurves by matching them with the corresponding effective, theoretical curves obtained fromthe viscoelastic model. As in UEQMA, the effective theoretical dispersion and attenuationcurves are obtained from the respective surfaces by averaging the complex effective phase

velocity ( )$ ,*V rR ω over the receiver spacings used in the actual test. CEQMA isimplemented in MATLAB code named ViscoRay that performs the simultaneous inversionof the experimental complex dispersion curve using Occam’s algorithm.

ViscoRay uses a MATLAB mex-file called Rayleigh written in FORTRAN 77 that solvesthe complex eigenvalue problem of Rayleigh waves in viscoelastic multi-layered media.Rayleigh computes the effective complex Rayleigh phase velocity and its partial derivativeswith respect to the complex shear and compression wave velocities of the medium layersusing the variational formulation of Section 3.8.


The effective theoretical dispersion and attenuation curves and the effective partialderivatives of Rayleigh phase velocity with respect to medium parameters computed byRayleigh, reflect the contribution of all the Rayleigh modes of propagation. It should benoted that the experimental attenuation coefficients are calculated iteratively using thegeometric spreading function Gyv ( , )r ω that depends on the shear wave velocity profile of

the current iteration.

4.7.4 Coupled Effective Multi-Mode Analysis

This is the most sophisticated algorithm presented in this study for the inversion ofsurface wave data. It combines the features of the simultaneous inversion with a thoroughconsideration of the multi-mode nature of Rayleigh wave propagation in multi-layeredmedia, which is reflected in the spatial dependence of the effective phase velocity andattenuation coefficient. Unfortunately, most of the difficulties associated with theexperimental determination of the dispersion and attenuation surfaces discussed in Section4.6.4 in the context of UEFMA also apply to CEFMA.

As it will be shown in Chapter 5, the difficulties of the algorithm CEFMA are related tothe experimental determination of the effective wavenumber, which is obtain from an

unstable process of numerical differentiation of the complex phase angle ( )Ψβ ωv r* , ,0 with

respect to the source-to-receiver distance. As mentioned in section 4.6.4, a smoothnumerical fitting of the experimental data, in this case the displacement transfer functions

( )T r,ω , may mitigate some of these difficulties. Once the experimental dispersion surface

( )[ ]$ ,*

exp.V rR ω has been defined, the objective of the algorithm CEFMA involves the task of

finding the smoothest complex shear wave velocity profile ( )VS* ω whose theoretical

complex dispersion surface matches ( )[ ]$ ,*

exp.V rR ω .

As shown in Fig.4.4, the algorithm CEFMA can be implemented following a differentstrategy. Because the quantity measured experimentally is the displacement transfer function

( )T r,ω , it follows that the most natural inversion of surface waves data should involve this

response function. The effective phase velocity ( )$ ,*V rR ω is obtained from ( )T r,ω via a

roughing filter operation, and hence the inversion of ( )$ ,*V rR ω data is necessarily an ill-conditioned operation. The inversion of surface waves data based on the use ofdisplacement response functions is not a common practice to this day.

However, in the opinion of the writer the direct inversion of displacement spectra

( )w r,ω or transfer functions ( )T r,ω constitutes the most rational approach to theinterpretation of active surface waves tests. The major difficulty of its practicalimplementation is the computation of the partial derivatives of the displacement spectra


with respect to the medium parameters. Currently, these partial derivatives have to becomputed numerically because no closed-form solutions, functions of the unperturbedmedium parameters, are available. Recent works in theoretical seismology are attempting toaddress the solution of this difficult problem (Zeng and Anderson, 1995).

129

5 RAYLEIGH PHASE VELOCITY AND ATTENUATIONMEASUREMENTS

5.1 Overview

The most important aspects of surface wave measurements are reviewed in this chapter.Surface wave tests are often called Spectral-Analysis-of-Surface-Waves (SASW) tests in theengineering literature (Nazarian, 1984; Stokoe et al., 1989; Tokimatsu, 1995). Althoughsurface waves include both Rayleigh and Love waves, most of the methods currently used tonear-surface soil and rock properties focus exclusively on the observation of Rayleigh waves.

The SASW test is a relatively young in-situ seismic technique (compared to moretraditional seismic tests such as cross-hole and down-hole tests) that was developed byStokoe and co-workers during the early 1980’s (Nazarian et al., 1983; Nazarian, 1984;Sánchez-Salinero, 1987; Rix, 1988; Stokoe et al., 1989). The technique evolved from theSteady-State-Vibration Technique used by the US Army Corps of Engineers WaterwaysExperiment Station (WES) during the early 1960’s (Richart et al., 1970). The SASW test isbecoming increasing popular in the geotechnical engineering community, primarily becauseit is a non-invasive field technique, and hence does not require the use of boreholes orprobes. This attractive feature may be crucial for certain types of geotechnical investigations.

Surface wave tests were originally developed for the determination of the elastic moduliprofile of soil deposits and pavement systems (Nazarian et al., 1983; Stokoe et al., 1989).More recently, Rix et al. (1998a) developed a technique for using the SASW test todetermine the material damping ratio profile of a layered soil deposit. So far, however, thetwo problems of determining the stiffness and material damping ratio profiles of a site havebeen treated separately (i.e., uncoupled). Section 4.7 described several algorithms for thesimultaneous (i.e., strongly coupled) inversion of surface wave dispersion and attenuationdata. Section 4.7 also noted that the dispersion and attenuation data should not only beinverted but also measured simultaneously. One goal of this chapter is to illustrate anexperimental procedure for the simultaneous measurement of both dispersion andattenuation data.

Chapter 5 is organized in two main sections: Section 5.2 reviews the methods that arecurrently used in conventional surface wave measurements. Section 5.3 proposes a newapproach to surface wave measurements where consistency between measurementtechniques and data interpretation is emphasized. Echoing the inversion analyses of Chapter4, the new methodology is developed following two strategies. For the uncoupled inversionalgorithms presented in Section 4.6, the dispersion and attenuation data should be obtainedfrom uncoupled measurements, a topic that is discussed in Section 5.3.1. Conversely, theuse of the simultaneous inversion algorithms discussed in Section 4.7 suggests that the

130 Rayleigh Phase Velocity and Attenuation Measurements

dispersion and attenuation data be measured simultaneously. Coupled measurements ofsurface wave data are discussed in Section 5.3.2. The chapter concludes with someconsiderations concerning the statistical errors associated with surface wave measurements.

5.2 Conventional Measurements Techniques

Surface wave methods are traditionally divided into active and passive methods(Tokimatsu, 1995). In the active methods Rayleigh waves are generated by either animpulsive or a vertically oscillating harmonic source applied at the free surface of a verticallyheterogeneous medium. The ensuing particle motion is recorded by an array of receiversplaced on the ground surface in line with the source. Since active surface wave methodshave a penetration depth that is typically on the order of 15 to 20 meters, they are wellsuited for near-surface site characterization. The major obstacle to greater penetrationdepths is the difficulty of generating lower frequency (i.e., longer wavelength) with portablesources.

Passive methods overcome this limitation because they do not involve generation ofwave energy with artificial sources. They are based on the observation of short- and long-period ground motion induced by cultural noise and microtremors. Passive methods requirethe particle motion to be recorded by a large number of sensors arranged in two-dimensional arrays over the ground surface. Penetration depths with passive methods canrange from less than 50 meters with short-period microtremors to several kilometers withlong-period microtremors (Tokimatsu, 1995). This study focuses exclusively on theinterpretation of active surface wave methods, particularly those involving harmonicsources.

The receivers used in surface wave tests for near-surface site characterization are usuallyvertical velocity transducers with natural frequencies ranging from 1 to 4.5 Hz (Stokoe et al.,1989). The recording device is usually a Fast Fourier Transform (FFT) dynamic signalanalyzer that is capable of performing real-time spectral analyses of the particle velocity timehistories measured at the receivers. Figure 5.1 shows a typical configuration of theequipment used in SASW testing during Rayleigh phase velocity measurements.

Sources are classified as transient and harmonic according to the time-variation of thedynamic force. Transient sources are typically impulsive; common examples includesledgehammers and dropped weights. They generate surface waves containing a broad rangeof frequencies. Lack of repeatability and poor signal-to-noise ratios are their most importantlimitations. Common harmonic sources include hydraulic vibrators and electro-mechanicalshakers that sweep through a pre-selected range of frequencies. For SASW tests performedin soil deposits, the frequency range is approximately 5 to 200 Hz (Rix, 1988). Harmonicsources are more repeatable and capable of higher signal-to-noise ratios than transientsources (Rix, 1988; Spang, 1995). Moreover, the use of harmonic sources greatly simplifies

Rayleigh Phase Velocity and Attenuation Measurements 131

the interpretation of surface wave data since the analysis of Rayleigh wave propagation invertically heterogeneous media is more difficult for transient than for harmonic waves.

5.2.1 Phase Velocity Measurements

Figure 5.2 shows a typical source-receiver configuration used during conventional phasevelocity measurements (i.e., the two-station method). Rayleigh waves are generated by aharmonic source oscillating at a circular frequency ω . Two receivers located at distances r1

and r2 from the source detect the vertical particle motion that is recorded by the signal

analyzer in the form of particle velocity spectra ( )V r,ω defined to be the Fourier transform

of the particle velocity ( )& ,w r t w t= ∂ ∂ .

From the velocity spectra ( )V r,ω , two other important spectral quantities are

calculated: the auto-power spectrum ( )G rr ω of each receiver and the cross-power spectrum( )G r r1 2ω of the two receivers:

( ) ( ) ( )

( ) ( ) ( )G V r V r

G V r V r

rr

r r

ω ω ω

ω ω ω

= ⋅

= ⋅

, ,

, ,1 2 1 2

(5.1)

ReceiversSource

RecordingDevice

VerticalParticleMotion

Figure 5.1 Typical Configuration of the Equipment Used in SASW Testing


Rayleigh Wave

F eyi t⋅ ω

Geophones

r1

r2

1 2

( )w 1 r t1 , ( )w 2 r t2 ,

Figure 5.2 Source-Receivers Configuration in SASW Phase Velocity Measurements

The symbol ( )⋅ in Eq. 5.1 denotes complex conjugation. The time delay between

receivers as a function of the circular frequency is given by ( )[ ]arg G r r1 2ω ω . Hence, the

phase velocity ( )VR ω of the propagating Rayleigh wave can be computed from:

( )( )

( )[ ]Vr r

GR

r r

ωω

ω=

−2 1

1 2arg

(5.2)

Equation 5.2 yields the experimental dispersion curve associated with the pair ofreceivers located at r1 and r2. This procedure is then repeated for different receiver spacings,and the individual dispersion curves are combined together to form the compositedispersion curve of the site (Nazarian, 1984).

Receiver positions are chosen according to one of two schemes: the common sourcearray and the common receiver midpoint array. Both schemes are illustrated in Figs.5.3 (a)and (b). Local stratigraphy and portability of the source control the selection of the receiverarray. Sometimes the SASW test is performed using both forward and reverse arrays(Stokoe et al., 1989). The receiver positions remain the same, but the source is moved to theopposite end of the array. The results of the forward and reverse arrays are combinedtogether in an attempt to mitigate the effects of lateral inhomogeneities and/or localdiscontinuities.

In Fig.5.2 the receivers are typically spaced with a ratio r r2 1 2= (Sánchez-Salinero,1987). The receiver spacing should also chosen to eliminate spatial aliasing. According to the


Nyquist’s criterion, spatial aliasing is avoided if the receiver spacing is chosen to be smallerthan half the wavelength one wants to measure.

An important spectral quantity that is also calculated during SASW phase velocitymeasurements is the ordinary coherence function, which is defined by:

F e i t⋅ ω

Receivers

1 2

F e i t⋅ ω

Receivers1 2

F e i t⋅ ω

Receivers1 2

Figure 5.3(a) SASW Arrangement Using Common Receiver Midpoint Array

F e i t⋅ ω

Receivers

1 2

F e i t⋅ ω

Receivers1 2

F e i t⋅ ω

Receivers1 2

near field

Figure 5.3(b) SASW Arrangement Using Common Source Array


( )( )[ ] ( )[ ]( )[ ] ( )[ ]γ ωω ω

ω ωr r

r r r r

r r r r

G G

G G1 2

1 2 1 2

1 1 2 2

2 =⋅

⋅(5.3)

The ordinary coherence function gives a measure of how the measured particle velocity

( )& ,w r t1 is related to ( )& ,w r t2 . It can be shown (Bendat and Piersol, 1986) that 0 11 2

2≤ ≤γ r r

with the upper bound γ r r1 21= corresponding to a situation where there is an exact linear

relationship between ( )& ,w r t1 and ( )& ,w r t2 . Low coherence values may be attributed to thepresence of signal noise or more generally to the situation where the measured particlevelocities at the receiver r1 and r2 are not linearly related. From Eq. 3.28 with

( ) ( )u r w ry , , ,0 ω ω= it can be easily shown that the theoretical coherence function (i.e., the

coherence function computed with synthetic particle velocities) is equal to one at allfrequencies. Therefore, since Eq. 3.28 has been derived by neglecting the body wave field,an additional cause for observed low values of the coherence function may be the near-fieldeffects.

In the context of Section 3.3, the Rayleigh phase velocity computed with Eq. 5.2 is the

average effective Rayleigh phase velocity ( )$ ,V rR ω over the distance ( )r r2 1− . This quantity,which is sometimes called the apparent phase velocity (Tokimatsu, 1995), reflects thecontribution of several modes of propagation of Rayleigh waves, and its magnitude dependson the location where it is measured as demonstrated by Eq. 3.23. Therefore, the currentprocedure based on Eq. 5.2 eliminates important information from the experimentalmeasurements, that is the functional dependence of the measured Rayleigh phase velocityon the receiver position. As discussed in Chapter 4, the well posedness of the Rayleighinverse problem can be improved by supplying more information to constrain the solution.Thus, a substantial improvement is expected by accounting for the dependence of themeasured phase velocity on the source-to-receiver distance.

5.2.2 Attenuation Measurements

The equipment configuration used in surface wave attenuation measurements is thesame as that shown in Fig. 5.1. The common source array shown in Fig. 5.3b is typicallyused. Rayleigh attenuation coefficients ( )α ωR are obtained from measurements of the

vertical displacement amplitudes w r( , )ω at several receiver offsets over a specified range of

frequencies.

The relevant spectral quantity is now the particle velocity auto-power spectrum ( )Grr ωcalculated at each receiver spacing. Because ambient noise may be important, particularly at


large receiver offsets, the experimental particle velocity spectra are corrected to account forthe noise effects (Rix et al., 1998a; Rix et al., 1998b):

( ) ( ) ( )G Grr sr rrω γ ω ω= ~ ~2 (5.4)

where ( )~Grr ω is the measured auto-power spectrum that is presumed to contain non-

coherent noise. The quantity ( )~γ ωsr2 is the ordinary coherence function between the

harmonic source and the measured vertical particle velocity. Computation of ( )~γ ωsr2

requires the use of an accelerometer at the source to monitor the motion of the harmonicoscillator.

From Eq. 5.4 the experimental vertical particle displacement spectrum is readilycomputed by:

( ) ( )w r

V r

C

G

Crr

,,

( )

( )

( )ω

ω

ω ωω

ω ω=

⋅=

⋅(5.5)

where ( )C ω is a frequency dependent calibration factor that converts the output of thevelocity transducer (volts) into engineering units (e.g., cm/sec).

Once the vertical displacement amplitudes ( )w r,ω have been computed, the Rayleigh

attenuation coefficients ( )α ωR can be determined from Eq. 3.76 by assuming

( ) ( )Ψyv r K r* *, ,0 ω ω≈ ⋅ where ( ) ( ) ( )[ ]K V iR R* ω ω ω α ω= + . This assumption is

equivalent to assuming that the complex phase angle ( )Ψyv r* , ,0 ω is dominated by the

fundamental mode of propagation. This hypothesis applies only to the phase angle and not

to the modulus of the vertical displacement ( )w r,ω . In other words, the attenuationcoefficients are calculated using a hybrid approach in which the material attenuation isassumed to be dominated by the fundamental mode, but the geometric attenuation iscorrectly computed by accounting for all the modes of propagation. With these assumptionsEq. 3.76 gives:

( ) ( ) ( )w r F r eyrR, ,ω ω α ω= ⋅ ⋅ − ⋅G (5.6)


where ( ) ( )w r u ry, , ,ω ω= 0 and ( ) ( )G Gr ryv, , ,ω ω= 0 . Equation 5.6 forms the basis of a

non-linear regression analysis to determine the frequency-dependent attenuation coefficients

( )α ωR from the experimental displacement amplitudes ( )w r,ω . The geometric spreading

function ( )G r ,ω can be computed from Eq. 3.25b since ( ) ( )G r ry, $ , ,ω ω= U * 0 . For weakly

dissipative media ( ) ( )$ , , $ , ,U Uy yr r* 0 0ω ω≈ and ( )G r ,ω can be computed from the elastic

solution of the Rayleigh forward problem.

Figures 5.4(a) and (b) show the results of two experimental attenuation coefficients( )α ωR computed at the Treasure Island National Geotechnical Experimentation Site

(NGES) at two different frequencies. The magnitude of the harmonic force Fy may be

considered as an additional parameter to be determined from the regression. In this case thetwo parameters ( )α ωR and Fy may be determined using a partitioned non-linear regression

algorithm that eliminates the linear parameter Fy from the non-linear regression for

( )α ωR . It can be shown (Lawton and Sylvestre, 1971) that this procedure is beneficial inaccelerating and stabilizing the convergence of the algorithm.

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

0 10 20 30 40 50 60

Distance (m)

Ver

tical

Par

ticle

Dis

plac

emen

t (m

)

Experimental

Theoreticalf = 30 Hz

αR = 0.0377 1/m

Figure 5.4(a) Attenuation Coefficient Computation at Treasure Island Site


1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

0 10 20 30 40 50 60

Distance (m)

Ver

tical

Par

ticle

Dis

plac

emen

t (m

)

Experimental

Theoreticalf = 68 Hz

αR = 0.0236 1/m

Figure 5.4(b) Attenuation Coefficient Computation at Treasure Island Site

Alternatively, the dynamic force Fy can be calculated by measuring the acceleration of

the harmonic oscillator, and hence it can be introduced in Eq. 5.6 as a known term; itshould be noted that in general ( )F Fy y= ω .

Obviously, the procedure of computing the ( )α ωR by specifying the force magnitude( )F Fy y= ω is preferred. The experimental attenuation coefficients ( )α ωR determined by

Eq. 5.6 are average values of the effective attenuation coefficient ( )$ ,α ωR r over the receiver

locations. As in the case of phase velocity measurements, the dependence of ( )$ ,α ωR r on

the source-to-receiver distance is eliminated via this averaging procedure. However, the

averaging procedure differs for ( )$ ,V rR ω and ( )$ ,α ωR r .

In phase velocity measurements, a two-station method is used to determine ( )VR ω forvarious receiver spacings. The experimental composite dispersion curve is then obtained byaveraging the individual dispersion curves obtained at each receiver spacing. A two-stationmethod applied to attenuation measurements, although in principle possible, would likelyyield inaccurate results because of the combined effect of material and geometricattenuation. Over short receiver-to-receiver distances such as those commonly used insurface wave testing, geometric attenuation controls the spatial decay of surface wavesgenerated by point sources; material attenuation is difficult to extract. A multi-station


method overcomes this problem because material attenuation is more easily observed andmeasured experimentally over larger distances.

The averaging procedure involved in the determination of the attenuation coefficient( )α ωR is based on the assumption that ( ) ( )Ψyv r K r* *, ,0 ω ω≈ ⋅ in Eq. 3.76. This

corresponds to using a lumped attenuation coefficient ( )α ωR in place of ( )$ ,α ωR r .

5.3 New Measurement Techniques

In this section a new approach is proposed to interpret surface wave measurements.The first and most important justification for the new approach is that it providesconsistency between phase velocity and attenuation measurements. As shown in Section 5.2the conventional procedures used to determine the experimental dispersion and attenuationcurves are quite different. The dispersion curve is obtained from the repeated application ofthe two-station method in which the results of several different receiver spacings are averagedto obtain a single composite dispersion curve. Conversely, the experimental attenuationcurve is determined from a procedure based on the application of a multi-station method,where the individual attenuation coefficients are calculated from a non-linear regressioninvolving measured displacement amplitudes. One of the objectives of the next two sectionsis to illustrate a procedure in which the experimental dispersion and attenuation curves areboth determined with the multi-station method.

The second reason motivating a new interpretation approach comes from theformulation of the coupled inversion presented in Section 4.7. As mentioned at thebeginning of this chapter, in a truly consistent approach to surface wave testing, thedispersion and attenuation curves should be both measured and inverted simultaneously.The objective of Section 5.3.2 is to illustrate a technique for the simultaneous determinationof the dispersion and attenuation curves from surface wave measurements.

5.3.1 Uncoupled Measurements

From Eq. 3.76 the vertical displacement ( )w r,ω induced in a linear viscoelastic

vertically heterogeneous medium by a harmonic source Fy ⋅ e i tω located at the ground

surface is given by:

( ) ( ) ( )[ ]w r F r eyi t r, ,

* ,ω ω ω ω= ⋅ ⋅ −G Ψ (5.7)


where ( ) ( )w r u ry, , ,ω ω= 0 , ( ) ( )G Gr ryv, , ,ω ω= 0 , and ( ) ( )Ψ Ψ* *, , ,r ryvω ω= 0 . In Section

3.7 it was shown that the term ( )[ ]Ψ *

,,r

rω can be interpreted as an effective complex

wavenumber and written as ( )[ ] ( ) ( )Ψ *

,,

$ ,$ ,r

V ri r

rR

Rωω

ωα ω= + ⋅

. By decomposing

( )Ψ * ,r ω into its real and imaginary parts, Eq. 5.7 can be rewritten as follows:

( ) ( ) ( ) ( )[ ]w r F r e eyr i t r, , , ,ω ω ω ω ω= ⋅ ⋅ ⋅− −G Ψ Ψ2 1 (5.8)

where ( ) ( )[ ]Ψ Ψ1 r r, ,*ω ω= ℜ , and ( ) ( )[ ]Ψ Ψ2 r r, ,*ω ω= ℑ . Considering only the spectral

part of Eq. 5.8 gives:

( ) ( ) ( )

( )[ ] ( )

w

arg w

r F r e

r r

yr, ,

, ,

,ω ω

ω ω

ω= ⋅ ⋅

= −

−G Ψ

Ψ

2

1

(5.9)

Since ( ) ( )[ ]$ ,

,,

V rr

R

r

ωω

ω=

Ψ1

and ( ) ( )[ ]$ , ,,

α ω ωR rr r= Ψ2 from Eq. 5.9 it is possible to

obtain:

( )( )[ ]( )

( ) ( )( )

$

,

$ , ln,

,

V r, =

rarg w

w

R ωω

∂∂

ω

α ω∂∂

ω

ω

−

= −

r

rr

r

rR G

(5.10)

Equation 5.10 suggests an interesting geometrical interpretation. If the angular frequency

ω is constant, the effective Rayleigh phase velocity ( )$ ,V rR ω is proportional to the inverseof the slope of the displacement phase plotted as a function of the source-to-receiverdistance r. Moreover Eq. 5.10 also shows that at constant frequency, the effective Rayleigh

attenuation coefficient ( )$ ,α ωR r can be interpreted as the slope (with changed sign) of a


corrected displacement amplitude, the latter considered again as a function of the source-to-receiver distance r. The corrected displacement amplitude is defined by the natural

logarithm of ( ) ( )w r r, ,ω ωG . Figure 5.5 illustrates this important geometrical

interpretation of the effective Rayleigh phase velocity ( )$ ,V rR ω and attenuation coefficient

( )$ ,α ωR r . Equation 5.10 is important because it suggests a procedure for the experimental

determination of ( )$ ,V rR ω and ( )$ ,α ωR r .

r*

( ) ( )

ω

ωω

ω

=

=

constant

$ ,$ ,

*

*

V rk r

R

R

r

arg[

w(r

,ω)]

$kR

1

Figure 5.5(a) Geometrical Interpretation of Effective Rayleigh Phase Velocity

This procedure requires the measurement of displacement phase and amplitudes in alinear receiver array (which are given by V r C( , ) [ ( )]ω ω ω⋅ ), for a specified set offrequencies. The measured displacement phase is then plotted at each frequency versus the rat each frequency. If the number of receivers is sufficiently large to define the curve well,Fig. 5.5(a) shows how to determine the effective Rayleigh phase velocity ( )$ ,V rR ω .

A plot of corrected displacement amplitude versus source-to-receiver distance can be

used to determine the effective Rayleigh attenuation coefficient ( )$ ,α ωR r as shown in Fig.5.5(b).


r*

$αR

1

( )ω

α α ω

=

=

constant

$ $ ,*R R r

r

ln[|

w(r

,ω)|

/G(r

,ω) ]

Figure 5.5(b) Geometrical Interpretation of Effective Rayleigh Attenuation Coefficient

In the latter case however, it is assumed that the ( )$ ,V rR ω data have already been

inverted so that the geometric spreading function ( )G r ,ω required to calculate thecorrected displacement amplitudes is known (for weakly dissipative media).

Although this procedure is in principle correct, its practical implementation may be verydifficult due to the unstable numerical differentiation required by Eq. 5.10. This problemassociated with the experimental determination of the dispersion and attenuation surfaceshas already been discussed in Section 4.6.4, where a possible solution strategy has also beenpresented.

Equation 5.10 demonstrates that the effective Rayleigh phase velocity and attenuationcoefficient are indeed derived quantities that are obtained from differentiation of theexperimentally measured displacement phases and amplitudes. This consideration is thebasis of a new interpretation of the SASW test where the inverted quantities aredisplacement phases and amplitudes rather than phase velocities and attenuationcoefficients. The details of this new interpretation were illustrated in Sections 4.6.4 and

4.7.4. If in Eq. 5.7 the complex phase angle ( )Ψ * ,r ω is approximated by

( ) ( )Ψ * *,r K rω ω≈ ⋅ with ( ) ( ) ( )[ ]K k iR R* ω ω α ω= + , Eq. 5.9 becomes:


( ) ( ) ( )

( )[ ] ( )

w

arg w

r F r e

r k r

yr

R

R, ,

,

ω ω

ω ω

α ω= ⋅ ⋅

= − ⋅

− ⋅G(5.11)

Equation 5.11 suggests a method for estimating the average effective Rayleigh phasevelocity and attenuation coefficient over the same linear array of receivers. The methodsimply consists of computing ( )kR ω and hence ( )VR ω from a linear regression involving

the experimental displacement phases and then inverting ( )VR ω to obtain the shear wave

velocity profile and ( )G r ,ω . The attenuation coefficient ( )α ωR is obtained from a non-linear regression involving the displacement amplitudes. The procedure for determining

( )α ωR is the same as the conventional method presented in Section 5.2.2. The key

difference is that the same configuration of receivers is used to obtain both ( )VR ω and( )α ωR , and therefore the method suggested by Eq. 5.11 provides consistency between

surface wave phase velocity and attenuation measurements.

5.3.2 Coupled Measurements

In this section an alternate approach to velocity and attenuation measurements in SASWtesting is presented. This approach is based on the concept of a displacement transfer functionthat allows the simultaneous determination of both Rayleigh phase velocity and theattenuation coefficient.

In a linear system, which in this case corresponds to a linear viscoelastic soil deposit, theratio between an output and an input signal in the frequency domain is called the frequencyresponse function or the transfer function of the system (Oppenheim and Willsky, 1997). Inthe typical SASW test configuration shown in Fig.5.1, the input signal is the harmonic forceapplied by the vertically oscillating source F ey

i t⋅ ω , while the output signal is the vertical

displacement w r( , )ω measured at a distance r from the source.

From Eq. 5.7, the displacement transfer function ( )T r,ω between source and receiveris given by:

( ) ( ) ( ) ( )T rw r

F er e

yi t

i r,,

,* ,ω

ωωω

ω=⋅

⋅ − ⋅= G Ψ (5.12)

Because the dynamic signal analyzer allows direct measurement of ( )T r,ω , Eq. 5.12 canbe used as a basis of a non-linear regression analysis to determine the complex phase angle


( )Ψ * ,r ω from the experimentally measured displacement transfer functions ( )T r,ω . From

the knowledge of the phase angle ( )Ψ * ,r ω the effective Rayleigh phase velocity ( )$ ,V rR ω

and attenuation coefficient ( )$ ,α ωR r may be computed from the relation

( )[ ] ( ) ( )Ψ *

,,

$ ,$ ,r

V ri r

rR

Rωω

ωα ω= + ⋅

. Note that in this procedure ( )$ ,V rR ω and

( )$ ,α ωR r are determined simultaneously.

As in the case of the uncoupled measurements, this procedure, although in principlecorrect, is difficult to implement from a practical point of view because of the need toperform unstable operations of numerical differentiation. Furthermore, there is an

additional difficulty in simultaneously measuring ( )$ ,V rR ω and ( )$ ,α ωR r because the

geometric spreading function ( )G r ,ω in Eq. 5.12 is unknown.

In vertically heterogeneous, linear viscoelastic media ( )G r ,ω is a complicated functionof the medium parameters that the SASW test seeks to determine. The problem can beovercome by an iterative strategy that combines Eq. 5.12 with an inversion algorithm, andwhich starts from a tentative profile of material parameters and proceeds until convergence.

Since in most cases the magnitude of ( )G r ,ω is controlled by a term proportional to 1 r(which is the Rayleigh geometric spreading law in homogeneous media), the iterativescheme requires relatively few iterations to converge.

The simultaneous measurement of dispersion and attenuation data may be considerably

simplified by assuming ( ) ( )Ψ * *,r K rω ω≈ ⋅ in Eq. 5.12. In this case the experimental

displacement transfer functions ( )T r,ω could be inverted using the equation

( ) ( ) ( )T r r e i K r, ,*

ω ω ω= ⋅ − ⋅ ⋅G to yield ( ) ( ) ( )KV

iR

R* ω

ωω

α ω= +

directly.

5.4 Statistical Considerations

5.4.1 Overview

Uncertainty in surface wave measurements is an important consideration. Inconventional SASW testing the measured quantities are the auto-power spectrum ( )G rr ωof each receiver location and the cross-power spectrum between a pair of receivers

( )G r r1 2ω . In the new coupled measurement technique presented in the last section, the

measured quantity is the particle velocity transfer function ( )H r,ω between source and


receiver, which is defined by ( ) ( ) ( )H r w r F e ryi t, & , ,ω ω ω ωω= ⋅ = i T . These experimental

quantities are assumed to be normally distributed.

The validity of this assumption may be arguable, but the uncertainties associated withthe methods used to interpret surface wave measurement do not justify the recourse tomore refined statistical distributions in the opinion of the writer. The statistics of normallydistributed experimental data are the expected value and variance. Bias and other systematicerrors in the measurements are not considered.

The recording device used in surface wave testing (generally a dynamic signal analyzer)

determines an estimate of the expected values of ( )G rr ω , ( )G r r1 2ω , and ( )H r,ω , which are

indicated by ( )[ ]E G rr ω , ( )[ ]E G r r1 2ω , and ( )[ ]E H r,ω . It is also possible to compute the

variances of these measured quantities using the following relations (Bendat and Piersol,1986):

( )[ ] ( )

( )[ ] ( )( ) [ ] ( )

Var GG

n

Var GG

Var arg G

rrrr2

d

r r

r r

r r1 2

1 2

1 2

ωω

ωω

γω

γ

γ

=

=⋅

≈−

2

2

2

2

1 2

1 2

1 2

1

2nd r r

r r

r r

;

(5.13a)

( )[ ] ( ) ( )

( ) [ ] ( )

Var HH

Var arg H

rr

n

r

sr

d sr

sr

sr

,,

,

ωγ ω

γ

ωγ

γ

≈− ⋅

⋅

≈−

1

2

1

2

2 2

2

2

2

(5.13b)

where ( )γ ωsr2 represents the ordinary coherence function between the harmonic source

and the receiver output signal. The number nd is the number of independent averages usedto estimate the spectral quantities.

The variance of the ordinary coherence function ( )γ ωr r1 2

2 or ( )γ ωsr2 can be estimated

by the following relation (Bendat and Piersol, 1986):


[ ] ( )Var γ

γ γr r

r r r r

dn1 2

1 2 1 22

2 22

2 1≈

−(5.14)

5.4.2 Statistical Aspects of Conventional Measurements

Once expected values and variances of the experimental measurements have beendefined, the tools of statistical analysis can be used to estimate the expected values andvariances of the quantities depending on them. In conventional phase velocitymeasurements the first of these quantities is the Rayleigh phase velocity ( )VR ω that is

obtained from the phase of the cross-power spectrum Gr r1 2 via Eq. 5.2. Because ( )VR ω is

a non-linear function of ( )arg Gr r1 2, ( )VR ω is in general non-Gaussian distributed. However,

by assuming the experimental data to be characterized by small variances, it is possible to

expand Eq. 5.2 in a Taylor’s series about ( )[ ]E arg G r r1 2 (provided ( )[ ]E arg G r r1 2

0≠ ), and

truncate it to first-order terms only (Papoulis, 1965). If it is further assumed that

[ ] [ ]Var Var r rω = − =2 1 0 (i.e. frequency of excitation and receiver spacing are considereddeterministic rather than random variables), the following results are obtained:

( )[ ] ( )( )( )[ ]

( )[ ]( )( )[ ] ( )[ ]

( )( )[ ]

E Vr r

E G

Var VVar G r r

E G

R

r r

R

r r

r r

ωω

ω

ωω ω

ω

≈−

≈⋅ −

2 1

2 1

2

4

1 2

1 2

1 2

arg

arg

arg

(5.15)

Since the experimental dispersion curve is constructed by averaging Rayleigh phase

velocities ( )[ ]VR jω ( )j nR= 1, obtained over nR different receiver spacings, and since

every linear combinations of normal distributions is itself a normal distribution (Papoulis,1965), the expected value and variance of the experimental dispersion curve ( )VR ω aregiven by:


( )[ ] ( )[ ]

( )[ ] ( )[ ]

E Vn

E V

Var Vn

Var V

RR

R jj

n

RR

R jj

n

R

R

ω ω

ω ω

=

=

=

=

∑

∑

1

1

1

21

(5.16)

In Eq. 5.16 the variance of ( )VR ω has been computed under the assumption that the

individual ( )[ ]VR jω are statistically uncorrelated.

In conventional attenuation measurements, the displacement amplitudes ( )w r,ω are

calculated from the experimental auto-power spectra via Eq. 5.5. Using the same procedureillustrated above for ( )VR ω , namely expanding Eq.(5.5) in a truncated Taylor’s series about

[ ]E G rr and assuming ( )[ ]Var C ω = 0 , it is possible to obtain:

( )[ ] [ ]

( )[ ] [ ][ ] [ ]

E w rE G

C

Var w rVar G

E G C

rr

rr

rr

,( )

( )

,( )

( ) ( )

ωω

ω ω

ωω

ω ω ω

≈⋅

≈⋅ ⋅

4

2

(5.17)

The expected value and variance of the Rayleigh attenuation coefficient ( )α ωR must becomputed from the non-linear regression based on Eq. 5.6. Unfortunately, because of the

non-linearity of the relationship between ( )w r,ω and ( )α ωR , one can only estimate

( )[ ]E Rα ω and ( )[ ]Var Rα ω . If the attenuation coefficient is estimated from Eq. 5.6 usinga standard non-linear least-squares algorithm (e.g. the Gauss-Newton method or theLevenberg-Marquardt method), the uncertainty associated with this estimate can beapproximately calculated with the following relation (Menke, 1989):

( )[ ] ( ) ( )[ ] ( )Var Covlast # last #

T

αRT T T T

R R R R R Rω ωα α α α α α≈

− −J J J w J J J

1 1

(5.18)

where ( )[ ]Cov w ω is an n nT T× matrix containing the covariances of the experimental

displacement amplitudes at the nT different receiver locations at a given frequency ω .


Since it is assumed that the data ( )w ,r ω are uncorrelated, the data covariance matrix is

diagonal with the non-zero elements equal to the variances of ( )w ,r ω given by Eq. 5.17.

The term JαR is a nT × 1 vector whose components ( )JαR k

( )k nT= 1, are defined from

Eq. 5.6 by:

( ) ( )[ ] ( ) ( )J Jw ,

r w ,α α ω∂ ω

∂αω

R Rk k

k

Rk k

rr= = = − ⋅ (5.19)

The subscript last # outside the brackets of Eq. 5.18 indicates that the terms inside theparentheses, essentially the vector JαR

, refer to the last iteration in the solution of the non-

linear regression based on Eq. 5.6.

Equation 5.18 completes the calculation of the expected values and variances of theexperimental Rayleigh phase velocities ( )VR ω and attenuation coefficients ( )α ωR . Thenext task is to determine how the uncertainties of these data are mapped into uncertainties ofthe estimated model parameters, which are the shear wave velocity and shear damping ratioprofiles of a given soil deposit. This and other related topics will be discussed in Sections5.4.4 and 5.4.5.

5.4.3 Statistical Aspects of New Measurement Techniques

5.4.3.1 Uncoupled Analysis

In the new approach to surface wave measurements, the average Rayleigh phase velocity( )VR ω in the simplified uncoupled analysis is computed from a linear regression of the

displacement phases versus receiver distance at constant frequency. More precisely, themethod consists of determining the Rayleigh wavenumber ( )kR ω from a linear regression

based on Eq. 5.11b, and then obtaining ( ) ( )V kR Rω ω ω= . By estimating ( )[ ]E kR ω usinga standard linear least-squares algorithm, it is a straightforward matter to calculate theuncertainty associated with this estimate using the following relation (Menke, 1989):

( )[ ] ( )[ ] ( )( )[ ] ( )[ ]Cov Covm G G G w G G Gω ω=− −

rT

r rT

rT

r rT

T1 1arg (5.20)


where ( )( )[ ]Cov arg w ω is an n nT T× diagonal matrix (assuming the data ( )[ ]arg w ,r ω to

be uncorrelated) whose elements are the variances of the experimental displacement phases

( )[ ]arg w ,r ω at the nT different receiver locations at a given frequency ω .

The term ( )[ ]Cov m ω is a 2 2× diagonal matrix whose two non-zero elements are the

variances of the intercept and the slope (i.e., ( )[ ]Var kR ω ) of the linear regression. Finally,

the term G r is a nT × 2 matrix defined by Gr r r

r

nT

T=

1 2

1 1 1

L

L where rk ( )k nT= 1,

denotes the source-to-receiver distance for receiver k.

Having defined ( )[ ]E kR ω and ( )[ ]Var kR ω , the expected value and variance of

( ) ( )V kR Rω ω ω= are determined using the Taylor’s series expansion described in theprevious section. The results are given by the following expression:

( )[ ] ( )[ ]

( )[ ] ( )[ ]( )[ ]

E VE k

Var VVar k

E k

R

R

R

R

R

ωω

ω

ωω ω

ω

≈

≈

2

4

(5.21)

Expected values and variances of the Rayleigh attenuation coefficient ( )α ωR aredetermined with the same procedure used for the conventional measurements; in particularEq. 5.18 gives a measure of the uncertainty associated with the estimate of ( )α ωR .

5.4.3.2 Coupled Analysis

In coupled measurements of dispersion and attenuation data, the quantity measured

experimentally is the particle velocity transfer function ( )H r,ω between source and

receiver, and defined by ( ) ( ) ( )H r w r F e ryi t, & , ,ω ω ω ωω= ⋅ = i T . The expected value and

variance of the modulus and phase of ( )H r,ω are defined by Eq. 5.13b. The corresponding

statistical quantities for the displacement transfer function ( )T r,ω are:


( )[ ] ( )[ ] ( )( )[ ] ( )( )[ ]

( )[ ] ( )[ ] ( )( )[ ] ( )( )[ ]

E T r E H r E T r E H r

Var T r Var H r Var T r Var H r

, , ; arg , arg ,

, , ; arg , arg ,

ω ω ω ω ωπ

ω ω ω ω ω

= = +

= =

2

2

(5.22)

Now, since the complex wavenumber ( ) ( ) ( )KV

iR

R* ω

ωω

α ω= +

is determined from

the non-linear regression ( ) ( ) ( )T r r e i K r, ,*

ω ω ω= ⋅ − ⋅ ⋅G , it is first necessary to calculate

( )[ ]E T r,ω and ( )[ ]Var T r,ω in order to compute expected value and variance of ( )K * ω .

It is convenient to write ( ) ( ) ( )[ ]T r T r iT r, , ,ω ω ω= +1 2 , from which it is simple to obtain:

( )[ ] ( )[ ] ( )( )[ ] ( )[ ] ( )[ ] ( )( )[ ]

( )[ ] ( )[ ] ( )[ ]

E T r E T r E T r

E T r E T r E T r

Var T r Var T r Var T r

1

2

1 2

, , arg ,

, , arg ,

, , ,

ω ω ω

ω ω ω

ω ω ω

≈ ⋅

≈ ⋅

= −

cos

sin(5.23)

where:

( )[ ] ( )( )[ ] ( )[ ]( ) ( )( )[ ] ( )( )[ ]

( )[ ] ( )( )[ ] ( )[ ]( ) ( )( )[ ] ( )( )[ ]

Var T r T r Var T r

T r T r Var T r

Var T r T r Var T r

T r T r Var T r

12

2 2

22

2 2

, cos arg , ,

, sin arg , arg ,

, sin arg , ,

, cos arg , arg ,

ω ω ω

ω ω ω

ω ω ω

ω ω ω

≈ +

+

≈ +

+

(5.24)

The uncertainty of the complex wavenumber ( )K * ω is finally computed from the non-

linear regression ( ) ( ) ( )T r r e i K r, ,*

ω ω ω= ⋅ − ⋅ ⋅G following a procedure that is formally identical

to that used for determining the uncertainty of the Rayleigh attenuation coefficient ( )α ωR .The result is:


( )[ ] ( ) ( )[ ] ( )Var Covlast # last #

H

Κ∗ ω ω≈

− −J J J T J J J

K

H

K K

H

K

H

K K

H* * * ** *

1 1(5.25)

where ( )[ ]Cov T ω is an n nT T× matrix representing the covariances, at a given frequency

ω , of the experimental displacement transfer functions at the nT receiver locations. Since

it is assumed that the data ( )T r,ω are uncorrelated, the matrix ( )[ ]Cov T ω is diagonal

with the non-zero elements equal to the variances of ( )T r,ω and given by Eqs. 5.23 and

5.24. The term JK* is an nT × 1 complex-valued vector whose components ( )J *K j

( )j nT= 1, are defined from the relation ( ) ( ) ( )T r r e i K r, ,*

ω ω ω= ⋅ − ⋅ ⋅G and are equal to:

( ) ( )[ ] ( ) ( )J J,

r ,* * *K j K j

j

j j

T r

Ki T r= = = − ⋅ω

∂ ω

∂ω (5.26)

Like Eq. 5.18, the subscript last # outside the brackets of Eq. 5.25 indicates that theterms inside the parenthesis refer to the last iteration in the solution of the non-linear

regression ( ) ( ) ( )T r r e i K r, ,*

ω ω ω= ⋅ − ⋅ ⋅G .

Once ( )[ ]E K * ω and ( )[ ]Var K * ω are computed, the expected value and variance of

the complex Rayleigh phase velocity ( )VR* ω are obtained from the complex extension of

Eq. 5.21, namely:

( )[ ] ( )[ ]

( )[ ] ( )[ ]( )[ ]

E VE K

Var VVar K

E K

R

R

R

R

R

*

*

*

*

*

ωω

ω

ωω ω

ω

≈

≈

2

4

(5.27)

Equation 5.27 completes the statistical analysis of surface wave measurements whendispersion and attenuation data are determined simultaneously from the particle velocity

transfer functions ( )H r,ω . However it may also be important to compute ( )[ ]E VR* ω and


( )[ ]Var VR* ω when surface wave data, namely ( )VR ω and ( )α ωR , are obtained

independently. The solution to this problem is obtained from the application of the theoryof random variables (Papoulis, 1965) to Eq. 2.34.

In particular, this equation can be interpreted as a mapping Φ: R C2 → that assigns a

complex random variable VR* to a pair of independent random variables VR and DR . If

the variables VR and DR are normally distributed and the mapping ( )Φ V DR R, is

sufficiently smooth, ( )Φ V DR R, may be expanded in a Taylor series about the point

( ) ( )[ ]E V E DR R, . Again assuming small variances for the variables VR and DR , this series

is truncated to first-order terms only, yielding:

( )[ ] ( )[ ]( )[ ] ( )[ ]( )

( )[ ] ( )[ ]( )[ ] ( )[ ]

( )[ ] ( )[ ] ( )[ ]( )[ ]( )[ ]( )

( )[ ]

E VE V

E DiE D

Var ViE D

E DVar V

E V E D i E D

E DVar D

R

R

R

R

R

R

R

R

R R R

R

R

*

*

ωω

ωω

ωω

ωω

ω ω ω

ωω

≈+

⋅ −

≈−

+

⋅ +

+⋅ + −

+

⋅

11

1

1

2 1

1

2

2

2

2

22

2

(5.28)

where:

( )[ ] ( )[ ] ( )[ ]

( )[ ] ( )[ ] ( )[ ] ( )[ ] ( )[ ]

E DE E V

Var DE V

VarE

Var V

R

R R

R

R

R

R

R

ωα ω ω

ω

ωω

ωα ω

α ωω

ω

≈

≈

⋅ +

⋅

2 2

(5.29)

In deriving Eqs. 5.28 and 5.29 it was assumed that the random variables VR and DR areuncorrelated.


In summary, despite the fact that the relationship between directly measured quantities

(i.e., ( )G rr ω , ( )G r r1 2ω , and ( )H r,ω ) and the derived surface wave data (i.e., ( )VR ω ,

( )α ωR , and ( )VR* ω ) is non-linear in most cases, the assumption of small variances allows

one to obtain explicit results for the expected values and variances of ( )VR ω , ( )α ωR , and

( )VR* ω .

5.4.4 Statistical Aspects of Uncoupled Rayleigh Inversion

Having defined the statistics of the experimental data, the next task is to determine howthe uncertainty of the measurements is mapped into uncertainty of the estimated modelparameters, which are the shear wave velocity VS and shear damping ratio DS profiles. Thefirst objective will be to obtain an approximation of the uncertainty associated with theinverted shear wave velocity profile VS . The problem of inverting an experimental

dispersion curve can formally be written as ( )V V VR R S= , where VR is a nF × 1 vector of

Rayleigh phase velocities ( )VR ω associated with nF different frequencies, and VS is anL × 1 vector whose components are the unknown shear wave velocities of an nL layers inthe soil deposit.

From a statistical point of view, [ ]E SV and [ ]Var SV are the quantities that are of

interest. Because the relationship ( )V V VR R S= is non-linear, only approximate results can

be obtained. If ( )V V VR R S= is inverted using Occam’s algorithm, it is possible to show

that the uncertainty associated with the estimated [ ]E SV can be computed with thefollowing relationship:

[ ] ( ) ( )

[ ] ( ) ( )

Cov

Cov

last #

last #

T

V W J W J W J W

V W J W J W J W

ST

V S

T

V S V S

T

V

RT

V S

T

V S V S

T

V

R R R R

R R R R

≈

⋅

⋅ ⋅

−

−

µ

µ

∂∂ ∂∂

∂∂ ∂∂

+

+

1

1(5.30)

where [ ]Cov VR is an n nF F× matrix of covariances of the experimental Rayleigh phase

velocities at nF different frequencies ω . It is assumed that the data ( )VR ω are

uncorrelated, and hence the matrix [ ]Cov VR is diagonal with the non-zero elements equal

to the variances of ( )VR ω . The variances are given by Eq. 5.16 for conventionalmeasurements and by Eq. 5.21 for the new measurement procedure.


[ ]Cov VS is a n nL L× diagonal matrix whose elements are the variances of the

estimated shear wave velocities ( )VS j with j nL= 1, . The term WVR

is a n nF F× diagonal

matrix defined by ( ) ( ) ( )WVR=

diag V V V nR R RF

1 1 11 2

/ , / ,..... , /σ σ σ where

( ) ( )σ Vk

R kRVar V= + with k nF= 1, and represents the standard deviations associated with

the experimental data VR . The parameter µ and the symbol ∂∂ have already been definedin Section 4.5; µ is the inverse of the Lagrange multiplier and ∂∂ is the two-point-centralfinite difference operator.

Finally, the term J S is the n nF L× Jacobian matrix whose elements ( )JS kj are defined

by the partial derivatives ( ) ( ) ( )JS kj R k S j= ∂ ∂V V . The subscript last # outside the brackets

of Eq. 5.30 indicates that the terms inside the parentheses (in essence the Jacobian matrixJ S ) should be computed with respect to the last iteration in the solution of the non-linear

problem ( )V V VR R S= .

The determination of the uncertainty associated with the estimated damping ratio

profile DS is simpler than determining [ ]Cov VS because the inversion of the experimental

attenuation curve ( )α ωR to obtain the damping ratio profile is linear. The problem cansymbolically be written as GDS = αα R . In this equation G is a n nF L× matrix formed bythe partial derivatives of Rayleigh phase velocity with respect to the shear and compressionwave velocities of the soil layers and defined by Eqs. 3.45 and 3.63b. The term αα R is anF × 1 vector of Rayleigh attenuation coefficients calculated for nF different frequenciesand DS is a nL × 1 vector containing the unknown shear damping ratios of the nL layerssoil deposit. If the linear equation GDS = αα R is inverted using Occam’s algorithm, theuncertainty of the estimated shear damping ratio profile DS can be computed from theuncertainty of the measured attenuation coefficients αα R using the following relation:

[ ] ( ) ( )

[ ] ( ) ( )

Cov

Cov

D W G W G W G W

W G W G W G W

ST

T T

RT

T TT

R R R R

R R R R

=

⋅

⋅ ⋅

−

−

µ

µ

α α α α

α α α α

∂∂ ∂∂

αα ∂∂ ∂∂

+

+

1

1(5.31)


where [ ]Cov αα R is an n nF F× diagonal matrix whose non-zero elements (the ( )α ωR are

assumed to be uncorrelated) are the variances of ( )α ωR at nF different frequencies ω ; the

variances can be calculated from Eq. 5.18. [ ]Cov DS is a n nL L× diagonal matrix

containing the variances of the estimated shear damping ratios ( )DS j with j nL= 1, .

Finally, the term WαR is a n nF F× diagonal matrix defined by

( ) ( ) ( )Wα α α ασ σ σR

=

diagR R R

Fn1 1 1

1 2/ , / ,..... , / where ( ) ( )σ ααR k

R kVar= + with

k nF= 1, and represents the standard deviations of the experimental data αα R .

5.4.5 Statistical Aspects of Coupled Rayleigh Inversion

The procedure used to calculate an approximation of the uncertainty associated with theinverted complex shear wave velocity profile VS

* is identical to that described in the

previous section to obtain [ ]Cov VS . The only difference is that the computation of

[ ]Cov VS* requires a systematic use of the complex formalism.

The non-linear relation between a complex-valued vector VS* of nL unknown shear

wave velocities (all referred to a reference frequency ω ref ), and a nF × 1 vector of complex

Rayleigh phase velocities VR* for nF different frequencies is denoted by ( )V V VR R S

* * *= . Asfor the real case, the non-linearity of this relationship allows one to obtain only approximate

results for [ ]Cov VS* . If the equation ( )V V VR R S

* * *= is inverted using Occam’s algorithm,

the uncertainty of the estimated VS* profile can be computed from the complex version of

Eq. 5.30, namely:

[ ] ( ) ( )

[ ] ( ) ( )

Cov

Cov

R*

R*

R*

R*

R*

R*

R*

R*

last #

last #

H

V W J W J W J W

V W J W J W J W

ST

V S

H

V S V S

H

V

RT

V S

H

V S V S

H

V

* * * * * * * *

* * * * * * * *

≈

⋅

⋅ ⋅

−

−

µ

µ

∂∂ ∂∂

∂∂ ∂∂

+

+

1

1(5.32)

where [ ]Cov VR* is an n nF F× diagonal matrix (for uncorrelated ( )VR

* ω data) whose

elements are the (complex) variances of the experimental ( )VR* ω at nF different


frequencies and given by Eq. 5.27. [ ]Cov VS* is a n nL L× diagonal matrix formed by the

variances of the estimated complex valued shear wave velocities ( )VS j

* ( )j nL= 1, .

The term WV

*

R* is a n nF F× diagonal matrix defined by

( ) ( ) ( )WV V V V n

diagF

R*

R*

R*

R*

* * * */ , / ,..... , /=

1 1 11 2

σ σ σ where ( ) ( )σV

kR k

Var VR*

* *= + with

k nF= 1, and represents the complex-valued standard deviations of the experimental VR* .

Finally, the term J S* is the n nF L× complex-valued Jacobian matrix whose elements ( )JS kj

*

are defined by the partial derivatives ( ) ( ) ( )JS kj R k S j

* * *V V= ∂ ∂ .

157

6 VALIDATION OF THE ALGORITHMS

6.1 Overview

The objective of this chapter is the validation of the algorithms developed during thisresearch study, and based on some of the theoretical concepts presented Chapters 3 and 4.The validation of these algorithms is carried out using the following procedure. For alayered medium characterized by a set of material parameters, synthetic displacement data atseveral receiver offsets are generated from the numerical solution of the Rayleigh forwardproblem. These surface wave displacement data, which simulate SASW field measurements,are then used to calculate synthetic dispersion and attenuation curves with the sametechniques used in a real SASW test and illustrated in chapter 5.

The simulated dispersion and attenuation curves are then inverted using the Rayleighinversion algorithms described in Chapter 4 in an attempt to recover the original shear wavevelocity and shear damping ratio profiles of the medium. A comparison between theoriginal and the inverted SV and SD profiles will provide the means for assessing theperformance of the algorithms. This validation procedure is carried out for stratified mediahaving different layering and material properties.

This chapter is organized in two main sections, describing the procedures adopted forthe validation of the uncoupled and coupled inversion algorithms, respectively. In each of thesesections, the algorithms associated with the fundamental mode analysis (i.e. UFUMA andCFUMA) are presented first, followed by those based on the equivalent multi-mode analysis (i.e.UEQMA and CEQMA).

6.2 Lamb’s Problem

Before proceeding with the analyses for layered soil deposits, a few results will bepresented for the solution of the Rayleigh boundary value problem in homogeneous elasticand viscoelastic media. For this simple case, an exact solution is available which can be usedto verify the results obtained from the algorithms developed for the solution of the elasticand viscoelastic Rayleigh forward problem.

The problem of determining the displacement field induced by a vertical harmonic pointload applied at the free surface of an homogeneous, isotropic, linear elastic half-space, wasfirst solved by Lamb (1904) in a classical paper entitled “On the Propagation of Tremors over theSurface of an Elastic Solid”. Lamb used the tools of complex variable theory to find thesolution of what today is known as the Lamb’s problem, which can be considered as thedynamic analogue of another classical problem of linear elasticity: Boussinesq’s problem.

158 Validation of the Algorithms

Although Lamb developed a solution for an arbitrary time variation of the source, onlythe solution for a harmonic source is presented here. The harmonic solution can be writtenas follows:

( ) ( ) ( )w rFe

iGk k H k rR

i t

R R R, ( )ωω

= ⋅ ⋅ ⋅2 0

2Φ (6.1)

where ( )w rR ,ω is the Rayleigh vertical displacement at the free surface of a homogeneous

elastic half-space at a distance r from a vertically oscillating harmonic source Fe i tω . The termG is the shear modulus of the elastic medium, kR is the Rayleigh wave-number, and the

symbol ( )H02( ) ⋅ denotes the Hankel function of the second kind of zero order (i.e.

( ) ( )H z J z iY z02

0 0( ) ( )= − where J z0 ( ) and ( )Y z0 are the Bessel functions of the first kind

and second kind, respectively, of zero order).

The function ( )Φ kR is defined as follows:

( ) ( )Φ kk k k

R kR

S R P

R

= −−

′

2 2 2

(6.2)

where kP and kS are the wave-numbers of the P-wave and S-wave, respectively. Finally, the

function ( )R kR is given by the following expression:

( ) ( ) ( )( )R k k k k k k k kR R S R R P R S= − − − −2 42 2 2 2 2 2 2 2 (6.3)

where k VR R= ω and VR is the frequency-independent Rayleigh phase velocity to bedetermined from the solution of the Rayleigh dispersion equation in homogeneous media,namely:

( )ξ ξ χ ξ χ3 28 8 1 2 16 0− + + − = (6.4)

where ( )ξ = V VR S

2, ( )χ = −1

2V VS P , and V kγ γω= with γ = P S, .

Validation of the Algorithms 159

Lamb’s solution was calculated for three different elastic media, and the results werecompared with those obtained with the new algorithm. Table 6.1 shows the materialproperties and frequencies used for the comparison test.

Table 6.1 Medium Properties and Frequencies Used for Validation of the ElasticLamb’s Problem

Case No.VP

[m/s]

VS

[m/s]

MassDensity

[t/m3]

Frequency

[Hz]

1 500 250 1.8 5

2 600 400 1.8 50

3 1000 600 1.8 100

The results of the validation test are shown in Fig. 6.1(a) through Fig. 6.1(c), wherevertical displacement amplitudes and phases are plotted versus the distance from the sourcefor a specified frequency.

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1x 10

-7

Distance [m]

Dis

plac

emen

t Am

plitu

de [m

] Lamb Solution Numerical Method

0 20 40 60 80 100-6

-4

-2

0

2

4

6

Distance [m]

Dis

plac

emen

t Pha

se [r

ad]

Lamb Solution

Numerical Method

Figure 6.1(a) Comparison of Solutions for the Elastic Lamb’s Problem (Case 1)

The magnitude of the harmonic source Fe i tω has been assumed equal to one. In allthree cases the agreement between the closed-form solution given by Lamb and thatobtained with the numerical algorithm is excellent.


0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1x 10

-7

Distance [m]

Dis

plac

emen

t Am

plitu

de [m


0 20 40 60 80 100-6

-4

-2

0

2

4

6

Distance [m]

Dis

plac

emen

t Pha

se [r

ad]

Lamb Solution Numerical Method

Figure 6.1(b) Comparison of Solutions for the Elastic Lamb’s Problem (Case 2)

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1x 10

-7

Distance [m]

Dis

plac

emen

t Am

plitu

de [m


0 20 40 60 80 100-6

-4

-2

0

2

4

6

Distance [m]

Dis

plac

emen

t Pha

se [r

ad]


Figure 6.1(c) Comparison of Solutions for the Elastic Lamb’s Problem (Case 3)

The solution of the Lamb’s problem in viscoelastic media can be obtained from theapplication of the elastic-viscoelastic correspondence principle to the corresponding elasticsolution. As a result, in the viscoelastic solution the terms k kP S, , kR and G appearing inEq. 6.1 are complex-valued. Table 6.2 shows the frequencies and material parameters usedfor the validation of the numerical solution of the Lamb’s problem in viscoelastic media.

Figures 6.2(a) through 6.2(c) show the results of the comparison. It is apparent from theplots that the closed-form and numerical solutions also agree for the viscoelastic case.


Table 6.2 Medium Properties and Frequencies Used for Validation of the ViscoelasticLamb’s Problem

Case No.VP

[m/s]

VS

[m/s]

DP

[%]

DS

[%]

MassDensity

[t/m3]

Frequency

[Hz]

1 500 250 1 3 1.8 5

2 600 400 2 4 1.8 50

3 1000 600 5 5 1.8 100

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1x 10

-7

Distance [m]

Dis

plac

emen

t Am

plitu

de [m

]

Lamb Solut ion Numer ical Method

0 20 40 60 80 100-6

-4

-2

0

2

4

6

Distance [m]

Dis

plac

emen

t Pha

se [r

ad]


Figure 6.2(a) Comparison of Solutions for the Viscoelastic Lamb’s Problem (Case 1)

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1x 10

-7

Distance [m]

Dis

plac

emen

t Am

plitu

de [m

]


0 20 40 60 80 100-6

-4

-2

0

2

4

6

Distance [m]

Dis

plac

emen

t Pha

se [r

ad]


Figure 6.2(b) Comparison of Solutions for the Viscoelastic Lamb’s Problem (Case 2)


0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1x 10

- 7

Distance [m]

Dis

plac

emen

t Am

plitu

de [m

]

Lamb Solu t ion Numer ica l Method

0 20 40 60 80 100-6

-4

-2

0

2

4

6

Distance [m]

Dis

plac

emen

t Pha

se [r

ad]


Figure 6.2(c) Comparison of Solutions for the Viscoelastic Lamb’s Problem (Case 3)

6.3 Numerical Simulations

For vertically layered media the validation of the inversion algorithms was conductedaccording to the following procedure. Three soil profiles were selected as simplifiedexamples of actual soil deposits. Tables 6.3 through 6.5 show the material properties andlayer thicknesses that were chosen for the three soil profiles. Case 1 represents a regular soilprofile where the stiffness of the layers increases with depth. Cases 2 and 3 represent twoirregular soil profiles. Case 2 represents the common situation in which a soft layer istrapped between two stiffer layers. Finally, Case 3 is an example of a medium characterizedby a stiff surface layer. The soil profile of Case 1 belongs to the category of normally dispersivesoil profiles (Tokimatsu, 1995), while Cases 2 and 3 are classified as inversely dispersive soilprofiles.

As discussed in Section 3.4, near-field effects decay very quickly with distance from thesource in normally dispersive media (they are negligible at distances larger than one-half awavelength). In inversely dispersive media, such effects are considerably more important(they may extend up to distances that are four times larger). Furthermore, in normallydispersive media the propagation of surface waves is to a great extent dominated by thefundamental mode, while in inversely dispersive soil deposits the response of the medium ismore often controlled by higher modes of propagation, particularly at high frequencies(Gucunski and Woods, 1991). For these three layered media, a numerical simulation wasconducted to compute synthetic dispersion and attenuation curves.

This computation was performed using the computer programs developed for thesolution of the Rayleigh forward problem and the methods for surface wave datainterpretation described in Section 5.3. The synthetic displacement field was computed at 20receiver locations, with the closest receiver located at a distance equal to one to twowavelengths from the source. The numerical simulation was performed for a set of 50


frequencies ranging from 5 to 100 Hz, which is a typical frequency range used in SASWtests. Once the synthetic dispersion and attenuation curves of the three soil deposits weredetermined, the last step of the validation procedure consisted of inverting these curves toobtain the corresponding shear wave velocity and shear damping ratio profiles. Acomparison of the latter with the original VS and DS profiles served as the basis forassessing the performance of the inversion algorithms.

Table 6.3 Medium Properties Used for the Validation of the Inversion Algorithms(Case 1)

LayerNo.

Thickness

[m]

VP

[m/s]

VS

[m/s]

DP

[%]

DS

[%]

MassDensity

[t/m3]

1 5 400 200 2.0 3.5 1.7

2 10 600 300 1.5 3.0 1.8

3 10 800 400 1.0 2.5 1.8

Half-Space

∞ 1000 500 1.0 2.0 1.8


LayerNo.

Thickness

[m]

VP

[m/s]

VS

[m/s]

DP

[%]

DS

[%]

MassDensity

[t/m3]

1 10 800 400 1.0 3.0 1.8

2 5 600 300 1.5 3.5 1.7

3 10 800 400 1.0 3.0 1.8

Half-Space

∞ 1000 500 1.0 2.5 1.8


The dispersion curves of the first four modes of propagation for the Case 1 soil profileare shown in Fig.6.3, together with the synthetic dispersion curve. As expected for thisnormally dispersive profile, there is a good agreement between the synthetic dispersioncurve and the dispersion curve associated with the fundamental mode of propagation.

0 2 0 4 0 6 0 8 0 1 0 0 1 2 01 5 0

2 0 0

2 5 0

3 0 0

3 5 0

4 0 0

4 5 0

5 0 0

F requency [Hz ]

Ray

leig

h P

hase

Vel

ocity

[m/s

]

F u n d a m e n t a l M o d e

S e c o n d M o d e

T h i r d M o d e

F o u r t h M o d e

S y n t h e t i c C u r v e

Figure 6.3 Rayleigh Dispersion Curves for Case 1 Soil Profile


LayerNo.

Thickness

[m]

VP

[m/s]

VS

[m/s]

DP

[%]

DS

[%]

MassDensity

[t/m3]

1 5 1000 500 1.0 2.5 1.8

2 10 800 400 1.5 3.0 1.8

3 10 1000 500 1.0 2.5 1.8

Half-Space

∞ 1200 600 1.0 2.0 1.8


However, Fig. 6.4 also shows that the agreement between the synthetic dispersion curveand the effective dispersion curve is even better, particularly at frequencies less than 15 Hz. Theeffective dispersion curve was computed by averaging the effective Rayleigh phase velocity

( )$ ,V rR ω over the receiver locations at each frequency. Figure 6.4 shows that, even innormally dispersive media, the simulated experimental dispersion curve compares betterwith the effective dispersion curve than with the modal dispersion curve.

0 2 0 4 0 6 0 8 0 1 0 01 5 0

2 0 0

2 5 0

3 0 0

3 5 0

4 0 0

F r e q u e n c y [ H z ]

Ray

leig

h P

hase

Vel

ocity

[m/s

] S y n t h e t i c C u r v e

E f f e c t i v e C u r v eF u n d a m e n t a l M o d e

Figure 6.4 Rayleigh Effective Dispersion Curve for Case 1 Soil Profile

Figure 6.5 shows modal and synthetic attenuation curves for the Case 1 soil profile. Theirregularities of the synthetic attenuation curve are mostly due to numerical noise affectingthe computation of the attenuation coefficients and caused by the coupling betweenmaterial and geometric attenuation.

Because the displacement amplitude decay with distance is largely governed bygeometric attenuation, the evaluation of the Rayleigh attenuation coefficients is sensitive tonumerical noise. Both seismic and electrical noise affect experimental amplitudemeasurements. Improvements in the accuracy of amplitude measurements can be achievedby using noise corrections procedures and other signal processing techniques (Spang, 1995;Rix et al., 1998a). Figure 6.5 shows that, despite the irregularities, the synthetic attenuationcurve compares fairly well with the attenuation curve associated with the fundamental modeof propagation. As expected, the modal and synthetic attenuation coefficients increase inmagnitude as the frequency increases.


0 20 40 60 80 1 0 0 1 2 00

0 .05

0 .1

0 .15

0 .2

Frequency [Hz ]

Ray

leig

h A

ttenu

atio

n C

oeffi

cien

t [1/

m] F u n d a m e n t a l M o d e

S e c o n d M o d e

T h i r d M o d e

F o u r t h M o d e


Figure 6.5 Rayleigh Attenuation Curves for Case 1 Soil Profile

The modal and synthetic dispersion curves corresponding to the soil profile of Case 2are illustrated in Fig.6.6, where only the first four modes of propagation have been plotted.

0 2 0 4 0 6 0 8 0 1 0 0 1 2 03 0 0

3 5 0

4 0 0

4 5 0

5 0 0

5 5 0

F requency [Hz ]

Ray

leig

h P

hase

Vel

ocity

[m/s

]


S e c o n d M o d e

T h i r d M o d e

F o u r t h M o d e




Case 2 profile is a soil deposit with a soft layer trapped between two stiffer layers. Theirregularity in soil stratification is reflected in modal dispersion curves which contain, asshown in Fig.6.6, three jumps discontinuities at frequencies of about 50, 82 and 98 Hz. It isimportant to note that modal dispersion curves need not be continuous. Depending on themedium stratification and on the frequency range, these curves may display one or morejump discontinuities. The current dispersion curves are computed using a set of 50frequencies ranging from 5 to 100 Hz. A finer frequency discretization would result insmoother dispersion curves, but the jump discontinuities are features that are independentof the frequency discretization. Note that the synthetic dispersion curve also has a jumpdiscontinuity at a frequency of about 35 Hz.

Figure 6.6 shows that in the frequency range of about 35 to 50 Hz the syntheticdispersion curve deviates from the fundamental mode dispersion curve; it mostly followsthe second mode of propagation. However, the fundamental mode still dominates thesynthetic dispersion curve for frequencies outside this range.

Figure 6.7 shows a comparison between the effective, the synthetic and the fundamentalmode dispersion curves. The ability of the effective dispersion curve to follow the irregularpattern of the synthetic dispersion curve over the frequency range of 35-50 Hz is readilyapparent. The results shown in Fig. 6.7 are remarkable. As mentioned earlier in this chapter,irregular layered media such as the profile of Case 2 are inversely dispersive and theirresponse is no longer governed by the fundamental mode of propagation as in normallydispersive media. The synthetic dispersion curve, which simulates the dispersion curve thatwould be obtained experimentally, is determined from measurements of displacementphases. Because the displacement field is obtained from the superposition of all of theRayleigh modes of propagation, the synthetic dispersion curve inherently reflects theconsequences of multi-mode wave propagation.

As a result, the agreement between a synthetic or an experimental dispersion curve andany modal dispersion curve is expected to be poor in media where the response iscontrolled by more than one mode of propagation. The effective dispersion curve, on theother hand, is built on the concept of effective phase velocity, which is the velocity ofpropagation of a superposition of harmonic waves with the same frequency and differentwave-number. It is not surprising therefore, that the effective and synthetic dispersioncurves illustrated in Fig.6.7 are almost identical.

The synthetic and the modal attenuation curves associated with the Case 2 soil profileare shown in Fig.6.8. Because the modal attenuation curves are very similar, it is difficult toidentify a predominant mode of propagation. However, the synthetic attenuation curveappears to intersect all four modal attenuation curves in several frequency ranges.


0 2 0 4 0 6 0 8 0 1 0 0 1 2 03 0 0

3 5 0

4 0 0

4 5 0

5 0 0

F requency [Hz ]

Ray

leig

h P

hase

Vel

ocity

[m/s

]


E f f e c t i v e C u r v e F u n d a m e n t a l M o d e


0 2 0 4 0 6 0 8 0 1 0 0 1 2 00

0 .01

0 .02

0 .03

0 .04

0 .05

0 .06

0 .07

0 .08

Frequency [Hz ]

Ray

leig

h A

ttenu

atio

n C

oeffi

cien

t [1/


S e c o n d M o d e

T h i r d M o d e

F o u r t h M o d e




The dispersion curves for the Case 3 soil profile are shown in Fig. 6.9. Case 3 is anexample of a soil deposit containing a stiff thin layer that may represent an overconsolidatedcrust overlying an otherwise regular soil medium; this soil stratification is encountered atmany geotechnical sites. The modal dispersion curves associated with this soil deposit arediscontinuous, emulating a pattern already observed for the Case 2 soil profile. However, themodal dispersion curves of Case 3 have only one jump discontinuity that occurs at afrequency of about 75 Hz.

0 2 0 4 0 6 0 8 0 1 0 0 1 2 03 5 0

4 0 0

4 5 0

5 0 0

5 5 0

6 0 0

6 5 0

F requency [Hz ]

Ray

leig

h P

hase

Vel

ocity

[m/s

]


S e c o n d M o d e

T h i r d M o d e

F o u r t h M o d e



Two discontinuities, one at about 41 Hz and the other at about 65 Hz, characterize thesynthetic dispersion curve. For frequencies less than 41 Hz, the synthetic curve is in goodagreement with the fundamental mode dispersion curve. However, for frequencies greaterthan 41 Hz, the second mode of propagation controls the synthetic dispersion curve.

A comparison between the synthetic, the fundamental mode, and the effectivedispersion curves is shown in Fig. 6.10. As for the Case 1 and Case 2 soil profiles, theagreement between the synthetic and the effective dispersion curves is excellent. The extentof the match between the two curves is indicated by the identical values of frequency wherethe jump discontinuities occur.

Figure 6.10 confirms the value of the effective dispersion curve as the most appropriateresponse function to be compared with the synthetic dispersion curve. Because the


procedure used to determine the synthetic dispersion curves was the same as that used inSASW measurements, these findings are expected to be validated by experimental surfacewave data.

0 2 0 4 0 6 0 8 0 1 0 0 1 2 03 5 0

4 0 0

4 5 0

5 0 0

5 5 0

6 0 0

6 5 0

F requency [Hz ]

Ray

leig

h P

hase

Vel

ocity

[m/s

]


E f f e c t i v e C u r v e F u n d a m e n t a l M o d e


0 2 0 4 0 6 0 8 0 1 0 0 1 2 00

0 .01

0 .02

0 .03

0 .04

0 .05

0 .06

Frequency [Hz ]

Ray

leig

h A

ttenu

atio

n C

oeffi

cien

t [1/


S e c o n d M o d e

T h i r d M o d e

F o u r t h M o d e




The synthetic and modal attenuation curves for the Case 3 soil profile are illustrated inFig.6.11. This figure displays features that are similar to those of the Case 2 soil profile; inparticular note that the modal attenuation curves are closely spaced over the entirefrequency range.

6.3.1 Uncoupled Analyses

6.3.1.1 UFUMA Inversion Algorithms

This section will present the results of the uncoupled inversions of the syntheticdispersion and attenuation curves for the Case 1, Case 2 and Case 3 soil profiles using theUFUMA (Uncoupled, Fundamental Mode Analysis) algorithm. Figure 6.12 shows thesequence of iterations required for the algorithm to converge for the Case 1 soil profile.From the figure the theoretical dispersion curve corresponding to the first iteration can beeasily identified.

0 2 0 4 0 6 0 8 0 1 0 0 1 2 01 0 0

1 5 0

2 0 0

2 5 0

3 0 0

3 5 0

4 0 0

F requency [Hz ]

Pha

se V

eloc

ity [m

/sec

]

I t e r . # 1

I t e r . # 7

Theore t i ca lSyn the t i c

Figure 6.12 Fundamental Mode Theoretical and Synthetic Dispersion Curves for Case 1Soil Profile

The theoretical curve associated with the seventh and final iteration is in goodagreement with the synthetic dispersion curve. The corresponding sequence of shear wavevelocity profiles is illustrated in Fig.6.13 where the dashed line indicates the initial modelused in the inversion. The final shear wave velocity profile is shown with a bold line. Finally,Fig. 6.14 shows the convergence of the algorithm in terms of the Root-Mean-Square (RMS)error between the synthetic and the theoretical dispersion curve.


0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0

0

5

1 0

1 5

2 0

2 5

3 0

3 5

S h e a r W a v e V e l o c i t y [ m /sec ]

Dep

th [m

]

I t e r . # 1

I t e r . # 7

Figure 6.13 Shear Wave Velocity Profile from UFUMA Inversion Algorithm for Case 1Soil Profile

0 2 4 6 8 1 0 1 20

0 . 5

1

1 . 5

2

2 . 5

3

I t e r a t i o n #

Exa

ct R

MS

Mis

fit

Figure 6.14 Convergence of UFUMA Inversion Algorithm for Case 1 Soil Profile

As described in Section 4.3.2, the uncoupled inversion of the experimental attenuationcurve to obtain the shear damping ratio profile DS is a linear problem which can be written


as GDS = αα R . In the UFUMA inversion algorithm, G is the matrix formed by the partialderivatives of the fundamental mode Rayleigh phase velocity with respect to the shear andcompression wave velocities of the soil layers (see Eqs. 3.45 and 3.63b). Figure 6.15 showsthe shear damping ratio profile obtained from the inversion of the equation GDS = αα R forthe Case 1 soil profile.

0 1 2 3 4 5

0

5

1 0

1 5

2 0

2 5

3 0

3 5

S h e a r D a m p i n g R a t i o ( % )

Dep

th (

m)

0 20 40 60 80 100 1200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Frequency (Hz)

Atte

nuat

ion

Coe

ffici

ent (

1/m

)

Theoretical

Synthetic

Figure 6.15 Shear Damping Ratio Profile and Theoretical Attenuation Curve fromUFUMA Inversion Algorithm for Case 1 Soil Profile

1 0- 1 0

1 00

1 01 0

0 .5

0 .6

0 .7

0 .8

0 .9

1

S m o o t h i n g P a ra m e t e r

rms

Err

or

Figure 6.16 Attenuation Curves RMS Misfit Error using UFUMA Inversion Algorithmfor Case 1 Soil Profile


Figure 6.15 also compares the synthetic and the theoretical attenuation curves obtainedfrom the solution of the forward problem GDS = αα R using the inverted shear dampingratio profile. The agreement between the two curves is satisfactory and is confirmed by theRMS error as a function of the Lagrange multiplier shown in Fig.6.16.

0 2 0 4 0 6 0 8 0 1 0 0 1 2 02 0 0

2 5 0

3 0 0

3 5 0

4 0 0

4 5 0

5 0 0

F requency [Hz ]

Pha

se V

eloc

ity [m

/sec

]

I t e r . # 1

I t e r . # 8


Figure 6.17 Fundamental Mode Theoretical and Synthetic Dispersion Curves for Case 2Soil Profile

The theoretical and synthetic dispersion curves for the Case 2 soil profile are shown inFig.6.17. The non-linear inversion algorithm required eight iterations to converge. However,the agreement between the synthetic and the theoretical dispersion curve of the finaliteration is not very satisfactory. As mentioned in the previous section, in irregular soilprofiles the effects of higher modes of propagation can no longer be neglected, andtherefore it is unrealistic to expect good agreement between a synthetic (or experimental)dispersion curve and a theoretical curve containing only the fundamental mode ofpropagation. The numerical simulation shown in Fig.6.17 confirms this hypothesis.

The sequence of shear wave velocity profiles during the iterative inversion process isillustrated in Fig.6.18 where the final profile is again denoted with a bold line. The RMSerror misfit between the synthetic and the simulated dispersion curve as a function of theiteration number is shown in Fig.6.19.


0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0

0

5

1 0

1 5

2 0

2 5

3 0

3 5

S h e a r W a v e V e l o c i t y [ m / s e c ]

Dep

th [m

]I t e r . # 8

I t e r . # 1

Figure 6.18 Shear Wave Velocity Profile from UFUMA Inversion Algorithm for Case 2Soil Profile

0 1 2 3 4 5 6 7 8 90

0 . 5

1

1 . 5

2

2 . 5

3

3 . 5

4

I t e r a t i o n #

Exa

ct R

MS

Mis

fit

Figure 6.19 Convergence of UFUMA Inversion Algorithm for Case 2 Soil Profile

The results of the uncoupled inversion of the synthetic attenuation curve are shown inFig. 6.20 and Fig. 6.21. The agreement between the synthetic and the fundamental mode


attenuation curves is acceptable but not very satisfactory. This observation is consistent withthe results from the inversion of the synthetic dispersion curve shown in Fig.6.17.

3 3 . 2 5 3 . 5 3 . 7 5 4

0

5

1 0

1 5

2 0

2 5

3 0

3 5


Dep

th (

m)

0 20 40 60 80 100 1200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Frequency (Hz)

Atte

nuat

ion

Coe

ffici

ent (

1/m

)

Theoretical

Synthetic


10- 1 0

100

101 0

0 .65

0 .7

0 .75

Smoo th i ng Pa rame te r

rms

Err

or



The last soil profile to be investigated is Case 3 which is a soil deposit characterized bythe presence of a thin, stiff surface layer overlying an otherwise regular shear wave velocityprofile. Figure 6.22 shows the sequence of theoretical dispersion curves obtained from theinversion of the synthetic dispersion curve. For this soil profile, the UFUMA inversion

0 20 40 60 80 100 120200

300

400

500

600

Frequency [Hz]

Pha

se V

eloc

ity [m

/sec

]

I ter . # 1

I ter . # 11

TheoreticalSynthetic

Figure 6.22 Fundamental Mode Theoretical and Synthetic Dispersion Curves forCase 3 Soil Profile

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0

0

5

1 0

1 5

2 0

2 5

3 0

3 5


Dep

th [m

]

I t e r . # 1I t e r . # 1 1

Figure 6.23 Shear Wave Velocity Profile from UFUMA Inversion Algorithm for Case3 Soil Profile


algorithm failed to converge. Figure 6.23 shows the shear wave velocity profiles associatedwith the theoretical dispersion curves of Fig. 6.22. Figure 6.24 shows the RMS error misfitas the number of iterations progresses; it is apparent from the oscillatory behavior of theRMS error that the solution did not converge.

The results obtained for Case 3 soil profile emphasize the limitations of the fundamentalmode approach to the interpretation of surface wave data. In irregular soil deposits wherehigher modes of propagation govern the response of the medium, the inversion of theexperimental dispersion curve may be either inaccurate, as occurred in the Case 2 soilprofile, or it may be unable to converge to a solution, as shown for the Case 3 soil profile.

Better results were obtained from the inversion of the synthetic attenuation curve asillustrated by Fig. 6.25 and Fig. 6.26. A more regular shear damping ratio profile (comparedwith the corresponding shear wave velocity profile) is the main reason for this result. Infact, Fig. 6.11 shows that as a consequence of a smoothly varying DS profile, the modalattenuation curves for Case 3 are closely spaced, and hence it is not surprising that afundamental-mode-based inversion of the equation GDS = αα R yields satisfactory results.

6.3.1.2 UEQMA Inversion Algorithms

In this section the synthetic dispersion and attenuation curves for the Case 1, Case 2 andCase 3 soil profiles are inverted to obtain the shear wave velocity and shear damping ratioprofiles using the UEQMA (Uncoupled, Equivalent-Multi-Mode Analysis) algorithm. Asdescribed in Section 4.3.3, this inversion algorithm accounts for all the Rayleigh modes ofpropagation via the concept of effective phase velocity.

0 2 4 6 8 1 0 1 20

1

2

3

4

I t e r a t i o n #

Exa

ct R

MS

Mis

fit

Figure 6.24 Non-Convergence of UFUMA Inversion Algorithm for Case 3 SoilProfile


Fig.6.27 shows the synthetic and the theoretical dispersion curves for the Case 1 soilprofile. Only five iterations were required for the algorithm to converge. From Fig.6.27 itcan be seen that the agreement between the theoretical effective and the syntheticdispersion curves is excellent for the final iteration.

2 2 . 5 3 3 . 5 4

0

5

1 0

1 5

2 0

2 5

3 0

3 5


Dep

th (

m)

0 20 40 60 80 100 1200

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Frequency (Hz)

Atte

nuat

ion

Coe

ffici

ent (

1/m

)

Theoretical

Synthetic


10-10

10-5

100

105

1010

0.25

0.3

0.35

0.4

Smooth ing Parameter

rms

Err

or



0 2 0 4 0 6 0 8 0 1 0 0 1 2 01 0 0

1 5 0

2 0 0

2 5 0

3 0 0

3 5 0

4 0 0

F r e q u e n c y [ H z ]

Pha

se V

eloc

ity [m

/sec

]

I t e r . # 1

I t e r . # 5


Figure 6.27 Effective Theoretical and Synthetic Dispersion Curves for Case 1 SoilProfile

The sequence of shear wave velocity profiles is shown in Fig.6.28 where the dashed lineis used to denote the starting profile, and the bold line corresponds to the final profile.

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0

0

5

1 0

1 5

2 0

2 5

3 0

3 5


Dep

th [m

]

I t e r . # 1

I t e r . # 5

Figure 6.28 Shear Wave Velocity Profile from UEQMA Inversion Algorithm for Case 1Soil Profile


The RMS error misfit between the theoretical effective and the synthetic dispersioncurves as a function of the iteration number is shown in Fig.6.29. In the UEQMA inversionalgorithm the shear damping ratio profile is obtained from the inversion of experimentalattenuation curve via the equation $GDS = αα R . The matrix $G is formed by the partialderivatives of the effective Rayleigh phase velocity with respect to the shear andcompression wave velocities of the soil layers averaged over the receiver spacings (see Eq.3.58).

0 2 4 6 8 1 0 1 20

0 . 5

1

1 . 5

2

2 . 5

3

I t e r a t i o n #

Exa

ct R

MS

Mis

fit

Figure 6.29 Convergence of UEQMA Inversion Algorithm for Case 1 Soil Profile

2.5 3 3.5 4 4.5 5

0

5

10

15

20

25

30

35

Shear Damping Rat io (%)

Dep

th (

m)

0 20 40 60 80 100 1200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Frequency (Hz)

Atte

nuat

ion

Coe

ffici

ent (

1/m

)

Theoretical

Synthetic

Figure 6.30 Shear Damping Ratio Profile and Theoretical Attenuation Curve fromUEQMA Inversion Algorithm for Case 1 Soil Profile


1 0- 1 0

1 00

1 01 0

0 . 7

0 . 8

0 . 9

1

1 . 1

S m o o t h i n g P a r a m e t e r

rms

Err

or

Figure 6.31 Attenuation Curves RMS Misfit Error using UEQMA Inversion Algorithmfor Case 1 Soil Profile

Figure 6.30 shows the resulting shear damping ratio profile and theoretical attenuationcurve for the Case 1 soil profile. At high frequencies the theoretical attenuation curvebecomes very irregular emulating the synthetic attenuation curve (see also Fig.6.5). The RMSerror misfit of the synthetic and predicted attenuation curve is illustrated in Fig.6.31.

The results for the Case 2 soil profile are presented in Fig.6.32, which shows thesuccession of theoretical effective dispersion curves required for the UEQMA inversionalgorithm to converge. The results obtained are remarkable: the cusped synthetic dispersioncurve is almost perfectly matched by the theoretical curve corresponding to the finaliteration. Figure 6.32 also shows the stability and convergence of the inversion algorithm.

As illustrated in the previous section, the fundamental-mode-based approach was unableto yield a theoretical dispersion curve that matches the cusped synthetic curve well.Conversely, this difficult task has been successfully solved by the effective, multi-modeapproach of the UEQMA algorithm. The sequence of shear wave velocity profilesassociated with the theoretical dispersion curves of Fig. 6.32 is illustrated in Fig. 6.33. Thefinal profile is denoted using a bold line.

The convergence of the algorithm in terms of the RMS error misfit between theoreticaland synthetic dispersion curves is shown in Fig. 6.34. Ten iterations were required for thealgorithm to converge.


0 20 40 60 80 100 120200

300

400

500

600

F requency [Hz]

Pha

se V

eloc

ity [m

/sec

]

I t e r . # 1

I t e r . # 1 0

Theoret ica lS ynthet ic


0 2 0 0 4 0 0 6 0 0 8 0 0

0

5

1 0

1 5

2 0

2 5

3 0

3 5


Dep

th [m

]

I t e r . # 1

I t e r . # 1 0



0 2 4 6 8 1 0 1 20

0 . 5

1

1 . 5

2

2 . 5

3

I t e r a t i o n #

Exa

ct R

MS

Mis

fit


Figures 6.35 and 6.36 summarize the results obtained from the inversion of the syntheticattenuation curve. The agreement between the theoretical and the synthetic attenuationcurves is satisfactory.

3 3 . 2 3 . 4 3 . 6 3 . 8 4

0

5

1 0

1 5

2 0

2 5

3 0

3 5


Dep

th (

m)

0 20 40 60 80 100 1200

0.01

0.02

0.03

0.04

0.05

0.06

Frequency (Hz)

Atte

nuat

ion

Coe

ffici

ent (

1/m

)

Theoretical

Synthetic



10- 1 0

100

101 0

0.61

0 .62

0 .63

0 .64

0 .65

0 .66

Smoo th ing Pa ramete r

rms

Err

or


Figure 6.37 illustrates the sequence of theoretical dispersion curves successfullyconverging to the synthetic dispersion curve for the Case 3 soil profile. Figure 6.37 showsthat the performance of the inversion algorithm UEQMA for the Case 3 soil profile is alsoremarkable. It should be recalled that the UFUMA algorithm failed to converge for Case 3.With the UEQMA algorithm, convergence is achieved after only six iterations.

0 20 40 60 80 100 120200

300

400

500

600

700

Frequency [Hz ]

Pha

se V

eloc

ity [m

/sec

]

I t e r . # 6

I t e r . # 1

Theoret ica lSynthet ic



Figure 6.38 illustrates the sequence of shear wave velocity profiles corresponding to thesix iterations. The RMS error misfit between the theoretical and the synthetic dispersioncurves as a function of the iteration number is shown in Fig. 6.39. Finally, the results of thelinear inversion of the synthetic attenuation curve are presented in Fig.6.40 and Fig.6.41.

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0

0

5

1 0

1 5

2 0

2 5

3 0

3 5


Dep

th [m

]

I t e r . # 1

I t e r . # 6


0 1 2 3 4 5 6 70

2

4

6

8

I t e r a t i o n #

Exa

ct R

MS

Mis

fit



2 2.5 3 3.5

0

5

10

15

20

25

30

35

Shear Damping Rat io (%)

Dep

th (

m)

0 20 40 60 80 100 1200

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Frequency (Hz)

Atte

nuat

ion

Coe

ffici

ent (

1/m

)

Theoretical

Synthetic


1 0- 1 0

1 00

1 01 0

0 . 3 5

0 . 4

0 . 4 5

0 . 5

0 . 5 5

0 . 6

S m o o t h in g P a ra m e t e r

rms

Err

or



6.3.2 Coupled Analyses

6.3.2.1 CFUMA Inversion Algorithms

In the previous two sections, the inversion of the synthetic dispersion and attenuationcurves was performed independently using the fundamental mode (Section 6.3.1.1) or themulti-mode approach (Section 6.3.1.2). However, in this and the next sections, thedispersion and attenuation curves are inverted simultaneously. In particular, this section willillustrate the results of the coupled inversion of these curves for Case 1, Case 2 and Case 3soil profiles using the CFUMA (Coupled, Fundamental Mode Analysis) algorithm.

Figure 6.42 shows the sequence of fundamental mode dispersion and attenuation curvesfor Case 1 soil profile. The agreement between the theoretical dispersion curvecorresponding to the sixth and final iteration and the synthetic dispersion curve is good forfrequencies greater than 20 Hz. However, at frequencies lower than 20 Hz the inversionalgorithm was unable to obtain a satisfactory match between these two curves.

0 20 40 60 80 100 120100

200

300

400

500

600

Frequency [Hz]

Pha

se V

eloc

ity [m

/sec

]

Iter. # 1

Iter. # 6

0 20 40 60 80 100 1200

0.05

0.1

0.15

0.2

Frequency [Hz]

Atte

nuat

ion

Coe

ffici

ent [

1/m

]

Iter. # 1

Iter. # 6

Theoretical

Synthetic

Theoretical

Synthetic

Figure 6.42 Fundamental Mode Theoretical Dispersion and Attenuation Curves for Case1 Soil Profile


The agreement between the synthetic and theoretical attenuation curves for the finaliteration is good at all frequencies. Figure 6.43 shows the sequence of shear wave velocityand shear damping ratio profiles corresponding to the dispersion and attenuation curves ofFig. 6.42. The dashed and the bold lines denote the initial and final profiles, respectively.

0 200 400 600 800

0

5

10

15

20

25

30

35

Shear Wave Velocity [m/sec]

Dep

th [m

]

Iter. # 1

Iter. # 6

0 2 4 6

0

5

10

15

20

25

30

35

Shear Damping Ratio [%]

Dep

th [m

]

Iter. # 1

Iter. # 6

Figure 6.43 Shear Wave Velocity and Shear Damping Ratio Profile from CFUMAInversion Algorithm for Case 1 Soil Profile

0 2 4 6 8 1 0 1 20

2

4

6

8

1 0

1 2

I t e r a t i o n #

Exa

ct R

MS

Mis

fit

Figure 6.44 Convergence of CFUMA Inversion Algorithm for Case 1 Soil Profile


A measure of the goodness of the simultaneous inversion is provided also in this case bythe RMS error misfit between the complex valued theoretical and synthetic dispersioncurves. Figure 6.44 shows the RMS error as a function of the iteration number for Case 1soil profile. The inversion algorithm required six iterations to converge.

The results of the simultaneous inversion of the synthetic dispersion and attenuationcurves for Case 2 are presented in Figs. 6.45 through 6.47. In particular, Fig. 6.45 shows thetheoretical and synthetic dispersion and attenuation curves for this soil profile. Theagreement between the synthetic dispersion and attenuation curves and the correspondingtheoretical curves of the sixth and last iteration is satisfactory. By comparing Fig. 6.45 withFigs.6.17 and 6.20, it appears that for the fundamental mode-based algorithms, thesimultaneous inversion furnishes more accurate results than the uncoupled inversion for thissoil profile. It should also be noted that the simultaneous inversion required a fewer numberof iterations to converge than the corresponding uncoupled inversion (six vs. eight).

0 20 40 60 80 100 120100

200

300

400

500

Frequency [Hz]

Pha

se V

eloc

ity [m

/sec

]

Iter. # 1

Iter. # 6


0 20 40 60 80 100 1200

0.05

0.1

Frequency [Hz]

Atte

nuat

ion

Coe

ffici

ent [

1/m

]

Iter. # 1

Iter. # 6




Figure 6.46 shows the sequence of shear wave velocity and shear damping ratio obtainedfrom the simultaneous inversion as the number of iteration progresses, and finally Fig. 6.47illustrates the convergence of the algorithm in terms of the RMS error misfit.

0 200 400 600 800

0

5

10

15

20

25

30

35


Dep

th [m

]

Iter. # 1

Iter. # 6

0 2 4 6

0

5

10

15

20

25

30

35


Dep

th [m

]

Iter. # 1

Iter. # 6


0 2 4 6 8 1 00

1

2

3

4

I t e r a t i o n #

Exa

ct R

MS

Mis

fit



Theoretical and synthetic dispersion and attenuation curves for Case 3 are shown in Fig.6.48. The agreement between the synthetic and theoretical curves for the last iteration issatisfactory. Recall that the fundamental mode, uncoupled inversion of the syntheticdispersion curve had failed to converge.

0 20 40 60 80 100 120300

400

500

600

Frequency [Hz]

Pha

se V

eloc

ity [m

/sec

]

Iter. # 1

Iter. # 8


0 20 40 60 80 100 1200

0.05

0.1

Frequency [Hz]

Atte

nuat

ion

Coe

ffici

ent [

1/m

]

Iter. # 1Iter. # 8



The sequence of shear wave velocity and shear damping ratio profiles for Case 3 isillustrated in Fig. 6.49. Finally, the RMS error misfit between the synthetic dispersion andattenuation curves is shown in Fig.6.50. The inversion algorithm required eight iterations toconverge.

6.3.2.2 CEQMA Inversion Algorithms

The CEQMA inversion algorithm combines the features of the simultaneous inversionwith those of the multi-mode propagation via the concept of effective velocity. This is themost sophisticated of the inversion algorithms developed in this study. Unfortunately, theresults obtained thus far using this technique indicate only moderate success. It is believedthat most of the problems exhibited by the algorithm are due to an inability of the


algorithm to determine, at certain frequencies, the correct sequence of Rayleigh modes(particularly the higher modes).

200 400 600 800

0

5

10

15

20

25

30

35


Dep

th [m

]

Iter. # 8

Iter. # 1

0 2 4 6

0

5

10

15

20

25

30

35


Dep

th [m

]

Iter. # 8

Iter. # 1


0 2 4 6 8 1 0 1 20

1

2

3

4

I t e r a t i o n #

Exa

ct R

MS

Mis

fit



Because the strategy of the CEQMA inversion algorithm is based on matching thecomplex-valued effective phase velocity with the complex-valued synthetic dispersion curve,an error in computing the correct sequence of modes (even at one frequency) will ultimatelycause the effective phase velocity to be incorrect. Figures 6.51 through 6.53 shows theresults for Case 1 obtained from the application of the CEQMA algorithm.

0 20 40 60 80 100 120100

200

300

400

500

600

Frequency [Hz]

Pha

se V

eloc

ity [m

/sec

]

Iter. # 1

Iter. # 6

0 20 40 60 80 100 1200

0.05

0.1

0.15

0.2

Frequency [Hz]Atte

nuat

ion

Coe

ffici

ent [

1/m

]

Iter. # 1

Iter. # 6

Theoretical

Synthetic

Theoretical

Synthetic

Figure 6.51 Effective Theoretical Dispersion and Attenuation Curves for Case 1 SoilProfile

The instability of the algorithm is indicated by irregular pattern of the effectivedispersion and attenuation curves at several iterations. The sequence of shear wave velocityand shear damping ratio profiles is illustrated in Fig. 6.52. Finally, Fig. 6.53 shows the RMSerror misfit.

The results of the simultaneous inversion of surface wave data for Case 2 soil profile areshown in Figs.6.54 through 6.56. For this soil profile, the theoretical dispersion andattenuation curves also exhibit an irregular pattern, particularly in the case of the attenuationcurves.


0 200 400 600

0

5

10

15

20

25

30

35


Dep

th [m

]

Iter. # 1

Iter. # 6

0 2 4 6

0

5

10

15

20

25

30

35

Shear Damping Ratio [%]D

epth

[m]

Iter. # 1

Iter. # 6

Figure 6.52 Shear Wave Velocity and Shear Damping Ratio Profile from CEQMAInversion Algorithm for Case 1 Soil Profile

0 2 4 6 8 10 120

2

4

6

8

10

12

Iterat ion #

Exa

ct R

MS

Mis

fit

Figure 6.53 RMS Error Misfit of CEQMA Inversion Algorithm for Case 1 Soil Profile


0 20 40 60 80 100 120200

300

400

500

600

Frequency [Hz]

Pha

se V

eloc

ity [m

/sec

]Iter. # 7

Iter. # 1

0 20 40 60 80 100 1200

0.05

0.1

Frequency [Hz]Atte

nuat

ion

Coe

ffici

ent [

1/m

]

Iter. # 1

Iter. # 7

Theoretical

Synthetic

Theoretical

Synthetic


200 400 600

0

5

10

15

20

25

30

35


Dep

th [m

]

Iter. # 1

Iter. # 7

0 2 4 6 8

0

5

10

15

20

25

30

35


Dep

th [m

]

Iter. # 1

Iter. # 7



0 2 4 6 8 1 0 1 20

1

2

3

4

5

I t e r a t i o n #

Exa

ct R

MS

Mis

fit


The final results are those for the Case 3 soil profile that are presented in Figs. 6.57through 6.59. They are characterized by the same irregular behavior of the previous two soilprofiles, which is also reflected by the oscillatory behavior of the RMS error misfit.

0 20 40 60 80 100 120100

200

300

400

500

600

Frequency [Hz]

Pha

se V

eloc

ity [m

/sec

]

Iter. # 1

Iter. # 6

Theoretical

Synthetic

0 20 40 60 80 100 1200

0.05

0.1

Frequency [Hz]

Atte

nuat

ion

Coe

ffici

ent [

1/m

]

Iter. # 1Iter. # 6

Theoretical

Synthetic



0 200 400 600 800

0

5

10

15

20

25

30

35

Shear W ave Velocity [m/sec]

Dep

th [m

]

I t e r . # 1

I t e r . # 6

0 2 4 6

0

5

10

15

20

25

30

35

Shear Damping Rat io [%]

Dep

th [m

]I t e r . # 1

I t e r . # 6


0 2 4 6 8 1 0 1 20

1

2

3

4

I t e r a t i o n #

Exa

ct R

MS

Mis

fit



6.3.3 Results and Discussion

This section will summarize and discuss the results obtained from the inversion of thesynthetic surface wave data for the Case 1, Case 2, and Case 3 soil profiles. One should notexpect, even in a numerical simulation, to obtain shear wave velocity and shear dampingratio profiles matching the original stratigraphy exactly. This is because the numericalsimulation was performed using the same data reduction procedures employed inexperimental SASW measurements. Therefore, the inversion of synthetic dispersion andattenuation curves will inevitably suffer the limitations that are inherent in the SASWmethodology, particularly the problem of non-uniqueness (see Chapter 4).

Nevertheless, the four algorithms UFUMA, UEQMA, CFUMA, and CEQMA are basedon different inversion strategies and it is instructive to compare and evaluate theirperformance. The results of the algorithm CEQMA have not been compared with those ofthe other algorithms because, based on the discussion in the previous section, they areconsidered unreliable. Figures 6.60 and 6.61 show the shear wave velocity and sheardamping ratio profiles corresponding to Case 1.

The RMS error misfit between the original and the predicted VS and DS profiles, which is

defined by L2

predS

origS nVV − and L2

predS

origS nDD − respectively (nL is the number

of layers), was also computed. The results of this calculation for the four algorithms arereported in Table 6.6.

100 200 300 400 500 600

0

5

10

15

20

25

30

35

Shear W ave Veloc i ty [m /sec]

Dep

th [m

]

U F U M A

U E Q M A

C F U M A

O R IG I N A L

Figure 6.60 Inverted Shear Wave Velocity Profiles for Case 1 Soil Stratigraphy


0 1 2 3 4 5 6

0

5

10

15

20

25

30

35

Shear Damp ing Ra t i o [% ]

Dep

th [m

]U F U M A

U E Q M A

C F U M A

O R IG I N A L

Figure 6.61 Inverted Shear Damping Ratio Profiles for Case 1 Soil Stratigraphy

Table 6.6 Inversion Algorithms RMS Error Misfit for Case 1 Soil Profile

UFUMA UEQMA CFUMA

VS 57.1 37.9 46.8

DS 0.66 1.01 0.45

From this table it appears that the algorithms providing the most accurate predictions ofthe original profile are the uncoupled, multi-mode analysis (UEQMA) and coupled,fundamental mode inversion (CFUMA). In terms of the shear wave velocity profile, theUEQMA algorithm is the most accurate (i.e., the lowest RMS error misfit); the uncoupledfundamental mode-based UFUMA algorithm yielded the worst prediction. In terms of theshear damping ratio profile the lowest RMS error was attained by the coupled inversionwith the CFUMA algorithm.


For the Case 2 soil profile, the results of the inversions are summarized by Figs. 6.62 and6.63 and by Table 6.7. Case 2 is characterized by an irregular stratigraphy where a soft layer istrapped between two harder layers. As mentioned several times in this chapter, such layeredmedia are inversely dispersive and their response includes contributions from higher modesof propagation. This feature is reflected in the performance of the inversion algorithms.

200 300 400 500 600

0

5

10

15

20

25

30

35

Shear W ave Veloci ty [m /sec]

Dep

th [m

]

U F U M A

U E Q M A

C F U M A

O R I G I N A L


0 2 4 6 8 10

0

5

10

15

20

25

30

35


Dep

th [m

]

UFUMA

U E Q M A

CFUMA

O R I G I N A L




UFUMA UEQMA CFUMA

VS 18.5 14.0 38.3

DS 0.72 0.62 1.72

The lowest RMS error misfit for both shear wave velocity and shear damping ratioprofiles was achieved by the effective, multi-mode analysis of the UEQMA algorithm. Theresults obtained by the fundamental mode based algorithms UFUMA and CFUMA were inboth cases less accurate, particularly those associated with the algorithm CFUMA. Thisresult confirms the inadequacy of the fundamental mode approach for irregular media, andreinforces the need for an approach based on multi-mode wave propagation such as theeffective Rayleigh phase velocity.

Finally, Figs. 6.64 and 6.65 and Table 6.8 illustrate the results for the Case 3 soil profile.For this irregular stratigraphy characterized by the presence of a thin and stiff surface crust,the analyses based on the effective Rayleigh phase velocity again provided the best overallresults. The multi-mode UEQMA inversion algorithm exhibited the best performance interms of RMS error misfit for the prediction of the shear wave velocity profile. Theperformance of the uncoupled fundamental mode algorithm UFUMA was the leastaccurate. In terms of the prediction of shear damping ratio profile, both the UEQMA andUFUMA algorithms performed well, whereas CFUMA algorithm exhibited a higher RMSerror.

Table 6.9 summarizes the performance of the algorithms UFUMA, CFUMA, andUEQMA in this numerical simulation. Even though more numerical simulations arerequired for a definitive validation of these algorithms, the results obtained in this sectionare important, and they lead to the following observations. Overall, the multi-mode-basedapproach of the algorithm UEQMA is resulted to be the most accurate inversion algorithm,particularly for determining the shear wave velocity profile. The fundamental mode-basedalgorithms UFUMA and CFUMA are generally less accurate than the algorithm UEQMA inpredicting both the shear wave velocity and the shear damping ratio profiles. There arehowever two exceptions to this general rule. The first exception is the prediction of theshear damping ratio for Case 1 soil profile. Here the algorithm CFUMA has yielded the mostaccurate results, whereas the algorithm UEQMA yielded the highest RMS error misfit. The


second exception is the prediction of shear damping ratio for Case 3 soil profile where bothUFUMA and UEQMA algorithms yielded approximately the same RMS error.

300 400 500 600 700

0

5

10

15

20

25

30

35

Shear W ave Velocity [m/sec]

Dep

th [m

]U F U M A

U E Q M A

C F U M A

O R I G I N A L


0 1 2 3 4 5 6 7

0

5

10

15

20

25

30

35

Shear Damping Rat io [%]

Dep

th [m

]

U F U M A

U E Q M A

C F U M A

O R I G I N A L




UFUMA UEQMA CFUMA

VS 47.3 5.6 18.9

DS 0.51 0.54 1.03

Table 6.9 Inversion Algorithms Performance in Terms of RMS Error Misfit

VS DS

Min [RMS] Max [RMS] Min [RMS] Max [RMS]

Case 1 UEQMA UFUMA CFUMA UEQMA

Case 2 UEQMA CFUMA UEQMA CFUMA

Case 3 UEQMA UFUMA UFUMA/

UEQMACFUMA

It should be remarked that both exceptions occurred in the prediction of the shear-damping ratio DS . This is not surprising since a) the attenuation properties of a dissipativemedium affect the overall response of a medium to a lesser extent than stiffness, and b) in theboundary value problem considered in this study (i.e. a harmonic point load over the surfaceof a vertically heterogeneous viscoelastic half-space) material attenuation is coupled withgeometric attenuation. The combination of the effects of a) and b) makes the computationof the Rayleigh attenuation coefficients and therefore, the shear damping ratio, verysensitive to numerical noise.


Although the numerical simulation presented in this study is not statistically significantand should be substantiated with other simulations using different profiles and materialparameters, it is still possible to draw two general conclusions. The first and most importantis that a multi-mode inversion analysis yields more accurate results than a fundamental modeanalysis. This is particularly true for irregular media (i.e. Case 2 and Case 3), and forpredictions of the shear wave velocity structure.

The second conclusion is that in regular soil deposits (i.e. Case 1) a fundamental modeanalysis may yield sufficiently accurate results for both shear wave velocity and sheardamping ratio determination. However, the results obtained from the numerical simulationseem also to indicate that the coupled inversion (more precisely, the strongly coupledinversion), where the shear wave velocity and shear damping ratio are determinedsimultaneously (CFUMA), is more accurate than the corresponding uncoupled analysis(UFUMA).

207

7 EXPERIMENTAL RESULTS

7.1 Overview

The algorithms developed in this research study for the interpretation of surface wavemeasurements will now be applied to experimental data. Rayleigh phase velocity andattenuation measurements were performed at the Treasure Island National GeotechnicalExperimentation Site (California, USA), where independent in situ and laboratorymeasurements of shear wave velocity and shear damping ratio are available for comparison.

After a brief description of the geotechnical characteristics of the site, the followingsections will illustrate the results of both the coupled and uncoupled inversion of theexperimental dispersion and attenuation curves. The chapter ends with a comparison ofthese results with those obtained from independent cross-hole and laboratorymeasurements performed at the site.

7.2 Treasure Island Naval Station Site

Treasure Island is a man-made island constructed of hydraulic fill soils in the easternportion of San Francisco Bay. The area investigated for this study is along the westernproperty margin of the National Geotechnical Experimentation Site (NGES) (see Figure7.1). Extensive geotechnical characterization of the site was performed for the EPRI (1993)

Parking

Storage

Fire StationBuilding 157

Current Test Area

B1

B2

B3

B4 B5

EPRI Soil Borings

N

Note: Not to scale

Figure 7.1 Treasure Island National Geotechnical Experimentation Site (After Spang, 1995)

208 Experimental Results

study and included soil borings, penetration tests, and in situ seismic tests. The approximatelocations of the borings used for cross-hole tests in the EPRI study are shown in Figure 7.1.

The soil conditions consist of approximately 10 m of loose, fine to medium sandunderlain by 12 to 18 m of soft clay (Bay Mud). Beneath the Bay Mud are stiffer soils (OlderBay Mud) which are underlain at depths on the order of 80 m by bedrock. The water tableis approximately 1.5 m below the ground surface. A generalized soil profile of the TreasureIsland site is presented in Figure 7.2. The cone penetration (CPT) and standard penetration(SPT) test data shown in Figure 7.2 were obtained from the EPRI study.

0 10 20

Cone Tip Resistance (MPa)

0

3

6

9

12

0 5 10 15

SPT N ValueSoil Profile

Gravelly Sand

Fine

To

Medium

Sand

(Bay Mud)

Soft Clay

Dep

th (

m)

Figure 7.2 Soil Profile and Properties at the Treasure Island NGES (After Spang, 1995)

Surface wave tests were performed at Treasure Island NGES to determine theexperimental dispersion and attenuation curves at the site. Rayleigh waves were generated bya vertically oscillating, electrodynamic force generator operating in swept-sine mode. Thedynamic force provided by the shaker is frequency-dependent, with the maximum dynamic

Experimental Results 209

force provided at lower frequencies. Frequencies used in field tests ranged from 5 to 100Hz, so that the dynamic force supplied by the electromechanical ranged from about 90 to500 N. Rayleigh waves were recorded by vertical transducers (geophones) having a naturalfrequency of 1 Hz at various offsets (r) from the source.

The geophones were spaced at 1.5-m intervals up to an offset of 9 m, at 3-m intervalsfrom 9 to 30 m, and at 6-m intervals from 30 m to the maximum offset of 60 m. Tenparticle velocity spectra were averaged in the frequency domain at each geophone offset.The acceleration of the shaker mass was measured with a piezoelectric accelerometer toensure that the source was repeatable throughout the test. Particle velocity spectra werecorrected in order to mitigate the effects of ambient noise on attenuation measurements.

The experimental dispersion and attenuation curves were obtained using theconventional techniques described in Section 5.2. In particular, the Rayleigh phase velocitywas determined using the so-called two-station method, whereas the Rayleigh attenuationcoefficients were calculated with the multi-station method and using a geometric spreadingfunction accounting for multi-mode Rayleigh wave propagation.

7.3 Uncoupled Inversion

This section will present the results of the uncoupled inversion of the experimentaldispersion and attenuation curves obtained at Treasure Island NGES. The stratigraphy usedfor the inversion analyses was selected based on the interpretation of the geotechnical datashown in Fig.7.2.

In this section, the results associated with the fundamental mode analysis will beillustrated first (i.e. the UFUMA algorithm), followed by those based on the equivalentmulti-mode analysis (i.e. the UEQMA algorithm). Figure 7.3 shows the sequence ofdispersion curves required for the UFUMA algorithm to converge.

A homogeneous profile with s/m60VS = was used as the initial estimate of the layershear wave velocities. Five iterations were required for the algorithm to converge. Figure 7.4shows the corresponding sequence of shear wave velocity profiles obtained from thefundamental-mode-based inversion. The dashed line indicates the initial profile whereas thebold line shows the final shear wave velocity profile. The Root-Mean-Square (RMS) errorbetween the experimental and the theoretical dispersion curve as function of the iterationnumber is illustrated in Fig.7.5.


0 10 20 30 40 50 60 70 800

50

100

150

200

250

Frequency [Hz]

Pha

se V

eloc

ity [m

/sec

]

I ter. # 1

Iter. # 5

Theoretical

Experimental

Figure 7.3 Fundamental Mode Theoretical and Experimental Dispersion Curves atTreasure Island NGES

0 50 100 150 200 250

0

5

10

15


Dep

th [m

]

Iter. # 1

Iter. # 5

Figure 7.4 Shear Wave Velocity Profile from UFUMA Inversion Algorithm at TreasureIsland NGES


0 2 4 6 8 100

1

2

3

4

5

6

7

8

Iteration #

Exa

ct R

MS

Mis

fit

Figure 7.5 Convergence of UFUMA Inversion Algorithm at Treasure Island NGES

The results of the fundamental mode uncoupled inversion of the experimentalattenuation curve are presented in Figs. 7.6 and Fig.7.7. In particular, Fig. 7.6 illustrates theshear damping ratio profile, and the theoretical attenuation curve obtained from thesolution of the forward problem GDS = αα R .

The RMS error between the theoretical and the experimental attenuation curves as afunction of the smoothing parameter is shown in Fig.7.7. The value selected for thesmoothing parameter is the smallest value that yields a solution vector composed of non-negative shear damping ratios. For this site the adopted value was 4108.2 ⋅ which resulted ina RMS error of 1.38.

The results of the uncoupled equivalent multi-mode analysis conducted with thealgorithm UEQMA will now be illustrated. Figure 7.8 shows the experimental dispersioncurve and the succession of theoretical dispersion curves as the iterations progress. Thesame homogeneous profile with s/m60VS = was assumed as the initial estimate of theshear wave velocities.

Figure 7.9 shows the sequence of shear wave velocity profiles obtained from the non-linear inversion. The bold line indicates the final shear wave velocity profile. The algorithmrequired five iterations to converge. The RMS error misfit between the experimental andthe theoretical dispersion curves as a function of the iteration number is shown in Fig.7.10.


0

5

10

15

0.0 1.0 2.0 3.0Shear Damping Ratio [%]

Dep

th [m

]

0.00

0.01

0.02

0.03

0.04

0.05

0 20 40 60 80

Frequency [Hz]

Att

enua

tion

Coe

ffic

ient

[1/m

]

Experimental

Theoretical

Figure 7.6 Shear Damping Ratio Profile and Theoretical Attenuation Curve fromUFUMA Inversion Algorithm at Treasure Island NGES

1.38

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

1.E-06 1.E-03 1.E+00 1.E+03 1.E+06 1.E+09 1.E+12

Smoothing Parameter

Rm

s E

rror

Figure 7.7 Attenuation Curves RMS Error using UFUMA Inversion Algorithm atTreasure Island NGES


0 10 20 30 40 50 60 70 800

50

100

150

200

250

Frequency [Hz]

Pha

se V

eloc

ity [m

/sec

]

I ter . # 1

I ter . # 5

Theoretical

Experimental

Figure 7.8 Effective Theoretical and Experimental Dispersion Curves at TreasureIsland NGES

0 50 100 150 200 250

0

5

10

15


Dep

th [m

]

Iter. # 5

Iter. # 1

Figure 7.9 Shear Wave Velocity Profile from UEQMA Inversion Algorithm at TreasureIsland NGES


0 2 4 6 8 1 00

1

2

3

4

5

6

7

8

I t e ra t i on #

Exa

ct R

MS

Mis

fit

Figure 7.10 Convergence of UEQMA Inversion Algorithm at Treasure Island NGES

Figure 7.11 and 7.12 summarize the results obtained from the linear inversion of theexperimental attenuation curve. For the selected smoothing parameter, the RMS errorbetween the theoretical and the experimental attenuation curves is equal to 3.41.

0 1 2 3

0

5

10

15


Dep

th [m

]

0 20 40 60 800

0.01

0.02

0.03

0.04

0.05

Frequency (Hz)

Atte

nuat

ion

Coe

ffici

ent (

1/m

)

Experimental

Theoretical

Figure 7.11 Shear Damping Ratio Profile and Theoretical Attenuation Curve fromUEQMA Inversion Algorithm at Treasure Island NGES


10-10

10-5

100

105

1010

2

2.5

3

3.5

4

Smoothing Parameter

rms

Err

or

3.41

Figure 7.12 Attenuation Curves RMS Misfit Error using UEQMA Inversion Algorithmat Treasure Island NGES

0 20 40 60 800

0.02

0.04

0.06

0.08

0.1

Frequency [Hz]

Atte

nuat

ion

Coe

ffici

ent [

1/m

]

I ter . # 1

I ter . # 6

Theoret ical

Exper imental

0 20 40 60 800

50

1 0 0

1 5 0

2 0 0

2 5 0

Frequency [Hz]

Pha

se V

eloc

ity [m

/sec

]

I ter . # 6

I ter . # 1

Theoret ical

Exper imental

Figure 7.13 Fundamental Mode Theoretical and Experimental Dispersion andAttenuation Curves at Treasure Island NGES


7.4 Coupled Inversion

This section will illustrate the results of the coupled inversion of the dispersion andattenuation curves measured at Treasure Island NGES. They were obtained using thefundamental-mode-based analysis of the CFUMA algorithm.

Figure 7.13 illustrates the convergence of the inversion algorithm in terms of dispersionand attenuation curves. The theoretical curves corresponding to the sixth and final iterationare in reasonable agreement with the experimental curves.

The corresponding sequences of shear wave velocity and shear damping ratio profilesare shown in Fig.7.14 where the dashed lines indicate the starting models used in theinversion. The final shear wave velocity and shear damping ratio profiles are shown usingbold lines.

0 50 100 150 200 250

0

5

10

15


Dep

th [m

]

Iter. # 6

Iter. # 1

0 2 4 6

0

5

10

15


Dep

th [m

]

Iter. # 6

Iter. # 1

Figure 7.14 Shear Wave Velocity and Shear Damping Ratio Profile from CFUMAInversion Algorithm at Treasure Island NGES

Finally Fig.7.15 shows the convergence of the algorithm in terms of the RMS errorbetween the experimental and theoretical complex phase velocities; the convergence of thealgorithm after six iterations is apparent.


0 2 4 6 8 1 00

1

2

3

4

5

I te ra t ion Num b e r

Exa

ct R

MS

Mis

fit

Figure 7.15 Convergence of CFUMA Inversion Algorithm at Treasure Island NGES

7.5 Results and Discussion

The results obtained from the uncoupled and coupled inversion of the experimentaldispersion and attenuation curves illustrated in Section 7.3 and Section 7.4 are nowcompared with independent in-situ and laboratory measurements of shear wave velocity andshear damping ratio. The cross-hole shear wave velocity data presented in Fig.7.16 are fromfive different source-to-receiver travel paths between the boreholes shown in Fig. 7.1.

The in situ shear damping ratios shown in Fig.7.17 were obtained from analysis of cross-hole seismic data utilizing a seismic waveform matching technique (Tang, 1992). Theyrepresent the mean values and the associated uncertainties measured between three differentpairs of the boreholes shown in Fig.7.1.

Figures 7.16 and 7.17 show a comparison between shear wave velocity and sheardamping ratio profiles determined from the inversion of surface wave data, and the valuesof these parameters obtained from independent in situ and laboratory measurements.

The values of shear wave velocity obtained from the interpretation of surface wave testcompare reasonably well with cross-hole measurements. Because Treasure Island NGES is arelatively homogeneous site, there are small differences between the shear wave velocityprofiles predicted by the algorithms UFUMA, UEQMA, and CFUMA. As mentionedseveral times in this dissertation, in regular soil profiles the response of the medium ismainly controlled by the fundamental mode of propagation. It is therefore natural to expectvery similar results from the UFUMA, UEQMA, and CFUMA algorithms. From Fig.7.16


the increasing lack of resolution with depth that is inherent to surface wave tests should alsobe noted, particularly when using relatively small sources. This limitation can be partiallyovercome by using more powerful sources capable of generating lower frequencies (< 5Hz).

0

3

6

9

12

15

100 125 150 175 200


Dep

th [m

]

CFUMA

UFUMA

UEQMA

Cross-Hole

Figure 7.16 Comparison at Treasure Island NGES of Shear Wave Velocity from SurfaceWave Test Results with Other Independent Measurements


0

3

6

9

12

15

0% 2% 4% 6% 8% 10%

Shear Damping Ratio

Dep

th [m

]

CFUMA

UFUMA

UEQMA

Cross-Hole

Resonant C.

Torsional S.

Figure 7.17 Comparison at Treasure Island NGES of Shear Damping Ratio fromSurface Wave Test Results with Other Independent Measurements

Concerning the predictions of shear damping ratios, Fig.7.17 shows that the valuesobtained from the interpretation of surface wave tests are generally less than those obtainedfrom cross-hole measurements. These differences can be attributed to three possible causes:

1. Frequencies used in cross-hole tests are usually on the order of several hundred Hertz ormore. The shorter wavelengths associated with these higher frequencies are moresusceptible to apparent attenuation due to scattering and other phenomena than are thelong wavelengths associated with surface wave testing at lower frequencies (i.e., 5 to 100Hz).


2. At higher frequencies involved in cross-hole tests, soil damping may be strongly affectedby fluid flow losses in addition to frictional losses as described earlier. These fluid lossesmay cause the damping to increase and become frequency dependent at higherfrequencies.

3. There are substantial differences in the volume of soil “sampled” by the two methods.The borehole spacing in the cross-hole measurements was approximately 3 m while thesurface wave measurements were performed with geophone offsets as large as 60 m.Thus, the cross-hole measurements yield localized attenuation properties, and thesurface wave measurements yield properties which are averaged over a much largervolume of soil. The extent to which the measurements differ will depend on theheterogeneity of the site.

Laboratory-measured shear damping ratios obtained from resonant column andtorsional shear tests are also available for comparison at the Treasure Island NGES site.Laboratory values of shear damping were measured on “undisturbed” soil specimens atdepths of 5.3 and 9.1 m from the borings drilled for the cross-hole tests. The laboratory-measured values of shear damping are also shown in Fig.7.17. At 5.3 m, the agreementbetween the laboratory and surface wave values of shear damping ratio is excellent. At 9.1m, the laboratory values are slightly greater than those determined from surface wave tests.For the resonant column tests, the damping ratios were measured at frequencies ofapproximately 40 to 50 Hz. The torsional shear tests were performed at 0.5 Hz.

221

8 CONCLUSIONS AND RECOMMENDATIONS

8.1 Conclusions

Surface wave tests are non-invasive field techniques that can be used to determine thelow-strain dynamic properties of soil deposits. In this dissertation a different approach tothe conventional interpretation of surface waves measurements has been presented. Thenew approach is developed around three fundamental ideas.

First, the definition of the so-called dynamic properties of soils is revisited within theframework of a consistent theory of mechanical behavior. Often in the geotechnicalliterature, the terms stiffness and damping ratio are used, in a loose sense, as if they wereintrinsic properties of soils. In reality, the definition of these terms is inherently linked to anassumed constitutive model to which these behavioral properties are ascribed. Differentidealizations of material behavior will result in different types of behavioral properties.Accordingly, although it is customary in geotechnical engineering to take for granted thesignificance of terms like stiffness and damping ratio, it is important to reexamine thedefinitions of these important mechanical parameters in the context of a consistent theoryof material behavior.

In Chapter 2 the definition of the low-strain dynamic properties of soils is formulatedwithin the framework of the linear theory of viscoelasticity. Experimental evidencedemonstrates that soils subjected to dynamic excitations at strain levels below the linear cyclicthreshold strain exhibit the ability to both store and to dissipate strain energy over a finiteperiod of time. At these low-strain levels, phenomena of instantaneous energy dissipationare usually negligible. A measure of the energy dissipated within the soil mass during theprocess of deformation is the internal entropy density production, which from experimentalobservations appears to be a rate-independent quantity within the seismic frequency band.These features of low-strain dynamic behavior of soils can accurately be modeled by thetheory of linear viscoelasticity.

The limitations and inconsistencies of two constitutive models commonly used ingeotechnical earthquake engineering and soil dynamics, the viscous Kelvin-Voigt model andthe rate-independent Kelvin-Voigt model, were noted. Both models fail to reproducefundamental features of soil behavior such as the instantaneous elastic response exhibited bya soil specimen subjected to a suddenly applied stress and the important phenomenon ofstress relaxation. Furthermore, it was pointed out that the rate-independent Kelvin-Voigtmodel violates both the time-translation invariance hypothesis of linear viscoelasticity, andwhen applied to wave propagation problems, the most elementary principles of causality.

222 Conclusions and Recommendations

A consistent viscoelastic constitutive model requires the definition of a set of materialresponse functions that can be specified in either the time or the frequency domain. In thiscontext an important distinction has been made between the conventional Low-StrainDynamic Properties of Soils (LS-DPS), which are stiffness and material damping ratio, and theLow-Strain Viscoelastic Properties of Soils (LS-VPS), which are the truly fundamental parametersof a linear viscoelastic model. The relationships between these two sets of model parametershave been established, and a new definition of material damping ratio has been introduced.The necessity for a new definition of damping ratio is justified from the need to overcomethe limitations of the conventional definition when applied to strongly dissipative media.

For wave propagation problems it was found convenient to introduce another set ofviscoelastic parameters which are the body wave phase velocities and attenuationcoefficients collectively denoted as the Low-Strain Kinematical Properties of Soils (LS-KPS). Thethree sets of model parameters, LS-VPS, LS-DPS, and LS-KPS, are equipollent; they simplyprovide alternate ways to characterize the mechanical response of linear viscoelasticmaterials.

One of the results obtained from reformulating the definitions of the low-straindynamic properties of soils is the recognition that stiffness and material damping ratio arenot independent parameters even in weakly dissipative media. The coupling betweenstiffness and material damping ratio is the natural consequence of material dispersion, aphenomenon that is formally defined by the Kramers-Krönig relations. Material dispersionis an intrinsic feature of any type of viscoelastic materials. As a result of their mutualdependence, a correct experimental procedure for determining stiffness and materialdamping ratio of soils should determine these two parameters simultaneously. However, in thecurrent practice of geotechnical engineering testing, stiffness and material damping ratio aredetermined separately using different measurement techniques. Based on theseconsiderations, the final part of Chapter 2 is dedicated to illustrating the principles of a newexperimental procedure to be conducted in laboratory with the resonant column test for thesimultaneous determination of soil stiffness and material damping ratio. The procedure isalso suitable for the investigation of the frequency dependence laws of these important soilparameters.

After reexamining the definition of the low-strain dynamic properties of soils within theframework of the theory of linear viscoelasticity, the focus of this dissertation shifted toprocedures used to determine these important soil parameters from the interpretation ofsurface wave measurements. With regards to this subject, two new ideas were developed andpresented in this study. Together they form the basis of a new approach to surface wavemeasurement and interpretation. The first contribution was the derivation of an explicitrepresentation of the effective Rayleigh phase velocity, which was defined as the phasevelocity of a superposition of a finite number of harmonic waves having the same frequencyand different wavenumbers. The effective Rayleigh phase velocity corresponds to the phase

Conclusions and Recommendations 223

velocity measured in surface wave testing with harmonic source if the near field effects areneglected.

Surface waves propagating in vertically heterogeneous media are dispersive, and theirvelocity of propagation is a multi-valued function of the frequency of excitation. Thisphenomenon, known in the literature as geometric dispersion, arises from a condition ofconstructing interference among rays that are either bent or reflected/refracted by theheterogeneity of the medium. Geometric dispersion is responsible for the existence ofseveral modes of propagation each traveling at a different phase and group velocity in avertically heterogeneous medium. Geometric dispersion also affects the geometric spreadingof Rayleigh waves in homogeneous media.

Mathematically, the modes of propagation are obtained from the solution of adifferential eigenproblem, where the boundary conditions are 1) the vanishing of tractionsat the free surface of the half-space and 2) the radiation condition at infinity. The continuityof stress and displacement fields must also be enforced in vertically heterogeneous mediawhere the material properties vary discontinuously with depth (i.e. multi-layered media). Atany given frequency of excitation, non-trivial solutions of this differential eigenproblem areobtained only for special values of the wavenumber, which are called eigenvalues. In an elasticmedium, each eigenvalue is associated with a real-valued function called the eigenfunction thatgives the depth-variation of the displacement and stress fields for each mode ofpropagation. Solution of the differential eigenproblem corresponds to the solution of thefree vibration or homogeneous problem of Rayleigh waves where no sources or initialconditions are specified. A natural question then arises about the solution of theinhomogeneous Rayleigh wave problem, particularly with regard to the velocity of propagationof the ensuing surface wave field. This is relevant in surface wave testing because phasevelocity measurements are the basis for determining, via an appropriate inversionprocedure, the shear wave velocity profile at a site.

For Rayleigh waves generated by harmonic sources, the various modes of propagationof surface waves are superimposed as in a spatial Fourier series. The phase velocity of theresulting waveform was named effective Rayleigh phase velocity. In Chapter 3 an explicitexpression for the effective Rayleigh phase velocity has been derived. An important featureof this kinematical quantity is its local nature, which makes its current value to be a functionof the spatial position where it is measured. If the contribution of the body wave field isneglected (a valid assumption at distances of more than one to two wavelengths from thesource), then the effective phase velocity is the phase velocity that would be measured at apoint during a surface wave test. For measurements at two or more receiver locations, themeasured phase velocity is equal to the averaged effective phase velocity over the receiverarray. The notion of effective phase velocity and its explicit representation formed the basisof a new interpretation of surface wave measurements where the modal dispersion curvesare replaced by the effective dispersion curve. The latter is obtained from the frequencydependence law of the effective phase velocity averaged over the receiver array.


This new approach overcomes a major problem affecting the conventionalinterpretation of surface wave testing: the inconsistency of matching an experimentaldispersion curve that reflects, in general, the contributions of several modes of propagationwith a simulated dispersion curve for only a single mode of propagation. The currentstrategy used by several researchers to overcome this inconsistency is to perform anumerical simulation of the actual experiment. With this approach the theoretical phasevelocities are computed from phase differences of theoretical displacements that arecalculated at locations that emulate those used in the actual SASW test. Although thismethod is exact, it has the disadvantage of requiring the computation of the displacementfield, which is computationally more expensive than merely solving the Rayleigheigenproblem if the body wave field is included. Another disadvantage of this method isthat the partial derivatives required for the solution of the non-linear inverse problem haveto be computed numerically, a task that is computationally more expensive than usinganalytical partial derivatives and also potentially inaccurate.

In the new approach to surface wave interpretation, the experimental dispersion curve iscompared and matched with the effective dispersion curve, which accounts for multi-modeRayleigh wave propagation. In Chapter 3 closed-form analytical expressions for the partialderivatives of the effective Rayleigh phase velocity with respect to the shear andcompression wave velocities of the layers have been derived by employing the variationalprinciple of Rayleigh waves. The major advantage offered by the analytical over thenumerical partial derivatives, is that the former are computed using the solution of theRayleigh eigenproblem referred to the original and not the perturbed profile of mediumparameters.

The second contribution in the area of surface wave propagation presented in this studyis a numerical technique for the solution of the Rayleigh eigenvalue problem in linearviscoelastic media. One immediate application of this result was the development of asystematic and efficient procedure for simultaneously determining the low-strain dynamicproperties of soil deposits from the interpretation of surface wave measurements. In theconventional interpretation of surface wave data, the shear wave velocity and the sheardamping ratio profiles at a site are determined separately from the inversion of anexperimental dispersion and attenuation curve. The simultaneous inversion of surface wavedata offers several advantages over the corresponding uncoupled analysis.

First, it explicitly recognizes and accounts for the inherent coupling existing betweenseismic wave phase velocity (which is directly related to stiffness) and material damping ratioas a consequence of material dispersion. Secondly, the simultaneous inversion is a better-posedmathematical problem (in the sense of Hadamard). The solution of the (strongly) coupledRayleigh inverse problem is based on the use of a complex formalism where the Rayleigh

phase velocity is viewed as an holomorphic function ( )V VR S* * of the complex-valued shear

wave velocity. Thus, the simultaneous inversion takes full advantage of the internal constraint


constituted by the Cauchy-Riemann equations satisfied by the analytic function ( )V VR S* * .

Thirdly, the simultaneous inversion eliminates some of the errors affecting thecorresponding uncoupled analysis, where some of the input data required for the inversionof the attenuation measurements are obtained from the inversion of the dispersion data,and thus they are affected by the uncertainties associated with the latter process. In thesimultaneous inversion, this problem is overcome by inverting both dispersion andattenuation measurements in a single, complex-valued, inversion procedure.

It should be noted that the simultaneous inversion presented in this study isconceptually different from other types of simultaneous inversions available in the literature.A fully coupled Rayleigh inversion requires the ability to solve the complex eigenproblem inlinear viscoelastic media, where Rayleigh phase velocity and attenuation depend upon boththe body wave velocities and material damping ratios of the medium. This type of inversion,here denoted as strongly coupled, has been implemented in this study by introducing a newtechnique for the solution of the complex Rayleigh eigenproblem.

In general, most of the difficulties associated with the solution of an eigenproblem(differential or algebraic) are related with the determination of the eigenvalues. In the caseof the complex Rayleigh eigenproblem, this task involves computing the roots of thecomplex-valued Rayleigh dispersion equation. This is not a trivial problem, particularlybecause the Rayleigh dispersion equation is highly non-linear and is known only numerically.The technique used in this work to accomplish this task is an elegant procedure based onthe use of Cauchy residue theorem of complex variable theory. Once the roots of thedispersion equation are calculated, the correspondence principle of linear viscoelasticity isinvoked for the computation of the eigenfunctions, the effective Rayleigh phase velocity,the modal and the effective partial derivatives of Rayleigh phase velocity with respect to thecomplex-valued body wave velocities of the medium.

Often in the seismological literature the term simultaneous inversion is used to denotean approximate procedure that, based on the assumption of weak dissipation, does notactually require the solution of the complex eigenproblem. Conversely, it uses a variationalapproach combined with the results obtained from the solution of the elastic eigenproblemto find an approximate solution to the complex eigenproblem. This type of approach ishere denoted as the weakly coupled inversion. The technique presented in this study for thesolution of the complex eigenvalue problem is not restricted to weakly dissipative media.

The theory of modal and effective Rayleigh waves propagation in elastic and viscoelasticmedia developed in Chapter 3 has been used in Chapter 4 to develop four types ofinversion algorithms named UFUMA, UEQMA, CFUMA, and CEQMA. For a given pairof experimental dispersion and attenuation curves, these algorithms determine the shearwave velocity and shear damping ratio profiles of a soil deposit. The inversion procedureused in these four algorithms is based on a constrained least squares algorithm known asOccam’s algorithm. Its main objective is to enforce maximum smoothness on the resulting shear


wave velocity and shear damping ratio profiles while attaining a specified error misfitbetween the experimental and the simulated dispersion and attenuation curves. The choiceof this inversion strategy was motivated by the need of minimizing the dependence of theinverted shear wave velocity and shear damping ratio profiles upon the assumed number oflayers, which is, in general, an additional unknown in the interpretation of surface wavetesting.

The algorithms UFUMA and UEQMA refer to the uncoupled fundamental mode anduncoupled equivalent multi-mode inversion analysis, whereas the algorithms CFUMA andCEQMA were designed to perform the coupled fundamental mode and coupled equivalentmulti-mode inversion analysis, respectively.

In Chapter 6 these algorithms were tested in a systematic numerical simulation involvinga homogeneous medium and three simplified, stratified media. Case 1 was a regular soilprofile where the stiffness increases regularly with depth, whereas Case 2 and Case 3 weretwo different types of irregular soil profiles. In this numerical simulation, the syntheticdispersion and attenuation curves were determined using the same procedures used in anactual SASW experiment and illustrated in Chapter 5. From the numerical simulation, theresults obtained with the algorithm CEQMA were considered unreliable due to an inabilitydisplayed by this algorithm to determine the correct sequence of Rayleigh modes at certainfrequencies.

Concerning the results obtained with the inversion algorithms UFUMA, UEQMA, andCFUMA, they can be summarized as follows:

1. Overall, the UEQMA inversion algorithm based on the concept of effective phasevelocity yielded the most accurate results, particularly for the prediction of the shearwave velocity profile. The fundamental mode based algorithms UFUMA and CFUMAwere generally less accurate in predicting both the shear wave velocity and sheardamping ratio profile;

2. For regular soil profiles (Case 1) the fundamental mode based inversion yieldedsatisfactory results. However the results obtained with the coupled inversion algorithmCFUMA are more accurate than those obtained with the corresponding uncoupledanalysis performed with the algorithm UFUMA;

3. For irregular soil profiles (Case 2 and Case 3) the results of the algorithm UEQMA werealways more accurate than those obtained with the algorithm UFUMA and CFUMA.

The results obtained from the numerical simulation are consistent with the expectationsfrom the theory, and also with the results of other independent studies (Gucunski andWoods, 1991; Tokimatsu, 1995). Whereas in regular soil profiles the fundamental mode ofpropagation governs the response of a layered medium to a dynamic excitation, irregular soil


profiles are inversely dispersive, and as such the response of the medium includescontributions from higher modes of propagation.

8.2 Recommendations for Future Research

Surface wave propagation with the associated inverse problem are fascinating andchallenging subjects whose need for further investigation is justified by reasons that extendfar beyond the practical applications. Areas of research where some of the ideas presentedin this study may be further expanded include:

1. Implementation of a routine for the subdivision of the region of the complex planecontaining the roots of the Rayleigh secular function into a series of smaller subregions.As explained in Section 3.6.1, the computation of the roots of the Rayleigh secularfunction via a high-degree polynomial becomes an ill-conditioned problem as thenumber of modes increases. As a result these roots are computed with a decreasingdegree of accuracy. The instabilities exhibited by the code CEQMA during thecomputation of higher modes of propagation are most likely due to this problem. Thetechnique of subdividing the region containing the roots should stabilize theperformance of the algorithm CEQMA.

2. Implementation of a new method for the construction of the Rayleigh secular functionin linear viscoelastic media to be integrated with the root-finding technique presented inthis study. The actual technique for constructing the Rayleigh secular function is basedon the method of reflection and transmission coefficients. This method, originallydeveloped for elastic media, does not seem to be very accurate when applied toviscoelastic systems, particularly at high frequencies and for large number of layers. Thespectral element method, the boundary element method and numerical integration arepossible alternative methods.

3. In surface wave tests the quantities measured experimentally are displacement spectra ordisplacement transfer functions. In this study it was shown that Rayleigh phase velocityand attenuation coefficient are derived quantities that are obtained from the displacementspectra or transfer functions via an unstable process of numerical differentiation. Basedon this observation it would be interesting to attempt the construction of an algorithmfor determining the medium parameters from the direct inversion of the displacementspectra or transfer functions.

4. Procedures should be developed for solving the Rayleigh inverse problem by adoptingGlobal-Search-Techniques such as genetic algorithms, fractal inversion, neural networkinversion, or Monte Carlo simulation. As mentioned in Chapter 4, these methods aremore robust and accurate than Local-Search-Techniques such as Occam’s algorithm,even though they are computationally more expensive.


5. Investigate if it is advantageous to re-cast the current interpretation of surface wavetesting using a wavelets-based analysis. From a theoretical point of view investigate thepossibility of expanding the solution of the Rayleigh eigenproblem using a waveletsmulti-resolution analysis.

6. Solution of the boundary value problem of surface waves using more sophisticatedconstitutive laws other than classical linear viscoelasticity. Examples include binaryporous media theories, non-local and polar theories, doublet-mechanics. Attempt torelax the usual assumptions of small displacements/displacement gradients.

229

APPENDIX A - ELLIPTIC HYSTERETIC LOOP IN LINEARVISCOELASTIC MATERIALS

A.1 Harmonic Constitutive Relations

The stress-strain relationships of linear viscoelastic materials subjected to harmonicexcitations assume a particularly simple form:

( ) ( ) ( )σ ω ω ε ωγ γ γ= ⋅G* (A.1)

where ( )ε ω εγ γω= ⋅

0e i t , ε γ 0

∈ R , ( )G γ ω* is the complex modulus, and γ = P S, is a

subscript denoting the irrotational and the equivoluminal (shear) components of theassociated tensorial quantity. By considering the real part of ( )ε ωγ , Eq. A.1 can be rewritten

as:

( ) ( ) ( )[ ]σ ω ω ε ω ϕ ωγ γ γ γ= ⋅ ⋅ −G t* cos0

(A.2)

where ( )[ ] ( )[ ]tan arg *ϕ ω ωγ γ= G is the loss angle. Using trigonometric identities Eq. A.2 can

be rewritten as:

( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( )σ ω ω ε ω ϕ ω ω ε ϕ ω ωγ γ γ γ γ γ γ− ⋅ ⋅ = ⋅ ⋅ ⋅G G t* *cos sin sin0

(A.3)

Equation A.3 combined with the relationship ( ) ( )ε ω ε ωγ γ= ⋅0

cos t gives:

ε

ε

σ ε

εγ

γ

γ γ γ

γ γ0

21

2 0

2

1

+

− ⋅

⋅

=

G

G

( )

( )

(A.4)

230 Appendix A

where G( )1 γ and G( )2 γ

are the real and the imaginary parts of the complex modulus ( )G γ ω* .

Equation A.4 is the equation of an ellipse rotated by an angle ( )ψ γ ω with respect to the

strain axis (see Fig. 2.9). It represents the stress-strain hysteretic loop exhibited by a linearviscoelastic material subjected to harmonic oscillations.

A.2Energy Dissipated in Harmonic Excitations

The area enclosed by the elliptic hysteretic loop can be interpreted as the amount ofenergy (per unit volume) dissipated by the material during a cycle of harmonic loading. In a

stress-controlled test, this area, here denoted by ( )∆Wdissipγ ω , is defined as:

( ) ( ) ( )∆W dW ddissipγ γ γ γω σ ε= = ℜ ⋅ℜ∫ ∫l l

(A.5)

where the symbol ( )ℜ ⋅ denotes the real part of a complex quantity, and l is length of thehysteretic loop. In Eq. A.5 the term dWγ represents the work done by the stress (per unit

volume of the material) for an infinitesimal variation of the strain. Considering Eq. A.2 and

the fact that ( ) ( )ℜ = ⋅ ⋅ + ⋅d t dtε ω ε ω πγ γ 02cos , Eq. A.5 can be rewritten as:

( ) ( ) ( )∆W G t t dtdissipγ γ

π ω

γ γω ω ε ω ϕ ω π= ⋅ ⋅ ⋅ − ⋅ + ⋅∫ * cos cos0

2

0

2 2 (A.6)

Using trigonometric identities this integral can be simplified to:

( ) ( ) ( )∆W G t dtdissipγ γ γ γ

π ωω ω ε ϕ ω= ⋅ ⋅ ⋅ ⋅∫* sin sin

0

2 2

0

2

(A.7)

which can be easily solved to give:

( )∆W Gdissipγ γ γω π ε= ⋅ ⋅( )2

2

(A.8)

Appendix A 231

A.3Principal Axes of the Elliptic Hysteretic Loop

To obtain the inclination of the ellipse principal axes, it is convenient to rewrite Eq. A.4as follows:

11

21

0

2

1

2

2

21

0 2

2

2

0 2

2εε

εσ ε

σ

εγ

γ

γ

γγ

γ γ

γ γγ

γ γ

⋅ +

⋅ −⋅

⋅ +⋅

=G

G

G

G G

( )

( )

( )

( ) ( )

(A.9)

Then, utilizing the result that the coefficients of a quadric centered at the origin and written inthe form A x xik i k = 1 represent the components of a second order tensor (Finzi and Pastori,1961), it is easily recognized from Eq. A.9 that:

A

G

G

G

G

G

G G

=

⋅ +

−⋅

−⋅

⋅

11

1

0

2

1

2

2

1

0 2

2

1

0 2

2

0 2

2

εε

ε ε

γ

γ

γ

γ

γ γ

γ

γ γ γ γ

( )

( )

( )

( )

( )

( ) ( )

(A.10)

The problem of finding the ellipse principal axes has been transformed in a problem of linearalgebra, namely of finding the eigenvectors of the matrix A. Because this matrix is real andsymmetric, its eigenvalues are also real and the corresponding eigenvectors, as expected, areorthogonal. A straightforward computation yield for the eigenvalues λ1 2, :

( )λ ωε

γ γ γ

γ γ

1 2

2 2 2

22

0 2

2

1 1 4

2,

* *( )

( )

=+

± +

−

⋅ ⋅

G G G

G(A.11)

The inclination of the ellipse principal axes ( )ψ ωγ1 = ψ and ( )ψ π γ2 = 2 + ψ is obtained

from the components of the eigenvectors; the result is:

232 Appendix A

( )tan ,

* *( )

( )

ψ ωγ γ γ

γ

1 2

2 2 2

22

1

1 1 4

2=

−

+

−

⋅

G G G

G

m

(A.12)

233

APPENDIX B - EFFECTIVE RAYLEIGH PHASEVELOCITY PARTIAL DERIVATIVES

In Section 3.5.2 it was shown that the first variation of the effective Rayleigh

phase velocity δ β$V ( )β = r y, can be written as:

δ∂

∂δ

∂

∂δβ

β β$$ $

VV

VV

V

UU

jj

jj= +

(B.1)

where the summation convention is implied over the index j ranges from one to thenumber of Rayleigh modes of propagation associated with the frequency ω (cf.Eq.3.49). Explicit results will now be obtained for the terms δVj and δU j of Eq. B.1.

From the relations G VS= ρ 2 , ( )λ ρ+ =2 2G VP and Eq. 3.41, the following result isobtained:

[ ]δ δ δV P V Q V dyj j S j P= +∞

∫0

(B.2)

where:

( ) ( )P yV

k UIkr

dr

dykr

dr

dyjS

j j

,ωρ

= ⋅ −

−

2

42

1

21

2

12 (B.3a)

( ) ( )Q yV

k UIkr

dr

dyjP

j j

,ωρ

= ⋅ +

2 21

12

2

(B.3b)

The strategy to obtain an explicit relation for δU j is a bit more laborious. Application

of Rayleigh principle to Eq.3.33a gives δI1 0= , hence Eq.3.50 simplifies as follows:

234 Appendix B

δ δδ δ

ωU

I

V IV

I

V I

I

Ij

j

= −⋅

+⋅

−

22

1

2

1

3

12(B.4)

The first variations δI2 and δI3 can be calculated from Eq.3.33 considering again

Rayleigh principle and the relations G VS= ρ 2 and ( )λ ρ+ =2 2G VP ; the result is:

( )δ ρ δ δI V r V V r V dyP P S S2 12

22

0

= +∞

∫ (B.5a)

δ ρ δ δI V rdr

dyV V r

dr

dyr

dr

dyV dyP P S S3 1

22

11

2

0

2 2= − +

∞

∫ (B.5b)

Substitution of Eq.B.5 and Eq.B.2 in Eq.B.4 yields:

[ ]δ δ δU V V dyj j S j P= +∞

∫ Π Ω0

(B.6)

where:

( ) ( ) ( )Π j

j

S

jj jy

IV kr r

dr

dyr

dr

dyk I P,ω

ωωρ= ⋅ − −

− ⋅

12

21

22

21

12 2

2 (B.7a)

( ) ( ) ( )Ω j

j

P

jj jy

IV kr r

dr

dyk I Q,ω

ωωρ= ⋅ −

− ⋅

12

1

12

12 2

2 (B.7b)

In light of Eq.B.2 and Eq.B.6, Eq.B.1 can be rewritten as follows:

δ∂

∂

∂

∂δ

∂

∂

∂

∂δβ

β β β β$$ $ $ $

VV

VP

V

UV

V

VQ

V

UV dy

jj

jj S

jj

jj P= +

+ +

∞

∫ Π Ω0

(B.8)

which suggests the following result (cf. Eq.3.42):

Appendix B 235

δ∂

∂δ

∂

∂δβ

β

ω

β

ω

$$ $

, ,

VV

VV dy

V

VV dy

SV

SP

V

P

P S

=

+

∞ ∞

∫ ∫0 0

(B.9)

where:

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

β

ω

β β

β

ω

β β

$ $ $

$ $ $

,

,

V

V

V

VP

V

U

V

V

V

VQ

V

U

SV

jj

jj

PV

jj

jj

P

S

= +

= +

Π

Ω

(B.10)

Equation B.10 provides an explicit relationship for the partial derivatives of the

effective Rayleigh phase velocity ( )$ , ,V r yβ ω with respect to the medium parameters

VP and VS . A complete definition of this relation however, requires specification of

the terms jVV ∂∂ β and jUV ∂∂ β , whose calculation will be the objective of the rest

of the Appendix.

If in Eq.3.47 the modal group velocity U k ( )k i j= , is held constant, Eq.3.46 can

be differentiated (in the Gateaux sense) with respect to Vk , to yield:

236 Appendix B

( )[ ]

( ) ( ) ( ) ( )

( ) ( )

δ ω

ωδ δ

δ δ

δ δω

β

β

β β

β

β

β β

$ , ,V r y

A

C rV

V

V

VV V D V V V V V V

V V

B

A

DV

V

V

Vr C

V V

U

ij iji

i

j

ji j ij ij i j j i i j

i jj

M

i

M

ij iji

i

j

jij ij

i j

=

− − +

− +

−

− −

− +

−

==∑∑2

1

2

2

1 1

2 2

1

2

11

2

2 2

Φ Φ

Φ Φ

( ) ( ) ( )

− +

+

+

⋅ +

==

−

==

∑∑

∑∑

δ δ

δ δβ

β

β

V

V

V

VV V

V V

B

A

DV V

V V V V V V

V V

i

i

j

ji j

i jj

M

i

M

ij iji j

i j j i i j

i jj

M

i

M

2 2

11

2

1

2

11

2

1 1 1

2Φ

(B.11)

where:

( )( ) ( ) ( )[ ]

A β

β

ωω

r yV V r V V

V V

ij i j i j

i jj

M

i

M

, ,cos

=+ −

==

∑∑Φ 1 1 1 1

11

(B.12a)

( )( ) ( )[ ]

Bβ

β

ωω

r yr V V

V V

ij i j

i jj

M

i

M

, ,cos

=−

==

∑∑Φ 1 1

11

(B.12b)

and:

( ) ( )[ ]C ij i jr r V V, sinω ω= −1 1 (B.13a)

( ) ( )[ ]D ij i jr r V V, cosω ω= −1 1 (B.13b)

Appendix B 237

After some algebra, Eq.B.11 can be reduced to the following form:

( )[ ] ( ) ( ) ( ) ( )δ ω δ δββ

β β β β β β β β$ , ,V r y

AA B V A B V

U ij ij i ij ij jj

M

i

M

= −

− −

==∑∑2

211

Λ Χ Σ Ξ (B.14)

where:

( )( ) ( )

( )Λ

Φβ

β ω

ij

ij ij i ij j

i j i j

rC V D V

V V V V=

−2

2

2

2 (B.15a)

( )( ) ( ) ( )[ ]

( )Χ

Φβ

β ω ω

ij

ij j ij i j ij i ij i ij i j j

i j i j

rV C V V D rV C V D V V V

V V V V=

− + − +2

2

2

3 (B.15b)

( )( ) ( )

( )Σ

Φβ

β ω

ij

ij ij j ij i

i j i j

rC V D V

V V V V=

+2

2

2

2 (B.15c)

( )( ) ( ) ( )[ ]

( )Ξ

Φβ

β ω ω

ij

ij j ij i j ij i ij j ij i j i

i j i j

rV C V V D rV C V D V V V

V V V V=

+ + + +2

2

2

3 (B.15d)

Equation B.14 can be re-written in a more compact form by further setting:

( )( ) ( )

( )( ) ( )

ΓΛ Χ

ΓΣ Ξ

β

β β β β

β

β

β β β β

β

ij

ij ij

ij

ij ij

A B

A

A B

A

=−

=−

2

2

2

2~

(B.16)

238 Appendix B

so that:

( )[ ] ( ) ( )δ ω δ δβ β β$ , , ~V r y V V

U ij i ij jj

M

i

M

= −

==

∑∑ Γ Γ11

(B.17)

By replacing Eq.B.2 for the terms δVi and δVj in Eq.B.17, the latter becomes:

( )[ ] ( ) ( )

( ) ( )

δ ω δ

δ

β β β

β β

$ , , ~

~

V r y P P V dy

Q Q V dy

U ij i ij jj

M

i

M

S

ij i ij jj

M

i

M

P

= −

+

−

==

∞

==

∞

∑∑∫

∑∑∫

Γ Γ

Γ Γ

110

110

(B.18)

Now the next task is the computation of ( )[ ]δ ωβ$ , ,V r y

V. At this purpose Eq.3.46

is differentiated with respect to U k while keeping the modal phase velocity Vk

constant, yielding:

( )[ ]

( )

( )

δ ω

δ δ

δ δ

β

β

β

β

β

β

$ , ,V r y

A

DU

U

U

U

V V

B

A

DV V

U

U

U

U

V V

V

ij iji

i

j

j

i jj

M

i

M

ij iji j

i

i

j

j

i jj

M

i

M

=

− +

−

− +

+

==

==

∑∑

∑∑

2

2

1 1

11

211

Φ

Φ

(B.19)

Equation B.19 can be reduced, after some algebra, to a form analogous to that ofEq.B.17, namely:

Appendix B 239

( )[ ] ( ) ( )δ ω δ δβ β β$ , , ~V r y T U T U

V ij i ij jj

M

i

M

= +

==

∑∑11

(B.20)

where:

( ) ( )

( ) ( )

T U W

T U W

ij j ij

ij i ij

β β

β β

=

=~(B.21)

and:

( )( )

WA

BV V

AD

U U V Viji j

ij ij

i j i jβ

ββ β

β

= +

−

⋅

2 1 12

Φ(B.22)

If the terms δU i and δU j in Eq.B.20, are replaced by Eq.B.6, the former

becomes:

( )[ ] ( ) ( )

( ) ( )

δ ω δ

δ

β β β

β β

$ , , ~

~

V r y T T V dy

T T V dy

V ij i ij jj

M

i

M

S

ij i ij jj

M

i

M

P

= +

+

+

==

∞

==

∞

∑∑∫

∑∑∫

Π Π

Ω Ω

110

110

(B.23)

Combining the results of Eq.B.18 and Eq.B.23 in light of Eq.B.1:

240 Appendix B

( )[ ] ( )[ ] ( )[ ]

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

δ ω δ ω δ ω

δ

β β β

β β β β

β β β β

$ , , $ , , $ , ,

~ ~

~ ~

V V Vr y r y r y

P P T T V dy

Q Q T T

U V

ij i ij j ij i ij jj

M

i

M

S

ij i ij j ij i ij jj

M

i

M

= + =

−

+ +

+

−

+ +

==

∞

==

∑∑∫

∑∑

Γ Γ Π Π

Γ Γ Ω Ω

110

11

∞

∫ δV dyP0

(B.24)

Finally, comparison of Eq.B.24 and Eq.B.9 yields:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

∂

∂

∂

∂ω

∂

∂

∂

∂ω

β

ω

ββ β β β

β

ω

ββ β β β

$ $, , ~ ~

$ $, , ~ ~

,

,

V

V

V

Vr y P P T T

V

V

V

Vr y Q Q T T

SV

Sij i ij j ij i ij j

j

M

i

M

PV

Pij i ij j ij i ij j

P

S

= = −

+ +

= = −

+ +

==∑∑ Γ Γ Π Π

Γ Γ Ω Ω

11

==

∑∑j

M

i

M

11

(B.25)

241

APPENDIX C - DESCRIPTION OF COMPUTER CODES

C.1 UFUMA (Uncoupled-Fundamental-Mode-Analysis)

Dispersion is the first module of the algorithm UFUMA used to perform theFundamental-Mode-Uncoupled-Inversion of experimental surface wave data at a site. Inparticular, the module Dispersion is a MATLAB m-file designed to implement the non-linearinversion of the experimental dispersion curve. The non-linear inversion is performed by usinga constrained-least-squares-algorithm called the Occam’s algorithm, which enforces maximumsmoothness to the resulting shear wave velocity profile of the site, while attaining a specifiederror misfit between the experimental and the simulated dispersion curves.

The shear damping ratio profile is calculated independently by a constrained-linearinversion of the experimental attenuation data also based on the Occam’s algorithm. This task isimplemented by the MATLAB m-file Damping, which constitutes the second module of thealgorithm UFUMA.

Dispersion uses a mex-file called Rayleigh which is a FORTRAN 77 written routine thatsolves the eigenvalue problem of surface Rayleigh waves in elastic vertically heterogeneousmedia (modified from Hisada, 1995). This code computes also the modal partial derivatives ofRayleigh phase velocity with respect to the shear and compression wave velocities of the soillayers using a variational formulation and the G-matrix formed by these partial derivatives. TheG-matrix and the Green’s function associated with the Rayleigh wave-displacement field,which is also computed by Rayleigh, are used by the module Damping for the inversion ofthe experimental attenuation measurements.

The theoretical dispersion and attenuation curves computed by Dispersion andDamping, as well as the modal partial derivatives of Rayleigh phase velocity with respect tomedium parameters, are calculated with respect to the fundamental mode of propagation ofRayleigh waves.

All INPUT data required by the mex-file Rayleigh are imported via a data file fromDispersion. The experimental phase velocities and attenuation coefficients are imported intoDispersion and Damping from external ASCII files.

EXAMPLE DATA FILE

PARAMETERS USED IN THE NON-LINEAR INVERSION ALGORITHM

242 Appendix C

itmax = 10; Maximum Number of Iterations Allowedimumax = 2; Maximum Number of Sub-Iterations AllowedtolVS = 1e-2; Termination Criterion for Convergence

SITE INVESTIGATED: Treasure Island

Number of LayersNL = 9;

Thickness of Layers (the half-space is denoted by 0.0)THK = [1.52 1.52 1.98 2.44 3.05 9.14 10.00 10.00 0.0]';

Mass Density of LayersDNS = [1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75]';

Compression Damping Ratio of Layers (need DP/DS in forming the G-matrix)DP = [1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0]';

Elastic Poisson's Ratio of LayersNU = [0.25 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30]';

Shear Damping Ratio of Layers (need DP/DS in forming the G-matrix)DS = [1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0]';

Initial Guess Shear Wave Velocity of LayersVS0 = [60.0 60.0 60.0 60.0 60.0 60.0 60.0 60.0 60.0]';

FREQUENCY DATA (Hz)

Type of Frequency Spacing [1 = logarithmic] [0 = linear arithmetic]IFREQ = 1;

Number of FrequenciesNF = 83;

Initial FrequencyIOM = 8.453;

Final FrequencyNOM = 70.823;

Frequency Increment (only for linearly spaced frequencies)DOM = 0.0;

SOURCE PARAMETERS (SPATIAL POSITION AND MAGNITUDE)

Depth of the SourceDPH = 0.0;

Magnitude: X-ComponentFXX = 0.0;

Magnitude: Y-ComponentFYY = 0.0;

Magnitude: Z-ComponentFZZ = 1.0;

RECEIVER PARAMETERS (Spatial Position in Cylindrical Coordinates)

Azimuth angle (Degrees)

Appendix C 243

PHI = 0.0;

Depth of the receiversZH = 0.0;

Number of ReceiversNP = 100;

Position First ReceiverINP = 1.00;

Position Last ReceiverNNP = 100.00;

Increment of Receiver PositionDNP = 1.00;

PARAMETERS USED TO COMPUTE ROOTS OF RAYLEIGH SECULAR FUNCTION

Specified Tolerance (TOL < 0.1)TOL = 0.001;

Number of Partitions to Search Roots from CMIN to CMAXNCC = 400;

Number of Sub-Partitions for Roots Close to VP and VSNSC = 10;

Range of Velocity to Use Sub-PartitionsDCR = 20.0;

IMPORT EXPERIMENTAL SURFACE WAVE MEASUREMENTSload ExpPhase.datload ExpAlpha.dat

END OF DATA FILE

C.2 UEQMA (Uncoupled-Equivalent-Multi-Mode-Analysis)

Dispersion is the first module of the algorithm UEQMA used to perform the Equivalent-Multi-Mode-Uncoupled-Inversion of experimental surface wave data at a site. In particular, themodule Dispersion is a MATLAB m-file designed to implement the non-linear inversion ofthe experimental dispersion curve. The non-linear inversion is performed by using aconstrained-least-squares-algorithm called Occam’s algorithm, which enforces maximumsmoothness to the resulting shear wave velocity profile of the site, while attaining a specifiederror misfit between the experimental and the simulated dispersion curves.

The shear damping ratio profile is obtained independently by a linear inversion of theexperimental attenuation data also based on the Occam’s algorithm. This task is implemented bythe MATLAB m-file Damping, which constitutes the second module of the algorithmUEQMA.

Dispersion uses a mex-file called Rayleigh which is a FORTRAN 77 written routine thatsolves the eigenvalue problem of surface Rayleigh waves in elastic vertically heterogeneous

244 Appendix C

media (modified from Hisada, 1995). This code also computes the effective Rayleigh phasevelocity and its partial derivatives with respect to the shear and compression wave velocities ofthe soil layers using a variational formulation. The effective G-matrix formed by these partialderivatives is also computed. The effective G-matrix and the Green’s function associated withthe Rayleigh wave-displacement field, which are also computed by Rayleigh, are used by themodule Damping for the inversion of the experimental attenuation measurements.

The effective theoretical dispersion and attenuation curves computed by Dispersion andDamping are calculated by taking into account all the modes of propagation of Rayleighwaves. Averaging the effective Rayleigh phase velocities and attenuation coefficients over thefrequency dependent receiver offsets eliminates the dependence of these quantities on thereceivers’ location.

All INPUT data required by the mex-file Rayleigh are imported via a data file fromDispersion, which is the same data file used by the code UFUMA. The experimental phasevelocities and attenuation coefficients are imported into Dispersion and Damping viaexternal ASCII files.

C.3 CFUMA (Coupled-Fundamental-Mode-Analysis)

ViscoRay is the module of the algorithm CFUMA used to perform the Fundamental-Mode-Coupled-Inversion of the experimental dispersion and attenuation curves at a site. Thenon-linear simultaneous inversion is performed by applying the complex formalism to aconstrained-least-squares-algorithm, called Occam’s algorithm, which enforces maximumsmoothness to the resulting complex shear wave velocity profile of the site, while attaining aspecified error misfit between the complex-valued experimental and simulated dispersioncurves.

ViscoRay is a MATLAB computer code interfaced with a mex-file called Rayleigh whichis a FORTRAN 77 routine that solves the complex eigenvalue problem of Rayleigh waves inlinear viscoelastic vertically heterogeneous media. This code also computes the partialderivatives of the (modal) complex Rayleigh phase velocity with respect to the complex shearand compression wave velocities of the soil layers using a variational formulation.

The theoretical dispersion and attenuation curves computed by the program ViscoRay, aswell as the modal partial derivatives of Rayleigh phase velocity with respect to mediumparameters, are referred to the fundamental mode of propagation of Rayleigh waves. However,the frequency dependent attenuation coefficients are calculated using a geometric spreadingfunction that accounts for all the modes of propagation (Rix et al., 1998a).

All INPUT data required by the mex-file Rayleigh are imported via a data file fromViscoRay, which is the same data file used by the codes UFUMA and UEQMA. The

Appendix C 245

experimental phase velocities and attenuation coefficients are imported into ViscoRay viaexternal ASCII files.

PARAMETERS USED TO COMPUTE THE ZEROS OF THE RAYLEIGH SECULAR FUNCTION

Number of Points in the Complex Plane used for Contour Integration withthe Gauss-Legendre Quadrature Formulae

Number of Points along curves DR = constantNCC = 50;

Number of Points along curves CR = constantNDD = 50;

C.4 CEQMA (Coupled-Equivalent-Multi-Mode-Analysis)

ViscoRay is the module of the algorithm CEQMA used to perform the Equivalent-Multi-Mode-Coupled-Inversion of the experimental dispersion and attenuation curves at a site. Thenon-linear simultaneous inversion is performed by applying the complex formalism to aconstrained-least-squares-algorithm called Occam’s algorithm which enforces maximumsmoothness to the resulting complex shear wave velocity profile of the site, while attaining aspecified error misfit between the complex-valued experimental and simulated dispersioncurves.

ViscoRay is a MATLAB computer code interfaced with a mex-file called Rayleigh, whichsolves the complex eigenvalue problem of Rayleigh waves in viscoelastic verticallyheterogeneous media. This code also computes the effective complex Rayleigh phase velocityand its partial derivatives with respect to the complex shear and compression wave velocitiesof the soil layers using a variational formulation.

The effective theoretical dispersion and attenuation curves computed by ViscoRay as wellas the effective partial derivatives of Rayleigh phase velocity with respect to mediumparameters, are calculated by taking into account all the modes of propagation of Rayleighwaves. Averaging the effective Rayleigh phase velocities and the effective partial derivativesover the frequency dependent receiver offsets eliminates the dependence of these quantities onthe receivers’ location. The experimental attenuation coefficients are calculated iteratively usinga geometric spreading function that also accounts for all the modes of propagation of Rayleighwaves (Rix et al., 1998a).

All INPUT data required by the mex-file Rayleigh are imported via a data file fromViscoRay, which is the same data file used by the code CFUMA including the parametersused to compute the zeros of the Rayleigh secular function.

246 Appendix C

247

BIBLIOGRAPHY

Abd-Elall, L.F., Delves, L.M., Reid, J.K. (1970). “A Numerical Method for Locating theZeros and Poles of a Meromorphic Function.”, from Numerical Methods for Non LinearAlgebraic Equations, by P. Rabinowitz, Ed. Gordon and Breach Science Publishers, pp. 47-59.

Abo-Zena, A.M. (1979). “Dispersion Function Computations for Unlimited FrequencyValues.”, Geophys. J. R. Astr. Soc., 58, 91-105.

Achenbach, J.D. (1984). “Wave Propagation in Elastic Solids.”, North-Holland, Amsterdam,Netherlands, pp. 425.

Aki, K., and Richards, P.G. (1980). “Quantitative Seismology: Theory and Methods.”, W.H.Freeman and Company, San Francisco, 932 pp.

Anandarajah, A. (1996). “Discrete Element Method for Platy Colloidal Particles.”, Symposium,Computational and Experimental Methods for Particulate Materials, ASME, Johns HopkinsUniversity, Baltimore, MD, June 12-14, 1996.

Anderson, D.L., and Archambeau, C.B. (1964). “The Anelasticity of the Earth.”,J. GeophysicalResearch, 69(10), 2071-2084.

Anderson, D.L., Ben-Menahem, A., and Archambeau, C.B. (1965). “Attenuation of SeismicEnergy in the Upper Mantle.” J. Geophysical Research, 70, 1441-1448.

Azimi, S.A., Kalinin, A.V., Kalinin, V.V., and Pivovarov, B.L. (1968). “Impulse andTransient Characteristics of Media with Linear and Quadratic Absorption Laws.”,Izvestiya, Physics of the Solid Earth, February 1968, pp.88-93.

Baran, P.A., and Sweezy, P.M. (1968). “Monopoly Capital: An Essay on the AmericanEconomic and Social Order (Harmondsworth: Penguin Books).

Båth M. (1968). “Mathematical Aspects of Seismology.”, Elsevier Publishing Company,Amsterdam, pp.415.

Bellotti, R., Ghionna, V.N., Jamiolkowski, M., and Robertson, P.K. (1989). “DesignParameters of Cohesionless Soils from In-Situ Tests.”, Specialty Session on In-Situ Testing ofSoil Properties for Transportation Facilities, Sponsored by Committee A2L02-Soil and RockProperties, National Research Council, Transportation Research Board, Washington, January,1989.

248 Bibliography

Bendat, J. and Piersol, A. (1986). “Random Data - Analysis and Measurement Procedures.”,2nd Ed., John Wiley & Sons, New York, 566 pp.

Ben-Menahem, A., and Singh, S.J. (1981). “Seismic Waves and Sources.”, Springer-Verlag,New York, 1108 pp.

Biot, M.A. (1955). “Theory of Elasticity and Consolidation for a Porous Anisotropic Solid.”Journal of Applied Physics, 26, 182-185.

Biot, M.A. (1956). “Theory of Propagation of Elastic Waves in a Fluid-Saturated PorousSolid.”, I. Lower Frequency Range; II. Higher Frequency Range, J. Acoust. Soc. Am., 28,168-178; 179-191.

Boore, D.M. (1972). “Finite Difference Methods for Seismic Wave Propagation inHeterogeneous Materials.” Methods of Computational Physics, Vol.11, Ed. Bolt, B.A.,Academic press, New York, pp.1-36.

Bowen, R.M. (1982). “Compressible Porous Media Models by Use of the Theory ofMixtures.” International Journal of Engineering Science, Vol. 20, No. 6, pp. 697-735.

Bracewell, R. (1965). “The Fourier Transform and its Applications.”, McGraw-Hill Co.

Casagrande, A. (1932). “Research on the Atterberg Limits of Soils.”, Public Roads 13 (8),121-130 and 136.

Chen, X. (1993). “A Systematic and Efficient Method of Computing Normal Modes forMultilayered Half Space.” Geophysics J. Int., Vol. 115, pp. 391-409.

Christensen, R.M. (1971). “Theory of Viscoelasticity - An Introduction.” Ed. AcademicPress, 245 pp.

Christoffersen, B., Nemat-Nasser, S., and Mehrabadi, M.M. (1981). “A MicromechanicalDescription of Granular Material Behavior.”, Journal of Applied Mechanics, 48, 339-344.

Cole, K.S., and Cole, R.H. (1941). “Dispersion and Absorption in Dielectrics. I. AlternatingCurrent Characteristics.”, J.Chem.Phys., 9, 341-351.

Constable, S.C., Parker, R.L., and Constable, G.G. (1987). “Occam’s Inversion: A PracticalAlgorithm For Generating Smooth Models From Electromagnetic Sounding Data.”Geophysics, 52, 289-300.

Cundall, P.A., and Strack, O.D.L. (1979). “A Discrete Numerical Model for GranularAssemblies.”, Geotechnique, Vol. 29, No.1, pp. 47-65.

Bibliography 249

De Boer, R. (1996). “Highlights in the Historical Development of the Porous Media Theory:Toward a Consistent Macroscopic Theory.” Applied Mechanics Review, ASME, Vol.49,No.4, 201-262.

Delves, L.M., and Lyness, J.N. (1967). “A Numerical Method for Locating the Zeros of anAnalytic Function.”, Math. Comp., 21, 543-560.

Dobry, R. (1970). “Damping in Soils: Its Hysteretic Nature and the Linear Approximation.”Research Report R70-14, Massachusetts Institute of Technology, 82p.

Dobry, R., and Vucetic, M. (1987). “Dynamic Properties and Seismic Response of Soft ClayDeposits.”, Proceedings, International Symposium on Geotechnical Engineering of Soft Soils, MexicoCity, Vol.2, pp.51-87.

Drnevich, V.P. (1985). “Recent Developments in Resonant Column Testing.”, Proceedings,Richart Commemorative Lectures, Sponsored by Geotechnical Engineering Division, in Conjunctionwith ASCE Convention, Detroit, Michigan, October 23, 1985.

Electric Power Research Institute. (1991). Proceedings: NSF/EPRI Workshop on DynamicSoil Properties and Site Characterization. Report NP-7337, Vol. 1, Research Project 810-14.

Electric Power Research Institute. (1993). “Guidelines for Determining Design BasisGround Motions” Vol. I: Methods and Guidelines for Estimating Earthquake Ground Motion inEastern North America. EPRI TR-102293 Project 3302 November 1993.

Engl, H.W. (1993). “Regularization Methods for the Stable Solution of Inverse Problems.”Surveys on Mathematics for Industry, Vol. 3, pp. 71-143.

Eringen, A.C. and Suhubi, E.S. (1964). “Nonlinear Theory of Simple Microelastic Solid, Iand II, Int. J. Eng. Sci. 2, 189-204, 389-404.

Eringen, A.C. and Kafadar C.B. (1976). “ Polar Field Theories.” Continuum Physics Vol. IV,Part I, Edited by A.C. Eringen, Academic Press, p.274.

Ewing, W.M., Jardetzky, W.S., and Press, F. (1957). “Elastic Waves in Layered Media.”,McGraw-Hill, 380p.

Faccioli, E., Maggio, F., Quarteroni, A., and Tagliani, A. (1996). “Spectral-DomainDecomposition Methods for the Solution of Acoustic and Elastic Wave Equations.”,Geophysics, Vol.61, No.4, pp.1160-1174.

Fam, M.A., and Santamarina, J.C. (1996). “Coupled Diffusion-Fabric-Flow Phenomena: AnEffective Stress Analysis.”, Canadian Geotechnical Journal, Vol.33, No.3, pp.515-522.

250 Bibliography

Ferrari, M., Granik, V.T., Imam, A., and Nadeau, J.C. (1997) “Advances in DoubletMechanics.”, Springer-Verlag, Berlin, 213 p.

Ferry, J.D. (1980). “Viscoelastic Properties of Polymers.”, 3rd Edition, John Wiley, NewYork, pp.641.

Finzi, B., and Pastori, M. (1961). “Calcolo Tensoriale e Applicazioni”, Zanichelli, Bologna,p.508 (in Italian).

Frost, J.D., and Kuo, C.Y. (1996). “Automated Determination of the Distribution of LocalVoid Ratio from Digital Images.”, ASTM Geotechnical Testing Journal, Vol.19, No.2,pp.107-117.

Fung, Y.C. (1965). “Foundations of Solid Mechanics.”, Prentice-Hall, New Jersey, pp.525.

Goldstein, H. (1980). “Classical Mechanics.”, Addison-Wesley Publishing Company, 2nd Ed.,pp.672.

Goodman, M.A., and Cowin, S.C. (1972). “A Continuum Theory for Granular Materials.”Archive for Rational Mechanics and Analysis 44: 249-266.

Granik, V.T., and Ferrari, M. (1993). “Microstructural Mechanics of Granular Media.”Mechanics of Materials, 15:301-322.

Green, A.E., and Rivlin, R.S. (1964). “Multipolar Continuum Mechanics.”, Arch. Rat. Mech.Anal. 17, 113-147.

Gucunski, N., and Woods, R.D. (1991). “Use of Rayleigh Modes in Interpretation of SASWTests,” Proceedings, 2nd International Conference on Recent Advances in Geotechnical EarthquakeEngineering and Soil Dynamics, Vol. 2, pp. 1399-1408.

Gurtin, M.E. (1963). “Variational Principles in the Linear Theory of Viscoelasticity.”, Arch.Ration. Mech. Anal., 13, 179.

Hall, J.R., and Richart, Jr.F.E. (1963). “Dissipation of Elastic Wave Energy in GranularSoils.”, Journal of Soil Mechanics and Foundations Division, ASCE, Vol.89, No.SM6, pp.27-56.

Hardin, B.O., and Drnevich, V.P. (1972). “Shear Modulus and Damping in Soils:Measurement and Parameter Effects.” J. Soil Mechanics and Foundation Engineering, ASCE,98(SM6), 603-624.

Hardin, B.O. (1978). “The Nature of Stress-Strain Behavior of Soils.”, Proceedings, EarthquakeEngineering and Soil Dynamics, ASCE, Pasadena, California, Vol.1, pp.3-89.

Harvey, D. (1981). “Seismogram Synthesis using Normal Mode Superposition: the LockedMode Approximation.”, Geophys. J. R. Astr. Soc., 66, 37-70.

Bibliography 251

Haskell, N.A. (1953). “The Dispersion of Surface Waves on Multilayered Media.”, Bulletin ofthe Seismological Society of America, 43, 17-34.

Henrici, P. (1974). “Applied and Computational Complex Analysis.”, Vol. 1, John Wiley &Sons, New York, pp. 682.

Herrmann, R.B. (1994). “Computer Programs in Seismology”, User’s Manual, Vol.II,St.Louis University, Missouri.

Hille, E. (1973). “Analytic Function Theory.”, Vol. 1, Chelsea, New York, 2nd Edition.

Hisada, Y. (1994). “An Efficient Method for Computing Green’s Functions for a LayeredHalf-Space with Sources and Receivers at Close Depths,” Bulletin of the Seismological Societyof America, 84(5), 1456-1472.

Hisada, Y. (1995). “An Efficient Method for Computing Green’s Functions for a LayeredHalf-Space with Sources and Receivers at Close Depths (Part 2).”, Bulletin of theSeismological Society of America, 85(4), 1080-1093.

Holzlohner, U. (1980). “Vibrations of the Elastic Half-Space Due to Vertical SurfaceLoads.” Earthquake Engineering and Structural Dynamics, 8, 405-414.

Idriss, I.M., and Sun, J.I. (1991). “Users Manual for SHAKE91.” University of California atDavis.

Ishibashi, I. (1992). Discussion to “Effect of Soil Plasticity on Cyclic Response.”, by M.Vucetic and R. Dobry, Journal of Geotechnical Engineering, ASCE, Vol.118, No.5, pp.830-832.

Ishibashi, I., and Zhang, X. (1993). “Unified Dynamic Shear Moduli and Damping Ratios ofSand and Clay.”, Soils and Foundations, Vol.33, No.1, pp.182-191.

Ishihara, K. (1996). “Soil Behaviour in Earthquake Geotechnics.”, Oxford SciencePublications, Oxford, UK, pp. 350.

Iwasaki, T., Tatsuoka, F., and Takagi, Y. (1978). “Shear Modulus of Sands Under TorsionalShear Loading.”, Soils and Foundations, Vol.18, No.1, pp.39-56.

Jamiolkowski, M., Leroueil, S., and Lo Presti, D.C.F. (1991). “Theme Lecture: DesignParameters from Theory to Practice.”, Proceedings, Geo-Coast-91, Yokohama, Japan, pp.1-41.

Jamiolkowski, M., Lancellotta, R., and Lo Presti, D.C.F. (1994). “Remarks on the Stiffness atSmall Strains of Six Italian Clays.”, International Symposium on Pre-FailureDeformationCharacteristics of Geomaterials, IS-Hokkaido, Sapporo, Japan, 1994.

252 Bibliography

Jenkin, C.F. (1931). “The Pressure Exerted by Granular Materials: an Application ofPrinciple of Dilatancy.” Proc. R. Soc. A. 131, 53-89.

Johnston, D.H., Toksöz, M.N., and Timur, A. (1979). “Attenuation of Seismic Waves In Dryand Saturated Rocks: II. Mechanisms.” Geophysics, 44(4), 691-711.

Jones, T.D. (1986). “Pore Fluids and Frequency-Dependent Wave Propagation in Rocks.”,Geophysics, Vol.51, No.10, pp.1939-1953.

Jongmans, D. (1990). “In-Situ Attenuation Measurements in Soils.”, Engineering Geology, 29,99-118.

Kausel, E. (1981). “An Explicit Solution For The Green Functions For Dynamic Loads InLayered Media..” Massachusetts Institute of Technology, Research Report R81-13, 79pp.

Kausel, E., and Roësset, J.M. (1981). “Stiffness Matrices for Layered Soils.”, Bulletin of theSeismological Society of America, 71, 6, 1743-1761.

Keilis-Borok, V.I., (1989). “Seismic Surface Waves in a Laterally Inhomogeneous Earth”,Kluwer Academic Publishers, 304 pp.

Kennett, B.L.N. (1974). “Reflections, Rays, and Reverberations.”, Bulletin of the SeismologicalSociety of America, 64, 1685-1696.

Kennett, B.L.N., and Kerry, N.J. (1979). “Seismic Waves in a Stratified Half-Space.”,Geophys. J. R. Astr. Soc., 57, 557-583.

Kennett, B.L.N. (1983). “Seismic Wave Propagation in Stratified Media.”, CambridgeUniversity Press, UK, pp.342.

Kjartansson, E. (1979). “Constant Q-Wave Propagation and Attenuation.”, J. Geophys. Res.,Vol.84, pp.4737-4748.

Knopoff, L. (1964). “A Matrix Method for Elastic Wave Problems.”, Bulletin of theSeismological Society of America, 54, 431-438.

Kokusho, T. (1980). “Cyclic Triaxial Test of Dynamic Soil Properties for Wide StrainRange.”, Soils and Foundations, Vol.20, No.2, pp.45-60.

Komatitsch, D., and Vilotte, J.P. (1998). “The Spectral Element Method: An Efficient Toolto Simulate the Seismic Response of 2D and 3D Geological Structures.” Bulletin of theSeismological Society of America, 88(2), 368-392.

Bibliography 253

Koppermann, S.E., Stokoe, II K.H., and Knox, D.P. (1982). “Effect of State of Stress onVelocity of Low Amplitude Compression Waves Propagating Along Principal StressDirections in Sand.”, Geotechnical Engineering Report, GR82-22, University of Texas, Austin,Texas.

Kramer, S.L. (1996). “Geotechnical Earthquake Engineering.”, Prentice-Hall, New Jersey,pp.653.

Krantz, S.G. (1982). “Function Theory of Several Complex Variables.”, John Wiley & Sons,New York, pp.437.

Kuo, C.Y., and Frost, J.D. (1997). “Initial Fabric and Uniformity of a Sand Specimen – AnImage Analysis Approach.”, Proceedings, of ASCE Symposium on Mechanics of Deformation andFlow of Particulate Materials, Evanston, pp. 214-227.

Kuraoka, S., and Bosscher, P.J. (1996). “Parallelization of the Distinct (Discrete) ElementMethod (DEM).”, Symposium, Computational and Experimental Methods for ParticulateMaterials, ASME, Johns Hopkins University, Baltimore, MD, June 12-14, 1996.

Lai, C.G. (1997). “On the Theory of Mixtures of Porous Media.”, Master Report inEngineering Science and Mechanics, The Georgia Institute of Technology, Atlanta,Georgia.

Lamb, H. (1904). “On the Propagation of Tremors over the Surface of an Elastic Solid.”,Philosophical Transactions of the Royal Society of London A203:1-42 pp.

Lambe, T.W., and Whitman, R.V. (1969). “Soil Mechanics.”, John Wiley & Sons, New York.

Lanczos C. (1970). “The Variational Principles of Mechanics.”, University of Toronto Press,Toronto, 4th Edition.

Lawson, C.L., and Hanson, R.J. (1974). “Solving Least Squares Problems.”, Prentice-Hall,340 pp.

Lawton, W.H. and Sylvestre E.A. (1971). “Elimination of Linear Parameters in Non-LinearRegression.” Technometrics, 13(3), 461-467.

Lee, W.B., and Solomon, S.C. (1979). “Simultaneous Inversion of Surface-Wave PhaseVelocity and Attenuation: Rayleigh and Love Waves over Continental and OceanicPaths.” Bulletin of the Seismological Society of America, 69(1), 65-95.

Leurer, K.C. (1997). “Attenuation in Fine-Grained Marine Sediments: Extension of the Biot-Stoll Model by the Effective Grain Model (EGM).” Geophysics, Vol.62, No.5, 1465-1479.

254 Bibliography

Liu, H.P., Anderson, D.L., and Kanamori, H. (1976). “Velocity Dispersion due toAnelasticity; Implications for Seismology and Mantle Composition.”, Geophys. J.R. Astr.Soc. 47, 41-58.

Lockett, F.J. (1962). “The Reflection and Refraction of Waves at an Interface betweenViscoelastic Materials.”, J. Mech. Phys. Solids, 10, 53.

Logan, J.D. (1997). “Applied Mathematics.”, John Wiley & Sons, 2nd Ed., pp.476.

Lo Presti, D.C.F. (1987). “Behavior of Ticino Sand During Resonant Column Tests.”, Ph.D.Thesis, Politecnico di Torino, Torino, Italy.

Lo Presti, D.C.F., Jamiolkowski, M., Pallara, O. and Cavallaro, A. (1996). “Rate and CreepEffect on the Stiffness of Soils.”, Proceedings, Conference on Measuring and Modeling TimeDependent Soil Behavior, Held in Conjuction with the ASCE National Convention, November 10-14, 1996, Washington, D.C.

Lo Presti, D.C.F, and Pallara, O. (1997). “Damping Ratio of Soils from Laboratory and In-Situ Tests.”, Proceedings, 14th International Conference on Soil Mechanics and FoundationEngineering, Hamburg, Germany, 6-12, September, 1997.

Lubliner, J. (1990). “Plasticity Theory.”, Macmillan Publishing Company, New York, pp.495.

Luco, J.E. and Apsel, R.J. (1983). “On the Green’s Function for a Layered Half-Space.” PartI, Bulletin of the Seismological Society of America, 73, 909-929.

Lysmer, J. and Drake, L.A. (1972). “A Finite Element Method for Seismology.” Methods ofComputational Physics, Vol.11, Ed. Bolt, B.A., Academic press, New York, pp.181-215.

Lysmer, J. and Waas, G. (1972). “Shear Waves in Plane Infinite Structures.”, J. Eng. Mech.Div., ASCE, 18, 859-877.

Malagnini, L. (1996). “Velocity and Attenuation Structure of Very Shallow Soils: Evidencefor Frequency-Dependent Q.” Bulletin of the Seismological Society of America, 86(5), 1471-1486.

Malagnini, L., Herrmann, R.B., Biella, G., and De Franco, R. (1995). “Rayleigh Waves inQuaternary Alluvium from Explosive Sources: Determination of Shear-Wave Velocityand Q Structure.” Bulletin of the Seismological Society of America, 85, 900-922.

Malagnini, L., Herrmann, R.B., Mercuri, A., Opice, S., Biella, G., and De Franco, R. (1997).“Shear-Wave Velocity Structure of Sediments from the Inversion of Explosion-InducedRayleigh Waves: Comparison with Cross-Hole Measurements.” Bulletin of the SeismologicalSociety of America, 87(6), 1413-1421.

Bibliography 255

Malvern, L.E. (1969). “Introduction to the Mechanics of a Continuous Medium.” Prentice-Hall, Inc., New Jersey, 713 p.

Manolis G.D., and Beskos, D.E. (1988). “Boundary Element Methods in Elastodynamics.”,Unwin Hyman, London, pp. 282.

Masad, E., Muhunthan, B., and Chameau, J.L. (1997). “Stress-Strain Model for Clays withAnisotropic Void Ratio Distribution.”, International Journal for Numerical and AnalyticalMethods in Geomechanics, Submitted for Publication (Revised June 1997).

Menke, W. (1989). “Geophysical Data Analysis: Discrete Inverse Theory.”, Revised Edition,International Geophysics Series, Vol.45, Academic Press, 289 pp.

Mindlin, R.D. (1964). “Microstructure in Linear Elasticity.”, Arch. Rat. Mech. Anal. 16, 51-78.

Mitchell, J.K. (1976). “Fundamentals of Soil Behaviour.”, John Wiley & Sons, New York.

Muhunthan, B. (1991). “Micromechanics of Steady State, Collapse and Stress-StrainModeling of Soils.”, Ph.D Thesis, Purdue University, Lafayette, Indiana, 221 pp.

Muhunthan, B., and Chameau, J.L. (1996). “Void Fabric Tensor and Ultimate State Surfaceof Soils.”, Journal of Geotechnical Engineering Division, ASCE, 123, No.2.

Nazarian, S., Stokoe, K.H., and Hudson, W.R. (1983). “Use of Spectral Analysis of SurfaceWaves Method for Determination of Moduli and Thicknesses of Pavement Systems.”,Transportation Research Record 930, Transportation Reseacrh Board, Washington, D.C.,pp.38-45.

Nazarian, S. (1984). “In Situ Determination of Elastic Moduli of Soil Deposits andPavement Systems by Spectral Analysis of Surface Waves Method.”, Ph.D. Dissertation,The University of Texas at Austin.

Nemat-Nasser, S., and Mehrabadi, M.M. (1983). “Stress and Fabric in Granular Mass.”,Mechanics of Granular Materials: New Models and Constitutive Relations (Jenkins, J.T., andSatake, M. Eds.), Elsevier, pp. 1-8.

O’Connell, R.J., and Budiansky, B. (1978). “Measures of Dissipation in Viscoelastic Media.”,Geophys. Res. Lett., Vol.5, pp.5-8.

Oda, M. (1972). “ Initial Fabrics and their Relations to Mechanical Properties of GranularMaterials.”, Soils and Foundations, Vol.12, No.2, pp.1-18.

Oppenheim, A., and A. Willsky, A. (1997). “Signals and Systems.”, Prentice-Hall, EnglewoodCliffs, New Jersey, 540 pp.

256 Bibliography

Papoulis, A. (1965). “Probability, Random Variables, and Stochastic Processes.”, McGraw-Hill, New York, 576 pp.

Parker, R.L. (1994). “Geophysical Inverse Theory.”, Princeton University Press, New Jersey,pp.386.

Passman, S.L., Nunziato, J.W., and Walsh, E.K. (1984). “ A Theory of Multiphase Mixtures.”Appendix 5C, 286-325, Rational Thermodynamics, C. Truesdell, Springer-Verlag, Berlin,578 pp.

Pipkin, A.C. (1986). “ Lectures on Viscoelasticity Theory.”, 2nd Edition, Springer-Verlag,Berlin, pp.188.

Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P. (1992). “NumericalRecipies in Fortran - The Art of Scientific Computing.”, Cambridge University Press, 2nd

Ed., pp. 963.

Read, W.T. (1950). “ Stress Analysis for Compressible Viscoelastic Materials.”, J. Appl. Phys.,21, 671.

Richart, F.E., Jr., Woods, R.D., and Hall, J.R. (1970), “Vibrations of Soils and Foundations.”,Prentice-Hall, Englewood Cliffs, New Jersey, 414 pp.

Rix, G.J. (1988). “Experimental Study of Factors Affecting the Spectral Analysis of SurfaceWaves Method.”, Ph.D. Dissertation, The University of Texas at Austin, pp.315.

Rix, G.J., Lai, C.G., Spang, A.W.,Jr. (1998a). “In-Situ Measurement of Damping Ratio UsingSurface Waves.” Accepted for publication to ASCE Journal of Geotechnical andGeoenvironmental Engineering, 29 pp.

Rix, G.J., and Lai, C.G. (1998). “Simultaneous Inversion of Surface Wave Velocity andAttenuation,” Geotechnical Site Characterization, Edited by P.K. Robertson and P.W.Mayne, Vol. 1, pp.503-508, Proceedings of the First International Conference on SiteCharacterization – ISC’98/Atlanta, Georgia, USA, 19-22 April 1998.

Rix, G.J., Lai, C.G., Foti, S., and Zywicki D. (1998b). “ Surface Wave Tests in Landfills andEmbankments“, Proceedings, 3rd ASCE Conference on Soil Dynamics and Earthquake Engineeringand Soil Dynamics Conference, Seattle, Washington, USA, August, 3-6,1998.

Roësset, J.M., Chang, D.W., Stokoe, K.H. II (1991). “Comparison of 2-D and 3-D Modelsfor Analysis of Surface Wave Tests.” 5th International Conference on Soil Dynamics andEarthquake Engineering, Karlsruhe, Germany, 1991, pp. 111-126.

Rothenburg, L. (1980). “ Micromechanics of Idealized Granular Systems.”, Ph.D Thesis,Carleton University, Ottawa, Canada, 332 pp.

Bibliography 257

Sànchez-Salinero, I. (1987). “Analytical Investigation of Seismic Methods Used forEngineering Applications.”, Ph.D. Dissertation, The University of Texas at Austin, pp.401.

Satake, M. (1982). “ Fabric Tensor in Granular Materials.”, IUTAM Symposium On Deformationand Failure of Granular Materials, Delft, pp. 63-68.

Schwab, F., and Knopoff, L. (1970). “Surface-Wave Dispersion Computations.”, Bulletin ofthe Seismological Society of America, 60, 321-344.

Schwab, F., and Knopoff, L. (1971). “Surface Waves on Multilayered Anelastic Media.”,Bulletin of the Seismological Society of America, 61, 4, 893-912.

Schwab, F., and Knopoff, L. (1972). “Fast Surface Wave and Free Mode Computations.”,Methods of Computational Physics, Vol.11, Ed. Bolt, B.A., Academic press, New York,pp.87-180.

Seed, H.B., and Idriss, I.M. (1970). “Soil Moduli and Damping Factors for DynamicResponse Analyses.”, Report EERC 70-10, Earthquake Engineering Research Center,University of California, Berkeley.

Shibuya, S., Mitachi, T., Fukuda, F., and Degoshi, T. (1995). “Strain Rate Effects on ShearModulus and Damping of Normally Consolidated Clay.” Geotechnical Testing Journal, 18(3),365-375.

Spang, A.W., Jr. (1995). In Situ Measurements of Damping Ratio Using Surface Waves.Ph.D. Dissertation, Georgia Institute of Technology, pp. 347.

Stoll, R.D. (1974). “Acoustic Waves in Saturated Sediments.” Physics of Sound in MarineSediments, Plenum Press, 19-39.

Stokoe, K.H. II, Rix, G.J., and Nazarian, S. (1989). “In Situ Seismic Testing with SurfaceWaves.”, Proceedings, 12th International Conference on Soil Mechanics and Foundation Engineering,Rio De Janeiro, 13-18 August, pp. 331-334.

Takeuchi, H. and Saito, M. (1972). “Seismic Surface Waves.” Methods of Computational Physics,Vol.11, Ed. Bolt, B.A., Academic press, New York, pp.217-294.

Tang, X.M.. (1992). “A Waveform Inversion Technique for Measuring Elastic WaveAttenuation Using Cylindrical Bars.” Geophysics, 57, 854-859.

Thomson, W.T. (1950). “Transmission of Elastic Waves through a Stratified Solid Medium”,J.Appl.Phys., 21, 89-93.

Tikhonov, A.N. and Arsenin, V.Y. (1977). “Solutions of Ill-Posed Problems.”, Winston &Sons, Washington D.C., pp.258.

258 Bibliography

Ting, J.M., Corkum, B.T., Kauffman, C.R., and Green, C. (1989). “Discrete NumericalModel for Soil Mechanics.”, J. Geotechnical Engineering, ASCE, 115 (3), 379-398.

Ting, J.M., Khwaja, M., Meachum, L.R., and Rowell, J.D. (1993). “An Ellipse-Based DiscreteElement Model for Granular Materials.”, International Journal for Numerical and AnalyticalMethods in Geomechanics, V. 17, pp. 603-623.

Tokimatsu, K. et al. (1992). “Effects of Multiple Modes on Rayleigh Wave DispersionCharacteristics.” J. Geotechnical Engineering, ASCE, 118(10), 1529-1543.

Tokimatsu, K. (1995). “Geotechnical Site Characterization using Surface Waves.” Proceedings,First International Conference on Earthquake Geotechnical Engineering, IS-Tokyo '95, Tokyo,November 14-16, Balkema, Rotterdam, 1333-1368.

Truesdell, C. (1957). “ Sulle Basi della Termomeccanica.”, Accademia Nazionale dei Lincei,Rendiconti della Classe di Scienze Fisiche, Matematiche e Naturali (8), 22, 33-88, 158-166 (inItalian).

Truesdell, C. and Noll W. (1992). “The Non-Linear Field Theories of Mechanics.”, Springer-Verlag, 2nd Edition, p.591.

Tschoegl, N.W. (1989). “The Phenomenological Theory of Linear Viscoelastic Behavior -An Introduction.”, Springer-Verlag, Berlin, pp.769.

Visintin, A. (1994). “Differential Models of Hysteresis.”, Springer-Verlag, Berlin, pp.407.

Vrettos, C. (1991). “Time-Harmonic Boussinesq Problem For a Continuously Non-Homogeneous Soil.”, Earthquake Engineering and Structural Dynamics, 20, 961-977.

Vucetic, M., and Dobry, R. (1991). “Effect of Soil Plasticity on Cyclic Response.” J.Geotechnical Engineering, ASCE, 117(1), 89-107.

Vucetic, M. (1994). “Cyclic Threshold Shear Strains in Soils.”, Journal of GeotechnicalEngineering, ASCE, Vol.120, No.12, pp.2208-2228.

Wilmanski, K. (1996). “The Thermodynamical Model of Compressible Porous Materialswith the Balance Equation of Porosity”, Journal of Non-Equilibrium Thermodynamics, 21, pp.1-30.

Winkler, K.W., and Nur, A. (1979). “Pore Fluids and Seismic Attenuation in Rocks.”,Geophysical Research Letters, Vol. 6, 1-4.

Zeng, Y., and Anderson, J.G. (1995). “A Method for Direct Computation of the DifferentialSeismogram with Respect to the Velocity Change in a Layered Elastic Solid.”, Bulletin ofthe Seismological Society of America, Vol.85, No.1, pp.300-307.

simultaneous inversion of rayleigh phase velocity and attenuation

Documents