Surface Wave Methods · Prof. Luigi Sambuelli, my supervisor, continuously guided and supported my...

Claudio Strobbia

Surface Wave Methods Acquisition, processing and inversion

Dottorato di Ricerca in Geoingegneria Ambientale

POLITECNICO DI TORINO

Acknowledgements

I would like to thank all the people that supported, inspired, corrected this work, the people with whom I have worked,

and the people with whom, without working, I have spent my time and shared my space.

Prof. Luigi Sambuelli, my supervisor, continuously guided and supported my work, mitigating and tolerating my anarchist way of working. Gigi stimulated this work with his always high, unifying

and open point of view, with his extraordinary ability in doubting and searching. Dr. Valentina Socco suggested the subject of this dissertation, and introduced me to geophysics, in

front of a sliding door. Then Valentina has been a necessary source of corrections, suggestions, ideas, and she reviewed all this work with attention, care and patience, putting order in my chaos. Dr. Sebastiano Foti has been an indispensable guide, with his rigorous approach to the problems,

his suggestions and ideas. Moreover I have to thank Sebastiano for all what he gave me to read, and for his enthusiasm.

Dr. Alberto Godio gave me the chance to work with him and shared his experience in different fields. Alberto’s irony reached his maximum in our discussion about inverse problems.

Dr. Isabel Lopes read this work when it was a draft, and her suggestions and questions strongly influenced this final text.

The other colleagues with whom I have worked, Gianluca, Andrea, Stefano, Cesare, and the all the students that helped in the acquisitions.

My family, all my friends.

A, for her presence, and her mouse.

Table of Contents

i

1. INTRODUCTION

1.1 MOTIVATION AND RESEARCH OBJECTIVES 2 1.2 OVERVIEW OF THE SURFACE WAVE METHOD 3

1.2.1 Definition of the surface wave method 4 1.2.2 Rayleigh waves, dispersion, modal superposition (chapter 2) 4 1.2.3 Modelling the propagation of Rayleigh Waves: simulation of the dispersion curve 7 1.2.4 Testing with surface waves: measuring the dispersion curve at a site 8 1.2.5 Inversion of the dispersion curve (chapter 6) 8 1.2.6 The three steps of the method 9

1.2.6.1 Acquisition (chapter 3) 10 1.2.6.2 Processing (chapter 4) 11 1.2.6.3 Inversion (chapter 5) 12

1.2.7 Limits and Advantages of the SWM (chapter 6) 13

1.3 OPTIMISATION OF THE TEST 14

2. SURFACE WAVES THEORY AND MODELLING OF THE TEST

2.1 INTRODUCTION 16

2.1.1 Body waves and surface waves 16 2.1.2 Body waves in seismic exploration 18 2.1.3 The use of surface wave for geotechnical characterization 20

2.2 SURFACE WAVES IN A HOMOGENEOUS MEDIUM 21 2.2.1 Introduction 21 2.2.2 Basic equation for elastic wave propagation 21

2.2.2.1 Newton's equation of motion 21 2.2.2.2 Hamilton's principle and Euler-Lagrange equation 23

2.2.3 A simple solution of the equation of motion in an infinite medium: body waves 24 2.2.4 Potentials for a general solution 25 2.2.5 Plane waves 27 2.2.6 General solution of the wave equation for the half space 28

2.2.6.1 Separation of variables 28 2.2.6.2 Potentials as a Fourier transform 30

2.2.7 Rayleigh waves as solution of the wave equation for the half space 30 2.2.8 Characteristics of Rayleigh waves 31

2.2.8.1 Velocity 31 2.2.8.2 Displacements 33 2.2.8.3 Energy 24 2.2.8.4 Geometric spreading 25

2.2.9 Conclusions 26

2.3 RAYLEIGH WAVES IN LAYERED MEDIA: THEORY 37 2.3.1 Introduction 37 2.3.2 Rayleigh problem as a linear eigenproblem 38 2.3.3 Solutions of the Rayleigh problem 39 2.3.4 The matrix formulation 40 2.3.5 Dissipative phenomena: linear viscoelasticity 45

Table of Contents

ii

2.4 SIMULATION OF THE RAYLEIGH WAVE PROPAGATION IN LAYERED MEDIA 48 2.4.1 Computational problem for the modal curves computation 48

2.4.1.1 The problem 48 2.4.1.2 Dunkin’s restatement 49 2.4.1.3 Details for the computation 51

2.4.2 Computation of modal curves 54 2.4.3 Properties of the modal curves 56

2.4.3.1 Example 1 58 2.4.3.2 Example 2 58 2.4.3.3 Example 3 59 2.4.3.4 Example 4 60

2.4.4 Scaling Properties of modal curves 61 2.4.4.1 Introduction 61 2.4.4.2 Layer Velocities 61 2.4.4.3 Layer Thickness 63

2.4.5 The group velocity 66 2.4.6 Eigenfunctions: displacements and stresses 66 2.4.7 Computation of the displacements at the free surface 70 2.4.8 Geometrical attenuation and modes superposition 71 2.4.9 Simulation of a real test 73 2.4.10 Apparent dispersion curve 74 2.4.11 Computation of synthetic seismograms 76 2.4.12 Partial derivatives 79

2.5 COMPUTER PROGRAM FOR THE FORWARD MODELLING 80

3. ACQUISITION

3.1 INTRODUCTION 82 3.2 PHYSICAL NOISE 84

3.2.1 Random noise 84 3.2.1.1 Nature of random noise 84 3.2.1.2 The stacking procedure to increase the signal to noise ratio 85 3.2.1.3 Use of the coherence function to evaluate the signal to noise ratio 87

3.2.2 Coherent noise 87 3.2.2.1 Coherent noise, model and information

873.2.2.2 Types of coherent noise 88 3.2.2.3 Energy of coherent noise 88 3.2.2.4 Body waves as coherent noise 89 3.2.2.5 Scattered surface waves as coherent noise 91 3.2.2.6 Lamb Waves as coherent noise 92 3.2.2.7 Air Blast 96 3.2.2.8 Near Field and Far Offset 96 3.2.2.9 Lateral Variations 97 3.2.2.10 Higher modes: coherent noise or useful information 99

3.2.3 Filtering 102

3.3 ACQUISITION LIMITS 104

3.3.1 Acquisition parameters and recorded information 104 3.3.2 The ideal and real acquisition 104 3.3.3 General limits and consequences of the spatial sampling 106

3.3.3.1 Wavenumber resolution 107 3.3.3.2 Maximum wavenumber and spatial aliasing 109 3.3.3.3 Zero padding and windowing 111 3.3.3.4 Spatial Windowing 112

Table of Contents

iii

3.4 DESIGN OF THE ACQUISITION PARAMETERS 114 3.4.1 Acquisition Layout 114

3.4.1.1 Receiver Spacing 114 3.4.1.2 Array length 115 3.4.1.3 Source off-set 116

3.4.2 Strategies to face the limitations 118 3.4.2.1 More acquisition with different spacings 119 3.4.2.2 Moving the source 119 3.4.2.3 Non even-spaced arrays 119

3.4.3 Time sampling 121

3.5 EQUIPMENT AND TECHNOLOGY 121 3.5.1 The source 121 3.5.2 Impulsive sources 125

3.5.2.1 Effects of the source depth 126 3.5.3 Vibrating sources 128

3.5.3.1 Sweep 128 3.5.3.2 S/N in sweep signals 130 3.5.3.3 Monochromatic signals 131

3.5.4 Sensors 133 3.5.4.1 Response of the receivers 133 3.5.4.2 Coupling 135 3.5.4.3 Use of two components receivers 136

3.5.5 Digital recorder 137 3.5.6 Description of the equipment used for the experimental work 137

4. PROCESSING

4.1 BASIC PRINCIPLES OF THE PROCESSING OF SWM DATA 140 4.2 DIFFERENT APPROACHES AND TECHNIQUES 141

4.2.1 Overview 141 4.2.2 Steady state Rayleigh method 141 4.2.3 Spectral Analysis of Surface Waves (SASW) 142

4.2.3.1 Signal quality evaluation 146 4.2.4 ω-p transform 148

4.2.4.1 The slant stack in seismic processing 148 4.2.4.2 Computing the t-p transform 150 4.2.4.3 The t-p transform for SWM processing 151

4.2.5 f-k transform 153 4.2.5.1 The f-k transform in seismic processing 153 4.2.5.2 Computing the f-k transform 154 4.2.5.3 The f-k transform and SWM processing 155

4.2.6 Crosscorrelation 156 4.2.7 Transfer function 157 4.2.8 Multi offset Phase inversion 158 4.2.9 Time frequency analysis 161

Table of Contents

iv

4.3 COMPARISON OF THE PROCESSING TECHNIQUES 164 4.3.1 Are they so different? 164 4.3.2 Sensitivity to random noise 165

4.3.2.1 Uncertainty of the recorded traces 165 4.3.2.2 Effects of the Array length and the number of receivers on the uncertainty of the wavenumber 170 4.3.2.3 Effects of the amplitudes of the traces on the estimated K 174 4.3.2.4 Effect of the offset of the error on the estimated wavenumber 176 4.3.2.5 Amplification on the phase velocity 177

4.3.3 Sensitivity to coherent noise 180 4.3.4 Mode separation possibility 181 4.3.5 Coherency with a simple model for inversion 183

4.3.5.1 Model parameters and test parameters 183 4.3.5.2 The forward model for SWM and the test parameters 183

4.3.6 Choice of the f-k processing 184

4.4 THE MULTI OFFSET PHASE INVERSION 185 4.4.1 Statistical analysis of the phase and amplitude 186 4.4.2 Processing of the phase 188

4.4.2.1 Unwrapping of the phase 188 4.4.2.2 Least squares estimation of the wavenumber 189 4.4.2.3 Weighted least squares estimation of the wave number 191

4.4.3 Testing the model 194 4.4.4 Identification of the lateral variations 195 4.4.5 Processing the amplitude 196 4.4.6 Influence of the higher modes 197

4.5 THE F-K PROCESSING 198 4.5.1 The f-k spectrum: zero padding and windowing 198

4.5.1.1 time 199 4.5.1.2 offset 199 4.5.1.3 Windowing 200

4.5.2 Search of maxima 202 4.5.2.1 Maxima position in the spectrum 202 4.5.2.2 Simple search of maxima 203 4.5.2.3 Spectrum deconvolution for modes identification 206

4.5.3 F-X filtered f-k 207 4.5.4 Uncertainty estimate 209

4.6 THE COMPUTER PROGRAM FOR THE PROCESSING 211

5. INVERSION

5.1 INTRODUCTION 214 5.1.1 The inversion as the last step of SWM 214 5.1.2 Importance of the inversion 216

5.2 BASIC CONCEPTS ABOUT INVERSION 217 5.2.1 Statement of the problem 217 5.2.1.1 Linear problems 217 5.2.1.2 Non linear problems 217

Table of Contents

v

5.2.2 Solution techniques 218 5.2.2.1 Different approaches for the solution of linear problems 218 5.2.2.2 Solution of non linear problems 220

5.2.3 Uncertainty of data 223 5.2.4 Information in data 223

5.2.4.1 Number of Measured Data 223 5.2.4.2 Independence of data 226 5.2.4.3 Singular value decomposition (Lanczos decomposition) 227 5.2.4.4 Effects of the lack of information in data 231

5.2.5 Resolution and variance 233 5.2.6 Parameterisation and Ockham’s razor 235 5.2.7 Data error and modelisation error 236 5.2.8 Testing the model 237

5.3 INVERSION OF SURFACE WAVE DATA 239 5.3.1 Inversion techniques for SWM 239 5.3.2 Factors influencing the solution 240 5.3.3 Considerations on the data to be inverted 240

5.3.3.1 Phase velocity or group velocity dispersion curve 240 5.3.3.2 Modal curves 241 5.3.3.3 Apparent dispersion curve 243 5.3.3.4 Energy picks 244 5.3.3.5 How many points are necessary in the dispersion curve? 246

5.3.4 Considerations on the parameters to be inferred 246 5.3.4.1 Physical and geometric parameters 246 5.3.4.2 Layers 249

5.3.5 Possible errors 253 5.3.5.1 Example 1 254 5.3.5.2 Example 2 257 5.3.5.3 Example 3 258

5.3.6 Estimate of the uncertainty of the model parameters 259 5.3.7 Observations on the low frequencies 260

5.4 THE COMPUTER PROGRAM FOR THE INVERSION 262

Extract from PhDThesis - Claudio Strobbia. 1

1

CHAPTER 1

Introduction

This chapter is an introduction to the dissertation and to the surface wave methods, it is a schematic description of all the main theory and operating steps involved, and is completely qualitative. The following chapters(2-6) will deal in detail with the theory of the propagation, the acquisition, the processing, the inversion, and will finally present a selection of experimental results. The main aim of the chapter 1 is hence to explain how the subjects that are developed later are related each other, and why they are necessary.

In this chapter

The qualitative theory of the surface wave propagation: the geometric dispersion, the modal superposition, the apparent dispersion curve. The possibility of simulating the surface wave propagation, and obtaining a synthetic dispersion curve for a given stratigraphy.

The acquisition of vibrations and the processing of these data to get an experimental dispersion curve at a site.

The inversion of the experimental dispersion curve to find the layered earth model.

The three steps of the method: acquisition, processing and inversion.

How each step can increase the accuracy and reliability of the result: information, variance, parameterisation.

Chapter 1.1 MOTIVATIONS AND RESEARCH OBJECTIVES


1.1 MOTIVATION AND RESEARCH OBJECTIVES

The characterisation of the medium is one of the most important step in geotechnical engineering: in situ seismic techniques allow the identification of soil properties at large scale, under undisturbed conditions, at very low strain. The analysis of seismic wave propagation can be a tool for the identification of soil properties: the elastic and dissipative characteristics are obtained in dynamic conditions.

Such parameters are necessary for analysing dynamic problems (site response evaluation, liquefaction potential evaluation and earthquake engineering, foundation of vibrating machines), but they can be very important in the characterization of hard-to-sample deposits, since they correlate with other mechanical properties at higher strain level.

The propagation of seismic body waves has been widely used for the characterization of the subsoil for different purposes, with a great number of different techniques. Compressional and shear waves are employed in seismic reflection and refraction, in vertical seismic profiling, in up-hole, down-hole, cross-hole tests, in seismic tomography, sonic log and so on.

Surface waves, often called the ground roll, have usually been considered as a coherent noise masking useful signals; something to be filtered out.

The use of surface waves for the characterization of the shallow subsurface has recently, in the last decades, become of growing interest among geotechnical engineers and geophysicist. Nevertheless, the test are often performed adapting equipment, acquisition techniques, modelling and inversion algorithms conceived for other applications. The different approaches that have been developed will be described in the following chapters.

This dissertation has a geophysical point of view, and focuses on the test itself, with the aim of studying the best procedure to get a more accurate and reliable estimate of the soil properties with a reasonable effort. The final aim is hence the optimisation of the test.

In the different steps of the surface wave test the specific requirements and limitations are emphasized, different approaches are presented and compared, in order to select the most suitable.

A series of algorithms have been implemented, for the data processing, for the simulation of the test and for the inversion. They are included in a software package, PoliSurf, result of the work of the surface wave group at Politecnico di Torino, which allow an interactive automated interpretation of surface wave data.

Many field tests have been performed at different sites: the effectiveness of both the method and the procedures have been checked, the limits have been confirmed. The comparison with other geophysical seismic techniques is also supported by a wide set of experimental data.

Chapter 1.2 OVERVIEW OF THE SURFACE WAVE METHOD


1.2 OVERVIEW OF THE SURFACE WAVE METHOD

The use of surface waves for the characterization of the subsurface has been introduced by seismologists for the investigation of the earth crust and upper mantle (Ewing et al, 1957; Dorman et al, 1960; Dorman and Ewing, 1962; Bullen, 1963; Knopoff 1972; Kovach 1978; Keilis-Borok et al., 1989; Mokhart et al., 1988; Al-Eqabi and Herrmann, 1993; Herrmann and Al-Eqabi, 1991).

At an opposite very small scale, ultrasonic Rayleigh waves have also been used for the material characterisation and for the identification of surface defects (Viktorov, 1967).

The engineering applications to an intermediate scale are maybe the youngest. Jones (1958, 1962) and Ballard (1964) developed the first measurement system. A strong impulse came from the introduction of the SASW method (Nazarian and Stokoe 1984, 1986; Stokoe and Nazarian, 1985; Stokoe et al., 1988, 1994; Roesset et al., 1991; Gukunski and Woods 1991; Tokimatsu 1992a ), that has been used for a wide variety of applications. The use of multi-station techniques allowed a more robust and stable estimation of the subsoil properties. Many authors developed the acquisition, processing and inversion techniques, and described applications for different purposes (McMechan and Yedlin, 1981; Gabriels et al., 1987; Tokimatsu 1995; Tselentis and Delis, 1990; Park et al., 1999; Xia et al., 1999; Foti, 2000). The use of Rayleigh wave at the small scale for the determination of the soil damping has been discussed by Lai (1998) and Rix et al. (2001). Passive methods for the subsoil characterisation using microtremors are presented by Horike (1985), Tokimatsu et al. (1998b), Zwicky and Rix (1999).

Jongmans and Demanet (1993) discussed the importance of surface wave analysis for the estimation of the dynamic characteristics of soils. Applications of analysis of Rayleigh, Stoneley, and Love waves to a variety of engineering problems is discussed by Glangeaud et al. (1999) , the use of Scholte- Rayleigh waves in marine applications is for instance reported by Shtivelman (1999). Comparison of surface wave results with other geophysical results can be found in Abbiss (1988), Hiltunen and Woods (1988), Shtievelmann (1999), Foti et al. (2002). The use of multi-mode nature of surface waves is reported by Szelwis and Behle (1984), by Gabriels et al (1987), by Socco et al. (2002). Application of surface waves for the processing of body waves data is discussed by Mari (1984)

An introductory overview of the surface wave method is necessary to understand the motivations of the different chapters, each one dealing with a specific step of the test (acquisition, processing, modelling, inversion). The final aim of the study is of course the most accurate and reliable estimate of the subsoil properties, and this introduction would explain how the different step of the test affect the result, and how they influence each other.

The general description provided in this first chapter is on purpose schematic and qualitative, without the mathematics that will be necessary later to explain and proof the details of the aspects that are here just outlined: also the references will be given in the following chapters.



1.2.1 Definition of the surface wave method

The surface wave method (SWM) is a seismic characterisation method based on the analysis of the geometric dispersion of surface waves, by which the vertical distribution of the dynamic shear modulus in the subsoil can be obtained: the procedure consists of estimating the dispersive characteristics at a site, (by means of acquisition and processing of seismic data), and then of inverting these data to estimate the subsoil properties. The obtained result is the vertical profile of the shear wave velocity (Figure 1.1).

Figure 1.1. The result of a SWM test is the vertical profile of the shear wave velocity (or of the stiffness G0).

The use of Rayleigh waves allow a simpler acquisition and is widely more diffused than that of Love Waves.

1.2.2 Rayleigh waves, dispersion, modal superposition (chapter 2)

Surface waves are seismic waves propagating parallel to the earth’s surface without spreading energy through the earth’s interior: their amplitude decreases exponentially with the depth, and most of the energy propagates in a shallow zone, roughly equal to one wavelength. The solicited zone is so bounded in depth, and consequently the wave propagation is influenced by the properties of this limited portion of soil (Figure 1.2). Surface Rayleigh waves often are dominant events in seismic records, because of their greater energy and because of a geometrical spreading that is lower than that of body waves, that spread energy in all directions (spherical spreading in a homogeneous medium).

Figure 1.2. Energy propagating in the horizontal direction, as a function of the depth normalized to the wavelength

(homogeneous half space). The energy does not propagate in the vertical direction.



The propagation of surface waves in a vertically heterogeneous medium shows a dispersive behaviour. Dispersion means that different frequencies have different phase velocities: in particular the geometric dispersion, in opposition to the intrinsic dispersion due to the material, depends on the geometry of the tested subsoil.

In a homogeneous medium, the different wavelengths “sample” different depths of the subsoil, but being always the same material, all the wavelengths have the same velocity (Figure 1.3 left)

If the medium is not vertically homogeneous, for instance it is layered, with layers having different mechanical properties, the behaviour of surface waves is different. The different wavelengths “sample” different depths to which different mechanical properties are associated: each wavelength propagates at a phase velocity depending on the mechanical properties of the layers involved in the propagation (Figure 1.3 right)

Figure 1.3. In a homogeneous medium (left) all the wavelengths sample the same material, and the phase velocity is

constant. When the properties changes with the depth (right) the phase velocity depends on the wavelength.

So the surface wave does not have a single velocity, but a phase velocity that is function of the frequency: the relation between frequency and phase velocity is called dispersion curve (Figure1.4), and depends on the stratigraphy. At high frequency the phase velocity is the Rayleigh velocity of the uppermost layer: at low frequency the effect of deeper layers become important, and the phase velocity tends asymptotically to the Rayleigh velocity of the deepest material, as if it extends infinitely in depth (the half space).

Figure 1.4. The dispersion curve describes the dispersion of surface waves:

usually is depicted as phase velocity versus frequency.



The dispersion curve can be experimentally measured and hence it is possible to interpret this curve in order to estimate the subsoil properties. It is a worthwhile effort to understand first which parameters actually influence the dispersion curve that can be measured.

The Rayleigh wave propagation in vertically heterogeneous media, is actually a multi-modal phenomenon: for a given stratigraphy, at each frequency different wavelength can exist (Figure 1.5).

Figure 1.5. Different modes of vibration means different wavelength, and so different velocities, for the same

frequency.

Hence different phase velocities are possible at each frequency, each of them corresponding to a mode of propagation, and the different modes can exhibit simultaneously (Figure 1.6).

Figure 1.6. In a usual frequency range for engineering testing, many modes can exist.

Higher modes exist only above their cut-off frequencies.

The different modes, except the first one, exist only above their cut-off frequency, which is for each mode the lower limit frequency at which the mode can exist: with a finite number of layers, in a finite frequency range, the number of modes is limited. At very low frequency, below the cut-off frequency of the second mode, only the first mode, known as the fundamental Rayleigh mode, exists.

Modes are not just theory or just mathematical possible solutions; they are often observed in experimental data, also in the frequency ranges of interest for a test for engineering purposes.

frequency

phas

e ve

loci

ty

1st mode

2nd mode3rd mode

4th mode

fC2 fC3 fC4 frequency

phas

e ve

loci

ty

1st mode

2nd mode3rd mode

4th mode

fC2 fC3 fC4



The energy associated to the different modes depends on many factors, the stratigraphy at first, but also the depth and the kind of source.

The first mode is sometimes dominant in wide frequency ranges, but in many common situations higher modes play important roles and are actually dominant, and so they cannot be neglected.

The different modes have different phase velocity, and so they separate at great distance from the source: otherwise they superimpose on to one another, and the mode identification can be impossible.

At the engineering scale the modal superposition is critical: the effective Rayleigh phase velocity deriving from the modal superposition is only an apparent velocity, also depending on the observation layout, on the source orientation and position (Figure 1.7).

Figure 1.7. The apparent dispersion curve due to modal superposition can be influenced by different modes: it also

depends on the observation layout. They grey lines are modes, the black line is the apparent velocity.

1.2.3 Modelling the propagation of Rayleigh Waves: simulation of the dispersion curve

When modelling the propagation of this kind of waves, some considerations have to be made: the modal curves are a characteristic of the site and can be theoretically computed considering only the properties of the layered model.

On the other hand, what is actually observable in a real test is only an apparent dispersion curve depending on the modal superposition. If the apparent dispersion curve has to be simulated, the energy distribution, among modes and within each mode as a function of the frequency, and the modal superposition have to be taken into account. The energy distribution depends also on the source, and the modal superposition affecting the apparent dispersion curve depends on the observation layout. So, if the apparent dispersion curve has to be predicted, all these factors have to be taken into account: the modelling algorithm has then to consider the modal superposition in computing the displacements in a real test configuration.

If the subsoil is schematised as a stack of homogeneous linear elastic layers of known properties and for each layer four parameters are needed. The thickness (apart from the last layer, the half space), the mass density, and two mechanical parameters: the shear and compressional velocities can be used, or the Lame’s constant λ and G, or the shear and bulk moduli G and K, or any other possible couples, for instance the shear velocity and the Poisson ratio (Figure1.8). The use of complex velocities or complex moduli allows to describe the visco-elastic behaviour of the materials, taking into account for the dissipative phenomena.



Figure 1.8-a. The parameterisation of the subsoil. For each layer the thickness H, the shear

velocity Vs, the Poisson Ratio, and the mass density

Figure 1.8-b. The modelling allows the prediction of the surface wave

propagation properties for a given layered model.

1.2.4 Testing with surface waves: measuring the dispersion curve at a site

It is possible to estimate the dispersive characteristics at a site analysing the data of the propagation, and this is the basis of the SWM. An acquisition of seismic data is followed by a processing procedure that estimates the phase velocity as a function of the frequency.

Figure 1.9. The processing of acquired seismic data can estimate the experimental dispersion curve at a site.

1.2.5 Inversion of the dispersion curve (chapter 6)

As it has been said, it is possible, given a layered medium of known properties, to compute theoretically the Rayleigh wave dispersion, considering all the parameters influencing the measurable apparent dispersion curve. On the other hand, the dispersion curve can be experimentally measured at a site. The surface wave methods are then based on the interpretation of this curve: if the dispersive characteristics at a site can be evaluated, we can estimate the correspondent soil properties. A common approach for the characterisation, in geophysics, is the



statement of an associated inverse problem: the unknown model parameters are determined from measured quantities, assuming a model that relates the two.

The inversion consist of estimating the parameters of a postulated model of the subsoil from a set of observation, trying to fit a schematic response to a finite set of actual observations. The quantitative description of complex systems needs a scheme or model that simplifies the reality, and allows a quantitative finite description: the approach cannot be complete, the geometry and the physics have to be simplified and often coupled physical aspects have to be uncoupled.

1.2.5.1 The SWM inverse problem Given a layered system, the Rayleigh waves show a dispersive behaviour: the different frequencies propagate at different velocities. The experimental data set consists then of the phase velocities as function of the frequencies and depends on the unknown properties of the system: this can be used to make inference about the unknown properties of the system.

Once the data are collected, they are inverted to find the model parameters, assuming a scheme of the subsoil (Figure 1.8) and a forward model (Figure 1.10).

Figure 1.10. The forward problem (a) and the inverse problem(b) for the surface waves method.

1.2.6 The three steps of the method

It is common to consider the method as a batch process of three phases: acquisition, processing and inversion (Figure 1.11).

The acquisition consists of gathering raw data of the seismic wave propagation containing Rayleigh waves in a wide frequency band. From these data the processing extracts the information about the dispersion of Rayleigh waves, which is then used by the inversion to estimate the model parameters, i.e. the mechanical properties of the subsoil. The inversion procedure is based, of course, on the forward modelling that is able to compute the Rayleigh wave propagation for a known model.

a)

b)



Figure 1.11. Scheme of the three steps of the method, with the corresponding result.

1.2.6.1 Acquisition (Chapter 3) The acquisition is usually performed generating seismic waves with an active source and recording the full waveform. The equipment can be very different: impulsive or vibrating sources, or even noise, in passive methods, can be used to generate surface seismic waves; vertical and horizontal geophones or accelerometers to detect the vibrations; digital recorders such as seismographs or signal analysers can be used for the recording or the analysis of the seismic waves.

The number of receivers, the sampling parameters, the layout geometry, and all the acquisition parameters are important: first, for the acquisition of high quality data, then for their effects on the processing, that have to be carefully taken into account

I. ACQUIRING GOOD-QUALITY DATA The acquisition is simply the observation and recording of the propagation of Rayleigh waves in time and space: raw data should contain Rayleigh waves in a wide frequency range.

The acquisition should be designed and performed so that the influence of the other events, such as body waves, non source-generated Rayleigh waves, back scattered Rayleigh waves is reduced in records. An acquisition allowing the identification and removal of such coherent noise should be preferred.

Even if surface waves are usually high energy dominant events, great care is needed not to deteriorate signal to noise ratio that can be poor in certain frequency range, and signal enhancement techniques can be needed where the environmental noise level is high compared with the signal.

Since a 1D model is usually assumed, the phenomenon is supposed to be stationary in the horizontal direction, and so the space sampling should not be critical: the observation in two points, from a theoretical point of view, is enough, when a single mode is dominant. On the other hand, a multi-station approach can be proven to be necessary to reduce the variance of the estimated dispersion curve (chapter 4), to increase the amount of extracted information and to recognize and reduce the effects of coherent and random noise: thus allowing a better model parameter estimate.



In order to infer the value of the phase velocity at different frequencies, different approaches can be followed: either monochromatic signals are produced and recorded, or transient signals are recorded and then decomposed, in the processing step, into their monochromatic components.

II. CONSIDERING THE ACQUISITION LIMITATIONS A proper design of the acquisition parameters can increase the signal quality: moreover, from a theoretical point of view, it is very important to analyse the effects of all the acquisition parameters on the final result of the processing. In order to compare the experimental curve with the predicted curve a coherent modelling is needed: all the acquisition parameters influencing the curve have to be taken into account in the modelling. So we have to model the loss of information, or the limitations, due to the acquisition.

Some limitations are due to the used technology (source, receivers, digital recorder…), but some limitations come directly from the sampling in space and time: in particular the mode separation possibility and the maximum number of identifiable modes directly depend on the spatial sampling.

1.2.6.2 Processing (chapter 4) The processing consists of extracting from the raw data the dispersive characteristics, i.e. the phase velocity as a function of the frequency. The spectral analysis of data is then a common aspect of the different processing techniques: a real wave train generated, for instance, by an impulsive source contains different frequency components, that can be easily extracted by the Fourier Transform.

On decomposed signals, the phase velocity is computed from the time delay, that can be obtained in many different ways, based on the common hypothesis that surface waves are associated to energy maxima. The problem is complicated by the presence of other seismic events, and of lateral variations, by the attenuation of the signal with the distance, and by the multi-modal nature of the Rayleigh wave propagation in vertically heterogeneous media

Some techniques directly extract the phase lag or the time shift between the main event of acquired seismic traces: so they do not allow the selection of interesting events, or of different modes of propagation, or the identification of the velocity of other propagation phenomena.

Other techniques, on the contrary, transform the whole wave field into another domain, where the propagating energy is mapped as a function of the frequency and the velocity, or the slowness, or the wavenumber. These transforms are invertible, and no loss of information occurs: in these domains the different events are identified, filtered, and then the dispersive characteristic are extracted.

The redundancy of data of multi-station techniques allows also to identify lateral variations, and to perform a test on the assumption of one-dimensional model.



1.2.6.3 Inversion (chapter 5) The object to be studied has firstly to be simplified and reduced to a scheme. A finite minimal set of model parameters is assumed to be able to describe the system, at least for the physical aspect that has to be investigated: this is the parameterisation (Figure 1.12a).

Then a set of mathematical relations is used to theoretically predict the physical behaviour of the system, when model parameters are known: then, when a known input is applied to the system, a set of data constituting the output can be predicted (Figure 1.12b)

What is called the forward problem is simply the application of these relations. For instance the subsoil is schematised as made up of homogeneous layers, whose properties are known and which completely describe the system: a set of relations allows to predict the displacement in time in each point of the medium, given as input the seismic source.

The opposite situation is when the output is known and the system unknown: in the inverse problem, the task is identifying the model parameters when the output is known (Figure 1.12c).

Then, when a set of measurable data depends on a set of model parameters by means of a known forward model, we can try to estimate the model parameters by inverting the measured data. The solution of inverse problems is based on the comparison between the measured values and the predicted values, and the goal is finding, by inversion, the set of parameters, subject to some constraints, that minimises the prediction error.

Referring to the example above, if we have measured the displacements, we can try to estimate the mechanical properties of the subsoil.

Figure 1.12. Forward and inverse problems. (a) The system is firstly parameterised. The forward problem (b) consist of predicting data using a model when model parameters are known. The inverse problem (c) consist in identifying model

parameters, when data are known.



1.2.7 Limits and Advantages of the SWM (chapter 6)

The comparison with other geophysical techniques shows advantages and limitations of the SWM.

The main limitation of SWM is that the assumed model is 1D, and so the result is one-dimensional: the 2D information present in data can be used only to give warnings on the inversion. Nevertheless with short arrays the lateral variations are often not critic.

On the other hand, the SWM presents many advantages.

The SWM overcome some intrinsic limitations of the refraction technique (hidden layer, velocity inversion, gradual variations), affecting both P-wave and S-wave refraction. It can increase the reliability of the results, and it can be applied in situations where the refraction does not work, when a stiff top layer due to a pavement is present.

The SWM shares with the S-waves surveys the advantages on the P-wave techniques: in saturated materials the sensitivity of P-waves to the mechanical properties of the solid skeleton can sometimes be very low, and P-wave surveys do not provide useful informatio. In such situations, the techniques that investigate the shear properties have to be used.

The acquisition of P-wave refraction and SWM data is very similar, and can even be performed simultaneously: some synergies between the two technique can be found.

Chapter 1.3 OPTIMISATION OF THE TEST


1.3 OPTIMISATION OF THE TEST

It is not a straightforward task to say directly the way in which each step of the test contributes to the result, and which are the optimal values of the different parameters of the test. Anyway, the final step is an inversion and produces the result that is the answer to our questions about the subsoil. Optimising the test means obtaining the best result, the most accurate and reliable vertical profile, with the minimum effort. The best result is the one showing small variances of the sought soil parameters within the desired depth.

Optimising the test means choosing the parameters of the acquisition, the processing and the inversion that allow to get that best result, with a reasonable effort.

The inversion has to give a low variance result: this is possible if the data that we invert have a low variance and a high information content. So the acquisition and the processing have this task: to give to the inversion a set of data with high information and low variance.


2CHAPTER 2

Surface Waves Theory and Modelling of the Test

This chapter illustrates the properties of the surface wave propagation and the algorithms that allow the simulation of the test and that will be used to solve the forward problem in the inversion. The dispersion of surface waves in vertical heterogeneous media is the physical principle on which the Surface Waves Methods are based: in this chapter the characteristics of the propagation of such waves in layered media are shown. The first aim is discussing the details of such propagation phenomenon, with a particular attention to the modal superposition at the small scale of engineering problems. The final objective and result of the chapter is a series of algorithms that allow the simulation of the test in a layered medium.

In this chapter

Some notes on the use of seismic waves for the characterisation.

The theory of surface wave propagation in homogeneous half space: this introduces the notation, the approach with potentials, and allow the description of the properties of surface waves in a very simple case.

The theory of Rayleigh wave propagation in layered media: the model of layered soil is described, the problem is stated, and the solution in matrix form is presented. The dissipative phenomena are considered with a linear visco-elastic model.

The solution of the Rayleigh problem is developed with the aim of simulating the test: the computational problem of the matrix formulation are presented, then the computation of the modal curves is obtained, the properties of the modal curves are described, then the stress and displacements are computed, and the displacements at the free surface. The effects of the modal superposition are presented, and the simulation of a real test can be performed.

Chapter 2.1 INTRODUCTION


2.1 INTRODUCTION

This short introduction contains some general ideas about seismic techniques based on body waves and surface waves. The differences between the two phenomena are just outlined, and the principles of body wave exploration are briefly mentioned. A general description of the surface wave methods closes this section, explaining why there is the need of a reliable and efficient modelling algorithm that simulate the Rayleigh wave propagation in layered media.

Actives geophysical techniques use a controlled propagation phenomenon in the investigated medium; the artificial impulse is generated, sent into the body, then observed and recorded to extract information about the unknown properties of the medium.

Seismic techniques are based on the study of the propagation of seismic waves: different kind of waves and different physical phenomena are involved and are used to get information about the interior of a body from the measures taken at its boundary. The most widely used body waves techniques are based on the reflection and refraction of such waves, while the surface waves methods use the dispersion that occurs in heterogeneous media.

2.1.1 Body waves and surface waves

A sudden application or variation of a force on a body produces stresses and strains that propagate within the medium as waves. Several kind of waves exist, depending on the material properties and on the geometry of the body. Body waves in homogeneous media propagate in all directions with spherical wave front, with a propagation velocity depending in a simple way on the elastic moduli of the medium: compressional waves and shear waves exist (Figure 2.1).

P-waves are compressional waves, (also called irrotational waves) with a particle motion parallel to the direction of propagation; in a elastic medium the propagation velocity of P-waves is(see for instance White, 1983)

( )ρ

µα

3/4+==

KVP ( )

ρµλ 2+

= (2.1)

where ρ is the mass density, K is the bulk modulus, and µ the shear modulus, λ is the Lame’s parameter. The relations between the elastic parameter and the corresponding expression for the seismic velocities are given in appendix A.

S-waves are shear waves (also called transverse waves, or distorsional waves, or equivoluminal waves) inducing a particle motion perpendicular to the direction of propagation; in an elastic medium the propagation velocity of S-waves is

ρµ

β == SV (2.2)

S-waves can be polarized, in such a way that the particle motion is along a definite direction perpendicular to the direction of propagation: SV-waves are polarized in the vertical direction, SH-



waves are polarized in the horizontal direction. It is obvious from (2.1 and (2.2 that the shear wave velocity is lower than the compressional wave velocity. As it can be seen from appendix A, the two body wave velocities are a pair of parameters that completely describes the behaviour of linear elastic medium.

Figure 2.1. Body waves P and S propagate inside the medium.

When the medium is bounded by a free surface, also another kind of phenomena occurs: surface waves are generated and propagate near the surface (Figure 2.2). They induce a different particle motion, and their velocity is a more complicated function of the moduli, as it will be shown below. With a point source they have cylindrical wavefronts, and they do not irradiate energy toward the earth’s interior,

Rayleigh waves induce a particle motion in the vertical plane that contains the direction of propagation, and in a homogeneous body is, at the free surface, a retrograde elliptic orbit (Figure 2.2, left). R-wave velocity in a homogeneous medium is less than the shear wave velocity. If there are variations of the elastic properties with depth, R-waves become dispersive: the different wavelength propagate with different velocities.

Love waves are surface waves existing only in special stratigraphic conditions, when the shear velocity in the upper layer is lower than that in the substratum; the particle motion is transverse to the direction of propagation and parallel to the free surface (Figure 2.2, right). They cannot exist in homogeneous media, and in layered media Love waves are dispersive: the velocity tends to the shear velocity of the upper layer at high frequency, and tends to the shear velocity of the substratum at low frequency.

The Stoneley waves are interfacial waves existing under special conditions at the interface between two media.



Figure 2.2. Surface waves, Rayleigh waves and Love waves, propagate parallel to the body surface

without spreading energy towards the interior.

A compressional source at the free surface imparts most of the energy in surface waves and moreover the geometric spreading of surface waves is lower than that of body waves. Therefore in active seismic techniques, Rayleigh waves actually dominate at great distance. Considering only the geometric attenuation, the energy is distributed on increasing surfaces, and so it decreases following the shape of the wavefront: with spherical wavefronts the energy decrease with the square of the distance, with cylindrical wavefront it decreases linearly with the distance. The energy is proportional to the square of the displacements, thus body waves have a decrease of amplitudes proportional to the distance from the source and Rayleigh waves proportional to the square root of the distance. Nevertheless, as shown by Richart et al. (1970), at the free surface, because of leaking of energy into the free space for body waves the displacement attenuation goes with the square of distance.

2.1.2 Body waves in seismic exploration

The use of body waves for the subsurface exploration is a well established practice for different applications and target, for very different depths (see for instance Yilmaz, 1997; Nolet, 1987, Hardage, 1983). Seismic techniques are exploration tools to measure the seismic velocity in the subsoil and to identify the boundaries between formations with different elastic properties.

Reflection and refraction of body waves that occur at seismic interfaces are the physical phenomena on which the seismic prospecting is based. Seismic reflection and seismic refraction are based on the analysis of travel times of different seismic events, observed at the surface or in boreholes: the seismic energy goes back to the surface by complex path along which the variations of acoustic impedance of the subsoil produce reflections and refractions (Figure 2.3).



Figure 2.3. The refraction and reflection of body waves are the physical principle used for seismic prospecting of the subsoil with measures taken at the surface.

P waves are widely used for the seismic prospecting: the use of S waves, in particular SH waves, for the geotechnical characterization is gaining interest because of a series of advantages. A greater resolution can be achieved due to the shorter wavelengths, and also a higher reflectivity and hence information especially in soft soil, and in particular in saturated media. Moreover the use of SH waves allows the same processing techniques and theoretical background with a different acquisition. The following figure (2.4) shows a seismogram of SH waves acquired with a single shot of a polarized source, with sensors Swyphones (Sambuelli et al., 2001) : a refraction is clearly visible from the 11th trace

Figure 2.4. A raw seismogram for SH waves shows a quite evident refraction.

In seismic reflection surveys, one of the main source of noise is the ground roll, a low velocity event that can mask late reflections. Ground roll is mainly composed by surface waves, and depends on the shallow structure at sites: it is a low frequency and high energy event created by the energization, and cannot be reduced by a stacking procedure being a coherent noise. To avoid its superposition to signals adequate acquisition geometry are used, and to enhance reflections many filtering techniques have been developed.

On the other hand surface waves can be successfully used to infer the properties of the first tens of meters: in seismic reflection processing this can be useful to compute the static corrections (Mari, 1984). Moreover, for the geotechnical characterisation, the properties of this seismic event are very interesting and surface waves can be considered an useful signal, and no more a noise: the same



processing techniques that have been developed to filter out the ground roll can be used to extract, enhance and analyse it.

2.1.3 The use of surface wave for geotechnical characterization

The analysis of the seismic propagation velocity and the identification of discontinuities by means of reflections and refractions are the tools of classical seismic surveying.

With surface waves methods the analysed phenomenon is the geometrical dispersion of Rayleigh waves in heterogeneous media. The principle is that different frequencies produce particle motion and deformation that are significant and not negligible within a limited depth depending on the wavelength, and so the propagation of the different frequencies occurs down to different depths below the surface. In vertical heterogeneous media the mechanical properties vary with the depth; the different wavelength propagate in layers of different properties and hence have different propagation velocities. This phenomenon is called geometric dispersion, and the relationship between the phase velocity and the frequency is referred to as dispersion curve.

It is possible to model the dispersion when the seismic stratigraphy is known. The characterization method is based on the inverse problem: when the dispersion characteristics have been experimentally evaluated at a site, we try to estimate the corresponding seismic model.

The first step of the test is then the acquisition: surface waves are produced (with active methods) or simply observed (passive methods) at a site, by measuring the particle velocity or acceleration at the free surface in different positions. From these data the phase velocity of different frequencies is estimated with a proper processing, and the apparent dispersion curve of the site, for the given acquisition geometry, is finally obtained.

The experimental apparent dispersion curve can be used as objective function for the inversion: the forward model is a 1D model of a layered medium. For each layer, using an elastic or viscoelastic model, 4 mechanical parameters are used (ρ, Η, VS VP): the problem is a linear differential eigenproblem, and has a multi-modal solution.

The main aim of this chapter is to obtain a forward modelling algorithm and finally a computer program able to simulate the Rayleigh wave propagation in layered media. A preliminary step is the description of the properties of Rayleigh waves in a homogeneous medium: propagation velocity, particle motion, energy, attenuation. Then the theory and the equations for a layered medium are presented. Finally the solution in layered media is discussed, the properties of Rayleigh waves in layered media are illustrated, and in the end the simulation of the real test is obtained.

Chapter 2.2 SURFACE WAVES IN A HOMOGENEOUS MEDIUM


2.2 SURFACE WAVES IN A HOMOGENEOUS MEDIUM

In this section the basic wave equations are stated, and the approach with displacement potentials for a general solution is presented. Then the solution of the equations for a medium with a free surface is obtained: the Rayleigh wave. The properties of the Rayleigh waves in a homogeneous half space are discussed, focusing on the velocity, on its sensitivity to mechanical parameters, on the decay of the particle motion with the depth.

2.2.1 Introduction

Surface waves are waves propagating in directions parallel to earth's surface: their amplitude decreases exponentially with depth and the most of the energy is confined near the earth's surface, so the propagation is influenced mostly by shallow materials.

The amplitude distribution is stationary with horizontal coordinates, apart from a geometrical spreading factor accounting for the increase of the surface of the wavefront. The energy in this kind of propagation spreads only in the horizontal direction, so that the geometrical spreading factor is smaller than for body waves, which spread energy both horizontally and vertically down into the earth interior.

Here we will describe the basic properties of surface waves in a homogeneous medium bounded by a free surface: this is useful to introduce the more interesting case of layered media.

The analytical approach followed here leads to discuss the properties of pure surface Rayleigh waves: obviously in a real propagation phenomenon they are superposed to body waves with direct, refracted and reflected paths.

2.2.2 Basic equation for elastic wave propagation

2.2.2.1 Newton's equation of motion At first we derive the properties of waves travelling in a homogeneous, isotropic, elastic body with a classical approach using the Newton’s law. A medium is homogeneous if its properties do not change in space, isotropic if they not depend on the direction, and elastic if the relationship between stress and strain is bi-univocal.

Using rectangular coordinates, in which the displacement u of a point has components in the three coordinates, xu , yu and zu . The six components of strain in terms of displacements are

xux

xx ∂∂

=ε y

uyyy ∂

∂=ε

zuz

zz ∂∂

=ε (2.3)

xu

yu yx

xy ∂∂

+∂∂

=ε zu

xu xz

zx ∂∂

+∂∂

=ε yu

zu zy

yz ∂∂

+∂

∂=ε



Forces distributed through the body and varying in time and space produce strains: in a linear elastic material we can express stresses acting on an elementary volume as a linear combination of strains (Hooke's law).

In general we can state klijklij a εσ = where a is a 4th order tensor:

considering the symmetry of stress and strain tensor, and with the hypothesis of isotropy we need only two elastic moduli as constants of proportionality. Different couples of parameters can be chosen (appendix A). Young's modulus and Poisson's ratio are often used, here we will use Lamè’s coefficients λ and µ to write the constitutive laws

( ) zzyyxxxxp λελεεµλ +++= 2 (2.4)

( ) zzyyxxyyp λεεµλλε +++= 2

( ) zzyyxxzzp εµλλελε 2+++=

xyxyp µε= xzxzp µε= yzyzp µε=

where p are the stress components

The acceleration of the element can be obtained from unbalanced forces, including linearised stress variations between nearby points: from the equilibrium of an elementary volume, in the X direction we get

zyxz

py

px

p

yxpzx

ppzxpy

yp

pzypxx

pp

xzxyxx

xzxz

xzxyxy

xyxxxx

xx

∆∆∆

∂

∂+

∂

∂+

∂∂

=

∆∆

−∆

∂∂

++∆∆

−∆

∂

∂++∆∆

−∆

∂∂

+

(2.5)

Assuming the body force B acting throughout the elementary volume (gravity or other fields) and introducing acceleration and mass zyx ∆∆∆ρ , we get the three components

2

2

tuB

zp

yp

xp x

xxzxyxx

∂∂

=+∂

∂+

∂∂

+∂

∂ρ (2.6)

2

2

tu

Bz

py

px

p yy

yzyyyx

∂∂

=+∂

∂+

∂∂

+∂

∂ρ

2

2

tuB

zp

yp

xp z

zzzzyzx

∂∂

=+∂

∂+

∂∂

+∂

∂ρ

Differentiating equation 2.4 and substituting in 2.6 we can eliminate stresses and express strains in term of displacement: we set body forces equal to zero and we get the Navier’s equation of motion

( ) ( ) 2

222

2

2

2

2

2

2

2tu

zxu

yxu

zu

yu

xu xzyxxx

∂∂

=

∂∂∂

+∂∂

∂++

∂

∂+

∂∂

+∂∂

+ ρµλµµλ (2.7)

( ) ( ) 2

222

2

2

2

2

2

2

2tu

zyu

yxu

zu

xu

yu yzxyyy

∂

∂=

∂∂

∂+

∂∂∂

++

∂

∂+

∂

∂+

∂

∂+ ρµλµµλ

( ) ( ) 2

222

2

2

2

2

2

2

2tu

zyu

zxu

yu

xu

zu zyxzzz

∂∂

=

∂∂

∂+

∂∂∂

++

∂∂

+∂∂

+∂∂

+ ρµλµµλ



that can be compactly written in vector notation as

( ) ( ) 2

22

tdivgrad

∂∂

=++∇uuu ρµλµ (2.8)

2.2.2.2 Hamilton's principle and Euler-Lagrange equation The equation of motion can be derived from the more general Hamilton's variational principle.

This is useful since in a compact way the same relation can be obtained, and it allows to take into account heterogeneities and other constitutive models: later it will help in a quick definition of the equations for a vertical heterogeneous body.

Hamilton's Principle is an integral principle that consider the entire motion of a system between time 1t and 2t . The instantaneous configuration of the system is described by the values of n generalized coordinates. As the system evolves, the system point moves in this hyperspace (very different from the physical three-dimensional space ) tracing out a curve, the path of motion of the system. A system described by spatial coordinates and velocities has a 6-dimensional configuration space.

Hamilton's principles consider the motion of a system described by a scalar potential function of the coordinates, velocities and time. The integral is a one dimensional form: it is referred to as the action given by

∫=2

1

),',(t

t

txxLA (2.9)

where L is the Lagrangian which, in a conservative system, is equal to the kinetic energy minus the potential energy

UTL −= (2.10) The dependence of x on t is not fixed, so x(t) is unknown and the exact path of integration is not known.

The Hamilton's principle states that the motion equation of a particle is equivalent to the minimization of the action

This implies the Euler-Lagrange equation

0'=

∂∂

∂∂

−∂∂

xL

td

xL

(2.11)

For a linear elastic material the potential energy is the elastic strain energy, and so the Lagrangian density is

( ) ijijiijii uuuuL εσρ21

21, , −= &&& (2.12)

where iu i=1,2,3 are the displacements, ijσ and ijε are the component of stress and strain, ρ is the mass density assumed constant with time.

Considering the Hooke's law compactly written as

ijijkkij µεδλεσ 2+= (2.13)



where λ and µ are Lame's moduli (in general function of the coordinates) and δ is the Kronecker's symbol the Lagrangian density is then

( )

+−= ijijkkiijii uuuuL εµελερ 2

, 21

21, &&& (2.14)

and the Euler - Lagrange equation yield, in vector notation,

( ) ( ) 2

22

tdivgrad

∂∂

=++∇uuu ρµλµ (2.15)

that is the Navier’s equation of motion for the indefinite medium.

2.2.3 A simple solution of the equation of motion in an infinite medium: body waves

If the particle displacement is parallel to the X axis and it is independent from y an z axis, we get from 2.15

( ) 2

2

2

2

2tu

xu xx

∂∂

=∂∂

+ ρµλ (2.16)

and a simple solution is

)/()/( αα xtgxtfux ++−= (2.17)

being f and g arbitrary functions and the propagation velocity is

ρµλ

α2+

= (2.18)

This is a compressional wave (of any time dependence) travelling in the X directions inducing particle motion in the X direction: the stresses in the different directions are of such a magnitude that lateral deformation are prevented.

If the particle displacement is in the Y axis direction and yu is independent from y and z, the equation 2.15 reduces to

2

2

2

2

tu

xu yy

∂∂

=∂∂

ρµ (2.19)

and a simple solution can be found as

)/()/( ββ xtgxtfu y ++−= (2.20)

being f and g arbitrary functions and the propagation velocity is

ρµ

β = (2.21)

In this case the particle motion is perpendicular to the direction of propagation. The strain is simple shear.



2.2.4 Potentials for a general solution

For the solution of the above equation, the use of potentials is very useful: by means of potential compressional and shear waves can be described, and this allows to select a combination of plane waves to satisfy a boundary condition.

The following theory is based on the Helmoltz’s theorem, which states that a vector field can be decomposed into the gradient of a scalar field and the curl of a vector field with zero divergence.

Ψu ×∇+Φ∇= with 0=⋅∇ Ψ (2.22)

and so the equation 2.15 can be written as

( )

=∇Φ=Φ∇+

ΨΨ &&

&&

ρµρµλ

2

22 (2.23)

The equation of motion can then be solved by means of displacement potentials.

Let’s considering the first term of 2.22 and the first equation of 2.23. If the three components of the displacements are obtained by a scalar potential Φ

xux ∂

Φ∂=

yuy ∂

Φ∂=

zuz ∂

Φ∂= (2.24)

the found displacements satisfy the equation of motion if the potential satisfies the wave equation

2

2

22

2

2

2

2

2 1tzyx ∂Φ∂

=∂

Φ∂+

∂Φ∂

+∂

Φ∂α

(2.25)

that is

2

2

22 1

t∂Φ∂

=Φ∇α

(2.26)

Equation 2.26 is exactly the first equation of 2.22, remembering the definition of α (2.18) . The simple case of compressional wave is obtained with Φ independent from y and z. The above equation admit as solution

−=Φ

αxtf (2.27

A compressional plane wave in an arbitrary direction can be obtained with the following solution of the above (2.25) wave equation for the scalar potential:

++−=Φ

αγλγ zyx zyx

tfcoscoscos

(2.28)

being xγ , yγ and zγ the angles between the direction of propagation and the three axis.

Let’s consider now the second term of 2.22, and the second equation of 2.23.

A solution of equation of motion (2.15) is found also introducing a vector potential Ψ with the three components xΨ , yΨ and zΨ such that



zyu yz

x ∂Ψ∂

−∂Ψ∂

= (2.29)

xzu zx

y ∂Ψ∂

−∂Ψ∂

=

yxu xy

z ∂Ψ∂

−∂Ψ∂

=

This displacements satisfy the equation of motion if the vector potential from which they are obtained satisfies the second vector equation 2.23 (remembering the definition of β 2.21)

2

2

22

2

2

2

2

2 1tzyx

xxxx

∂Ψ∂

=∂

Ψ∂+

∂Ψ∂

+∂

Ψ∂β

(2.30)

2

2

22

2

2

2

2

2 1tzyx

yyyy

∂Ψ∂

=∂

Ψ∂+

∂Ψ∂

+∂

Ψ∂β

2

2

22

2

2

2

2

2 1tzyx

zzzz

∂Ψ∂

=∂

Ψ∂+

∂Ψ∂

+∂

Ψ∂β

But this is not sufficient, since the components of the vector Ψ are non independent. To ensure that they are compatible with the request 0=⋅∇ Ψ we can derive them from two scalar functions χ and γ so that

yx ∂∂

=Ψγ (2.31)

xy ∂∂

−=Ψγ

χ=Ψz

The simple shear wave is obtained if xΨ , yΨ are taken equal to zero and zΨ is independent from y and z: the solution for equation (2.15) is a plane shear wave travelling in the positive x direction and inducing particle displacement along y

−=Ψ

βxtfz (2.32)

with the scalar functions satisfying

2

2

22

2

2

2

2

2 1tzyx ∂

∂=

∂∂

+∂∂

+∂∂ γ

βγγγ (2.33)

2

2

22

2

2

2

2

2 1tzyx ∂

∂=

∂∂

+∂∂

+∂∂ χ

βχχχ (2.34)



The result is that the three displacements xu , yu and zu (scalar functions) can be described by a combination of scalar and vector potentials which need only three scalar potentials Φ , χ and γ These potentials allows us to specify plane waves, shear or compressional, with any direction and any waveform: a combination of such solution is still a solution. This will be useful when we shall have to select a combination of plane waves to satisfy a boundary condition.

2.2.5 Plane waves

With the potential formulation of the wave equation in rectangular coordinates, solutions are exponential functions of space and time, with real, imaginary or complex coefficients in the exponents. For each boundary geometry there are some limitations for these coefficients, and in general the potentials must remain finite

Figure 2.5. The geometry of the problem and the coordinate system.

In order to simplify the discussion the solution are considered in the form of plane waves, rather than considering the waves diverging from a point source: this involves no loss of generality, since the point source solution may be developed by integration of plane wave solutions. For the case of Rayleigh waves it can be shown (Aki and Richards, 1980) that the z-dependence of plane wave and cylindrical waves is the same.

For simplicity let us consider a 2D geometry of a stress free plane XZ normal to the Y axis (Figure 2.5), lying in an elastic medium. Any compressional wave travels in the XZ plane, and the particle motion coincides with the direction of propagation, , so 0=yu . Reflections of P waves can generate shear waves: anyway shear waves will lie in the same plane, and also their associated particle motion. So we can consider solutions independent on y and for which 0=yu . This will include shear waves with motion in the XZ plane (called SV if the plane is bounded by the horizontal free surface of earth)

The equations written above for the potentials can be combined to give the wave equation to be satisfied by the potentials and to express stresses that must vanish at the free surface.

We have two wave equations for the two scalar potential,

2

2

22

2

2

2 1tzx ∂Φ∂

=∂

Φ∂+

∂Φ∂

α (2.35)

2

2

22

2

2

2 1tzx ∂

∂=

∂∂

+∂∂ γ

βγγ (2.36)



and the relation between the vector potential and the scalar potential γ ,

0=Ψx xy ∂

∂−=Ψ

γ 0=Ψz (2.37)

When the potentials have been found, the displacements can be then obtained summing the two displacements (from Φ and from Ψ ),

zxu y

x ∂Ψ∂

−∂Φ∂

= (2.38)

0=yu

xzu y

z ∂Ψ∂

+∂Φ∂

=

and the same holds for stresses:

∂∂Ψ∂

−∂

Φ∂−

∂Φ∂

=zxzt

p yxx

22

2

22

2

2

22 ββρ (2.39)

0=xyp

∂Ψ∂

+∂

Ψ∂−

∂∂Φ∂

= 2

2

2

22

22 22

tzzxp yy

zx ββρ

This set of equations can be used to generate solutions imposing the different boundary conditions.

2.2.6 General solution of the wave equation for the half space

2.2.6.1 Separation of variables

The equation above can be solved by separation of variables assuming that

( ) ( ) ( )tfzfxf 321=Φ and ( ) ( ) ( )tgzgxg 321=γ (2.40)

The dependence on x, z, t must be exponential for both the scalar potentials. From 2.37 it can be noticed that the dependence of yΨ has to be an exponential function of x, z, t, so it can be written

tMzKx eeeA Ω=Φ 0 (2.41)

tNzKxy eeeB Ω=Ψ 0 (2.42)

The coefficient M, N, K and Ω are in general complex, and from equations 2.35, 2.36 and 2.37 they have to satisfy the two relationships

2

222

αΩ

=+ MK (2.43)

2

222

βΩ

=+ NK (2.44)



The time is allowed to range between ∞− and ∞+ , and the same is for the elastic half-space in the x direction: to obtain finite potentials in the whole domain, Ω and K has to be pure imaginary, and hence we can write

ωi=Ω (2.45)

ikK = (2.46)

with ω and k real quantities ranging over positive and negative values.

With K and Ω pure imaginary M and N has to be either real or pure imaginary since their square 2M and 2N are the sum of real values (2.43 and 2.44), so they are real positive or negative.

( ) ( ) 2222222 // αωαω −=+−= kiikM (2.47)

( ) ( ) 2222222 // βωβω −=+−= kiikN (2.48)

Their square roots have to be either real or pure imaginary depending on the sign of the above differences

If 22222 // βωαω <<k M and N are both imaginary

If 22222 // βωαω << k M is real and N is imaginary

If 22222 // k<< βωαω M and N are both real

The definitions of M and N subject to the above differences can be compactly written

immaa

iaa

M +=

−+

+=

2/12/1

2sgn

2ω (2.49)

innbb

ibb

N +=

−+

+=

2/12/1

2sgn

2ω (2.50)

where 222 /αω−= ka (2.51) 222 / βω−= kb

The potentials satisfying the wave equations, being also solution of the equation of motion, may be written as:

( ) tiikxMzMz eeeAeA ω−+=Φ 21 (2.52)

( ) tiikxNzNz eeeBeB ω−+=Ψ 21 (2.53)

with the coefficients 1A , 2A , 1B , 2B functions of k and ω



2.2.6.2 Potentials as a Fourier transform

We can also usefully express the potentials as Fourier Transforms: the coefficients 1A , 2A , 1B , 2B can be functions of k andω , and the sum of terms for different values of k andω is still a solution.

Expressing the summation as integration over all values of k andω we get

( )( )

( ) ( )[ ]∫∫∞

∞−

−∞

∞−

+=Φ ωωωπ

ω dkdeeekAekAtzx tiikxMzMz ,,2

1,, 212 (2.54)

( )( )

( ) ( )[ ]∫∫∞

∞−

−∞

∞−

+=Ψ ωωωπ

ω dkdeeekBekBtzx tiikxNzNzy ,,

21,, 212 (2.55)

In the expressions above, the bracketed quantities are the Fourier transform of the functions of x, z and t which constitute the desired solution, and can be written as

( ) ( ) MzMz ekAekA −+= ωωϕ ,, 21 (2.56)

( ) ( ) NzNz ekBekB −+= ωωψ ,, 21 (2.57)

In equation 2.54 and 2.55 the variables x and t appear in the exponentials only, multiplied by k and ω respectively : this combination of variables describe a travelling wave with apparent velocity of

kc /ω= (2.58)

Substituting kc /ω= we get ( )cxtitiikx eee /−= ωω (2.59)

and introducing the same substitution into the definition of M and N

)/1(/ 2222222 ααω ckkM −=−= (2.60)

)/1(/ 2222222 ββω ckkN −=−= (2.61)

The sign of 2M and 2N , depending on k and ω , depends on the ratio between c and α and c and β . So, as we said before, M and N can be real or imaginary depending on the relation between c, α and β . Different possible solutions can be found depending on the ratio between the different velocities.

2.2.7 Rayleigh waves as solution of the wave equation for the half space

Considering the definitions of M2 and N2 rearranged in 2.60 and 2.61, there are three cases, c<< αβ , αβ << c and αβ <<c . The third case, the only one giving real values of M and

N, occurs when the propagation velocity is lower than the P and the S velocities and leads to Rayleigh surface waves

If the phase velocity have to be less than the shear velocity, both N and M are real and can be expressed (2.49, 2.50) as

mM = and nN =

and the solution for potentials can be written



( )( )

( ) ( )[ ]∫∫∞

∞−

−∞

∞−

+=Φ ωωωπ

ωω dkdeeekAekAtzx ticxizmzm /212 ,,

21,, (2.62)

( )( )

( ) ( )[ ]∫∫∞

∞−

−∞

∞−

+=Ψ ωωωπ

ωω dkdeeekBekBtzx ticxiznzny

/212 ,,

21,, (2.63)

where 2/1

22

11

−=

αω

cm

2/1

22

11

−=

βω

cn (2.64)

In order to determine the coefficients A and B we can impose the null potentials at infinity and the zero stresses at the free surface. The coefficient 1A and 1B must be zero to avoid exponential growth with depth and infinite value of Φ and Ψ . Then from each equation imposing vanishing stresses at the surface, a ratio between the two others coefficients 2A and 2B (equation 7) is obtained:

( )( )

Lic

icAB

⋅−=−

−−= ω

β

ωβ sgn/12

sgn/22/122

22

2

2 (2.65)

( )( ) Li

cic

AB

⋅−=−

−−= ω

βωα sgn

/2sgn/12

22

2/122

2

2 (2.66)

where

( )( )

( )( )22

2/122

2/122

22

/2/12

/12/2

ββ

β

βc

cccL

−−

=−

−= (2.67)

From the above equation (2.67) the equation (2.68) for the phase velocity can be derived

01142 2

22/1

2

22

2

2

=

−

−−

−

βαβccc (2.68)

A real root of this equation gives the speed of Rayleigh wave Rc , first described by Lord Rayleigh in 1885 .

2.2.8 Characteristics of Rayleigh waves

2.2.8.1 Velocity

For the case of elastic, isotropic, homogeneous body, the velocity doesn't depend on ω , i.e. there is no dispersion. If we define p and q as

2

21638

αβ

−=p 2

28027272

αβ

−=q (2.69)

we get the velocity as

( )[ ] ( )[ ] 2/13/12/1323/12/132 3/827/4/2/27/4/2/

++−−+++−= pqqpqqcR

β (2.70)



when 0274

22

>

−

pq

and as

( ) ( )2/12/1321

2/1 3/83

4/27coscos3/2

+

−−−−=

− pqpcR πβ

(2.71)

when 0274

22

<

−

pq

Both p and q are functions of the ratio βα / ( PV=α and SV=β ), that can be expressed as a function of the Poisson ratio υ

2/1

2/11

−−

=υ

υβα (2.72)

Rayleigh wave velocity depends on a β and υ : it mainly depends on the shear velocity β , being about 0.9 times β , and it is much less sensitive to the variations of υ .

The ratio between the P-wave velocity and the S-wave velocity, and between the Rayleigh wave velocity and S-wave velocity as a function of the Poisson ratio are illustrated in figure 2.6.

Figure 2.6. The ratio VP/VS and VR/VS as a function of the Poisson ratio.

The minimum ratio between VP and VS is 2 , and the ratio tends to infinity when the Poisson ratio tends to 0.5: this because in such condition the shear velocity should be zero.

The exact range of variation of the Rayleigh velocity is

( ) ( )5.00

96.087.0

==

<<

υυS

R

VV



2.2.8.2 Displacements To obtain the displacements produced by Rayleigh waves let's discuss the steady-state oscillation at the angular frequency 0ω in a homogeneous half space.

In equations 2.62, 2.63, 2.64 we set 01 =A 01 =B and

( ) ( )[ ]002 ωωδωωδπ −++= AA (2.73)

which means that we only have amplitude A for the angular frequency 0ω . The potentials resulting from the integrals are

( )Rzm cxtAe /cos 0 −=Φ − ω (2.74)

( )Rzn

y cxtLe /sin 0 −=Ψ − ω (2.75)

where L is defined in 2.67, the corresponding displacements are, following equations 2.38

( )[ ] ( )Rzn

Rzm

z cxtAecLemu /cos/ 00 −+−= −− ωω (2.76)

( ) ( )[ ] ( )Rnzzm

Rx cxtAenLecu /sin/ 00 −⋅−= −− ωω (2.77)

With horizontal free surface, zu is vertical and the direction of propagation x is horizontal: the particle motion for a fixed x and y is a retrograde ellipse, since the two components have a phase difference of 2/π exactly, and the vertical component is greater than the horizontal one.

The amplitude decreases with depth exponentially, but with different rates for the two components: zu remains positive, xu becomes negative (fig 2.7), and so the ellipse becomes prograde below a

certain depth (about π2/1 times the wavelength) where the displacement trajectory collapse to a vertical segment.

The particle motion decreases exponentially with depth and becomes quickly negligible (Figure 2.7), so the wave propagation affects only the shallowest materials, and it is influenced only by mechanical properties of such layers. We can plot the dimensionless displacements against the dimensionless depth (Figure 2.7)

( ) AuAcuU zRzz /2/ 0 πλω == ( ) AuAcuU xRxx /2/ 0 πλω == λ/zz = (2.78)

Figure2.7. Rayleigh wave displacements with dimensionless depth.



2.2.8.3 Energy The energy of Rayleigh waves is also concentrated near the surface: in the following the expressions give intensity, kinetic-energy density and potential-energy density

xxzzzzz vpvpI −−= (2.79)

xxxzxzx vpvpI −−= (2.80)

( ) 2/22zx vvKE += ρ (2.81)

( ) ( )[ ] 2/42222zzxxzxzzxxPE εεερβεερα −++= (2.82)

KEPEE += (2.83)

With the symbols defined in 2.3, 2.4, and being v the velocity

Using the above obtained displacements (2.76 and 2.77) we find zI proportional to

tt 00 cossin ωω ,

so the vertical energy flow has a frequency which is twice the frequency, and its average is zero: there is no energy flow towards the earth's interior.

The flow xI is proportional to tt 02

02 cossin ωω and so its time average is not zero.

the average flow in the vertical z direction and in the horizontal x direction can be derived.

0=zI (2.84)

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

zsrR

zrRx ecsrecrssrsrecrII 222222/12222

0 /43//121)(//41 −+−− +−++++−+= βββ

(2.85)

with

( ) 2/122 /1 αRcr −= ( ) 2/122 /1 βRcs −=

RcAI 2/240 ρω= λπ /2 zz =

The horizontally travelling energy as a function of the depth and of the Poisson’s ratio is plotted in figure 2.8



Figure 2.8. Horizontal average intensity Ix/Io against normalized depth, for different

poisson ratio nu. Shear velocity is constant.

2.2.8.4 Geometrical spreading The energy of a source is imparted into the body and propagates from the source on wave fronts of increasing surface: the energy hence decreases with the distance from the source also for a geometrical reason, and this is called the geometrical spreading.

The geometrical spreading depends on the geometry of the body and of the propagation: for instance wave-fronts of Stoneley waves in fluid filled borehole don’t increase their surface, and so there is not geometrical attenuation.

With regards to body waves, a buried source in a infinite medium produces spherical wave fronts, with surfaces increasing with the square of the distance and the amplitude, into the body interior, decreases with the distance.

It is more interesting to evaluate the geometrical attenuation of body waves along the free surface: because of leaking of energy in the free space (Richart et al., 1970), the displacement attenuation goes with the square of the distance.

Surface waves do not spread energy towards the earth’s interior, as it has been shown above (2.84), and wave fronts are cylindrical. The energy reduces as the wavefront surface increases (linearly with the distance): the amplitudes decreases with the square root of the distance. As it will be shown later, Rayleigh waves propagate with several modes; in this case each of them follow this geometric damping law, while the overall amplitude due to the superposition behaves in more complicated ways.



2.2.9 Conclusions

Rayleigh wave equation in a homogeneous half-space have been derived to compute their velocity, displacement and energy.

The Rayleigh wave velocity in a homogeneous isotropic half space is not dependent on the frequency, i.e. there is no dispersion, and it is mostly influenced by the shear-wave velocity.

The velocity of R waves is lower than the shear-wave velocity, so, for a given frequency, surface waves have the shorter wavelength.

The displacements, the stresses and the energy have an exponential decay with depth, and so the surface waves affect only a shallow zone.

The influenced layer is limited to a depth roughly equal to one wavelength: this means that the wave propagation is conditioned by the mechanical properties of this zone, then for a given wavelength, deeper materials are not influent. As a consequence, the half-space has not to be strictly infinite to allow the application of the above theory. If a shallow homogeneous layer is thick enough for a given wavelength, the latter will propagate such as in an infinite half-space.

So, the half space has not to be strictly infinite, in order to have such phenomenon.

The Rayleigh waves do not spread energy into the body, and so, not considering the intrinsic attenuation, it is clear that being the geometrical spreading smaller, the surface waves are preponderant on body waves at a great distance from the source.

Chapter 2.3 RAYLEIGH WAVES IN LAYERED MEDIA: THEORY


2.3 RAYLEIGH WAVES IN LAYERED MEDIA: THEORY

In this section the equations of the Rayleigh wave propagation in layered media are obtained, and the matrix approach for the solution is presented. The computational techniques and the properties of the solution will be discussed later.

2.3.1 Introduction

If in a homogeneous half space the Rayleigh wave problem has a quite simple solution, for heterogeneous media it becomes very complex: the solution exists for transverse isotropic media with the free surface parallel to the isotropy plane.

In a homogeneous body there is a unique velocity for all frequencies, and so for all wavelength. In a layered medium the different wavelength “sample” different portions of soil, and so they are influenced by different materials, and the geometric dispersion arises. This dispersion is called geometric because it is due to geometric variations of elastic properties, and it is not intrinsic (material dispersion).

Figure 2.9. The geometric dispersion in a layered medium.

As it is easy to see, the greater wavelength, corresponding to the lower frequencies, are influenced also by deeper layers (Figure 2.9), while shorter wavelengths are influenced only by shallow layers.

The solution must satisfy the free surface condition at z = 0, the vanishing of amplitude at infinity, the continuity at each boundary. A nontrivial solution exists, for each frequency, for special values of the wavenumber, and so for fixed phase velocities: given a frequency more than one solution can exist. Many modes of propagation can be contemporarily present.

The dispersion also allows the existence of a group velocity different from the phase velocity. Each mode has its relationship between frequency and phase velocity: the velocity of a wavetrain, composed by several single frequency signal, is the group velocity, while the phase velocities are related to each frequency.



2.3.2 Rayleigh problem as a linear eigenproblem

The Rayleigh problem is a eigenproblem: this means that a limited number of solutions is possible, and the propagation has to follow one or more of them.

Using again the Lagrangian density, assuming a linear elastic material for which the potential energy is the elastic strain energy, introducing Hooke's law with Lame's parameters we can write the Euler-Lagrange equation for a vertically heterogeneous medium (Lai, 1998).

The Lagrangian density is (in analogy with the 2.14, introducing the z dependence)

( ) ( ) ( )

+−= ijijkkjii zzuuzL εεµελρ 2

, 21,'

21 (2.87)

and the Euler - Lagrange equation yields, in vector notation, to:

( ) ( ) 2

22 2

tdzdcurle

dzdGdiv

dzdedivgradGG zz ∂

∂=

+×++++∇

uuuuuu ρλ

λ (2.88)

where the symbol µ has been replaced by G to indicate the stiffness.

To find the solution of the above equation the displacement field u(z,t) for harmonic Rayleigh waves is assumed in the form

( ) ( ) ( ) ( ) kxtikxti ekziruuekzruu −− ==== ωω ωω ,,;0;,, 23211 (2.89)

This is a plane wave in the x direction 02 =u , but the z dependence is the same for a cylindrical wave (Aki and Richards, 1980)

We can state

( ) ( )kxtizr ekzr −= ωωτ ,,3 (2.90)

( ) ( )kxtizz ekzir −= ωωτ ,,4 (2.91)

and stating

( ) [ ]Trrrrzf 4321 ,,,= (2.92)

we get

( ) ( )zfzAdzdf

= (2.93)

This is a linear differential eigenvalue problem with displacements eigenfunctions

( )ω,,1 kzr and ( )ω,,2 kzr

and stress eigenfunctions

( )ω,,3 kzr and ( )ω,,4 kzr

For a given frequency ω , non trivial solutions exist only for special values of the wavenumber k, ( ) ( ) ( )ωωω nkkk ,...,, 21 , the eigenvalues. The corresponding functions 4321 ,,, rrrr solutions of the

equation are the eigenfunctions.



The relation ( )ωjkk = is known in implicit form

( ) ( ) ( )[ ] 0,,,, =ωρλ jR kzzGzF (2.94)

known as the Rayleigh dispersion equation, or the Rayleigh secular function: in vertically heterogeneous media the velocity of propagation is a multi-valued function of frequency. Each function

),,(, ωjij kzrk

defines a mode of propagation (Figure 2.11).

Figure 2.10. The modal curves are the possible non trivial solution of the problem.

Once the eigenvectors are found (the roots of the dispersion equation) the eigenfunctions for each mode of propagation can be computed. The Rayleigh eigenfunctions are the stresses and displacements as a function of the depth, of frequency and of the mode.

2.3.3 Solutions of the Rayleigh problem

In the formulation of the equation (2.94) there is no hypothesis on the law of variation of mechanical properties with depth. Assuming a layered model a solution can be found, and many solution techniques have been proposed.

There are many ways to solve this eigenvalue problem (Lai, 1998): numerical integration, finite difference, finite elements, boundary elements, spectral element. The propagator matrix methods (Gilbert and Backus, 1966) are the most frequently used.

The Thomson-Haskell method of the transfer matrix (Thomson 1950, Haskell 1953) is a special case of the propagator matrix method applicable to a stack of homogeneous layers overlying a halfspace. The Dunkin's modification (Dunkin, 1965) will be later described in details since it has been implemented to solve the forward problem of Rayleigh wave propagation in layered system. Many other authors have worked on the subject proposing modifications and optimisations of the matrix method (section 2.4).

The dynamic stiffness matrix method proposed by Kausel and Roesset (1981) is a finite element formulation derived from the Thomson&Haskell algorithm. The method of reflection and transmission coefficients (Kennet 1974), derived from the Thomson&Haskell method, is an



efficient iterative algorithm for computing the Rayleigh dispersion equation, and explicitly models the constructive interference that leads to the formation of surface wave modes.

2.3.4 The matrix formulation

Each algorithm used for the solution of the Rayleigh eigenvalue problem has to construct the Secular function and then to find its roots as a function of frequency: for this purpose the matrix formulation of Thomson and Haskell is the basis of the solution techniques for the layered medium.

The transmission of plane elastic waves through a stratified medium can be treated with the matrix formulation proposed by Thomson (1950) and corrected by Haskell (1953): we search for the solutions of equation of motion that satisfy the source-free condition everywhere, and the conditions of vanishing forces at the free surface. Moreover we impose the continuity of stresses and strains at interface where elastic properties have a discontinuity, otherwise the discontinuity would act as a seismic source.

The computation of the solution in terms of modal contribution requires finding the roots of the Rayleigh secular function (Rayleigh dispersion function), to evaluate this function the matrix products among layer matrices have to be formed.

In the original formulation, modal calculations exhibit numerical difficulties, especially at high frequencies, caused by the computation of squares of exponential terms that cancels numerically causing loss of significant figures in the secular function. We follow here the approach of Dunkin (1965), who proposed a matrix formulation that avoid these difficulties and gives natural decomposition into receiver and source parts, by the use of development by minors. Knopoff (1964) also proposed a method to avoid the numerical problems rising in the computation.

The adopted model consists of L-1 homogeneous and isotropic layers overlying the half space. Each layer is characterized by the compressional wave velocity nα , the shear wave velocity nβ , the density nρ and the thickness nd . The nth layer is bounded by the n-1 and the n interfaces. The halfspace is the Lth element (Figure 2.11).

Figure 2.11. Geometry of the layered medium with α, β, ρ for each layer.

Using again the Helmoltz’s decomposition, we can say that the displacements in the thn layer can be obtained by the dilatational and shear potential nΦ and nΨ of the nth layer



zxu

ny

nn

x ∂Ψ∂

−∂Φ∂

= (2.95)

xzu

ny

nn

z ∂Ψ∂

+∂Φ∂

= (2.96)

and the equation of motion in each layer is satisfied if nΦ and nyΨ are solutions of the appropriate

wave equation (eq. 2. 23)

2

2

22

2

2

2

2

2 1tzyx ∂Φ∂

=∂

Φ∂+

∂Φ∂

+∂

Φ∂α

2

2

22

2

2

2

2

2 1tzyx

yyyy

∂Ψ∂

=∂

Ψ∂+

∂Ψ∂

+∂

Ψ∂β

As it has been illustrated above (section 2.2.6), the equations can be solved by separation of variables, and in each layer we have (eq. 2.52 and 2.53)

( ) tiikxMzMzn eeeAeA ω−+=Φ 21 (2.97)

( ) tiikxNzNzny eeeBeB ω−+=Ψ 21 (2.98)

the Fourier transformed potentials (section 2.2.7), when αβ <<c (Rayleigh surface waves condition) can be written as :

( )( )

( ) ( )[ ]∫∫∞

∞−

−∞

∞−

+=Φ ωωωπ

ωω dkdeeekAekAtzx ticxizmzmn /212 ,,

21,, (2.99)

( )( )

( ) ( )[ ]∫∫∞

∞−

−∞

∞−

+=Ψ ωωωπ

ωω dkdeeekBekBtzx ticxiznznny

/212 ,,

21,, (2.100)

If the phase velocity is lower than shear wave velocity, both N and M are real and can be expressed (15,16) as mM = and nN = and the solution for potentials give

2/1

2211

+=

ncm

αω (2.101)

2/1

2211

+=

ncn

βω (2.102)

So, the Fourier Transform can be written as the sum of an up-going and down-going term as ( ) ( ) −+−−− +≡+= −−

nnzzm

nzzm

nnnnnn eBeA ϕϕϕ 11 (2.103)

( ) ( ) −+−−− +≡+= −−nn

zznn

zznnn

nnnn eDeC ψψψ 11 (2.104)

where 2222 / nn km αω−= (2.105)

2222 / nn kn βω−= (2.106)



We can define two sets of vectors: the first is

( ) ( ) ( ) ( ) ( )[ ]zzzzz nnnnn−−++≡Φ ϕψψϕ ,,, (2.107)

containing the potentials in the layer nth, and the second is

( ) ( ) ( ) ( ) ( )[ ]zpzpzuzuzS nzx

nzz

nz

nxn ,,,≡ (2.108)

containing the displacements and the stresses in the same layer nth ( nzzp and n

zxp are the two stress components in the nth layer).

All components are functions of the transformed variables ω,k

The displacements and the stresses can be obtained from the potentials (2.38 and 2.39) as

−+−+ +−+=∂

∂−

∂Φ∂

= nnnnnny

x nnikikzx

u ψψϕϕψ

(2.109)

−+−+ ++−=∂

∂−

∂∂

= nnnny

z ikikmmxz

u ψψϕϕψϕ (2.110)

∂∂Ψ∂

−∂

Φ∂−

∂Φ∂

=zxzt

p yxx

22

2

22

2

2

22 ββρ (2.111)

∂Ψ∂

+∂

Ψ∂−

∂∂Φ∂

= 2

2

2

22

22 22

tzzxp yy

zx ββρ (2.112)

At any point in the nth layer we can compactly write the above relations between potentials and stresses and displacements as:

( ) ( )zTzS nnn Φ= (2.113)

and the inverse relation is of course

( ) ( )zSTz nnn1−=Φ (2.114)

The matrix T contains the parameters of the relations 2.109-2.111,

−−−−

−−

=

nnnnnnnn

nnnnnnnn

nn

nn

n

akmialminikaia

ikmikmniknik

T

µµµµµµµµ

222ln2

(2.115)

with 222 /2 nka βω+=

Within a layer the z-dependence of the potentials can be summarized and compactly written between the two interfaces as

( ) ( )1−Φ=Φ nnnnn zEz (2.116)

Considering that for each individual potential there is an exponential relation such as, for instance

( ) ( )zedz nndmn

++ =+ ϕϕ



and similarly for the others components, we can write:

( )

=

−

−

nn

nn

nn

nn

dn

dm

dn

dm

nn

ee

ee

dE

000000000000

(2.117)

The boundary conditions (stress free surface, continuity at layer interfaces, zero potential at infinity), in terms of transformed quantities are:

( ) 01 0 SS = (2.118)

( ) ( )nnnn zSzS 1+= n=1,2,…,L-1

0==++

LL ψϕ

Thomson and Haskell define a layer matrix G that continues the solution from a layer to the following

( ) ( )11 −+ = nnnnn zSGzS (2.119)

This layer matrix can be obtained from the following equations starting from the stress and strain in the n+1th layer, at the interface zn with the upper nth layer :

a) ( ) ( )nnnn zSzS 1+=

Starting from the layer n+1, the continuity of stress and strain at the interface n allow to get the stress and strain at the same interface, but in the upper nth layer

b) ( ) ( )nnnnn zSzT 1+=Φ

The relation between potential and stress-strain in the nth layer gives the potentials at depth zn in the layer nth

c) ( ) ( )nnnnnn zSzET 11 +− =Φ

The z dependence of the potentials in the nth layer allow the continuation of the potentials from zn up to zn-1

d) ( ) ( )11

1 −−

− =Φ nnnnn zSTz

The relation between potential and stress-strain in the nth layer, already used for the step b), gives stress and strain at the depth zn-1 in the layer nth, which is the final result of equation 2.119

Summarizing the four steps the result is

( ) ( )nnnnnnn zSzSTET 111

+−− = , that gives the G matrix of equation 2.119

1−= nnnn TETG (2.120)

The matrix nG continues the stress displacement solution downward through a layer and across an interface. S is converted to Φ , then continued by E from the top to the bottom, then converted to S and equated to the S of the layer below.



Considering the disturbance generated at the top surface, the solution is continued from the top surface to the half space by

( ) 001211

11 SRSGGGTz LLL ⋅==Φ −−

− L (2.121)

If R is partitioned into 2nd order matrices

=

2221

1211

RRRR

R (2.122)

Then, the boundary condition of no radiation at L requires that

[ ] [ ] 0,, 00120011 =+= τσRwuR (2.123)

being 0000 ,,, τσwu the components of 0S

When stresses are applied at the surface, then the displacements

[ ] [ ]00121

1100 ,, τσRRwu −−= (2.124)

and if we let

11R represent the transpose of the cofactor matrix of 11R then

[ ] [ ]001211

1100 ,

det, τσR

RRwu −= (2.125)

The expression 11det R is the function whose roots give rise both to leaking and propagating modes:

0det 11 =R must be true for the characteristic oscillations for the problem in which the upper surface is free by setting [ ] 0, 00 =τσ . A solution can exist for pairs of ( )ω,k satisfying

( ) 0det, 11 == RkF ω (2.126)

The solution can exist only for those couples of ( )ω,k : they are the eigenvalues, and on the ( )ω,k are the modal curves.

It is interesting to notice that some of the more commonly encountered modal equations are expressible in terms of subdeterminants of the matrix just defined: for the homogeneous half-space we have

( ) 0det 111

1 =−T (2.127)

The displacements of a point at depth z in the layer n is

( ) ( ) 00111 SMSGGzzGzS nnnn =−= −− L (2.128)

where:

( )1−− nn zzG means that nd is replaced by 1−− nzz in nG .



2.3.5 Dissipative phenomena: linear viscoelasticity

The solution of the wave propagation problem depends on the constitutive model used to represent the mechanical response of the medium to dynamic excitation. In order to obtain the phase velocity of the Rayleigh wave propagation, as it has been illustrated above, a simple elastic model can be used. This does not allow to take into account for dissipative phenomena occurring in the wave propagation through soils, even at very low strain, (Lai and Rix, 2002), due to the friction between particles and to the motion of pore fluid.

The soil response to dynamic excitation is influenced by the magnitude of the strain and stress applied. The experimental evidence shows that even at very small strain levels energy dissipation occurs. Vucetic (1994) describes the behaviour introducing different thresholds of strain: at very low shear strain there is no stiffness reduction, the behaviour is linear, and there is no degradation of the properties with the number of cycles, nevertheless the hysteretic loop of the stress strain relation causes loss of energy.

Figure 2.12. Hysteretic loop of stress strain at very low strain, after Foti (2000).

The behaviour of soil below this limit of very small strain is called linear viscoelastic, and the value is called the linear cyclic threshold shear strain. The linear viscoelastic model is necessary to describe the dissipation at very low strain, which is the case of seismic methods . This model can be schematised with an elastic spring and a viscous dashpot (Maxwell model, Kelvin-Voigt model), and the stress strain relation involves complex moduli. A complex modulus has an imaginary part that makes the stress and strain out of phase (from which the elliptic hysteresis).

For the wave propagation this leads to complex velocities.

( )ρ

µα

**** 3/4+

==KVP (2.129)

ρµ

β*

** == SV (2.130)

The meaning of the complex velocities in terms of spatial attenuation can be easily illustrated: if a complex velocity is used, a complex wavenumber is introduced in the wave equation (Schwab and Knopoff, 1971). The solution has then to consider so an attenuation factor accounting for spatial decay of a wave propagating through a dissipative medium.



The simple case of one dimensional harmonic S-wave propagation can be used as example

2

2

2*2

2 1tu

xu yy

∂

∂=

∂

∂

β has as solution ( )txkiAetxu ω+=

*

),( with ** / βω=k

if the wavenumber *k is complex, it can be written as )Im()Re(* kikk += the solution can be written as

( )[ ] [ ] [ ] xtkxixkitxkitxkiki eAeeAeAetxu αωωω −+−+++ === )Im()Re()Im()Re(),( (2.131)

Finally, to completely describe the wave propagation in dissipative media, assuming a linear viscoelastic model, a complex velocity can be used, or alternatively a real phase velocity and an attenuation coefficient.

The material damping ratio D and the quality factor Q, Q = 1/2D, are the parameters usually used as a measure of energy dissipation, is the material damping ratio D, and the quality factor. The damping ratio can be defined, from the hysteretic loop, as a ratio between the energy dissipated during the cycle and the maximum energy reached during the cycle (figure 2.12).

( ) ( )( )ωπω

ωWWD

4∆

= (2.132)

The quality factor used by seismologists is simply (Aki and Richards, 1980)

DQ

21

= (2.133)

The solution for weakly dissipative media can be obtained directly from the solution of the linear elastic eigenvalue problem substituting in the relevant expressions the complex velocities. In other cases, when no assumption is made on the damping factor, a coupled computation is needed (Lai and Rix, 2002).

The attenuation is quite easy to be described in linear viscoelastic soils, using complex moduli and complex velocities, when body waves are considered. Surface waves, as combination of P and SV waves, have an attenuation due both to shear wave attenuation and longitudinal wave attenuation. Viktorov (1967) shows that the Rayleigh wave attenuation is mainly due to the shear wave attenuation: the attenuation of Rayleigh waves can be obtained as a weighted average of the attenuation of shear and compressional waves. The weights for the shear-wave attenuation is A, the weight for the compressional wave velocity is (1-A). The value of A only depends on the Poisson ratio. (Figure 2.13)



Figure 2.13. Weights for the shear wave attenuation (A) and the compressional wave attenuation (1-A) as a function of

the Poisson ratio, after Viktorov (1967).

The use of complex moduli and velocities makes the phase difference between the two displacements at the free surface other than π/2. This means that the elliptic trajectory of the particle at the free surface has no vertical and horizontal axes, but is inclined.

Chapter 2.4 SIMULATION OF THE RAYLEIGH WAVE PROPAGATION IN LAYERD MEDIA


2.4 SIMULATION OF THE RAYLEIGH WAVE PROPAGATION IN LAYERED MEDIA

In this section the computational steps for the simulation of the test are described, and the main properties of the Rayleigh wave propagation in layered media are discussed. The first step is computing the eigenvalues, i.e the possible solutions for each frequency: the eigenvalues are the modal curves. Then the displacements and stresses for each frequency, at each depth, and each mode are computed, and finally with the Green’s function the displacements at the free surface are computed for a point source. The synthetic seismograms are obtained, in frequency-offset domain and in the frequency wavenumber domain. Hence, given a testing layout, the displacements induced at the free surface at known distance from the source are predicted.

2.4.1 Computational problem for the modal curves computation

2.4.1.1 The problem The first step in the computation of R-waves is the determination of eigenvalues, i.e. the modal curves. This requires finding the zeros of the secular function of Rayleigh dispersion equation

0)det( 11 =R (equation 2.126). The matrix R is obtained from the product of the different layer matrices, containing exponential terms. Individual terms become very large and subtractions can reduce the number of significant digits (Dunkin, 1965). In the original (Haskell-Thomson) formulation, modal calculation exhibit numerical difficulties, specially at high frequency, caused by the computation of squares of exponential terms that cancel numerically causing loss of significant figures in the secular function.

The limit, according to Schwab and Knopoff (1970), with 16 digit precision can be expressed in terms of minimum wavelength λ , given a depth of the halfspace H and an accuracy in phase velocityσ , by the relation

( ) 4.12/3.14103/ σλ −⋅=H The wave propagation in layered media has been a subject of intensive study, and many authors have proposed computational strategies for the fast and efficient solution of the Rayleigh dispersion relation.

Thomson (1959) laid the theoretical groundwork extended by Haskell (1953). Dorman et al. (1960) and then Press et al. (1961) described the use of an electronic computer to calculate surface wave dispersion for a multi layered elastic half space using the Thomson&Haskell approach. The computations of high frequencies has been made possible by the works of Knopoff (1964), Dunkin (1965), Thrower (1965), who analytically eliminated the numerical problems of the original T&H formulation. Gilbert and Backus (1966) used a more general approach that ultimately degenerated to the previous method under certain conditions. Randal (1967) gave a detailed description of the application of the Knopoff’s formulation to a computer program. Watson (1970) proposed an efficient modification of the Dunkin’s approach, rearranging the indices of the elements of the computation and saving computer time.

I follow here the approach of Dunkin (1965), who proposes a matrix formulation that avoid these difficulties and gives natural decomposition into receiver and source parts, by the use of development by minors, and the rearrangement of Watson (1970).



2.4.1.2 Dunkin’s restatement The use of complex arithmetic can be avoided with this approach: it is only applicable to elastic materials, while the linear viscoelastic behaviour can be simulated only with the complex arithmetic. The coupling between the two aspects in strongly dissipative media has been discussed in Lai and Rix (2002).

Dunkin presents a restatement of the matrix formulation of Thomson and Haskell that avoids the numerical problems inherent in the Thomson-Haskell form, and uses the real arithmetic.

To keep off the numerical computation of the layer matrices product, the second order subdeterminant of the product is obtained by the second order subdeterminants of the individual matrices, that are not directly multiplied.

Let's define the second order subdeterminant of P that involves rows i and j and columns k and l. ji

lk

ij

lk

ji

kljkiljlikij

klpppppppp =−=−=−= (2.134)

For P equal to the product nAAAP L21= , being A the individual layer matrices in the Dunkin’s notation

we have

uvkl

nstuv

nmnop

ijmn

ij

klaaaap 121 −= L (2.135)

choosing any m<n, but never a couple such as cd and dc.

Let's decompose the total modal solution in the nth layer, nS , into separate modal contributions.

( ) ( ) ξπ

ξ dezSzS xitsmn

mn

m

m

+

Γ∫∑=

21 (2.136)

being mΓ the region of the real ξ axis for which the mth pole is included and

( )

=

0

012

11

21

11

τσ

Rf

RMM

zSs

mn (2.137)

[ ]112111

12111

1112

122211 rr

rr

rrrrr

R −

−−=

−= −

(2.138)

So we can write

( ) [ ]

−

−

−= −

0

0121121

11

12

21

11111

1τσ

Rrrfr

rMM

rzSs

mn (2.139)

or, considering the receiver (R) and the source (W) function mm

nmn WRS 0= (2.140)

where m is not summed, for the mth mode mW0 is the transform of the vertical displacement at the free surface associated with the mth mode

and caused by the distribution of surface stresses 0σ and 0τ , while mnR are the normalized



transforms of the displacements and stresses in the nth layer associated with the mth mode, normalized by the vertical displacement at the free surface.

Considering the secular function 2.126 in terms of subdeterminant

( ) 0det, 12

1211 === rRkF ω

we can write

( ) pqlcd

ef

lab

cd

l

ab

l gggtrRkF12

2211212

1211det, −−− ⋅⋅⋅⋅==≡ Lω (2.141)

expression which is linear in the second order determinants belonging to any nG : this expression contain no exponential terms.

The source term is

( ) ( )[ ] 10241121140231113210

−−−−= sm frrrrrrrrW τσ

[ ] 10

12

41012

310−+= s

m frrW τσ

that can be written as,

1041

1031

12112

0−−

+⋅⋅⋅⋅= s

tuturs

tu

ab

cd

L

ab

Lm fggggtW τσL (2.142)

Let's calculate the four components of ( )azRmn ,

the result is

( ) ( ) ( ) efab

cdnn

nn

vbL

rsL

lrmn gzzgzzggtrazR

21

11

1111, LL −

−− −−= (2.143)

Finally we have, summarizing:

Secular function

( ) pqlcd

ef

lab

cd

l

ab

l gggtrRkF12

2211212

1211det, −−− ⋅⋅⋅⋅==≡ Lω

Source term (it contains f)

1041

1031

12112

0−−

+⋅⋅⋅⋅= s

tuturs

tu

ab

cd

L

ab

Lm fggggtW τσL

Receiver term

( ) ( ) ( ) efab

cdnn

nn

vbL

rsL

lrmn gzzgzzggtrazR

21

11

1111, LL −

−− −−=

2.4.1.3 Details for the computation The secular function is the most important point of the computation since its roots give the modal curves



( ) pqlcd

ef

lab

cd

l

ab

l gggtrRkf12

2211212

1211det, −−− ⋅⋅⋅⋅==≡ Lω

where ab

cd

ng

are the subdeterminants of the layer matrix

12

ab

Lt

are the subdeterminants of the matrix 1−LT , of the Lth layer, which is the half space. The matrix 1−

LT is simply the inverse of the T matrix (equation 2.115) of the half space.

−−−−−−−

=−

nnnnnnnnn

nnnnnnnn

nnnnnnnnn

nnnnnnnnn

nnn

nn

nmikmnkimaiknnmnaai

nmikmnkmimaiknnmnankmi

nmT

µµµµµ

µµµµ

ωµβ

22

22

2 2

21

where we have defined 2222 / nn km αω+= 2222 / nn kn βω+= 222 /2 nn ka βω+=

It can be stated, as proposed by Dunkin,

k/ωλ = 222 //2 ωβγ nn k=

2221 /~nn kkm αω+= − 2221 /~

nn kkn βω+= −

nnn dmkmSM ~sinh~ 1−= nnn dnknSN ~sinh~ 1−=

nndmkCM ~cosh= nndnkCN ~cosh=

The layer matrix components are

( )CNCMgg nnnn 14411 ++−== γγ

( )[ ]SNnSMigg nnnnn 23412

~1 γγ −+==

( ) ( )CNCMkigg nnn −==

−122413 λρ

( ) ( )SNnSMkg nnn 21214

~+−=−

λρ

( )[ ]SNSMmigg nnnnn 1~24321 +−== γγ

( ) CNCMgg nnnn γγ −+== 13322

( ) ( )SNSMmkg nnn −=

− 21223

~λρ

( )( )CNCMkigg nnnnn −+== 124231 γγλρ

( )[ ]SNnSMkg nnnnn 222232

~1 γγλρ −+=

( )[ ]SNSMmkg nnnnn 222241 1~ +−−= γγλρ

The terms that are needed for the computation of the secular function are the second order subdeterminants listed in the following:



( ) ( ) ( )[ ]SNSMnmCMCNgg nnnnnnnn2222234

34

12

12~~112212 γγγγγγ ++−++++−==

( ) ( )SMCNmCMSNkgg nn21224

34

12

13~+−==

−λρ

( ) ( )( ) ( )[ ]SNSMnmCMCNkigggg nnnnnn221223

34

14

34

12

23

12

14~~1112 γγγλρ +++−+−====

−

( ) ( )CMSNnSMCNkgg nn21213

34

12

24~−==

−λρ

( ) ( ) ( )[ ]SMSNnmCMCNkg nnn222212

34~~112 ++−−=

−λρ

( ) ( )[ ]SMCNCMSNnkgg nnnn222234

24

13

121~ +−−== γγλρ

CMCNgg == 24

24

13

13

( )[ ]CMSNnSMCNigggg nnn223

24

14

24

13

23

13

14~1 γγ −+====

SMSNng n213

24~−=

( ) ( )( )( ) ( )[ ] SMSNnmCMCNkigggg nnnnnnnn2233234

23

34

14

23

12

14

12~~11121 γγγγγλρ +++−++−====

( )[ ]SMCNmCMSNigggg nnn224

23

24

14

23

13

14

13~1 γγ −+−====

( )( ) ( )[ ]SMSNnmCMCNgg nnnnnn222223

23

14

14~~11121 γγγγ +++−++==

114

14

23

14

14

23−== ggg

( )( )[ ]SMCNmCMSNkgg nnnn222234

13

24

12~1 γγλρ −+−==

SMSNmg n224

13~−=

( ) ( ) ( ) ( )[ ] SMSNnmCMCNkg nnnnnnn2244222234

12~~1112 γγγγλρ +++−+=

Some authors (Watson, 1970) simplify the notation assigning a unique index to each unique combination of the subdeterminant indices, and this also save computer time. Details can be found in Watson (1970), Wang and Herrmann (1980).

The computation of the modal curves is the heavier task in the R-wave solution computation. Different approaches have been proposed by many authors: for instance see Gilbert and Backus (1966), Thrower (1965), Schwab and Knopoff (1970). Computer Programs in Seismology of prof. Herrmann has a complete set of routines for the surface waves simulation. Other approaches are described by Lai (1998), who used the algorithm of Hisada (1994, 1995)

The algorithm that has been implemented in this work in a Matlab™ code, mainly follows the Dunkin’s approach for the computation of the secular function. The following figure shows the behaviour of the real secular function and the complex secular function computed at fixed frequency: the complex secular function has been computed with the program of Lai and Rix, that implements the algorithms of Hisada (1994,1995).

The modal wavenumbers are the wavenumbers at which this function is zero. The real secular function is bounded in (-1, 1) and the zeros correspond to change of sign (Figure 2.14 top). Considering the complex secular function, the zeros are either minima of its modulus or simultaneous zeros of the real and the imaginary part. In addition to this, the range of the modulus is wide, and sometimes for the numerical problems described above, the zeros are not present at all.



Figure2.14. Secular function a the same frequency for the same stratigraphy, two expressions (real and complex).

When a reliable and efficient technique allows the computation of the secular function, its roots have to be found to get the modal curves. However the secular function is a rapidly oscillating function, and so the root searching techniques may fail. The following figure (2.15) shows the values of the secular function, at fixed frequency, as a function of the wavenumber in the range 0-9 rad/m. The number of zeros is not known, and roots can be very close each other.

Figure2.15. The secular function for the different wavenumber at fixed frequency. The behaviour can be rapidly

oscillating, and the root finding is not simple. Two zeros can be close each other.



2.4.2 Computation of modal curves

Modal curves are the values of wavenumber for which a solution of the problem can exist with the imposed boundary conditions. They are the zeros of the secular function in the acceptable domain: for each frequency there is an acceptable range of k, defined by velocity limits. We find the minimum and the maximum Rayleigh wave velocity (calculating the Rayleigh wave velocity for each layer considered as an homogeneous half-space) and we calculate

( ) minmax / Rck ωω = (2.144a)

( ) maxmin / Sck ωω = (2.144b)

In this range, between the two lines of slope min/2 Rcπ and max/2 Scπ the zeros of the secular function constitute the modal curves. The roots are found with a bracketing-bisection technique (Press, 1992)

The example below shows the position of the zeros the in frequency-wavenumber domain, and the value of the secular function for two frequencies.

Figure 2.16-a. Modes in frequency wavenumber domain. They are found searching

at each frequency the zeros in the admissible range. Model: 400m/s, 200m/s, 500m/s; 1m, 2m.

Figure2.16b. The value of the secular function at the frequency of 40 Hz (left) and at the frequency of 280 Hz.



Figure2.17. Secular function, lines of zeros, region of acceptable values between the

two straight diverging lines (2 layers : 100 m/s, 250 m/s; 2m).

The computation is hence heavy: at each frequency the secular function has to be computed to find its zeros. A bracketing-bisection procedure (Press et al., 1992) has been implemented: the more critic step is the bracketing, that may fail due to the rapidly oscillating behaviour of the function. An adaptive bracketing based on the intervals between zeros at higher frequency has been implemented. The computation starts from the highest frequency in the range of interest, where the number of modes and zeros is maximum. A very fine discretization is adopted for the bracketing at that frequency, and the position of the roots is used to predict the roots at the following step at lower frequency. The discretization is coarser where roots are far from each other, and it is fine where they are close: the following figure (2.18) shows the secular function with two roots very close each other: the coarse discretisation (left) with step of 0.001 rad/m in this case fail the identification.

Figure 2.18. Bracketing of the zeros of the secular function. The horizontal line of zero is not crossed but the samples

of the coarse discretization (left), while two roots are identified with the finer discretization (right).

Modal curves are easily transformed in phase velocity as a function of the frequency, being simply

kfv π2= . This is the result of the first step of the modelling: given a soil model (layer thickness

and properties), and given a set of frequencies, at each frequency the modal phase velocities are computed: these are the velocities at which the energy can propagate (Figure 2.19).

Figure 2.19. Modal curves of the example above, in frequency wavenumber and in frequency-phase velocity

(2 layers : 100m/s, 250 m/s; 2m). They are discrete curves computed numerically.

2.1.1 Properties of the modal curves

At a given frequency there are M normal modes; M can be in general finite or infinite: with a finite number of homogeneous layers overlaying a homogeneous half space M is always finite.

The limit frequency at which a zero line exits from the acceptable range or stops existing, is called the cut-off frequency of that modal curve: the mth mode exists only for f>fcut m, and at their cut-off frequency each mode has a phase velocity equal to the maximum shear-wave velocity in the layered system.

The cut-off frequencies have a quite regular distribution: for a two layer model, the cut-off frequencies are (Aki and Richards, 1980).

2/1

22

211 1

2)1(

−

−⋅

⋅−=

βββ

Hnfcn (2.145)

where n indicates the number of the mode, H is the thickness of the first layer, 1β and 2β are the shear-wave velocities of respectively the first and the second layer. When a greater number of layers is present, the distribution of the cut-off frequencies is no more so regular.

The shape of a single modal curve, apart the minimum and maximum velocities, does not have other special constraint: the phase velocity can increase or decrease with the frequency: usually an



increase of phase velocity with increasing frequency is associated to a velocity inversion, i.e. a layer with low velocity underlying a layer with higher velocity (Figure 2.20).

It is intuitive that at high frequencies, when f tends to infinity, the surface wave should have a infinitesimal wavelength, and then it should propagate only in the first layer, having indeed a phase velocity equal to the Rayleigh wave velocity of the first layer. Nevertheless this refers to the macroscopic behaviour, that could be due to a sequence of different modes, and not to a single modal curve.

The relative excitation of different modes depends on the nature and on the depths of the source: these aspects will be discussed later in 2.4.6. In general we can say that in order to separate the different modes we have to record at a great distance from the source, where they arrive separate in time due to their different propagation velocities.

Figure 2.20. Modal curves when a stiff top layer is present.

The first mode, often called the fundamental mode, is the one having the lowest phase velocity, and sometimes it is referred to as mode 0. The modal curve can have a quite strange behaviour (Figure 2.21), and can show apparent intersection, that make the computation particularly difficult.



Figure 2.21. The behaviour of modal curves can be quite strange, with apparent intersection.

Here a strong velocity inversion (200m/s 300m/s 1000m/s 300m/S 1100m/S).

A series of example is given hereafter to discuss some properties of the modal curves. When it is not differently specified, in the examples below the velocities of the layers are shear wave velocities, and the Poisson ratio is always 0.33.

2.4.3.1 Example 1 A simple 2 layer normally dispersive model is considered at first: the first layer is 20 m thick and has a shear velocity of 200 m/s, the half space has a shear velocity of 300m/s. (Figure 2.22)

Figure 2.22. Modal curves of a two layer model (20m at 200 m/s, halfspace 300 m/s). 14 modes exist at 90 Hz. The first

mode tends to the Rayleigh Wave velocity of the first layer, the others to the shear velocity of the first layer.

The first mode tends to the Rayleigh wave velocity of the first layer, which is 186.5 m/s, while all the others tends to the shear wave velocity of the first layer (200 m/s). The cut-off frequencies are evenly spaced with a spacing of 6.70 Hz, from equation 2.145, and the limit phase velocity equals



the shear velocity of the half space (300 m/s). At high frequency, where only the first layer rule the propagation, the first mode is dominant.

2.4.3.2 Example 2 The spacing of cut-off frequencies is regular only for simple cases. In the following example (Figure 2.23), where 3 layers are present (200 m/s 300 m/s 400 m/s, 20 m, 20m, half space ), the cut-off frequencies are not even-spaced. Furthermore three asymptotic velocities are possible: for the first mode the Rayleigh wave velocity of the first layer (186.5 m/s), for the other modes the shear wave velocity of the first layer (200 m/s). Also the shear wave velocity of the intermediate layer (300 m/s) appears an alignment as made up of different modes that flatten at that phase velocity.

At high frequency the Rayleigh wave velocity of the first layer is given by the first mode.

Figure 2.23. Modal curves of a three layer model (20m at 200 m/s, 20 m at 300 m/s then 400 m/s).

2.4.3.3 Example 3 An inversely dispersive model is here presented: two layers ( the first with shear velocity of 250 m/s and 10 m thick, the second with shear velocity of 150 m/s and 10 m thick) lay on a half space at 300 m/s of shear wave velocity (Figure 2.24)



Figure2.24. An other example of modal curves. All modes tend at high frequency to the minimum shear velocity. At high frequency higher modes have an horizontal branch at the Rayleigh velocity of the first layer. The cut-off

frequencies are associated to the shear velocity of the half-space. Model: 250 m/s, 10m; 150 m/s, 10m; half-space 300 m/s.

In the inversely dispersive model of the figure above, the modal curves tend at high frequency to the shear velocity of the intermediate layer (lower velocity in the model): it can be noticed that the modal curves flatten, at high frequency, at the Rayleigh velocity of the uppermost layer, 233 m/s, producing a continuous line that allows the existence of a physically acceptable solution at high frequency. The surface wave should be confined, at high frequency, in the uppermost layer: the phase velocity has to be equal to the Rayleigh wave velocity of the uppermost layer for the shortest wavelengths. For instance above 40Hz, also the maximum possible wavelength (300 m/s /40 Hz = 7.5 m) is almost confined in the first layer. The phase velocity hence should tend to 233 m/s for all the wavelengths greater than this value, and this behaviour is possible if the energy at different frequencies follows different modes.

2.4.3.4 Example 4 In the following example a small variation in the shear velocity of one layer change the asymptotic behaviour of the first modal curve (Figure 2.25). In this earth model, the velocities are expressed in km/s and the thickness in km.

Layer Thickness [km]

Shear velocity [km/s]

Compressional velocity [km/s]

density

1 1 3.8 6.5 2.7

2 1 3.5 (left) 3.3 (right)

6.0 2.5

3 1 4.4 7.5 3.1

4 1 4.1 7.0 2.9

Half space ∞ 4.7 8.0 3.3

It can be noticed that for both the models the first mode at low frequency has the Rayleigh velocity of the half space (4.31 km/s), then the phase velocity decreases for the influence of the low velocity



layer, and increases again because of the uppermost layer. At high frequency the surface wave should be confined in the first layer and have a phase velocity of 3.49 km/s, which is the Rayleigh velocity of the first layer Vr1: this can be obtained following one or more of the modal curves.

In the figure it can be easily noticed that with the lower velocity inversion (left) the first mode at high frequency tends to Vr1: with the higher velocity inversion, on the other hand , it tends to the Vs2, and the other modes tends to the Vr1.

Figure 2.25. A little variation in the model can change the asymptotic behaviour of modal curves.

2.4.4 Scaling Properties of modal curves

2.4.4.1 Introduction The modal curves depend on the velocities and the thickness of the layers, the relation being quite complicated and computationally heavy with the complex models that are often needed to describe the very shallow subsurface. If the curves of one model could be obtained simply from the curves computed for other models, a great save in computing time would be obtained: when the procedure is used in the forward modelling for an inversion, a fast simplified computation could be very useful. The scaling properties of the modal curves are discussed here: when the velocities or the thickness of all layers change of the same factor, the modal curves simply scale. The applications of these properties will be presented in chapter 5.

2.4.4.2 Layer Velocities

The first consideration to make is about the velocities β and α , that are the shear velocity SV and the compressional velocity PV of the layers in the model.

In the expressions of the secular function, f we can see that the frequency and the shear and compressional velocities of a layer are related in a special manner.

In the terms m,n and a , in fact 2222 / nn km αω+= 2222 / nn kn βω+= 222 /2 nn ka βω+=

and so the frequency ω appears only in combination with 2β or 2α .



The other terms in which the frequency or the velocities are present are factors that multiply the expression: for instance µ (related to β by ρµβ /= ) can appear as a factor with exponent –1, 0,+1. In the term λ the frequency is isolated k/ωλ = , but this term again can only be a factor.

Fixing the thickness and the densities, fixing the ratio between the velocities β and α , βα ⋅= C we can express the secular function as a function of the ratio between velocity and frequency

( ) ( )kfconstkf ,/),(,, βωβωβω ⋅=

This express the scaling property of the secular function with layer velocities. The secular function has exactly the same value, when the velocities are multiplied by a factor F, provided that also the frequencies are multiplied by the same factor F.

In figure 2.26 the secular functions calculated, using the Dunkin’s normalized expression are shown, for two normal dispersive six layers model, the second with all the velocities being a half of the first (200m/s 500m/s 1000m/s 2000m/s 3000m/s; 2m 2m 2m 2m ):

Figure2.26. Secular function for two scaled models.

It can be noticed that the image is stretched only along the frequency axis of a factor 2.

This scaling of the frequency has a simple effect on the phase velocity, obtained as a function of frequency as

kkfv ωπ

==2

The scaling of the frequency becomes directly a scaling of the phase velocity of Rayleigh waves. Finally, if two model have the same thickness and densities, multiplying the layer velocities by a factor F, the Rayleigh phase velocities and the frequencies are multiplied by the same factor.



With simple algebraic passages one can get the modal curves. In the following figure (2.27) the modal curves for 5 layers model. (400m/s 600m/s 800m/s 1200m/s; 1m-1m-1m: 200m/s 300m/s 400m/s 600m/s; 1m-1m-1m)

Figure 2.27. Scaling of Modal curves: the figure below refers to a model with doubled velocities

compared to the one of the figure above. Both the velocity and the frequencies are stretched.

2.4.4.3 Layer Thickness Then we have to consider the thickness d. The layer thickness appear only in exponential term, and the secular function can be restated as

( ) ( )dkdfdkf ⋅⋅= ,,, ωω

being d a scaling factor for both k and ω . If the thickness of all layers is multiplied by a factor F, the secular function assumes exactly the same values if the frequency and the wavenumber are divided by the same factor F.

In figure 2.28 the secular functions for a five layers model and a double thickness model are depicted. In the second case all the thickness have been doubled (200m/s 500m/s 1000m/s 2000m/s 3000m/s; 2m 2m 2m 2m 200m/s 500m/s 1000m/s 2000m/s 3000m/s; 4m 4m 4m 4m )



Figure2.28. Secular function for the two scaled models. In the second model (bottom) the secular function has been

stretched both in the wavenumber and in the frequency axis. The point f =100 k=3of the top image is stretched to the point f=50 k=1.5 of the second image.

Being the velocity simply obtained as

kkfv ωπ

==2

the velocity is not scaled by a scaling of thickness (fig 2.29). Finally, the scaling of the thickness scale the frequency but not the velocity of the modal curves: is just a scaling of the wavelength.



Figure 2.29. Modal curves for the scaled model. The bottom figure is obtained

from a model with doubled layer thickness.

The two scaling properties can of course be considered simultaneously, and then from a starting model, we can easily obtain all the model with the same ratio between the velocities and the thickness: a normalization is possible.

In logarithmic axis, the scaling become a translation, and all modal curves, for a ratio of d and v can be obtained with a translation from a normalized model

Figure2.30. Modal curve depicted in logarithmic axis the scaling becomes a translation. Grey line (100 200 100 400

m/s: 2m,3m,2m) . Black line: velocities multiplied by 1.4, thickness divided by 1.1.



2.4.5 The group velocity

In dispersive media each frequency has its own velocity: a real waveform is composed by different frequency components, which travel with different velocities. The shape is then modified while it propagates, because the phases of the different frequency components change, and in general the time duration of the signal increases with the offset (broadening of the pulse).

If we consider a signal having a central frequency f we can define the phase velocity of each frequency and the group velocity of the wave train. The group velocity U is coincident with the phase velocity V in non dispersive media, but in general can be computed as (Sheriff and Geldart 1995)

( ) λλ

ωω

ddVV

dfdVfV

dkdU −=+== (2.146)

From the above equation it is easily seen that if phase velocity V increases with frequency, the group velocity U is greater. If it decreases with increasing frequency, as it happens at normally dispersive sites, V is greater than U. The figure below (Fig 2.31) shows for a linear phase velocity, respectively decreasing and increasing with the frequency, the relations between group velocity and phase velocity.

Figure 2.31. Phase velocity and group velocity vs. frequency for a decreasing (left) and increasing (right) phase velocity.

2.4.6 Eigenfunctions: displacements and stresses

The modal curves are only possible solutions, possible velocities at which some energy could propagate. Often only one of them has a relevant energy at each frequency, it is not necessarily the same mode in the whole frequency range, and not necessarily the first mode. For a complete and significant simulation of the Rayleigh wave propagation, the contribution of each mode has to be evaluated. Going back to the definition of the problem (equation 2.98), it can be seen that the unknowns actually are the displacements and the stresses as a function of the depth (the eigenfunctions): the modal curves are simply the eigenvalues, a mathematical constraint, a condition of non-trivial solution of the problem. When the modal curves have been computed, the eigenfunctions are straightforwardly computed.

The example below shows the different steps of the computation for a simple two layers normal dispersive model (200m/s 400m/s; 6m) we show the results for the eigenvalues and the



eigenfunction. The first part of the computation consists of calculating the Rayleigh dispersion equation and finding its roots. For each frequency we search for the zeros, and we find the Rayleigh wavenumbers )( fkk jj = , which are the eigenvalues of the problem. Each curve corresponds to a Rayleigh mode (Figure 2.32).

Figure 2.32. Modal curve are the eigenvalues in f- k plane, only possible solutions.

We can see different modal curve, lower modes with higher values of the wavenumber, and so lower modes with lower velocity of propagation. The number of modes increases with increasing frequency, and each mode, except the first, exist only for Cjff > being Cjf the cut-off frequency of the jth mode: at the cut-off frequency the propagation velocity equals the maximum shear-wave velocity, and the ratio between k and f is ( )SVk max/f2 =π , in this case 400 m/s.

It is easy to transform the modal curves in term of frequency and phase velocity simply calculating the phase velocity from each wavenumber (Figure 2.33).

Figure2.33. Eigenvalues in f- v plane.

Then, with the found eigenvalues, we can calculate the eigenfunctions, which express, for each frequency, for each mode, displacements and stresses with depth.



Figure 2.34 shows the vertical and horizontal displacements eigenfunction for a homogeneous half-space. For a layered medium the decay of amplitude with the depth is still exponential, with different rate in the different layers, and with the imposed continuity at the interfaces. In figure 2.33 vertical and horizontal displacements associated to the first mode are plotted for the frequency of 20 Hz. It is easy to recognise the presence of the interfaces a depth of 6m.

Figure2.34. Vertical and horizontal displacement induced by Rayleigh waves in a layered medium.

If we plot the same eigenfunctions of the first mode for different frequencies we see that the amplitude decay with depth depends on frequency, so it depends on wavelength λ .

But we notice also that the amplitudes depends on frequency and for certain frequencies amplitudes are much greater (Figure 2.35).

Figure 2.35. Vertical and horizontal displacement for the first mode at different frequencies.



If we plot the values of displacement at the free surface for different frequencies we can recognise the presence of maxima.

We can also see the depth at which the horizontal displacement is zero, and so the elliptic trajectory of the particle collapse on a vertical trajectory.

It is evident that the energy distribution with depth is strongly influenced by the elastic properties of the layers, and the theoretical assumption of the half-space are no more valid.

If we analyse eigenfunctions of other eigenvalues we can compare amplitudes of motion, rate of decay and so on for the different modes. For a frequency of 100 Hz, greater than the cut-off frequency of the 6th mode, we can plot for all modes horizontal and vertical displacements with depth (Figure 2.36).

Figure 2.36. Displacement with depth for the first six modes.

2.4.7 Computation of the displacements at the free surface

The computation of the displacements at the surface is the step that allows to simulate the real test: the synthetic seismograms can be obtained, with the aim of being compared with the experimental ones, eventually after some processing step able to extract the relevant information.

In a homogeneous medium only one mode can propagate, with a propagation velocity that does not depend from the frequency. In a vertically heterogeneous medium, at a given frequency, several modes can exist. For a given frequency, the different existing modes are superimposed on one another.

It can be shown (Aki and Richards, 1980) that the displacement is independent from the azimuth angle and so the two components are

( ) ( )[ ] ( )4/

1

.,,,, πωωω −−

=∑= rkti

j

M

jrr

jezrAzru (2.147)



( ) ( )[ ] ( )4/

1.,,,, πωωω +−

=∑= rkti

j

M

jyz

jezrAzru (2.148)

where rA , zA and jk are the amplitudes and the wavenumber associated to the jth mode.

The horizontal coordinate here is r, the radial distance from the source. It can be shown (Aki and Richards, 1980) that these expressions hold for the plane waves described above.

The actual particle displacement is obtained by taking either the real or imaginary part of the expression above: with trigonometric identities the equation of the wave front can be obtained, and an analytic expression of the effective phase velocity can be derived (Lai, 1998).

To compute the particle displacements due to a harmonic unit point source the displacement Green’s function has to be computed. The vertical displacement, considering the mode superposition, is given in the expression below. More details can be found in Lai (1998).

( ) ( ) ( )[ ]ωωωω ,,,,,,ˆ zrtizz

zezrUzru Ψ−⋅=

(2.149)

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

5.0

1 1

2222 cos,,,,241,,

−

= ∑∑= =

M

i

M

j jjjiiiji

jisjsijiz IUVIUVkk

kkrzkrzkrzkrzkrr

zrUπ

ω

(2.150)

( )

( ) ( )( )

( )

( ) ( )( ) ( )

+⋅

+⋅

=Ψ

∑

∑

=

=−M

jj

jjjj

sjj

M

ii

iiii

sii

z

rkIUVk

zkrzkr

rkIUVk

zkrzkr

zr

1

22

1

22

1

4/cos,,

4/sin,,

tan,,π

π

ω

(2.151)

where V, U and k are the phase velocity, group velocity, and wavenumber, I is the first energy integral. It has to be noticed that because of the modal superposition, the spatial variation of the displacement is no longer harmonic, even with a harmonic source. When the source function is known ( ti

z eF ω⋅ ), the real displacement can be easily computed as

( ) ( )ωω ,,ˆ,, zruFzru z ⋅= (2.152)

2.4.8 Geometrical attenuation and modes superposition

The factor ( )ω,, yrU y is the Rayleigh geometrical spreading function and models the geometrical attenuation in vertical heterogeneous media. In homogeneous media, the cylindrical wave fronts correspond to an attenuation by a factor proportional to 5.0−r , the superposition of several modes in heterogeneous media alter this simple law.

The different modes have different velocities, their phase difference changes with the distance: the overall amplitude strongly depends on the phase difference between the modes at each distance. The example below shows these ideas: let us consider a two layer model, with a first layer 2m thick with shear wave velocity of 200 m/s and a half space of 300m/s of shear wave velocity. Figure 2.37 shows the modal curves.



Figure2.37. Modal curves for a simple model (200m/s 300 m/s; 2m).

The only possible velocity at 14 Hz is 265 m/s of the first mode: at 74 Hz two phase velocities, associated to the two existing modes, are present with the phase velocities of 191 m/s and 284 m/s. The amplitude at 14 Hz has a decay proportional to the square root of the distance from the source, simply due to the geometric spreading (Figure 2.38, left). At the frequency of 74 Hz two modes are superimposed, and the amplitude has a different behaviour (Figure 2.38, right)

Figure 2.38. Overall amplitudes with the distance, for a single mode and for the superposition of two modes. Left, at 14

Hz, where only one mode exists. Right, amplitudes at 74 Hz, where two modes exist and are superimposed. Same model 200m/s 300 m/s; 2m.

At the frequency of 74 Hz two modes exists with different velocities: at the source the two modes are in phase, increasing the distance their phase difference (time delay) increases as a consequence of their different velocities.

The wavenumbers are the phase rotation with the distance, and they are 2.435 rad/m and 1.637 rad/m, (obtained as vfk /2 ⋅= π ) which corresponds to a difference of 0.798 rad/m. The sum is maximum with a zero phase difference and minimum with a π phase difference: the sum is periodic with period 2π/0.798 = 7.87 m (Figure 2.39)



Figure 2.39. Amplitude as a function of the distance, after the geometrical spreading correction. Same model.

When two modes have the same amplitude at a certain frequency, the resulting sum is zero at certain distances from the source. An example with real data is given in chapter 3 .

When more than two modes are present, the shape is more complicated (Figure 2.40), but the principle is the same.

Figure2.40. Amplitude as a function of the distance when many modes are superimposed.

The displacement due to a unit source (Fz = 1) can be obtained from the expressions above in space-frequency domain: then, at each distance, the complex spectrum can be obtained.

Figure2.41. Amplitude and phase of a trace at 10 m from the source, with unit source.



2.4.9 Simulation of a real test

The simulation of the test is an essential step for the inversion, based on the comparison between the experimental results of a testing procedure and the numerical simulation of the test. In the simulation, all the parameters influencing the result have to be considered: the dispersive characteristic that can be observed are just apparent dispersive characteristics, and they are also influenced by the observation layout. In the simulation, hence, the receiver array has to be considered, and one possibility is to compute a synthetic seismogram

The complex wave field in frequency offset domain can be computed: assuming a unit flat band source in the desired frequency range the complex displacement is computed in every point where the receivers will be placed or have been placed. The figure below (Figure 2.42) shows the amplitude spectra of 24 traces, two meters spaced, in the range 5-90 Hz, for the same model that have been considered in the above (2m 200m/s, half space 300 m/s). It is evident that below 48 Hz only the first mode exist, and at fixed frequency the amplitude simply decreases with the offset: above this limit cut-off frequency of the second mode, two modes exist (see Figure 2.37), and the amplitude depends on the modal superposition.

Figure2.42. Trace amplitude as a function of the offset and the frequency. The cut-off frequency of the second mode can be clearly identified: above 48 Hz two modes superimpose producing oscillation of the amplitude, at a fixed frequency,

as a function of the offset. Same model: 200m/s, 300 m/s; 2m.

The complex spectrum has been computed, even if only the amplitude spectrum has been depicted: an inverse transform of the complex displacement, transforming the frequency into time, could give the seismograms. On the other hand, from these displacements in f-x, with a 1D Fourier transform in the space dimension, the f-x domain is transformed into f-k domain (Figure 2.43). Of course, if the f-k spectrum is to be compared with experimental data, all the padding and windowing parameters have to be the same.



Figure 2.43. The f-k spectrum is simply computed Fourier transforming the f-x. Same model: 200m/s, 300 m/s; 2m.

This synthetic spectrum is computed considering also all the acquisition parameters and layout of a real test: the relative maxima at each frequency and the absolute maxima are searched, and the whole modelling procedure is consistent with the test. The aspect that is not considered in this kind of modelling is the material attenuation.

2.4.10 Apparent dispersion curve

The synthetic seismograms can be processed with the same procedure that is used for experimental data: for instance, if a f-k processing is chosen, the f-k spectrum is analysed to find absolute maxima and eventually relative maxima related to secondary modes.

The apparent dispersion curve can be defined as the phase velocity associated to the maximum spectral density at each frequency, that is, at each frequency, the phase velocity at which most of the energy propagates.

This phase velocity can be a modal phase velocity or not: the propagation actually is possible only following modal curves (see 2.3.2), but with a finite array length, if more than one mode is present, the above definition of apparent curve leads to a velocity that can be different.

So the apparent dispersion curve is not an intrinsic property of the site, but depends on the observation, and depends on the processing, and actually is due to the modal superposition. These aspects are discussed from a theoretical point of view in chapter 4.

In the following example the possibility of simulating the apparent dispersion curve taking into account the array is shown. For the models whose modal curves are depicted in figure 2.22 and figure 2.24, here the spectra and the apparent dispersion curves are given (Figures 2.44, 2.45, 2.46).



Figure 2.44. Model: 200m/s, 300m/s; 20 m; 48 m array.

Figure2.45. Model: 250m/s, 150m/s, 300m/s; 10m, 10m; 48 m array.

Figure2.46. Model: 250m/s, 150m/s, 300m/s; 10m, 10m; 4m array.

Modal curves f-k spectrum Apparent curve





It is important to notice that the effect of higher modes are not only important at inversely dispersive site: it can be easily seen that higher modes can become important, even at low frequency, and even at normally dispersive sites without strong contrasts of soil properties. The example below (Figure 2.47) shows a normally dispersive site where the second mode is dominant at low frequency. This feature is very important, because below the cut-off frequency of the second mode less energy is propagated, and often experimentally the curve is obtained only above that limit value. In this contest, the high velocity of the apparent curve, due to the second mode, can produce a significant error in the model estimate if it is interpreted as a first mode curve. These aspects will be discussed in chapter 5.

Figure2.47. Spectrum, modal curve and apparent curve for the model 100m/s, 120m/s, 250m/s; 1m, 1m. The second mode has a relevant energy approaching to his cut-off frequency and strongly condition the apparent phase velocity,

which is however due to the modal superposition. (array 24 m long).

2.4.11 Computation of synthetic seismograms

The computation of synthetic seismograms in time-offset domain can be useful to test the processing algorithms on synthetic data and to better understand the features of the surface waves propagation. To compute synthetic seismograms the source spectrum has to be known.

When the source is known tiz eF ω⋅ , the real displacement can be easily computed as

( ) ( )ωω ,,ˆ,, zruFzru z ⋅=

The displacements are computed for a unit point source and easily computed for a known source spectrum, summing the displacements for each spectral component of the source. When the source spectrum is known, the synthetic wave field can be computed.

Here an experimental force signal of a sledge hammer (PCB Piezotronics, 8kg) is presented. The time duration of the impulse is few milliseconds (Figure 2.48).



Figure 2.48. Force signal of the sledgehammer in time domain.

The spectrum of the signal is shown in the following figure (2.49)

Figure 2.49. Force signal of the sledgehammer un frequency domain: amplitude

spectrum (top) and phase spectrum(bottom).

The zoom of the low frequency portion of the amplitude spectrum is depicted in the following figure 2.50

Figure 2.50. Amplitude spectrum of the hammer: zoom on the low frequencies.



The coupling of the source with the soil is not considered here, so the signal that is actually transmitted into the earth from the source is supposed to be exactly identical to the force signal recorded on the source.

The following figure 2.51 shows the modal curve, the f-k spectrum, and the synthetic seismograms for a simple 3 layers model with the source described above.

Figure 2.51. Modal curves, f-k spectrum, and synthetic seismogram for a simple

3 layer model 80m/s, 120m/s, 500 m/s; 2m, 2m.

The algorithm can be tested in different situations: for instance the following figure 2.52 shows the differences between a homogeneous medium and a layered dispersive medium.

The homogeneous medium is not dispersive, and every frequency component of the signal propagate with the same phase velocity. The wavelet propagate without being deformed: a simple translation in time offset is obtained (Figure 2.52, left).

In a layered medium (a first layer with a shear wave velocity of 200 m/s on a half-space of 400 m/s) surface waves are dispersive, and the different frequencies propagate with different phase velocities, so at a certain distance from the source they arrive at different times. This produce the broadening of the impulse with the distance from the source that can be easily noticed in figure 2.52 (right).

Figure 2.52. Synthetic seismograms for a homogeneous medium (Vs=200 m/s) (left) and a 2 layer model (200 m/s, 400

m/s; 2m)(right). The dispersion in the layered model produces the broadening of the impulse.



2.4.12 Partial derivatives

The partial derivatives of the Rayleigh modal phase velocities are very useful in the inversion of surface wave. Applying the variational principle of Rayleigh waves, Lai (1998) obtains the closed-form expression for the partial derivatives of the modal phase velocity respect to the layer body wave velocities.

The partial derivatives respect to the shear and compressional wave velocities of the layer jL are given by the following expressions

dzdzdrkr

dzdrkr

UIkV

VV j

jj

z

z

S

LS

R ∫−

−

−=

∂∂

1

21

21

21

2 42

ρ

dzdzdrkr

UIkV

VV j

jj

z

z

P

LP

R ∫−

−=

∂∂

1

22

11

22ρ

where the symbols have the same meaning than above.

This quantities can be useful to perform a sensitivity analysis, to chose the most important parameters in the inversion, to compute the Jacobian matrix for a linearised inversion.

Chapter 2.5 COMPUTER PROGRAM FOR THE FORWARD MODELLING


2.5 COMPUTER PROGRAM FOR THE FORWARD MODELLING

The computer program for the forward modelling has been implemented in Matlab. The algorithms are those described in the above. Many useful programming indications have been found in the Fortran programs of Herrmann (Computer programs in Seismology), and the Matlab routines of Lai and Rix.

All the examples give in this chapter have been computed using this program: a series of benchmark solutions test have been performed to confirm the correctness of the result.

The input data are the shear and compressional wave velocities, the densities, and the thickness of the layers, the position of the source and receivers: the possible output are the modal curves, in terms of phase velocity or wavenumber, the stress and displacements as a function of the depth, the site frequency response, the displacement at the free surface (or at a known depth) at the offsets given in input, the power spectra in f-x domain and in f-k domain.

“When the half space has not the highest velocity…”


3

CHAPTER 3

Acquisition

This chapter describes the acquisition of seismic data for the surface wave methods; particular attention is paid to the design of the acquisition in such a way to obtain the maximum of information and the minimum of uncertainty, stressing the intrinsic limitations of the sampling in time and space. The different sources of noise are discussed, considering the procedures to reduce their effect, then the intrinsic limits of the acquisition are presented. This allows to give practical indications on the acquisition parameters and on the characteristics of the equipment, and gives warning about the artefacts that the processing will have to face.

In this chapter

The physical noise in records: the random noise and the coherent noise. Different kind of coherent noise: body waves, scattered surface waves, Lamb waves, air blast, near field and far field effects, lateral variations.

The intrinsic limits of the acquisition: the effects of the sampling on the acquired data. The effects of the spatial sampling, wavenumber resolution, maximum wavenumber and spatial aliasing, spatial windowing. Effects upon the frequency ranges, on the mode separation, on the modes identification accuracy.

The design of the acquisition parameters: the layout geometry - receiver spacing, array length, first receiver offset - and the strategies to face the limitations – multi array, unequal spaced array, multi array simulation.

The characteristics of the equipment: the source in general, impulsive sources and vibrating sources, the sensors, the digital recorder. Finally the description of the equipment that has been used for the experimental part.



3.1 INTRODUCTION

Many acquisition techniques have been used in surface waves surveying, depending on the different applications, depth of investigation, and scale of acquisition. The elements to be discussed are the kind of source that generate the surface waves, the receiver that detect the vibration or pressure field, and the scale of the observation. The sources and the instruments can be very different, and the scale of observation can vary from thousands of kilometres to some centimetres.

In earthquake seismology surface waves have long been used to infer shear-velocity models of the crust and upper mantle (Ewing et al, 1957; Dorman et al, 1960; Dorman and Ewing, 1962; Bullen, 1963; Knopoff 1972; Kovach 1978): these surface waves propagating in the crust and in the upper mantle, known as long period oscillations, involves periods between 5-10 s and 300-500 s, sometimes up to 800 s (Keilis-Borok et al., 1989). They are of course observed and acquired with wide arrays of seismometers at almost continental scales.

Surface waves have also been used for the determination of the “shallow structure”, in the seismological meaning referring to the first hundreds of meters of the crust. Examples of acquisitions with arrays of tens of kilometres length are given by Mokhart et al. (1988), Al-Eqabi and Herrmann (1993), Herrmann and Al-Eqabi (1991). The surface wave data consist in general of some tens of traces ( with time window of tens of seconds) within array length of some tens of kilometres, gathered with standard low-frequency vertical geophones (less than 2 Hz ) and explosive as source: wavelengths of some kilometres are acquired.

The engineering scale is smaller: some hundreds of meters or even some tens of meters. The equipment and technology that is used has to be light, portable, and cheap: the receivers are geophones, the sources are sledgehammers, weight drop, small vibrators, and also noise can be used. The one and two station acquisition pioneered by Jones (1958, 1962) and Ballard (1964) developed to the SASW test (Nazarian and Stokoe 1984; Stokoe et al., 1994) has widely been used for the characterisation of pavement systems and soil deposits (Abbiss, 1981, Hiltunen and Woods, 1988, Gukunski and Woods, 1991, Rix and Lai, 1998; Tokimatsu et al., 1998a). The use of two components receivers for the acquisition of both the component of the Rayleigh wave displacement have also been proposed (Tokimatsu et al., 1998). Active multi-station techniques use arrays of geophones (12-48 and more) and different sources, without substantial difference from the shallow crust acquisition (Gabriels et al., 1987; Tselentis and Delis 1991; Schtievelman 1999; Park et al., 1999; Foti 2000). Passive techniques usually observe vertical motion of microtremors using a two-dimensional array of sensors distributed on the ground surface, in the attempt to go deeper using the great wavelengths of microtremors (for instance Kanai and Tanaka 1961, Horike, 1985, Tokimatsu et al., 1998b): some techniques use the power spectra and other the phase velocity of microtremors. Multi-station applications have also been preformed in marine environment, with the use of hydrophones to detect the pressure field related to the surface wave propagation on the sea bottom, and explosive or airguns as sources (McMechan and Yedlin, 1981; Shtievelmann 1999)

Going to the smallest scale, the use of ultrasonic surface waves and the techniques to generate and acquire surface waves are described for instance in the book of Viktorov (1967).

The seismic acquisition is the observation of the effects of the propagation of seismic waves in the time and in the space. The complex vibration field produced by a seismic source can be detected by



different kind of transducers, placed on the surface or in boreholes and each acquisition should try to enhance the phenomenon that has to be observed, reducing the other seismic events.

Surface wave methods are based upon the inversion of dispersive characteristics of such waves, obtained by the processing of full-waveform data. The acquisition step hence has to record the effects of the Rayleigh wave propagation in the most complete way, reducing the presence of other phenomena in raw data.

In general, the acquisition is the observation of the surface wave within a full wave field propagation: as it has been mentioned, the source, the receiver type, number and positions, the recording instrument can be very different. All these aspects influence the result: the data quality depends of course on the acquisition, but the adopted layout and equipment also influence the information that can be extracted by raw data.

The acquisition step is here discussed in the view of the data quality and of the effects of the acquisition parameters on the information contained in the field data.

The acquisition of good quality data is one of the main aspects of all geophysical techniques, and directly influences the final result. It is a rather diffused opinion that the acquisition of SWM is quite simple, since surface waves are high energy, dominant events in records, and this is actually quite true. However the analysis is performed in frequency domain, and the signal to noise ratio has to be high in the whole frequency band where the information is needed, and this requirement is not simple to be satisfied. Surface waves actually usually dominate in records, but only in certain frequency ranges, mostly depending on the site, which sometimes are not sufficient to extract the desired information.

Furthermore, when an inversion has to be performed, it is fundamental that the forward problem considers all the factors that influence the inverted data. The acquisition parameters have then to be considered carefully, in order to take into account all their effects and to properly design the test so that the inverted data contain the maximum of coherent information and the minimum of coherent noise.

The aim of the acquisition is observing and recording surface waves in time and space. The limits of the acquisition are due to physical noise and to intrinsic observation limitations.

The former depends on the fact that the measured particle vertical velocity is supposed to be due only to the source-generated propagating surface wave: actually other source-generated phenomena are present, and moreover other types of noise (seismic noise non-source generated, electrical and electronic noise) affect records.

For the latter, the observation of the propagation within a limited window in time and space and in a limited number of discrete points induces other limits, and produces loss of information.

Chapter 3.2 PHYSICAL NOISE


3.2 PHYSICAL NOISE

This section discuss the different kind of noise that affects seismic record, and in particular the surface wave data acquisition. Random noise and coherent noise are considered, and field data example are shown.

The acquisition should try to reduce the presence of external events in records, by an adequate design and control of the experiment. The noise in records can be classified in random noise and coherent noise, according to its origin and repeatability.

3.2.1 Random noise

3.2.1.1 Nature of random noise The ‘random noise’ is not necessarily strictly random in the statistical meaning, but it is random with respect to the test: it is not directly produced by the source, having an origin completely external from the test. In each acquisition this random component is summed to the coherent signal.

There can be seismic noise (any type of vibrations, produced by traffic, human activities, vibrating machines, wind…) and electric noise in receivers, cables, and in the seismograph.

The level of this kind of noise can be evaluated on records acquired without energizing: the spectrum of the noise can be analysed to see its frequency content and sometimes this allows to identify its origin.

When a narrow band noise is present, i.e. a frequency component is dominating, it’s usually due to human activities, and it is quite easy to filter it out (Figure 3.1, Figure 3.2).

Figure 3.1. The amplitude spectrum of a noise with dominating frequency at 30 Hz. Other isolated peaks are present.

Frequently peak are at 50Hz and its multiples (60Hz in the USA) because of the electrical net.



Figure 3.2. The amplitude spectrum of a broad band noise.

3.2.1.2 The stacking procedure to increase the signal to noise ratio Considering the random noise, the signal to noise ratio (S/N) can be improved by increasing the energy of the signal, i.e. either by directly increasing the power of the source, or by a stacking procedure. Summing different acquisitions, if the starter system is reliable, the S/N should increase with the square root of the number of acquisition, as shown below.

The coherent signal can be decomposed in its monochromatic components, and repeating the test each of them should have the same amplitude and the same phase: if the source is not repeatable, at least the phase should be identical. The random noise can be stationary referring to amplitudes, but the phase should change in a random way among the different acquisitions: the phase difference with the signal changes in a random way.

The stacking can then increase the S/N, due to the fact that the signal, summed in a synchronised way increases more than the noise. Considering a single frequency, if the phase is constant (synchronized sum), with the stacking the phase difference is zero and the sum is maximum: the sum of the amplitudes increase as the number of stacked signal n.

If the phase is random, as in a frequency component of the random noise, the phase is uniformly distributed in ( )π2,0 ; the absolute value of the phase difference between two summed component is linearly distributed in ( )π,0 and this makes the sum lower than its maximum; considering a

constant amplitude, with the stacking it increases as n . The SNR then increases as nn

n = .

(Figure 3.3)



Figure 3.3. The amplitude of the sum when it’s synchronized and with a certain phase difference.

Often the different shot gathers are summed by the instrument and recorded in a single file: the preview of traces in time domain allows to roughly evaluate the data quality and to select a sufficient number of stacks.

It has to be mentioned that to improve the signal quality by stacking it is strictly necessary to have a perfect synchronisation of acquisition, and that the sum of time shifted data can even deteriorate the signal quality: the trigger performance is hence very important in stacking.

A constant time shift does not affect neither the amplitude of spectra nor the phases; if all the records are shifted by the same quantity respect to the shot instant, the result is identical: a pre-trig can be used without negative effects, except an increase in noise energy (see 3.6.3).

On the contrary, if the trigger is not reliable and does not assure a perfect synchronization of records, i.e. the delay is not constant, it is better to save the data in different files. In such a way, data can be pre-processed, for instance cross-correlated to compute and correct the trigger shift between the different records, and then they can be summed after being synchronized.

The S/N does not increase summing shifted traces (Figure 3.4): the amplitude of the f-k spectrum is not affected by the phases of the traces, and the effect of such a stacking is simply an overall scaling of the spectrum, like multiplying a single seismogram, the noise and the signal, by a constant value.

In an analogous way, if the stacking is performed in f-k domain, the complex spectra have to be used: for the linearity of the Fourier Transform, the stacking in time-offset or in frequency-wavenumber domain give exactly the same result.

Figure 3.4. Synthetic seismograms: the right seismogram is obtained summing time shifted traces: the time shift affects

only the phase, the amplitude of the f-k spectrum of the two is identical, except a scaling factor.



3.2.1.3 Use of the coherence function to evaluate the signal to noise ratio Another reason to record separately the different acquisitions instead of stacking them directly in the field, is the possibility to evaluate the signal to noise ratio comparing statistically the different acquisitions.

One possibility, usually adopted in the 2-station processing, is the use of the coherence function. The coherence function is a spectral quantity that allows a quantitative assessment of the signal quality, and evaluates the presence of random noise in the signals. It is discussed in chapter 4.

Other approaches, such as the statistical analysis of the phase distribution (chapter 4), allow the direct assessment of the signal quality as a function of the frequency and the offset, and requires separate acquisitions of the different shots.

3.2.2 Coherent noise

3.2.2.1 Coherent noise, model and information The coherent noise cannot be suppressed by repetition of acquisition, since it repeats itself too, being source-generated or test-generated: it is due to an interaction between the test and stationary entities. Not considering possible instrument problems, the main reasons of coherent noise are in the tested subsoil. The coherent noise, hence, is the part of information that we don’t want to interpret, or that we are not able to interpret.

Many geophysical techniques for subsoil characterization lead to inverse problems: a set of parameters is assumed to completely describe the tested domain, and a theoretical model, from these parameters, is assumed to be able to compute the set of data that will be measured. From the measured data, by inversion, the parameters are found.

What is considered coherent noise depends strongly on the adopted model. No model can represent completely the reality and the experimental data depend on a large set of parameters: the events not considered by the model but present in the real propagation phenomenon can be considered coherent noise. If they are not identified and filtered out, their effects on acquired data are interpreted as due to the model, and introduce errors in the inferred model. Two cases can occur: either no model can fit reasonably the data, or a wrong model fit them (Figure 3.5). These aspects will be discussed in detail in chapter 5.

Figure 3.5. Examples of coherent noise/model errors. left) The best linear model has a great misfit with quadratic data.

right) Experimental data (dots) actually correspond to the second mode(solid black line). If they are fitted by a first mode (grey dashed) there is great error even with a little misfit.



In SWM the subsoil scheme is a stack of homogeneous elastic layers: the model parameters are the geometrical and mechanical parameters of each layer, and the modelling, with the algorithm illustrated in chapter 2, allows to simulate the multi-modal surface wave propagation as horizontally travelling plane wave. So, no other soil characteristics is accounted for , and no other phenomenon is considered.

3.2.2.2 Types of coherent noise Surface waves are considered coherent noise in reflection seismic acquisition, while in SWM whatever is not surface wave is noise.

In particular body waves are present in records from direct, refracted and reflected paths, deriving from the heterogeneities of acoustic impedance in the subsurface. If the stratigraphy allows the existence of other seismic events, such as Lamb waves, these have to be considered coherent noise. But also surface waves can be considered noise: the forward modelling consider only source generated surface waves propagating as plane waves in a laterally homogeneous medium. In the linear configuration of the test, the horizontal dimension is neglected: all the effects of the horizontal dimension, for instance due to variations of the subsoil properties, have to considered coherent noise. Back scattered surface waves can be present with horizontal discontinuities of natural or artificial origin, and this can also distort the dispersion curve.

Surface waves that propagate a short distance from the source cannot be treated as horizontally travelling plane waves, and this effect is known as near field effect (Stokoe et al., 1994). High frequency surface waves attenuate rapidly with distance and can not dominate records at great offset: this is known as far-offset effect (Park et al., 1999).

The lateral variations have a strong effect on acquired data, and on the inferred dispersion characteristics: their presence should try to be identified and considered. Other problems can depend on the multi-modal nature of the surface wave propagation: the energy associated to higher modes depends on the stratigraphy and on the source, but sometimes, specially in inversely dispersive sites, higher modes can be dominant in experimental data. If the adopted model uses only the first mode, higher modes have to be considered coherent noise and have to be carefully filtered out.

When the coherent noise depends mostly on the subsoil properties, it often carries information and, if the adopted model of the phenomenon allows its simulation, it is not recommendable to filter and discard it. Higher modes, for instance, are often considered as a noise, but they are present in the theoretical model, they can hence be interpreted and inverted (for instance Gabriels et al., 1987). Nonetheless this can be a difficult task, due to the modal superposition that occurs when the array length does not allow the identification of the velocity differences between modes. (chapter 4)

The most important thing is the identification of coherent noise: when the coherent events are recognized in a certain domain, they can be filtered out, or analysed and interpreted.

3.2.2.3 Energy of coherent noise The usual assumption in SWM is that records are dominated by surface waves: this hypothesis is almost always applicable. More than two third of the seismic energy is imparted into Rayleigh waves (Richart et al., 1970) when a compressional source is used at the free surface. Moreover, the geometrical attenuation of body waves is greater, and surface waves should be dominant in records.



Actually scattered surface waves can have a great energy with great reflection coefficient, and higher modes can be dominant, even in normal dispersive sites and even without abrupt variations of soil properties. Moreover it has to be considered that surface waves should dominate record at each frequency: this hypothesis is less true, and P-waves can, for instance, dominate a certain band at high frequency (see figure 3.7). The separation of coherent noise can be therefore important.

The identification of other phenomena can be done, when multi offset data are gathered, by coherency pattern, arrival time, velocity, attenuation.

In the following an overview of different types of coherent noise is given, showing experimental data and discussing their properties.

3.2.2.4 Body waves as coherent noise Body waves in records can be P and S waves propagating from the source to receivers with different paths: these two wave phenomena have different energy and velocity. In particular their propagation velocity can be very different, especially in soft saturated soils, where the P-wave velocity is mainly dependent on the water.

Considering the relationship between P and S waves, both the propagation velocity and the attenuation have to be analysed. The propagation velocity can be very different, sometimes allowing the separation even with short arrays: an example is shown in figures 3.6 and 3.7. The site stratigraphy is organic clay on a hard bedrock, with the water table at a depth of 0.5 m. The P-wave velocity is more than 30 times higher than the R-wave velocity.

Figure 3.6. Data acquired at a very soft site (a deposit of soft organic clay). The 24 traces seismogram (left) and the last trace (right). The body waves can be clearly seen at early times. (P wave velocity due to the saturating water).

Sledgehammer, vertical geophones.

The other aspect refers to the attenuation. The geometric attenuation is different for the two phenomena (square root of the distance for surface waves and square of the distance for the body waves at the surface), and should make R-waves dominant in the far offset. This is the usual situation, however the material attenuation can play a strong role in the P/R amplitude ratio.

In the example above the amplitudes of body waves are not important in comparison to Surface waves amplitudes (seismograms with true amplitude, normalization of traces). In a similar situation,

P

R



the ratio between the amplitude of the different phenomena can be very different, and so can not respect the usual assumptions.

Body wave damping is influenced by the saturating water, while the Rayleigh wave attenuation, in such a material, is completely due to the shear wave damping (Viktorov, 1967): moreover P-wave wavelength are much higher, and this reduces the attenuation.

The intrinsic attenuation of Rayleigh waves is hence much higher, and the dominance of body waves with the distance is obtained. Fortunately, body waves can be here easily recognized and filtered out, in all domains where data are analysed. The following example (Figure 3.7) shows data gathered at a site with soft saturated organic clay, where P waves become dominant in far offset.

Figure 3.7. Field data. Seismograms with a evident body waves and then S waves and R waves. Saturated Organic

clays, hammer.

The P-wave velocity is 1600 m/s, the shear wave velocity, estimated by the inversion of the Rayleigh wave dispersion, is less than 35 m/s. The Poisson ratio, consequently, can be estimated as

4998.02

21

22

22

=−

−=

SP

SP

VVVV

ν

and hence the Rayleigh wave attenuation is almost equal to the shear wave attenuation (Viktorov 1967; see section 2.1.1).

By the analysis of the experimental decay of the spectral amplitudes as a function of the offset, the coefficient α of the expression ( ) xfe ⋅−α can be experimentally estimated, for the frequencies between 5 and 12 Hz, between 0.06 and 0.12, and the damping ratio varies in this range between 0.05 and 0.055: this values refers to a material that cannot be considered weakly dissipative (Lai and Rix, 2002).

It is possible to see that in this situation, the intrinsic attenuation compensate the different geometric attenuation and make the P-wave dominant in the far offset.

It is possible sometimes to filter this source of noise before the processing procedure. In the this example above, body waves can be recognised both in time-offset and frequency-wavenumber domains. The following figure (Figure 3.8) shows the time-frequency analysis of the 10th trace of the seismogram depicted above (Figure 3.7). The P-waves can be recognised for both the early



arrival time and for the higher frequency content respect to the R-waves. In figure 3.8b the trace is depicted as energy density as a function of the time and of the frequency (see chapter 4.2.9.)

Figure 3.8. The time-frequency analysis (b) of the 10th trace(a) shows that the differences in time and in frequency

between R and P waves are present (P-wave: early times higher frequencies). R waves are dominant, but not at high frequency (more than 30 Hz). (10th trace, 20 m from the source).

The presence of body waves can be important even at sites where the velocities are not so different. The following example (Figure 3.9) refers to a site of sands and gravels.

Figure 3.9. An other example of a seismogram with strong body waves before the surface wave. The frequencies here

are quite similar, and the amplitude not so different.

P-waves

P-waves

R-waves

R-waves

R-waves

P-waves

a)

b)



3.2.2.5 Scattered surface waves as coherent noise The hypothesis of a subsoil of homogeneous layers of constant thickness can be violated by the presence of lateral variations of the soil properties, by dipping layers, or by the presence of natural or artificial abrupt variations, such as edges, cracks, faults, building foundations, retaining walls, trenches, cavities and so on. At a surface discontinuity an incident Rayleigh wave is partitioned among a reflected R-wave, a transmitted R-wave, and a reflected body wave (Richart et al., 1970)

Many theoretical studies have been even recently conducted to find transmission and reflection coefficients of Rayleigh waves from edges, fractures, inclined contact between elastic wedges (Viktorov, 1958; Blake and Bond, 1990, 1992; Budaev and Bogy 1995, 1996, 2000; Gautesen 2002a, 2002b, Pecorari 2001), mainly with the aim of using ultrasonic surface wave for the sizing of defects. In most of this works the medium is anyway supposed to be homogeneous, and the main interest is focused on the defect shape orientation and properties. For the seismological applications, the problem of surface wave propagation across a vertical contact between two layered structure is discussed for instance by Keilis-Borok et al. (1989).

The scattering of Rayleigh waves can be used to recognize the presence of underground obstacles and cavities, but the reflection of Rayleigh waves is not the only interesting aspect. The effects of such obstacles on the Rayleigh wave dispersion can produce misinterpretation of the shear wave velocity profile: Gucunski et al. (1996) investigated the effects of underground obstacles on a Rayleigh wave dispersion curve by a transient response analysis of an axisymmetric finite element method. If the obstacle is located apart from the measuring alignment, an apparent velocity is obtained for the scattered surface waves.

The example below (Figure 3.10) shows a reflection of Rayleigh waves.

Figure 3.10. Seismogram with back scattered surface waves.

3.2.2.6 Lamb Waves as coherent noise When a stiff top layer such as a pavement lays on a very soft layer, we could imagine the possibility of the existence of pseudo-Lamb waves in the top layer acting as a free plate.

Lamb waves, from a theoretical point of view, are guided waves propagating in plates with free boundaries, for which the displacements occur in the direction of wave propagation and perpendicular to the plate (Viktorov, 1967). The solution of the problem of the Lamb wave

Back scattered R-waves



propagation gives two groups of waves, each one satisfying the equation of motion and the boundary conditions, that can propagate independently one from the another (Viktorov, 1967). They are the symmetric and antisymmetric modes, for which respectively the motion is symmetric or antisymmetric respect to the mid plane of the plate (Figure 3.11).

Figure 3.11. Antisymmetric Lamb waves (up) and symmetric Lamb waves.

Both symmetric and antisymmetric Lamb waves have many modes of propagation, and they both are dispersive; the first symmetric and antisymmetric modes exist at low frequency, and they tends to the Rayleigh wave velocity at high frequency.

The excitation of the different modes depends on the spectrum of the source, on its position, and on the excitation function of the plate itself: often the first antisymmetric mode dominates in a wide range.

The Lamb waves are formed by a combination of multiple reflected, reinforcing SV waves when VP>VL>VS , SV and P when VL>VP>VS (Ewing et al, 1957)

The example below shows the experimental seismogram (Figure 3.12a) and the dispersion curve (Figure 3.12b) of Lamb waves in a concrete floor with void below: probably only the first antisymmetric mode is involved

Figure 3.12. Seismogram (a) and dispersion curve (b) experimentally obtained on a concrete floor with void below

When a stiff top layer lays on a soft material, the boundary conditions can be close to that of Lamb waves: nevertheless, the lower boundary is not free, and the model used for the Rayleigh wave propagation in layered media rest adequate to describe the system, but also Lamb waves can exist.

a)

b)



In the example below, a stiff top layer (0.30 m thick) lays on a very soft soil (Figure 3.13): the Rayleigh wave modal curves are depicted with the energy distribution (Figure 3.14).

Figure3.13.: A stiff top layer lays on a very soft layer (VS = 600m/s and VP= 1200 m/s on VS = 60 m/s and VP = 80 m/s)

Figure 3.14. The f-k spectrum with modal curves in f-k domain (left) and modal curve with apparent dispersion curve in

f-v(right).The Rayleigh wave solution exist for such a stratigraphy

The modal curves shown in figure 3.14.left with the energy distribution show that the energy is concentrated at few frequencies. The array for which the spectrum has been computed is 48 m long, and allows a good mode separation, as it can be seen in figure 3.14.right

The displacements associated to modes are illustrated in figure 3.15. The particle motion in the soft layer is not negligible, while the half space is almost undisturbed.



Figure 3.15. The displacements as a function of the depth for the first three Rayleigh modes of the model of figure 3.13

For the same model, the Lamb solution for the first layer view as a free plate gives the Lamb modal curves shown in figure 3.16 up to 2000 Hz. The Lamb wave solution has been obtained implementing in Matlab the equations given by Ewing et al. (1957)

Figure 3.16. Lamb waves modal curves for a free plate 600m/s 1200m/s 0.3 m. The first two symmetric modes and

antisymmetric mode exist below 2000 Hz.

In most of cases, especially with pavements, the cut off frequencies of higher modes are high respect to SWM acquisition, and so only the first symmetric mode and the first antisymmetric mode can exist.

The first symmetric mode, at low frequency, tends to the velocity of longitudinal waves in two dimensional medium, that is function of the Poisson’s ratio and is smaller than VP; at high frequency tends to the Rayleigh wave velocity of the plate.

The first antisymmetric mode tends, at low frequency, to zero, and always increases monotonically asymptotically to the Rayleigh velocity of the plate.



Lamb waves can be generated in SWM testing, when a velocity inversion is present, and are identical to pure Lamb waves for wavelength greater than 7 times the rigid plate thickness. The inverse dispersion associated to the first antisymmetric mode is easily recognized, and eventually filtered.

The possibility of existence of Lamb waves, however, does not affect the presence of Rayleigh waves: the two phenomena can superimpose, each one with its own energy.

3.2.2.7 Air Blast The air blast is a coherent event in seismic records due to the effects of the sound of the shot (even for a sledgehammer shot): the frequencies are quite high and the velocity almost constant (Figure 3.17) and equal to the sound velocity in air. Its velocity is potentially troublesome for SWM since superimpose on a Rayleigh wave velocity typical of some soil deposits. Hence a velocity filter can have problems in separating them from the signal; the energy, nevertheless, is usually lower, and the frequency higher.

Figure3.17. A seismogram with evident air blast with a sledgehammer shot.(diagonal, from 0m-0s to 35m-0.1s).

The amplitude is comparable with P waves, the frequency is higher

No useful information about the subsoil is contained in airblast.

3.2.2.8 Near Field and Far Offset Rayleigh waves can be regarded as plane waves only beyond a certain distance from the source (Richart et al., 1970) that is beyond the radius of what could be thought as a near field. In two station techniques the measurement are usually considered affected by near field effects if the first receiver is placed at a distance from the source less than half the considered wavelength (Stokoe et al., 1994).

Moreover in general the great attenuation of high frequencies makes the record lacking of coherence in the high frequency bands in the far offset (Figure 3.18).

Air blast



Figure3.18. Amplitude spectrum in frequency offset domain. It is evident the great attenuation of high frequencies in far offset

The near field and far offset effects can be reduced respecting some empirical rules in acquisition, or by a processing that considers these limitations (see chapter 4.3, 4.4).

3.2.2.9 Lateral Variations Being the assumed model one-dimensional, each lateral variation in the tested subsoil at the scale of the acquisition violates the basic hypothesis and produce effects that cannot be correctly interpreted. Nevertheless the effects of lateral variations can be identified and they give a warning on the applicability of the method. Data should not be inverted using an inadequate forward model , and only qualitative information should be extracted when lateral variations are present.

The identification of lateral variations is then an essential step of processing: the multi offset layout allows different techniques to test this basic hypothesis of one-dimensionality. The details will be discussed in chapter 4.

The first possibility to evaluate the presence of lateral variations is the use of two conjugate end-off acquisition and the comparison of the two obtained dispersion curves: if the two dispersion curves show important differences, probably lateral variations occur, and anyway some hypotheses are violated (Figure 3.19). On the contrary, if the two are quite similar, the inversion can be performed.



Figure 3.19. The two opposite dispersion curves with significant differences (Field data)

The lateral variations can be identified also by comparison of the information extracted from different portions of the array. When the overall f-k spectrum is computed as the sum of elementary 2 traces spectra (chapter 4.5.3), the degree of similarity among them can indicate the presence of 2D effects.

Another tool that allows the detection of lateral variations is the processing of local phase differences between traces (chapter 4.4). The following examples (Figures 3.20, 3.21) show the phase velocity for a fixed frequency in the different portions of the array.

Figure 3.20. Field data. Phase difference as a function of the offset for a single frequency. The good linearity of the

phase as a function of the offset indicate the absence of lateral variations.



Figure 3.21. Other field data. The phase difference as a function of the offset don’t have a good linearity. The variance

of the phases does not justify the variations, so lateral changes of the properties are suspected.

3.2.2.10 Higher modes: coherent noise or useful information Higher modes are often present in experimental data, but they can be considered as useful information and not as noise: their phase velocity depends on the subsoil properties, and they can be theoretically simulated by the forward modelling described in chapter 2.

A multi offset acquisition is necessary to properly identify the different modes, and anyway with some processing techniques the effects of higher modes in raw data or in frequency decomposed data can be misinterpreted.

For instance, jumps of the phase and the oscillating behaviour of the amplitude with the offset can be due to the superposition of different modes. The example below refers to a site where a higher mode has a significant energy in the acquired frequency band.

The seismograms acquired with an impulsive source and 24 vertical geophones is shown in figure 3.22 (left). In time offset the presence of two events with different velocities can be recognised: the two modes can also be separated in the f-k spectrum (Figure 3.22, right). This means that the f-k transform in this case allows the separation of the two modes (see 3.4.4), and hence the identification of the properties of both.



Figure 3.22. Field data. Seismograms (sledgehammer as source, 24 geophones), and its f-k spectrum.

Two modes are clearly recognized in the spectrum

The case here presented is quite interesting, since the two modes have almost the same amplitude in a wide frequency range. In particular, the section at 27.5Hz, depicted in figure 3.23, shows the presence of two modes with the same amplitude: the wavenumbers of the two modes can be estimated from the position of the two main maxima of the spectrum.

Figure 3.23: Field data. A section of the f-k spectrum at 27.5 Hz shows the presence of two lobes, corresponding to two modes of similar energy.

At this site, with the used array length, and with the adopted processing technique, the higher mode is not a coherent noise, but on the contrary can increase or confirm the information carried by the first mode.

At the same site, with the same receiver array, a different processing or acquisition can lead to a different conclusion: the higher mode could be not identified and could even corrupt the information of the first mode.

Considering the same frequency of figure 3.23 (27.5Hz), if a monochromatic signal is recorded, the presence of the two modes is no more easily identified. The seismogram acquired with a vibrating source (Figure 3.24, left) shows abrupt variations around 8 m and 16 m: the amplitude has an oscillating behaviour, and the phase has jumps of π radians at the same offsets (Figure 3.24, right).



Figure 3.24: Field data. Left) Monochromatic seismogram, Right ) The spectral amplitude and the phase vs. the offset

It is quite easy to show that this is the correct behaviour due to the modal superposition. With a spectrum like the one showed in figure 3.22, the energy travels with two phase velocities, and two wavelengths. The two superimpose and the distance from the source influences the amplitude and the phase: a scheme is presented in figure 3.25.

Figure 3.25. Scheme of the modal superposition with two modes at a single frequency. The phase difference between the two modes increases with the distance from the source (a). The resultant (b) has a phase and an amplitude that are

strongly influenced by the superposition.

If we consider a simplified spectrum with two spikes at the two wavenumber (0.7606 and 1.4477 rad/m), it is easy to see that the phase difference between the two modes varies with the distance. The wavenumber is the phase difference occurring in 1m, and so the amplitude has a periodic phase difference that increases of k1-k2 radians per meter, i.e. 0.6871 rad/m. Hence the amplitude is periodic with period mk 15.9/2 =∆π , and the phase rotates every 9.15m.

As a test synthetic data are generated, without intrinsic attenuation and corrected for the geometric spreading, and they are depicted in figure 3.26.

a)

b)



Figure 3.26. Synthetic monochromatic seismogram (left) and trace amplitude and phase as a function of the offset

(right) . The data are obtained with two identical modes with different wavenumbers.

It is evident that if the presence of higher modes is not recognized and only few traces are collected and processed, the errors induced by higher modes can be dramatic in the phase velocity and attenuation estimate. This is one of reasons to prefer a multi station approach.

The importance of higher modes for a correct interpretation of the SWM is discussed in detail in chapter 5.

3.2.3 Filtering

A pre-processing phase of filtering can help in noise removal.

The filtering is the separation of data in its components, using the differences in frequency, velocity, wavenumbers: dealing with a two variable process, the differences between the two variables can be used to enhance the signal relative to the noise. The two dimensional impulse response function of some theoretical filters in 2D (see for instance Buttkus, 2000) are shown in figure 3.27.

The pass band can be, for 1D filters, in frequency, in wavenumber or in velocity, and for 2D filters in limited region of f-v, f-k, v-k spaces

Figure 3.27. Frequency- wavenumber response functions for different types of filter: (a) frequency filter (b)wavenumber

filter (c) velocity filter (d) f-v filter (e) f-k filter (f) v-k filter. (adapted after Buttkus,2000)



These filters can be effective when the event to remove or to enhance is separated in one of the mentioned domains.

Frequency filtering can be used to filter out noise, when it affects only a limited range of frequencies. Nevertheless, since the analysis and the processing are conduced in frequency domain, the filtering operation has a reduced utility; since it is possible to discard directly the information of noisy frequencies.

Velocity filtering can be useful to remove coherent events of velocity different from the surface wave velocity, but the random noise often has apparent velocities in a wide range, and hence it is not possible to remove it all. If there is not a single source of noise the velocities can be positive and negative and very different.

The example below shows an application of a 2D filter in f-k domain. The same data shown above (Figure 3.22) are processed to remove the higher mode. The original spectrum is shown in figure 3.28 left: the portion of the spectrum corresponding to the higher mode is muted with a smoothed edges filter, then the obtained spectrum is inverse-transformed to give the filtered seismogram of figure 3.29 right.

Figure 3.28. Left) original f-k spectrum right) Filtered spectrum, the higher mode has been blanked

Figure 3.29. Field data. Left) Seismogram Right) filtered seismogram obtained by the spectrum of figure 3.28right

Chapter 3.3 ACQUISITION LIMITS


3.3 ACQUISITION LIMITS

This section discusses the intrinsic limits of the acquisition. The finiteness of the time and space window of acquisition has a series of consequences that particularly affect the modal superposition. These limitations do not depend upon the processing technique that is adopted, and are here discussed considering the effects of the sampling in f-k domain.

3.3.1 Acquisition parameters and recorded information

The mechanical properties of the subsoil are inferred with an inversion process: the uncertainties of the model depend on the uncertainties on data and on the amount of information.

The data used for the inversion are the dispersion characteristics extracted by the processing from acquired traces: as in each mix determined inverse problem it is not a straightforward task to evaluate the real quantity of information in data, but it is evident that the loss of information in acquisition and processing affects the efficiency of the method in the model parameter estimate.

Many processing technique exist, and the details of processing will be discussed in chapter 4: the quality of dispersion data can be evaluated after the processing, however it depends mostly on acquisition.

Here the effects of sampling parameters are discussed in f-k domain, because the theoretical description of the Rayleigh wave field is usually given in f-k domain (Aki and Richards, 1980), and because this allows a rigorous derivation of the acquisition limitations using the theorems of sampling and spectral analysis.

3.3.2 The ideal and real acquisition

As it is shown in chapter 2, the equation of wave propagation in a linear elastic layered medium with zero stress boundary conditions on the free surface has surface waves as non-trivial solution. Considering the corresponding eigenvalue problem, for each frequency ω, uniquely determined wave numbers k1(ω), …, kn(ω), and phase velocities v1=ω/k1,…, vn=ω/kn can be found. Each eigenvalue corresponds to a mode of propagation. At low frequency (ω→0) only the fundamental mode exists. Increasing ω, also higher modes exist, each one beyond his cut-off frequency: at its cut-off frequency each mode has phase velocity equal to the maximum shear wave velocity in the layered system (Aki and Richards, 1980) (Figure 3.30).



Figure 3.30. modal curves in fk and fv domain

The energy associated to each mode depends on the site but also on the type and depth of the seismic source. The modal velocities are a characteristic of the mechanical layered system and modal curves are continuous and regular functions of model parameters.

The wavefield associated to surface wave propagation, in a layered medium, can be written in terms of mode superposition as (Aki and Richards, 1980)

ωωπ

ωω dexStxs xkti

mm

m ))((),(21),( −

∞

∞−∫ ∑=

so the energy can be only in correspondence of modes.

The ideal Surface Wave Method would record only surface waves to compute experimental modal curves and would invert them to obtain the model parameters.

The limits of a real discrete sampling, within a limited time and space interval, prevent from achieving this ideal situation. The wave field in its theoretical expression written above cannot be observed, because a finite window and a discretization are introduced.

( ) ( ) ( )txwtxstxrOBS ,,, ⋅=

and so

( ) ( ) ( )kfWkfSkfROBS ,,, ∗=

The multiplication in time-offset domain is a convolution in frequency-wavenumber domain with the spectrum of a 2D window function. The box-car window and its spectrum are depicted in figure 3.31.



Figure 3.31.:A 2D box car window and its fk spectrum

This introduce a series of limitations that have important consequences on the inferred dispersion curves. The following section about the spatial sampling discuss the effect of the parameters on the dispersion curve, and is useful to give indications for the design of the acquisition.

3.3.3 General limits and consequences of the spatial sampling

In the following the limits introduced by the acquisition are presented. These limits are discussed and demonstrated analysing the properties of data in f-k domain: this does not mean that the obtained limits only affect the f-k analysis. On the contrary, the f-k transform do not produce any loss of information, and data in f-k are completely equivalent to data in t-x domain, so these limits are intrinsic and don’t depend on the processing.

Independently from the chosen processing technique, the effects of the survey layout and acquisition parameters over the data are so strong that only an “apparent” dispersion curve can be obtained, mainly resulting from difficulties in separating the energy associated to different modes: as far as the modal superposition is concerned, the effects of the sampling are of crucial importance. This is the reason of the main differences between surface wave methods for seismological and engineering purposes.

The sampling parameters can be critical for different aspects and can strongly affect the result of the acquisition. The main aspects are related to the spatial sampling of surface waves: the receiver spacing and the total spatial length of the record. The windowing and the interpolation of the zero padding influence the results in terms of resolution, accuracy, maximum and minimum investigation depth both in multi-station and in two receivers layout. This influence is evident even when the first mode is dominant, but has heavy consequences in stratigraphic situations where higher modes of propagation relevant (Foti et al., 2000).

• the limit on the maximum k may be critical with wide bandwidth dispersion curves, that can be aliased;

• the resolution in k affects the precision of the phase velocity determination and the possibility of discriminating different modes;

• the k interpolation produced by zero padding increases the accuracy of maxima positioning while the capability of peak separation depends only on the actual array length.

• windowing introduces in the k spectrum ripples that can mask the presence of secondary modes.



These sampling issues can lead to a quite irregular and misleading behaviour of the experimental dispersion curve if they are not correctly assessed in the design of the testing array.

Moreover, this limitations have to be taken into account also in the inversion. Only the modal curves should be considered a characteristic of the site: the experimental dispersion curve is strongly affected by mode superposition effects and depends also on the acquisition parameters, it can then be considered as an apparent curve. Since the dispersion characteristics are not an intrinsic property, to correctly compare synthetic and experimental data within an inversion algorithm, it is necessary to use a modelling that reproduces closely the testing procedure, taking in account the acquisition parameters (chapter 2).

To evaluate the ideal total information carried by surface waves one should analyse modal curves in a certain frequency range: Rayleigh waves can exist only for the velocities of modal curves.

Time sampling (in t) is a minor concern since the sampling frequency with all instruments can be set high enough for surface waves usual frequencies and window length can be easily increased (particularly when using vibrating sources the actual signal length can be increased too).

On the other hand spatial sampling (in x) can pose some problems mainly due to: the wavenumber resolution, the maximum achievable wavenumber and the effects of spatial windowing.

3.3.3.1 Wavenumber resolution When simulating the usual test conditions more than one eigenvalue are obtained; this means that more than one mode exists. Whenever the fundamental mode is not dominant, modal superposition plays an important role in experimental data (Foti et al. 2000).

Theoretically if we take a section at constant frequency of the f-k spectrum the energy related to the different modes should be ideally located exactly in correspondence of the wavenumbers k1, k2,… kn, associated to the modes 1,2,…,n. Actually the resolution in k space, derived from the spatial windowing of acquisition (geophones layout length), allows only for smeared picks features (Figure 3.32). This windowing effect can prevent from an effective discrimination of modal curves.

Figure 3.32. Global effect of sampling on f-k spectrum. An ideal section for a fixed frequency (left) and the real

situation (right).

k

A

k1 k2 k3 k

A

k1 k2 k3



Figure 3.33: Spectra calculated with 3 array length (24, 48 , 96 m). Two picks are present but they both are retrieved only by large arrays. (blue: 24m, magenta: 48m, red: 96m).

Two aspects have to be discussed separately: the actual wavenumber resolution depends on the array length, which conditions the shape of the spectrum of the window, and the discretization of the spectrum which depends on the numerical impossibility of a infinite zero padding. It can be demonstrated that the f-k transform gives a correct position of maxima only when it is computed with an infinite zero padding: all finite number of samples give a discretization of the spectrum.

The interpolation does not change the actual modal separation capability of the record, while great importance has the macroscopic effect of the actual array length, as shown in figure 3.34. Two spectra have been computed for different array lengths for a stratigraphy in which higher modes play an important role. The obtained synthetic dispersion curves show that modal superposition leads to a dispersion curve not corresponding to any real modal curve, because of insufficient resolution. The absolute maximum of the spectrum is due to the superposition of the contributions of two modes, and hence the resulting dispersion curve shows a smooth passage to higher modes with increasing frequency (Figure 3.34, right). When the array length is sufficient to separate modes, the maximum follows with jumps the mode with the highest energy at each frequency (Figure 3.34, left)

Figure 3.34 : Two dispersion curves are computed for the same stratigraphy using different array length: 48m (left), 12m (right), with the same zero padding (2048 traces).

0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

1

2

3

4

5

6

7

x 10-6

wavenumber [rad/m]



It is hence easily shown that, consequently, the dispersion curve, computed by picking maxima of the f-k spectrum, also depends on sampling parameters and is not only a characteristic of the system itself: it is an “apparent dispersion curve”. This misleading identification of the true dispersion characteristics is due to the finiteness of the spatial window. This problem is well established in the literature concerning 2-station SASW test (Sanchez-Salinero, 1987; Gucunski and Woods, 1992, Tokimatsu, 1995).

Indeed in that configuration, a single estimate of the phase velocity is obtained and it is affected by mode superposition if higher modes are relevant in the propagation phenomenon. As a matter of fact the limitations imposed by the 2-station technique can be interpreted from the above considerations regarding multi-station methods. Indeed it can be shown that the processing adopted for the 2-station method, based on the phase difference between the two signals, is formally equivalent to an f-k analysis with infinite zero padding .

The real spectral resolution and the mode separation strongly change in the different frequency band of the 2 receiver dispersion curve: the distance is reduced increasing the frequency, and so the modal superposition occurs specially at high frequency.

Long arrays seem to be suitable, since increasing the spatial window, i.e. the array length, simplifies the mode identification. On the other hand with a short array there are less problems with lateral variations, S/N ratio, high frequency attenuation and, being fixed the number of channels, with spatial aliasing.

3.3.3.2 Maximum wavenumber and spatial aliasing A wide range dispersion curve is needed for an accurate inversion, and the maximum wavenumber that can be identified with a real array depends on the receiver spacing..

The 2D Fourier transform of a real 2D signal is symmetric with A(f,k)=A(-f,-k). The maximum k is, following Nyquist sampling theorem, kNyq=2π (0.5/ ∆x). All the energy associated to k*>kNyq will be aliased in k*-2kNyq. (Figure 3.35)

Figure 3.35. Scheme of the f-k spectrum

In end off gathers, all the coherent energy travels in the positive direction and is associated to positive wavenumbers. In the negative quadrant only noise and aliased events are present, so it is possible to recover the aliased information laying between –kNyq and 0 with a 2kNyq horizontal unwrapping without introducing additional noise (Figure 3.37).



The spatial aliasing produce indeed apparent negative velocities, that can be identified also in raw data (Figure 3.36). In the example below (Figure 3.36), a monochromatic signal is produced by a source in x = 0, but the low velocity produce aliasing and an apparent negative velocity

Figure 3.36. The spatial aliasing produce an apparent negative velocity

In the computation of the f-k spectrum, it is then possible to shift the negative quadrant and increase the maximum wavenumber (Figure 3.37). This computational aspect shows that is possible to go beyond the pure spectral limitations

Figure 3.37. The f-k spectrum computed from real traces (3m spacing, 24 geophones, weight drop). Spatial aliasing

recovering with a 2kNyq translation of negative quadrant.



3.3.3.3 Zero padding and windowing The ideas expressed above have been discussed in f-k domain but are valid in general: what has been said about the acquired data does not depends on the processing that is adopted. In chapter 4 it will be shown that in general it is possible even to separate superimposed modes, with the spectral deconvolution.

The array length influence the mode separation and the spacing influence the maximum wavenumber, that is the minimum wavelength. The other aspect discussed in the following are more related to the processing. The “interpolated” resolution due to the zero padding and the effects of the windowing are concerned when the phase velocity is estimated by a search in the wavenumber space.

This aspect are discussed here to avoid possible confusion between the actual wavenumber resolution depending on the array length, and the discretization of the spectrum depending on the number of samples in which it is computed. The computation of the spectrum needs a zero padding: it can be shown that the maxima of the f-k spectrum are in the exact position of modes only when the spectrum is evaluated in an infinite number of points within the range, that is only if an infinite zero padding is performed. Every numerical computation with a finite zero padding gives a spectrum which is just a discretization of the theoretically continuous spectrum.

Using N receivers the spectral distribution is obtained at N values of k between –kNyq and +kNyq for each frequency. This is only a computational aspect, but it is very important when the velocities are estimated searching in the wavenumber space.

In standard surveys, the relatively small number of geophones leads very often to a poorly defined spectrum in the wavenumber domain (Figure 3.38). Such a poor definition prevents from the accurate localisation of peaks, which is required to get the experimental curve. To increase the details of the wavenumber spectrum a zero padding procedure in space domain can be applied (Figure 3.38). The resolution, with P zero padding, that is adding P-N fictitious null traces, is then ∆k*=1/(P∆x). It is important to remark that the number of modes that can be sorted out depends only on the actual resolution in k, that is defined by the actual length of the geophone array.

0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

2.5

3

3.5

4

x 10-6

wavenumber [rad/m] Figure 3.38. Section of the f-k spectrum at a fixed frequency (red). The effects of zero padding consists of a better

maximum positioning (blue).

Figure 3.39 shows the dispersion curve obtained using the same synthetic traces and different options for zero padding to compare the accuracy of maxima positioning. Note that when no zero padding is applied it is not possible to obtain a reasonable estimate of the dispersion curve to be used in an inversion process. This is due to the lack of accuracy in the detection of maxima



location: for a certain frequency range the same location of the maximum is found, hence the estimated velocity (ω/k|max) is a linear function of frequency in that range.

Figure 3.39. Synthetic dispersion curves obtained with different zero padding applied to the same traces: 2048 traces

(continuous blue line), 512 traces (green dots), 24 traces, that is no padding, (red asterisks).

The same considerations can be made about field data (Figure 3.40): The following figure shows the f-k spectra of the same real seismogram, computed without zero-padding(left) and with a 2048 zero-padding

Figure 3.40: On real data the effects of zero padding in space to increase the interpolated resolution in wavenumber

3.3.3.4 Spatial Windowing The use of a finite length geophone layout implicitly involves a windowing of data in space domain. The resulting spectrum can be found by convolution of the real spectrum, ideally having only spikes corresponding to modes, with the window spectrum. The latter has a main lobe and often ripples



and secondary lobes. Spatial windowing introduces a fictitious spreading of energy in the fk spectrum due to the main lobe of the window spectrum and also the secondary lobes often prevent higher mode to be identified, hiding the energy maxima associated to them (Linville and Laster, 1966).

This effect can be reduced using an appropriate windowing in the space domain, for example a Hamming window. In Figure 3.41 the spectrum on the left is computed using a box window, the one on the right using a Hamming window. A generalized Gaussian window, without ripples, help in discriminating secondary maxima due to higher modes and those due to ripples.

Data are acquired obviously without any special windowing, this aspect refers more to processing: the details of windowing will be discussed in chapter 4.

Figure 3.41. The f-k spectra computed from synthetic seismograms. Using a box window (left) evident ripples are

produced. Using a Hamming window (right) the ripples have lower amplitude, but the main lobe is larger.

Charter 3.4 DESIGN OF THE ACQUISITION PARAMETERS


3.4 DESIGN OF THE ACQUISITION PARAMETERS

In this section practical indications on the acquisition parameters for a multi-station active test are obtained: the receiver spacing, the array length, the source offset are discussed. Some field techniques to face the limitations of a finite number of receivers are also presented.

As it has been mentioned above, many techniques have been proposed for the acquisition of surface wave data. The parameters that characterise the acquisition are: the number and position of the receivers; the array length and the spacing; the offset of the source; the kind of receivers and their response; the kind of source and the signal that can be produced; the parameters of the time sampling. In the following the different parameters are discussed, and the equipment that can be used.

3.4.1 Acquisition Layout

The acquisition parameters have to be designed depending on the aim of the test, i.e. the desired investigation depth and the resolution for shallow layers. The parameters that have to be designed are the array length and the receiver spacing, the use of a non uniformly spaced array, the source-receiver distance and the time sampling parameters (Figure 3.44).

Figure 3.44. Scheme of the receivers layout.

3.4.1.1 Receiver Spacing The properties of the first layer are obtained from the velocity associated to the shorter wavelength, so the smallest recorded wavelength affect the identification of the first layer. This minimal wavelength is usually associated to the highest recorded frequency, and the maximum wavenumber limit can be critical to record high frequencies, particularly at low velocity sites.

If the receiver spacing is ∆X, we get:

XXK nyq ∆

=∆

=ππ2

21 , that becomes

XK MAX ∆

=π2 if we recover the negative quadrant of the spectrum (see 3.4.3.2); so the minimum

wavelength is X∆=minλ



Nevertheless the real limit is often due to the attenuation of the high frequencies.

3.4.1.2 Array length The actual array length affects the mode separation possibility, and the resolution of the spectrum. The dependence of the maximum wavelength on the array length is not so pressing as it could seem. In general, from spectral analysis limits, the maximum wavelength should be equal to the array length, if we have record in space at a fixed time: actually this limit can be overcome and greater wavelengths can be analysed. Considering two receivers only, if the signal is monochromatic (or it has been decomposed into its monochromatic components) the phase velocity can be computed from the time shift between the two traces (Figure 3.45)

Figure 3:45. With two traces, given a distance, the velocity is computed from the time shift, and hence no limits on the

maximum wavelength are imposed.

The time shift can be computed for instance by means of the cross correlation between the two traces, and the smaller identifiable time shift is equal to one time sample. For a fixed frequency

*f , if the spacing between the two traces is X∆ and the sampling rate is t∆ , the maximum velocity is

tXVMAX ∆

∆=

and the maximum wavelength is

tfXfVMAXMAX ∆⋅

∆== *

*/λ

There is no upper limit on the wavelength imposed by the array length, and it is worthwhile to mention that the maximum wavelength depends more on the site conditions, and the frequencies that can propagate, given also a source spectrum.

These considerations don’t want to suggest in general the use of short arrays: on the contrary, as it will be shown in chapter 4, short arrays produce higher uncertainties in the estimate. Intuitively, the greater is the array length, the greater is the phase difference given a frequency and its phase velocity: the estimate will be less sensitive to the errors on the phases. However, for a given receiver layout, it is possible to perform the analysis considering also wavelength greater than the array length without violating any sampling theorem: this would not be possible trying to guess the wavelength from the analysis of the velocities at different offsets, simultaneously in a single time sample.



As an example a real data set (Figure 3.46) and its processing are shown. By processing both the whole 24 traces seismogram (46 m long) and a shorter subset of 6 traces (10 m long), one gets similar spectra: the search of the maxima of the spectra gives the two dispersion curve depicted in figure 3.46 (right), which reaches similar wavelengths. In particular, with the short array, a wavelength more than 4 times greater that the array length can be analysed. The uncertainties of the wavenumber, however, are 20 times greater (see chapter 4)

Figure 3.46. On the left the whole seismogram and (grey coloured) a subset of six traces; on the right the

corresponding dispersion curves in wavelength-phase velocity (solid line: whole seismogram. Dots: subset of 6 traces)

3.4.1.3 Source off-set The distance between the source and the first receiver is used in the modelling, according to the hypothesis of plane Rayleigh waves, only for the geometrical spreading correction, and does not change the dispersion characteristics: however ‘near field’ effects are present and should be either taken into account or removed from experimental data.

At short distance from the source the hypothesis of plane waves is no more valid, and records are affected by effects that are not taken into account in a conventional modelling. This distance depends on the wavelength, and empirical rules are suggested by Stokoe et al. (1994).

Actually, using a great source offset, it will be difficult to record the high frequencies, that will be greatly attenuated over a certain distance, depending on the site (Figure 3.47).

It is preferable to use a small offset, and eventually, to filter the low frequencies in the near offset. It is possible to separately analyse the different frequencies, in order to see if the phase velocity shows significant variations with the distance from the source. In that case, the signals affected by the near field can be filtered and discarded.



Figure 3.47. Frequency attenuation, in dB, with distance from the source. To record the high frequencies short

distances from the source have to be used

The example below shows two data sets acquired at the same site, with the same receiver layout, changing only the shot position. The source offset equals the receiver spacing, that is 2m, in the first case (figure 3.48, left), and is 12 m in the second case (Figure 3.48, right). In this way half of the seismogram can be overlaid, to correct eventual shifts.

A series of differences are present between the two acquisitions. The shorter source offset introduce in the seismograms, and also in the f-k spectra, more high frequencies.

Part of this high frequency energy is carried by body waves, and actually introduces noise in the spectrum: but it can also be noticed that first mode can be followed up to higher frequencies (Figure 3.49, left). The two dispersion curves are almost identical in the whole range, apart a greater noise in the high frequencies for the acquisition with the shorter source offset.

Figure 3.48. At the same site, with the same receiver layout, two shot positions are acquired.

The left seismogram has a 1m source offset, the right a 12 m source offset.



Figure 3.49. The two f-k spectra are depicted (normalized amplitudes). The left correspond to 1 m source offset, and has higher frequencies.

3.4.2 Strategies to face the limitations

The survey layout for surface wave testing must be designed adequately to account for the above limitations imposed by spatial sampling and its implication on spectral estimates. The following considerations can be helpful to this regard. A small geophone spacing avoids spatial aliasing in low velocity sites and allows an adequate detection of high frequency surface waves, which are strongly attenuated at great distance. Long receiver arrays are needed to obtain the necessary wavenumber resolution and to sample longer wavelengths more accurately.

The survey layout has moreover to consider the theoretical 1D model used to interpret data: the longer are the layouts, the higher will be in general the probability of lateral variations.

The limited number of receivers does not allow a long array with small spacing and this trade-off must be taken into account. The typical solution is to perform different acquisitions in the same site with different receiver spacing. Nevertheless it implies an increase in field testing time. A short array with little spacing samples the high wavenumber and the high frequencies, which furthermore have a greater attenuation. A long array gives spectral resolution to better separate modes at low frequency.

Different strategies allows to overcome the limitations due to the number of receivers. The use of different acquisitions with different spacings joins the resolution of long arrays with the high frequencies and short wavenumber of short arrays. The source or the receivers can be moved to increase the number of traces (Gabriels et al., 1987): of course moving the source is faster but introduce an artefact that can be dangerous if lateral variations are present. The use of non-even spaced arrays reduces the field work with a good compromise.



Figure 3.50 Different strategies for the acquisition: a) use of different spacings b) moving the source c) moving the

receivers d) non even spaced array

3.4.2.1 More acquisition with different spacings The first way to overcome the limitations due to the finite number of receivers, and so the trade-off between resolution and high-frequencies is the use of several layouts. For instance with 24 geophones one layout with spacing 1m and one layout with spacing 3 m can be used.

3.4.2.2 Moving the source Moving the source is a fast way to acquire seismograms where traces are placed at different distances from the source.

An example is given in the following: the two records of figure 3.48 have been reattached to give the seismogram depicted in figure 3.51. The partial superposition allow to verify the absence of time shifts.

Figure 3.51. The two seismograms of figure 3.48 are re-attached. The second one is shifted 12m, and a 36 trace

seismogram is obtained. The traces from 13 to 24 are present in the two records



3.4.2.3 Non even-spaced arrays An interesting alternative is given by the use of non-uniform receivers spacing. For instance the testing layout adopted in Figure 3.52 allows the selection of three subset of traces from the same shot gather of 24 geophones. Each subset of 12 traces can be processed independently using a standard f-k methods with a suitable zero padding and the results can be eventually combined.

Figure 3.52. Example of nonuniform array. From this 24 channel record 3 set of uniformly spaced traces are extracted.

Figure 3.53 . Three sets of traces can be extracted (left), and the corresponding f-k spectra are computed(right)

The processing of non uniformly spaced array can be performed even without extracting subset of uniformly spaced traces, with the algorithm illustrated in chapter 4.



3.4.3 Time sampling

Usually the sampling in time of traces is not critical. The sampling rate can be designed simply considering the sampling theorem and considering the desired highest frequency. Frequencies greater than 100 Hz are rarely recorded, even if higher frequencies can be useful when a pavement is present. Usually a sampling rate of 0.5ms or 1 ms is sufficient; spectra can be respectively computed up to 1000 or 500 Hz.

The time window length has to be long enough to record the whole surface wave on all traces: with long array at low velocity sites few seconds can be needed ( T = L / Vmin ). If seismograms are truncated in time because of a too short window, a portion of the low velocity energy is lost, and velocities can be over estimated, or anyway the associated energy is reduced.

Since the spectral resolution in frequency depends on the window length, a long window with a pre-trig can be used to avoid zero padding: it has to be mentioned that, in this way, additional noise is very likely introduced in data, it is therefore preferable to perform a numerical zero padding. In fact, an impulsive source produces a time limited signal, lasting TS, with a certain S/N, while noise acts during the whole duration of the recording window TW. Hence the signal to noise ratio in frequency domain is reduced by a factor Ts/Tw: this does not happen with a numerical zero padding.

The total length of the signal cannot be controlled with an impulsive source, and on the contrary is very important for the S/N of vibrating sources.

Computing the Short Time Fourier Transform (STFT) of a single trace, recorded with 2 seconds of pre-trig-window (Figure 3.54), it is evident that in some frequency bands, the noise level can be quite high compared with the signal (Figure 3.55), then, in these frequency bands, the noise in the pre-trig window can deteriorate the coherence of the signal.

Figure 3.54. A single trace with 1.9s of pre-trig window. Before 1.9 s only noise is present, and this pre-trig seems

quite silent: the overall signal quality is quite good.



Figure 3.55. Time frequency analysis of the trace of figure 3.54. The three figures depict the

energy density as a function of the time for three frequencies. It can be seen that at low frequency the signal level above the noise is not very high.

NOISE SIGNAL NOISE

Charter 3.5 EQUIPMENT AND TECHNOLOGY


3.5 EQUIPMENT AND TECHNOLOGY

In this section the characteristics of the equipment are discussed, referring to a multi-station active test for engineering purposes. The receivers and the sources are considered for their effect on the acquired data quality.

After the acquisition parameters have been designed, the test is performed. The site should be flat, free of obstacles, and the geology one-dimensional. In active measurements the surface wave is generated with an active source, the vibrations are detected by sensors and acquired with a digital recorder (Figure 3.56). The technical characteristics of the three main elements of the recording equipment affect the quality of data and the range in which the dispersion curve can be evaluated.

Figure 3.56. The equipment used for active tests.

3.5.1 The source

The source can be a standard seismic source, even if the requirements of a source for surface waves test are slightly different. In reflection and refraction surveys the high frequency content of the propagating signal affect directly the resolution at all depths, and so the source is required to produce high frequencies.

In SWM, the source has to produce surface waves with a high signal to noise ratio in a wide frequency band: this task is often more difficult in the low frequency part of the interesting band, which is needed to achieve a greater investigation depth.

There is a great deal of sources that can be used for SWM, depending on the investigation depth and on the target, on the environments and on the tested materials.

A signal at a certain distance x from the source, can be written as:

∑−

⋅⋅=m

xf

xefSitefSourcefxs

)(

)()(),(α

The spectrum of the source is so linearly related to the frequency content of the signal: however the effect of the site can be dominant, since the site response function can be much higher in some frequency bands. At certain sites, the site response strongly cut certain frequency bands, the source spectrum slightly help in increasing the signal quality in that bands. The sites often have a strong and bandpass filtering signature

The synthetic example below shows the site response in a case where a strong contrast exist: 4 meters of sediments (2 layers, each 2m thick, with 100 m/s and 120 m/s of shear-wave velocity) on



a bedrock (450 m/s of shear-wave velocity). The low frequencies have very little energy, and the propagation is almost confined in the upper layers that acts as a waveguide. The figure 3.57 (left) shows the site response in dB: it can be noticed that at 8 Hz the response is 100 times lower than at 15 Hz. The f-k spectrum shown on the right evidences the lack of energy in the low frequencies

Figure 3.57. Site response( left) for a site with a high contrast of velocity (100m/s 120 m/s, 450 m/s; 2m, 2m). The synthetic f-k spectrum(right) shows the lack of energy in the low frequencies.

The field data example below (Figure 3.58) shows the overall spectrum of two acquisitions at different sites with the same source (sledgehammer on plate, 20 stacks to average the differences)

Figure 3.58. Spectrum of the signal produced by the same source at two different sites (normalized amplitudes)

Nevertheless the use of an adequate source is necessary, in combination with the design of the array length.

It is important to mention that the low frequency content of the signal itself is not sufficient to have a great penetration depth: for instance, we can consider the two cases depicted in the figure 3.58.

In the case where low frequency are present, the velocities are low and the wavelength is relatively small (20m); in the other case, to the greater velocities correspond grater wavelengths (80m).



In general the signal to noise ratio in SWM acquisition is less critical than in other seismic methods, since the energy associated to Rayleigh waves is greater. It is possible to gather good quality data even at very noisy sites as urban areas or industrial plants (Xia et al., 2002).

3.5.2 Impulsive sources

An impact sources like a sledgehammer is easy to use, cheap and in many geological conditions and noise level it has sufficient energy for a 50-100 m array. It is easy to be triggered with an inertia switch on the hammer, and the reliability of this system allows enhancement by in situ stacking: as it has been shown, the signal-to-noise ratio increase with stacking with the square root of the number of shots.

The weight of the hammer conditions the frequency content of the generated pulse, and for high frequencies a light hammer is preferable. The use of a metallic plate increase the frequency of the signal. When striking on a pavement, very high frequency can be produced (more than 1000 Hz)

If more energy is needed, a weight drop can be used. The use of a metallic baseplate usually increase the frequency content of the source. The triggering of these systems can be obtained by circuit closing, when a metallic plate is used, with inertia o piezoelectric switch, or with a starter geophone. A single shot with a 130 kg A single shot with a 130 kg weight dropping from 3m has more energy than 10 sledgehammer shots. The frequency content of the signal is not so different from the sledgehammer. The two sources were compared in a site with the same array: 10 stacks of sledgehammer and a single shot of drop weight. The frequency content is very similar (figure 4.59).

Figure 3.59. Spectra of the signals recorded at the same site with 8 kg sledgehammer, 10 stacks (solid blue line) and

drop weigth 130 kg 3m (red dashed line)

Other impulsive sources use bullets propelled by small charges, shot in a metal pipe inserted in a small hole in the ground (minibang). The energy is greater than the one of a sledgehammer and the frequency content is similar.

Explosives located in holes can produce enough energy in one shot: there are often problems for authorizations and the need of licensed explosive-handlers. An other important aspect about the use of explosive in holes is the depth of the source, that has to be considered in shallow surveying.

The following example shows the seismograms and the spectra acquired at the same site, with the same receiver array, using two different sources. Figure 3.60 (left) shows the seismogram acquired with a sledgehammer, figure 3.60 (right) the seismogram acquired with 100g of explosive in hole (1m deep). The spectra are depicted in figure 3.61



Figure 3.60. Seismograms with sledgehammer(left) and explosive in hole (right)

Figure 3. 61: Spectra of the two signals (black explosive, grey sledgehammer)

3.5.2.1 Effects of the source depth The source depth influences the energy distribution within frequency bands and within different modes. A buried source produces a greater content in low frequencies.

The example below shows the effects of the source depth on the energy distribution, by the comparison of the synthetic seismograms and the spectra that are simulated for two different positions of the source. The source is acting vertically, with a flat band in the range between 5Hz and 100Hz, the receiver layout is 24 traces 1 meter spaced. The stratigraphy is a two layers model, with 2m at 200 m/s of shear wave velocity over a half space at 500 m/s of shear wave velocity.

The source is placed at the surface (seismogram in Figure 3.62 left, modal curves and apparent curve Figure 3.63left), and at a depth of 1m (seismogram in Figure 3.62 right, modal curves and apparent curve in Figure 3.63 right).



Figure 3.62. Synthetic seismograms Source at the free surface(left) and at 1m depth(right).

Same model (200m/s,500m/s; 2m), same array layout, identical frequency content of the source(flat band in 5-100 Hz): only Rayleigh waves are simulated

The modal curve are of course the same, being a characteristics of the site stratigraphy: if the positions of the maxima of energy are plotted in frequency-phase velocity domain together with the modal curves, the different energy distribution can be noticed (Figure 3.63). In the highest frequency band with the buried source, the apparent phase velocity corresponding to the maximum of energy follow the second mode (Figure 3.63 right), while with the source at the surface it follows the first mode (Figure 3.63 left).

This little difference can be very important, because the apparent curve often is the only characteristics that can be extracted from experimental data, and so the influence of the source position should be taken into account.

Figure 3.63.The modal curves and the apparent curves, with source at the free surface and source 1m deep

Also the overall energy distribution as a function of the frequency is affected by the position of the source: the amplitude spectra at 10 m from the source for the two source depths are depicted in figure 3.64right, the source spectrum which is flat below 80 Hz is also shown (Figure 3.64, left).



Figure3.64. The source spectrum(left) and the trace spectrum at 10 m (right),

with source at the free surface and source at 1 m depth

3.5.3 Vibrating sources

Instead of producing a wide spectrum with a short impulse, a vibrating plate can be used. Vibrating source are used for seismic reflection with sweep signals and correlation analysis: for SWM acquisition the main goal is producing different frequencies with a sufficient energy. Sweep signals or monochromatic frequencies can be produced and recorded.

The recorded amplitude at each frequency depends on the source and on the site response, if we operate in the flat band of the receivers. The spectrum of the source is a characteristic that has to be considered.

3.5.3.1 Sweep A sweep signal is a non stationary function that can have the general form

( ) ( )( )ϕπ += tftAA 2sin

A linear sweep has a linear increase of the frequency versus time, that is

( ) tctf ⋅=

With such a frequency function, the different frequencies are differently sampled, i.e. a different number of periods is recorded for each of them. With a discrete sweep, if a constant number of period is recorded for each frequency, the interval related to each frequency is the inverse of the frequency, and so a logarithmic series of intervals is used (Figure 3.65).



Figure 3.65. A discrete sweep in time frequency domain: two period for each frequency are recorded

To better sample the low frequencies, a logarithmic or quadratic sweep are preferable.

A logarithmic sweep has the following function of frequency with time

( ) tbftf ⋅+= 100

a quadratic sweep has the following function of frequency with time

( ) 20 tbftf ⋅+=

A sweep can be used to record a wide frequency range in a fast way: the amplitude at the different frequencies can be quite different, depending on the site response and on many factors.

It is interesting to notice that when geophones are used, the vibration velocity is recorded: the response curve of receivers, and so the flat band correspond to the velocity: the flat band of the source often refers to force, and hence acceleration.

When recorded signals are depicted in time domain, even at small distance from the source, and even in the theoretically flat band of the source, very different amplitudes are observed: a linear sweep of force signal at constant amplitude produces a velocity signal which has not a constant amplitude, but an amplitude which decreases with the frequency. To have a constant velocity a linear increase of the acceleration amplitude has to be used (Figure 3.66)

Figure 3.66. Synthetic signals: linear sweep in acceleration (left) and corresponding velocity(right).

A constant amplitude velocity needs a linear increase of amplitude in acceleration



Figure 3.67. A recorded signal with a sweep. The source spectrum, the site and

the receivers are responsible for the amplitudes.

3.5.3.2 SNR in sweep signals In sweep signals the signal to noise ratio is then very different in the different frequency band: both the signal and the noise have different amplitudes in the different frequency bands . If the analysis is performed on the whole record, in certain ranges the S/N can be very low with such non stationary signals

In figure 3.68 the time frequency analysis of a single trace is depicted. The signal is a linear sweep, that appears as a narrow band of energy whose frequency increases with the time. It can be easily noticed that around 10Hz, almost in the whole record, the noise level is higher. At each frequency the signal duration is limited, while the noise acts during the whole recorded interval. The signal to noise ratio is so diminished of a factor depending on the duration of the noise and of the signal.

This effect is not dramatic in the frequency ranges where the signal level is high (for instance 30Hz), but can corrupt the information in the low signal energy bands (for instance 10Hz).

Figure 3.68. Time frequency analysis of a single trace, with a linear sweep of 15s between 5 and 40 Hz

Sweep Signal

Noise



3.5.3.3 Monochromatic signals A valid alternative approach is the use of a set of monochromatic signals, each one of constant amplitude in the whole window (Figure 3.69).

In the spectrum only certain frequencies have to be considered, and the dispersion curve is discretised at the used frequencies.

Figure 3.69. Raw data with monochromatic source (30Hz)(left), and narrow band-pass filtered (right).

Note the jump on the tenth trace, due to the presence of two modes.

Different monochromatic signals can be stacked on the same file: in time domain the signal will appear blurred, as in the stacking procedure the phase is random, but in frequency domain the energy will emerge clearly (Figure 3.70). This procedure, nevertheless, deteriorates the signal to noise ratio of individual records. The signal at a specific frequency is present in one file only, while the noise at that frequency affects the whole set of stacked files.

Figure 3.70. Spectrum of a trace made up stacking 5 monochromatic signals (20, 25, 30, 35, 40 Hz).

The amplitude refers to the velocity of vibration from a geophone.

The main advantage on the sweep is that the time duration of the signal and the noise is the same, and so the signal energy at each frequency is not diluted in a larger window. In the frequency band where the data quality is critical this advantage may be very important.



With a given stationary instantaneous SNR, the SNR of the recorded signal increases with the record length: if the noise sources are random, the length of the signal act as a sort of stacking. Hence long record have a better signal quality.

A signal enhancing technique that can be used to increase the signal to noise ratio is an internal stacking applied to each file. A single trace is divided into portions of the same length corresponding to an integer number of periods of the frequency that want to be enhanced: this guarantees the synchronization in the sum of the different portions. In the example below (figure 3.71), with a 4 Hz signal, even if the initial phase is not known, the partitioning into 0.5 s windows produces 8 synchronised signals.

Figure 3.71. A coherent stationary signal is divided in sub-window containing a fixed number of periods.

The other frequencies, unless the multiples, are not enhanced: neither the noise is enhanced, if it is not stationary (Figure 3.72). Moreover this allow the selection of the portions of the time windows where there is less noise: this selection of the sub-windows to be stacked allow to discard impulsive noise events.

The figure shows a single interval of the original seismogram (Figure 3.72, left), and the result of the stacking (Figure 3.72, right).

Figure 3.72. A monochromatic signal is split into sub-windows which are stacked, after removal of noisy sub-

windows.. The signal quality depends on the time duration of the signal



3.5.4 Sensors

3.5.4.1 Response of the receivers Usually the transducers used to detect the induced vibrations are geophones, electrodynamic velocity transducers.

Most of geophones are of the moving coil type, with a small coil suspended by a spring in a magnetic field produced by a permanent magnet fastened to the casing. The vibration of the soil cause a displacement of the geophone case and magnet: the relative movement of the coil, due to its inertia, produce a small voltage proportional to the relative velocity. The maximum of the response is when the coil is parallel to the vibration direction: vertical and horizontal geophones exists.

A geophone is an oscillating system characterized by its mass, spring and damping. The theoretical complex response is

( ) ωωωωω

nn ih222

2

+− being nω the natural angular frequency and h the fraction of the critical

damping. From these expressions the theoretical amplitude and phase can be plot (Figure 3.73).

Figure 3.73. Theoretical response amplitude and phase response for different damping factor

The damping strongly influences the curve, and it is recommended to set it such as the response is as much as possible flat in the frequency band of interest. The damping of a geophone can easily be modified by inserting a shunt resistor across the coil terminals: the current circulating in the coil, depending on the shunt resistance, produces a magnetic field opposing to the coil movement and damps it. Hence the shunt resistor is chosen such that the damping factor is about equal to 0.6, to flatten the amplitude above the natural frequency within a large band. The loss of signal output that results is easily made up by additional gain of the amplifier.

The real response curve that can be measured on a shaking table is different, and measures also all the aspects that make a real geophone different from an ideal forced damped oscillator. If we look at an experimentally measured response curve (Figure 3.74), at higher frequencies, the geophone



records can be contaminated with spurious resonance noise (due to undamped motions of the geophone’s coil normal to the principle axis or rotation or noise in the spring).

Figure 3.74. Spurious frequencies for a 100Hz geophones tested on a shaker (thanks to L.Sambuelli et al)

The receiver response is important because it influences directly the recorded signal srec. In frequency domain the recorded signal is the product of the complex spectrum of the true signal Strue by the receiver complex spectrum R.

( ) ( ) ( )fRfSfS truerec ⋅=

that is, considering the amplitude A and the phase ϕ ,

( ) Rtruerec AAfA ⋅=

( ) Rtruerec f ϕϕϕ +=

The spectra of the different receivers affect hence for both the amplitude and the phase of the recorded signals. The amplitude are attenuated following the receiver response curve, but this does not influences the inferred phase velocity, which is estimated from the phases. Geophones can be used also out of their flat band, and below their natural frequency, getting attenuated signals.

The phase of the receiver does not have a strong effect if the receivers are all identical. Otherwise it introduces an error in the recorded phases differences, and an error in the estimated phase velocity.

If we examine two monochromatic traces, if two receivers are used, the real phase lag is modified of a quantity equal to the difference of the phases of the receivers at that frequency

( )21:2,1:2,1 RRtruerec ϕϕϕϕ −−∆=∆

The difference of the phases of the receivers, if they are the same receiver type, is zero if the theoretical curves (Figure 3.73) are considered: when the receiver are tested, it can be seen that the phase difference can be also important. The following figure 3.75 shows the phase difference as a function of the frequency for two geophones tested on a shaking table (Experimental data: courtesy of Luigi Sambuelli).



Figure 3.75. Phase difference between two 100 Hz tested geophones. Note the increase of

phase lag close to spurious frequencies

The additional phase lag due to the receivers affect the phase velocity, which can be computed as

ϕπ

∆⋅∆⋅

=fXV 2 , where f is the frequency, ϕ∆ is the phase difference, ∆X the distance

and the uncertainty on V can be estimated as

ϕσϕ

πσ ∆⋅∆

⋅∆⋅= 2

2 fXV

It can be seen that the uncertainty can be high for little phase differences, and so increasing the spacing D is necessary with great wavelength: increasing ∆X increases also the true ϕ∆ , but it appears with power 2. (the true ϕ∆ indeed can be obtained as Xk ∆⋅=∆ϕ )

For example, velocity of 600 m/s at 150 Hz and 1m spacing would give 2πϕ = . If the error is 10°, i.e. 18π , the obtained velocity is 675 m/s. This effect is strongly reduced when a multi offset record is acquired, and the single phases are considered simultaneously (chapter 4).

3.5.4.2 Coupling The coupling between geophones and soil is important for signal quality: removing the uppermost layer to improve coupling is necessary for some applications. In SWM acquisitions the high energy of the signal allows a fast positioning of receivers.

On pavements and hard ground the use of steel plate allows to simply lay the geophones on the ground without deteriorating the signal quality too much. A series of test, comparing the signals from geophones planted whit spikes and laid on the ground with base plate, showed very similar signals



When other connecting systems are used, the risk of introducing spurious frequencies is high. A strong resonance is evident in data of figure 3.76, acquired with geophones coupled to the wall of a marble pillar with an L-shaped metallic support.

Figure 3.76 Experimental seismogram acquired with geophones fastened on L shaped steel support.

The resonance of the support introduces a ringing in every signals.

3.5.4.3 Use of two components receivers The use of two component receivers allows the detection of the two components of the particle motion due to the Rayleigh wave propagation. The acquisition of the two components of Rayleigh waves has been proposed for instance by Tokimatsu et al. (1992a).

The example below (Figure 3.77) shows the results with both a vibrating source and an impulsive source. In this example vertical velocity transducers of natural frequency of 11 Hz and horizontal velocity transducers of natural frequency of 14 Hz were used. The two geophones were mounted on a small and rigid steel support, in the view of coupling their movements.

With the vibrating source, the particle velocity is quite stationary, and the path is nice shaped. With impulsive sources, the path can be confused, also because of the time separation among the different events that arrives at the measuring point at different times.

Figure 3.77. Velocity composition with a vibrating source(left) and an impulsive source(sledgehammer)(right)



3.5.5 Digital recorder

The electric signal produced by geophones is recorded by a seismograph, which is a multichannel (12, 24, 48 or more) digital recorder.

The dynamic between Rayleigh waves at different frequencies and between different modes can be quite high. The dynamic range of the seismograph affects the quality of data, the detectable frequency range and the detectable modes. Digital Instantaneous Floating Point (DIFP) gain control increases the dynamic of the instruments, that can be up to 24 bits.

3.5.6 Description of the equipment used for the experimental work

The equipment that have been used for the experimental part of this work is described in the following.

Sledgehammer: 8 kg, piezoelectric switch.

Controlled sledgehammer (PCB Piezotronics). (Courtesy of prof. Signanini, Chieti University)

Weight Drop: 180 kg up to 3m, mounted on a car trolley

Vibroseis: The used vibrator is an electro-mechanical vibratory shaker (Model 400 Electro-Seis Shaker manufactured by APS Dynamics, Inc.), a long stroke electrodynamic shaker: its thrust axis may be oriented at any angle between vertical and horizontal with no degradation of performance.

The force envelope is depicted in figure 3.78 (the amplifier that has been used for the test is a model 144).

Figure 3.78. The frequency response of the Vibrator APS400

Geophones: Natural frequency of 4.5 Hz , damping ratio of 0.6

Digital recorder. The seismograph is a 24 channels ABEM model TERRALOC MK6.

The frequency band (flat within 3dB) that can be covered by the instrument is comprised between 2 and 4000 Hz.



Figure 3.79. Photographs


4 CHAPTER 4

Processing

This chapter describes the basic principles of the processing of surface wave data, presents and compares the different approaches and processing techniques, and finally gives the details of the ‘f-k’ processing and of the ‘Multi Offset Phase Inversion’. The basic common principles of the processing techniques are presented. Both the two receivers and multi-offset techniques are described to give an overview on different processing techniques. A comparison shows that many of them are equivalent, and is based on the sensitivity to different kind of noise and uncertainty of data. The main differences depend on the acquisition parameters. Then two different processing are presented in detail: first the multi offset phase inversion, which allows a robust estimation of the phase velocity and attenuation considering also the different uncertainties. Then the details of the f-k processing are discussed, and the algorithms are presented: this is the processing technique that have been finally selected.

In this chapter

Basic Principles of the processing of Surface Wave Data

An overview of the different existing approaches and techniques: the concepts and the main relations of 2 receiver and multi offset approaches are presented

The comparison between the different processing techniques: sensitivity to different kind of noise, mode separation possibility, coherency with a simple model for the inversion, logistic

The Multi Offset Phase Inversion: statistical evaluation of the uncertainty of the experimental phases, weighted estimate of the wavenumber, test of the model.

The details of the f-k processing: computation of the spectrum, muting, padding and different windowing, search for maxima, spectrum deconvolution for the separation of superimposed modes. The f-x filtering before the computation of the f-k spectrum.

Chapter 4.1 BASIC PRINCIPLES OF THE PROCESSING OF SWM DATA


4.1 BASIC PRINCIPLES OF THE PROCESSING OF SWM DATA

In the Surface Wave Methods, the processing step has to estimate the dispersive characteristics from the acquired raw data. From the records of the particle motion at different positions, the processing algorithms infer the dispersion curve, i.e. the phase velocity as a function of the frequency (Figure 4.1). When monochromatic data are analysed, a single phase velocity can be obtained: to get the whole dispersion curve, several monochromatic data set have to be acquired and then processed. On the contrary, impulsive transient data can be decomposed into their frequency components, and a single record can give information on a wide frequency band.

Figure 4.1. The processing estimates the dispersion curve from raw seismic data

The data that are processed are full-wave records: the phase velocity of Rayleigh waves at each frequency has hence to be estimated from data that contain also the effects of other seismic events.

Two operations are usually needed in the processing of full-wave form data: the selection of the events to analyse, and the identification of their properties (velocity and attenuation as a function of the frequency).

The usual assumption is that the main events, at each frequency, are related to the propagation of the Rayleigh waves, which are then supposed to be dominant. Under this hypothesis, the phase velocity at each frequency can be obtained measuring the distances and the travel times, the latter obtained for instance by searching the position of the energy maxima in data.

Some processing techniques transform the whole data set from the time-offset domain where it has been acquired into domains where the energy is mapped as a function of other parameters, such as frequency, slowness, wavenumber. Local maxima of energy density can hence be identified in the new domain, and they correspond to different seismic events that are present in the record. The coordinates of the energy density maxima, in the new domain, give the properties of the propagation of the corresponding event.

Some methods of processing allow a direct evaluation of the signal quality, in order to select the more reliable information. Some processing techniques allow a weighting of traces and frequency ranges according either to their energy or to a statistical evaluation of the signal quality. Sometimes this is implicit, i.e. the traces with higher energy have a greater weight in the overall result, sometimes, as in the phase inversion, each trace is evaluated, at each frequency, to compute the variance to be used as weight.

The choice of the processing algorithm affects in some way the result even if it depends mostly on the acquisition parameters. On synthetic data without any noise the result of all the processing techniques should be the same. Important differences can be in the sensitivity to random and coherent noise, in modal separation possibility (and consequently amount of extracted information), in the possibility of a simple procedure for a modelling coherent with the test.



4.2 DIFFERENT APPROACHES AND TECHNIQUES

This section is an overview of the different approaches and processing techniques that have been used for the processing of surface wave data. The different mathematical tools are briefly described, trying to stress the common aspects.

4.2.1 Overview

Surface waves have been used for the characterisation at very different scales: from the ultrasonic tests (Viktorov 1967) to the continental studies of the earth crust and upper mantle (Ewing et al, 1957; Dorman et al, 1960; Dorman and Ewing, 1962; Bullen, 1963; Knopoff 1972; Kovach 1978). Different processing techniques have been used to measure the phase and group velocity from data, and to infer the attenuation. Several parameters affecting the processing depend on the application that is considered: the required resolution, the amount and quality of data, the frequencies that have been acquired, the complexity of the model that has to be reconstructed and so on.

Multiple filter analysis, introduced by Dziewonski et al. (1969) has been used for the determination of the group velocity as a function of the frequency from a dispersed wavefront. Herrmann (1973) studied the use of multiple filter analysis to estimate the spectral amplitudes of various models. The use of the cross-power spectrum of two-station data (Dziewonski and Hales, 1972) has been adopted by many authors working with the SASW approach. Nolet and Panza (1976) presented the f-k wave field transform for an unambiguous investigation of higher modes, McMechan and Yedlin discussed the use of another wavefield transform, the ω-p obtained from a slant stack. The frequency time analysis (for instance Keilis-Borok et al., 1989), the correlation (Park et al., 1999), and other approaches, also for multicomponents data (Glangeaud et al., 1999) have been used for the processing of surface wave data.

In the following some of the techniques that can be used for the processing of surface wave data for engineering purposes are outlined: the main aim of the following overview on processing methods is discussing the different tools in the view of the comparison that will be carried on in section 4.3.

After the comparison, two processing technique will be presented in detail: the f-k processing and the multioffset phase difference inversion.

The different algorithms that are discussed below have been implemented in Matlab codes: this made possible a series of tests on synthetic data and real data. In the computer program that have been implemented for the processing of surface wave data, many of the algorithms have been included, even if the approach that has been finally selected is the f-k processing.

4.2.2 Steady state Rayleigh method

The first technique used for soil characterization with surface waves at small scale was the Steady State Rayleigh Method (Jones, 1958, 1962; Ballard, 1964). Nowadays it appears rough and time consuming, but the original idea was used by other techniques.

In particular the field equipment is composed of a mechanical vibrator and a single receiver. Source and receiver direction are chosen in order to record Rayleigh waves or Love waves. The steady state method uses a stationary monochromatic signal produced by the vibrator. The receiver



is moved away from the source along a radial direction: points with zero phase difference from the source are searched.

The distance between two points in which the recorded signals are in phase is assumed to be equal to the main wavelength at the energising frequency, and hence to the wavelength associated to the relative harmonic of surface waves (Figure 4.2). Different positions are measured, and either the mean distance is considered or a regression on the sequence is performed (Figure 4.3).

From the estimated wavelength value, the velocity is computed as

fV ⋅= λ (4.1)

Repeating the process for different frequencies the dispersion curve over a certain frequency range could be obtained. This technique joins acquisition and processing in the field work, because directly estimates the phase velocity.

Figure 4.2. In the Steady state Rayleigh method the wavelength is directly measured in the field

Figure 4.3. The wavelength is estimated measuring different in-phase distances

4.2.3 Spectral Analysis of Surface Waves (SASW)

This technique has been introduced and diffused by the work of Stokoe and his co-workers (Nazarian and Stokoe 1984, 1986; Stokoe and Nazarian, 1985; Stokoe et al, 1988, 1994) and its processing is based on the techniques presented by Dziewonski et al. (1972). The spectral analysis of surface waves uses the main idea of the SSRM, applies spectral analysis techniques to reduce the acquisition time and work, and decompose a transient signal in its components to obtain a broad band dispersion curve. The basic principle is the same as SSRM, but instead of searching directly the wavelength by moving the receiver position, the wavelength is computed from the analysis of



the two signals observed at known measured distance. The phase difference between two points is used to compute the wavenumber, and the phase velocity. The kind of processing that is performed on the pairs of traces that are analysed strongly condition the acquisition.

The acquisition is performed with two receivers only, with spacing equal to the source offset: several acquisition are performed for each geometrical configuration, to improve the S/N by stacking. With a given configuration, only a limited range of frequencies can be obtained, mainly due to spectral limitations, but also to attenuation and near field effects. Short spacing and high frequency energising are used to infer the dispersion at higher frequencies, i.e. for shorter wavelengths, great spacing and low frequency energising are used to infer dispersion at lower frequencies, i.e. greater wavelengths. The receivers are then moved, to sample the whole dispersion curve, according to different schemes: common receivers midpoint or common source (Figure 4.4). It has to be mentioned that the investigated portions of soil depend on the position of the receivers and on the distance between them.

Figure 4.4. The common source array(left) and the common receiver mid point array(right) for the 2station sasw test

The source position can of course be reversed, and this should reduce the effects of the phase distortion of the measuring system (see chapter 3).

Measure of the phase difference The analysis of the two signals is carried on in frequency domain: if monochromatic sources are used, a single frequency should be present, but the noise always adds other frequencies to the recorded signal. If transient signals are generated, for instance by impulsive sources, these signals are decomposed into their frequency components by means of the Fourier Transform.

The velocity at each frequency can be obtained by the time delay between the two measuring stations: the furthest signal is shifted in time, due to its greater distance from the source. In monochromatic signals, the time shift corresponds to a phase difference (Figure 4.5). This is the basis of the processing proposed by Dziewonski and Hales (1972).



Figure 4.5. Time shift between two monochromatic signals

The phase difference can be computed directly from the two complex spectra )(

1111)()()( fiF efAfYty ϕ=→ (4.2)

)(222

2)()()( fiF efAfYty ϕ=→ (4.3)

∆ϕ = ϕ2 - ϕ1 (4.4)

and it is, actually, the phase of the cross-power spectrum )(

21211212)()()( ϕϕ −=⋅= ieAAfYfYfY (4.5)

This latter was frequently used in two-receiver SASW since the adopted instruments gave it directly as result of the acquisition-processing

Estimate of the velocity

Given two receivers at a known distance X∆ , the measured phase difference is π2 when the propagating wave has a wavelength that equals the distance X∆ : this is the principle of the SSRM. More in general, the ratio between the phase difference and π2 equals the ratio between the distance and the wavelength.

λπϕ /2/ X∆=∆ (4.6)

If the distance between the receivers and the phase difference between the two corresponding signals have been measured, the wavelength can be computed as

ϕπλ ∆⋅∆= /2X (4.7)

The velocity is then obtained, given the frequency f, as

fV ⋅= λ (4.8)

Another way to show the same result, is to consider that the wavenumber is simply

Xk

∆∆

=ϕ and the velocity is (4.9)



kfV π2

= (4.10)

The velocity is then computed from the phase difference between the signals at the two receivers. If the wavelength is shorter than the receiver spacing, the phase difference is greater than π2 , and cannot be obtained directly from the analysis of data. All the points at a distance of XN ∆+λ have the same phase difference (Figure 4.6)

Figure 4.6. To a single phase difference ϕ∆ may correspond the distance x∆ or any xN ∆+λ

When analysing a certain frequency range, either one has to be sure that the wavelength is greater than the spacing, or the number of entire additional wavelength has to be known. Sometimes this can be done unwrapping the phase on a wide frequency range (Figure 4.7). With a single geometrical configuration, if the signal quality is good and the phase difference has a continuous behaviour, the increase of the phase difference with the frequency can be followed up to values greater than π2 . The phase of the cross power spectrum, from a modulo π2 can be unwrapped to a full phase representation (Poggiagliolmi et al, 1982): each jump of the phase corresponds to a wavelength. In the unwrapped phase difference, the low frequencies, with greater wavelengths, have a small phase difference, while the high frequencies, with shorter wavelengths, have a great phase difference.

Figure 4.7. By unwrapping the phase difference (phase of the cross power spectrum) also the wavelengths shorter than

the receiver spacing can be identified, after Foti (2000)



4.2.3.1 Signal quality evaluation It is obvious that the uncertainty on the estimated velocity depends on the uncertainty on the estimated phases of the two signals.

In field data the uncertainty on the phase difference depends on the uncertainty in the two phases, mostly due to the noise in records. The signal quality is so a key point in dispersion curve evaluation and in the assessment of the accuracy of the result: this aspect will be evaluated in section 4.3

The aspects of the propagation of the uncertainty will be discussed later, in section4.3. Here only some consideration on the signal quality evaluation is done.

When a monochromatic source is used, the signal quality at each frequency can be evaluated directly in time domain. It is not straightforward to evaluate the signal quality as a function of the frequency when the decomposition is made with spectral analysis tools: either the energy at each frequency is considered as indicator of good quality, or a statistical processing is needed

In the case of two receivers SASW, it is also interesting the use of a function that evaluate the signal quality as a function of the frequency, in order to select the frequency ranges where the information has to be considered reliable. In the traditional SASW processing this is obtained by means of the coherence function. The coherence function is a spectral function obtained comparing different registrations, that is a measure of the degree by which input and output signals are linearly correlated as a function of the frequency (Figure 4.8).

Figure 4.8. Example of spectral quantities of a couple of traces: (a) Phase of the cross-power spectrum (CPS); (b) Coherence function; (c) Auto-power spectrum(APS) (first receiver); (c) Auto-power spectrum (second receiver).

(after Foti, 2000)



The coherence function evaluates for a couple of traces the ratio between the random component and the coherent component, but it does not allow to identify the nature of the coherent part of the signal.

Given two traces y1 and y2, Y1 and Y2 are the Fourier transformed signal, and

)()( 1111 fYfYG = the auto power spectrum of the first trace (4.11)

)()( 2222 fYfYG = the auto power spectrum of the second trace (4.12)

)()( 2112 fYfYG = the cross power spectrum (4.13)

The coherence function is

2211

1212212 GG

GGY⋅⋅

= (4.14)

and is computed using the average spectral quantities

∑∑

∑∑⋅

⋅=

nn

nn

GN

GN

GN

GNY

2211

12122

12 11

11

(4.15)

being N the number of acquisition, that is

∑∑∑∑

⋅

⋅=

nnnn

nnnn

YYYYYYYY

Y2211

2121212 (4.16)

If only one record is considered, the coherence function loses significance because it simplifies to 1.

If y1 and y2 are uncorrelated noise the coherence function, calculated with n pairs of signals, converge to 0 as n/1 . So the value of the coherence function decreases increasing the number of summed shots. When the signal repeats itself identically the coherence function remain equal to 1 for any number of repetitions.

In real cases a value close to unity is an index of good correlation and hence of good data quality. In a certain frequency range, the coherence function gives the idea of the contribution of coherent signal and uncorrelated noise in the overall signal. The difference in the coherence function value between good and poor quality frequency bands increases with the number of repetitions: the value decreases slightly in the formers and strongly in the latter. The limit value, index of absence of correlation, for n shots, is n/1 .

It is interesting to notice that the coherence function evaluate the system, i.e. gives an idea of the coherence of the system that relates the couple of traces: the coherence can be high even when the signal are different, if the system response is linear.

If for example y2 is obtained from y1 via a LTI filtering operation

)()()()(*)()( 1212 fYfHfYtythty ⋅=⇒= (4.17)

The coherence function reduces to



2

2

1111

1111

1111

1111

2211

2121212 H

HYHYHYYYHYYHY

YHYHYYYHYYHY

YYYYYYYY

Ynnnn

nnnn

nnnn

nnnn

nnnn

nnnn =⋅⋅

⋅⋅=

⋅⋅⋅

⋅⋅⋅=

⋅

⋅=

∑∑∑∑

∑∑∑∑

∑∑∑∑ (4.18)

and does not depend on the series of acquired Y1 and Y2, but only on their relation: the signals can be very different. The coherence function, so, does not judge the relations between the signals, that can be completely independent from each other: if they repeat themselves with low variations the coherence will be high.

4.2.4 ω-p transform

The wavefield transformations are basic tools in the seismic data processing (Yilmaz, 1987), since they allow the separation and identification of different seismic event. The application of wavefield transformation to seismic common shot gather, in order to analyse dispersive wave field, is the basis of the multi-channel surface wave methods.

By means of the wave field transformation (ω−p, f-k) the dispersive waves can be transformed into images of the dispersion curve of each mode in the data: the use of global transforms of the data set for the analysis of the dispersion has been discussed by Nolet and Panza (1976).

The use of the ωp transform for the analysis of dispersive waves has been proposed by McMechan and Yedlin (1981), who discussed the use of the slant stack to image dispersive waves in general. The application of wave field transforms has the advantage that all data contribute to the final image, and the subjective selection of data (picking peaks, windowing, selection of additional number of wavelengths, etc…) can be avoided. In the transformed domain, often modes can be separated even when their presence is not visually detectable in the untransformed data.

4.2.4.1 The slant stack in seismic processing The first transform used in surface wave multistation data processing is a basic tool of seismic processing, the slant stack or τ−p transform; that allows the decomposition of a wavefield in its plane components. It is a particular case of the Radon transform defined as

dxxpxupu ),(),( += ∫+∞

∞−

ττ (4.19)

where τ is the delay time and p is the ray parameter. It is a stack of the wavefield for each couple of τ and p. In general the ray parameter p is defined, in a medium with vertical velocity v(z), as

)()sin(

zvp ϕ

= (4.20)

where ϕ is the angle of incidence of the wavefront that change along the raypath. According to the Snell law, the ray parameter p is constant along the ray. At the source it is

)0()sin( 0

vp

ϕ= (4.21)

On recorded signals, it can be determined from traveltime data of adjacent geophones if the velocity at the surface is known.

0sin)0( ϕxtvs ∆=∆= (4.22)



xt

vp

∆∆

==)0(

)sin( 0ϕ (4.23)

So p is the reciprocal of the apparent velocity at the Earth’s surface. The delay time τ can be seen as the intercept time rearranging τ+= pxt and obtaining a straight line with slope p and intercept time τ (Figure 4.9).

Figure 4.9. Meaning of τ and p in the slant stack

and so p can be obtained differentiating the traveltime curve, p = dt/dx. Hence the t-p transform of a record in time offset domain, stacks the wavefield along a straight line of slope p, for each value of t (Figure 4.10).

Figure 4.10. The τp transform sum the amplitudes of the signal over lines of slope p and intercept τ

A straight line in time offset has constant τ and p. Each linear event in the x-t domain with a slope p and intercept τ maps to a single point in the τp plane. An inverse transform exists, it is hence possible to go back into the initial t-x domain. In seismic prospecting the slant stack is widely used: for the analysis of seismic velocities and filtering, for improvement of SNR and suppression of multiples, for migration, inversion, and interpolation of traces increasing the spatial sampling rate in the inverse transform. (see Yilmaz 1987, Buttkus 2000)

The separation and the analysis of the different types of waves is the most important application of the τ−p transform in SWM.



4.2.4.2 Computing the τ−p transform

This paragraph gives some indications on the computation of the τ-p transform: the main aim is showing the close relationship with the f-k transform.

The τp transform is computed stacking on straight lines of constant p for each value of t, and on real data the integral is then a sum

),(),(1

jjkl

N

jkl xxpupu += ∑

=

ττ with ττ ∆= ll and pkpk ∆= (4.24)

For the computation, the minimum and maximum value of p (pmin and pmax), and the ∆p, have to determined. pmin and pmax are easily estimated, ∆p depend on the frequency content of the signal: to avoid aliasing the following sampling condition has to be satisfied

maxmax21

xfp

π≤∆ being fmax and xmax respectively the maximum frequency and the length of the

transformed data in the x direction. It is important to observe that the τ-p transform can be easily computed using the two-dimensional Fourier transform, i.e. the f-k spectrum. (Buttkus, 2000)

dxdtextukfU kxfti )(2),(),( −−∞

∞−

∞

∞−∫∫= π (4.25)

and the inverse

dkdfekfuxtU kxfti )(2),(),( −∞

∞−

∞

∞−∫∫= π (4.26)

if ),( kfU is computed along a straight line k = fp for the fixed slowness value p, then

dxdtextufpfU pxtfi )(2),(),( −−∞

∞−

∞

∞−∫∫= π (4.27)

substituting pxt −=τ one gets

ττ τπ dxdexpxufpfU fi2),(),( −∞

∞−

∞

∞−∫∫ += (4.28)

And remembering that

dxxpxupu ),(),( += ∫+∞

∞−

ττ (4.29)

we have

∫∞

∞−

−= ττ τπ depufpfU fi2),(),( (4.30)

which has the inverse Fourier transform

∫∞

∞−

−= dfefpfUpu fi τπτ 2),(),( (4.31)



showing finally that the t-p transform can be determined by first transforming the wavefield u(x,t) into the f-k domain and then calculating for each p-value the 1D inverse Fourier transform along a straight line k = fp. The details of the algorithm for direct and inverse tp transform can be found in Buttkus (2000)

Figure 4.11. Synthetic seismogram in time-offset and its τp panel

In figure 4.11 it is shown a synthetic seismogram obtained with a simple wavelet propagating at a constant velocity of 100 m/s. The slowness is 0.01 s/m, and also the time delay is depicted in the τ-p panel.

4.2.4.3 The τ-p transform for SWM processing

The limit of the τ−p transform is that, for SWM, the information about the frequency is needed, and so an additional step of processing has to be performed. Data in a common shot wavefield can be transformed into images of the dispersion curves of each mode by means of two linear transforms (McMechan and Yedlin, 1981). First a slant stack transform time-offset (t-x) data into intercept time-phase slowness (τ-p), in which the phase velocities are separated. Then a 1D Fourier transform is applied to the τ variable, and the frequency associated to each phase velocity is obtained.

The result is a linear invertible transform from time offset into frequency-slowness, or frequency velocity domain, and all data, of course, contribute to the final image. An example is given in figure 4.12: a seismogram is transformed into an image where the energy density is depicted as a function of the frequency and of the slowness.



Figure 4.12. The ωp transform of the synthetic seismogram of figure 4.11.

The synthetic data of figure 4.12 show that the ω−p is an image of the propagation that, even with a spectral smearing, show the position of seismic events in the new domain. The maximum of the ω-p spectrum, in all the frequency range where energy is present, has a slowness of 0.01 s/m, that correspond to the velocity of 100 m/s of the original data.

It can be demonstrated (McMechan and Yedlin, 1981) that the maxima of the new domain correspond to the modal curves when they are the only events. If the wavefield is described, in frequency-wavenumber domain by a relation

),(),(),( )(

ωω

ω ω

kDkNeddktxV tkxi∫∫ −= (4.32)

where x is the offset, t the traveltime, k is the wavenumber, N(k,ω) is a function of the source and D(k,ω) is the dispersion relation. Applying a slant stack to both sides we get

( ) dxkDkNeddkdxpxxVpU pxkxi∫ ∫ ∫∫

=+= +−

),(),(),(),( )(

ωω

ωττ τω (4.33)

Performing the x integration and the k integration we obtain

),(),(),(

ωωωω

ωτ ωτ

pDpNedpU i−∫= (4.34)

and finally, Fourier transforming τ,

),(),(),(

ωωωω

ωpDpNpU = (4.35)

In the transformed wavefield ),( ωpU , the points that satisfy the dispersion relation [ ]0),( =ωωpD send U to infinity; we have maxima of U in those points, and so the dispersion curve can be found searching these maxima.

The demonstration given by McMechan and Yedlin (1981) assumes a wavefield described by equation 4.32, that is a wavefield only containing the desired dispersive event.



It can also be noticed that the transform has to be computed with an infinite zero padding when the wavefield is recorded within a limited window.

4.2.5 f-k transform

Another wave field transform that is used for the analysis of dispersive waves in multi—channel data is the two-dimensional Fourier transform, that is the f-k transform. This technique allows the reconstruction of an image of the dispersion of dispersive events and is particularly suitable to process surface wave data. This transform is a linear operator that has similar properties to the ω-p transform described above. It has been proposed for the processing of surface waves by Nolet and Panza (1976), while an interesting application is presented by Gabriels et al. (1987) that used this transform to process multi-mode Rayleigh waves identifying 6 modes. The f-k transform has also been used for the processing of 2D arrays for passive methods (Horike, 1985; Zwicky and Rix, 1999)

4.2.5.1 The f-k transform in seismic processing The 2D fourier transform is a linear operator that, applied to time-offset data, gives a frequency-wavenumber (f-k) spectrum.

( )∫∫∞

∞−

−−∞

∞−

= dxdtetxukfU kxfti )(2,21),( π

π (4.36)

This transform decomposes the original wavefield in its components at constant frequency and wavenumber: the seismic data are transformed into an image of the energy density as a function of the frequency and of the wavenumber.

This transform is used in seismic processing because it allows to separate and filter events having different frequencies, wavenumbers and apparent velocities (Figure 4.13).

Figure 4.13. The f-k transform allows to identify and to filter different seismic events in records.

One of the main applications of the f-k transform is the f-k dip filtering. Different wave phenomena are separated and can be muted and suppressed: in particular, in seismic processing, the ground roll is viewed as high energy coherent noise, and its removal is an important step.

If data are analysed in f-k domain and maxima are picked, the filtering is not necessary, since the surface wave is usually the main event.



4.2.5.2 Computing the fk transform The 2D Fourier transform of a discrete signal can be computed using the 2D-DFT

( ) ( ) NlNjexmtnxklfjXN

n

M

m

Mml

Nnji

,...,01,...,0,,1

0

1

0

2=−=∆∆=∆∆ ∑∑

−

=

−

=

−− π

(4.37)

where ∆f=1/T=1/(Ν∆t) and ∆k=1/L=1/(M∆x).

This is easily implemented with a succession of two 1D Fourier transforms in the two dimensions:

= ∑∑

−

=

−−

=

− 1

0

21

0

2 M

m

Mmli

nm

N

n

Nnji

jl exeXππ

(4.38)

The expression above implies an infinite zero padding: all the terms out of the summation range are supposed to be zeros. Each of the two DFT can easily computed with a common FFT algorithm.

For each one-dimensional transform all the properties and limitations of the DFT apply. The maximum wavenumber and frequency depends on the Nyquist sampling theorem, and spatial and time aliasing can occur. The spectral resolution, in frequency and wavenumber, depends on the window length, and leakage effects due to the finite observation are present in spectra (for instance, Buttkus, 2000).

An example of application is shown in the following. A synthetic seismogram in time offset domain(Figure 4.14, left), with a single non dispersive event, is transformed into f-k domain: the resulting image (Figuer 4.14, right) represent the energy density as a function of the frequency and of the wavenumber. A single velocity correspond to a straight line from the origin of the plane; the line slope increases with the velocity.

Figure 4.14. The same synthetic seismogram shown above is here f-k transformed

The main limitation imposed on acquisition is that traces have to be uniformly spaced to use a conventional spectral estimate: this may be overcome using other techniques (Duijndam, 1999).



4.2.5.3 The f-k transform and SWM processing The f-k transform allows to map data in a domain where the frequency and the velocity of each point are easily computed, and the associated energy is directly depicted. Hence it is quite easy to compute, at each frequency, the velocity of each seismic event: main events, that should be associated to surface waves, can be recognized for their energy, and can be analysed in terms of frequency and phase velocity.

Furthermore, it can be demonstrated that, if the propagating wavefield is composed only by Rayleigh waves, the f-k transform gives an image of the multi-modal propagation (Tselentis and Delis, 1998).

The wavefield associated to surface wave propagation, in a layered medium, can be written in terms of mode superposition as (Aki and Richards, 1980)

ωωπ

ωω dexStxs xkti

mm

m ))((),(21),( −

∞

∞−∫ ∑= (4.39)

where

( ) ( ) ( )( )

xeRPIxS

x

mmm

m ωα

ωωωω−

=),( (4.40)

m is the mode number, ( )ωI the instrument response, ( )ωmP the source spectrum, ( )ωmR the path response, ( )ωα m the attenuation coefficient.

If we have N aligned geophones with spacing x∆ , we can write the phase difference for the mode m as ( ) xkm ∆ω

The relation between the wavenumber, the phase velocity c and the slowness p, for the mode m is

( ) ( ) ( )ωωω

ωω m

mm p

ck == (4.41)

If we perform the slant stack on the N traces

( ) ( ) ( ) ( )

( ) ωωπ

ωωπ

ττ

ωωωτ

ωτω

deexS

dexSpxxspS

nmn

nmn

xkpxiiM

mnm

N

n

xkpxiM

mnm

N

n

N

nnn

))((

11

))((

111

,21

,21,,

−∞

∞− ==

−+∞

∞− ===

∫ ∑∑

∫∑∑∑

=

==+=

(4.42)

Expanding S

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

ωωωωπ

ττ ωωωτωα

deex

eRPIpxxspS nmnnm

xkpxiiM

m

x

mmm

N

n

N

nnn

))((

111 21,, −

∞

∞− =

−

==∫ ∑∑∑ =+= (4.43)

and compensating for attenuation one gets S independent from x

The influence of the geometric attenuation can be easily removed by multiplying each contribution by the square root of the source-receiver distance and it will be therefore neglected in the following, assuming that such correction is applied on the original data. As the material attenuation is concerned, its contribute can be taken out from the expression of S, so that this last quantity becomes a function of frequency only.



( ) ( ) ( )( ) ( )

( ) ( )( ) ( ) ωωπ

ωωπ

τ

ωαωωτ

ωαωωωτ

deeeS

deeSpS

N

n

xxkkxiim

M

m

N

n

xxkpxim

M

m

mnmn

mnmn

∫ ∑∑

∫ ∑∑∞

∞− =

−−

=

∞

∞− =

−−+

=

=

==

11

11

21

21,

(4.44)

It is then easy to recognize in the above equation the integral of the fourier transform

ωωπ

τ ωτ deSS i∫∞

∞−

= )(21)( (4.45)

taking the Fourier transform of both sides of 4.44 we get

( ) ( ) ( )( )

= −

=

−

=∑∑ nmm xfkki

N

n

xfM

mm eefSpfS

11),( α (4.46)

and we see that it is a function of f and k, that can be written as

( ) ( )( )∑=

−⋅=M

mmmm fkkWfSkfF

1

),( (4.47)

If we neglect the material attenuation contribution, it is evident that differentiating the quantity in the square bracket in 4.46 with respect to k and setting the results equal to zero, the maximum of the energy spectra is obtained for

k=km; (4.48)

Furthermore it can be shown that also if the above differentiation is performed without neglecting the material attenuation the conclusion is the same, i.e. the accuracy is not conditioned by material attenuation (Tselentis and Delis, 1998).

The details of the computation are shown in section 4.4, where the f-k procedure for the processing of surface wave data is illustrated.

4.2.6 Crosscorrelation

The identification of the shift between signals is the main step of the processing: it can easily be done by means of the crosscorrelation. The cross correlation is an operation that allows to determine the relations between signals and to compare events occurring at different times. It is based on the cross-covariance function, defined as the time average of the product of a function at time t (after subtracting its mean value) and another function at t + τ .

( ) ( ) ( )( ) ( ) ( )( )dttytytxtxT

RT

TTxy −+⋅−= ∫

−∞→

ττ21lim (4.49)

The autocovariance is a statistical measure of the relationship between the different parts of a random process, the crosscovariance function describes statistically the interdependence of two processes (Buttkus, 2000)

The correlation is used for the determination of differences in signal arrival times, to evaluate for instance the travel times of different waves. The vibroseis method also is based on this procedure: a sweep signal is generated with a vibrator, thus many reflections and events can be superimposed in recorded signals. The recording is correlated with the emitted signal to compute the travel times of



different events, being the autocovariance function of a sweep very short (Figure 4.15). More details about the operator and its applications the can be found in Buttkus (2000).

Figure 4.15. The crosscorrelation easily identify the time shift even in noisy data. A sweep is depicted with its shifted

noisy copy: the crosscorrelation produce an evident peak.

The simpler application of correlation can be the determination of the time shift between two monochromatic or narrow bandpass filtered signals recorded at different distances.

In correlation study, if the velocity is known to be positive, the time shift can be followed up to more than a half a period, even if the maximum becomes negative.

Figure 4.16 Two time shifted sinusoids and their discrete crosscorrelation

4.2.7 Transfer function

Remembering the expression for the vertical displacement at the free surface of a layered medium, viewed as a LTI system, one can define the transfer function, that is the ratio, in frequency domain, between the output and the input. The input has to be known, thus the source has to be monitored, and the output are the signals measured at different offset: the transfer function is so experimentally estimated, and the properties of the soil are inferred using a model of the system.

The model is a stack of viscoelastic layers, described in chapter2, that allow to write the vertical displacement as

( ) ( ) ( ) ( )[ ]ωωωωω ,,, rtiZZ erGFrU Ψ−= (4.50)



where ZF is the dynamic force transmitted by the vibrator, and ( )ω,rG the geometric attenuation law. The term ( )ω,rΨ is the complex phase angle depending on the complex wavenumber, on the velocity and on the intrinsic attenuation. The displacement transfer function can be computed dividing the measured displacement by the measured force of the vibrator

( ) ( )( ) ( ) ( )ω

ω ωω

ωω ,,,, ri

tiZ

Z erGeF

rUrT Ψ−== (4.51)

When a single mode is assumed to be present, the dependence of the complex phase angle from the offset become linear, and may be written as

( ) ( ) ( ) riKerGrT ⋅−= ωωω ,, (4.52)

and the complex wavenumber ( )ωK may be estimated from the data (Foti et al., 2001; Foti 2000)

This approach is a powerful tool for a coupled estimate of the phase velocity and the attenuation, but it needs monitoring the source, whose force depends also on its coupling to ground (Ven der Veen et al., 1999)

4.2.8 Multi offset Phase inversion

A technique based on the inversion of the phase difference between the spectral components of the signal at each receiver is here proposed. It can be thought as an extension of the two receiver processing, but the information of each trace is weighted, and the uncertainty of the velocity at each frequency is estimated. The principle is the same used in the transfer function method proposed by Foti (2000), but the source is not measured, and impulsive sources may be used: the details are shown later (section 4.4), and here just an outline of the principle and the main steps are given.

The wavefield associated to surface waves in a dissipative medium can be written, in frequency domain, as the sum of the different modes as (Aki and Richards, 1980)

( ) ( ) ( )( )xkti

mm

mexAxs ωωωω −∑= ,, (4.53)

where ( )ωmk is the wavenumber of the mth mode as a function of the frequency, mA is the amplitude, considering the source and the path.

The expression can be rearranged separating the space dependent and frequency dependent part of the amplitude as

( ) ( ) ( )( )

( ) ( )( )ωϕωωωα

ωωω Imm

xktix

mm

ex

eRIxs +−−

∑=, (4.54)

The amplitude has a dependence on the input source I, on the response of the site R, on the geometric attenuation and on the intrinsic attenuation.

This complex expression, for each mode, has amplitude

( ) ( )( )

xeRI

xωα

ωω−

(4.55)

and phase



( ) ( )ωϕω Ixk +⋅− (4.56)

and then, considering a single mode, for a single frequency, the amplitude and the phase have the simple behaviours of figure 4.17

figure 4.17. Theoretical behaviour of the phase and the amplitude for one frequency as a function of the offset

When experimental data of amplitude and phase as function of the offset and the frequency have been gathered, the two expressions can be used to estimate the unknown parameters of the propagation. The source component of the amplitude can be unknown: anyway, even with controlled sources, the ground coupling of the source can play a dominant role in the spectral energy actually transmitted to the ground, and it is difficult to be evaluated. The experimental phases and amplitudes that are needed to estimate the wavenumber (from equation 4.56) and the attenuation coefficient (from equation 4.55) can be easily obtained by a single Fourier transform of multi-offset data. For a single frequency, with n receivers at the offsets

( )nxxxX ,...,, 21=

we will have two vectors of data: the amplitudes

( )naaaA ,...,, 21= (4.57)

and the phases

( )nϕϕϕ ,...,, 21=Φ (4.58)

An example of experimental amplitude and phases is given in the following figure 4.18



Figure 4.18. Experimental amplitude and phase, for the frequency of 20 Hz, with a 24 receiver array, sledgehammer

To get this result, different shots are collected separately, in order to allow a statistical comparison of them in frequency domain, and the data can be filtered in frequency offset domain. Then the phases at each frequency are considered: the phase difference is computed for each frequency and each shot, and the statistical distribution of the phases is assessed. The experimental phases are finally unwrapped to a full-phase representation, such as that of figure 4.18 (right).

For each frequency, the phase as a function of the offset (Figure 4.19) is then analysed: a weighted least squares inversion is performed, to compute the velocity considering directly the data quality of each trace. The weights have to be computed considering that high values of the phase variance are related to lack of information.

Figure 4.19. Experimental phases vs. the offset, for six different frequencies. The higher frequencies, with greater

slopes, show higher uncertainties in far offset



Figure 4.20. The less reliable points should have lower weights in the estimate of the slope

This approach has the main advantage that data quality is considered and evaluated in a statistical way: local variations can moreover be identified, to give warning about the presence of lateral variations.

The main limit is that only the properties of the most energetic event are estimated, and a preliminary filtering is necessary when several modes are present.

4.2.9 Time frequency analysis

The acquired signal are strongly non stationary, and their representation in time frequency domain (TF) can be very interesting. A signal in time domain can be transformed into a energy density function of the time and the frequency for instance with a Short Time Fourier Transform (STFT).

In this procedure each signal is analysed by a moving window that extract portions of the signal and associates the frequency domain representation of the window (Figure 4.21). This analysis is also referred to as Frequency Time ANalysis (FTAN).



Figure 4.21 The time frequency analysis is based on the Short Time Fourier Transform, i.e. the computation of the FT in short moving windows. A 1D signal(in time) gives a 2D spectrum (spectral density in time and frequency)

The moving window can overlap: long windows produce higher frequency resolution but lower time resolution. The Short Time Fourier Transform can be used to see in a signal the time depending behaviour of the different frequencies.

A simple example in figure 4.22 shows the meaning of the STFT. On the left the time domain representation of a signal s1, a signal s2, and their sum. On the right the same representation in time-frequency domain. The Time Frequency analysis, hence, shows the presence of different frequencies and also shows when they are present in the signal. A signal is represented as energy density as a function of the time and of the frequency.

A transient signals can be processed without loosing the time information, as it would happen with a simple Fourier transform of the whole signal. The problem is the well known reciprocity principle: a high resolution in time correspond to a low resolution in frequency (Oppenheim and Shaefer, 1975; Buttkus, 2000)

Figure 4.22. The time-frequency analysis can apply to transient signals.

When different signals are recorded at different distances, they can be analysed in time frequency domain in order to compute the time delay at each frequency. The time delay can be obtained analysing the energy distribution in time at each frequency, for the recorded signals: the barycentre, in time, of the energy distribution for each frequency can be used (Audisio et al., 1999).

In the figure below (figure 4.23) are depicted two traces in time domain and in time-frequency domain. The position of barycentre of the energy at the frequency of 20 Hz is shown in the TF spectra.



Figure 4.23. The time-frequency analysis of two traces obtained with an impulsive source. If we cut the two TF spectra at constant frequencies, the travel times can be extracted.

For the surface waves analysis, this tool can be used to extract the travel times of the different frequency components of the signals at different offsets.

Similar approaches can be obtained with the time-scale analysis with the wavelets, that are able to infer the local properties of signals.

Chapter 4.3 COMPARISON OF THE PROCESSING TECHNIQUES


4.3 COMPARISON OF THE PROCESSING TECHNIQUES

This section aims to compare the different processing techniques, that however are very similar from a theoretical point of view. The comparison is carried out considering the sensitivity to coherent and random noise, and trying to obtain the probability density function of the desired propagation properties (phase velocity or wavenumber) considering the experimental uncertainties and their propagation through the processing.

4.3.1 Differences

The different processing techniques are different mathematical tools to compute the phase velocity at each frequency from field data in time-offset domain. Actually they are not so different from each other: they all are able to identify the properties of the propagation, and when applied to ideal synthetic data they all should give the same result. Moreover most of the different techniques are mathematically identical. The conventional SASW processing is completely identical to a f-k processing of two traces, the different wavefield transforms are related by linear invertible transforms (appendix C). The vertical flexibility coefficient at the top boundary of the surface layer used by Al-Hunaidi (1994, 1998) is the f-k spectrum at a single frequency of the two traces seismogram.

The different processing techniques, hence, have to be compared with regards to the way in which they face the problems of real data. Different aspects have to be considered:

a) real data do not contain pure Rayleigh waves, but also other coherent phenomena

b) real data are noisy

c) real data are limited in time and space

Two operations are usually needed in the processing of full-wave form data: the selection of the events to analyse, and the identification of their properties (velocity and attenuation as a function of the frequency). One important difference among the processing techniques is the possibility of selecting at each frequency different events, and not only the most energetic: this allows the analysis also of secondary events. It has to be remarked that sometimes the secondary events are the most interesting, and anyway this possibility increases the information that is extracted.

The noise is always present in field data and cannot be disregarded: the processing should reduce the effect of noise on the inferred propagation properties. A better processing technique produce a dispersion curve that is less contaminated by noise. However the main differences are in the number of data that are processed. The accuracy of the result, for a given signal quality, mainly depends on the number of traces that is processed and on the array length, but also on the possibility of identifying and discarding the low-quality information. Some techniques actually allow the control of the data quality, the selection of the information to process, and also a weighting of the information based on its reliability.

The limited extent of the observation can be critic at the small scale of the engineering applications: the main aspect is that the several modes of the Rayleigh wave propagation do not separate, but often completely superimpose within the observation window. As a consequence only apparent properties can often be obtained: an apparent phase velocity, and also an apparent attenuation. Some processing techniques better face this problem, and allow a better mode separation than others.



In this section the different processing techniques are compared with regard to their sensitivity to random noise and to coherent noise, and to their mode separation capability.

4.3.2 Sensitivity to random noise

The first aspect to be considered is the sensitivity to noise of the different processing techniques, i.e. the way in which the noise distorts the inferred dispersion curve.

Some steps are involved and have to be discussed. The single traces are recorded, then analysed: the phase of the traces for each frequency is extracted, then processed to compute the wavenumber. (if the phase difference is extracted, the wavenumber is simply Xk ∆∆= /ϕ ). Then the velocity can be

obtained simply as kfV π2

= .

If we want to reduce the uncertainty on the velocity, two things have to be done: reduce the uncertainty on the phases, and reduce the amplification of this errors on the estimated wave number. The final step of computing the velocity is not necessary, and the inversion to infer the subsoil properties could be performed with dispersion curve in frequency-wavenumber domain.

So, the following factors are of crucial importance

• Effects of noise on the single recorded signals, and in particular on their phases

• Propagation of the uncertainty of phases on the estimate of the wavenumber

• If the phase velocity is needed, amplification of the uncertainty of the wavenumber on the estimated phase velocity.

4.3.2.1 Uncertainty of the recorded traces In order to reduce the uncertainty on the phases, which are the experimental data, an optimal acquisition should be performed: the array length and the spacing have to be designed depending on the wavelength that have to be acquired, and a careful acquisition have to gather high SNR data.

What is random noise has been illustrated in chapter 3. Considering here a single frequency (the signal can be obtained as a sum of monochromatic functions) the random noise is a monochromatic signal, at the same frequency of the useful signal, with random phase.

With regard to the effects of random noise, only the phase of traces is considered at each frequency; the velocity, for each frequency, is estimated from the different processing by means of different algorithms, but in most of them the amplitude of the traces has no influence (Appendix C)

The effect of the noise is hence a phase rotation of the recorded signal, due to the superposition of signal and noise: with a given random phase shift, the phase rotation decreases increasing the ratio between the amplitude of the signal and the noise, that is the S/N (Figure 4.24).

Since the amplitude of traces decreases with the distance, the S/N also decreases, and the phase rotation produced by noise increases.



Figure 4.24. The phase rotation ∆φ induced by the noise on the signal can be shown in the complex plane. For the same

phase difference between S and N, the phase rotation depend on the Signal to Noise ratio

From a statistical point of view, the distribution of the resulting phase from the sum of a zero phase signal and a random phase signal, can be obtained considering that the noise is

ϕieN ⋅

with random ϕ uniformly distributed in ( )π2,0 and that the signal is simply

S

The resulting signal (S+N) has hence a real part

SN +ϕcos (4.59)

and an imaginary part ϕsinN (4.60)

The phase of the recorded signal is hence distributed as

+

=

+

−−

NSSNN

/cossintan

cossintan 11

ϕϕ

ϕϕ (4.61)

This means that when the S/N is higher than 1 the distribution is bounded between two limits, depending on the S/N, that never exceed the interval ( )2/,2/ ππ− . When the SNR is lower than 1, the distribution does not have such limits, and tends to an uniform distribution in ( )ππ ,− when the S/N tends to zero (Figure 4.25).

Figure 4.25. The theoretical distribution of the phase for different signal to noise ratio (the phase of the noise is a

random variable uniformly distributed in –π +π)

S/N=2 S/N=1

S/N=0.5 S/N=0.9



This kind of distribution is obtained when the S/N is constant: if this is not true, a more complicated distribution is obtained. It can be obtained intuitively summing different distribution corresponding to different S/N.

In the expression

+

−

SNN

ϕϕ

cossintan 1

the phase is distributed uniformly in ( )π2,0 , the amplitude of the noise N can vary. If N has a normal distribution with mean Nµ and variance Nσ , the signal to noise ratio is distributed as

( )22

/1

21

N

Nx

N

e σ

µ

πσ

−−

(4.62)

If for instance the mean S/N is 3 and the variance of the noise is the 20% of the mean value of the noise itself the resulting distribution for the S/N is not symmetric and is depicted in figure 4.26

Figure 4.26. The distribution of the Signal to Noise Ratio, with constant signal amplitude and normal noise amplitude

For the distribution of the phases, with the same mean S/N and with different variances of the noise (ranging from 0 to 0.5 the mean value of the noise) we get the distributions in figure 4.27



Figure 4.27. The distributions of the phase, with a fixed mean Signal to Noise Ratio and different variances of the SNR

In real acquisition also the source is not perfectly constant, even with controlled sources. A statistical prediction of the phases is then quite difficult, but of course little variations around the average value mean good quality, while high dispersion means poor quality. It has been experimentally observed, in a series of experiments performed during this work, that the phase distribution, in real acquisitions with a great number of shots, tends to be a normal distribution for high values of the S/N (Figure 4.30).

When real data are gathered averaging different acquisitions, if the phase of each trace at each frequency is analysed, the data quality can be inferred. The trace depicted in frequency domain and time domain (Figure 4.28) is obtained averaging 9 different acquisitions. The amplitude spectrum shows that the highest signal level is in the band 40-70 Hz: in this range the S/N is also the highest, also because in this case the main frequencies of the noise are lower. The single phases of the 9 acquisitions are shown (Figure 4.28) for the two frequencies of 5 Hz (low S/N) and 50 Hz (high S/N).

σS/N = 0.3 σS/N = 0.4 σS/N = 0.5

σS/N = 0 σS/N = 0.1 σS/N = 0.2



Figure 4.28. A single trace obtained averaging 9 acquisitions, in time domain(top) and frequency domain (bottom). A

high energy content is within 35 Hz and 80 Hz

Figure 4.29. The 9 experimental phases, at the frequency of 5Hz (left), where the signal is weak, and at 50 Hz, where the signal has the maximum of energy.

The obtained distributions of the experimental phases are in good agreement with the S/N: at 50Hz the phase distribution as a much lower variance than at 5Hz.

A greater number of repetitions allow a better evaluation of the phase distribution. The following figure 4.30 is obtained from the analysis of 100 acquisitions: the histograms have 15 classes between the minimum and maximum values. In a frequency band where the signal is low, the phases are distributed in a quite uniform way in all the range. Where the signal level is high compared with the noise, the phases are concentrated around a single value, with a little variance.

f = 5 Hz LOW S/N f = 50 Hz HIGH S/N



Figure 4.30. Phase distribution for two frequency value (5Hz top, 60 Hz bottom). 100 shots.

4.3.2.2 Effects of the Array length and the number of receivers on the uncertainty of the wavenumber

The acquisition is responsible for the uncertainty on each single trace, and the reduction of these uncertainties is one of the main task of a careful, properly designed and performed acquisition. In the design of the array the effects depending on the horizontal dimension (near field, far offset, etc.) described in chapter 3 have to be taken into account.

Apart this, the choice of the array is mainly due to the need of reducing the propagation of the single uncertainties of the phases. The uncertainty on the estimated wavenumber depends on the uncertainty on data, of course, but also on their distribution, that is the number and the position of recorded signals.

In particular let’s consider the phase of the traces at different distances. Considering a single frequency component, it is evident that the length of the array and the number of receivers have a great influence on the estimated wavenumber (and hence the phase velocity), and it is intuitive that a multi offset estimate can average the errors of the single traces and give a more reliable estimate, even with the same array length. The example below (Figure 4.31) shows that if only two signals are recorded, the wavenumber is directly obtained as the slope of the line through the two points. If more signals are recorded, the line can be obtained with a more robust estimate, reducing the effect of single errors on data: this is a general concept not depending on the adopted processing.



Figure 4.31. The number of receivers influence the uncertainty on the estimated phase velocity, even with the same

array length. With two traces (left), the wavenumber is directly the slope of the line through the two points (in phase-offset). With several traces (right) single uncertainties are averaged and reduced.

The first step, so, is to evaluate the error amplification due to the array dimension and to the number of receivers.

A possible approach to evaluate the propagation of the uncertainty of the phases of the single traces on the final estimate of the velocity can be done by the unit covariance matrix (Menke, 1989).

If the physical model is one dimensional, the phase is a linear function of the offset: considering also the possibility of near field effects, or ignoring the exact position of the source, the linear slope holds only in a certain range of offsets, in which the relation is

ii xk ⋅+= 0ϕϕ (4.63)

where iϕ is the phase measured at the offset ix , k is the wavenumber, and 0ϕ the intercept phase (the details are given in section 4.4).

The wavenumber k is estimated considering n measured signals. The system of n equations 4.63 can be compactly written in matrix form as

[ ]TkG 0ϕ⋅=Φ (4.64)

where Φ is the vector of the phases and G the kernel matrix, depending on the number of points and on their distribution, which can be written as

=

nx

xx

GMM2

1

1

11

(4.65)

The matrix G contains the information about the data distribution, and not on their measured values: it can be evaluated a priori, before the acquisition of data, and allows to compute the amplification of uncertainty of data. As it can be noticed in the schematic example below, with the same uncertainty of data, the final uncertainty depends on their position and number (Figure 4.32). The uncertainty of the slope increases reducing the number of points and the length of the array.

.



Figure 4.32. The array length and the number of receivers influence the uncertainty of the estimated phase velocity

The unit covariance matrix can be written as ( ) 1cov −= GGT

u and contains the variance of the estimated k. Some further detail about this approach are given in chapter 5, and a complete treatment can be found for instance in Menke (1989).

The figure 4.33 shows the variance on the wavenumber as a function of the total array length and the number of receivers: the uncertainty of the phases are supposed constant.

Figure 4.33. Uncertainty on the wavenumber for unit uncertainty on the measured phases. It is evident that for the

same array length a great number of receivers reduces the error

For each array length, a greater number of receivers reduces the amplification of the uncertainty: the use of 24 receivers reduces the uncertainty of 4 times and the use of 48 receivers of more than 8 times in comparison to the use of 2 receivers. Given a certain ratio between the number of receivers, for instance 2 and 24, the ratio between their amplification does not change with the array length.

For each number of receivers, the use of longer array reduces the amplification of the uncertainty: a 20 m array reduce the uncertainty of more than 4 times respect to a 10 m array, and a 40 m array of



more than 18 times, for any number of receivers. However, the increase in the array length is not always possible: moreover, with a longer array the data quality is probably lower, due to the greater attenuation. Increasing the array length also the uncertainty of the phase can not be supposed constant; furthermore the risk of lateral variations is increased, increasing the array length.

The most interesting aspect hence is that the same amplification can be obtained with a shorter array, if more receivers are used. This fact has a series of advantages, as shown in chapter 3, and moreover allow a better data quality: there will be the same theoretical amplification, but a shorter array will have lower uncertainties to be amplified.

Figure 4.34. With the same array length, the use of a different number of receivers change the accuracy of the result

The following figure 4.35 shows the comparison between 2 receivers and 24 receivers arrays: half the length can be used with 24 receivers, and the source offset can exclude the risk of near field effects.

Figure 4.35. The amplification for different array length for a 2receiver array and for a 24 receivers array.

A two-receiver array needs a double array length compared to a 24 receivers to have the same variance amplification.

Of course the choice of the array is also conditioned by the wavelengths that we want to sample: the maximum spacing is conditioned by the spatial aliasing. This is, however, an other advantage of a multi-offset, that can be used to sample a wide range of wavelength without changing the configuration. These considerations on the averaging effect of a multi-offset array hold also if a coherent noise is considered, particularly when it affect a single portion of the array (as for instance the near field).



4.3.2.3 Effects of the amplitudes of the traces on the estimated K In the phase velocity estimate, most of the techniques do not use the information of the trace amplitude, and only the phases are processed. Also with the f-k processing the position of maxima at a fixed frequency depends only on the phases of traces at that frequency. However the amplitudes of the traces with the offset are not constant at all: the geometrical and intrinsic damping, and the modal superposition, produce strong variations of trace amplitudes.

This means that there is not an intrinsic weighting function in the estimate, and so all traces give the same contribution, independently from their energy. The traces with greater amplitudes probably will have a greater S/N, and a more reliable phase information, the contrary holds for far traces having little amplitude.

To introduce these weights, with the aim of reducing the effects of bad data on the final estimate, it is not possible to act on trace amplitudes: only a complete muting is effective.

This is easily shown considering that if the traces are scaled, only a windowing effect is introduced. The weighting function is multiplied in offset domain, and its spectrum is convolved in wavenumber domain. If only one mode is present, the position of the maximum in f-k domain will not be modified by the window: only a different shape of the lobe will be obtained. On the other hand, when several modes are present, the window spectrum can change the modal superposition. The following example (Figure 4.36) shows the effect of a weighting with weights decreasing with the offset (Figure 4.36, left): the spectrum of this weighting function is depicted in figure 4.36, right.

Figure 4.36. If weights are introduced, only an additional windowing effect is obtained. Left, weights for the traces. Right: spectrum of the weighting function. The resulting spectrum is simply the original spectrum convolved with the

window spectrum

When these weights are applied to real data, the modification of the modal superposition appears in the spectra. In figure 4.37 a seismogram is depicted (normalization of each trace) with its spectrum.

In figure 4.38 the spectrum of the original seismogram is compared with the spectrum of the weighted seismogram: the bandwidth (in wavenumber) of the main lobe of the window is increased, and between 20 and 30 Hz the superposition of the two modes is stronger.

The two dispersion curves obtained by searching the maximum at each frequency are shown in figure 4.39: the main differences are due to the change of modal superposition caused weighting window spectrum, and are in the frequency band 20-30 Hz.



Figure 4.37. Real seismogram and its f-k spectrum. Two modes are easily identified

Figure 4.38. The traces are weighted, and the two spectra are compared

Fgure 4.39. The two dispersion curves are compared here: the distortion is produced by the shape of the window

spectrum that affect the modal superposition. Black: no weighting. Grey: weighting.



4.3.2.4 Effect of the offset of the error on the estimated wavenumber It is interesting to asses which is the effect of a single error on a trace upon the final estimate of the wavenumber: in particular it is interesting to see if the position of the error within the array is influent or not.

If the phase of a spectral component of a trace is modified of a fixed quantity, the effects on the spectrum mainly depends on its position. A fixed phase rotation is applied to a single trace of a synthetic seismogram: the following figure 4.40 shows an example where the third trace has been perturbed. The phase rotation can be due to experimental uncertainties or to underground anomalies, such as obstacles and cavities. (Gukunski et al, 1996)

Figure 4.40. On synthetic monochromatic data, the phase of a single trace is modified, to evaluate the effect of the

position of this error on the estimated wavenumber

The error on the estimated wave number depends on the distance from the centre of the distribution: the first traces and the last traces have a greater effect on the estimated wavenumber, either using the f-k processing, or all of with all the equivalent techniques, or the phase difference inversion.

The figure 4.41 shows an example with 24 traces: a phase rotation of a single trace is produced is synthetic data: the error of the estimated wavenumber is plot vs. the position of the modified trace.

Figure 4.41. The same phase rotation has different effect depending on the position of the trace .The greater the

distance from the centre of the array, the greater the effect on the estimated wavenumber



It is interesting to notice that the amplitude of the effects of a phase shift on a trace depends on the array length and on the position: the far traces produce a greater change on the estimated velocity, even if their energy is little. Furthermore, for their little energy, they can be easily influenced by noise: in seismic data with end-off shot, the far traces have a lower SNR. Their phases can be easier distorted by noise, and the effect of this uncertainty is amplified by the distance. It is important, hence, to reduce the weights of low SNR traces in the final estimate. In particular, with many processing techniques, the weights are difficulty controlled, hence the corrupted frequencies of each trace should be discarded and excluded from the analysis.

4.3.2.5 Amplification on the phase velocity The uncertainty on the phase velocity, thus, depends on the uncertainty on the phases of the acquired data, on the amplification on the wavenumber, and finally on the amplification from the latter to the phase velocity. The relation between the wavenumber and the phase velocity is

kfV π2

=

and the uncertainty can be estimated, with a first order approximation, as kk

fV ∆=∆ 2

2π .

If we consider a single velocity of 1000 m/s in a wide frequency range, and a constant wavenumber variation, we get the two curves of phase velocity that diverge at low frequency (Figure 4.42).

Figure 4.42. Amplification of the wavenumber uncertainty on the phase velocity.

A constant uncertainty of the wavenumber produces a great amplification.

It is evident that the same uncertainty on the wavenumber produces very different uncertainties on the phase velocity.

In the f-k plane, the same variation of velocity correspond to very different intervals for the wavenumber (Figure 4.43): this means that the energy density of the same event is different in the two domains, since the spreading around the maximum is different: the energy density, as it is discussed later, can be considered as a probability density.



Figure 4.43. The relations between the f-k and fv planes are not linear. Equal areas in f-v correspond to very different areas in f-k domain. The energy spread is uniform in f-k

The non-linear relation between the wavenumber and the phase velocity and the amplification of the uncertainty of the phase velocity have to be carefully considered. This effect can be also analysed considering a theoretical wave-field written (as in eq. 4.39)

ωωπ

ωω dexStxs xkti

mm

m ))((),(21),( −

∞

∞−∫ ∑=

and the acquired wave field, observed within a limited time and space window, that has a spreading of the energy of the different modes due to these acquisition windows.

In the f-k domain, the ideal data have the energy that is concentrated in the mode positions, while the energy of acquired data follows the window spectrum. Considering the energy density as a probability density function, real data have a probability density depending on the window spectrum. The probability density function of the wavenumber, hence, is the same at each frequency.

In f-v domain, on the contrary, the probability density of the velocity depends on the frequency.

The following figure 4.44 shows the frequency wavenumber (f-k) and the frequency-slowness (f-p) representation of the same synthetic seismogram (a non dispersive event at 100 m/s in a frequency band of 0-90 Hz): the energy spreading, hence the uncertainty, only depends on the windowing.



Figure 4.44. A non dispersive event 100 m/s in 0-90 Hz has a uniform spreading in f-k and not in f-p domain

This is not a wrong result: the probability density of the slowness has a greater variance than that of the wavenumber. However this can not be neglected, and the uncertainty of the phase slowness has always to be associated to its mean value, and has to be used in the inversion process that estimates the subsoil properties.

A further aspect is that the phase velocity distribution, obtained from the slowness distribution, is no more symmetric (Figure 4.45): the event associated to the highest probability is not the median of the distribution. The use of the Gaussian distribution is no more possible.

Figure 4.45. Distribution of the phase slowness and of the phase velocity. The latter is not symmetric



4.3.3 Sensitivity to coherent noise

Coherent noise can be present in records in different forms, due to the effects of different seismic events or due to a model error (chapter 3).

The way in which the coherent noise affects the inferred dispersion characteristics extracted by the processing depends on many factors, referring to acquisition and processing itself.

Of course, as for random noise, the number of receivers strongly influence the final result. Let’s consider for instance the near field effects: if only two receivers are used, and the first is affected by near field effects, the final result will be strongly contaminated. If a greater number of receivers is used, the presence of the near field can be recognized, and anyway the effect on the final estimate is strongly reduced.

The following figure 4.46 shows the experimental phases as a function of the offset for three different frequencies (24, 30 and 75 Hz) corresponding to three different wavelengths (40, 27, 10 m): they have been obtained from the 1D Fourier transform of a seismogram with a single dominant Rayleigh mode, and hence the phases refers to this main event, even if they may be distorted by other seismic events. The used array is a 24 geophones, 2 m spaced, with 2m of source offset, and the experimental phases show a good linearity.

Figure 4.46. The experimental phases are depicted for three frequencies, with the associated wavelength. There is no

evidence of some slope change between near field and far field

Analysing the lower of the three frequencies of figure 4.46, hence the greater wavelength, it can be seen that only the first receiver seems to be out of the trend (Figure 4.47). This means that near field effects are not so strong to modify the phase velocity of the main event



Figure 4.47. A zoom on the greater wavelength (40m) shows that the experimental phases have a good linearity in all

the offsets, apart the first trace(2m from the source)

4.3.4 Mode separation possibility

The multi modal nature of the Rayleigh wave propagation has been discussed in detail in chapter 2. In chapter 3 the effects of higher modes on the phase and amplitude of acquired traces has been illustrated. The number of modes, their relative energy, their phase velocities depends on the site and the source: their separation depends on the array which is used for the acquisition. At the same site, with the same source, the array length conditions the mode separation or superposition. This is hence an important parameter as far as modal separation is concerned.

But the processing technique too strongly conditions the possibility of extracting these information, and of recognizing the presence of modes avoiding misinterpretations. To show the possible errors coming from a misinterpretation of the modal superposition the example of real data shown in figures 3.22-3.24 can be very useful. At 27.5 Hz two modes superimpose with the same energy: the two wavelength are 4 m and 8 m. If we place two receivers only, at 6 m and 12 m respectively, we see an increase of energy and an apparent velocity which is far from the two modal velocities (Figure 4.48).



Figure 4.48. Monochromatic seismogram: the phase rotations are due to the destructive interference between two mode with similar energy (see chapter 3, figures 3.22-3.24).

On the contrary, the analysis of the whole set of traces could indicate the presence of two distinct events, whose properties can be identified: the example of the f-k processing of this single frequency, already given in chapter 3, is reported also in the following figure 4.49.

Figure 4.49. Energy density as a function of the wavenumber k. Two separate event are identified

The mode separation possibility can be an important parameter for assessing the performance of a processing technique, because the amount of information extracted and the filtering possibility depend on it.

The processing techniques can be classified in two classes considering the mode separation: some of them transform data and show the distribution of energy as a function of other parameters (velocity and frequency) allowing also the identification of local maxima; the others extract directly the main event, i.e. the absolute maximum of energy, and give its velocity as a function of frequency.

Considering the processing techniques allowing the analysis of several modes, the maximum number of maxima and the mode separation possibility depend on the acquisition parameters. With the f-k processing, for instance, there is a maximum number of modes that are identifiable with a fixed number of receivers, as it has been illustrated in chapter 3.



4.3.5 Coherency with a simple model for inversion

4.3.5.1 Model parameters and test parameters In each inverse problem the coherence of the model is a key point: model parameters have to describe completely the tested system for the investigated physical aspect, within the uncertainties related to the measurements. Sometimes, to reduce the number of unknowns, a sensitivity study can show which parameters can be neglected. But if the experimental data are not an intrinsic property of the system, but they are the result of a measuring procedure, also each parameter of the testing procedure (acquisition and processing) having an influence on data has to be considered.

This is the case of SWM: as it has been shown in chapter 2 the dispersion curve is an apparent curve resulting from the modal superposition and strongly depending on some acquisition parameters.

The forward model hence has to consider the these acquisition parameters to produce synthetic data comparable with the experimental ones. In the forward model no noise is present, and so the number of receiver does not affect the result: on the contrary, the actual array length is very important since it affect the mode separation or superposition.

The synthetic data, then must be produced considering this parameter: if different arrays are used in order to obtain different ranges of the curve, the same procedure has to be followed also in forward modelling.

However, if one mode can be clearly and surely identified in experimental data, the forward model can be simplified and has to simulate only modal curves, which are a characteristic of the site not depending on acquisition. Anyway, this situation is not so frequent in experimental data.

4.3.5.2 The forward model for SWM and the test parameters In the forward model used for SWM inversion (chapter 2), the modal curves are produced first (eigenvalues); then the displacement and stresses as a function of depth and frequency (eigenfunctions). Finally the Green function allows to compute the displacements due to modal superposition, at the free surface at every distance from the source. In such a way each testing configuration can be simulated: even-spaced arrays, non uniformly spaced arrays, SASW data with different configurations. In all these cases, the synthetic data are processed with the same procedure of experimental displacement measured in the field.

For instance, in SASW test, the mode separation increases increasing the spacing: at high frequency, with little spacing, the modal superposition is usually strong (Figure 4.50).

Figure 4.50. If different arrays are used to get the whole desired range, and different frequency ranges are estimated

with different array length, the modal superposition will change with the frequency



4.3.6 Choice of the f-k processing

The comparison of the different processing techniques points out that most of them are equivalent, and that the most important parameters influencing the result are related to the acquisition.

In particular a multi-offset technique has a series of advantages: more information with a lower variance is obtained; the different modes, other events and coherent noise can be identified. The redundancy of the data allows a stable estimate, and even a control of the hypothesis of one-dimensional medium. The same receiver layout can supply information in a wide frequency range, and hence the required acquisition is simpler and faster.

The f-k transform is the natural tool for the Rayleigh wave analysis: this kind of wave can be easily written in f-k domain as a sum of different modes, and the spectral spreading of the wavefield is constant in f-k domain. The properties of Rayleigh waves are properly recognised when field data are transformed into the f-k domain. Of course other equivalent 2D transforms can be used with similar results.

The main limitations of the f-k processing come from the global nature of the estimate, obtained by the whole set of traces. The horizontal coordinate is neglected, and hence all 2D effects can be misinterpreted, not properly considered or simply not recognised. For instance, the attenuation of the high frequencies can be considered a 2D effect: a global estimate does not take into account that part of the information is actually not reliable. When lateral variations are present, the f-k processing gives a mean behaviour that is difficulty interpreted. To overcome these limits, a local evaluation of the information should be introduced. The procedure tested to be effective is a combination of local and global estimate: the first is mainly adopted to assess the applicability of the latter.

In the following part of the chapter the details of two different processing will be discussed: the multi offset phase inversion and the f-k processing.

The first is a quite new approach allowing the estimate of the phase velocity and the attenuation from the local properties of the propagation. It allows the control of the uncertainties from the raw data to the final estimate, it can weight the information according to their uncertainty, and directly estimates the uncertainty of the dispersion curve.

In the following the f-k processing is discussed in detail, showing the different steps of the computation and of the search in the spectrum. The modal superposition and its effects are shown, and the possibility of distinguishing superimposed modes by means of the spectrum deconvolution is mentioned. Finally a processing technique allowing the use of a local estimate and a local filtering procedure within a f-k analysis is presented.

Chapter 4.4 THE MULTI OFFSET PHASE INVERSION


4.4 THE MULTI OFFSET PHASE INVERSION

In this section the MOPI is described in detail. Its limits and advantages are shown, and a field data example is presented.

In the spectral decomposition of the seismograms the information is the amplitude and the phase of each trace at each frequency, and this information is treated differently from the different processing techniques. The signal quality evaluation is performed only to select the frequency ranges where the information is supposed to be reliable, and some empirical rules are used during acquisition to avoid undesired effects depending on the near offset or on the far offset.

A direct evaluation of the quality of this elementary information, i.e. the amplitude and the phase of each spectral component at each offset, is performed: the reduction of the uncertainty of the final estimate of the velocity and attenuation needs a filtering and weighting of the data in frequency – offset domain. The estimate of the propagation properties is made considering the local properties, and so the lateral variations can be identified.

This procedure (Strobbia and Foti, 2003) can be considered an extension of the 2 receivers phase difference analysis (Dziewonski and Hales, 1972) and is based on the simple relationship of the phase and the amplitude as a function of the offset. As it has been mentioned above (equation 4.54), the wavefield of Rayleigh waves can be written in terms of modal amplitude and phase as a function of the offset and the frequency

The amplitude is written as

( ) ( ) ( )( )

xeRIxA

xωα

ωωω−

=, (4.63)

and this relation allows the estimate of the attenuation coefficient α from the experimental amplitude and offsets, for each frequency.

The linear relation between phase and wavenumber

( ) ( ) ( )ωϕωωϕ Ixkx +⋅−=, . (4.64)

can be used to estimate the unknown wavenumber at each frequency, from the experimental phases and offsets.

The processing can be described as follows:

• different records of different acquisitions are transformed into frequency domain

• the statistical distribution of the phase and of the amplitude is evaluated, at each offset and at each frequency

the phases and the amplitude are processed separately

• the phase is unwrapped to a full-phase representation

• the wavenumber is estimated with a weighted least squares inversion of the experimental phases: the weights are computed from the statistical distribution of the phases

• the relation between phases and offset is used to test the hypothesis of one-dimensional medium: the residuals of the inversion can be used for a statistical test



• the attenuation coefficient is estimated with a weighted least squares inversion of the experimental amplitudes

4.4.1 Statistical analysis of the phase and amplitude

The analysis of different records has the aim of evaluating the effects of random noise on data: the coherent noise, as it has been shown, cannot be evaluated or reduced by repeating the acquisition since it repeats itself too.

Different separate acquisitions are needed: the different seismograms are at first transformed into frequency domain (Figure 4.51). The same acquisition parameters should be used in acquisition, and the same spectral estimate in the transform: the obtained f-x spectra should have the same dimensions.

Figure 4.51. The raw seismograms (time- offset) and their amplitude spectra (frequency- offset)

The complex spectra are then analysed: at each frequency and at each offset the amplitude and the phase are considered. Both for the amplitude and the phase the mean and the standard deviation are computed.

This task is almost straightforward for the amplitude data, even if a normalization considering the source variability has to be performed: on the other hand the periodic behaviour of the phase leads to some complications.

The experimental phase is obtained in a modulo-2π representation from which an unwrapped phase vs. offset function has to be obtained. If the unwrapping is applied before the statistical analysis to the different records, the resulting distribution of the unwrapped phase can be strongly influenced by the unwrapping algorithm, which can amplify some little differences. On the other hand, to process phase data its periodicity has to be considered, especially with low-variance data: otherwise the real distribution can be misinterpreted. For instance, the two distributions of figure 4.52 apparently have the same mean value if the periodic behaviour of the phase is not accounted for.



Figure 4.52. The phase has to be analysed considering its periodic behaviour: the risk is, for cases such as in the right

one, to compute a wrong mean and overestimate the variance.

The positive and negative values of the phase can be analysed separately, and the parameters of the two distribution are compared with those of the overall set of data. If the two variances are significantly lower than the overall variance, the distribution is corrected by shifting the positive or the negative values.

The standard deviation of the raw phases can be directly plotted and analysed: the resulting image (Figure 4.53) shows the region of the f-x domain where the phase information is reliable. High values of the standard deviation of the phases correspond to poor information, and low reliability.

In particular, a uniform distribution of the phase in the interval (-π, +π) , corresponding to a complete lack of coherence and random behaviour of the phase, would give a variance of 3/2π . Values of the variance close to this limit threshold indicate a poor information of the traces in that frequency range.

Figure 4.53. Variance of the phase. The attenuation of the high frequencies in far offset produce a

poor signal quality, and a high phase uncertainty

The mean value of the phases cannot be plotted as it is, since an unwrapping is necessary. The following figure (4.54) shows the measured phases between 17 and 23 Hz for the 24 traces. It is clear that the oscillating behaviour of the phase does not allow a clear identification of the velocities



Figure 4.54. Experimental phases as a function of the frequency and the offset.

4.4.2 Phase processing

4.4.2.1 Phase unwrapping The unwrapping of the experimental phases is an important step of the processing: this step may be difficult with noisy data (Poggiagliolmi et al, 1982).

An unwrapping algorithm is hence applied to the mean values of the phases: common unwrapping algorithms usually change absolute jumps greater than π radians to their complements to 2π. A variety of unwrapping algorithms have been reported in the literature (Oppenheim and Schaefer, 1975). When the phase difference between adjacent receivers is not too big, this simple definition works well. The example below (Figure 4.55) shows the unwrapping of the phases at 20Hz of the spectrum in figure 4.54.

Figure 4.55. Experimental raw phase (left) and unwrapped phase (right) at the

frequency of 20Hz of the spectrum of figure 4.54



The phase difference between two adjacent receivers depends on the wavenumber and on the spacing:

xk ∆⋅=∆ϕ

and can be greater than π radians when

π>∆⋅ xk

Hence x

k∆

>π correspond to wavenumbers greater than the critical Nyquist wavenumber

corresponding to a spatial sampling of x∆ (see chapter 3).

If the wavenumber is smaller than this critical value, the unwrapping procedure is applied straightforwardly. In cases such as the one depicted in figure 4.56, the unwrapping must be applied with the constraint of positive velocity (and wavenumber).

This allows to reach NYQUISTMAX kk 2= , which is the same result obtained before (chapter 3), recovering the quadrant of the negative wavenumbers of the f-k spectrum.

Figure 4.56. The wavenumber at this frequency is slightly greater than π/2. The phase difference between adjacent receivers is slightly greater than π

4.4.2.2 Wavenumber least squares estimation The unwrapped phase (Figure 4.57) is the analysed: an inversion of the experimental phases is performed considering that the relation between the phases and the offsets is linear in a laterally homogeneous medium (equation 4.64).



Figure 4.57. The experimental phases are inverted using a linear model

The linear equation that is considered is

oii xk ϕϕ +⋅= (4.65)

where the phase iϕ at the offset ix depends on the wavenumber k

and the above relation can be written for all the N experimental points.

+⋅=

+⋅=+⋅=

0

022

011

ϕϕ

ϕϕϕϕ

NN xk

xkxk

M (4.66)

In matrix form this is

MG ⋅=Φ

where [ ]TNϕϕϕ ,...,, 21=Φ is the vector of the experimental phases

[ ]TkM 0,ϕ= is the vector of the unknown model parameters, and

=

1

11

2

1

MM

Nx

xx

G (4.67)

is the data kernel matrix (Menke, 1989).

The model parameters can be estimated in the least squares sense by the pseudo-inverse gG −

Φ⋅= − gGM



( ) TTg GGGG ⋅⋅=−− 1 .

This simple procedure is very similar to other processing techniques used for surface wave analysis. When only two traces are processed, the pseudo-inverse is simply the inverse matrix, and the estimated wavenumber is directly obtained as

Xk

∆∆

=ϕ

that is the classical 2-station phase difference approach (Dziewonski and Hales, 1972) traditionally used in the SASW method. The relation of this procedure with the f-k processing is shown in appendix C.

The great advantage of the proposed procedures the possibility of reducing the effects of low quality traces on the estimate of the phase velocity: far traces, in the high frequency ranges, do not have any coherent signal, and so do not contribute to the information, but degrade the information.

As it can be seen in figure 4.57, far traces have higher uncertainties, because of the lower energy and lower S/N: the amplitudes of the traces do not affect directly the estimate, and so weights have to be introduced to control the variance.

4.4.2.3 Weighted least squares wave number estimation Figure 4.58 depicts the experimental phases as a function of the offset for 6 different frequencies: it is evident that the variance is strongly dependent on the distance. Again it can be noticed that higher frequencies (greater slopes) show higher uncertainties in the far offset. This property of the data can be used to weight the least square estimate so that unreliable data do not corrupt the final result.

Figure 4.58. The phases as a function of the offset for 6 different frequencies. The high frequencies with greater slopes

shows higher uncertainties in the far offset

f



The sum of the squared errors that is minimized does not consider that the distance from an experimental point has different meanings depending on the uncertainty on that point. The uncertainty on the estimate is reduced, as it can be intuitively understood, if the weight of unreliable points is lower than the weight of reliable points. This avoids that unreliable points, lying out of the trend, pull the interpolating line in their direction. This is not, of course, only the case of the outliers: the use of weights, from a statistical point of view, is the way to get the best estimate of model parameters. When the distribution of the phase values assumed by each experimental point is normal, it can be demonstrated (Menke, 1989) that the best estimate is obtained using as weights the inverse of the variances of the experimental phase values measured at each point.

In the case of a distribution such as that of the phase, a simple definition of weights is not possible: when the variance is 3/2π radiances, the point has zero reliability.

A preliminary filtering can be made, in order to assign a zero weight to the unreliable portions of the f-x phase spectrum. Referring again to the example shown before, the following figure (4.59) shows a example of muting: the processing algorithm actually automatically excludes the traces that, at each frequency, has a standard deviation greater than a fixed threshold .

Figure 4.59. Experimental variances of the phase, and manual filtering of unreliable portions

Another possible approach is the use of the spectral amplitude above the noise level, that has a good correlation with the variance of the phase.

After the muting procedure, the weight are introduced into the computation, and the pseudo-inverse g

WG − can be obtained as (Tarantola, 1987)

( ) WWGGWWGG TTTTgW ⋅⋅⋅⋅⋅⋅=

−− 1 (4.68)

where W is the diagonal matrix containing the weights.

This approach reduces the variance of the estimated parameters (here the wavenumber). Since the uncertainty of the experimental data are known, the uncertainty of the estimated parameters can be estimated considering that

Φ⋅= − gWGM (4.69)

implies, under the assumption of normally distributed and independent data, that



22 2 Φ− ⋅= σσ g

WM G . (4.70)

where 2Mσ and 2

Φσ are respectively the covariance matrices of the model parameters and of the data, and g

WG −2 is the matrix containing the squares of the elements of gWG − (Santamarina and

Fratta, 1998).

Adopting the above procedure, the mean value of the wavenumber and the associated uncertainty canbe estimated at each frequency of interest (figure 4.60)

Figure 4.60. The wavenumbers and their uncertainty with the frequency

It has to be remarked that the above variances do not consider the effects of coherent noise in the raw seismic data, and hence the uncertainties can be underestimated.

The wavenumber is then used to compute the phase velocity V, and whose uncertainties can be estimated with a first order approximation

kfV ⋅

=π2 (4.71)

KV kf

σπ

σ 2

2 ⋅= (4.72)

or more accurately considering the probability density of the phase velocity corresponding to a normal probability density of the wavenumber (Figure 4.45).

The following figure 4.61 shows a dispersion curve with the associated uncertainties, obtained by direct propagation of the variances of the experimental phases. The standard deviation, however, does not describe the shaper of the probability density function of the phase velocity.



Figure 4.61. The dispersion curve is obtained from the best estimate and the experimental uncertainties are statistically

estimated

4.4.3 Testing the model

The forward model considered for the interpretation of surface wave data assumes a layered subsoil, where the properties can vary only in the vertical direction. Lateral variations of the soil properties cannot be fitted in such a framework, hence it is very important to have the possibility of checking this assumption; to this aim, several test can be implemented within the proposed method.

The analysis of the residuals of the regression process may help in understanding if some hypotheses have been violated: the easiest test concerns the value of the misfit function at its minimum (Tarantola, 1987). If this minimum is large, compared with the individuals uncertainties of the phases, there is the risk of model errors: the phases are not a linear function of the offset.

A qualitative approach is based on a visual examination of the result: for example, the presence of lateral inhomogeneities produces deviations of the experimental phases from the theoretical linear behaviour. Experimental data of figure 4.62 show an abrupt change of the slope of the experimental phases: they are likely associated to a shallow lateral variation of the soil properties in a test site.



Figure 4.62. Experimental data showing an abrupt change of the slope of the phases in the middle of the array

More rigorous statistical test can be implemented. If each of the N individual prediction error is a Gaussian independent random variable x, each with zero mean and unit standard deviation, the sum of their squares is a random variable with chi-squared probability density with N degrees of freedom.

∑=

=N

iix

1

22χ

If the different error have different variances, the overall misfit can be predicted in a different way, weighting with the covariance matrix C and including the vector of the means µ , to give a random variable with the same probability density

( ) ( )µµχ −−= xCx T2

Too large values of the experimental misfit, with respect to the probability density defined above, may suggest the violation of some hypothesis.

4.4.4 Identification of the lateral variations

This approach could also allow the identification of lateral variations: the experimental estimate of the lateral variation could be a first step of a two-dimensional surface wave method for engineering applications. The main problem in this context is in the interpretation: an arbitrary lateral inhomogeneity can be handled only by numerical models. When the lateral variation is much smaller than the vertical, a procedure for the surface wave computation is proposed by Keilis-Borok et al. (1989).

The identification of lateral variations in dispersion data has been described by Al-Eqabi and Herrmann (1993) for a large scale data set. When lateral variation of the dispersion are present the array is partitioned into spatial segment of uniform dispersion, and the result is a dispersion curve specific to each segment: the dispersion curve are inverted separately. Al-Equabi and Herrmann



(1993) showed the effectiveness of their reconstruction generating synthetic seismograms for the obtained laterally varying earth model, under the assumption that no mode conversion occurs at the segment boundaries (Keilis-Borok et al., 1989).

The approach here proposed, from a theoretical point of view, allows the partitioning among segment of uniform wavenumber: the segments can be of different extents depending on the wavelength. A kind of phase velocity pseudo-section could be obtained: the phase velocity can be plot as a function of the wavelength and of the coordinate along the array, as it is sketched in figure 4.63

Figure 4.63. Pseudosection of phase velocity. If lateral variations are present, it is possible to partition the array in

segments of homogeneous properties. A kind of pseudo-section can be obtained.

As it has been mentioned, the interpretation (or inversion) of these data cannot be performed with a one-dimensional forward modelling. But if lateral variation are present, the global processing of data can give wrong results, as it will be shown in chapter 5. When there are sharp lateral changes the global information obtained by the processing of the whole array may yield two dispersion curves (Mockart et al, 1988). If the changes are gradual, the processing will give a dispersion curve not clearly defined, that is an average dispersion curve.

4.4.5 Amplitude Processing

In the amplitude processing, the possible variation of the source power has to be taken into account: this can be the main reason for the variations of spectral amplitudes. A normalization procedure is at first performed on raw data, then the single values of amplitude are analysed. Figure 4.64 shows the attenuation of the high frequencies in far offset and the correspondent increase of the uncertainty.



Figure 4.64. The mean (left) and percent standard deviation (right) of the amplitude as a function of frequency and

offset. It is evident the attenuation of the high frequencies in far offset

The amplitude are then inverted assuming as model the exponential law

( ) ( )( )( )

11

1

,/,xx

exAxAxx

−=

−− ωα

ωω

and estimating the attenuation coefficient α at each frequency. The procedure is quite similar to that describe above for the phase, but an iterative inversion has be performed. The noise level has moreover to be included, and the additive nature of the noise amplitude has to be considered.

4.4.6 Influence of the higher modes

In the above only a single mode of propagation has been considered: indeed the described approach holds only if one mode is strongly dominant. The interference between different modes, when they have similar energy in a certain frequency band, can strongly change the relation between the phase and the offset, and between the amplitude and the offset. An example has been given in figure 4.47: the amplitude may be zero and the phase have jumps at certain distances from the source.

If the array is long enough to separate the different modes, a preliminary filtering allows the processing of the single modal contributions: the procedure can be repeated for the different modes. If the array length does not allow the separation, each processing technique will give an apparent dispersion curve influenced by the modal superposition, and the same result can be obtained by the phase difference inversion.

Chapter 4.5 THE f-k PROCESSING


4.5 THE f-k PROCESSING

This section is devoted to the f-k processing of surface wave data. The different aspects of the processing are discussed, the parameters for the computation are given. In particular the search of maxima in the spectrum is discussed: the spectrum- window deconvolution is proposed as a technique to separate superimposed modes.

The f-k processing has been introduced in section 4.2 and the limitations of the acquisition have been presented in chapter 3 referring to an f-k processing; here some further details are discussed, and the parameters used for the computation are presented. Then an alternative approach for the computation as sum of elementary contributions is presented.

The flow chart of the f-k processing is presented in figure 4.65.

Figure 4.65. The flow chart of the f-k processing of surface waves

4.5.1 The f-k spectrum: zero padding and windowing

The f-k transform is simply a 2D Fourier transform: a discrete function of two variables, time and offset, is transformed into a discrete function of two other variables, frequency and wavenumber. The two transforms in the two directions are independent, and the properties and parameters of each of them can be discussed separately.

crosscorrelation time shift and

DATA (f-x) N records

f-x FILTER DESIGN

DATA (f-x) 1 stack record

f-x DATA FILTERED

FILTERING f-x

DATA (t-x) N records

WINDOWING 1,2, …, m

FFT (t f)

DATA (f-k) 1 stack record

MAXIMA SEARCH

SPECTRUM DECONVOLUTION

DISPERSION CURVE Phase Velocity-

Frequency-

AMPLITUDE CORRECTION

DISPERSION CURVE Phase Velocity-Energy -

Frequency

Comparison of different records

Selection of records to process

Muting in time-offset

DATA (t-x) N records

DATA (t-x) 1 stack record

FFT (t f)

FFT (x k)



4.5.1.1 time The time is transformed into frequency: this first transform has the aim of separating the different spectral component of the signal, to compute then the properties of each of them.

The zero padding increases the frequency resolution of the spectrum, when the signal length is not sufficient, and this can be useful specially at low frequency.

A frequency resolution of 0.25 Hz, for instance, can be obtained with a 4 s record: padding can be necessary with shorter records. When the time length of the record is not sufficient, the number of samples in the padded signal can be computed as

tfN pad ∆⋅∆

=1

where f∆ is the desiredfrequency resolution and t∆ the sampling interval.

The number of samples can be very high with little t∆ , the dimension of the spectrum increases, and the computation becomes heavy. An under-sampling of traces, or decimation, can often be performed because the very high frequencies are usually not necessary.

If the signal is not monochromatic, the spectrum has a frequency content which does not shows spikes: the observation on a limited time window does not introduce strong distortion of the actual behaviour. The windowing in time domain often is not strictly necessary, if the signal is not artificially truncated. The figure 4.66 shows a signal recorded at 6 m from the source, a sledgehammer, and its amplitude spectrum computed without any windowing.

..

Figure 4.66. A trace in time domain and in frequency domain, transformed without any windowing

4.5.1.2 offset The transform of the offset has the aim of identifying the wavenumber with which the energy propagate at each frequency. This transform can be quite problematic, since the theoretical behaviour has spikes (because the propagation admits only few modal wavenumber at each frequency), and needs long arrays to be observed. The effect of the limited window has been



discussed in chapter 3, and here the aspects that can be useful in the processing are discussed in detail.

From a theoretical point of view, in a laterally homogeneous medium, the spectrum should have M spikes corresponding to M modes: if a window of length L has been used, the convolution of the real spectrum with the window spectrum can be observed. The mode separation possibility depends on the actual array length, that influences the spread of the window spectrum.

However the computation requires a zero padding, even if this does not increase the resolution. The f-k spectrum can be evaluated numerically only in a limited number of points, that is in a limited number of wavenumber: the computation with a finite zero padding is a discretization of the theoretical spectrum obtained with an infinite zero padding (chapter 3.4.3, figures 3.41 and 3.42 ). The computation in 2048 values is usually enough to get a continuous behaviour of the dispersion curve.

4.5.1.3 Windowing If data are simply transformed without any other preliminary operation, as it has been mentioned before, a boxcar window is introduced. To recognize these effects, different kind of window can further be applied, to compare the position of secondary maxima. The results can be compared to assess if secondary events are due to the window.

Two windows that reduce the amplitudes of the ripples are the Hanning window and the Hamming window.

The Hanning window of n samples is defined by

[ ] 1,...,1,01

2cos15.01 −=

−−=+ nk

nkkw π

The Hamming window of n samples is defined by

[ ] 1,...,1,01

2cos46.054.01 −=

−⋅−=+ nk

nkkw π

Figure 4.67 show three windows in time domain and in frequency domain. Many other windows have been developed and can be used (see for instance Oppenheim and Schaefer, 1975).



Figure 4.67. Different windows, in time domain (top) and in frequency domain (bottom)

Hamming and Hanning windows are quite similar in time domain, but the former has lower amplitude ripples. In order to evaluate the width of the main lobe and the amplitude of the secondary lobes, it is better to plot the spectra of the windows in dB (Figure 4.68)

Figure 4.67 Comparison between the spectra of three windows

To recognize if in a experimental spectrum an event is due to a ripple, it is possible to apply different windowing, and then to compare the position of the secondary events.



4.5.2 Search of maxima

4.5.2.1 Maxima position in the spectrum In the experimental spectrum the position of maxima depends mainly on the correct position of modes in the f-k plane: this is strictly true if only one mode is present. The modal superposition, as it has been mentioned, can move the maxima position. This has to be taken into account when many modes are present, specially if they have comparable energies.

Each event introduces an energy contribution that move the position of the adjacent events. In the example below (Figure 4.69) a synthetic spectrum is computed, with three modes: a section of the spectrum is plotted, with the position of the modes and the energy density that is recorded with a 24 m array. The spectrum indicates the position of the main events.

Figure 4.69. Three modes (spikes) and the spectrum obtained with a 24 m array

However even the first maximum is moved from its correct position, and the shape of the window is distorted by the contribution of the second maximum, as it can be noticed from the zoom of the previous figure (Figure 4.70)

Figure 4.70. Zoom of the previous figure: even if modes are separated,

the position is modified by the modal superposition



The effect is of course stronger if the distance between the two modes is smaller, and stronger interferences between the windows ripples and main lobes occur: the modal superposition, as it has been illustrated occurs when the sum of the windows has a maximum which is different from the mode positions (Figure 4.71).

Figure 4.71. With the same array the modal superposition is stronger if the distance between the mode is smaller

4.5.2.2 Simple search of maxima The search of maxima, hence, has to consider these effects. Simple searching techniques may be used: if only the absolute maxima are searched, the aim is finding only the main event, and so a window having a narrow main lobe has to be preferred. In the example of figure 4.72, the same synthetic data are processed with two different windows. The boxcar window (top) has higher ripples but allow the separation of the two modes, while the Hanning window (bottom) has lower ripples but its larger main lobe produce a complete superposition of the two events

Figure 4.72. The same array length, with different windows, has a different mode resolving power. A Boxcar window

has a narrower main lobe, and separate the two modes. The position of the maxima is not correct



If, on the other hand, relative maxima are searched, a window with lower ripples has to be used: this procedure can be effective when modes are well separated and the main problem is the identification of ripples.

The following example of processing shows that a simple search of relative maxima can be effective in a case when several modes are present. The data have been acquired with 24 vertical geophones, 2 m spaced, with 2m of source offset: the source is a sledgehammer.

The seismogram is plotted in figure 4.73: different events are present and separate within the limited array length, with a broadening of the signal that requires 0.6 s to be completely recorded at the furthest receiver

Figure 4.73. Example of f-k processing. The raw seismogram, 24 receivers 2m spaced

The f-k spectrum has been computed with a 2048 traces zero padding, and recovering the quadrant of the negative wavenumbers (chapter 3.4): the maximum wavenumber is hence π radiant. (figure 4.74)

Several coherent events are present in the spectrum, corresponding to several modes, and the absolute maximum at each frequency, does not follows a single mode. A first jump can be noticed at around 30 Hz, and a second jump at around 47 Hz.



Figure 4.74. Example of f-k processing. The f-k spectrum with the position of the absolute maximum for each frequency

The search of the relative maxima gives the curves that are depicted in frequency-velocity domain in figure 4.75. It can be noticed that the absolute maximum lays on different modes in the different frequency bands.

Figure 4.75. Example of f-k processing. The position of the absolute maximum (blue asterisks) and of relative

maxima(black dots)



4.5.2.3 Spectrum deconvolution for modes identification Often modes are not well separated, and the modal superposition can play an important role: an alternative procedure can be the spectrum deconvolution.

The f-k spectrum that can be obtained processing data is the result of the convolution of the real spectrum with the spectrum of the window. Since the window is exactly known, and the spectrum is quite simple, having spikes in correspondence of the wavenumbers, the deconvolution of the spectrum can be performed.

The idea is finding the position of spikes that produce, convolved with the window spectrum, the experimentally observed spectrum. This means that, given a single maximum, it is possible to recognize if it is due to a single event or to different close events analysing its shape. Referring again to the example of figure 4.71, where two spikes superimpose, it is evident that the shape of the “bell” does not match with the shape of a single window (Figure 4.76)

Figure 4.76. Principle of the deconvolution: the window spectrum shape is known, and allow to identify superimposed

windows

A simple spectral deconvolution can be applied: the spectrum and the window are transformed, then they are divided, and finally inverse-transformed. With synthetic data this procedure is efficient

The principle is that the real unknown spectrum is convolved with the window spectrum to give the observed experimental spectrum, and this operation becomes a multiplication when dealing with transformed quantities: the unknown real spectrum can be obtained with a division between the transformed quantities. Referring to figure 4.77, the experimental spectrum A and the window spectrum B are transformed to give SA and SB .These are spectra of the spectra. The spectral ratio SA / SB is computed, and this latter function is inverse transformed to give the real spectrum.



Figure 4.77. Scheme of the deconvolution of the spectrum

The window has to take into account all the effects influencing real data: geometric and intrinsic attenuation at first. With real data some other steps are necessary (whitening for instance), but the first results that have been obtained in the processing of field data encouraging.

4.5.3 f-x filtered f-k spectrum

The overall seismogram, as we mentioned before, has different regions where the reliability of the data is different: a processing taking also the offset in account is needed to enhance the data quality and reduce the effects of poor traces. As it has been mentioned, the far traces can be particularly noisy, especially in the high frequencies. One possibility is to split the seismograms in elementary contributions of two traces seismograms (Strobbia, 2002).

The Fourier transform is a linear operator, hence F2D(d1(x,t))+F2D(d2(x,t))=F2D(d1(x,t)+d2(x,t))

If we decompose the original seismic gather in the sum of two traces seismograms, without any overlap, we can compute the overall f-k spectrum as sum of the complex elementary spectra (Figure 4.78) after the geometrical spreading correction. This procedure is computationally heavier than transforming directly the whole data set, in particular when a long zero padding in space domain is needed, but it allows a better control and data quality assessment: bad quality traces can be discarded, or considered only in the frequency range where their contribution is not noisy.



Figure 4.78. (a) Two-traces seismograms: offset (m) and time (s). (b) Elementary spectra. Wavenumber(rad/m) and frequency (Hz)

The elementary seismograms and elementary spectra can be analysed by means of the coherence function, in order to asses the signal quality of pairs of receivers: for each couple of traces, the coherence function is computed, and gives indications about the frequency ranges that have to be excluded. Another possibility, as shown above, is the evaluation of the spectral amplitude as a function of the frequency and of the offset: this procedure indicate the regions where the signal quality is high.

This approach of selecting couples of traces is completely equivalent to a muting performed in f-x domain before the f-k transform.

The example below shows the effectiveness of the procedure in increasing the quality of the estimate at high frequency. The seismogram is depicted in figure 4.79, left: the acquisition has been performed with 24 vertical geophones, 0.5 m spaced, 1m of source offset. The seismogram is transformed into f-x domain (figure 4.79, right), where the energy is depicted in dB and clearly show the attenuation of the high frequencies in the far offset.

Figure 4.79. Seismogram in time offset and frequency offset domain. The poor signal regions are easily identified

a)

b)



In the new domain, the data can be filtered, so that only coherent event are retained. In the high frequency band, of course, the far offset traces are discarded. The second transform is then applied, and a filtered f-k spectrum is obtained. The seismogram here depicted is processed with the overall spectrum and with the filtered spectrum, and the two resulting dispersion curves are depicted in figure 4.80. In this case the main advantage is in the high frequencies, which are mainly influenced by shallow layers: here with the filtered spectrum a stiff top layer (pavement) can be well identified, while the uncorrelated noise does not allow the identification of its properties when all the traces are considered. Even if the pavement is not the target of the test, a reliable estimate of its properties is necessary also for the inversion of the low frequency portion of the dispersion curve.

Figure 4.80. The dispersion curve is estimated from the whole seismogram (left) and from the f-x filtered

seismogram(right). The filter has muted the high frequencies in the far offset.

4.5.4 Uncertainty estimate

One possible approach to estimate the uncertainty of the dispersion curve is the separate processing of the different shots, for each shot position. If the acquisitions are recorded separately, it is possible, after the processing, to apply all the steps to each file, with the same parameters, and to compare the results.

What is obtained is an indication of the dispersion of the phase velocities with the repetition of the acquisition: the distribution of the processed phase velocity is obtained (figure 4.80).



Figure 4.81 The dispersion curve of the averaged seismogram is depicted together with the individual dispersion curve

of each single shot.

It is interesting to notice that in the low frequency band, the experimental distribution of the phase velocity seems to be asymmetric (Figure 4.80): this is in good agreement with the theoretical distribution of the phase velocity, as discussed in paragraph 4.3.2.5, and as shown in figure 4.44.

The phase velocity of the average seismogram is different from the average of the phase velocities.

Chapter 4.6 THE COMPUTER PROGRAM FOR THE PROCESSING


4.6 THE COMPUTER PROGRAM FOR THE PROCESSING

In this section is briefly described the computer program, implemented in Matlab, that allows the interactive processing of surface waves with the different techniques described in the above.

The program allows the processing of seismic data to extract the dispersion characteristics and the attenuation. The different steps of the processing are visualised and saved as Matlab variables, and can be exported in numeric and graphic format.

• Raw seismogram are imported and converted into matrices (each trace a column), in the time-offset domain. Records can be analysed by crosscorrelation to compute and correct the trigger time shift if any.

• Single traces and the whole seismogram can be transformed into frequency domain, time-frequency domain, frequency-wavenumber domain. The slant-stack can be performed, and the ω-p transform is available too. Different windows can be applied before the transformation.

• One-dimensional and two-dimensional filtering can be performed: manual muting in the different domains is possible, manual picking.

• Statistical analysis of the different records, to evaluate the distribution of different spectral quantities (amplitude and phase as a function of the offset and the frequency).

• A chosen processing procedure can be applied to an averaged seismogram, and the same procedure is automatically applied to the single acquisition to evaluate the uncertainty.

• Automated search of absolute and relative maxima within assigned boundaries can be performed in the different domains.

• A complete f-k processing is easily performed: data are transformed into f-x, analysed to identify good quality regions, filtered, different windowing are applied and the f-k transform is completed, then different search of maxima are used to extract the maximum of information about modal curves.

• The multi offset phase inversion is implemented, following the steps described in 4.3.


5 CHAPTER 5

Inversion

This chapter, after introducing the basic concepts of the inverse problem theory, discusses the inversion of surface waves data, the reliability of the results, and the effects of possible errors. The inverse problem theory is presented focusing the attention manly on the uncertainty and information content of data, the resolution and variance of the result, the parameterisation of the model, the data error and model error. Then, for the surface wave data inversion, some considerations about the data to be inverted and the model are presented: which set of data is the better for a certain target, and given a set of experimental data which parameters can be reliably estimated. Some possible dramatic errors coming from the misinterpretation of some data feature, particularly due to the multi mode nature of surface waves, are shown.

In this chapter

The basic concepts about the inversion: solution techniques for inverse problems; uncertainty and information in data; evaluation of the information content; resolution of data; resolution of model, and variance of the model parameters; parameterisation of the model and simplicity, data error and modelisation error; testing the model.

The inversion of surface wave data: inversion techniques for the surface wave data; the factors influencing the solution; considerations on the data to invert to increase the information and reduce the uncertainty, without replicating the same information, considerations on the model parameters, to find an optimal set of parameters to estimate with the maximum reliability. The errors deriving from the misinterpretation of data, especially due to the multi mode nature of surface waves

The use of constraints and the joint inversion of surface waves with other data



5.1 INTRODUCTION

5.1.1 The inversion as the last step of SWM

In geophysics the term inversion means the “estimation of the parameters of a postulated earth model from a set of observations” (Lines and Treitel, 1984).

The inversion is the last step of the SWM, and is the step that produces the final result. It consists of estimating the parameters of a layered model from the dispersion curve measured at a site (Figure 5.1). The inversion transforms the properties of the propagation into the properties of a layered soil model. The SWM inverse problem is non-linear, and often mix-determined.

Figure 5.1. Scheme of the inversion of SWM: the soil properties are inferred from

the dispersive properties of Rayleigh waves

The unknown system is parameterised and is assumed to be completely described by a finite set of M model parameters: the model allows to predict the values of N measurements, that are called, in the following, data (Menke, 1984). The model is the set of relations describing the observed phenomenon: the model parameters are the value that allow a quantitative description of the object that is studied.

The unknown model parameters are the geometric and mechanical properties of the layers constituting the subsoil: for each layer i , the shear velocity, the Poisson ratio, the mass density, and the thickness, iiiSi hV ,,, ρν .

[ ]LLSLSS VhVhVm ρνρνρν ,,;...;,,,;,,, 22221111= (5.1)

The number of unknown model parameters, when L layers are considered, is 14 −= LM , because the last layer is the half space and its thickness is assumed a priori as infinite.

The measured data are the Rayleigh wave velocities for N discrete values of frequency,

NRRRR VVVVd ,...,,, 321= (5.2)

The model parameters and the data are functionally related as illustrated in chapter 2, and for each frequency a function of the model parameters gives the Rayleigh velocity. The forward problem is solved with the series of algorithms allowing the simulation of the Rayleigh wave propagation that has been described in chapter2.



( )( )

( )LLSLSSNRN

LLSLSSR

LLSLSSR

VhVhVGV

VhVhVGVVhVhVGV

ρνρνρν

ρνρνρνρνρνρν

,,,...,,,,,,,,

,,,...,,,,,,,,,,,...,,,,,,,,

22221111

2222111122

2222111111

=

==

M (5.3)

These relations are the mathematical statement of the forward problem, that consist of computing the data when the model parameters are know, in the case of SWM computing the Rayleigh velocities when the soil properties are known (Figure 5.2).

The inverse problem, on the contrary, consists of estimating the model parameters when the Rayleigh velocities have been measured.

Figure 5.2. Model parameters and data for the surface wave inverse problem

The set of model parameters described above completely represents the layered model: the velocities can be in general complex, thus allowing to account for material attenuation and viscoelastic behaviour. All these parameters have to be specified to solve the forward problem: however, the solution is more sensitive to some of them, in particular the shear velocity and the layer thickness. Therefore in the inversion, mass density and Poisson ratio are often assumed a priori on the basis of other information The influence of earth parameters on the dispersion curve is treated for instance by Aki and Richard (1980). The relations of the SWM are not linear: the problem is hence a non-linear inversion.

The simpler interpretation, which is not an inversion, of the dispersion curve is the 3/λ or 2/λ interpretation: it simply consists of assuming that the shear velocity is the 110% of the Rayleigh velocity, and the depth is equal to one third or one half of the wavelength (for instance Abbiss, 1981). An other approach that has been proposed (Stokoe et al. 1984), based on a proper forward modelling, is the trial-and-error inversion. Starting from an initial guess, the theoretical dispersion curve is compared to the experimental dispersion curve, and the profile is modified until a good fitting is reached. Automated iterative inversions have been used with the least squares criterion by many authors, with different updating criteria, damping and weighting (Herrmann, 1994; Lai 1998; Xia et al, 1999; Schtievelman 1999). Genetic algorithms (Al-Hunaidi, 1998), joint inversion with other techniques (Hering et al, 1995; Comina et al., 2002), and coupled inversion of phase velocity and attenuation (Lai, 1998) have been applied.

Often only the first mode is considered, but interesting examples of inversion of several modes (Gabriels et al., 1987) and apparent curve have been discussed.



5.1.2 Importance of the inversion

The data to be inverted may be quite obscure, and they are only a synthetic description of the way in which surface waves propagate in the tested site: it is not a straightforward task to say a priori which are the properties of data that allow the best knowledge of the soil.

The model parameters, on the contrary, are the answer to our questions about the subsoil, and it is clear which properties this answer should have: it should be unique, complete and reliable. These are the requirements of the test and should be the properties of the result of the inversion.

From the desired properties of the result, considering the inversion, the properties that the data should have, in terms of variance and information, can be obtained. Some aspects of the design of the acquisition and processing can be demonstrated.

On the other hand, when data have been collected, it is possible to evaluate the reliability of the inferred model parameters, the applicability of the model, the parameterisation, both for the limits of the domain (in depth) and the discretization within these limits (the number of layers). Well known problems in the inversion, when only data are considered, are the related items of non-uniqueness, ill-posedness, and instability.

Furthermore, we don’t want only to get the best model from data, but we want also to evaluate the uncertainty of the result. “A parameter estimation or inversion procedure is incomplete without an analysis of uncertainties in the result” (Duijndam, 1988). The result of the inversion, in the Bayesian approach, is the a posteriori probability density function of the parameters, but anyway some indications on the uncertainty is necessary.

Considering an inversion procedure able to produce the best model, considering the data, their uncertainty, the eventual constraints and the a priori information, the questions to be answered are the following:

• Which data allow the best estimate of model parameters?

• For a certain desired target at a site, which frequency range is needed? Within a frequency range, which is the minimum number of frequency points that carries the whole information, without replicating it? Which is the best sampling of the data space?

When data have been collected, and their uncertainty has been estimated, the main question refers to the uncertainty of the model parameters. Different aspects should be considered.

• Which is the uncertainty of thickness and shear velocity?

• Which is the investigation depth that has actually been obtained?

• Some model parameters are often estimated a priori, for instance the density and the Poisson ratio: is this approach always correct, and which can be the consequences of errors on the estimate of these parameters?

• The number of layers that has been chosen influences the result, and different models can be equivalent. Which is then the best parameterisation, given a set of data?

Before discussing the case of the surface wave data inversion, some of the basic concepts of the inversion will be briefly presented, to introduce the tools that will be used later.

Chapter 5.1 BASIC CONCEPTS ABOUT INVERSION


5.2 BASIC CONCEPTS ABOUT INVERSION

In this section the basic concepts of the inversion are briefly presented, to introduce the tools that will be used later to discuss the surface wave data inversion. In particular the assessment of the information content of data, the independence of data, the resolution and variance of the result, the data and modelisation error, the test of the model and the Occam’s razor are outlined.

5.2.1 Statement of the problem

5.2.1.1 Linear problems When the relations between model parameters and data are linear, i.e. each datum is a linear combination of model parameters, the coefficients ijG of the linear relations are known,

MNMNNN

MM

MM

mGmGmGd

mGmGmGdmGmGmGd

⋅++⋅+⋅=

⋅++⋅+⋅=⋅++⋅+⋅=

L

M

L

L

2211

22221212

12121111

(5.4)

The problem can be simply stated in matrix form as mGd ⋅= (5.5)

where G is the data kernel matrix, d is known and is the column vector of the N data. It is possible to solve this matrix equation to find the vector of the M model parameters m and, for this simple case, the solutions can be obtained with different approaches from different viewpoints: the length method, the generalized inverse, the maximum likelihood method (Menke, 1984).

“This ‘solution’ may be chosen in order to optimise the resolution provided by the data, the error in the solution, the fit to the data, the proximity of the solution to an arbitrary function, or any combination of the above” (Jackson, 1972).

The matrix G is very important both to find the solution by “inverting” it and to assess the information content, the resolution, the variance, as it will be shown later.

5.2.1.2 Non linear problems

The inverse problem of SWM is non linear: the equations above cannot be directly used for the inversion, and some other solution technique have to be used, based, for instance, on iteration and the local linearisation of the problem. A local linearisation of the problem about the best model, even if it has been obtained otherwise, can be used to discuss the properties of the solution.

If the functions are not linear, but vary smoothly enough, they may be expanded about some set of initial values of the model parameters m , and ignoring higher order terms, it may be written (for instance see Jackson, 1972)

j

xj

ii m

mG

dj

∆∂∂

=∆0

(5.5)



and the whole set of relations becomes

MM

NNNN

MM

MM

mmG

mmG

mmG

d

mmGm

mGm

mGd

mmGm

mGm

mGd

∆⋅∂∂

++∆⋅∂∂

+∆⋅∂∂

=∆

∆⋅∂∂

++∆⋅∂∂

+∆⋅∂∂

=∆

∆⋅∂∂

++∆⋅∂∂

+∆⋅∂∂

=∆

L

M

L

L

22

11

22

2

21

1

22

12

2

11

1

11

(5.6)

that can be written in matrix form as ∆mJ∆d ⋅= (5.7)

where d∆ is the vector of the increments of data, ∆m is the vector of the increments of the model parameters, and J is the Jacobian matrix. This equation correspond to equation 5.5 of linear problems.

This allows to write in matrix linear form the relation between an increment of the model parameters and an increment of data, and the same technique of linear problems can be used to get the pseudo inverse of this relation. Most of the considerations that are possible for linear inverse problems can apply to this linearization.

When a set of data has to be fitted to infer a set of model parameters, a first guess of the model m0 is generated, and the forward problem is solved to find a first theoretical data set d0. This data are compared with the experimental ones, and a vector of prediction errors 0

exp0 ddd −=∆ is obtained;

inverting equation 5.7, the correction 0m∆ to be applied to the initial estimate of m is obtained. This procedure can be repeated iteratively until the vector of the prediction id∆ error is small enough and hence a good fitting is obtained.

5.2.2 Solution techniques

5.2.2.1 Different approaches for the solution of linear problems Different approaches can be applied in theory to get the same solution technique (Menke, 1984). The aim is to obtain a meaningful solution, from a physical point of view, for the unknown properties of the model, which is obtained in such a way that the data are justified, according to an assumed physical law, and considering all the additional information available to the analyst. One possible way to reach the aforementioned task is the minimization of some length of the prediction error: the best model is assumed to be the one which predicts data as close as possible to observed data. Different norms of the vector of the prediction error can be used: high order norms weights large error preferentially, low order norms are less sensitive to outliers. The L2 norm corresponds to the Least Squares criterion, that minimize the sum of the squares of the prediction errors. (Tarantola, 1987)

Let’s consider the equation 5.5, (the same could be done considering equation 5.7),



mGd ⋅=

where d is the vector of N experimental data, m is the vector of the M unknown model parameters, and G is the data kernel matrix containing the coefficients Gij of the linear relations (equation 5.4).

If N data have been observed, we can define the prediction error e as estobspredobs Gmddde −=−=

where obsd is the vector of the observed values of the data, predd is the vector of the predicted data, obtained using the forward model G with the estimated model parameters estm

The different solution techniques are based on the minimisation of the vector e: the sum of the squared errors can be simply obtained with the expression eTe, and the least squares criterion is simply

[ ])()(min)min( estobsTestobsT GmdGmdee −⋅−=

which leads to the generalised inverse (Moore-Penrose inverse; Penrose, 1955),

( ) TTg GGGG 1−− =

that can invert the relation 5.5 giving the estimate of the model parameters from the observed data obsgest dGm −=

A weighted measure of length considers the a priori information about the uncertainty on single data. The application of probability theory, assuming a multivariate Gaussian distribution, allows to get the Maximum Likelihood Solution of the linear inverse problem as a minimization of a weighted measure of the prediction length, corresponding to the weighted least squares solution.

=

N

NW

eeeeσσσ

,...,,2

2

1

1

On the other hand this implies that is data are not normally distributed, then another measure of the prediction error may be more appropriate. (Menke, 1989).

Another approach focuses mainly on the operator, and not on the estimate: the linear matrix equation is inverted by means of a generalized inverse, whose properties are discussed to get the best solution, in terms of resolution and variance. The data resolution matrix and the model resolution matrix can be introduced to evaluate the properties of the solution and to design the best pseudo-inverse. The solution technique depends mainly on the information that data provide about the unknown model parameters and on the uncertainty of data: as it will be seen, also in linear problems, some of the model parameters can lack in information, and a global error minimization can produce a solution with high amplification of the variance of data. So, regularisation, weighting, and damping will be used to better control the inversion.

In the simples least-squares or “Gauss Newton” approach, the cumulative squared error is minimised. In order to reduce the difficulties and the amplification of the variance, a constraint of maximum energy of the parameter change vector can be introduced: this solution is known as damped least squares, or Marquardt-Levenberg approach, because it and was introduced by Levenberg (1944 ) and described by Marquardt (1963). The quantity that is minimized is

)min( 2 mmee TT η+

that gives the pseudo-inverse



( ) TT GIGG 12 −+η

the damping factor η bound the length of the vector of the model parameters: in iterative linearised techniques this limits the correction to the model parameters.

The details can be found for instance in Lines and Treitel (1984), Tarantola (1987), Menke (1989) and Santamarina and Fratta (1998).

These solution techniques give different results when the problems are actually mix-determined: when the model parameters are well constrained by the data, the solutions are very close.

5.2.2.2 Solution of non linear problems In non-linear problems the forward problem has non-linear relations between the model parameters and the data. It is possible to define an objective function to be minimized such as for linear problems, but it is not possible to find a closed-form solution for the minimization, and therefore the problem becomes a search problem in the model space. Two approaches can be followed: either a local search starting from an initial guess, or a global search in the whole model space. Global search procedures search a global minimum of the prediction error in the space of model parameters. Local search procedures, starting from an initial guess, generate a series of approximations that should converge to a minimum: the initial guess has however to be sufficiently close to the solution.

I. GLOBAL SEARCH

The trial and error procedure consists of: generating a solution from a first estimate; comparing the solution with experimental data; using different criteria and norms; updating the model on the basis of the residuals distribution. Of course this kind of approach can be used when a limited number of model parameters has to be estimated, and the updating criteria depend on the experience of the analysts.

If the forward model has some special properties, such as scaling properties, this can be very useful in the updating of the model, because a reduction of the unknown can be performed at some steps.

As it has been mentioned, the way by which the solution is found does not affect the possibility of assessing the properties of the solution by means of the analysis of the linearized formulation around the minimum.

The Monte Carlo approach, see for instance Tarantola (1987), search the solution with a random exploration of the model space. A large number of solutions is randomly generated, respecting bounds for each model parameter: these solutions are compared with experimental data to find the best model. This is a completely general approach, for linear and non-linear problems, for any distribution, that is not trapped in local minima, but can be expensive in terms of computation time.

From the field of artificial intelligence, Artificial Neural Networks and Genetic Algorithms are other way to solve this kind of problems (Santamarina and Fratta, 1998)

II. LOCAL SEARCH

The local search algorithm try to minimise the misfit searching around an initial estimate, and locally evaluating the misfit function, which is defined from the prediction error: the solution is moved along the direction corresponding to a reduction of the misfit.



The problem may be linearised around an initial estimate, and the solution with a Newton-type algorithm iteratively inverts the residuals. This local search algorithms try to minimize the misfit following the local decrease of the misfit function

mFd ⋅=

mJd ∆⋅=∆

dJm g ∆⋅=∆ −

The iterative procedure, starting from a initial guess, 0m , computes the final model with a simple expression

[ ]synti

gii ddJmm −=− −

+exp

1

[ ]ig

ii mFdJmm ⋅−=− −+

exp1

The pseudoinverse gJ − can be computed with different approaches: the Gauss-Newton approach minimises the cumulative squared prediction error, the Marquardt-Levenberg approach also constrains the sum of squares of the elements of the model parameter change vector. The Marquardt-Levenberg exploits the good properties of the Gauss-Newton, suitable with small errors, and the steepest descent method, more suitable with big errors (Smith and Shanno, 1971).

It is not possible to evaluate a priori the uniqueness of the solution, and the convergence is controlled by the surface of residuals.

The example below shows the “surface of residuals” for a simple case that can be easily represented. A two layer model is considered. The velocity of the first layer is known (as it could be obtained from the high frequency range of the dispersion curve) and hence only two parameters are unknown: the velocity and the depth of the bedrock. Figure 5.3 shows the right model and its dispersion curve, that is its response, with a solid black line: it also shows the boundaries of the domain of variation of the model, and the corresponding curves. The curves corresponding to models with the bedrock depth ranging from 1m to 6 m, and the velocity of the bedrock ranging from 210 m/s to 400 m/s, are depicted in figure 5.3, right.



Figure 5.3. Local search for a two parameters problem (The velocity and depth of the bedrock). In the left the right

model and the domain of variation of the parameters, in the right the obtained set of data and the one corresponding to the true model

The distance ( in the sense of L2 norm) between the right curve and the curves corresponding to the other models is computed and depicted in figure 5.3. The figure represent the surface of the residuals, and has a minimum for V=300m/s and H=3m, where it is zero.

This figure suggests the iterative approach to find the right model starting from an initial guess of the solution: following the local descent of the residuals, that is the misfit function, one can go toward a local minimum (Figure 5.4). If the initial guess is close enough to the right model, the procedure converge to the right model.

Figure 5.4. The surface of the residuals for the case of figure 5.3. Dark colours correspond to the highest residuals.



The path of the evolving solution during the iterative procedure depends on the used algorithm: the path of the Steepest-Descent, the Gauss-Newton, Marquardt-Levenberg are different, and in general the methods can lead to different solutions (see for instance Lines and Treitel, 1984).

Iterative techniques of local search use different approaches to correct the model parameters, either reducing the residuals or promoting a unit ratio: ART (Algebraic Reconstruction Techniques) SIRT (Sequential Image Reconstruction Techniques), MART (Multiplicative ART), are described by Herman (1980). The different model parameters can be updated simultaneously

Some of the updating techniques can be more suitable for SWM, where the model parameters referred to shallow layers are well determined from a limited range of the dispersion curve.

In general the solution of non-linear problems with iterative algorithm can find difficulties due to local minima: the surface of residuals may have several minima for non linear transformations, and convergence may be trapped in a local minimum.

The initial model is very important for all local search algorithms: the data pre-processing may permit the identification of the salient characteristics of the solution without formal inversion. Moreover the convergence to the right model is simpler if a way of generating reliable first model is available.

5.2.3 Uncertainty of data

All experimental data have an uncertainty that sometimes can be estimated, either statistically or on the basis of the knowledge of the measuring procedure. All the operations of the processing procedure propagate the uncertainties and produce uncertain quantities.

When uncertain data are used in an inversion process, their uncertainties have to be taken into account for two reasons: first of all, to propagate them on the final estimate of the model parameters, secondly, to use in the inversion weights that consider less important the “low quality” data.

When the shape of the probability density distributions of data is not to far from Gaussian, the standard deviation can be computed as overall indication of the uncertainties.

5.2.4 Information in data

The amount of information that is carried by measured data is one of the most important aspect in the inverse problems.

5.2.4.1 Number of Measured Data Even if the assessment of the information starts with the analysis of the number of measured points, a large number of measurements do not necessarily implies a large amount of information, as many of the data may duplicate the same information.

If N points have been measured, and the model have M parameters, for each measured point we can write one equation



MNMNNN

MM

MM

mGmGmGd

mGmGmGdmGmGmGd

⋅++⋅+⋅=

⋅++⋅+⋅=⋅++⋅+⋅=

L

M

L

L

2211

22221212

12121111

that is mGd ⋅=

The first evaluation of the problem simply consider the number of model parameters and the number of data, and so the dimension of the matrix G (N x M).

If N = M, then G is a square matrix, and the problem is even determined: it is such as searching the slope and intercept (2 model parameters) of a straight line that pass through two points (2 data). (figure 5.5)

The unique solution is simply obsest dGm ⋅= −1

Figure 5.5. A even-determined inverse problem: fitting a straight line through two points

When N > M, the problem is over-determined: this is the case of a straight line through 3 or more points. In general, if points are not aligned perfectly, this single straight line does not exist. Some criterion should be used to choose among the infinite possible straight line that don’t pass through the points (figure 5.6).

Figure 5.6. An over-determined inverse problem: fitting a straight line through 8 points

It is usual to assume that the right model parameters are those that produce the better estimate of acquired data: so the correct model is the one that minimize the prediction error, i.e. the overall distance between the predicted data and the observed experimental data.

[ ]predN

obsN

predobspredobs dddddde −−−= ,...,, 2211

If the L2 norm of this vector is minimized, this means that the quantity



( ) ( ) ( )GmdGmdeedd TT

i

predi

obsi −⋅−=⋅=−∑ 2

is minimized with respect to each mi , which means that (see for instance Menke, 1989)

0=−⋅ dGGmG TT

that is, if [ ] 1−GGT exists

( ) dGGGm TTest 1−=

This is the least squares solution of the overdetermined problem mGd ⋅= , the matrix ( ) TT GGG 1− is called the Moore-Penrose generalized inverse (Penrose, 1955)

When each datum id has its estimated variance, it is better to weight the single prediction error considering its uncertainty: the same prediction error has different meanings if the experimental point is accurate or not.

−−−=

N

predN

obsN

predobspredobs ddddddeσσσ

,...,,2

22

1

11

Figure 5.7 shows a set of experimental points with their uncertainties, and the two straight lines estimated with a weighted least squares and a conventional least squares solution.

Figure 5.7. Weighted least squares to consider the different reliability of the experimental data

From a statistical point of view the best estimate with Gaussian data is the weighted least squares (Menke, 1989): the use of the inverse of the variances as weights minimize the uncertainty of model parameters. Discussing the information content, it can be noticed that the very inaccurate points, actually, do not contribute to the information.



If N < M the problem is underdetermined: it is such as searching the straight line that passes through a single point (Figure 5.8). Infinite straight lines pass through that point, and a criterion to chose the best one has to be assumed. A priori information should be incorporated.

Figure 5.8. A under determined problem: fitting a straight line through only one point

5.2.4.2 Independence of data The number of the points is not sufficient to evaluate the information in data, because data may be not independent. A problem that apparently is over-determined may be actually underdetermined or mix-determined.

If the strict algebraic independence has to be evaluated, the rank of the data kernel matrix can be analysed. The matrix G has N rows, as the data, and M columns, as the unknown model parameters. The rank of the matrix should be maximum, i.e. M, to allow a ‘least squares’ solution.

If we look for a plane in the 3D space measuring N points that belong to the plane, we get N equations in the form

cybxaz

cybxazcybxaz

NNN +⋅+⋅=

+⋅+⋅=+⋅+⋅=

M222

111

that can be written in matrix form as

mGd ⋅=

where the matrix G (N x 3) can be written as

=

1

11

2

1

2

1

NN y

yy

x

xx

GM

When the data set consists of three points that do not coincide and are not aligned, a unique plane exist (Figure 5.9), given by the equation

dGm ⋅= −1

The condition of existence of the inverse matrix 1−G is that it has full rank: this means that points are distinct and not aligned.



Figure 5.9. An even determined problem with independent data

If we have more than 3 points, a best plane in the least squares sense can be found, but only if points are not aligned (Figure 5.10). The least squares solution exists if the inverse [ ] 1−GGT exist, and this happens if the rank of G is maximum (Santamarina e Fratta,1998). The rank is maximum in this case if there are 3 linearly independent rows, and so the matrix has rank 3.

Figure 5.10. An over-determined problem with independent data

If the points on the contrary are perfectly aligned, only two rows are linearly independent and the rank is not maximum, so the least squares solution does not exist (Figure 5.11). In this case data are not independent; even with a great number of points there is not enough information to constrain the plane, i.e. there is not enough information to determine the model parameters.

Moreover, the lack of information may not affect a single model parameter, but a linear combination of the model parameters: this is a mix-determined problem. Part of the model is over-determined, part is under-determined.



Figure 5.11. The problem is apparently over determined, but the data are not actually independent

The problem is in some way underdetermined, since the equation contains information only about some model parameters: the combination of model parameters that can be found lies in a subspace of the model parameters space Sp(m). The rest of the space is called the null space, and no information is provided by the data about these combinations of model parameters. No model parameter can span the whole space d: only a subspace Sp(d) can be satisfied by model parameters.

But, if points are almost aligned, the rank is still 3, and the solution, in the mathematical sense, exists. But data are not actually independent, since this case is not so different from the one illustrated above: the mathematical existence of the solution is not sufficient to make it significant.

In the case on the left side of figure 5.12, the points are perfectly aligned, and the solution does not exist: it is only possible to determine the slope of the plane in the direction of alignment. In the case depicted on the right side, the points are not perfectly aligned, but only almost aligned: the solution for the best plane exist, but is evident that the information carried by the points is mainly about the slope in the direction in which they are “almost” aligned.

Figure 5.12. The data can be dependent or almost dependent

Another criterion is hence needed to evaluate the information in data.



5.2.4.3 Singular value decomposition (Lanczos decomposition) The singular value decomposition is an eigenvalue decomposition which is very useful for the assessment of the linear problems and the evaluation of the information in data; it allows to identify the null space and to see how close is the problem to one of lower rank.

It extends the eigenvector analysis to the real NxM case (Lanczos, 1961). It is based on the decomposition of the kernel matrix G (N X M) in a product of three matrices

TVUG ⋅Λ⋅=

The matrix U (N x N) contains the eigenvector that span the data space S(d)

[ ]NuuuU ,..,, 21= being iu the individuals vectors, which are orthogonal to each other and may be chosen of unit length. The matrix V (M x M) contains the eigenvectors that span the model space S(m)

[ ]MvvvV ,...,, 21= being iv the individual vectors, which are orthogonal to each other and may be chosen of unit length

The matrix Λ (N x M) is a diagonal eigenvalue matrix whose diagonal elements are the singular values, usually arranged in decreasing size, corresponding to the eigenvectors of U and V.

=Λ

0000000000

0000000000000

3

2

1

Mλ

λλ

λ

L

M

L

Some of the eigenvalues can be zero, and they correspond to the eigenvectors that span the null space. The natural generalised inverse can be obtained for mix-determined problems using the singular value decomposition and considering only the non null eigenvalues.

This decomposition, moreover, allows to evaluate the information of each model parameter or combination of model parameters.

The eigenvectors represent combinations of model parameters, and the eigenvalues represent the information which is associated. This analysis of the data can for instance tell us that it is possible to determine the mean value of two parameters, but not their individual values. Considering a two parameters model, if a big singular value is associated to the sum of the two unknowns, and a little singular value is associated to their difference, this means that the sum is well determined, and the difference in not well determined.

The great advantage is that this kind of assessment is made a priori, when the position of points is known and without considering the measured values.

As an example let’s consider again the case of a plane in the 3D Space. The model parameters are a,b and c of the equation

cybxaz +⋅+⋅=

and the matrix G is



=

1

11

2

1

2

1

NN y

yy

x

xx

GM

and is decomposed with the SVD. The eigenvectors of the matrix V represent the combination of the model parameters a, b and c

Four cases are presented in the following example (figure 5.13): for all of them 6 experimental points are used to infer the 3 parameters of the plane. The four distributions of data differently constrain the plane, and the assessment of the information content is performed with the SVD.

I. CASE1

In the first case the data are well distributed, and contain information about the whole set of model parameters: the three eigenvalues are 6.068, 3.190 and 2.243.

II. CASE 2

In the second case the points are aligned, they all have the same y, and so it is not possible to determine uniquely a plane. Nevertheless in the direction x there is enough information, and so the parameter a could be identified. The lack of information affect the second parameter, i.e. the coefficient b. There is indeed a null eigenvalue, which correspond to the eigenvector (0, 1, 0), that means that the combination of the model parameters 0a+1b+0c is completely under-determined.

III. CASE 3

In the third case the points are again perfectly aligned, in the direction y = x. There will be a combination of the model parameters that cannot be determined. The null eigenvalue correspond to the eigenvector (0.707, –0.707, 0), which means that the combination 0.707 a- 0.707b is without information: in that direction, indeed, points gives no information at all.

IV. CASE 4

The fourth case is the most interesting, since points are almost aligned along the same direction as the third case. In such a more realistic situation eigenvalues are not zero, but are small. The eigenvector corresponding to the small eigenvalues says the direction where there is little information.



eigenvalues

6.068 0 0 4.434 0 0 6.209 0 0 6.425 0 0 0 3.19 0 0 2.311 0 0 2.334 0 0 2.312 0 0 0 2.243 0 0 0 0 0 0 0 0 0.11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

eigenvectors

V1 V2 V3 V1 V2 V3 V1 V2 V3 V1 V2 V3 0.597 -0.721 0.353 0.977 -0.215 0 0.701 -0.091 0.707 0.675 -0.192 0.713 0.798 0.579 -0.167 0 0 1 0.701 -0.091 -0.707 0.724 -0.0161 -0.690 0.084 -0.382 -0.921 0.215 0.977 0 0.129 0.992 0.000 0.144 0.981 0.128

Figure 5.13. Example of the information assessment by means of singular value decomposition

These concepts about the amount of information in data are very important because they allow the assessment of the reliability and uncertainty of the result. They will be applied to the SWM inversion to discuss the properties of the solution and the requirements of the data to invert. The model parameters, for the SWM inversion, will be the properties of the different layers, and this kind of analysis will allow to understand which layers can be resolved.

5.2.4.4 Effects of the lack of information in data The assessment of the information in data is an important aspect of the solution of inverse problems: in real data often the lack of information does not correspond to null singular values, but to small singular values. The risk is high: a solution, from a mathematical point of view, can be found, and so also a value for those parameters that actually are not constrained from the data.

The example of the following figure 5.14 shows an application to SWM: the singular values of a six layer model are shown sorted in descending order. They have been obtained considering as unknown only the shear velocities, and linearising the relations around the solution.



Figure 5.14. Example of the singular value for a six layer model with unknown velocities.

They decrease but are not zero

It is evident that some model parameter, or some combination of them, has few information: this result is very useful, since it says which model parameters will be difficult to determine, or which combination of them.

The reason why few information means difficulty in estimation depends mainly on the variance amplification. When the information regarding a model parameter is small, even a little uncertainty on data, results in an unacceptably large variance on the model parameter. The critical aspect is that a single parameter, or combination of parameters, without enough information can corrupt all the others.

For the kth model parameter, the variance can be estimated, referring to the SVD formalism, as (Jackson, 1972)

∑=

=

m

j j

kjk

Vm

1

2

varλ

where V are the eigenvectors and λ the singular values.

The relation above shows that little singular values can deteriorate the result producing an uncertain estimate of all model parameters: a different kind of solution should be used to extract al least the reliable information. In the case of the plane, if the points are almost aligned, we will find a little singular value relative to the direction perpendicular to the alignment. But the plane will have a degree of freedom, and the resulting estimate will show a great variance on both the model parameters a and b (Figure 5.15).

.



Figure 5.15. The lack of information can affect a combination of model parameters, not only single model parameters

It can then sometimes be useful to damp the little singular value (as mentioned in table 5.1) to reduce the variance: this damping produces, however, a decrease of resolution.

The condition number, for square symmetric matrices such as GGT , can be easily estimated as the ratio between the maximum and the minimum singular values (Santamarina and Fratta, 1998)

( )( )i

icλλ

minmax

=

A matrix is singular and cannot be inverted if ∞=k , and is said to be ill-conditioned when the condition number is very large (>1012): numerical inaccuracies occurs in the solution and errors are magnified.

5.2.5 Resolution and variance

When problems are mix-determined, i.e. simultaneously underdetermined and overconstrained some considerations about resolution and variance of the inversion are necessary: the problem can be approached with the concept of the generalized inverse.

The equation of the forward problem is always mGd ⋅=

and the solution can be found with a generalized inverse, a matrix operator gG − operating on both sides of the equation

mGGdGm ggst ⋅== −−

The generalized inverse, in the least squares sense, is

( ) TTg GGGG 1−− =

But in general, a good inverse should satisfy the following criteria (Jackson, 1972)

• The data resolution matrix should be close to the identity matrix, i.e. Ng IGG ≈⋅ −

• The model resolution matrix should be close to the identity matrix, i.e. Mg IGG ≈⋅−

• the unit covariance should be small



The data resolution matrix indicates how well data can be predicted, or how they are independent (Wiggins, 1972). When the model has been estimated with obsgest dGm −= we can predict data with the forward equation mGd ⋅= getting obsgestpred dGGmGd ⋅⋅=⋅= − .

This means that the data resolution matrix gGGS −⋅= (N x N) is the identity matrix it data are perfectly fitted; if S is not the identity matrix the prediction error is not zero. In real cases, the data resolution matrix is not the identity matrix, and each datum can be obtained as a weighted average of neighbouring real data.

The model resolution matrix indicates how well model parameters can be predicted (Backus and Gilbert, 1968). If a true set of model parameters exists, observed data can be obtained as

trueobs mGd ⋅= , and the estimated model parameters obtained with the generalized inverse are obsgest dGm −= , which means

truegest mGGm ⋅= −

Hence the estimate of model parameters can be thought as obtained from the real model parameters by means of a model resolution matrix GGR g ⋅= − (M x M) . If R is the identity matrix each model parameter is uniquely determined. In real cases R shows the resolution with which model parameters can be determined, or resolved, as weighted average of neighbouring model parameters.

Each row i of the model resolution matrix contains the weights of the weighted average of the M real model parameters giving the estimated model parameter i. An example is reported in the figure 5.33

The unit covariance matrix indicates how the covariance on data is mapped to model parameters: this propagation is function of the data kernel G and of the pseudoinverse gG − , not of the data themselves. If data are supposed to be uncorrelated and to have equal variance 2σ it can be written

[ ] [ ] gTggTgu GGGdGm −−−−− == covcov 2σ

The unit covariance matrix is independent on the actual values and variances of data.

The goodness of a generalized inverse can be measured in terms of spread of the data resolution and of the model resolution matrices, and in terms of size of the unit covariance matrix; the optimisation of the generalized inverse gives (Menke, 1989)

[ ] TTg GIGGG ⋅+=− 2η

which is the damped least squares relation, being ε the damping factor, that minimize a weighted combination of data resolution spread and covariance size.

The spread is defined as norm of the difference of the resolution matrices and the identity matrix: the spread is a measure of the off-diagonal elements.

Other considerations, for underdetermined, overdetermined, purely determined cases are presented by Backus and Gilbert (1968)

There is, however, a Trade-off between model resolution spread and variance size: one decreases increasing the other. In the minimization of the two, the weights are α and α−1 , (Menke, 1989)



( ) [ ]( )mR ucovsize1)(spread αα −+

Figure 5.16. The damping factor rules the trade off between resolution and variance (after Menke, 1989)

5.2.6 Parameterisation and Ockham’s razor

In all the above, the experimental data have been discussed to assess the information that they carry about a set of model parameters. It is however important to make some considerations about the model itself, which has only be postulated.

When a real system is tested in order to estimate its unknown properties, a mathematical forward model allowing to simulate the test in a simplified system has to be implemented. In the reality a lot of aspects and events can influence the behaviour of a system, and hence can influence the data that measure and describe its response.

When formulating inverse problems, in the parameterisation of the unknown medium, simplicity has to be preferred: the posterior probability of an hypothesis with fewer parameters is enhanced (Jefferys and Berger, 1992). In practice, the model should be the simpler, i.e. should contain only the parameters that can be estimated with the available data, considering their uncertainties. If a model parameter has a significant influence outside the measured range, or if within the measured range has an influence which is comparable with the noise effects, it should not be included. The model that can be determined depends on the measured data and on their uncertainty. Then our knowledge about the “Great Nature’s Book” (Galileo, 1632), at whatever scale, strongly depends on the kind and the quality of glasses we use to read it (L.Sambuelli, personal communication).

The evaluation of the uncertainties is important, because in such a situation, we could decide to complicate the model, increasing the number of parameters, in order to reduce the misfit: this can be a dangerous choice, since it can force the model to fit the errors of data. The obtained results can have a little significance: in the example below (Figure 5.17), with 8 experimental points, a linear law follows the trend minimizing the misfit, and moreover it filters the noise. A high order (7) polynomial perfectly fits the data: the null misfit, nevertheless, is not sufficient to give a reliable estimate of the 8 coefficients of the polynomial.



Figure 5.17. The same data set can be fitted by models of different complexity. Reducing the misfit does not always

means to increase the reliability of the result (modified, after Santamarina and Fratta, 1998)

Simple models filter data noise and extract the useful (and sometimes the only achievable) information contained in the measurements. More complicated models fit even better data points: reducing the misfit do not guarantee a higher significance of the result.

This criterion is general: in the case of SWM the parameterisation is essentially the number of layers, the depth of the half-space, the number of parameters for each layer. These aspects will be discussed in section 5.3.

5.2.7 Data error and modelisation error

In inversion problems there are two sources of errors in the system of equations: measured data are affected by experimental uncertainties and fluctuations, and the equations that relate the model parameters to the data only approximately represent the studied phenomenon. The estimated model parameters are hence obtained from noisy data with a model that only partially reflects the reality: the model parameters are then inferred from “inaccurate, insufficient, and inconsistent data”, said Jackson (1972). Two sources of errors on model parameters can be identified: data errors and modelization errors. These terms are referred to the errors in the estimation of the model parameters caused respectively by random fluctuations in data and an incorrect or uncompleted model definition. The errors of the model could be considered, with a excessively self confident approach, as coherent noise, saying that is the reality that does not follow the model (Aesopus, “The fox and the grapes”).

Data errors (Santamarina and Fratta, 1998) produce a higher misfit, and reduce the curvature around the minimum, diminishing the ability to resolve the optimum value. Model errors produce moreover difficulty in the inversion, decreases the convergence, and masks real features in data and bias the interpretation.

Sometimes experimental data cannot be “reasonably” fitted, for example if we try to fit quadratic data with a linear law (Figure 5.18).



Figure 5.18. When model errors are present, often data cannot be fitted

“Reasonable” fitting means that it is possible to test the model, when the uncertainties of data are known: if, with a given set of experimental data and a given model, the misfit is too big compared with the uncertainties, then the assumed model is not correct, as shown below.

A reliable estimate of the uncertainty is needed: in the example below (Figure 5.19), with little uncertainty the linear model is not probable (left), with greater uncertainties it passes the test. This kind of test (briefly described below) gives indications on the possibility of increasing the complexity of the model, considering the uncertainty of data.

Figure 5.19. The same experimental points are fitted by the same model: the value of the misfit is the same, but the

probability of correctness of the model is different. The right situation with higher uncertainties on data corresponds to a higher probability for the model (of course: in the night (the darkness of errors) all the cats (models) are black (true).

For instance, considering the SWM inversion, if a three layer situation is fitted with a two layers model, significant errors will be present: anyway the possibility of increasing the complexity of the model to reduce the misfit in data has to be carefully evaluated.

The risk of model errors due to the multi-mode nature of the surface wave propagation and the modal superposition is discussed in par. 5.3.5: if a branch of dispersion curve due to a second mode is inverted as a first mode, a high error is introduced, and the velocities are highly overestimated.

5.2.8 Testing the model

The analysis of the residuals may help in understanding if some hypothesis has been violated: the easiest test concerns the value of the misfit function at its minimum (Tarantola, 1987)

If each ei of the N individual prediction error is a normal independent random variable, each one with zero mean and unit standard deviation, the sum of their squares is a random variable following the “chi-squared” probability density with N degrees of freedom.



∑=

=N

iie

1

22χ

( ) ( )( )

( )2exp22

; 2

2

122

2 χχ

χ −Γ

=−

NNpdf N

N

If the different errors have different variances, the overall misfit can be computed in a different way. In fact, weighting the ei’s with the covariance matrix C and including the vector of the means µ , a random variable with zero mean and unit standar deviation is obtained; it then follows the same probability density

( ) ( )µµχ −−= − xCx T 12 .

The experimental misfit can be compared with a theoretically predicted misfit, computed considering the estimated uncertainties of the experimental data: too large values of the experimental misfit with respect to the probability density define above (for instance above the 95% of probability), may suggest the risk of violation of some hypothesis.

Chapter 5.2 INVERSION OF SURFACE WAVE DATA


5.3 INVERSION OF SURFACE WAVE DATA

In this section the inversion of surface wave data is discussed. Surface wave inversion is relatively straightforward once the dispersion is correctly defined (Herrmann and Al-Eqabi, 1991).

Many inversion techniques can be used to invert a dispersion characteristic, most of them quite easily converge to the solution, the properties of the layered medium. Some properties, can be directly obtained from the curve, without formal inversion, and the number of parameters is often quite small.

In the following more attention is given to the properties of the solution, to the quality and amount of information, to the effects of the possible misinterpretation of some data feature, to the parameterisation, to the adopted constitutive law: all these aspects are slightly influenced by the inversion algorithm, but strongly influence the result of the test. Even those users that don’t want to know how the inversion works have to know how to control it, and the errors produced by a wrong use of the inversion.

5.3.1 Inversion techniques for SWM

The first simplified approach that have been used for the interpretation of dispersion data is the 3/λ or 2/λ approach, which assumes that the shear velocity equals the 110% of the Rayleigh

phase velocity, and refers it to a depth equal to one third or one half of the wavelength (see for instance Abbiss, 1981). This approach of course leads to a smoothed shear velocity profile that cannot have sharp boundaries, and can give important errors if the experimental phase velocity dispersion curve is influenced by higher modes.

The inversion of dispersion data with the trial and error approach has been used and described by Stokoe et al. (1994). The number of parameters to be found actually is limited, and allows the use of this approach, based on the intuition of the physic of the problem by the analyst.

The Monte Carlo approach, at the planetary scale, has been used by Press (1968), that generated more than five millions Earth model to fit the data of long period oscillation of the Earth.

The widely more diffused approach is the linearised iterative least squares approach, that has been used by many authors, with some differences in the data, the seeked model parameters, the computation of the partial derivatives, the inversion strategies, the use of smoothness constraints, and so on. (Herrmann 1994; Herrmann and Al-Eqabi, 1991; Horike, 1985; Lai 1998; Foti, 2000; Xia et al., 1999; Tselentis and Delis, 1998; Mokhart et al,. 1988; Shtivelman, 1999; Gabriels, 1987 among the others)

Also genetic algorithms (Al-Hunaidi, 1998), the simulated annealing procedure (for instance, recently, Martinez et al., 2000) have been applied.

Joint inversion with other techniques, especially electric and electromagnetic techniques (Hering et al, 1995; Comina et al., 2002), and coupled inversion of phase velocity and attenuation (Lai, 1998) have been used.

Often only the first mode is considered, but interesting examples of inversion of several modes (Gabriels et al., 1987) and apparent curve have been discussed.



5.3.2 Factors influencing the solution

The algorithms used to perform the inversion are not as influent on the properties of the solution as the aspects regarding either the quality and the amount of information or the possible model errors produced by neglecting the modal superposition or by the misinterpretation of the dispersion characteristics. The information and variance of data are the main aspects of this inverse problem. The following part of the chapter deals with the problems of the inversion from two point of view.

First a more general discussion about the data to be inverted: how to increase the information; how to reduce the useless data that duplicate the same information; which other properties of the propagation, besides the velocity, can add useful independent information about the model. The importance of careful acquisition and processing is stressed showing the effects of a misinterpretation of data.

Second the properties of the solution are assessed in order to evaluate the reliability of the characteristics of each layer; the selected model, in terms of number of layers; the investigation depth.

5.3.3 Considerations on the data to be inverted

The data space d and the model parameter space m have to be considered. The Rayleigh wave relations map the elastic and geometric parameters of the model m space into the d space of the modal curve.

As it has been mentioned, the model parameters are in general the physical and geometric properties of the model, i.e. the shear-wave velocity, the Poisson ratio, the mass density and the thickness of the layers

[ ]LLSLSS VhVhVm ρνρνρν ,,,...,,,,,,,, 22221111= (5.1)

while the data are the Rayleigh–wave velocity as a function of the frequency.

NRRRR VVVVd ,...,,, 321=

but the data could also include other propagation properties, such as for instance the power spectra or the distribution of the energy as a function of the frequency along a mode

The attenuation and the velocity of the layers are physically coupled, and may be uncoupled in weakly dissipative media. In the following we consider an uncoupled inversion of experimental data to estimate the shear velocities and the thickness of the layers.

Usually the propagation velocity is considered, and the energy distribution is not used. The dispersion causes the propagation velocity to be a function of the frequency, and so a phase velocity and group velocity exist (chapter 2).

5.3.3.1 Phase velocity or group velocity dispersion curve The velocity of a wavetrain, composed by several single frequency signals, is the group velocity, in opposition to phase velocities that refers to each frequency. The group velocity U is coincident with the phase velocity V in non dispersive media, but in general can be computed as (Sheriff and Geldart, 1995)



( ) λλ

ωω

ddVV

dfdVfV

dkdU −=+==

From the above equation it is easily seen that if phase velocity V increases with frequency, the group velocity U is greater than V. If V decreases with increasing frequency, as it happens at normally dispersive sites, V is greater than U. The simple relation between wavenumber and group velocity shows the relation between the data space of the wavenumber and the data space of the group velocity.

5.3.3.2 Modal curves The problem of finding the model parameters from the inversion of the modal phase velocities is tipically mix determined. In particular there is a lot of information about the shallow layers, which are consequently over-determined, while there is few information about the deeper layers, which are under-determined. In particular all the wavelengths test the shallow layers, while deep layers are sampled only by greater wavelengths. The arrows of figure 5.20 show the dependence of the different frequencies from the different layers: in the inversion the path is inverted. It can be noticed that the high frequencies can be used to estimate the shallow depths, but the shallow depth influence all the frequencies.

Figure 5.20. Scheme of the relations between the model space and the data space

The sensitivity of the phase velocities modal curves is an aspect that can help to understand which data contain the information about the different model parameters.

The following figure 5.21 shows the partial derivatives of the first three modal phase velocities with respect to each layer shear velocity. The model is a three layer model; two layers, 1m thick each, on a half space, and with shear wave velocity being 100 m/s, 200 m/s, 400 m/s. Considering for instance the first mode, it is evident that the phase velocity shows sensitivities to the different layers that are a function of the frequency: at high frequency the first layer, at low frequency the half-space, and in the middle the second layer. Higher modes have behaviours more difficult to be interpreted.



Figure 5.21. Partial derivatives of the modal phase velocities respect to the layer shear velocities

The sensitivity can be seen as the way in which the variation of the properties of a layer influences a mode in a certain frequency band.

It is possible to see directly how the different layers influence the different modal phase velocities: the effects of the same variation of the shear-wave velocity of the first and the third layer are shown in the following figure 5.22. The right side of figure 5.22 confirms that the high frequencies are not sensitive to the deep layers, and they could be inverted separately. (same model of Figure 5.21)

Figure 5.22 Variation s of the modal curve corresponding to the same increase of the shear velocity respectively of the

first layer (left) and the third layer (right)

This approach could be used for the assessment of the properties of the solution and the design of experiments: however it has to be considered that this simple approach has a limit due to the energy distribution on modes, that can make some modes not detectable, and can influence the modal superposition. The modal curves are only possible solutions, and from an experimental point of view they can be not detectable: it can happen either that modes are well separated but some of



them are not identified, or that modes are superimposed in a average behaviour, depending on the modal phase velocities but also on the modal energies.

5.3.3.3 Apparent dispersion curve It is not possible, then, to make considerations about the information in modal curves without considering the possibility of detecting them: in the following figure (5.23) it is shown with black asterisks the apparent curve that could be detected experimentally, with the same model used for figure 5.22, and a 46 m array. It has to be noticed that the low frequency branch of the dispersion curve has a little energy, and even if is the only possible velocity at those frequencies, it is hardly discernable.

Figure 5.23. Modal curve (lines) and apparent curve (asterisks) corresponding

to the maximum of energy for each frequency

The modal superposition can play a strong role, both in normal dispersive and inversely dispersive sites. The array length influence the modal superposition in experimental data, and have to be considered in the inversion.

The apparent dispersion curve can be the result of the superposition of different modes in an experimental configuration. It depends on the site properties and on the array dimension. Of course all the influencing factors have to be considered in the inversion, and so a multi-modal modelling with modal superposition has to be used. These aspects about the simulation of the test, accounting for the modal superposition and the array length, have been discussed in chapter 2. In chapter 3 and 4 the effects of the modal superposition have been considered for both the acquisition and the processing.

The inversion of such a function, anyway, poses a series of problem due to its discontinuities: the jumps between different modes produce high variations of the data even for small variations of the model parameters. If branches of modes can be isolated, this allow a multi-modal inversion of different modes. However it is not possible to identify the mode number: when different modal curves are obtained in experimental data, their identification is not straightforward. Sometimes, for instance, only branches of the first and the third modes are visible and there are no evidences helping to assign the right mode number to the branches

In many cases, a trial and error inversion is the only way to get a stable result: it can be used, of course, as a first estimate for a local-search refinement. An example is given in chapter 6.



5.3.3.4 Resonance frequencies The phase velocities of modal curve are the eigenvalues of the problem, are only the way in which the energy can propagate at a site. Then the relations among different modes and the distribution of the energy along a single modal curve (Figure 5.24) depend on the eigenfunctions, i.e. the stresses and strains with depth. (see chapter 2).

Figure 5.24. The f-k spectrum showing the energy distribution

Several modes can exist, and they all can be dominant in different frequency ranges. Moreover, along a single modal curve, even with a flat frequency band source and not considering the attenuation, the energy distribution can be very inhomogeneous. When strong acoustic impedance changes occurs, resonant frequencies in the displacements at the free surface are easily identified.

The figure below (Figure 5.25, 5.26) shows the synthetic spectrum obtained simulating a test at a site with a layer 2m thick, with 100 m/s of shear-wave velocity, over a bedrock with shear-wave velocity of 400 m/s, with vertical source at the free surface. A resonant frequency is identified around 22 Hz with an input source at the free surface that has a unit amplitude spectrum between 5 and 50 Hz.



Figure 5.25. The displacements of the two modes at 23 Hz

Figure 5.26. Modal curves and f-k spectrum for a two layer stratigraphy 2m 100m/s, bedrock 400 m/s of shear velocity:

it can be noticed that the response is amplified around 22 Hz. (synthetic data)

As for the resonance frequency of layers for a simple SH – wave with vertical incidence on a stack of layers, the greater is the impedance contrasts, the greater the is amplitude of the resonance (Kramer, 1996): the amplification can be easily simulated in elastic and viscoelastic media, and is one of the results of the modelling described in chapter 2.

The experimental observation of resonant frequencies, at sites where high impedance contrast exists, is common, but a quantitative analysis requires a higher control of the amplitudes in throughout acquisition and processing. Many factors in the equipment and at the site can modify the amplitude distribution as a function of the frequency (source, receivers, attenuation…), and it is not simple to consider them all for a quantitative analysis of the amplifications: anyway the position of energy maxima increases the information.

If the experiment is designed and performed controlling the attenuation and the frequency response of the whole, these latter factors can be included in the modelling and also the energy distribution can be used in the inversion procedure.



5.3.3.5 How many points are necessary in the dispersion curve? The sampling of the dispersive characteristics in frequency domain is very important aspect in acquisition, processing and inversion: in the inversion it heavily influences heavily the computation time and the uncertainty of the final model. In the different frequency bands the information density is different, and so different sampling may be adopted. The example below (Figure 5.27) shows an experimental dispersion curve obtained from a 24 traces records, with uniform sampling in frequency. The inversion is performed assuming a 10layers model (the number of layers which can be reliably estimated) with a weighted damped least squares algorithm. The mass density and Poisson ratio are assumed a priori, and the layer thickness are assumed all equals. (their thickness as a function of the depth are discussed later).

Figure 5.27. Modal curve (dots experimental, solid line numerical) and the corresponding model.

The modal curve is uniformly sampled with 114 points.

The Jacobian matrix of the final model is used to compute the data resolution matrix is depicted in figure 5.28: the data set was a vector of 114 values of phase velocity at 114 different frequencies (from 4 to 33 Hz), and so the data resolution matrix is a square matrix 114 x 114. As it has been discussed in 5.2.5, independent data correspond to a diagonal matrix, and the relation between diagonal and off-diagonal element is a measure of the independence of data (the spread defined above).

It is evident that in the low frequency band the diagonal elements are much higher than the others, while in the high frequency ranges this is not true. This means that in the low frequency range the points are more independent, so each of them increases actually the information content. In the high frequency range, on the contrary, points are mutually dependent, and actually replicate the same information.



Figure 5.28. The data resolution matrix is depicted in the left and three cuts are shown in the right

The consequence is that in the high frequency range less points are needed. The same effect can be seen if the experimental data are analysed in wavelength vs. phase velocity: even if it is not straightforward to describe the relations between data space and model space, to adequately sample the different depths in the model space it is necessary to adequately sample the different wavelengths in the data space (Figure 5.29).

Figure 5.29. The same experimental dispersion curve in frequency – phase velocity(left) and wavelength – phase

velocity(right). A uniform sampling in frequency is not uniform in wavelength.

If the dispersion curve in the frequency – phase velocity domain were sampled in such a way to have an experimental point each meter of wavelength between the minimum and maximum values, the resulting graph would be like the one in figure 5.30.



Figure 5.30. The sampling of the dispersion curve which correspond to a uniform sampling of the wavelength

5.3.4 Considerations on the parameters to be inferred

5.3.4.1 Physical and geometric parameters The model parameterisation depends on the available forward modelling and on the necessity of reducing the number of unknowns. The formulation discussed in chapter 2 allows the simulation of the surface wave propagation in a layered medium. So the model parameters which are needed in the forward modelling are the properties of each layer, and the acquisition layout geometry. The latter of course is known, and has to be used in the modelling to coherently compare synthetic with experimental data. The layer properties on the contrary are not known and have to be determined by means of the inversion process. Some of the parameters can be easier identified than other, since their sensitivity and variation range are such that their effect on data is easily detected. In general, a model parameter can be estimated if within its variation range it produces detectable variations in the measured data, considering the uncertainties.

Parametric studies (Nazarian, 1984; Xia et al., 1999) show that the shear velocity and the thickness of the layers are the most important parameters to be inverted. The ranges of the density are small and the values can be a priori estimated. The Poisson ratio, moreover, for its lower influence and its small range of possible values, is usually assumed a priori, or neglected.

However the effect of the saturating water on the P-wave velocity, and so on the Poisson ratio, is high, and the abrupt variation of the compressibility of the medium can produce effects that should not be neglected (Foti and Strobbia, 2002). Therefore the information on the water table position should be used to avoid the possible over-estimation of the velocities in the inversion.



5.3.4.2 Layers As far as the layer is concerned, two aspects have to be considered: the number of layers and the depth of the half-space when the velocities only are inverted. These aspects refers respectively to the discretization and the boundaries of the investigated domain. If the layers are considered as a discretization of the studied domain, the thickness of the layers as a function of the depth has to be discussed: the loss of resolution and sensitivity with the depth has to be considered, and the thickness should then be increased as the depth increases.

The example below shows the effects of the number of layers on the result of the inversion of an experimental dispersion curve. The site presents a quite homogeneous sandy formation laying on gravels.

I. NUMBER OF LAYERS

The first inversion is performed with a model of 6 layer with constant thickness: the thickness of the layers has been obtained simply dividing the investigation depth, estimated from the maximum wavelength, by the chosen number of layers (Figure 5.31).

This model cannot fit well the high frequency portion of the dispersion curve for any number of iterations because of the limited number of layers in the shallow part. This could be considered a model error: no value of the parameters of the first two layers could fit the high frequency range of the experimental dispersion curve, and the number of layers has to be increased. The obtained stratigraphy has also a velocity inversion, that hence has to be considered an inversion artefact. Indeed it depends on the high frequency range that is not properly fitted. The limited number of layers, moreover, produces quite high impedance contrasts between the “layers”, that are unlikely for that site, characterized by a quite homogeneous formation in the shallow part. When the number of layers is increased, the site characteristics are better identified. There is not a geological layering with sharp boundaries, but a continuous variation of the mechanical properties of the soil.

Figure 5.31. Final Model (left) and fitting of the curve (right) for the best 6 layer model



The inversion with 10 layers (Figure 5.32) shows a better fitting in the high frequency portion, and a model with a quite smooth increase of shear velocity with depth, with a half space characterised by a higher velocity. The fitting is excellent and the misfit is very small if compared with the possible uncertainties of the dispersion curve. The thickness of the layers could be optimised, but with the chosen approach this result is satisfactory.

Figure5.32. Obtained model and modal curve fitting with a 10 layers model. The fitting is good and the model

reasonably compatible with the a priori information. The reliability of the different layers is not constant.

With a further increasing of the number of layers the model follows each oscillation of the dispersion curve, but there is the risk of fitting the noise (as mentioned in 5.2.6).

When the number of layers becomes high (Figure 5.33), of course decreases the reliability of the single shear velocities. A trend is identified, and it is better to invert for the coefficient of a velocity law, for instance a polynomial law, which is assumed to velocity variation with the depth. In particular the velocity inversion that is obtained in the uppermost part of the profile is simply due to the small change of the experimental phase velocity, that decreases more slowly above 27 Hz.



Figure 5.33. Obtained stratigraphy and modal curve fitting with a 35 layers model. The single shear velocities are not reliable, and inversion artefacts are introduced (shallow velocity inversion)

Considering the final model of the inversion with the 35 equally thick layer model, it can be shown that in data simply there is not the information to resolve each layer. The model resolution matrix shows that the estimated shear velocities are obtained as weighted averages of the real velocities, and the number of layers that are weighted increases with depth. Using a 35 parameter model, the model resolution matrix is the square 35 x 35 matrix depicted in figure 5.34, left. In the upper part of the model there is a higher resolution, in the deeper part a lower resolution.

Figure 5.34. The model resolution matrix and the plot of the diagonal elements for the 35 layer model

The resolution strongly decreases with depth as the sensitivity, and it is interesting to notice that the deepest layers are not resolved at all.

The singular value decomposition shows clearly this aspect.



The plot of the singular values (Figure 5.35, left) shows that there isn’t a clear threshold between big and small singular values, but a strong decrease that makes the fifth singular value to be one tenth of the first singular value.

Each singular value indicate a certain amount of information, and the associated eigenvector (V) describe which is the combinations of model parameters carrying that amount of information. The coefficient of the first 4 eigenvectors are plot in figure 5.35, right: it can easily be seen that they refers to weighted averages of layers.

In particular the first singular value has the following coefficients 0.1892 0.5981 0.6059 0.3843 0.2341 0.1454 0.0913 0.0603 0.0413 0.0296 …

As it has been mentioned in section 5.2.4.3, these combination means that the first singular value refers to the combination 0.1892VS1+0.5981VS2+0.6059VS3+0.3843VS4+0.2341VS4+0.1454VS5+0.0913 VS6+0.0603VS7+0.0413VS8+0.0296VS9… This, roughly speaking, means that the maximum of information that is present allows the estimate of the mean of the second and third layers, with also the contribution of the third the fourth, and the first, and other coefficients that are actually negligible. The coefficient of the first four singular values are depicted in figure 5.35, right: each line represent an eigenvector, with the coefficient associated to the 35 layer.

Figure 5.35. The singular value (left) and the first 4 eigenvectors, as coefficient of the different model parameters. The

first eigenvector is the almost mean of the second and third layer

The figure 5.35 also shows that the information and the resolution decreases with depth.

The singular value decomposition, showing the decrease of the resolution and information in the deepest part of the model, states a kind of “suppression principle”: if the singular value is associated to an eigenvector that average two layers, these ones may be substituted by a single average layer.

A complicated stratigraphy can be simplified to find an equivalence with an averaging stratigraphy: figure 5.36 shows two models (left), one with a great number of layers and the other with a small number of layers, and the corresponding responses (right). The two responses are in practice identical, and hence the two models are equivalent.



Figure 5.36. Two different models and the corresponding dispersion curve.

They are equivalent, in the sense that there is not enough resolution for the fine discretisation.

II. INVESTIGATED DOMAIN: MINIMUM AND MAXIMUM DEPTH

The minimum depth depends on the minimum sampled wavelength. If the acquisition high frequencies are reliably acquired, the minimum wavelength that can be observed depends on the spatial sampling, and equals the geophone spacing. The shortest propagating wavelength carries the information related to the shallowest layer, and gives the mean properties of the layers above its investigation depth.

These mean properties can still be influenced by a thinner layers that is not investigated: in this case it is of course necessary to incorporate a priori information about this layer, or to acquire shorter wavelengths.

The maximum depth depends on the maximum reliably estimated wavelength: different authors suggests, as a rule of thumb, to limit the maximum depth to one half of the maximum wavelength (Shtivelman, 1999), but sometimes it is possible to go deeper, down to one wavelength (Herrmann and Al-Eqabi, 1991). Actually the maximum investigated depth depends on the site: a rigorous approach should give a result with its uncertainties, and should also compare the results of different inversions with different numbers of layers and depths.

5.3.5 Possible errors

One of the main source of errors can be the misinterpretation of some features of data: in particular the effects of the multi-mode nature of the surface wave propagation can lead to important errors.

The three examples given in the following show how the modal superposition can produce errors that can be extremely important.



5.3.5.1 Example 1 The first example refers to a simple normal dispersive stratigraphy of three layers (Figure 5.37).

Figure 5.37. A simple 3 layer stratigraphy and its modal curve

The first two modal curves are quite close between 13 and 20 Hz: in wavenumber domain the difference between the two curves is less than 0.05 rad/m (Figure 5.38), and the second mode has more energy than the first. The two modes can be clearly separated only with an array hundreds of meters long, otherwise their energies will be superimposed (Figure 5.39).

Figure 5.38. The modal curves in wavenumber domain

If an more realistic array is used, the two modes are superimposed. The apparent curve lays between the two and tends to the second mode approaching its cut-off frequency. Below the cut-off



frequency of the second mode, of course the first mode only can exist, but the associated energy is small and probably difficult to be detected experimentally (5.39). A test performed at this site would probably give only the continuous branch of the apparent curve above 14 Hz: a similar case from the real world is given in chapter 6.

Figure 5.39. The synthetic f-k spectrum and the apparent curve. Very low energy is present below the cut-off frequency

of the second mode

If the continuous branch of the dispersion curve is considered and is misinterpreted as a first mode, a great error can be introduced. The mode weighted damped least square inversion considered a pure first mode gives a model that fit well the data (Figure 5.40)

Figure 5.40. The model that fit, with its first mode, the synthetic main branch of the dispersion curve of figure 5.38

Comparing the model that has been used to generate the data and the model that is obtained by the inversion (Figure 5.41), it can be noticed that in the upper part, above 7m, they are in good agreement: this portion of the model corresponds to the frequency band of the dispersion curve



which was actually due to the first mode. Going deeper, the data are influenced by the second mode, (which has a higher velocity than the first) but as they have been interpreted as a first mode: over-estimated velocities are obtained.

Figure 5.41 The difference between the true model (grey) and the one that come from the fitting of figure 5.39(black)

It must be stressed that it is not possible, from the experimental data alone, to recognize if the obtained dispersion pattern is due to a modal superposition or to a single mode and, moreover, if a single mode is the first one or a higher mode. It is so very important to evaluate the obtained result, because such an error can be quite dramatic.

A possible procedure to check the result is to use the final model as input of a multi-modal forward modelling, to asses the relative importance of the different modes.

If this procedure is applied to the result of the inversion above (Figure 5.40), it could be noticed that the first mode (which well fits experimental data) is not dominant in all the considered range: if the test was performed at such a site, the modal superposition would have been observed. The synthetic characteristic to be compared with the experimental, hence, cannot be the first mode.

Figure 5.42. The multi-modal modelling of the obtained stratigraphy is necessary when a first mode inversion is

performed. This allows at least to see if the first mode is dominant in the resulting model



5.3.5.2 Example 2 An other example of possible errors due to the modal superposition considers the presence of a higher mode which is close to the first mode in a central frequency band.

The effects described in chapter 4 about the modal superposition have to be taken into account: the presence of a mode close to another can locally move the position of the maximum that is identified by the processing (Figure 5.43).

Figure 5.43. The modal superposition can move the maximum from one mode to a average position between two modes

If a higher mode is present close to the first one, with a high energy content, it can move the energy maximum to higher velocities. Below the cut-off frequency of the second mode, or simply out of the frequency band where this higher mode has a great energy, the maximum will go back to its correct position, on the first mode, reducing its phase velocity. This local increase of phase velocity, if the presence of the higher mode is not recognised, can be misinterpreted. The example below (Figure 5.44) shows an example of such a dispersion curve: the shape of the curve can be interpreted as a local increase around 20 Hz, or a decrease between 15 and 20 Hz.

Figure 5.44. The decrease of the phase velocity going towards the low frequencies has to be well verified, because it

correspond to a velocity inversion. It has to be verified that it is not a modal superposition effect.



That small variation in the dispersion curve has a great importance in the model estimation: if it is a real feature it implies a velocity inversion. The example below (Figure 5.45) shows the models and the modal curves (first modes) for a simple case where a layer 2 m thick produces an effect on the curve that can be compared with the one seen above (Figure 5.44).

Figure 5.45. The local increase of the phase velocity in the dispersion curve has a great effect on the stratigraphy

5.3.5.3 Example 3 The same kind of effect can be produced by lateral variations. If the test is performed at a site where a homogeneous bedrock is overlaid by a layer of sediments showing an abrupt lateral variation (Figure 5.46), as in the example below, the behaviour of the dispersion curve is confused. When a sharp lateral variation is present, two dispersion curves could be obtained: if the variations are gradual or less strong, a confused dispersion curve is obtained (Mockart et al., 1988).

Figure 5.46. Scheme of a abrupt lateral variation in the sediments on a homogeneous bedrock



Figure 5.47 The two f-k spectra that may be obtained from two separate acquisitions

Figure 5.48 The spectrum that is obtained from a single measure and the corresponding curve

The different branches obtained by figure 5.48 could be incorrectly interpreted as different modes.

5.3.6 Estimate of the uncertainty of the model parameters

The uncertainties of the model parameters can be obtained considering the last step of the iterative inversion algorithm, and the matrix equation that has been solved, or inverted, as

dGm g ∆⋅=∆ −

implies that, with a first order Taylor’s approximation, if the data are independent, that is that covariance term are zero,

22 2 dg

m G σσ ⋅= − ,

where gG −2 is the matrix containing the squares of the elements of gG − (Santamarina and Fratta, 1998).

In general, if the solution is obtained with a linearised inversion, and only the shear velocity of the layers is considered, it can be written (Lai, 1998):



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

TSRlast

TSS

TSS JJJVJJJV #

1#

1 covcov −−⋅⋅⋅=

The application of this last equation to the inversion of the dispersion curve depicted in the following figure 5.48 (left)gives the uncertainties of the shear velocities of the model of figure 5.49 (right).

Figure 5.49 Uncertainties of the estimated model parameters. Left) experimental dispersion curve with standard deviations, right) shear velocity model with standard deviations of the layer velocities

5.3.7 Observations on the low frequencies

In the inversion process, it has to be considered that the different points do not have the same uncertainty: the experimental phase velocities should be weighted according to their experimental uncertainties. The example below shows the points of ten different acquisitions, and the solid line has been obtained stacking the ten record.



Figure 5.49. Dispersion curves of ten separate shots (points) and the curve obtained stacking the ten record. At each

frequency we have 10 experimental phase velocities, showing a kind of distribution of the phase velocity.

The problem of this inversion is that:

a) Low frequency points potentially add information about the interesting model parameters

b) Low frequency points have a small sensitivity respect to the deep interesting layers

c) Low frequency points have the highest uncertainties

Finally, we are interested in low frequency points because they allow to go deeper (a), but small variations of their phase velocities may correspond to great variations of the layer properties (b), and moreover these small variations can be due to experimental uncertainties.

Chapter 5.4 THE COMPUTER PROGRAM FOR THE INVERSION


5.4 THE COMPUTER PROGRAM FOR THE INVERSION

In this section is briefly described the computer program, implemented in Matlab, that allows the interactive inversion of surface waves with the different techniques described in the above.

The program allow the inversion of the modal dispersion curve to find the shear velocities of layers of assumed thickness, or to find thickness and shear velocities of layered model. The multi-modal forward modelling of the final model allows to assess the importance of higher modes in the obtained model.

The trial and error inversion of apparent curves due to the modal superposition is possible in a interactive manner, using the scaling properties of the Rayleigh waves.

• A branch of a modal curve can be inverted, assuming the mode number, with a weighted damped least squares algorithm.

• Several branches of different modes can be simultaneously inverted

• A trial and error procedure is possible, comparing the experimental curve and spectrum with the simulated ones. The applied global stretch of the numerical curve is used to compute the scaling of thickness and velocities.

references

REFERENCES

• Abbiss C.P., 1981, Shear wave measurements of the elasticity of the ground,

Geotechnique, 31, 91-104

• Aki, K., Richards, P.G., 1980, Quantitative seismology. Theory and methods, W.H. Freeman and Company

• Al-Eqabi G.I., Herrmann R.B., 1993, Ground Roll: A potential tool for constraining shallow shear wave structure, Geophysics, 58, 713-719

• Al-Hunaidi O., 1994, Analysis of dispersed multi-mode signals of the SASW method using the multiple filter crosscorrelation technique, Soil dynamics and Earthquake engineering, 13, 13-24

• Al-Hunaidi O., 1998, Evolution based genetic algorithm for the analysis of non-destructive surface wave test on pavements, NDT&F, 31, 273-280

• Audisio A., Bonani S., Foti S., 1999, Advances in spectral analysis for SASW test, in: Jamiolkowsky M., Lancellotta R., LoPresti D. (eds), Pre-failure deformation Characteristics of Geomaterials, Balkema, Rotterdam, 395-402

• Backus, G., Gilbert F., 1970, Uniqueness in the inversion of inaccurate gross earth data, Phil. Trans. R. Soc. Lond., 266A, 123-192

• Backus, G., Gilbert, F, 1968, The resolving power of gross earth data, Geophys. J. Roy. Astr. Soc, 16, 169-205

• Ballard R.F., 1964, Determination of soil shear moduli at depth by in situ vibratory techniques, Waterways Experiment Station, Miscellaneous Paper, No.4–691, December

• Blake R.J., Bond L.J., 1990, Rayleigh wave scattering from surface features: wedges and down steps, Ultrasonics, 28, 214-228

• Blake R.J., Bond L.J., 1992, Rayleigh wave scattering from surface features: up steps and toughts, Ultrasonics, 30, 255-265

• Bloch S., Hales A.L., 1968, New techniques for the determination of surface wave phase velocities, Bulletin of the Seismological Society of America, 58 (3), 1021-1034

• Budaev B.V., Bogy D.B., 1995, Rayleigh waves scattering by a wedge I, Wave Motion, 22, 239-257

• Budaev B.V., Bogy D.B., 1996, Rayleigh waves scattering by a wedge II, Wave Motion, 24, 307-314

• Budaev B.V., Bogy D.B., 2001, Scattering of Rayleigh and Stoneley waves by two adhering elastic wedges, Wave Motion, 33, 321-337

• Bullen, K.E., 1963, An introduction to the theory of seismology, Cambridge University Press

references

• Buttkus B., 2000, Spectral analysis and filter theory in applied geophysics, Springer, Berlin, 667 pp

• Chapman, N.R., 1991. Estimation of geoacoustic properties by inversion of acoustic field data, in: Hovem, J.M., Richardson,M.D., and Stoll,R.D. (Eds.), Shear Waves in marine sediments, Kluwer Academic Press, Dordrecht: 511-520

• Claerbout, J.F., 1985, Fundamentals of geophysical data processing, Blackwell Scientific Publication, California, 274 pp

• Comina C., Foti S., Sambuelli L., Socco L.V., Strobbia C., 2002, Joint inversion of VES and surface wave data, proc. SAGEEP, LasVegas

• Dorman J., Ewing M., 1962, Numerical Inversion of seismic surface waves dispersion data and crust-mantle structure in the New York-Pennsylvania area, J Geophysical Research, 67, 5227-5241

• Dorman J., Ewing M., Olivier J., 1960, Study of the shear-velocity distribution in the upper mantle by mantle Rayleigh waves, Bulletin of the Seismological Society of America, 50, 87-115

• Duijndam A.J.W., Schoneville M.A., 1999, Nonuniform Fast Fourier Transform, Geophysics, 64(2), 539-551

• Duijndam A.W.J., 1988, Bayesian estimation in seismic inversion. Part I: Principles, Geophysical Prospecting, 36, 878-898

• Duijndam A.W.J., 1988, Bayesian estimation in seismic inversion. Part II: Uncertainty analysis, Geophysical Prospecting, 36, 899-918

• Dunkin, J.W., 1965, Computation of modal solutions in layered, elastic media at high frequencies, Bulletin of Seismological Society of America, 55, 335-358

• Dziewonski A., Bloch S., Landisman M., 1969, A technique for the analysis of transient seismic signals, Bulletin of the Seismological Society of America, 59, 427-444

• Dziewonski A.M., Hales A.L., 1972, Numerical analysis of dispersive seismic waves, in: B.A. Bolt (ed), Methods in computational physics, , Vol.11, 271-295, Academic Press

• Ewing, W.M., Jardetzky, W.S., Press, F., 1957, Elastic waves in layered media, McGraw Hill Book Co. Inc., NewYork

• Foti S., 2000, Multi-Station Methods for Geotechnical Characterisation Using Surface Waves, PhD Diss., Politecnico di Torino, 229 pp

• Foti S., Lai C.G., Lancellotta R., 2002, Porosity of Fluid-Saturated Porous Media from Measured Seismic Wave Velocities, Geotechnique, 52 , 359-373

• Foti S., Lancellotta R., Sambuelli L., Socco L.V., 2000, Application of fk Analysis of Surface Waves for Geotechnical Characterization, Annali di Geofisica 43, N.6 pp.1199-1209.

• Foti S., Sambuelli L., Socco L.V., Strobbia C., 2001 Fast SWM for Site Characterisation Devoted to Trenchless Technology. Atti del 7th Meeting Environmental and Engineering Geophysics Society European Section, 2 – 6 settembre, Birmingham, 270 – 271.

references

• Foti S., Sambuelli L., Socco L.V., Strobbia C., 2002. Spatial Sampling Issues in fk Analysis of Surface Waves Atti del SAGEEP 2002, Las Vegas, Feb. 2002 CDROM.

• Foti S., Strobbia C., 2002, Some notes on model parameters for surface wave data inversion, proc. SAGEEP, LasVegas

• Gabriels P., Snieder R., Nolet G., 1987, In situ measurement of shear wave velocity in sediments with higher-mode rayleigh waves, Geophysical prospecting, 35, 187-196

• Gautesen A.K., 2002a, Scattering of a Rayleigh wave by an elastic quarter space – revisited, Wave Motion, 35, 91-98

• Gautesen A.K., 2002b, Scattering of a Rayleigh wave by an elastic three quarter space, Wave Motion, 35, 99-106

• Gilbert F., Backus G.E., 1966, Propagator matrices in elastic wave and vibration problems, Geophysics, 31(2), 326-332

• Glangeaud F., Mari J.L., Lacoume J.L., Mars J., Nardin M., 1999, Dispersive seismic waves in geophysics, Eur. J. Environ. Eng. Geophys., 3, 265-306

• Gucunski N., Woods R.D., 1991, Inversion of Rayleigh wave dispersion curve for SASW test, Proc. 5th Conf. On Soil Dyn. And Earthquake Engineering, Karlsuhe, 127-138

• Gucunski, N., Ganji, V., Maher, M.H., 1996, Effects of obstacles on Rayleigh wave dispersion obtained from SASW test, Soil Dynamics and Eathquake Engineering, 15, 223-231

• Gukunski N., Woods R.D., 1992, Numerical simulation of SASW test, Soil Dynamics and Earthquake Engineering, 11(4), Elsevier, 213-227

• Hardage, B.A., 1983, Vertical Seismic Profiling: Part A: principles, Geophysical Press, 552pp

• Haskell, N.A., 1953, The dispersion of surface waves on Multilayered Media, Bulletin of the Seismological Society of America, 43, 17-34

• Hering A., Misiek R., Gyulai A., Ormos T., Dobroka M., Dresen L., 1995, A joint inversion algorithm to process geoelectric and surface wave seismic data. Part I: basic ideas, Geophysical Prospecting, 43, 135-156

• Herman, G.T., 1980, Image reconstruction from projections, the fundamentals of computerized tomography, Academic Press, New York

• Herrmann R.B., 1973, Some aspects of band-pass filtering of surface waves, Bulletin of the Seismological Society of America, 63, 663-671

• Herrmann R.B., 1994, SURF: surface wave inversion program, R.B.Herrmann editor

• Herrmann R.B., Al-Eqabi G.I., 1991, Surface wave inversion for shear velocity, in: J.M.Hoven et al. (eds), Shear waves in marine sediments, Kluwer Academic Publisher, 545-556

• Herrmann, R.B., 1996, Computer programs in seismology: an overview on synthetic seismogram computation, User’s manual, StLouis University, Missouri

• Herrmann, R.B., Wang, C.Y., 1985, A comparison of synthetic seismograms, Bulletin of Seismological Society of America, 75: 41-56

references

• Hiltunen D.R., Woods R.D., 1988, SASW and crosshole test results compared, Earthquake Engineering,

• Hisada Y., 1994, An efficient method for computing Green’s function for a layered half space with sources and receivers at close depth, Bulletin of the Seismological Society of America, 84, 1456-1472

• Hisada Y., 1995, An efficient method for computing Green’s function for a layered half space with sources and receivers at close depth (part 2), Bulletin of the Seismological Society of America, 85, 1080-1093

• Horike M., 1985, Inversion of phase velocity of long-period microtremors to the s-wave velocity structure down to the basement in urbanized areas, J.Phys.Earth, 33, 59-96

• Jackson, D.D., 1972, Interpretation of Inaccurate, Insufficient and Inconsistent Data, Geophys. J. Roy. Astr. Soc, 28, 97-109

• Jefferys W.H., Berger J.O., 1992, Ockham’s razor and bayesian analysis, American Scientist, 80, 64-72

• Jones R.B., 1958, In-situ measurements of the dynamic properties of soil by vibration method, Geotechnique, 8, 1-21

• Jones R.B., 1962, Surface waves techniques for measuring the elastic properties and thickness of roads: theoretical development, British Journal of Applied physics, 13, 21-29

• Jongmans D., Demanet D., 1993, The importance of surface vibration study and the use of Rayleigh waves for estimating the dynbamic chacteristics of soils, Eng. Geol., 34, 105-113

• Kanai, K., Tanaka, T., 1961, On microtremors, VIII, Bull. Earthq.Res.Inst., 39, 97-114

• Kausel E., Roesset J.M., 1981, Stiffness matrices for Layered Soils, Bulletin of the Seismological Society of America, 71, 1743-1761

• Keilis-Borok V.I., Levshin A.L., Yanovskaya T.B., Lander A.V., Bukchin B.G., Barmin M.P., Ratnikova L.I., Its E.N., 1989, Seismic surface waves in laterally inhomogeneous earth, Kluwer Academic Publishers, 293 pp

• Kennet B.L.N., 1974, Reflections, Rays and Reverberation, Bulletin of the Seismological Society of America, 64, 1685-1696

• Knopoff L., 1964, A matrix method foe elastic wave problems, Bulletin of the seismological society of America, 54, 431-438

• Knopoff L., 1972, Observation and Inversion of Surface wave dispersion, Tectonophysics, 13,497-519

• Kovach R.L., 1978, Seismic surface waves and crust and upper-mantle structure, Rev. Geophys and Space Phys., 16, 1-13

• Kramer S.L., 1996, Geotechnical Earthquake Engineering, Prentice-Hall, New Yersey

• Lai and Rix, 2002, BSSA

• Lai C.G., 1998, Simultaneous Inversion of Rayleigh Phase Velocity and Attenuation for Near-Surface Site Characterization, PhD Diss., Georgia Inst. Of Techn., Atlanta (Georgia, USA) , 370 pp

references

• Lai C.G., Rix G.J., 1999, Inversion of multi-mode effective dispersion curves, in Pre-failure deformation Characteristics of Geomaterials, Jamiolkowsky M., Lancellotta R., LoPresti D. editors, Balkema, Rotterdam, 411-418

• Lamb, H., 1904, On the propagation of tremors over the surface of an elastic solid, Philos. Trans., vol.CCIII, pp. 1-42

• Lanczos C., 1961, Linear Differential operators, D.Van Nostrand Co., London

• Levenberg K., 1944, A method for the solution of certain nonlinear problems in least squares, Quarterly of Applied Mathematics 2, 164-168

• Lines R.L., Treitel S., 1984, A review of least squares inversion and its application to geophysical problems, Geophysical prospecting, 32, 159-186

• Love A.E.H., 1911, Some Problems of Geodynamics, Cambridge University Press

• Mari, J.L., 1984, Estimation of static corrections for shear wave profiling using dispersive properties of Love waves, Geophysics, 49, 1169-1179

• Marquardt D.W., 1963, An algorithm for least squares estimation of nonlinear parameters, Journal of the Society of Industrial Applied Mathematics, 2, 431-441

• Martinez M.D., Lana X., Olarte J., Badal J., Canas J.A., 2000, Inversion of Rayleigh wave phase and group velocity by simulated annealing, Physics of the Earth and planetary interiors, 122, 3-17

• McMechan G.A., Yedlin M.J., 1981, Analysis of dispersive wave by wave field transformation, Geophysics, 46, 869-874

• Menke W., 1989, Geophysical Data Analysis: Discrete Inverse theory, International Geophysics Series. San Diego: Academic Press. 289 pp

• Misiek R., Liebig A., Gyulai A., Ormos T., Dobroka M., Drese L., 1995, A joint inversion algorithm to process geoelectric and surface wave seismic data. Part II: applications, Geophysical Prospecting, 45, 65-85

• Mockart T.A., Herrmann, R.B., Russel R.D., 1988, Seismic velocity and Q model for the shallow structure of the Arabian shield from short-period Rayleigh waves, Geophysics, 54, 1379-1387

• Nazarian S., Stokoe K.H. II, 1986, Use of surface waves in pavement evaluation, Transp. Res. Rec, 1070, 132-144

• Nazarian S., Stokoes II K.H., 1984, In situ shear wave velocity from spectral analysis of surface waves, Proc 8th Conference on Earthquake engineering- St Francisco, vol. 3, Prentice Hall, pp.31-38

• Nolet G., Panza G.F., 1976, Array analysis of seismic surface waves: limits and possibilities, Pure and Applied geophysics, 114, 776-790

• Nolet, G., 1987, Seismic tomography, D.Reidel Publishing Co., Dordrecht, Holland, 386pp

• Oppenheim A.V., Schaefer R.W., 1975, Digital Signal Processing, Prentice-Hall, Inglewood Cliffs

references

• Park C.B., Miller R.D., Xia J., 1999, Multichannel analysis of surface waves, Geophysics, 64(3), 800-808

• Park C.B., Miller R.D., Xia J., 2001, Offset and resolution of dispersion curve in multichannel analysis of surface waves (MASW), proc. SAGEEP 2001

• Pecorari C., 2001, Scattering of a Rayleigh wave by a surface breaking crack with faces in partial contact, Wave Motion, 33, 259-270

• Penrose R., 1955, A generalized inverse for matrices, Proc.Camb. Phil. Soc., 51, 406-413

• Poggiagliolmi E., Berkhout A.J., Boone M.M., 1982, Phase unwrapping, possibilities and limitations, Geophysical Prospecting, 30, 281-291

• Press F., 1968, Earth models obtained by Monte Carlo inversion, Journal of Geophysical Research, 73, 5223-5234

• Press F., Harkrider D., Seafeldt C.A., 1961, A fast convenient program for computation of surface wave dispersion curves in multi-layered media, Bulletin of the seismological society of America, 51, 495-502

• Press W.H., Teukolsky S., Vetterling W., Flannery B, 1992, Numerical Recipes – The art of technical computing, Cambridge University Press, Cambridge, 2nd edition

• Randall M.J., 1967, Fast programs for layered half-space problems, Bulletin of the Seismological Society of America, 57, 1299-1316

• Rayleigh J.W.S., 1985, On waves propagated along the plane surface of an elastic solid, Proc. Lond. Math. Soc., 17, 4-11

• Richart F.E. Jr., Woods R.D., Hall J.R., 1970, Vibration of soil and foundations, Prentice-Hall, New Jersey

• Rix G.J., Lai C.G., Foti S., 2001, Simultaneous Measurement of Surface wave dispersion and attenuation curves, Geotechnical Testing Journal, ASTM, 24, 350-358

• Roesset J.M., Chang D.W., Stokoe K.H. II, 1991, Comparison of 2D and 3D models for analysis of surface wave tests, Proc. 5th Int. Conf. On Soil Dyn. And Earthq. Eng., Karlsruhe, vol.1, pp 111-126

• Sambuelli L, Deidda G.P., Albis G., Giorcelli E., Tristano G., 2001, Comparison of standard horizontal geophones and newly designed horizontal detectors, Geophysics, 66, 1827-1837

• Sanchez-Salinero I., 1987, Analytical investigation of seismic methods used for engineering applications, PhD Dissertation, University of Texas at Austin

• Santamarina J.C., Fratta D., 1998, Introduction to discrete signals and inverse problems in civil engineering, ASCE Press, New York, 327 pp

• Schwab F, Knopoff L., 1970, Surface wave dispersion computation, Bulletin of the seimological society of America, 60, 321-344

• Schwab F., Knopoff L., 1971, Surface waves on multilayered anelastic media, Bulletin of the seismological society of America, 61, 893-912

• Sheriff R.E., Geldart L.P., 1995, Exploration Seismology, University Press, Cambridge

references

• Shtivelman V., 1999, Using surface waves for estimating the shear-wave velocities in the shallow subsurface onshore and offshore Israel, European journal of environmental and engineering geophysics, 4, 17-36

• Smith F.B., Shanno D.F., 1971, An improved Marquardt procedure for nonlinear regression, Technometrics, 13, 63-75

• Socco L.V., Strobbia C. (2003) Extensive Modelling To Study Surface Wave Resolution Accettato per gli Atti del SAGEEP 2003, San Antonio, Apr. 2003 CDROM.

• Socco L.V., Strobbia C., Foti S., 2002, Multimodal Interpretation of Surface Wave Data Atti del 8th Meeting Environmental and Engineering Geophysics Society European Section, 8 – 14 settembre, Aveiro.

• Stokoe K.H. II, Nazarian S., 1985, Use of Rayleigh wave in liquefaction studies, Proc. Of the Measurement and Use of shear wave velocity for evaluating dynamic soil properties, ASCE, N.Y., 1-17

• Stokoe K.H., Wright S.G., Bay J., Roesset J.M., 1994, Characterization of geotechnical sites by SASW method, in Geophysical characterization of sites, (ISSMFE TC#10) by R.D. Woods (ed), Oxford & IBH Publ., pp. 15-25

• Stokoes K.H. II, Nazarian S., Rix G.J., Sanchez-Salinero I., Sheu J., Mok Y., 1988, In situ seismic testing of hard-to-sample soils by surface wave method, Eartquake Engineering and Soil Dynamics II- Recent advances in ground-motion evaluation-Park city, ASCE, pp-264-277

• Strobbia C., 2002, Weighted f-k for the surface wave analysis, proc. EEGS-ES meeting, Aveiro, Portugal

• Strobbia C., Foti S., 2003, Statistical Regression of Phase difference in surface wave data, Atti SAGEEP, SanAntonio, Texas, USA

• Szelwis R., Behle A., 1984, Shallow shear wave velocity estimation from multi-modal

Rayleigh waves, in: Shear Wave exploration, S.H. DanBom and S.N. Domenico (editors), Geophysical Development Series, vol 1, SEG, 214-226

• Tarantola, A., 1987, Inverse Problem Theory : Methods for data fitting and model parameter estimation, Elsevier Science Publisher, Amsterdam, 614 pp

• Thomson, W.T., 1950, Transmission of Elastic Waves through a Stratified Solid Medium, Journal of Applied Physics, 21, 89-93,

• Thrower E.N., 1965, The computation of the dispersion of elastic waves in layered media, J.Sound.Vib., 2, 210-266

• Tokimatsu K., 1995, Geotechnical Site characterisation using surface waves, Proc. 1st Int.Conf. on Eartq. Geotechn. Eng., IS-Tokio, Balkema, 1333-1368

• Tokimatsu K., Shinzawa K., Kuwayama S., 1992b, Use of short period microtremors for Vs profiling, Journal of Geotechnical Engineering, 118 (10), 1544-1558

• Tokimatsu K., Tamura S, Kojima H., 1992a, Effects of multiple mode on Rayleigh wave dispersion characteristics, Journal of Geotechnical Engineering, American Society of Civil Engineering, 118 (10) 1529-1543

references

• Tselentis G.A., Delis G., 1998, Rapid assessment of S-wave profiles from the inversion of multichannel surface wave dispersion data, Annali di Geofisica, 41, 1-15

• Van der Veen M., Brouwer J., Helbig K, 1999, Weighted sum method for calculating ground force: an evaluation by using a portable vibrator system, Geophysical Prospecting, 47, 251-267

• Viktorov, I.A., 1967, Rayleigh and Lamb waves: physical theory and applications, Plenum Press, NewYork, 154 pp

• Vucetic, M., 1994, Cyclic threshold shear strains in soils, J.Geotechnical Eng., 120, ASCE, 2208-2228

• Wang, C.Y., Herrmann, R.B., 1980, A numerical study of P-, SV-, SH- wave generation in a plane layered medium, Bulletin of Seismological Society of America, 70 1015-1036

• Watson T.H., 1970, A note on fast computation of Rayleigh wave dispersion in the multilayerd elastic half-space, Bulletin of the Seismological Society of America, 60, 161-166

• White J.E., 1983, Underground Sound. Application of seismic waves, Elsevier Science Publisher, 253 pp.

• Wiggins R., 1972, The general linear inverse problem: implications of surface waves and free oscillation of the earth structure, Rev. Geophys. Space Phys., 10

• Xia J., Miller R.D., Park C.B., 1999, Estimation of near-surface shear-wave velocity by inversion of Rayleigh waves, Geophysics, 64, 691-700

• Yilmaz O., 1987, Seismic data processing, Society of Exploration Geophysics, Tulsa

• Zwicky D., Rix G.J., 1999, Frequency wavenumber analysis of passive surface waves, Proc. Symp. On the Appl. Of Geophysics to Environ. And Eng. Problems, Oakland, 75-84

references

317

Modelling of the Rayleigh wave propagation

Aki and Richards, 1981; Blake and Bond, 1990; 1992. Budaev and Bogy, 1995, 1996, 2001; Dunkin, 1965; Gautesen 2002a, 2002b; Gilbert and Backus, 1966; Gucunski and Woods, 1992; Gucunski et al., 1996; Haskell 1953; Herrmann 1996; Herrmann and Wang, 1985; Hisada 1994, 1995; Kausel and Roesset, 1981; Keilis Borok et al., 1989; Kennet, 1964; Knopoff, 1964; Lai, 1998; Lamb, 1904; Love, 1911; Press et al, 1961; Randall, 1967; Rayleigh, 1885; Schwab and Knopoff 1970, 1971; Thomson, 1950; Thrower, 1965; Viktorov, 1967; Wang and Herrmann, 1980; Watson 1970; White 1970;

Processing of Surface waves Al-Hunaidi, 1994; Audisio et al., 1999; Bloch and Hales, 1968; Dziewonski et al., 1969; Dziewonski and Hales, 1972; Foti, 2000; Herrmann, 1973;Horike, 1985; McMecan and Yedlin, 1981; Nolet and Panza, 1976; Park et al, 1999, 2001; Rix et al., 2001; Strobbia, 2002; Strobbia and Foti, 2003; Tselentis and Delis, 1998; Tokimatsu et al, 1992a; Zwicky and Rix, 1999.

Inversion , surface wave and general Al-Hunaidi, 1998; Backus and Gilbert, 1970; Backus and Gilbert, 1968; Duijndam, 1988; Hering et al, 1995; Jackson, 1972; Lai, 1998; Lai and Rix, 1999; Lanczos, 1961; Levenberg, 1944; Lines and Treitel, 1984; Marquardt, 1963; Martinez et al., 2000; Menke, 1989; Misiek et al., 1995; Penrose, 1955; Press, 1968; Smith and Shanno, 1971; Tarantola, 1987; Wiggins, 1972; Xia et al., 1999

Books Aki and Richards, 1980; Buttkus, 2000; Claerbout, 1985; Ewing et al., 1957; Hardage, 1983; Keilis- Borok et al., 1989; Menke, 1989; Nolet, ; Oppenheim and Shaefer, 1975; Richart et al., 1970; Santamarina and Fratta, 1998; Tarantola, 1987; Viktotov, 1967; White, 1983; Yilmaz, 1987;

Surface Wave Methods · Prof. Luigi Sambuelli, my supervisor, continuously guided and supported my...

Documents

Transcript of Surface Wave Methods · Prof. Luigi Sambuelli, my supervisor, continuously guided and supported my...