Diss. ETH No. 21291

Array Processing for Seismic Surface Waves

A dissertation submitted toETH Zurichfor the degree ofDoctor of Sciences

presented by

Stefano Maranò

Laurea Specialistica in Ingegneria delle Telecomunicazioni, Università degli Studi di Trentoborn on December 23, 1983citizen of Italy

accepted on the recommendation ofProf. Dr. Donat Fäh, examinerProf. Dr. Hans-Andrea Loeliger, co-examinerProf. Dr. Domenico Giardini, co-examinerProf. Dr. Heiner Igel, co-examiner

2013

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Abstract

The analysis of seismic surface waves plays a major role in the under-standing of geological and geophysical features of the subsoil. Indeed seis-mic wave attributes such as velocity of propagation or wave polarizationreflect the properties of the materials in which the wave is propagating.

The analysis of properties of surface waves allows geophysicists togain insight into the structure of the subsoil avoiding more expensiveinvasive techniques (e.g., borehole techniques). A myriad of applicationsbenefit from the knowledge about the subsoil gained through seismic sur-veys. Microzonation studies are an important application of the analysisof surface waves with direct impact on damage mitigation and earthquakepreparedness.

This thesis aims at improving signal processing techniques for theanalysis of surface waves in different directions. In particular, the maingoal is to deliver accurate estimates of the geophysical parameters ofinterest. The availability of improved estimates of the quantities of in-terest will provide better constraints for the geophysical inversion andthus enabling us to obtain an improved structural earth model.

For a rigorous treatment of the estimation of wavefield parameterswe rely on tools from statistical signal processing. Wavefield parametersare estimated using the maximum likelihood (ML) method. A compu-tationally efficient implementation of such an estimator is obtained bymodelling seismic surface waves with a factor graph, a particular type ofprobabilistic graphical model. A theoretical bound on estimation accu-racy, the Cramér-Rao bound (CRB), enables us to quantify the sourcesof uncertainty and provides a benchmark for evaluating estimation algo-rithms.

One main contribution of this work is the development of a methodfor the analysis of seismic surface waves. The method is versatile enoughto model different types of waves and to handle measurements of dif-ferent type. All the wavefield parameters of Love waves and Rayleighwaves, together with all the measurements, are jointly modelled withinthe proposed framework. The method ensures an optimal usage of theavailable measurements according to the ML criterion.

The method also deals with the simultaneous presence of multiplewaves, possibly of different type. The proposed algorithm decomposesthe wavefield by gradually increasing the number of waves modelled anditeratively refining estimates of the parameters of each wave. Sensorswith different noise level are also accounted for and the estimation ac-counts for the possible different quality of the measurements.

Performance is assessed on field measurements of ambient vibrationsfrom sensor arrays. It is shown how the proposed method outperformsmethods in the literature in different ways, namely: Rayleigh wave el-lipticity is retrieved with increased accuracy, the retrograde/progradeparticle motion of the Rayleigh wave is retrieved for the first time, andthe simultaneous presence of multiple waves is considered. It is alsoshown that the implementation of the proposed method exhibits, fora sufficiently large signal-to-noise ratio (SNR), the smallest achievablemean-squared estimation error (MSEE) indicated by the CRB.

The joint processing of translational and rotational motions is testedon recordings from controlled explosions. We show the retrieval of Loveand Rayleigh wave parameters in several settings not considered in theliterature, both in the case of a single six-component sensor and the case

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of an array of three and six-component sensors. Analytic expressions ofthe CRB of each parameter of geophysical interest are derived. Theseexpressions allow us to quantify and understand the sources of uncer-tainty limiting the estimation accuracy of the wavefield parameters. Thestatistical models for Love and Rayleigh waves relying on translationalmeasurements, rotational measurements, and both translational and ro-tational measurements are considered.

The impact of array geometry on the estimation of parameters ofLove and Rayleigh waves is also investigated. It is explained in detailhow the array geometry affects the MSEE of parameters of interest, suchas the velocity and direction of propagation, both at low and high SNRs.A cost function suitable for the design of the array geometry is proposed,with particular focus on the estimation of the wavenumber of both Loveand Rayleigh waves. Several computational approaches to minimize theproposed cost function are presented and compared. Finally, numericalexperiments verify the effectiveness of the proposed cost function andresulting array geometry designs, leading to greatly improved estimationperformance in comparison to arbitrary array geometries, both at lowand high SNR levels.

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Kurzfassung

Die Analyse von seismischen Oberflächenwellen spielt für das Ver-ständnis der geologischen und geophysikalischen Eigenschaften des Un-tergrundes eine wichtige Rolle. Tatsächlich spiegeln Attribute der seismi-schen Welle, wie die Ausbreitungsgeschwindigkeit oder die Polarisierung,die Eigenschaften des Materials wieder, in dem sich die Welle ausbreitet.

Die Analyse der Eigenschaften von Oberflächenwellen erlauben Geo-physikern, einen Einblick in die Struktur des Untergrundes zu gewinnen,und dadurch teurere invasive Techniken (wie z. B. Bohrungen) zu ver-meiden. Die Kenntnisse über den Untergrund, welche durch seismischeMessungen gewonnen werden, kommen einer Vielzahl von Anwendungenzugute. Mikrozonierungen stellen eine wichtige Anwendung in der Ana-lyse von Oberflächenwellen dar, und haben einen direkten Einfluss aufdie Vorbereitung auf Erdbeben und die Begrenzung von Schäden.

Diese Dissertation zielt darauf ab, Signalverarbeitungstechniken zurAnalyse von Oberflächenwellen bezüglich mehrere Aspekte zu verbessern.Insbesondere besteht das Hauptziel darin, möglichst genaue Schätzun-gen der relevanten geophysikalischen Parameter zu bestimmen. Die Ver-fügbarkeit von verbesserten Schätzungen wird bessere Randbedingungenfür die geophysikalische Inversion bereitstellen und erlauben, verbessertestrukturelle Modelle der Erde zu erhalten.

Für eine rigorose Behandlung der Schätzung der Wellenfeldparametersetzen wir auf Werkzeuge aus der statistischen Signalverarbeitung. Wel-lenfeldparameter werden mit Hilfe der Maximum-Likelihood-Methodegeschätzt. Eine rechnerisch effiziente Umsetzung solcher Schätzer wirddurch Modellierung seismischer Oberflächenwellen mit einem Faktorgra-fen, einer bestimmten Art eines probabilistischen graphischen Modells,erhalten. Eine theoretische Grenze der Schätzgenauigkeit, die Cramér-Rao Ungleichung (CRU), ermöglicht es uns, die Quellen der Unsicher-heit zu quantifizieren und stellt einen Vergleichspunkt zur Bewertungder Schätzalgorithmen dar.

Ein Hauptbeitrag dieser Arbeit ist die Entwicklung eines Verfahrenszur Analyse von seismischen Oberflächenwellen. Das Verfahren ist flexi-bel genug, um verschiedene Arten von Wellen zu modellieren und unter-schiedliche Messmethoden zu handhaben. Alle Wellenfeldparameter derLove- und Rayleighwellen zusammen mit all den Messungen werden ge-meinsam modelliert. Das Verfahren gewährleistet eine optimale Nutzungder verfügbaren Messungen nach dem Maximum-Likelihood Kriterium.

Das Verfahren behandelt auch die gleichzeitige Anwesenheit von meh-reren Wellen, möglicherweise von unterschiedlichen Wellentypen. Dervorgestellte Algorithmus zerlegt das Wellenfeld durch allmähliches Er-höhen der Anzahl modellierter Wellen und verfeinert die Schätzung derParameter der einzelnen Wellen iterativ. Sensoren mit unterschiedlichemGeräuschpegel werden auch berücksichtigt und die Schätzung berück-sichtigt die mögliche unterschiedliche Qualität der Messungen.

Die Effizienz der Methode wird aufgrund von Feldmessungen der na-türlichen Bodenunruhe mit einem Array von Sensoren beurteilt. Es wirdgezeigt, dass das vorgeschlagene Verfahren den Methoden aus der Li-teratur in unterschiedlichen Aspekten überlegen ist. Insbesondere wirddie Elliptizität der Rayleighwelle mit verbesserter Auflösung wiederge-geben, die retrograde/prograde Partikelbewegung wird zum ersten Malhergeleitet, und die gleichzeitige Anwesenheit von mehreren Wellen wirdbetrachtet.

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Es wird auch gezeigt, dass die Umsetzung der vorgestellten Methodefür ein ausreichend grosses Signal-Rausch-Verhältnis (SRV) den kleinstenerreichbaren mittleren quadratischen Schätzungsfehler (MSEE) aufweist,welcher von der CRU vorgegeben wird.

Die gemeinsame Bearbeitung von Translations- und Rotationsbewe-gungen wird mit Aufzeichnungen von kontrollierten Explosionen gete-stet. Wir demonstrieren die Herleitung der Paramter der Love- und Ray-leighwellen in mehreren Konfigurationen, welche in der Literatur bis-her nicht berücksichtigt wurden, sowohl im Fall eines einzelnen sechs-Komponenten-Sensors und bei einer Anordnung von Sensoren mit dreiund sechs Komponenten. Es werden analytische Ausdrücke der CRU fürjeden geophysikalisch relevanten Parameter hergeleitet. Diese Ausdrückeermöglichen es uns, die Quellen der Unsicherheit zu verstehen, welche dieSchätzgenauigkeit der Wellenfeldparameter begrenzen. Es werden stati-stische Modelle für Love- und Rayleigh-Wellen berücksichtigt, welche aufTranslationsmessungen, Rotationsmessungen und gemeinsamen Messungder Translations- und Rotationsbewegungen beruhen.

Die Auswirkungen der Arraygeometrie auf die Schätzung der Love-und Rayleighwellenparameter wird untersucht. Es wird im Detail erläu-tert, wie die Arraygeometrie die MSEE der relevanten Parameter, wiedie Geschwindigkeit und Richtung der Ausbreitung, sowohl bei niedri-gem und hohem SRV beeinflusst. Eine Kostenfunktion wird hergeleitet,welche für die Optimierung der Arraygeometrie geeignet ist, dies miteinem besonderen Schwerpunkt auf die Schätzung der Wellenzahlen vonsowohl Love wie auch Rayleighwellen. Mehrere computergestützte Lösun-gen zur Minimierung der Kostenfunktion werden vorgestellt und mitein-ander verglichen. Numerische Experimente bestätigen die Wirksamkeitder vorgeschlagenen Kostenfunktion und die daraus resultierende Formder Arraygeometrie, was zu stark verbesserten Schätzungen verglichenmit beliebigen Arraygeometrien sowohl bei niedrigem und hohem SRVführt.

Contents

Abstract ii

Kurzfassung iv

Contents vi

1 Introduction 1

1.1 Related Work and Background . . . . . . . . . . . . . . . . . . 21.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Array Processing for Seismic Surface Waves Using Factor

Graphs 10

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Seismic Wavefield . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Analysis of Surface Waves . . . . . . . . . . . . . . . . . . . . . 162.4 Modelling Surface Waves with Factor Graphs . . . . . . . . . . 202.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . 242.6 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . 27

3 Seismic Waves Estimation and Wavefield Decomposition 40

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 System Model and Problem Statement . . . . . . . . . . . . . . 423.3 Proposed Technique . . . . . . . . . . . . . . . . . . . . . . . . 453.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 493.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 Processing of Translational and Rotational Motions of Sur-

face Waves 63

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.3 Theoretical Performance Analysis . . . . . . . . . . . . . . . . . 724.4 Processing Technique . . . . . . . . . . . . . . . . . . . . . . . . 804.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 814.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . 93

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Contents vii

4.A Derivation of Fisher Information Matrices . . . . . . . . . . . . 93

5 Sensor Placement for the Analysis of Seismic Surface Waves 96

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.3 Sources of Error . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.4 Problem Statement and Design Criterion . . . . . . . . . . . . 1075.5 Array Design Methods . . . . . . . . . . . . . . . . . . . . . . . 1105.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 1135.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.8 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.A Relationship Between Likelihood Function and Sampling Pattern 1215.B Some Remarks on the MOIs of an Array . . . . . . . . . . . . . 1245.C MIP Array Layouts . . . . . . . . . . . . . . . . . . . . . . . . . 125

6 Conclusions 129

6.1 A Method for the Analysis of Surface Waves . . . . . . . . . . . 1296.2 Sensor Placement for the Analysis of Seismic Surface Waves . . 1326.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

A Seismic Waves Estimation and Wavefield Decomposition with

Factor Graphs 135

A.1 Introduction and System Model . . . . . . . . . . . . . . . . . . 135A.2 Seismic Wavefield . . . . . . . . . . . . . . . . . . . . . . . . . . 136A.3 Proposed Technique . . . . . . . . . . . . . . . . . . . . . . . . 138A.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . 140A.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

B Multi-Sensor Estimation and Detection of Phase-Locked Si-

nusoids 143

B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143B.2 Computing Likelihoods with Factor Graphs . . . . . . . . . . . 145B.3 Connection with the Discrete Fourier Transform . . . . . . . . 148B.4 Noise Variance Estimation . . . . . . . . . . . . . . . . . . . . . 148B.5 Extension to Wave Superposition . . . . . . . . . . . . . . . . . 149B.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

C Estimation of Wavefield Parameters of a Single P Wave at

the Free Surface 151

C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151C.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151C.3 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . 153C.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 153C.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

D Sensor Placement for the Analysis of Seismic Surface Waves

- Supplement 159

D.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159Details of Optimised Arrays . . . . . . . . . . . . . . . . . . . . . . . 205

Contents viii

E Derivation of the Cramér-Rao Bounds for Love and Rayleigh

Waves 205

E.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205E.2 Derivation of Fisher Information Matrices . . . . . . . . . . . . 208E.3 Derivation of Cramér-Rao Bounds . . . . . . . . . . . . . . . . 217

Bibliography 222

Acknowledgements 230

About the Author 231

Short Biography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

Chapter 1

Introduction

The analysis of seismic surface waves plays a major role in the understandingof geological and geophysical features of the subsoil. Indeed seismic wave at-tributes such as velocity of propagation or wave polarization reflect the prop-erties of the materials in which the wave is propagating. Seismic surveyingmethods represent a valuable tool in oil and gas prospection (Sheriff & Gel-dart, 1995) and in geotechnical investigations (Tokimatsu, 1997; Okada, 1997).

The seismic wavefield at the earth surface is primarily composed of surfacewaves and body waves. Love waves and Rayleigh waves are surface waveswhich exhibit different polarization and, in general, propagate with differentvelocities. Both velocity of propagation and certain polarization attributestypically change with frequency. The relationship between phase velocity ofa wave and frequency is called dispersion relation or dispersion curve (Aki &Richards, 1980).

The knowledge of these wave properties allow geophysicists to gain insightinto the structure of the subsoil avoiding more expensive invasive techniques(e.g., borehole techniques). A myriad of applications benefit from the knowl-edge about the subsoil gained through seismic surveys. Seismic hazard studiesare an important application of the analysis of surface waves with direct impacton damage mitigation and earthquake preparedness.

Many different techniques have been developed for the analysis of surfacewaves. They can be classified according to several criteria including: the na-ture of the source of the waves employed; whether relying on a single sensoror on multiple sensors; type of sensors used, e.g. geophone, triaxial seismome-ters, or rotational sensors; assumptions on specific properties of the wavefield.Each method presents distinctive advantages and is more suited to a specificapplication than others.

In this thesis, we are interested in the development of signal processingmethods for the analysis of surface waves. A part of the present work is devotedto the analysis of ambient vibrations using array of sensors. Another part ofthis thesis deals with the joint processing of translational motion and rotationalmotion recordings of surface waves both from single sensor and array of sensors.The design of array geometries suitable for the analysis of surface waves is alsoaddressed in this thesis.

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1.1 Related Work and Background

In this section, selected techniques for the analysis of surface waves are intro-duced. Later in this section, concepts from statistical signal processing andestimation theory that are used in this thesis are presented.

One important application of analysis of surface waves is geotechnical sitecharacterization (Tokimatsu, 1997), used in engineering seismology. The goalof site characterization is to determine characteristics of the ground near thesurface of the earth in order to estimate amplification of an incident seismicwavefield. Of particular interest are amplification of shear waves and the prop-erties of seismic surface waves that are both related to the shear wave velocityprofile, i.e., how shear wave velocity changes with depth. The main moti-vation is that a site of soft sediments (exhibiting low seismic velocities) cangreatly amplify the wavefield generated by an earthquake compared to a rocksite (having high velocities). Therefore the knowledge of soil properties is ofgreat importance in the design of earthquake resistant buildings and in theassessment of existing infrastructure.

To the aim of site characterization, seismic waves travelling near the surfaceof the earth, surface waves, can be studied. The ground motion induced byseismic waves is recorded by seismic instrumentation. From such recordings,the frequency dependence of the wave velocity of surface waves, the dispersionrelation, is estimated using signal processing techniques. In a last step, astructural earth model matching the estimated properties of the wavefield isinferred. More precisely, a model of the seismic wave velocities within theupper layers of the earth, compatible with the observed dispersion relation, isfound by solving a non linear geophysical inverse problem (Tarantola, 2004).

Analysis of Seismic Surface Waves

A number of different approaches have been developed for the analysis of sur-face waves. Each approach has intrinsic features that make it suited for specificapplications. A comprehensive review and classification of surface waves anal-ysis can be found in Rost & Thomas (2002) and Wapenaar et al. (2006). Welimit ourselves to reviewing techniques related to this work.

Seismic Source

One way to differentiate techniques used in the analysis of seismic surfacewaves concerns the source of the waves analysed. Certain techniques employsan active, controlled, source of energy such as a sledgehammer or an explosion.Other techniques exploit seismic waves generated by an earthquake. A lattergroup of techniques use ambient vibrations, very low amplitude seismic wavesgenerated by natural sources and human activities (Bonnefoy-Claudet et al.,2006b).

Single Station Methods

The analysis of surface waves from a single sensor is of great practical interest,because of the simplicity of measurement operations. A widely-used singlestation method is the H/V ratio technique which has been widely used fordifferent purposes (Fäh et al., 2003; Bonnefoy-Claudet et al., 2006a). The

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H/V ratio is used as a proxy for Rayleigh wave ellipticity and to determinethe SH resonance frequency. Methods that attempt to estimate Rayleigh waveellipticity from a single station exist (Hobiger et al., 2009; Poggi et al., 2012).

Multiple Stations Methods

One approach used in the analysis of seismic surface waves involves the useof array of seismic sensors and array processing techniques (Van Trees, 2002).The use of array processing techniques in seismology has a long history. Earlyapplications of array processing techniques to seismology date back severaldecades (Aki, 1957; Capon, 1969; Lacoss et al., 1969). More recent develop-ments allow us to separate Love waves and Rayleigh waves (Fäh et al., 2008).The most important assumption necessary for all these techniques is the pres-ence of plane wave fronts, which implies an underlying layered earth model (1Dmodel).

Seismic interferometry methods, are another class of techniques that rely onthe cross-correlations between pairs of sensors (Wapenaar et al., 2006). Thesemethods exploit the relationship existing among the cross-correlations betweena pair of sensors and the Green’s function between the same two sensors (Claer-bout, 1968). This kind of techniques have now a successful history and havebeen applied in different settings including the analysis of coda waves (Campillo& Paul, 2003) and of ambient vibrations (Shapiro et al., 2005). Typically, im-portant requirements for this class of techniques are the diffuse nature of thewavefield and an even distribution of the sources at every azimuth.

Both array processing techniques and interferometry techniques lead to theretrieval of the dispersion relation of surface waves. In general, the first class oftechniques usually leads to the retrieval of a more accurate dispersion curve inshorter surveying times. Polarization properties of the wavefield may also beretrieved. However the applicability of these techniques is limited to sites wherethe 1D model assumption holds. In contrast, interferometric techniques are notconstrained by such limiting assumption. However they typically require thestacking of a significant number of cross-correlations and thus the availabilityof very long recordings. By means of a final tomographic reconstruction step,interferometric techniques allow the determination of velocity maps rather thanmere dispersion curves obtained with array techniques.

Rotational Seismology

The analysis and the study of rotational motions are, to a certain extent,less developed than other aspects of seismology due to the historical lack ofinstrumental observations. This is due to both the technical challenges involvedin measuring rotational motions and to the widespread belief that rotationalmotions are insignificant (Gutenberg, 1927; Richter, 1958).

In the last years the attention of the seismological community towards thestudy of rotational motion increased significantly (Lee et al., 2009; Igel et al.,2012). One reason being the availability of direct measurements of rotationalmotions.

The most striking feature of rotational motions is that, together with trans-lational motions, they enable us to estimate velocity of propagation of seismicwaves from a point measurement. The amount of rotational motion induced by

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a seismic wave is inversely proportional to the wavelength, and is thus relatedto the velocity of propagation. As a result, a six-components measurementof both translational and rotational motions at a single spatial location gathersufficient information to estimate the velocity of propagation of a seismic wave.This fact unleashes a myriad of potential applications.

The processing of both translational and rotational motions from a singlesensor location has been also addressed and it has been shown that the retrievalof Love wave velocity is possible (Igel et al., 2005; Ferreira & Igel, 2009).

Sensor Placement

Performance of a system for the analysis of surface waves may be substantiallyaffected by the physical arrangement of the sensors and their number. In arrayprocessing techniques, a poor array geometry may, for example, lead to a largeuncertainty in the retrieved dispersion curves.

The limitations of different array geometries have been investigated by dif-ferent authors, e.g., Woods & Lintz (1973); Asten & Henstridge (1984); Toki-matsu (1997); Kind et al. (2005); Wathelet et al. (2008). In particular, theinterest has been to identify a range of wavenumbers, or a related quantitysuch as velocity or slowness, where the result of the array processing is morereliable. The largest and the smallest resolvable wavenumbers have been re-lated either to the array aperture and the smallest interstation distance or tothe height of the sidelobes of the array response function.

Qualitative guidelines, based on empirical evidences, for array design areprovided in Rost & Thomas (2002) and in Kind et al. (2005).

Statistical Signal Processing

In the last decades the availability of seismic recordings increased in termsof quality, quantity, and diversity. Within a similar time frame, the steadyimprovements in digital electronics provide us with unprecedented computingcapabilities. These factors both create the need and enable the developmentof sophisticated data processing algorithms able to model complex systemsexploiting all the available measurements.

Statistical signal processing provides effective means to deal with the esti-mation of the values of parameters from measurements corrupted by randomnoise. This is a well developed subject, lying at the meeting point betweenstatistics and signal processing. Hereafter we introduce selected aspects of sta-tistical signal processing that relevant to this work and refer the interestedreader to Kay (1993) for a more detailed reading.

Parameter Estimation

The estimation of parameters of interest relying on noisy observations is acentral task in statistical signal processing. An estimator is a procedure thatrelies on the noisy measurements to provide an estimate of the parameter ofinterest. Mathematically, an estimator is a function g(·) of the measurements

y providing an estimate θ of the value of the unknown parameter θ, i.e., θ =g(y). The choices about how to design an estimator and the analysis of itsperformance are the main topics of estimation theory.

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Different design choices of an estimator are possible and they may lead todifferent performance. We observe that an estimator is a random quantity,indeed for different noisy observations of y different estimates θ are obtained.Therefore, rather than analysing the quality of a single estimate it is moremeaningful to evaluate performance of an estimator in an average sense. Atleast two properties of an estimator are desirable: unbiasedness and smallmean-squared estimation error (MSEE).

Bias is a measures of the average deviation of the estimate from the truevalue and is defined as

Bias(θ) = Eθ − θ , (1.1)

where E· denotes the expected value operator. An estimator is unbiased

whenever on the average it provides the right value, i.e., Bias(θ) = 0. Otherwisethe estimator is said to be biased.

The MSEE measures the average of the square of errors and is defined as

MSEE(θ) = E(θ − Eθ)2 . (1.2)

It is desirable for an estimator to have small MSEE as this implies smallerfluctuations around the expected value Eθ. As it will be clarified, the MSEEcannot be made arbitrarily small.

The Cramér-Rao bound (CRB) is a lower bound on the MSEE of an esti-mator. For a given statistical model, a fundamental limit exists and the MSEEcannot be made smaller than this bound. Such limit depends on the statisticalmodel and is independent of the estimation technique used and algorithmic im-plementation. The CRB is therefore a useful benchmark to validate the qualityof an estimation algorithm. An analytic expression of the CRB also allows usto understand how the different parameters of the system affect estimationaccuracy.

Maximum Likelihood Estimation

The maximum likelihood (ML) estimation method is a widely-used and wellknown estimation technique (Fisher, 1922). Its popularity is due to the factthat it is almost always possible to implement, at least numerically, an MLestimator. Notably, a ML estimator exhibits optimal performance under manysettings. In particular, for a sufficient sample size, the ML estimator is unbiasedand the MSEE achieves the smallest possible value indicated by the CRB. Themain steps in ML estimation are the following.

First, it is necessary to formalize mathematically the relationship betweenthe measurements, the parameters of interest, and potentially other unobservedquantities. This step accounts for establishing a statistical model for the mea-surements and, in particular, defining the probability density function (PDF)of the measurements. We denote with pY (y, θ) the PDF of the measurementswhere Y are random variables and θ is a deterministic parameter.

The second step is to compute the likelihood of the observations. Thelikelihood function (LF) is readily obtained from the PDF. The LF of theobservations is a function of the parameter θ and is defined as

ℓ(θ) = pY (y, θ) , (1.3)

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where y denotes the observed measurements.The final step, is to maximize the LF over the parameter space.

θML = argmaxθ

ℓ(θ) , (1.4)

where θML is the ML estimate of the parameter. The maximization in (1.4) maybe tackled by different means. At best, an analytic solution may be found. Inother cases, an exhaustive search, over a possibly multidimensional parameterspace, may be the only available option.

Factor Graphs

A complex system where a large number of random variables and statistical pa-rameters interact with complex relationships can be effectively represented by agraphical model (Jordan, 2004). Within the graphical model, observed randomvariables (measurements), unobserved random variables, and parameters of thestatistical model are represented in a unique framework together with the func-tional relationships occurring among them. In our approach we rely on factorgraphs, one flavour among many graphical modelling techniques (Kschischanget al., 2001; Loeliger, 2004; Loeliger et al., 2007).

The factor graph can be used to perform inference tasks in an efficientmanner. As an example, computing the likelihood of the observations andsubsequently ML estimation can be performed exploiting the structure of thegraph. Moreover, by inspecting the graph structure it is possible to understandthe relationship between the different parts of the stochastic system and then,for example, derive sufficient statistics which enable to efficiently compute sta-tistical quantities of interest.

To summarize, the factor graph enables us to define the statistical modelpY (y, θ) and to derive efficient algorithms to address inference tasks. Bothtasks are of crucial importance in practice and may be not trivial for complexsystems.

Spatial Sampling

The sampling of spatial signals exhibits some difficulties that are not present inthe more common setting of sampling signals in time. One important differenceis that when sampling spatially it is not possible to use anti-aliasing filters.Another peculiarity of spatial sampling is that, in certain applications, thenumber of sensors available to sample the wavefield is very limited and uniformsampling schemes cannot be used.

When sampling temporal signals, an analogue anti-aliasing filter is typicallyused before sampling in order to limit the bandwidth of the signal and preventaliasing in the sampled signal. Such filtered signal, can then be sampled ata sufficiently high rate according to the Nyquist-Shannon sampling theorem.This approach is however infeasible for spatial signals since the anti-aliasingfilter cannot be implemented.

When a large number of sensors is available uniform sampling schemes maybe used. For example, if the spatial frequency content of the signal is limitedto a known disk shaped region, hexagonal sampling is optimal in the sense thatthe density of the samples is minimized (Vaidyanathan, 1993). However, in a

7

setting where the number of available sensors is very limited a non-uniformsampling scheme may be necessary (Holm et al., 2001). The choice of thelocation of the sensors is a non-trivial task.

1.2 Motivation

This work aims at improving signal processing techniques for the analysis ofsurface waves in different directions. In particular, the main goal is to deliveraccurate estimates of the geophysical parameters of interest by optimally ex-ploiting the available measurements. The availability of improved estimates ofgeophysical quantities of interest will provide better constraints for the geo-physical inversion and thus an improved structural earth model.

This project started with the Swiss Commission for Technology and Inno-vation project no. 9260.1 PFIW-IW. The goal of the project is to develop a toolthat can be applied to analyses ambient vibration wavefields in areas of hydro-carbon reservoirs. This includes an optimization of existing methods for arraysignal processing, and the development of a real-time system for on-site pro-cessing. Specific objectives addressed in this thesis include: the identificationof different wave types from ambient vibration measurements, the identificationof small amplitude waves, and the optimization of array geometry.

Array processing techniques for the analysis of seismic surface waves cur-rently in use presents some limitations and therefore have potential for im-provements. In several recent applications of array processing to the analysisof ambient vibrations, only the vertical component of the seismometer has beenused (Cornou et al., 2003a; Wathelet et al., 2008). In other work, the horizon-tal and the vertical component are processed separately and Love waves aredistinguished from Rayleigh waves (Fäh et al., 2008). In Poggi & Fäh (2010)Rayleigh wave ellipticity is estimated, however without modeling the retrogradeor prograde particle motion. A rigorous joint treatment of all measurementsand all wavefield parameters is lacking in literature, and has been addressed inthis thesis.

The analysis of surface waves employing measurements of rotational mo-tions is presently limited to the analysis of Love waves from single sensors (Igelet al., 2005; Ferreira & Igel, 2009). In these works it is shown that the re-trieval of Love wave dispersion from a single sensor is possible. To the bestof our knowledge, there are no techniques for the analysis of Rayleigh wavesfrom single sensor and there are no techniques to jointly process rotationaland translational motions from array recordings. This is another main pointaddressed in the thesis.

The design of array geometries has not been addressed quantitatively by theambient vibrations community. It is necessary to develop quantitative criterionand algorithms for sensor placement and move beyond the current qualitativeguidelines presently in literature (Rost & Thomas, 2002; Kind et al., 2005).The issue of sensor placement is the third main point addressed in this thesis.

1.3 Contributions

The main contribution of this work is the development of a method for the anal-ysis of seismic surface waves. The method is versatile enough to model different

8

types of waves and handle measurements of different type. All the wavefield pa-rameters of Love waves and Rayleigh waves together with all the measurementsare jointly modelled within the proposed framework. The method ensures anoptimal usage of the available measurements according to the ML criterion. Inparticular, besides the estimation of Love wave and Rayleigh wave phase veloc-ity, the proposed method accounts for the retrieval of Rayleigh wave ellipticitytogether with the retrograde or prograde sense of rotation of the wave. Themethod also deals with the simultaneous presence of multiple waves, possiblyof different type. This enables to decompose the wavefield and to detect weakerwaves otherwise covered by stronger waves. The proposed method accounts forsensors with different noise level by merging measurements of different qualityproperly. This is critical when using together sensors of different kind such astranslational sensors and rotational sensors.

Performance of the proposed method are first evaluated on synthetic datasetsby means of Monte Carlo simulations. It is shown that the actual implementa-tion exhibits the smallest achievable MSEE indicated by the CRB for a suffi-ciently large signal-to-noise ratio (SNR). Performance are also assessed on fieldmeasurements of ambient vibrations from sensor arrays. The joint processingof translational and rotational motions is tested on recording from controlledexplosions. The case of a single six components sensor and the case of an arrayof three and six components sensors are considered.

The proposed method is flexible enough to be able to model different wavetypes with minor modifications. For example, it is shown on synthetic datahow it is, in ideal conditions, possible to retrieve both P and S wave velocityfrom measurements of a single incident P wave.

The impact of array geometry on the proposed method is also studied. Thedifferent sources of error related to the geometry of the sensor array are re-viewed in detail for the specific cases of Love waves and Rayleigh waves. Aquantitative design criterion is proposed together with sensor placement algo-rithms. Usefulness of the design criterion and effectiveness of the consideredsensor placement algorithms are validated by means of Monte Carlo simula-tions.

Other contributions of this thesis includes the derivation of analytical ex-pressions the CRBs of the quantities of geophysical interest. This providesa benchmark for the evaluation of algorithms and understanding on how thedifferent system parameters affect estimation accuracy.

1.4 Outline

This thesis is composed of a collection of independent manuscripts and is or-ganized as follows.

In Chapter 2, a tutorial style article introduces aspects of the analysis ofsurface waves and their modelling using factor graphs.

Chapter 3 presents a ML approach for the analysis of Love waves andRayleigh waves using translational triaxial seismometers. Applications to syn-thetic datasets and field measurements of ambient vibrations are shown. Addi-tional details concerning the design of the factor graph underlying the proposedmethod are given in Appendix A and Appendix B. Appendix C describes themodelling of a single incident P wave using the proposed method.

9

In Chapter 4, the ML method is extended to address the joint process-ing of translational and rotational motions. Applications to the analysis ofseismic surface waves and the joint processing of rotational and translationalmeasurements are shown.

In Chapter 5, the impact of array geometry on the estimation system isreviewed, sensor placement criterion are proposed, and an algorithm for sensorplacement is presented and evaluated. In Appendix D, a list of arrays obtainedusing the proposed sensor placement method for different numbers of sensorsis given.

Appendix E includes additional details of the derivations of Fisher infor-mation matrices for the statistical models considered in the thesis and theCramér-Rao bounds of the wavefield parameters of geophysical interest.

Chapter 2

Array Processing for Seismic Surface Waves

Using Factor Graphs:

An Application to the Analysis of Ambient Vibrations

Stefano Maranò1, Christoph Reller2, Donat Fäh1, and Hans-AndreaLoeliger2

1 ETH Zurich, Swiss Seismological Service, 8092 Zürich, Switzerland.2 ETH Zurich, Dept. of Information Technology & Electrical Engineering, 8092

Zürich, Switzerland.

Abstract

Using seismic surveying methods it is possible to gather informationabout geological and geophysical features of the subsoil. Depending onthe application, surveying methods consist of different measurement se-tups and involve different signal processing techniques. In this work, weare interested in the analysis of seismic surface waves for geotechnicalsite characterization. In particular, we focus on the processing of arrayrecordings from ambient vibrations. After reviewing array processingtechniques of current use for the processing of surface waves, we give anintroduction to factor graphs, tailored to the analysis of wavefields. Thefactor graph approach provides a framework for maximum likelihood pa-rameter estimation of any wave type in an efficient manner. In addition,in the same framework, we explain how to address wave superpositionby gradually decomposing the wavefield. We show numerical examplesof the processing of ambient vibrations recordings from arrays of seismicsensors.

2.1 Introduction

Analysis of the seismic wavefield enables us to gather knowledge of geologicaland geophysical features of the subsoil. Indeed, seismic wave attributes suchas velocity of propagation or polarization reflect the properties of the materialsin which the wave is propagating. Seismic surveying methods allow us to

The first two authors contributed equally to the writing of this chapter.

10

11

gain insight into the structure of the subsoil avoiding more expensive invasivetechniques (e.g., borehole techniques) and therefore represent a valuable toolin geophysical and geotechnical investigations (Tokimatsu, 1997; Okada, 1997)as well as in oil and gas prospection (Sheriff & Geldart, 1995).

One application of seismic surveying methods is geotechnical site charac-terization (Tokimatsu, 1997), used in engineering seismology. The goal of sitecharacterization is to determine characteristics of the ground near the surfaceof the earth in order to estimate amplification of an incident seismic wave field.Of particular interest are amplification of shear waves and the properties ofseismic surface waves that are both related to the shear wave velocity profile,i.e. how shear wave velocity changes with depth. The main motivation is thata site of soft sediments (low velocities) can greatly amplify the wavefield of anearthquake compared to a rock site (high velocities). Therefore the knowledgeof soil properties is of great importance in the design of earthquake resistantbuildings and in the assessment of existing infrastructure.

To the aim of site characterization, seismic waves traveling near the surfaceof the earth, surface waves, can be studied. The ground motion induced byseismic waves is recorded by seismic instrumentation. From such recordings,the frequency dependence of the wave velocity of surface waves, the dispersionrelation, is estimated using signal processing techniques. In a last step, astructural earth model matching the estimated properties of the wavefield isinferred. More precisely, a model of the seismic wave velocities within theupper layers of the earth, compatible with the observed dispersion relation, isfound by solving a non linear geophysical inverse problem (Tarantola, 2004).

Seismic surveying methods for site characterization can be classified intoactive and passive methods. Active methods require a controllable source ofenergy such as an explosion or a sledgehammer. Common active surface wavemethods are spectral analysis of surface waves (SASW) (Stokoe et al., 1994)and multichannel analysis of surface waves (MASW) (Park et al., 1999). Pas-sive methods exploit seismic waves, called ambient vibrations, generated bynatural sources such as ocean waves, wind or atmospheric changes and by an-thropogenic sources (Bonnefoy-Claudet et al., 2006b). The advantage of passivemethods is their applicability to urban areas and the ability to analyze lowerfrequencies that are very difficult to excite with active techniques, thus allowingto resolve deeper structures in the earth (Okada, 1997).

The purpose of this paper is two-fold. First, we introduce the reader to theseismic wavefield and to array processing techniques in use for the analysis ofsurface waves. Our interest is on passive surveying method for site characteri-zation. Nonetheless, parts of the material presented are valid in other settings.We focus on array processing techniques and we do not explain the details ofthe aforementioned geophysical inverse problem. Second, we provide an in-troduction to factor graphs with emphasis on the techniques necessary for theanalysis of wavefields. We show how the framework provided by factor graphsallows the statistical modeling of complex systems in an effective manner.

This paper is organized as follows. We give an introduction to the seismicwave field of surface waves and the sensors in use to measure it. We reviewthe methodology in use for the analysis of surface waves, in particular arrayprocessing methods used for the analysis of ambient vibrations. The secondpart of this paper focuses on the recent work of the authors (Reller et al., 2011;Maranò et al., 2011). This approach uses factor graphs to obtain a ML estimate

12

κ

(a) A Rayleigh wave with retrogradeparticle motion.

κ

(b) A Love wave.

Figure 2.1: A depiction of the displacement induced by surface waves. Bothwaves travel in the κ direction.

of wavefield parameters in an efficient way. We provide an introduction to factorgraphs, with a focus on the tools necessary for the study of wavefields, togetherwith an application to surface waves. Finally, we provide numerical examplesof the analysis of ambient vibrations.

In our notation, we write vector quantities in a slanted bold type (x, θ)and matrices in a bold Roman type, usually upper-case (H). We use (·)∗ forcomplex conjugation, (·)T for transposition, and (·)H =

(

(·)T)∗

for Hermitiantransposition. We loosely stick to the convention of using upper-case letters forrandom variables (Y , X) and lower-case letters for a realization of a randomvariable (y, x) or deterministic values.

2.2 Seismic Wavefield

Seismic waves are waves propagating through the earth (Aki & Richards, 1980).Consider a test particle at rest in absence of a seismic wave. The passage ofa seismic wave will cause the particle to oscillate and rotate around its restingposition. In an elastic medium, the particle will return to its original positionafter the passage of the wave. In general, the motion of a particle is completelydescribed by six quantities, namely the displacements along three orthogonalaxes and the rotations around these axes.

An important characteristic of the seismic wavefield is the simultaneouspresence of several waves of possibly different type. An important distinctionamong wave types is between body waves and surface waves.

Waves propagating in the interior of the earth are called body waves andare further classified as primary waves (P waves) or secondary waves (S waves).P waves are compressional waves like the acoustic waves: the particle motioninduced by a P wave is aligned with the direction of propagation. S waves,also known as shear waves, induce a particle motion perpendicular to the di-rection of propagation. In the elastic case, body waves are non-dispersive, i.e.,they propagate at each frequency with the same velocity. Anelasticity alsointroduces body wave dispersion.

Surface waves propagate on the surface of the earth, and their energy isconcentrated near the surface. The lower the frequency of a surface wavethe deeper is the penetration depth into the earth. Among surface waves wedistinguish between Rayleigh waves and Love waves. As explained in more

13

0

0

-5

-10

-15

-20

-25

-30500 1000 1500 2000

Velocity [m/s]

Dep

th[m

]

P wave velocity

S wave velocity

(a) P wave and S wave velocities for a structural model of a single layerover an infinite half-space.

00

200

400

600

800

1000

2 4 6 8 10 12

Vel

oci

tyω κ

[m/s]

Frequency [Hz]

Fundamental mode

First higher mode

Second higher mode

(b) Dispersion curve for Rayleigh waves.

00

200

400

600

800

1000

2 4 6 8 10 12

Vel

oci

tyω κ

[m/s]

Frequency [Hz]

Fundamental mode

First higher mode

Second higher mode

(c) Dispersion curve for Love waves.

Figure 2.2: A simple layered-earth model 2.2(a) and the corresponding disper-sion curves for Rayleigh waves 2.2(b) and Love waves 2.2(c).

14

detail below, Rayleigh waves and Love waves have a different polarization.Fig. 2.1 provides a graphical representation of surface waves.

Surface waves exhibit a dispersive behavior, i.e., the velocity of propagationdepends on the frequency. The retrieval of the dispersion relation (i.e., howvelocity depends on frequency) is often the main goal of seismic surveyingmethods using surface waves. Rayleigh waves and Love waves propagate ingeneral with different velocities. In addition, they propagate in modes, i.e.,at a given frequency the wave can propagate only with a discrete number ofspecific velocities.

Fig. 2.2(a) shows the P wave and S wave velocities for a structural modelconsisting of a 25 meters thick layer of soft sediments with low seismic veloc-ities and an underlying bedrock with higher velocities. Body wave velocitiestogether with the density of the materials (not shown) suffice to characterizea simple layer over half-space model. Such a model is representative of whatis typically found in river basins, alpine valleys or lake shores. Remember thatthe motivation to studying surface waves is that their properties, such as dis-persion curve and polarization, carry information about the structure of thesubsoil. The dispersion curves for Rayleigh waves (Fig. 2.2(b)) and Love waves(Fig. 2.2(c)) are computed numerically from the structural model of Fig. 2.2(a),using the propagator matrix method (Gilbert & Backus, 1966).

Modern sensors used in seismology are capable of measuring vector quan-tities such as translations and rotations of the ground particle. Most widelyused in the seismological community are triaxial velocimeters. A single triaxialvelocimeter is capable of measuring the ground velocity along the three orthog-onal directions. Therefore such sensors are sensitive to translation and not torotation. Seismological equipment can measure frequencies as low as few milli-hertz and record movements in the range of nanometers per second (Wielandt& Steim, 1986; Wielandt & Streckeisen, 1982). Rotational seismology is a rela-tively recent field, which was enabled by recent advances in sensor technology.Sensors used in rotational seismology are capable of measuring the groundrotations induced by the passage of a seismic wave (Lin et al., 2009).

In this paper, we restrict ourselves to the translational motions induced byseismic surface waves. To measure surface waves, an array of triaxial velocime-ters is deployed on the surface of the earth. We restrict our interest to smallaperture arrays (tens to hundreds of meters) and work with a flat earth model.Each sensor measures the ground velocity along the direction of the axes of thecoordinate system x, y, and z. Fig. 2.3 depicts the coordinate system we use.We provide wave equations of the displacement field u, despite the actual mea-surement is the velocity field ∂u

∂t . The displacement of the ground at positionp ∈ R3 and time t can be described by the vector field

u(p, t) ,(

ux(p, t), uy(p, t), uz(p, t))

: R4 → R3 . (2.1)

We now give the analytic expression for Rayleigh waves and Love waves.The wave equations we describe hereafter are valid only on the surface of theearth (i.e., for z = 0), for plane wave fronts, and for a monochromatic (i.e.,single frequency ω) wave. The direction of propagation of a surface wave is

given by the wave vector κ , κ (cosψ, sinψ, 0)T, whose magnitude κ is the

wavenumber. The velocity of propagation of the wave is ω/κ. We denote theamplitude and phase of the wave by α and ϕ, respectively.

15

x

y

z

p

ψ

Figure 2.3: Right-handed Cartesian coordinate system with the z axis pointingupward and the azimuth ψ measured counterclockwise from the x axis.

Rayleigh waves exhibit an elliptical particle motion confined to the verticalplane perpendicular to the surface of the earth and containing the directionof propagation of the wave. The particle displacement generated by a singleRayleigh wave is

ux(p, t) = α sin ξ cosψ cos(ωt− κTp+ ϕ)

uy(p, t) = α sin ξ sinψ cos(ωt− κTp+ ϕ) (2.2)

uz(p, t) = α cos ξ cos(ωt− κTp+ π/2 + ϕ) .

The ellipticity angle ξ determines the eccentricity of the elliptical motion andthe sense of rotation of the particle motion of a Rayleigh wave. If ξ ∈ (0, π/2),then the Rayleigh wave elliptical motion is said to be retrograde, cf. Fig. 2.1(a).If ξ ∈ (−π/2, 0), then the wave is said to be prograde. For ξ = 0 and ξ =±π/2 the polarization is vertical and horizontal, respectively. We parameterizeRayleigh waves with the parameter vector θ , (α, ϕ, κ, ψ, ξ). Fig. 2.1(a) depictsthe displacement induced by a Rayleigh wave.

Love waves exhibit a particle motion confined to the horizontal plane, theparticle oscillates perpendicular with respect to the direction of propagation.The particle displacement generated by a single Love wave is

ux(p, t) = −α sinψ cos(ωt− κTp+ ϕ)

uy(p, t) = α cosψ cos(ωt− κTp+ ϕ) (2.3)

uz(p, t) = 0 .

We parameterize Love waves with the parameter vector θ , (α, ϕ, κ, ψ). Fig. 2.1(b)depicts the displacement induced by a Love wave. We assume the wavefield tobe composed of the linear superposition of several Rayleigh and Love waves.

Only certain elements in the parameter vector θ are of interest to the geo-physicist. The quantities α, ϕ, and ψ depend on the source while the quantitiesκ and ξ are independent of the source and depend on the geology of the site,cf. Fig. 2.2. Therefore in a seismic survey the interest lies in the estimation ofsource-independent quantities which are directly related to the geology of thespecific site. In other applications, such as the seismic verification of nuclearexplosions, one is interested in deriving properties of the source.

16

2.3 Analysis of Surface Waves

The first step of the analysis of surface waves is the use of array processingtechniques for the estimation of physical quantities of interest such as velocity(or alternatively wavenumber) and polarization attributes. The processing isrepeated separately at different frequencies and in different time windows. Thenext step is to retrieve the dependence of these quantities on the frequency.Afterwards, these dependencies can be used in the last step, called geophysicalinversion, to infer a structural model.

Overview of Array Processing Techniques

For the analysis of surface waves, a planar sensor array is typically deployedand array processing techniques are used (Wathelet et al., 2008; Di Giulioet al., 2006). Array processing techniques used in geophysics are similar to thetechniques used in the direction-of-arrival (DOA) estimation problem (Krim &Viberg, 1996; Van Trees, 2002). The main difference is that, in the seismic case,the velocity of propagation is also a parameter to be estimated. This differentsetting is referred to as frequency-wavenumber analysis. In fact, velocity isoften the most relevant parameter for the geophysicist. Other distinctive traitsof the seismic wavefield are the wave polarization and the simultaneous pres-ence of several waves of different type. Early applications of array processingtechniques to seismology date back to the sixties (Capon, 1969; Lacoss et al.,1969). For decades, only the vertical seismometer component has been used,which however is not optimal for triaxial sensors.

We concentrate on the analysis of the seismic wavefield assuming that thewavefield does not change over the observation time. In addition, we restrictthe analysis to a specific frequency ω.

Consider a source emitting a monochromatic signal α cos(ωt + ϕ). Thesensor array observes L distinct signals at discrete times tk for k = 1, . . . , K as

Y k = Re(

aα ei(ωtk+ϕ))

+Zk , (2.4)

where Y k ,(

Y(1)k , . . . , Y

(L)k

)T. Measurements are corrupted by zero mean

additive Gaussian noise Zk with diagonal covariance matrix diag(σ21 , . . . , σ

2L).

The vector a ∈ CL is called the steering vector and models how the sourcesignal is scaled and delayed at each sensor. E.g., let the ℓ-th element of the

steering vector be ρℓ eiφℓ , then Y

(ℓ)k = αρℓ cos(ωtk+ϕ+φℓ)+Z

(ℓ)k . In general,

the steering vector depends both on properties of the wavefield (e.g., wavevelocity, polarization) and on characteristics of the array (e.g., position of thesensors, properties of the sensors).

The beamforming techniques usually employed in geophysics operate in thefrequency domain. From the ℓ-th signal the frequency snapshot Yℓ is computedusing the discrete-time Fourier transform as

Yℓ ,

K∑

k=1

Y(ℓ)k e−iωtk . (2.5)

The spectral spatial covariance matrix is defined as

R , E

[

Y YH]

, (2.6)

17

where Y =(

Y1, . . . , YL)T

. An estimate R of the spectral spatial covariancematrix can be obtained with the sample covariance matrix having elements

[R]j,ℓ ,1

W

W∑

w=1

y(w)j (y

(w)ℓ )∗ , (2.7)

where w indexes different observations yj taken at different time windows (VanTrees, 2002).

Now, we recall the DOA estimation problem in the simple case of a uniformlinear array (ULA). Consider a plane wave traveling with known velocity ω/κthrough a ULA of N scalar sensors (i.e. L = N) with inter-sensor distance d.The array steering vector a is defined as

a(ψ) ,(

1, e−iκd cosψ, . . . , e−i(N−1)κd cosψ)T. (2.8)

The complex argument of each element of the steering vector corresponds tothe phase delay with respect to the first element for an incoming wave withazimuth ψ.

In order to estimate the azimuth, the Bartlett beamformer method matchesthe phase delay experienced by the incoming wave at different sensors and thensum up the delayed signals to form the beamformer output (Krim & Viberg,1996). The Bartlett beamformer is

P (ψ) ,a(ψ)H Ra(ψ)

a(ψ)H a(ψ). (2.9)

The value of ψ that maximizes P (ψ) provides an estimate of the DOA. In-deed, it is possible to show that, in the noiseless case, for the steering vectorcorresponding to the true ψ the frequency snapshots Yℓℓ=1,...,L are summedin-phase in (2.9) and the beamformer output is thus maximized. This methodis also referred to as classical beamforming.

In the seismic case, often we do not restrict ourselves to ULAs, but usearbitrary planar arrays, and the velocity of propagation is not known. For anarray of N sensors at positions pnn=1,...,N the steering vector is defined as afunction of both azimuth ψ and wavenumber κ as

a(ψ, κ) ,(

e−iκTp1 , . . . , e−iκ

TpN)T. (2.10)

Note that the dependency on ψ and κ are hidden in the wave vector κ. Theoutput of this beamformer is

P (ψ, κ) ,a(ψ, κ)H Ra(ψ, κ)

a(ψ, κ)H a(ψ, κ). (2.11)

Again, the estimates of ψ and κ are found by maximizing P (ψ, κ). Note thatin both the acoustic setting and in the electromagnetic setting, the velocity ofpropagation of the wave is known and the maximization of P (ψ, κ) is performedonly on the azimuth.

Other array processing techniques widely known in the radar and antennacommunities such as the Capon method (Capon, 1969) and the MUSIC algo-rithm (Schmidt, 1986) have been used in the seismological community. The

18

Capon method, also known as minimum variance distortionless beamformer(MVDB), attempts to minimize the contribution of the noise and of interferingsignals. The output of the Capon beamformer is

P (ψ, κ) ,1

a(ψ, κ)H R−1 a(ψ, κ). (2.12)

The main advantage of the Capon beamformer is its ability to resolve closelyspaced sources which may be seen as a single source by the Bartlett beam-former (Krim & Viberg, 1996).

The multiple signal classification (MUSIC) algorithm is also able to dis-tinguish closely related sources. In the MUSIC algorithm the spectral spa-tial covariance matrix R is decomposed via eigenvalue decomposition. Letλℓℓ=1,...,L and vℓℓ=1,...,L be the eigenvalues, sorted in decreasing order,

and the eigenvectors of R, respectively. In presence of M signals, the first Meigenvalues are called signal-subspace eigenvalues and the remaining L−M arecalled noise-subspace eigenvalues. It is observed that the noise eigenvectors areorthogonal to the true steering vector a⋆, i.e., vHℓ a

⋆ = 0. The estimates forψ, κ are found maximizing the MUSIC pseudo spectrum

P (ψ, κ) ,1

a(ψ, κ)H Πa(ψ, κ), (2.13)

where Π ,∑L

ℓ=M+1 vℓ vHℓ . Moreover, a method for estimating the number of

waves M is required. In (Cornou et al., 2003a), the MUSIC algorithm is usedfor the analysis of ambient vibrations using the vertical component.

In a more general setting, we are interested not only in azimuth and wavenum-ber but also in further properties such as wave polarization. Moreover, we alsowant to process the signals from all the components of the sensors jointly.This setting, in which the number of signals L is a multiple of the numberof sensors N , has been referred to as vector-sensor beamforming (Hawkes &Nehorai, 1998). For a generalized beamformer, we define the steering vectora(θ), accounting for arbitrary scalings and delays, as a function of the wavefieldparameters θ. This beamformer output depends on the wavefield parametersand can be written as

P (θ) ,a(θ)H Ra(θ)

a(θ)H a(θ). (2.14)

As a concrete example, for a Love wave the steering vector a ∈ C2N has theform

a(ψ, κ) ,(

− sinψ e−iκTp1 , cosψ e−iκ

Tp1 , . . .)T

. (2.15)

Eq. (2.3) supports this definition of steering vector. The steering vector for aRayleigh wave can be similarly defined as a function of three parameters, i.e.,a(ψ, κ, ξ) ∈ C3N . The Capon method and the MUSIC algorithm can be usedalso in this setting.

In recent work (Donno et al., 2008), a vector-sensor technique for the anal-ysis of seismic waves using jointly the three components is proposed. First, po-larization and velocity parameters are estimated, then amplitude is estimatedby linear regression.

In (Fäh et al., 2008), a technique able to analyze also the horizontal com-ponents was proposed. The technique is applied to seismic surface waves to

19

distinguishing between Rayleigh and Love waves. Vertical and horizontal com-ponents of the sensors are however processed separately, which is sub-optimal.A method for the estimation of Rayleigh wave ellipticity is proposed in (Poggi& Fäh, 2010); also here the three components are not treated jointly.

Another method used in the analysis of ambient vibrations is called spatialautocorrelation method (SPAC) (Aki, 1957; Köhler et al., 2007). In SPAC,properties of the cross-correlation between the signals recorded at pairs of sen-sors are exploited. This method requires a large number of seismic sourcesuniformly distributed around the array.

Later in this paper we focus on a method for ML estimation of wavefieldparameters and wavefield decomposition that was proposed by the authorsin (Maranò et al., 2011). It can be shown that the general Bartlett beamformerof (2.14) is proportional to the likelihood function when the temporal samplingis uniform and the noise variances σ2

ℓℓ=1,...,L are equal.

Retrieval of Dispersion and Ellipticity Curves

We have shown that using array processing techniques, physical quantities ofinterest such as the wavenumber κ or ellipticity angle ξ can be estimated. Thenext goal is the retrieval of the dispersion curves (for both Rayleigh and Lovewaves) and of the ellipticity curve (only for Rayleigh waves). Such curvesdescribe how the physical quantity of interest changes with frequency. Theestimates of the wavenumber and ellipticity angle obtained at each frequencyand in each time window using array processing techniques must be combined.

Seismic wavefields found in nature have spectral characteristic depending onthe seismic source. Different sources are able to excite different wave types indifferent ranges of frequencies (Bonnefoy-Claudet et al., 2006b). The quantitiesof interest are often unknown functions of frequency (e.g., velocity or polariza-tion as a function of frequency) and different frequencies are typically processedindependently. As an example, Fig. 2.4 shows the outcome of processing thesame data (ten seconds) at six different frequencies and a plot summarizinghow wavenumber changes with frequency. Note that the dispersion relationcan be equivalently represented as in Fig. 2.2(b) (velocity vs. frequency) or asin Fig. 2.4(g) (velocity vs. wavenumber).

During a seismic survey, sources vary over time. Commonly, a long-timerecording is split into shorter windows. Inside each window the seismic sourcesare assumed to be constant. Each window is processed independently and theparameters estimated in the different time windows are jointly represented ingraphical form at the end of the processing by means of, as an example, anhistogram. An expert can individuate the dispersion or ellipticity curves byvisual inspection of such depictions.

Geophysical Inversion

The last stage of seismic surveying consists to finding a structural model thatis able to explain the observed properties of the seismic wavefield (Tarantola,2004). In the analysis of surface waves, these properties are the observed dis-persion curves and ellipticity curves (Wathelet et al., 2008; Wathelet, 2008; Fähet al., 2003). Often a layered-earth model is used. Such a model is described bythe thickness of the layers, by P wave and S wave velocities, and by the material

20

density (see also Fig. 2.2(a)). Finding the parameters describing the structuralmodel, based on the observed dispersion curves and ellipticity curves, poses anon-linear problem. In general, several models can fit the retrieved curves rea-sonably well. The expertise of a geologist and prior knowledge of the geology ofthe site can help to further constraint the inversion and to select a meaningfulmodel.

2.4 Modelling Surface Waves with Factor Graphs

Introduction

We quickly recall that the aim of array processing techniques in the contextof this paper is to find ML estimates of a wave parameter vector θ given themeasured data. For Rayleigh waves θ = (α, ϕ, κ, ψ, ξ) and for Love wavesθ = (α, ϕ, κ, ψ), where α and ϕ are the amplitude and the phase of the wave,κ is the wave number, ψ is the angle of incidence, and ξ is the Rayleigh waveellipticity.

Under certain conditions, the general beamformer output (2.14) is equiva-lent with a likelihood function for this parameter vector under the statisticalmodel in Eq. (2.4). In principle, this equivalence can be derived using classicaltools of statistics (Kay, 1993). In this section we present an alternative ap-proach to ML estimation (Maranò et al., 2011; Reller et al., 2011), in which weuse factor graphs and sinusoidal state-space models (see boxes “Factor Graphsand Likelihood” and “A State-Space Model for Noisy Sinusoids”).

In this framework, extensions to differing (or non-uniform) sampling rates,differing noise variances in each signal, and signal superposition are particularlyeasy to derive. Even more importantly, this approach allows us to highlightunderlying principles which may be used in related problems. Last but notleast, factor graphs provide a unified framework for many other signal process-ing tasks (Loeliger et al., 2007) and we believe that this approach will fosterfurther developments in various areas in statistical signal processing.

A Glue Factor for Sinusoidal State-Space Models

In the following we embark on jointly modelling the PDF of all signals from allsensors for a given frequency ω in a single factor graph. (See box “Factor Graphsand Likelihood” for an introduction to the notion of a factor graph.) This factorgraph, shown in Fig. 2.5(a), consists of one sinusoidal state-space model persignal (See box “A State-Space Model for Noisy Sinusoids”, cf. Fig. S.3), andone additional factor gθ, the glue factor. We name gθ a glue factor becauseit connects all the individual state space models at time t0. The only factordepending on the wave parameter vector θ is the glue factor gθ. This fact willenable us to derive a sufficient statistic for ML estimation of θ.

Let L = 3N be the total number of signals observed by N triaxial sensors.Recall that in our system model (2.4) we formulate each signal as a noisyversion of the scaled and phase shifted sinusoid α cos(ωtk +ϕ), where α and ϕare the amplitude and the phase of the wave. The ℓ-th signal takes the form

Y(ℓ)k = α ρℓ cos(ωtk + ϕ+ φℓ) + Z

(ℓ)k , (2.16)

21

where Z(ℓ)k is zero-mean white Gaussian noise with variance σ2

ℓ . For everywave type of interest we can define a mapping Γ : θ 7→ ρℓ, φℓℓ=1,...,L relatingthe wave parameter vector to the amplitude scalings and phase shifts. For aRayleigh wave as in Eq. (2.2), this mapping is

(ρ3n−2, φ3n−2) =(

sin ξ cosψ,−κTpn)

,

(ρ3n−1, φ3n−1) =(

sin ξ sinψ,−κTpn)

,

(ρ3n, φ3n) =(

cos ξ,−κTpn + π/2)

,

(2.17)

and for a Love wave as in Eq. (2.3), this mapping is

(ρ3n−2, φ3n−2) =(

− sinψ,−κTpn)

,

(ρ3n−1, φ3n−1) =(

cosψ,−κTpn)

,

(ρ3n, φ3n) = (0, 0) ,

(2.18)

where n = 1, . . . , N enumerates the sensors. In principle, such a mapping canbe defined for arbitrary sinusoidal waves. Note that in any such mapping, αand ϕ do not feature.

In the factor graph of Fig. 2.5(a),

X(ℓ)k = α ρℓ

(

cos(ωtk + ϕ+ φℓ)

sin(ωtk + ϕ+ φℓ)

)

(2.19)

is the state vector of the ℓ-th sinusoidal state space model. For ease of notation

we denote the state vectors at time t0 by Sℓ , X(ℓ)0 . Similarly, let

s(θ) , α

(

cosϕ

sinϕ

)

(2.20)

be the state vector of the wave at time t0. With these definitions we now modelthe coupling between the signals by means of Dirac delta constraints in the gluefactor

gθ(s1, . . . , sL) =L∏

ℓ=1

δ(

sℓ −Hℓ(θ) s(θ))

, (2.21)

where the coupling matrices

Hℓ(θ) , ρℓ

(

cosφℓ − sinφℓ

sinφℓ cosφℓ

)

(2.22)

depend on θ via the mapping Γ. The internals of the glue factor (2.21) aredepicted in Fig. 2.5(b). Note that we fix S = s(θ) since θ is a vector of fixed(but unknown) parameters.

The overall factor graph in Fig. 2.5(a) is cycle-free and represents the PDF

f(y,x|θ), where X contains the state vectors

X(ℓ)k

k=1,...,K,ℓ=1,...,Lof all the

state-space models and y contains all the measured signals

y(ℓ)k

k=1,...,K,ℓ=1,...,L.

22

Maximum Likelihood Estimate

The goal is to make an ML estimate

θ = argmaxθ

f(y |θ) (2.23)

of the parameter vector θ. The factor graph in Fig. 2.5(a) with Figs. S.3and 2.5(b) inserted allows us to derive the likelihood function f(y |θ) in termsof sum-product messages on any edge in this graph as in Eq. (S.4). We choosethe edge S in Fig. 2.5(b) and hence get

f(

y∣

∣θ)

=

∫

s

−→µS(s)←−µS(s) ds (2.24)

= ←−µS(

s(θ))

(2.25)

=

∫

· · ·∫

s1,...,sL

gθ(s1, . . . , sL)L∏

ℓ=1

←−µSℓ(sℓ) ds1 · · · dsL , (2.26)

where the second equality is due to the fact that s(θ) is fixed and the thirdequality is a direct application of the sum-product rule (S.3).

We see that for ML estimation of θ, we can first pre-process the observationsy into messages

←−µSℓ

ℓ=1,...,Lusing sum-product message passing, and then

use (2.26). In other words,

←−µSℓ

ℓ=1,...,Lis a sufficient statistic for estimating

θ. Note that this statement in principle applies to more general models, aslong as their PDF factorizes as in Fig. 2.5(a).

At first, Eq. (2.26) looks unwieldy, but for Gaussian messages ←−µSℓ andfactors as in Table 2.1, message update tables from (Loeliger et al., 2007) can

Local function Factor graph symbol

Gaussian PDF N(

x∣

∣0, σ2)

N (0, σ2)X

Equality con-straint

δ(x−y)δ(x− z)=

X

Y

Z

Sum constraint δ(z − x− y)+

XYZ

Linear con-straint

δ(y −Ax) AX Y

Table 2.1: Factor graph nodes for linear Gaussian graphs. We denote the(multivariate) Dirac delta by δ(·) and the Gaussian PDF with mean m andvariance σ2 by N

(

·∣

∣m,σ2)

.

23

be used. These tables show, how the inverse covariance matrix←−WS and mean

vector ←−mS of ←−µS can be expressed in terms of

Hℓ(θ),←−WSℓ ,

←−mSℓ

ℓ=1,...,L.

For the maximization of Eq. (2.25) over θ, we note that s depends onα and ϕ only, while H depends on the remaining parameters only. SinceEq. (2.20) formulates a one-to-one mapping between (α, ϕ) and s, we can findML estimates of (α, ϕ) by first finding an ML estimate of s followed by invertingthe mapping. The ML estimate of s is, however, simply the mean ←−mS of themessage ←−µS .

The resulting, partially maximized expression ←−µS(

←−mS

)

can still be written

in terms of

Hℓ(θ),←−WSℓ ,

←−mSℓ

ℓ=1,...,L, and can be shown to be proportional to

the general beamformer output (2.14) in the case of uniform sampling and if thenoise variances

σ2ℓ

ℓ=1,...,Lare equal for all signals. The final maximization

over the remaining parameters is non-convex and can be done using a gridsearch followed by a gradient ascent method.

Noise Variance Estimation and Superposed Waves

In this section, we consider two extensions which can have a considerable im-pact on the performance of the estimation of waves parameter vectors θ. Incontrast to the previous section, we now describe iterative algorithms for ap-proximate ML estimation. Despite their sub-optimality, these algorithms yieldgood results.

In general, the noise variances

σ2ℓ

ℓ=1,...,Lcan vary considerably among the

signals. In our setup, an increase in σ2ℓ can model not only measurement noise

but also sensor failure, sensor misplacement, interfering signals and weakersuperposed waves. In these circumstances, estimating σ2

ℓ in most cases leadsto an improved weighting of the observed signals and hence to better estimatesθ.

A sensible initial estimate of the noise variance σ2ℓ is simply the signal power

1K

∑K−1k=0

(

y(ℓ)k

)2. We propose to use cyclic maximization (Stoica & Selen, 2004)

to iteratively improve estimates σ2ℓ of σ2

ℓ . Specifically we alternate

θ = argmaxθ

f(

y∣

∣θ, σ21 , . . . , σ

2L

)

(2.27)

and

(

σ21 , . . . , σ

2L

)

= argmax(σ2

1 ,...,σ2L)f(

y∣

∣θ, σ21 , . . . , σ

2L

)

. (2.28)

The maximization (2.27) is the same as (2.23), while for (2.28) either thestandard ML estimates can be used or an approximation thereof as given in(Reller et al., 2011). The latter has the advantage that it depends only of oursufficient statistic

←−µSℓ

ℓ=1,...,Land the signal powers. In the single wave case,

the algorithm described by Eqs. (2.27) and (2.28) will be referred to as the MLmethod.

In a second extension, we consider a scenario where M waves are linearlysuperposed. For any given frequency ω, the measured signals still can be mod-elled as noisy sinusoids. Now there are, however, M parameter vectors θm

24

and M mappings Γm for m = 1, . . . ,M . Since we model Rayleigh and Lovewaves exclusively, each of the M mappings takes the form of Eq. (2.17) orEq. (2.18). It is straightforward to model all the waves simultaneously in theglue factor by using extended matrices

(

Hℓ(θ1), . . . ,Hℓ(θM ))

for ℓ = 1, . . . , L

and an extended wave state(

s(θ1)T, . . . , s(θM )T

)T. But the space over which

to maximize in Eq. (2.25) increases M fold.As an alternative, we again propose cyclic maximization. This algorithm is

started by settingM = 1 and estimating one wave as in the previous subsection.Assume now that we have estimated M wave parameter vectors. We increaseM by one and iterate the following. Pick some m ∈ 1, . . . ,M and updatethe estimate of θm while fixing θj = θjj∈1,...,M\m. This amounts tochanging the glue factor (2.21) to

gθm(s1, . . . , sL) =

L∏

ℓ=1

δ(

sℓ − s6mℓ −Hℓ(θm) s(θm)

)

, (2.29)

where s6mℓ ,

∑

j∈1,...,M\m Hℓ(θj) s(θj) is the estimated state due to all

the waves except for the m-th. The corresponding θm can again be found bymaximizing over θm in Eq. (2.25), where ←−µS is calculated using the glue fac-tor (2.29). In the multiple waves case, the algorithm described by Eqs. (2.27),(2.28), and (2.29) will be referred to as ML method.

2.5 Numerical Examples

In this Section numerical examples of the analysis of surface waves on bothsynthetic and real data are shown. First the estimator mean-squared error(MSE) is compared with the CRB. Next, an example of the functioning of theML method for modeling multiple waves is given using a simple monochromaticsynthetic dataset. Then, a more sophisticated synthetic wavefield is used toanalyze the performance of the estimators presented in this paper. At last, adataset acquired in a real seismic survey is considered.

Cramér-Rao Bound Analysis

We are interested in comparing the MSE of the estimation for different methodswith the theoretical limit given by the CRB Kay (1993). We restrict ourselvesto the analysis of the wavenumber κ as this is the parameter of most practicalinterest. For equal noise variance σ2 in all signals, the element of the Fisherinformation matrix corresponding to the wavenumber κ is

E

[

−∂2 ln f(y |θ)∂κ2

]

=α2K

∑Nn=1

(

∂κTpn∂κ

)2

2σ2. (2.30)

When sensors are arranged regularly spaced on a circle, the Fisher informationmatrix is diagonal. Therefore, the MSE of any unbiased estimator is lower-bounded as

E[

(κ− E[κ])2]

≥ 2σ2

α2K∑N

n=1

(

∂κTpn∂κ

)2 . (2.31)

25

We compare the MSE of the estimates from the ML method of the previoussection and from the Bartlett beamformer (2.11) with the CRB by means ofa numerical simulation. We consider an array of N = 7 sensors and a sin-gle Rayleigh wave with circular particle motion (i.e., ξ = π/4). In Fig. 2.6the MSEs of the ML method and the Bartlett beamformer of Eq. (2.11) arecompared with the CRB for different SNRs. We define SNR , α2/2σ2.

The Bartlett beamformer, which uses only the vertical component, is out-performed by the ML method as expected. The ML method exhibits smallerMSE and achieves the CRB for a sufficiently large SNR. Even for high SNRthe vertical component Bartlett beamformer does not achieve the CRB as thesignals on the horizontal components are disregarded. At low SNR, the MSEsaturates since the wavenumber estimate is constrained by the algorithm im-plementation to belong to a finite interval.

Monochromatic Seismic Wavefield

We present a the first example, in which we generate a synthetic wavefieldcomposed of two Rayleigh waves and two Love waves Maranò et al. (2011).The number of waves M = 4 is known. Waves are monochromatic at knownfrequency of 1 Hz. We use an array of 14 triaxial sensors, 500 samples, and 5 sec-onds of observation. Measurements are corrupted by additive white Gaussiannoise, with different variance in each channel. Noise variances and wavefieldparameters are unknown to the algorithm.

The ML method is used to estimate the model parameters. Fig. 2.7(a)shows how the estimates of the amplitudes α converge toward their true values(dotted lines) after a sufficient number of iterations. The factor graph modelsthe presence of a single Rayleigh wave from iteration 1 until iteration 5. Atiteration 6 the graph is enlarged to account for a second Rayleigh wave, there-fore accounting for the simultaneous presence of two waves. At iterations 10and 15 the graph is further enlarged to account for two additional Love waves.In the end, the graph accounts for four waves. The factor graphs accounts foran additional wave as the likelihood (not shown) converges to a stable value.Similarly, Fig. 2.7(b) shows the estimates of noise variance σ2

ℓ . Sudden decreasein estimated variance in the graph corresponds to the inclusion of an additionalwave in the graph.

Fig. 2.8 depicts the log-likelihood function of a Love wave as a function ofwavenumber κ and azimuth ψ in polar coordinates (κ, ψ). In Fig. 2.8(a) it ispossible to see one strong peak at ψ3 = −π/4 and no other strong peaks arevisible. At iteration 14, one Love wave remains in the wavefield and the associ-ated peak, located at ψ4 = π, is now clearly visible, as shown in Fig. 2.8(b). Atthe last iteration, no more waves remain in the residual wavefield, Fig. 2.8(c).

SESAME Synthetic Dataset

In our second example, we now use a more sophisticated synthetic wave-field developed in the SESAME project Bard, P.-Y. (2008); Bonnefoy-Claudetet al. (2006a). This synthetic dataset captures the complexity of the seismicwavefield, accounting for the simultaneous presence of several seismic sources,emitting both short burst of energy and longer harmonic excitations. It is awavefield of ambient vibrations, where the wavefield is dominated by surface

26

waves (i.e., Rayleigh and Love waves) but also other waves are present (e.g.,body waves). The synthetic wavefield is generated by modeling several seismicsources in a structural model with one layer with low seismic velocities over ahalf-space with higher velocities. The body wave velocity profile for this modelis depicted in Fig. 2.2(a). We use 38 triaxial sensors and solely 10 seconds ofrecording sampled at a rate of 100 Hz. Different frequencies are processed inde-pendently. Of practical interest is the velocity dispersion of surface waves Aki& Richards (1980).

We initially model a single Rayleigh wave, M = 1. In Fig. 2.9(a) the esti-mates of the wavenumber κ (black dots) suggest the Rayleigh wave dispersioncurves. For comparison, the theoretical dispersion curves are depicted by lines.Theoretical curves are computed from the known earth model used in produc-ing the synthetic dataset. Fig. 2.9(b), depicts the estimates for the ellipticityangle ξ for M = 1. Values of ξ ∈ (−π/2; 0) correspond to a retrograde particlemotion while ξ ∈ (0;π/2) to a prograde particle motion.

In Fig. 2.10(a), the number of waves modeled by the factor graph is in-creased to M = 3. It is shown that increasing the number of waves modeledallows to better retrieve the fundamental and the higher modes. Fig. 2.10(b),shows the estimates for the ellipticity angle for M = 3.

In Fig. 2.11, different techniques for the estimation of the wavenumber andellipticity angle of a Rayleigh wave are compared in the single wave setting. TheBartlett beamformer of Eq. (2.11) employing the sole vertical component, thethree components Bartlett beamformer as in Eq. (2.14), and the ML methodare compared. The estimates of the three components Bartlett beamformerand the ML method often belong to the higher modes while the vertical com-ponent Bartlett beamformer to the fundamental mode. This is probably dueto a different distribution of energy among the horizontal and vertical compo-nents for the different modes. To see this compare the theoretical pattern ofthe Rayleigh wave ellipticity depicted in Fig. 2.11(b) and the Rayleigh wavedescribed in Eq. (2.2) with the estimated wavenumbers among the modes inFig. 2.11(a). The analysis of the sole vertical component does not allow theretrieval of Rayleigh wave ellipticity. Indeed, the elliptical motion cannot beinferred without considering the three components. The small difference be-tween the estimates from the three component Bartlett beamformer and fromthe ML method are explained by the different treatment of noise variances.

The Brigerbad Survey

We now show an application of the analysis of ambient vibrations to the realsite of Brigerbad in Switzerland. The data was recorded during a seismic surveyperformed by the Swiss Seismological Service in 2010. The site is located in theRhone valley, a deep alpine valley, in southern Switzerland. An array of twelvetriaxial sensors is deployed in a flat area outside Brigerbad, with geometryas shown in Fig. 2.12. Every sensor has a GPS receiver collecting accuratetiming information and thus enabling the synchronization of all the recordings.Concerning the physical placement of each sensor, a small hole is dug with theaim of removing the first layer of grass and earth. The horizontal alignment ofeach sensor is obtained with a spirit level and the alignment toward north witha compass. The x, y, and z components are aligned with the east-west, north-south, and up-down directions, respectively. Once the setup is completed, the

27

sensors record ambient vibrations for the duration of one hour with a samplingfrequency of 100 Hz.

In Fig. 2.13, six signals from the Brigerbad measurement are shown. Thesignals depict 30 seconds of recording of the three components from sensornumber 1, the central sensor, and sensor number 2. The two sensors are lo-cated 9.8 meters apart. Each trace depicts the velocity of the ground motionat the sensor position along a certain axis. Bursts of energy typical of ambi-ent vibrations can be seen around 17 and 23 seconds. On the trace of the ycomponent of the second sensor (fifth trace from the top), a strong harmoniccomponent is particularly noticeable. This is occasionally found in urban envi-ronment where human activity can affect the measurements with narrowbandcontributions.

The recorded signals are split in windows of the duration of ten seconds.Windows are processed independently as described in the earlier sections. Weuse the Parzen window method Duda et al. (2001) to obtain a single graphicalrepresentation depicting the estimated parameters from the whole recording.

Fig. 2.14 shows the outcome of the processing of the Brigerbad dataset.Figs. 2.14(e) and 2.14(f) show the dispersion curve for Love and Rayleighwaves, respectively, obtained with the Bartlett beamformer. In Figs. 2.14(a)and 2.14(b) it is possible to see the Love wave and Rayleigh wave dispersioncurves obtained with the ML method. The ML method models a fixed num-ber of M = 3 waves. Compared to the Bartlett beamformer more outliers arepresent, but the first higher mode of Rayleigh waves is visible in Fig. 2.14(b).

Fig. 2.14(c), depicts estimates of the Rayleigh wave ellipticity angle ξ. Onlythe estimates associated with the fundamental mode are shown. In Fig. 2.14(d),the estimates associated with the first higher mode are shown. The retrievalof the ellipticity curves provides additional information about the structure ofthe subsoil. Ellipticity curves can be used in addition to the dispersion curvesin the formulation of the geophysical inverse problem.

In most of the pictures, it is possible to see the impact of a narrowbandinterferer, presumably some industrial machinery such as a pump, at around 2.5Hz. From such pictures we recognize the large number of outliers in real cases.However, a trained eye can identify the dispersion curves and the ellipticitycurves.

2.6 Conclusions and Outlook

We have reviewed key aspects of the seismic wavefield and different types ofwaves with particular emphasis on surface waves and their properties. The mostwidely used array processing techniques to analyze seismic surface waves havebeen reviewed. After an introduction to factor graphs tailored to likelihoodcomputation and state-space models, we have presented in more detail how itis possible to perform ML estimation of wavefield parameter of seismic surfacewaves. We have shown how to account for different noise variance on eachchannel, by properly merging the information from sensors with different noiselevel, and how to account for arbitrary sensor positions and arbitrary samplinginstants. In the same framework, the superposition of multiple waves has alsobeen addressed. We have compared the performances of different algorithmswith theoretical limits. We have provided concrete examples from the analysis

28

of seismic surface waves. We have shown results from the ambient vibrationswave field, both from high-fidelity synthetics and from a real seismic survey.

Further developments of the ML technique include the adaptive choice ofthe number of waves modeled. The goal is to find a trade off between goodnessof fit and model complexity. We conjecture that this will remove many outliers.

In the near future, modern sensors will enable to gather an increasing num-ber of diverse measurement. An example is the quickly developing field ofrotational seismology where sensors capable of measuring ground rotation areemployed. Using the factor graph approach it is possible to readily adapts tothis new type of measurement.

Acknowledgments

The authors wish to thank Dr. Jan Burjánek and Dr. Clotaire Michel formaking the recordings of the Brigerbad survey available.

This work is supported, in part, by the Swiss Commission for Technologyand Innovation under project 9260.1 PFIW-IW and by Spectraseis AG.

29

−0.05 0 0.05 −0.05

0

0.05

κ cosψ [m−1]

κsinψ

[m−1]

(a) 3 Hz

−0.05 0 0.05 −0.05

0

0.05

κ cosψ [m−1]

κsinψ

[m−1]

(b) 4 Hz

−0.05 0 0.05 −0.05

0

0.05

κ cosψ [m−1]

κsinψ

[m−1]

(c) 5 Hz

−0.05 0 0.05 −0.05

0

0.05

κ cosψ [m−1]

κsinψ

[m−1]

(d) 6 Hz

−0.05 0 0.05 −0.05

0

0.05

κ cosψ [m−1]

κsinψ

[m−1]

(e) 7 Hz

−0.05 0 0.05 −0.05

0

0.05

κ cosψ [m−1]

κsinψ

[m−1]

(f) 8 Hz

3 4 5 6 7 80

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Fundamental mode

First higher mode

Second higher mode

Est. wavenumber κ

Frequency [Hz]

Wav

enum

ber

[1/m

]

(g) Wavenumbers estimated at different frequencies.

Figure 2.4: The beamformer outputs P (ψ, κ) computed at different frequenciesare shown in Figs. 2.4(a)-2.4(f). Dark colors indicate high values while lightcolors indicate low values. The white crosses indicate the pairs (κ, ψ) maxi-mizing the beamformer output. In Fig. 2.4(g) the estimated wavenumbers areplotted versus frequency suggesting the dispersion curves.

30

Factor Graphs and Likelihood

Factor graphs (Kschischang et al., 2001; Loeliger et al., 2007; Loeliger, 2004)are one flavor among many graphical modeling techniques (Koller et al., 2007).Originally, factor graphs have emerged in the field of error correcting codesas an evident extension of Tanner graphs. We use Forney style factor graphs(Forney, 2001) and we borrow the notation from (Loeliger et al., 2007).

A factor graph represents the factorization of a function on many variablesinto several factors. Each edge in the graph represents a variable and eachnode represents a factor. In our setup, the variables are random variables andthe function a joint PDF of these variables.

Consider the example of a hidden Markov model (HMM) with hidden vari-ables X , (X0, . . . ,XK) and observable variables Y , (Y1, . . . , YK). For sucha model, the joint PDF f factors into conditional PDFs fkk=0,...,K as

f(x,y) = f0(x0)

K∏

k=1

fk(xk, yk |xk−1) . (S.1)

Fig. S.1 depicts the corresponding factor graph. For such a model, the likeli-hood of the fixed observations Y = y is

f(y) =

∫

x

f(x,y) dx , (S.2)

where the integration is over all values of x.The likelihood (S.2) can be computed by sum-product message passing. The

key observation is that the integrations can be “pushed” into the factorizingfunction f(x,y) due to the distributive law. This leads to the calculation ofintermediate results, which can be viewed as messages being sent along theedges of the graph. Such a message is a (potentially scaled and degenerate)PDF on the variable associated with the corresponding edge.

In our notation we use directed edges in the graph. For some edge Z, wedenote the forward message (the message sent in the same direction as theedge) by a left-to-right arrow −→µZ and the backward message (the message sentin the opposite direction) by a right-to-left arrow ←−µZ .

f0 f1X0

Y1 = y1

X1fkXk−1

Yk = yk

XkfKXK−1

YK = yK

XK

Figure S.1: Factor graph for a hidden Markov model.

Consider a factor graph containing the generic factor g(s1, . . . , sn, z) de-picted in Fig. S.2. By definition, the sum-product message −→µZ leaving thisnode on edge Z is calculated in terms of the incoming messages −→µSjj=1,...,n

and the local function g as

−→µZ(z) ,∫

· · ·∫

s1,...,sn

g(s1, . . . , sn, z)

n∏

j=1

−→µSj (sj) ds1 · · · dsn . (S.3)

Using this message update rule, we can send messages in any cycle-free factorgraph starting at the leaves of the graph, until we have calculated two messageson every edge: E.g., for edge Z in Fig. S.2 we have two messages −→µZ and ←−µZ .

31

Factor Graphs and Likelihood (Continued)

From the definition (S.3) of sum-product messages, it follows that the likeli-hood (S.2) can be calculated as

f(y) =

∫

xk

−→µXk(xk)←−µXk(xk) dxk (S.4)

for any edge Xk representing some hidden variable in the factor graph. Thisapplies to any cycle-free factor graph representing a PDF f(x,y).We now assume that the PDF in Eq. (S.1) is parameterized by a parametervector θ affecting only the factor f0:

f(x,y |θ) = f0(x0 |θ)K∏

k=1

fk(yk,xk |xk−1) . (S.5)

In this case the likelihood function f(y |θ) can be calculated as

f(y |θ) =∫

x0

−→µX0(x0 |θ)←−µX0(x0) dx0 , (S.6)

where only the backward message ←−µX0 depends on the data y. It followsimmediately that, for ML estimation of θ, the message ←−µX0 is a sufficientstatistic.

g

S1−→µ

S1

Sn−→µSn

Z−→µZ ←−µZ

Figure S.2: Sum-product message passing through a generic factor g.

gθX

(1)0

, S1

X(1)k−1

y(1)k

X(1)k

X(1)K−1

y(1)K

X(L)0

, SL

X(L)k−1

y(L)k

X(L)k

X(L)K−1

y(L)K

(a) Overall factor graph for f(x,y |θ). The interiors of theboxes are depicted in Figs. 2.5(b) and S.3.

=

S = s(θ)

H1(θ)S1

HL(θ)SL

gθ

(b) Details of the glue factorgθ.

Figure 2.5: Entirety and details of the factor graph.

32

A State-Space Model for Noisy Sinusoids

Consider the following discrete-time state-space model

Xk = AkXk−1

Yk = cXk + Zk(S.7)

for k = 1, . . . , K, where

Ak ,

[

cosΩk − sinΩk

sinΩk cosΩk

]

(S.8)

is the state transition matrix with discrete-time frequency Ωk , ω(tk − tk−1),and c , (1, 0). In this model, ω is the fixed continuous-time frequency, tk arethe (potentially non-uniform) sampling times, Xk ∈ R2 is the state vector, Zkis zero-mean white Gaussian noise with variance σ2, and Yk is the observableoutput. If we fix X0 = α(cosϕ, sinϕ)T then the output is a noisy sinusoid

Yk = α cos(ωtk + ϕ) + Zk . (S.9)

The model (S.7) can be viewed as an HMM with a PDF of the form (S.1). Afactor graph representation is given in Fig. S.1 with the internal factorization ofa single factor fk being depicted in Fig. S.3. The four types of nodes appearingin this graph are listed in Table 2.1 along with their respective functions.

In such a factor graph all messages have the form of a (potentially degenerateand scaled) multivariate Gaussian PDF. The forward message on some edge S

is thus parameterized by a mean vector −→mS , an inverse covariance matrix−→WS ,

and a scale factor. For the backward message we use the notation ←−mS ,←−WS .

For the types of local functions listed in Table 2.1, the message update rulesaccording to the sum-product rule (S.3) can be calculated explicitly in terms ofthe parameters of the messages. These update rules are derived and tabulatedin (Loeliger et al., 2007).

+

Yk = yk

N (0, σ2)Zk

c

=Ak

Xk−1 Xk

fk

Figure S.3: State-space model a for sinusoid.

33

−32 −30 −28 −26 −24 −22 −20 −18 −16 −1410

−7

10−6

10−5

10−4

10−3

MSE

[m−2]

SNR [dB]

Bartlett Beamformer

ML method

Cramér-Rao Bound

Figure 2.6: Comparison of the MSE of wavenumber estimates with the CRBat different SNR.

34

2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

α1

α2

α3

α4

Am

plitu

de

[m]

Iteration

(a) Estimated amplitude at different iterations. The graph accounts for anadditional wave at iteration 1, 6, 10, and 15. The dotted lines correspondto the true values.

2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ21

σ22

σ23

σ24

σ25

σ26

Nois

eVariance

[m2]

Iteration

(b) Estimated noise variance at different iterations. Only six channels areshown. The dotted lines correspond to the true values.

Figure 2.7: Estimated amplitudes and noise variances at different iterations.

−0.06 0 0.06 −0.06

0

0.06

κ cosψ [m−1]

κsinψ

[m−1]

(a) Iteration 1

−0.06 0 0.06 −0.06

0

0.06

κ cosψ [m−1]

κsinψ

[m−1]

(b) Iteration 14

−0.06 0 0.06 −0.06

0

0.06

κ cosψ [m−1]

κsinψ

[m−1]

(c) Iteration 18

Figure 2.8: Log-likelihood function of a Love wave.

35

2 3 4 5 6 7 8 9 10 11 120

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Fundamental mode

First higher mode

Second higher mode

Est. wavenumber κ

Frequency [Hz]

Wav

enum

ber

[1/m

]

(a) Rayleigh wave dispersion curve.

2 3 4 5 6 7 8 9 10 11 12

Frequency [Hz]

Ellip

tici

tyangleξ

[rad]

-π2

-π4

π2

π4

0

Est. ellipticity ξ

(b) Rayleigh wave ellipticity angle curve.

Figure 2.9: Estimates of κ and ξ obtained modeling a single (M = 1) Rayleighwave.

36

2 3 4 5 6 7 8 9 10 11 120

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Fundamental mode

First higher mode

Second higher mode

Est. wavenumber κ

Frequency [Hz]

Wav

enum

ber

[1/m

]

(a) Rayleigh wave dispersion curve.

2 3 4 5 6 7 8 9 10 11 12

Est. ellipticity ξ

Frequency [Hz]

Ellip

tici

tyangleξ

[rad]

-π2

-π4

π2

π4

0

(b) Rayleigh wave ellipticity angle curve.

Figure 2.10: Estimates of κ and ξ obtained modeling three (M = 3) Rayleighwaves.

37

2 3 4 5 6 7 8 9 10 11 120

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Vertical comp.

Three comp.

ML method

Frequency [Hz]

Wav

enum

ber

[1/m

]

(a) Rayleigh wave dispersion curve.

2 3 4 5 6 7 8 9 10 11 12

Three comp.

ML method

Frequency [Hz]

Ellip

tici

tyangleξ

[rad]

-π2

-π4

π2

π4

0

(b) Rayleigh wave ellipticity angle curve.

Figure 2.11: Comparison of different techniques for the analysis of a singleRayleigh wave.

38

Figure 2.12: Geometry of the sensor array used in the Brigerbad survey. Theinlet pinpoints the location of the array within Switzerland. The geographiccoordinates are Swiss coordinates (CH1903).

0 5 10 15 20 25 30

−10

0

10

−10

0

10

−10

0

10

−10

0

10

−10

0

10

−10

0

10

xx

yy

zzG

round

vel

oci

ty[µ

m/s]

Time [s]

Figure 2.13: The picture shows six signals of duration 30 seconds from theBrigerbad measurement. The top three traces belong to sensor 1 and thebottom three to sensor 2 (cf. Fig. 2.12).

39

2 4 6 8 10 12 140

0.05

0.1

Frequency [Hz]

Wavenum

ber

[1/m

]

(a) Love wave dispersion curve (ML method).

2 4 6 8 10 12 140

0.05

0.1

Frequency [Hz]

Wavenum

ber

[1/m

](b) Rayleigh wave dispersion curve (MLmethod).

2 4 6 8 10 12 14

Frequency [Hz]

Ellip

ticity

angleξ

[rad]

-π2

-π4

π2

π4

0

(c) Rayleigh wave ellipticity angle curve forfundamental mode (ML method).

2 4 6 8 10 12 14

Frequency [Hz]

Ellip

ticity

angleξ

[rad]

-π2

-π4

π2

π4

0

(d) Rayleigh wave ellipticity angle curve forthe first higher mode (ML method).

2 4 6 8 10 12 140

0.05

0.1

Frequency [Hz]

Wavenum

ber

[1/m

]

(e) Love wave dispersion curve (Bartlettbeamformer, horizontal components).

2 4 6 8 10 12 140

0.05

0.1

Frequency [Hz]

Wavenum

ber

[1/m

]

(f) Rayleigh wave dispersion curve (Bartlettbeamformer, vertical component).

Figure 2.14: Analysis of Love waves and Rayleigh waves at the Brigerbad siteusing the Bartlett beamformer and the ML method.

Chapter 3

Seismic Waves Estimation and Wavefield

Decomposition:

Application to Ambient Vibrations

Stefano Maranò1, Christoph Reller2, Hans-Andrea Loeliger2, and DonatFäh1

1 ETH Zurich, Swiss Seismological Service, 8092 Zürich, Switzerland.2 ETH Zurich, Dept. of Information Technology & Electrical Engineering, 8092

Zürich, Switzerland.

Published in Geophys. J. Int., vol. 191, no. 1, pp. 175–188, Oct. 2012.

Abstract

Passive seismic surveying methods represent a valuable tool in localseismic hazard assessment, oil and gas prospection, and in geotechnicalinvestigations. Array processing techniques are used in order to estimatewavefield properties such as dispersion curves of surface waves and ellip-ticity of Rayleigh waves. However, techniques presently in use often failto properly merge information from three-components sensors and do notaccount for the presence of multiple waves. In this paper, a techniquefor maximum likelihood estimation of wavefield parameters including di-rection of propagation, velocity of Love waves and Rayleigh waves, andellipticity of Rayleigh waves is described. This technique models jointlyall the measurements and all the wavefield parameters. Furthermore itis possible to model the simultaneous presence of multiple waves. Theperformance of this technique is evaluated on a high-fidelity syntheticdataset and on real data. It is shown that the joint modeling of all thesensor components, decreases the variance of wavenumber estimates andallows the retrieval of the ellipticity value together with an estimate ofthe prograde/retrograde motion.

40

41

3.1 Introduction

Analysis of the seismic wavefield enables us to gather knowledge of geologicaland geophysical features of the subsoil. Indeed seismic wave attributes suchas velocity of propagation or polarization reflect the properties of the struc-ture in which the wave is propagating. The analysis of these properties allowgeophysicists to gain insight into the subsoil avoiding more expensive invasivetechniques (e.g., borehole techniques). Seismic surveying methods representa valuable tool in oil and gas prospection (Sheriff & Geldart, 1995) and ingeophysical investigations (Tokimatsu, 1997; Okada, 1997).

In this paper we present an application to ambient vibrations of a recentlyproposed technique for the analysis of the seismic wavefield (Maranò et al.,2011). Ambient vibrations are seismic waves generated by natural or anthro-pogenic sources such as ocean waves, atmospheric changes or traffic (Bonnefoy-Claudet et al., 2006b) which are exploited in passive seismic methods. Theadvantage of passive methods is their applicability to urbanized areas and theability to analyze lower frequencies that cannot be excited with active tech-niques, thus allowing to resolve deeper structures in the earth (Okada, 1997).

In the case of a structure with low-velocity sediments above rock, the seismicwavefield of ambient vibrations is primarily composed of surface waves. Butother waves are present such as body waves and resonances. In particular, theseismic wavefield is composed of an unknown number of simultaneously presentwaves of different type. In this work we focus on the analysis of surface waves,the interest lies in estimating the frequency dependence of the velocity andwave polarization. Specifically, we are interested in retrieving the dispersionrelation for both Love wave and Rayleigh wave, and Rayleigh wave ellipticity.

In order to infer subsurface features of the earth, it is necessary to solvea geophysical inverse problem, see e.g. (Tarantola, 2004). The properties ofthe seismic wavefield deduced from seismic surveys are used in such an inverseproblem.

For the analysis of surface waves from ambient vibrations, a planar sen-sor array is typically deployed and array processing techniques are employed.Most of the array processing techniques in use assume planar wave fronts. Inparticular, frequency-domain beamforming techniques can be used. Central tothese methods is the estimation of the spectral spatial covariance matrix, seee.g., Van Trees (2002).

Two well-known techniques are the classical beamforming, or Bartlett method (La-coss et al., 1969), and the high-resolution beamforming, or Capon method (Capon,1969). In both techniques signals from different sensors are delayed and summedup. Delays are computed as a function of angle of arrival and velocity of prop-agation. The final estimates of these two parameters are the values that max-imize the sum. In the Capon method the sum is weighted by complex gainsin order to reduce the impact of noise and the disturbance from interferingsignals.

Another technique that found application in the seismological community isthe MUSIC algorithm (Schmidt, 1986). In this method an eigendecompositionof the spectral spatial covariance matrix is performed and properties of thenoise subspace are exploited. Cornou et al. (2003a) have used the MUSICalgorithm for the analysis of ambient vibrations.

42

Single station approaches employing the three components of a triaxial seis-mometer exist (Christoffersson et al., 1988). In array processing, however, fora long time only the vertical component has been used. In recent work (Fähet al., 2008), a technique was proposed to analyze also the horizontal compo-nents. The technique allows to distinguishing between Love waves and Rayleighwaves. Vertical and horizontal components are however processed separately,leading to sub optimal performances. Further work proposes a method for theestimation of Rayleigh wave ellipticity (Poggi & Fäh, 2010). Also this latterwork lacks of a joint treatment of all the three components.

In this paper, we describe a recently developed technique to perform MLparameter estimation of wave parameters (Maranò et al., 2011). This techniquemodels jointly the measurements from all components and all the parameters.It will be shown that this leads to a substantial improvement in the retrievalof the dispersion curves. In addition, an estimate of Rayleigh wave ellipticityincluding the sense of rotation of the particle is provided. We believe that thisnew information will provide a valuable additional constraint for the geophysi-cal inversion. The technique also allows to address the issue of multiple wavesby means of wave field decomposition within the same framework leading toa more accurate parameter estimation and the detection of weaker waves. Weassess the performance of the proposed technique on the ambient vibrationswave field, both on high-fidelity synthetics and on real data, and compare withclassical beamforming (Bartlett method).

The paper is organized as follows. In Section 2, we introduce our notationthereby recalling the wave equations of the displacement field induced by Lovewaves and Rayleigh waves, and we elaborate on the representation of Rayleighwave ellipticity. In the same section we also define the estimation problemaddressed in this paper. In Section 3 we present the technique central to thispaper emphasizing its novel contributions. In Section 4 we provide numericalexamples of the analysis of ambient vibrations from both high fidelity syntheticsand real data. In Section 5 we summarize our contributions.

3.2 System Model and Problem Statement

Seismic Surface Waves

To measure seismic waves, we deploy an array of triaxial seismometers on thesurface of the earth. We restrict our interest to small aperture arrays and workwith a flat earth model. We use a three-dimensional, right-handed Cartesiancoordinate system with the z axis pointing upward. The azimuth ψ is measuredcounterclockwise from the x axis. Each sensor measures the ground velocityalong the direction of the axes of the coordinate system x, y, and z. We saythat each sensor has three components, each component measuring the motionof the ground along a certain direction. For the sake of simplicity, we providewave equations of the displacement field u, despite the actual measurement isthe velocity field ∂u

∂t . The displacement field, at position p ∈ R3 and time tcan be described by the vector field

u(p, t) , (ux(p, t), uy(p, t), uz(p, t)) : R4 → R

3 .

In this paper, we study waves propagating near the surface of the earthand having a direction of propagation lying on the horizontal plane z = 0.

43

We consider the wavefield to be composed of the superposition of several Lovewaves and Rayleigh waves. The wave equations we describe hereafter are validfor z = 0 and for plane wave fronts. The direction of propagation of a waveis given by the wave vector κ = κ (cosψ, sinψ, 0)T, whose length κ is thewavenumber.

Love waves exhibit a particle motion confined to the horizontal plane. Theparticle oscillates perpendicular to the direction of propagation. The particledisplacement generated by a single Love wave at position and time (p, t) is

ux(p, t) = −α sinψ cos(ωt− κTp+ ϕ)

uy(p, t) = α cosψ cos(ωt− κTp+ ϕ) (3.1)

uz(p, t) = 0 .

Rayleigh waves exhibit an elliptical particle motion confined to the verticalplane perpendicular to the surface of the earth and containing the directionof propagation of the wave. The particle displacement generated by a singleRayleigh wave is

ux(p, t) = α sin ξ cosψ cos(ωt− κTp+ ϕ)

uy(p, t) = α sin ξ sinψ cos(ωt− κTp+ ϕ) (3.2)

uz(p, t) = α cos ξ cos(ωt− κTp+ π/2 + ϕ) .

We call ξ ∈ [−π/2, π/2] ellipticity angle of the Rayleigh wave. This quantitydetermines the eccentricity and the sense of rotation of the particle motion. Ifξ ∈ (−π/2, 0), the Rayleigh wave elliptical motion is said to be retrograde (i.e.,the oscillation on the vertical component uz is shifted by +π/2 radians withrespect to the oscillation on the direction of propagation). If ξ ∈ (0, π/2) thewave is said to be prograde. For ξ = 0 and ξ = ±π/2 the polarization is verticaland horizontal, respectively. The quantity |tan ξ| is known as the ellipticity ofthe Rayleigh wave.

We now explain in more detail the parametrization of ellipticity used tomodel Rayleigh waves as used in Eq. 3.2. Commonly, Rayleigh wave ellipticityis referred to as the ratio of the absolute values of the amplitude on the radialcomponent and on the vertical component, i.e., the H/V ratio. Consideringequation Eq. (3.2), and defining H=|α sin ξ| and V = |α cos ξ| it follows that

H

V=|α sin ξ||α cos ξ| = |tan ξ| .

Note that there is no information about the sense of rotation of the particle inthe H/V ratio as the sign of tan ξ is lost. By considering directly the ellipticityangle ξ it is possible to preserve this information and infer the sense of particlerotation.

Fig. 3.1 depicts the two different representations for Rayleigh wave elliptic-ity in the case of a layer over a half space and clarifies this idea. Namely, theSESAME structural model M2.1 (Bard, P.-Y., 2008; Bonnefoy-Claudet et al.,2006a) is used (see also Tab. 3.1). It is known that in such a model the mo-tion of the fundamental mode is retrograde at low frequencies (Malischewskyet al., 2008, 2006). At each singularity (i.e., H = 0 or V = 0) the sense ofrotation changes from retrograde to prograde or vice versa. First, we look at

44

Table 3.1: Details of the SESAME structural model M2.1.

vp vs Qp Qs ρ Thickness[m/s] [m/s] [kg/m3] [m]

Layer 1 500 200 50 25 1900 25Layer 2 2000 1000 100 50 2500 ∞

FundamentalFirst higherSecond higherThird higher

Ellip

ticity

H/V

Frequency [Hz]

0.01

0.1

1

100

2 4 6 8

10

10

(a) Rayleigh wave ellipticity curve in thecommon H/V representation.

FundamentalFirst higherSecond higherThird higher

Frequency [Hz]

Ellip

ticity

angleξ

[rad]

+π2

+π4

0

-π2

-π4

2 4 6 8 10

(b) Rayleigh wave ellipticity curve in the el-lipticity angle ξ representation.

Figure 3.1: Two different representations of Rayleigh wave ellipticity in a layerover a half-space model.

the fundamental mode (solid red line) in the H/V representation of Fig. 3.1(a).The particle motion is retrograde up to 2 Hz, where the first singularity occursand the particle motion is horizontally polarized. Between 2 Hz and 3.8 Hzthe particle motion is prograde, and at 3.8 Hz the wave is vertically polarized.Above 3.8 Hz the motion is again retrograde.

We stress that from this picture it is not possible to get any informationabout the sense of rotation of the particle and we are able to draw the aboveconclusions only because of our knowledge about the structural model. InFig. 3.1(b) the ellipticity is represented by means of the ellipticity angle ξ.As explained earlier in this section, the particle motion is retrograde whenξ ∈ (−π/2, 0) and it is prograde when ξ ∈ (0, π/2). The polarization is verticalfor ξ = 0 and horizontal for ξ = ±π/2. Similar considerations can be made forthe higher modes. This latter representation of Rayleigh wave ellipticity allowsto visualize the sense of rotation of the wave.

Problem Statement

Our end goal lies in the estimation of wavefield parameters θ based on noisymeasurements y from an array of seismometers. For a Love wave we definethe parameter vector θ(L)

, (α, ϕ, κ, ψ). For a Rayleigh wave we define θ(R),

(α, ϕ, κ, ψ, ξ). First, we are interested in computing the likelihood p(y|θ) of themeasurements y given a specific parameter vector θ. Second, these likelihoodcomputations enable us to perform ML parameter estimation.

The seismic wavefield is composed of multiple, simultaneously present,waves. This interference can downgrade the quality of the result of the anal-

45

ysis. In this work, we propose an approach, called wavefield decomposition,enabling us to separate the contribution of different waves and improving theaccuracy of the parameter estimation.

In addition, we assume the noise variance to be different on each sensorand on each sensor component. Therefore we are interested in estimating thesenoise variances. In the estimation of wavefield parameters, more weight is givento sensors with smaller noise variance and less weight to noisy sensors.

3.3 Proposed Technique

Overview

In the proposed technique we devise a statistical model of the seismic wave fieldthereby tackling the superposition of an unknown number of waves of differenttype. In this section we describe how the algorithm deals with:

• wave field parameters estimation in the single wave setting,

• wavefield parameters estimation in the multiple wave setting,

• wave type choice,

• determination of the number of waves,

• noise variance estimation.

In the final application, different frequencies are processed separately anda long recording is split in shorter time windows. The composition of thewavefield is allowed to change at different frequencies and in different timewindows. In this section, we describe the modeling of multiple monochromaticwaves with the same frequency. The wavefield composition (i.e., the numberand the type of waves) is assumed to remain unchanged within each timewindow.

An informal high-level description of the operating principle of the proposedmethod is provided in Algorithm 1.

Maximum Likelihood Parameter Estimation

Our interest lies in computing the likelihood of the observations y for a specificwave type and wave parameter vector θ. Then a maximization of the likelihoodfunction enables us to perform ML parameter estimation.

We rely on noisy measurements from L channels. In the case of N three-components sensors, we have L = 3N . In particular, on the ℓ-th channel the

measurements Y(ℓ)k at discrete instants tk for k = 1, . . . , K are

Y(ℓ)k = u(pℓ, tk) + Z

(ℓ)k ,

where u(pℓ, tk) is a deterministic function with unknown wavefield parameters

θ and Z(ℓ)k is zero-mean additive white Gaussian noise with variance σ2

ℓ . Withthis signal model, the PDF of the observations y is

p(y |θ) =L∏

ℓ=1

K∏

k=1

1√

2πσ2ℓ

e−(

y(ℓ)k

−u(pℓ,tk))2/2σ2

ℓ , (3.3)

46

where we have grouped all the measurement as y = y(ℓ)k k=1,...,Kl=1,...L .

Observe that, for a given wave type, u(pℓ, tk) is a deterministic function ofthe wavefield parameters θ for each ℓ and k. This function is written explicitlyin Eqs. (3.1) and (3.2). The ML estimate θ of the parameter vector θ isobtained by means of the following maximization

θ = argmaxθ

p(

y∣

∣θ)

.

This suffices to estimate wave parameters in the single wave setting. Moredetails on ML estimation can be found, e.g., in Kay (1993).

Wavefield Decomposition

In the seismic wavefield several waves of different functional form are presentsimultaneously. This superposition can severely downgrade the quality of theestimation process if not addressed appropriately. In this work, we propose anapproach, called wavefield decomposition, enabling us to separate the contribu-tion of different waves and improving the accuracy of the parameter estimation.

Assuming a linear medium, each sensor records the linear superposition ofsuch waves. Therefore, in presence of M waves, we have that the measurement

Y(ℓ)k is

Y(ℓ)k =

M∑

m=1

u(m)(pℓ, tk) + Zk ,

where u(m)(pℓ, tk) is the contribution of the m-th wave.It follows immediately, that in the multiple wave setting, the PDF (3.3)

should be altered by replacing u(pℓ, tk) by∑Mm=1 u

(m)(pℓ, tk). The PDF isnow parametrized by several wavefield parameter vectors, (θ1, . . . ,θM ).

In principle, also in the multiple wave setting it is possible to obtain waveparameter estimates by maximizing

(

θ1, . . . , θM)

= argmax(θ1,...,θM )

p(y |θ1, . . . ,θM ) .

Unfortunately, such a maximization is unfeasible, even for small M , becausethe parameter space is increased M -fold.

Therefore we propose a greedy algorithm that increases gradually the num-ber of waves modeled. The algorithm begins modeling a single wave and esti-mates the parameter vector θ1 of the first wave. This wave can be either a Lovewave or a Rayleigh wave. In a second step, the parameters of the first wave arekept fixed to θ1 while the maximization is performed over θ2. The number ofwaves modeled by the algorithm is increased gradually until a stopping crite-rion is reached. Each estimated parameter vector benefits from the estimationof the other waves as the parameter estimation is repeated iteratively.

Model Selection

Two questions arising naturally are how to choose the wave type and howmany waves should be modeled. Both questions pertains to model selection.We employ the Bayesian information criterion (BIC) for this task (Schwarz,1978).

47

The BIC is used both to select the wave type and to stop the from modelingadditional waves. Considering a set of possible models, differing for wave typeand number of waves, the model with the smallest BIC is selected. The BIC isdefined as

BIC = −2p(

y∣

∣θ1, . . . , θM)

+Np ln(LK) ,

where Np denotes the total number of estimated parameters of the model andLK is the number of measurements.

In order to limit the computational complexity of the proposed method, weset the number of waves jointly modeled to be at most Mmax.

Noise Variance Estimation

We assume the measurements to be corrupted by additive white Gaussian noisewith zero mean. However, we do not assume the noise variance to be equalin different sensors or components. The estimation algorithm properly weightsmeasurements from channels with different noise level.

An ML estimate of noise variance can be obtained with the following max-imization

(

σ21 , . . . , σ

2L

)

= argmax(σ2

1 ,...,σ2L)p(

y∣

∣θ1, . . . , θM , σ21 , . . . , σ

2L

)

,

where the wavefield parameters are kept fixed and the maximization is per-formed only on the σ2

ℓℓ=1,...,L. Because of the signal model, it is equivalentto perform L separate maximizations on σ2

ℓ for ℓ = 1, . . . , L.Since the wavefield parameter estimates are influenced by the different noise

variances we iteratively repeat the two maximizations(

σ21 , . . . , σ

2L

)

= argmax(σ2

1 ,...,σ2L)p(

y∣

∣θ1, . . . , θM , σ21 , . . . , σ

2L

)

and(

θ1, . . . , θM)

= argmax(θ1,...,θM )

p(

y∣

∣θ1, . . . ,θM , σ21 , . . . , σ

2L

)

.

Being the likelihood a finite value, this iterative maximization is guaranteed toconverge.

An initial estimate for the noise variance can be obtained from the signalenergy

σ2ℓ =

1

K

K∑

k=1

(y(ℓ)k )2 .

Additional Details

The description of this section provides a rigorous description of the functioningof the proposed method and makes an implementation of the method possibleusing tools widely used in statistics. However, in our implementation, insteadof computing (3.3) directly, we model the PDF of the observations by means ofa factor graph (Loeliger et al., 2007). The factor graph formalism allows to toderive a sufficient statistic and enables us to perform ML parameter estimationin a computationally attractive manner.

Further details of our implementation relying on factor graphs are givenin Reller et al. (2011) and in Maranò et al. (2011).

48

Algorithm 1 High-level description of the proposed method.

1. Mmax ← Maximum number of waves.Initial estimate for σ2

ℓ :2. for ℓ = 1 to L do

3. σ2ℓ = 1

K

∑Kk=1(y

(ℓ)k )2

4. end for

Increase the number of waves from 1 to at most Mmax:5. for m = 1 to Mmax do

6. Compute BIC for a model of m− 1 waves.For all the possible wave types (e.g., Rayleigh, Love) fit the m-th wave:

7. for all T = R,L do

8. repeat

9. θ(T)

m = argmaxθm p(y|θ1, . . . , θm−1, θ(T)m , σ2

1 , . . . , σ2L)

10.(

σ21 , . . . , σ

2L

)

= argmax(σ21 ,...,σ

2L)p(y|θ1, . . . , θm, σ

21 , . . . , σ

2L)

11. until convergence of p(y|θ1, . . . , θm, σ21 , . . . , σ

2L).

12. Compute BIC for a model of m waves.13. end for

14. Choose model with smallest BIC. Potentially, stop adding waves andexit.Refine estimation of existing waves:

15. repeat

16. for i = 1 to m do

17. θi = argmaxθi p(y|θ1, . . . , θi−1, θi, θi+1, . . . , θm, σ21 , . . . , σ

2L)

18. end for

19.(

σ21 , . . . , σ

2L

)

= argmax(σ21 ,...,σ

2L)p(y|θ1, . . . , θm, σ

21 , . . . , σ

2L)

20. until convergence of p(y|θ1, . . . , θm, σ21 , . . . , σ

2L).

21. end for

Summary of Contributions

The proposed method brings several improvements with respect to techniquescurrently in use.

• The proposed technique enables us to perform ML parameter estimationof wavefield parameter in a monochromatic wavefield relying on mea-surements corrupted by additive white Gaussian noise. The approachaccounts for all the measurements and all the parameters jointly. Ap-plicability of the proposed technique is not limited to the applicationpresented in this paper. In particular, the technique allows to combinemeasurements from different types of sensors, and is readily extensibleto waves with different polarization and to spherical wave fronts. Thetechnique can cope with different sampling rates in each sensor.

• Rayleigh wave ellipticity is retrieved including information about the pro-grade or retrograde particle motion. This is useful in mode separationand in the identification of singularities of the ellipticity (i.e., peaks andminima of the H/V representation of the ellipticity).

• The wavefield decomposition addresses the simultaneous presence of mul-

49

tiple waves. By accounting for multiple waves, the estimation accuracyof each wave increases as parameters are iteratively re-estimated. Thisleads to the decomposition of the wavefield and allows the detection ofweaker waves.

• The proposed technique estimates the noise variance in each channel.This brings about various advantages. It enables us to use sensors ofdifferent technology and therefore with different noise levels. A misplacedor badly working sensor, will exhibit a higher noise level and will beautomatically given less weight in the estimation process. Alternatively,it is possible to identify sensors having suspiciously high noise varianceand perform a target check on that specific sensor.

• The issues of spatial sampling and array geometry are outside the scopeof this work. However, it is known that the joint usage of all the sensorcomponents leads to a benefit in terms of spatial aliasing (Hawkes &Nehorai, 1998).

3.4 Numerical Results

Introduction

We present results of the proposed technique in different settings of increasingcomplexity. First, in Section 3.4, we compare the MSE of the proposed estima-tor with the CRB and the MSE of other estimators. In Section 3.4, we analyzea synthetic monochromatic wavefield with the aim of demonstrating the func-tioning of the algorithm in detail. In Section 3.4, we assess the performanceof the algorithm on high-fidelity synthetics of the ambient vibrations wavefielddeveloped during the SESAME project (Bonnefoy-Claudet et al., 2006a; Bard,P.-Y., 2008). At last, in Sections 3.4 and in 3.4, two applications to two sitesin Switzerland are presented. The data was recorded during seismic surveysperformed by the Swiss Seismological Service in 2011.

We now give some details about the processing. All frequencies are pro-cessed independently. We apply no filtering to the recordings other than meanremoval. The whole signal is split into blocks (time windows) of equal lengthwithin which the signal is assumed to be stationary. For comparison, we presentresults obtained using the three-components method for vertical, radial, andtransverse component proposed in Fäh et al. (2008) using the same windowlength. We will refer to the three-components technique simply as “classicalbeamforming”.

In the figures, dispersion curves are shown in wavenumber (in m−1), versusfrequency (in Hz). Ellipticity curves are shown both in ellipticity H/V and inellipticity angle ξ versus frequency.

From the processing of long recordings, a large quantity of estimated waveparameters is available. In order to obtain a single picture representative ofthe results from the whole recording, we use the Parzen window method (Dudaet al., 2001). The resulting gray-scale pictures depict with darker color param-eter values that are frequently estimated, with lighter color less frequent values.

Empirical array resolution limits are computed according to Asten & Hen-stridge (1984). Given the minimum and the maximum array inter-station dis-tance (dmin and dmax respectively), the minimum and maximum resolvable

50

wavenumber are defined as

κmin =2π

dmaxandκmax =

π

dmin.

Such resolution limits are depicted graphically as thin dashed black lines.

Cramér-Rao Bound Analysis

We are interested in comparing the MSE of different estimators with the theo-retical limit given by the CRB (Kay, 1993). The CRB is a lower bound on thevariance of unbiased estimators. We restrict ourselves to the analysis of thewavenumber κ as this is the parameter of most practical interest. For equalnoise variance σ2 in all signals, the element of the Fisher information matrixcorresponding to the wavenumber κ is

E

[

−∂2 ln p(y |θ)∂κ2

]

=α2K

∑Nn=1

(

∂κTpn∂κ

)2

2σ2. (3.4)

When sensors are arranged regularly spaced on a circle, the Fisher informationmatrix is diagonal. Therefore, the MSE of any unbiased estimator is lower-bounded as

E[

(κ− E[κ])2]

≥ 2σ2

α2K∑N

n=1

(

∂κTpn∂κ

)2 . (3.5)

We compare the MSE of three different estimators with the CRB by meansof a numerical simulation. We consider the vertical and the radial componentbeamforming of Fäh et al. (2008) and the ML method of Section 3. We consideran uniform circular array of N = 7 sensors and a single Rayleigh wave withelliptic particle motion defined by ξ = π/3. Such a wave has most of the energyon the horizontal components.

In Fig. 3.2 the MSEs of the ML method and the classical beamforming arecompared with the CRB for different SNRs, where we define SNR = α2/2σ2.At low SNR, where the noise dominates, the estimate is substantially random.The MSE saturates for decreasing SNR since the wavenumber estimate is con-strained by the algorithm implementation to belong to a finite interval. Asthe SNR increases, the ML method always exhibits smaller MSE. For suffi-ciently large SNR, the ML method achieves the CRB. Even for high SNR thevertical component beamformer and the radial component beamformer do notachieve the CRB as they disregard the energy on the horizontal componentsor on the vertical component. The radial component beamformer exhibits ingeneral smaller MSE than the vertical component beamformer because mostof the energy of the wave is on the horizontal components (i.e., H/V =

√3).

Monochromatic Wavefield

In the first example, we generate a synthetic wavefield composed of two Lovewaves and two Rayleigh waves. All waves are monochromatic with knownfrequency of 1 Hz. We use an array of 14 triaxial sensors, 500 samples, and5 seconds of observation. The measurements are corrupted by additive whiteGaussian noise, with different variance in each channel. The true wave field

51

−26 −24 −22 −20 −18 −16 −14 −12 −10 −8

10−7

10−6

10−5

10−4

MSE

[m−2]

SNR [dB]

Vertical component

Radial component

ML method

Cramér-Rao Bound

Figure 3.2: Comparison of the MSE of wavenumber estimates with the CRBat different SNR.

parameters are θ(R)1 = (0.9, 0, 0.03, π/4, π/4)T, θ

(R)2 = (0.7, π4 , 0.03, π/2, π/4)

T,

θ(L)3 = (0.8, π3 , 0.04,−π/4)T, and θ

(L)4 = (0.2, π, 0.04, π)T. The noise variances,

the wavefield parameters, and the number and type of waves are unknown tothe algorithm.

Fig. 3.3(a) shows how the estimates of the amplitudes αℓ converge towardtheir true values (dotted lines) after a sufficient number of iterations. The al-gorithm models additional waves at iterations 6, 11, and 14 as the likelihood(not shown) converges to a stable value. Similarly, Fig. 3.3(b) shows the es-timates of the noise variances σ2

ℓ . Sudden changes in estimated variance inthe graph correspond to the inclusion of an additional wave in the graph. Theimprovements in the estimated parameters between two wave inclusions, aredue to repeated ML estimation of wave parameters and noise variances.

For the same experiment Fig. 3.4 depicts the (normalized) log-likelihood(LL) of Love waves and Rayleigh waves, at different iterations, as a functionof wavenumber and azimuth. At iteration 1, the algorithm computes the like-lihood function for Love waves and Rayleigh waves, as seen in the two leftmostcolumns. Two strong peaks are visible for Rayleigh waves, at azimuths π/4 andπ/2. For Love waves, only one peak at azimuth −π/4 is visible. The algorithmchooses to model, as first wave m = 1, a Rayleigh wave. At iteration 6, thefirst two columns are again showing the likelihood of the data for Love wavesand Rayleigh waves also modeling the Rayleigh wave previously estimated. ForRayleigh waves, now only a single peak is visible as the contribution from thefirst wave is already modeled. The depiction of the likelihood function for Lovewaves appears to be substantially unchanged. The second wave modeled bythe algorithm is a Love wave. At iteration 11, an additional Rayleigh wave ismodeled. At iteration 14, only one Love wave remains in the wavefield (the

wave parametrized by θ(L)4 ) and the associated peak, located at ψ4 = π, is now

visible. In the last iteration, all the four waves are modeled by the algorithm.At each step, wave type choice and algorithm termination are performed usingthe BIC.

52

2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

α(R)1

α(L)3

α(R)2

α(L)4

Am

plitu

de

[m]

Iteration

(a) Estimated amplitudes at different iterations. The graph accounts for anadditional wave at iteration 1, 6, 11, and 14. The dotted lines show the trueamplitude of the waves.

2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ21

σ22

σ23

σ24

σ25

σ26

Nois

eVariance

[m2]

Iteration

(b) Estimated noise variances at different iterations. Only six channels areshown. The dotted lines show the value of the true variances.

Figure 3.3: Estimated amplitudes and noise variances at different iterations.

SESAME Model M2.1

We assess the performance of the algorithm on a synthetic model of a layerover a half-space developed during the SESAME project (Bonnefoy-Claudetet al., 2006a; Bard, P.-Y., 2008). The recording has a duration of 400 seconds.The whole recording is split in non-overlapping windows of 2.5 seconds, eachwindow is processed independently. An array of 14 sensors, with an aperture ofroughly 80 meters is used. The geometry of this array is depicted in Fig. 3.5(a).Table 3.1 shows the geophysical properties of the model analyzed (model M2.1of the SESAME dataset).

In the figures, the results for detected waves are overlaid with the theoreticaldispersion curves and ellipticity curves computed from the structural modelparameters in Table 3.1. For both dispersion and ellipticity curves, the redsolid line refers to the fundamental mode, the dashed blue line to the first

53

It.

LL

Lov

eW

ave

LL

Ray

leig

hw

ave

m=

1(R

ayle

igh)

m=

2(L

ove)

m=

3(R

ayle

igh)

m=

4(L

ove)

p(y

|θ(L))

p(y

|θ(R))

p(y

|θ1)

1B

IC=4.688×104

BIC

=4.587×104

p(y

|θ(L),θ

1)

p(y

|θ(R),θ

1)

p(y

|θ1,θ

2)

p(y

|θ1,θ

2)

6B

IC=4.123×104

BIC

=4.185×104

p(y

|θ(L),θ

1,θ

2)

p(y

|θ(R),θ

1,θ

2)

p(y

|θ1,θ

2,θ

3)

p(y

|θ1,θ

2,θ

3)

p(y

|θ1,θ

2,θ

3)

11

BIC

=4.037×104

BIC

=3.583×104

p(y

|θ(L),θ

1,θ

2,θ

3)

p(y

|θ(R),θ

1,θ

2,θ

3)

p(y

|θ1,θ

2,θ

3,θ

4)

p(y

|θ1,θ

2,θ

3,θ

4)

p(y

|θ1,θ

2,θ

3,θ

4)

p(y

|θ1,θ

2,θ

3,θ

4)

14

BIC

=3.553×104

BIC

=3.581×104

p(y

|θ(L),θ

1,θ

2,θ

3,θ

4)

p(y

|θ(R),θ

1,θ

2,θ

3,θ

4)

p(y

|θ1,θ

2,θ

3,θ

4)

p(y

|θ1,θ

2,θ

3,θ

4)

p(y

|θ1,θ

2,θ

3,θ

4)

p(y

|θ1,θ

2,θ

3,θ

4)

18

BIC

=3.555×104

BIC

=3.557×104

Figure 3.4: The normalized LL functions for Love waves and Rayleigh waves atdifferent stages of the algorithm are depicted in polar coordinates as a functionof wavenumber and azimuth (in each picture, the horizontal axis is κ cosψand the vertical axis is κ sinψ). The two leftmost columns show the LL ofthe residual wavefield, i.e. the LL function of an additional wave while theparameters of the waves estimated in previous iterations are kept fixed. Theother columns show the LL for the waves modeled by the algorithm. The BICvalues shown motivates the choice of wave type and the termination of thealgorithm.

54

2020 2040 2060 2080

2020

2040

2060

2080

x [m]

y[m

]

(a) Array layout with 14 sensors.

1 2 3 4 5 6 7 8 9 10 110

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Wavenum

ber

[1/m

]

Frequency [Hz]

(b) Vertical component.

1 2 3 4 5 6 7 8 9 10 110

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Wavenum

ber

[1/m

]

Frequency [Hz]

(c) Radial component.

1 2 3 4 5 6 7 8 9 10 110

0.01

0.02

0.03

0.04

0.05

0.06

0.07W

avenum

ber

[1/m

]

Frequency [Hz]

(d) Transverse component.

Figure 3.5: Rayleigh wave and Love wave dispersion curves obtained using themethod in (Fäh et al., 2008) for the model M2.1.

higher mode, the dot-dashed magenta line to the second higher mode, and thedotted green line to the third higher mode. Theoretical curves for Rayleighwave and Love wave modes are depicted with the same colors but never appearin the same picture.

Fig. 3.5 depicts results of the method in (Fäh et al., 2008). The Rayleighwave dispersion curves are seen on the vertical (Fig. 3.5(b)) and on the radial(Fig. 3.5(c)) components. Love wave dispersion curve is seen on the transversecomponent (Fig. 3.5(d)).

Fig. 3.6 depicts Love wave and Rayleigh wave dispersion curves as estimatedwith the ML technique. Fig. 3.6(a) and 3.6(c) refer to the algorithm modelingat most one wave (Mmax = 1). Fig. 3.6(b) and 3.6(d) refer to the joint modelingof at most three waves (Mmax = 3).

In general, we observe that the wavenumber estimates exhibit less scatterand less outliers when compared with the results depicted in Fig. 3.5. This isdue to the joint usage of the three components and the use of the BIC.

Fig. 3.7 shows the result of ellipticity estimation. Fig. 3.7(a) and 3.7(b)show the estimate of ellipticity in the H/V representation, for different Mmax.

55

1 2 3 4 5 6 7 8 9 10 110

0.01

0.02

0.03

0.04

0.05

0.06

0.07W

avenum

ber

[1/m

]

Frequency [Hz]

(a) Rayleigh wave dispersion curve, Mmax =1.

1 2 3 4 5 6 7 8 9 10 110

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Wavenum

ber

[1/m

]

Frequency [Hz]

(b) Rayleigh wave dispersion curve, Mmax =3.

1 2 3 4 5 6 7 8 9 10 110

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Wavenum

ber

[1/m

]

Frequency [Hz]

(c) Love wave dispersion curve, Mmax = 1.

1 2 3 4 5 6 7 8 9 10 110

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Wavenum

ber

[1/m

]

Frequency [Hz]

(d) Love wave dispersion curve, Mmax = 3.

Figure 3.6: Rayleigh and Love wave dispersion curves obtained using the MLtechnique for the model M2.1. Comparison between different number of waves.

Fig. 3.7(c) and 3.7(d) show the estimate of the ellipticity angle ξ. In thesefour figures all the estimated parameters are plotted and contribution of dif-ferent modes are not distinguished as easily as for dispersion curves. For thisreason, estimates corresponding to the first mode are isolated in the frequency-wavenumber plane and only the corresponding ellipticity estimates are shownin Fig. 3.8. In this latter figure the behavior of the ellipticity of the fundamentalmode can be understood more clearly. In in Fig. 3.8(c) and 3.8(d) it is possibleto clearly identify the frequency at which the sense of rotation is changing (i.e.,when ξ = 0). In addition, when comparing Fig. 3.8(c) with Fig. 3.8(d), it ispossible to appreciate how the modeling of multiple waves makes it easier tofollow the curves. Also, the estimated curve just above the resonance frequencyof 2 Hz appears more accurate.

In Fig. 3.9 we compare the estimated wavenumbers for Rayleigh waves atdifferent frequencies using different methods. Each figure shows the wavenum-ber estimates at a fixed frequency. The pictures in 3.9 can be compared withFig. 3.5(b), 3.5(c), 3.6(a), and 3.6(b). The theoretical wavenumbers are shownwith vertical lines. Each curve is normalized to have unit area. At 2 Hz,(Fig. 3.9(a)) the ML method better resolves the fundamental mode, which issubstantially undetected by the vertical beamforming and detected with some

56

1 2 3 4 5 6 7 8 9 10 110.01

0.1

1

10

100

Frequency [Hz]

Ellip

ticity

H/V

(a) Rayleigh wave ellipticity curve, Mmax =1.

1 2 3 4 5 6 7 8 9 10 110.01

0.1

1

10

100

Frequency [Hz]

Ellip

ticity

H/V

(b) Rayleigh wave ellipticity curve, Mmax =3.

1 2 3 4 5 6 7 8 9 10 11

Frequency [Hz]

Ellip

ticity

angleξ

[rad]

-π2

-π4

π2

π4

0

(c) Rayleigh wave ellipticity angle curve,Mmax = 1.

1 2 3 4 5 6 7 8 9 10 11

Frequency [Hz]

Ellip

ticity

angleξ

[rad]

-π2

-π4

π2

π4

0

(d) Rayleigh wave ellipticity angle curve,Mmax = 3.

Figure 3.7: Rayleigh wave ellipticity curves obtained using the ML technique forthe model M2.1. No selection on the wavenumber-frequency plane is performed.

bias by the radial beamforming. At 5.0 Hz, (Fig. 3.9(b)) the proposed methoddetects both the fundamental and the first higher mode. The two modes aredetected separately by the vertical and the radial beamforming due to the dif-ferent ellipticity of the different modes. At 8.0 Hz, (Fig. 3.9(c)) the fundamentalmode is more clearly resolved by the ML method. At 10.5 Hz, (Fig. 3.9(c))the ML method detects both the fundamental and the second higher mode.Note that the bias in estimation on the second higher mode is shared by all theestimators. Indeed the estimated mode might be a mixture of the second andthe third higher mode. In general, the proposed method also exhibit a smalleramount of outliers.

Brigerbad, Wallis

The Brigerbad site is located in the Rhone valley, a deep Alpine valley, insouthern Switzerland. An array of 12 Lennartz 5 seconds triaxial sensors isused. The layout of the array is depicted in Fig. 3.10(a). The whole recordingis 58 minutes long and it is split into 10 seconds windows which are processedindependently. Sampling rate is 200 Hz.

57

1 2 3 4 5 6 7 8 9 10 110.01

0.1

1

10

100

Frequency [Hz]

Ellip

ticity

H/V

1 2 3 4 5 6 7 8 9 10 11

0.01

0.02

0.03

0.04

0.05

0.06

0.07

(a) Rayleigh wave ellipticity curve for funda-mental mode, Mmax = 1.

1 2 3 4 5 6 7 8 9 10 110.01

0.1

1

10

100

Frequency [Hz]

Ellip

ticity

H/V

1 2 3 4 5 6 7 8 9 10 11

0.01

0.02

0.03

0.04

0.05

0.06

0.07

(b) Rayleigh wave ellipticity curve for funda-mental mode, Mmax = 3.

1 2 3 4 5 6 7 8 9 10 11

Frequency [Hz]

Ellip

ticity

angleξ

[rad]

-π2

-π4

π2

π4

0

1 2 3 4 5 6 7 8 9 10 11

0.01

0.02

0.03

0.04

0.05

0.06

0.07

(c) Rayleigh wave ellipticity angle curve forfundamental mode, Mmax = 1.

1 2 3 4 5 6 7 8 9 10 11

Frequency [Hz]

Ellip

ticity

angleξ

[rad]

-π2

-π4

π2

π4

0

1 2 3 4 5 6 7 8 9 10 11

0.01

0.02

0.03

0.04

0.05

0.06

0.07

(d) Rayleigh wave ellipticity angle curve forfundamental mode, Mmax = 3.

Figure 3.8: Rayleigh wave ellipticity curves obtained using the ML techniquefor the model M2.1. The fundamental mode is selected in the wavenumber-frequency plane.

Fig. 3.10 shows the results of the analysis performed using the methodin Fäh et al. (2008). The fundamental mode of the Rayleigh wave is visible onthe vertical component (Fig. 3.10(b)). The first higher mode of the Rayleighwave is visible on the radial component (Fig. 3.10(c)) and only weakly on thevertical component. The fundamental mode of the Love wave is visible on thetransverse component (Fig. 3.10(d)).

Fig. 3.11 shows the results of the analysis performed using the ML techniquedescribed in this paper. Both the fundamental mode and the first higher modeof the Rayleigh wave are visible in Fig. 3.11(b). The fundamental mode of theLove wave is visible in Fig. 3.11(a).

In Fig. 3.11 Rayleigh wave ellipticity curves are shown for different modesand different representations. The ellipticity of the fundamental mode is shownin Fig. 3.11(c) and 3.11(d). We emphasize how the zero of the H/V curve,just above 6 Hz, is very clearly identified by looking at the ellipticity anglerepresentation of Fig. 3.11(d). Analogously, in Fig. 3.11(e) and 3.11(f) theRayleigh wave ellipticity for the first higher mode is shown.

58

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Wavenumber [1/m]

VerticalRadialML (1)ML (3)

(a) 2.0 Hz.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

VerticalRadialML (1)ML (3)

Wavenumber [1/m]

(b) 5.0 Hz.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

VerticalRadialML (1)ML (3)

Wavenumber [1/m]

(c) 8.0 Hz.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

VerticalRadialML (1)ML (3)

Wavenumber [1/m]

(d) 10.5 Hz.

Figure 3.9: The estimated wavenumber for Rayleigh waves at different frequen-cies using different methods. Theoretical wavenumbers are shown with verti-cal lines. For the ML method, different values of maximum modeled wavesMmax = 1 and Mmax = 3 are shown.

Rheintal, St. Gallen

The Rheintal site is located in the Rhein valley, an Alpine valley, in easternSwitzerland. An array of 13 Lennartz 5 seconds triaxial sensors is used. Thelayout of the array is depicted in Fig. 3.12(a). The whole recording is almostsix hours long and it is split in 10 seconds windows which are processed inde-pendently. Sampling rate is 200 Hz.

Fig. 3.12 shows the results of the analysis performed using the methodin Fäh et al. (2008). The fundamental mode of the Rayleigh wave is visibleonly in the vertical component (Fig. 3.12(b)). The Love wave fundamentalmode is weakly visible on the transverse component. The analysis of the radialcomponent brings no clear information.

Fig. 3.13 shows the results of the analysis performed using the ML tech-nique described in this paper. The fundamental mode of the Rayleigh waveis visible in Fig. 3.13(b). The fundamental mode of the Love wave is visiblein Fig. 3.13(a). We note that the Love wave dispersion curve is now clearlyvisible with the ML method. Since the ML technique chooses between Loveand Rayleigh wave adaptively, the algorithm tends to model the stronger waves

59

(a) Geometry of the sensor array used in theBrigerbad survey. The inlet pinpoints the lo-cation of the array within Switzerland. Thegeographic coordinates are Swiss coordinates(CH1903).

2 4 6 8 10 12 140

0.05

0.1

Wavenum

ber

[1/m

]

Frequency [Hz]

(b) Vertical component.

2 4 6 8 10 12 140

0.05

0.1

Wavenum

ber

[1/m

]

Frequency [Hz]

(c) Radial component.

2 4 6 8 10 12 140

0.05

0.1

Wavenum

ber

[1/m

]

Frequency [Hz]

(d) Transverse component.

Figure 3.10: Rayleigh wave and Love wave dispersion curves obtained usingthe method in (Fäh et al., 2008) for Brigerbad survey.

first, then removes its contribution, allowing the detection of weaker signals (inthis instance the fundamental mode Love wave), and the final elaboration isimproved.

In Fig. 3.13(c) and 3.13(d) Rayleigh wave ellipticity curves of the funda-mental mode are shown in the different representations. We emphasize howthe zero of the H/V curve around 2.5 Hz is again clearly identified by lookingat the ellipticity angle representation of Fig. 3.13(d).

3.5 Conclusions

In this paper, we have presented an application to the analysis of surface wavesfrom ambient vibrations recording of a recently developed technique for arrayprocessing of the seismic wavefield.

The technique performs ML wavefield parameter estimation accounting forall the measurements and all the parameters jointly. The technique allowsto model the simultaneous presence of multiple waves. Notably, we provide

60

2 4 6 8 10 12 140

0.05

0.1

Wavenum

ber

[1/m

]

Frequency [Hz]

(a) Love wave dispersion curve.

2 4 6 8 10 12 140

0.05

0.1

Wavenum

ber

[1/m

]

Frequency [Hz]

(b) Rayleigh wave dispersion curve.

2 4 6 8 10 12 140.01

0.1

1

10

100

Frequency [Hz]

Ellip

ticity

H/V

2 4 6 8 10 12 14

0.05

0.1

(c) Rayleigh wave ellipticity curve for funda-mental mode.

2 4 6 8 10 12 14

Frequency [Hz]

Ellip

ticity

angleξ

[rad]

-π2

-π4

π2

π4

0

2 4 6 8 10 12 14

0.05

0.1

(d) Rayleigh wave ellipticity angle curve forfundamental mode.

2 4 6 8 10 12 140.01

0.1

1

10

100

Frequency [Hz]

Ellip

ticity

H/V

2 4 6 8 10 12 14

0.05

0.1

(e) Rayleigh wave ellipticity curve for firsthigher mode.

2 4 6 8 10 12 14

Frequency [Hz]

Ellip

ticity

angleξ

[rad]

-π2

-π4

π2

π4

0

2 4 6 8 10 12 14

0.05

0.1

(f) Rayleigh wave ellipticity angle curve forfirst higher mode.

Figure 3.11: Dispersion curves and ellipticity curves obtained using the MLtechnique for Brigerbad survey. These results are obtained from a single pro-cessing with Mmax = 3.

61

(a) Geometry of the sensor array used in theRheintal survey. The inlet pinpoints the lo-cation of the array within Switzerland. Thegeographic coordinates are Swiss coordinates(CH1903).

0.5 1 1.5 2 2.5 3 3.5 40

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Wavenum

ber

[1/m

]

Frequency [Hz]

(b) Vertical component.

0.5 1 1.5 2 2.5 3 3.5 40

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Wavenum

ber

[1/m

]

Frequency [Hz]

(c) Radial component.

0.5 1 1.5 2 2.5 3 3.5 40

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Wavenum

ber

[1/m

]

Frequency [Hz]

(d) Transverse component.

Figure 3.12: Rayleigh wave and Love wave dispersion curves obtained usingthe method in (Fäh et al., 2008) for the Rheintal survey.

an ML estimate of Rayleigh wave ellipticity and the sense of particle rotation(prograde vs. retrograde).

We evaluated the performance of this technique on high-fidelity syntheticdataset from the SESAME project and on real data from two surveys. Thismethod improves estimates of Rayleigh and Love waves dispersion curves, andallows for an estimate of Rayleigh wave ellipticity. We have also shown thatmodeling multiple waves enables us to detect weaker waves that are not visiblewith traditional methods.

Further developments of the method will include an adaptive window selec-tion and the extension to other wave types such body waves and resonances.

3.6 Acknowledgments

The authors wish to thank Dr. J. Burjánek, Dr. C. Cauzzi, Q. Keeris, P.Galvez, Dr. C. Michel, Dr. V. Poggi, and Dr. J. Revilla for their invaluableassistance during the Rheintal measurement campaign. We also wish to thankSpectraseis AG for providing technical support during the same survey. Con-

62

0.5 1 1.5 2 2.5 3 3.5 40

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01W

avenum

ber

[1/m

]

Frequency [Hz]

(a) Love wave dispersion curve.

0.5 1 1.5 2 2.5 3 3.5 40

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Wavenum

ber

[1/m

]

Frequency [Hz]

(b) Rayleigh wave dispersion curve.

0.5 1 1.5 2 2.5 3 3.5 40.01

0.1

1

10

100

Frequency [Hz]

Ellip

ticity

H/V

0.5 1 1.5 2 2.5 3 3.5 4

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

(c) Rayleigh wave ellipticity curve for funda-mental mode.

0.5 1 1.5 2 2.5 3 3.5 4

Frequency [Hz]

Ellip

ticity

angleξ

[rad]

-π2

-π4

π2

π4

0

0.5 1 1.5 2 2.5 3 3.5 4

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

(d) Rayleigh wave ellipticity angle curve forfundamental mode.

Figure 3.13: Rayleigh wave and Love wave dispersion curves obtained usingthe ML technique for the Rheintal survey. Mmax = 3.

cerning the Brigerbad dataset, the authors wish to thank Dr. J. Burjánek andDr. C. Michel. This work is supported in part by the Swiss Commission forTechnology and Innovation under project 9260.1 PFIW-IW.

Chapter 4

Processing of Translational and Rotational

Motions of Surface Waves:

Performance Analysis and Applications to Single Sensor and

to Array Measurements

Stefano Maranò and Donat Fäh

ETH Zurich, Swiss Seismological Service, 8092 Zürich, Switzerland.

To appear in Geophys. J. Int.

Abstract

The analysis of rotational seismic motions has received considerableattention in the last years. Recent advances in sensor technologies allowus to measure directly the rotational components of the seismic wavefield.Today this is achieved with improved accuracy and at an affordable cost.The analysis and the study of rotational motions are, to a certain extent,less developed than other aspects of seismology due to the historical lackof instrumental observations. This is due to both the technical challengesinvolved in measuring rotational motions and to the widespread beliefthat rotational motions are insignificant.

This paper addresses the joint processing of translational and rota-tional motions from both the theoretical and practical perspective. Ourattention focuses on the analysis of motions of both Rayleigh waves andLove waves from recordings of single sensors and from an array of sensors.From the theoretical standpoint, analysis of Fisher information (FI) al-lows us to understand how the different measurement types contribute tothe estimation of quantities of geophysical interest. In addition, we showhow rotational measurements resolve ambiguity on parameter estima-tion in the single sensor setting. We quantify the achievable estimationaccuracy by means of Cramér-Rao bound (CRB). From the practicalstandpoint, a method for the joint processing of rotational and trans-lational recordings to perform maximum likelihood (ML) estimation ispresented. The proposed technique estimates parameters of Love wavesand Rayleigh waves from single sensor or array recordings. We support

63

64

and illustrate our findings with a comprehensive collection of numericalexamples. Applications to real recordings are also shown.

4.1 Introduction

The joint analysis of translational and rotational motions has the potential toimprove the estimation of important physical properties of the near-subsurface.The most striking feature of rotational motions is that, together with transla-tional motions, they enable us to estimate velocity of propagation of seismicwaves from a point measurement. The amount of rotational motion induced bya seismic wave is inversely proportional to the wavelength, and is thus relatedto the velocity of propagation. As a result, a six-components measurementof both translational and rotational motions at a single spatial location gathersufficient information to estimate the velocity of propagation of a seismic wave.This fact unleashes a myriad of potential applications (Igel et al., 2012; Leeet al., 2009).

Different approaches to estimate or directly measure rotational motionshave been developed in the past decades. In some early applications, groundrotations have been estimated from the spatial derivatives of the translationalmeasurements from an array of sensors (Niazi, 1986; Oliveira & Bolt, 1989;Spudich et al., 1995). In Nigbor (1994), one of the earliest direct measure-ment of rotational motions, employing a solid-state rotational velocity sensor,is found. Recent advances in sensor technology allow us to directly measurerotational motions with unprecedented accuracy and/or portability (Schreiberet al., 2009; Nigbor et al., 2009; Lee et al., 2012). A technology used in modernand portable rotational sensors relies on an electrochemical transducer wherethe motion of a fluid caused by an external acceleration is converted into anelectrical signal (Leugoud & Kharlamov, 2012). A comparison of array derivedrotational motions with direct measurements is found in Suryanto et al. (2006).

The direct measurement of rotational motions provides an additional in-dependent observation of the seismic wavefield. This is extremely valuable,for instance, in the analysis of surface waves. The measurements of rotationalmotions supplement the measurements of translational motions, potentiallyincreasing the accuracy of the estimation of the geophysical parameters of in-terest. Applications relying on the analysis of surface waves are numerous.Notably, the analysis of the seismic wavefield enables us to gather knowledgeof geological and geophysical features of the subsoil. Indeed seismic wave at-tributes such as velocity of propagation or polarization reflect the properties ofthe structure in which the wave is propagating. The analysis of these propertiesallow geophysicists to gain insight into the subsoil and the assessment of localseismic hazard relating to the near-surface (Tokimatsu, 1997; Okada, 1997).

The analysis of recordings from a single sensor is of great practical interest,for example, in engineering seismology. The estimation of seismic parame-ters from a single sensor is particularly desirable because of the simplicity ofmeasurement operations. Concerning the analysis of translational motions, awell-established single station method is the H/V ratio technique which hasbeen widely used for different purposes (Fäh et al., 2003; Bonnefoy-Claudetet al., 2006a). Other methods estimate Rayleigh wave ellipticity from a sin-gle station (Hobiger et al., 2009; Poggi et al., 2012). The processing of both

65

translational and rotational motions from a single sensor location has beenalso addressed and it has been shown that the retrieval of wave velocity ispossible (Igel et al., 2005; Ferreira & Igel, 2009).

When plane wavefronts can be assumed, array processing techniques areusually employed. The use of array processing techniques in seismology hasa long history. The earliest techniques analyzed a single component (Capon,1969; Lacoss et al., 1969). More recent developments allow us to separate Lovewaves and Rayleigh waves (Fäh et al., 2008) and to estimate Rayleigh waveellipticity (Poggi & Fäh, 2010). Recent work from the authors, includes a MLestimation technique accounting for all measurements and wavefield parametersjointly (Maranò et al., 2012). To the best of our knowledge, at this timethere are no applications to seismology of the joint processing of rotational andtranslational motions for an array of sensors.

In this paper, we are interested in the joint analysis of translational androtational motions induced by surface waves. We consider different aspects ofthe problem from a signal processing perspective. We investigate the potentialand the limitations introduced through joint processing of these two types ofmeasurement. A method exploiting all the available measurements is presented.Examples are provided to support the theoretical investigation and to show theapplicability of the proposed method.

One contribution of this paper is a method for the joint processing of trans-lational motion and rotational motion recordings of surface waves. All themeasurements are considered in a single framework and the algorithm providean ML estimate of the wavefield parameters. We extend a method proposed bythe authors in Maranò et al. (2012). The original method jointly accounts forthe measurement from three components translational sensors and the wave-field parameters of Love waves and Rayleigh waves. The simultaneous presenceof multiple waves is also accounted for.

The other main contributions of this paper are the following:

• We derive expressions of the Fisher information matrix (FIM) of eachstatistical models of interest. Fisher information (FI) enables us to gaininsight about the contribution of each measurement to the estimation ofdifferent wave parameters of interest.

• We show under which conditions, such as number of sensors, type ofsensors, and wave type, it is possible to identify wave parameters.

• We derive lower bounds on the achievable accuracy of the estimators forthe geophysical parameters of interest, namely wavenumber and ellipticityangle. This allows us to compare the performance of any algorithm witha an accuracy bound independent of estimation technique and algorithmimplementation.

• We provide a comprehensive collection of numerical examples illustratingthe potential and limitations of the joint processing of translational androtational motions.

• We show applications of the presented algorithm to two distinct realdatasets of the retrieval of Love wavenumber, Rayleigh wavenumber, andRayleigh ellipticity angle for both single sensor and array measurements.

66

The remainder of the paper is organized as follows. Section 4.2 presents thesystem model. The notation used in this manuscript is introduced and Lovewave and Rayleigh wave equations for translational and rotational motions areprovided. In Section 4.3, we analyze from a theoretical standpoint the limita-tions and the performance improvement achievable by including the rotationalmotions in the processing. In Section 4.4, we present the algorithm used inthis work for the estimation of wavefield parameters. Finally, several numer-ical results are provided in Section 4.5, including examples on synthetic dataand real applications on two datasets. Conclusions are drawn in Section 4.6.

4.2 System Model

In this paper, we are interested in modeling the seismic wavefield both in itstranslational motions and its rotational motions. In this section, we introducewave equations describing quantities of interest and a model of the measure-ments.

We describe the seismic wavefield, at position p ∈ R3 and time t with thevector field u(p, t) : R4 → R6

u(p, t) = (ux, uy, uz, ωx, ωy, ωz) (p, t) , (4.1)

where the first three components of the vector field describe the translationalmotions and the last three the rotational motions. For the sake of simplicity, weportray wave equations of the seismic wavefield in displacements and rotationsdespite the actual measurements may be velocities or accelerations.

We use a three-dimensional, right-handed Cartesian coordinate system withthe z axis pointing upward. The azimuth ψ is measured counterclockwise fromthe x axis. The sign of the rotations follow the right-hand rule.

Rotational Motions at a Free Surface

We now provide the derivation of rotational motions at the free surface. In thissection, we omit the dependence on (p, t) for conciseness of notation. Frommechanics (Aki & Richards, 2002), rotational motions (ωx, ωy, ωz) are relatedto the curl of translational motions (ux, uy, uz) as

(ωx, ωy, ωz) =1

2∇× (ux, uy, uz) (4.2)

=

∣

∣

∣

∣

∣

∣

∣

x y z

∂∂x

∂∂y

∂∂z

ux uy uz

∣

∣

∣

∣

∣

∣

∣

,

where x, y, z are the versors of the coordinate system and where |·| denotesthe determinant of the matrix. It follows directly from (4.2) that

67

ωx =1

2

(

∂uz∂y− ∂uy

∂z

)

ωy =1

2

(

∂ux∂z− ∂uz

∂x

)

(4.3)

ωz =1

2

(

∂uy∂x− ∂ux

∂y

)

.

Boundary conditions at the free surface require all the stress along z tovanish and leave the displacement unconstrained. Let T denote the Cauchystress tensor, we enforce the boundary conditions as Tz = (0, 0, 0)T. UsingHooke’s law for a linear elastic medium, the following conditions are found atz = 0

τx,z = µ

(

∂uz∂x− ∂ux

∂z

)

= 0

τy,z = µ

(

∂uz∂y− ∂uy

∂z

)

= 0 (4.4)

τz,z = λ

(

∂ux∂x

+∂uy∂y

+∂uz∂z

)

+ µ

(

∂uz∂z− ∂uz

∂z

)

= 0 ,

where λ and µ are the Lamé parameters. Comparing (4.3) with (4.4) it isapparent that the conditions τx,z = 0 and τy,z = 0 are influencing the rotationalmotions. From (4.4) it is found that

∂uz∂x

=∂ux∂z

and∂uz∂y

=∂uy∂z

, (4.5)

substituting into (4.3) we find the rotational motions at the free surface to be

ωx(p, t) =∂uz(p, t)

∂y

ωy(p, t) = −∂uz(p, t)

∂x(4.6)

ωz(p, t) =1

2

(

∂uy(p, t)

∂x− ∂ux(p, t)

∂y

)

.

Translational and Rotational Motions for Surface Waves

In this paper, we study waves propagating near the surface of the earth andhaving a direction of propagation lying on the horizontal plane z = 0. Thewave equations we describe hereafter are valid for z = 0 and for plane wavefronts. The direction of propagation of a wave is given by the wave vectorκ = κ (cosψ, sinψ, 0)T, whose magnitude κ is the wavenumber.

Love waves exhibit a translational particle motion confined to the horizontalplane, the particle oscillates perpendicular to the direction of propagation.The particle displacement generated by a single monochromatic Love wave at

68

position and time (p, t) is

ux(p, t) = −α sinψ cos(ωt− κ · p+ ϕ)

uy(p, t) = α cosψ cos(ωt− κ · p+ ϕ) (4.7)

uz(p, t) = 0 ,

where α ∈ R+ and ϕ denote the amplitude and the phase of the wave, respec-tively. The temporal angular frequency is denoted with ω. The azimuth ψindicates the direction of propagation of the wave.

From (4.7), using (4.6) the rotational motions induced by a Love wave arefound to be

ωx(p, t) = 0

ωy(p, t) = 0 (4.8)

ωz(p, t) =1

2ακ sin(ωt− κ · p+ ϕ) .

Rotational motions induced by a single Love wave are limited to the ωz compo-nent. Rotations are scaled of a factor κ/2 with respect to the wave amplitudeα.

We define the wavefield parameter vector for a Love wave as θ(L) = (α, ϕ, κ, ψ)T.Rayleigh waves exhibit a translational particle motion having an elliptical

pattern and confined to the vertical plane perpendicular to the surface of theearth and containing the direction of propagation of the wave. The particledisplacement generated by a single Rayleigh wave is

ux(p, t) = α sin ξ cosψ cos(ωt− κ · p+ ϕ)

uy(p, t) = α sin ξ sinψ cos(ωt− κ · p+ ϕ) (4.9)

uz(p, t) = α cos ξ cos(ωt− κ · p+ π/2 + ϕ) .

The angle ξ is called ellipticity angle of the Rayleigh wave and determines theeccentricity and the sense of rotation of the particle motion. If ξ ∈ (−π/2, 0),the Rayleigh wave elliptical motion is said to be retrograde (i.e., the oscillationon the vertical component uz is shifted by +π/2 radians with respect to theoscillation on the direction of propagation). If ξ ∈ (0, π/2) the wave is saidto be prograde. For ξ = 0 and ξ = ±π/2 the polarization is vertical andhorizontal, respectively. The quantity |tan ξ| is known as the ellipticity of theRayleigh wave. See Maranò et al. (2012) for a detailed description of thisparametrization.

From (4.9), using (4.6) the rotational motions for a Rayleigh wave are foundto be

ωx(p, t) = ακ sinψ cos ξ cos(ωt− κ · p+ ϕ)

ωy(p, t) = −ακ cosψ cos ξ cos(ωt− κ · p+ ϕ) (4.10)

ωz(p, t) = 0 .

Rotational motions induced by a single Rayleigh wave are limited to the ωx andωy components. When a Rayleigh wave is horizontally polarized (ξ = ±π/2),no rotational motions are generated.

We define the wavefield parameter vector for a Rayleigh wave as θ(R) =(α, ϕ, κ, ψ, ξ)T.

69

Measurement Model

The seismic wavefield is sampled at different spatial locations and time instantsby means of instrumentation able to measure the translational motions and therotational motions. At each location a sensor measures the ground translationalong the direction of the axes of the coordinate system x, y, z and the groundrotation around the same axes. We say that each sensor has six components.

To measure seismic waves, we deploy an array of Ns sensors on the surfaceof the earth positioned at locations pnn=1,...,Ns

. We restrict our interest tosmall aperture arrays and work with a flat earth model. Each signal is sampledat K instants tkk=1,...,K . The recording from the six components of the n-thsensor are grouped in six channels numbered from ℓ = 6n−5 to ℓ+5 = 6n and

ordered as u(6n−5)k = ux(pn, tk), u

(6n−4)k = uy(pn, tk), u

(6n−3)k = uz(pn, tk),

u(6n−2)k = ωx(pn, tk), u

(6n−1)k = ωy(pn, tk), and u

(6n)k = ωz(pn, tk). We let

L = 6Ns denote the total number of channels in an array of six componentssensors.

Each measurement is corrupted by independent additive Gaussian noise

Z(ℓ)k ∼ N (0, σ2

ℓ ). Noise variance is, in general, different on each channel

Y(ℓ)k = u

(ℓ)k (θ) + Z

(ℓ)k . (4.11)

The quantities u(ℓ)k (θ) are deterministic functions of wavefield parameters θ as

described in (4.7)-(4.10).It follows from the signal and measurement wave model that the PDF of

the measurements is

pY (y |θ) =L∏

ℓ=1

K∏

k=1

1√

2πσ2ℓ

exp

(

− (u(ℓ)k (θ)− y(ℓ)k )2

2σ2ℓ

)

, (4.12)

where we grouped the measurements as Y = Y (ℓ)k ℓ=1,...L

k=1,...,K.

Whenever a sensor has less than six components, the corresponding missingchannels are omitted from the product in (4.12).

70

An Example of the Estimation of a Sinusoid in Noise.

We review the statistical tools used in this article through a simple toy example.The estimation of amplitude and phase of a sinusoid from noisy measurementsis considered. For a comprehensive introduction to estimation theory, we referthe interested reader to Kay (1993).

System model. We consider a sinusoid with known angular frequency ω. Thesinusoid is a deterministic function of amplitude α and phase ϕ, which are un-known. Noisy measurement of the sinusoid are taken at K known time instantstkk=1,...,K . Each measurement Yk is corrupted by independent additive noiseZk as

Yk = α cos(ωtk + ϕ) + Zk ,

for k = 1, . . . , K. We assume the statistical properties of the noise to beknown. Specifically, the noise has Gaussian distribution with zero mean andknown variance σ2, i.e., Zk ∼ N (0, σ2).

Probability density function. Given the assumption of independent and iden-tically distributed noise, it is straightforward to write the probability densityfunction (PDF) of the measurements as

pY (y |θ) =K∏

k=1

1√2πσ2

exp

(

− (α cos(ωtk + ϕ) − yk)22σ2

)

.

The parameter vector θ = (α, ϕ) collects the parameters of the model that areunknown and need to be estimated.

Likelihood function. Given measurements y = ykk=1,...,K , the LF of theobservations is pY (y |θ). Observe that the likelihood function (LF) is a functionof θ and the measurements y are fixed. Given the observations y, the LFquantifies how likely are the parameters θ.

Maximum likelihood estimation. To obtain an estimate θ of the true unknownparameters, we choose to follow the maximum likelihood (ML) principle. Inthis view, it is necessary to find the vector θ that maximizes the LF

θ = argmaxθ

pY (y |θ) .

Such maximization can, in general, always be addressed numerically. However,faster and more accurate analytical solutions may be available.At least two properties are desirable for an estimator. First, the estimatorshould be unbiased. On the average we expect the estimator to provide thetrue value, i.e. Eθ − θ = 0, where E· denotes the expected value. Second,the estimator should be accurate. Or, in other words, the estimator varianceE(θ − θ)2 should be as small as possible. Under certain assumptions, MLestimators are unbiased and have smaller variance that any other unbiasedestimator.

71

An Example of the Estimation of a Sinusoid in Noise (con-tinued).

Fisher information. Fisher information (FI) quantifies the information we ob-tain about each parameter from our experiment. The Fisher information ma-trix (FIM) is defined as

I(θ) = E

−∂2 ln pY (y |θ)

∂θ2

,

which can be interpreted as the average Hessian matrix at the point θ of thenegative log-likelihood function (LLF). Observe that the FIM depends on thestatistical model and on the parameter vector θ but is independent of themeasurements.For our model, the FIM is

I(θ) =K

2σ2

(

1 0

0 α

)

.

Each element on the main diagonal represent the amount of information relatedto each element of the parameter vector. From the first element, we understandthat the FI about the sinusoid amplitude α is proportional to the number ofsamples K and inversely proportional to the noise power σ2. From the secondelement, we understand that the FI about the sinusoid phase ϕ is related to thenumber of samples and the noise power in the same way. In addition, the FIabout the sinusoid phase increase linearly with the amplitude of the sinusoiditself.

Identifiability. An important sanity check is whether the statistical modelconsidered is identifiable. Loosely speaking, a model is identifiable when theestimation problem is well-posed.Following our example, consider modeling the noisy measurement of the sinu-soid with an alternative statistical model as

pY(

y∣

∣θ′)

=

K∏

k=1

1√2πσ2

exp

(

− (α cos(ωtk + ϕ1 + ϕ2)− yk)22σ2

)

,

where the parameter vector is θ′ = (α, ϕ1, ϕ2). It is evident, that there is someambiguity in this parametrization since there are infinite ϕ1, ϕ2 pairs definingthe same sinusoid. Therefore two distinct parameter vectors defining the samePDF exist and thus the model is not identifiable.When such ambiguity is not immediately evident, another way to verifywhether a model is identifiable or not, is to test the singularity of the FIM.For this latter model, is found

I(θ′) =K

2σ2

1 0 0

0 α α

0 α α

,

which is a singular matrix, as expected for an unidentifiable model.

72

An Example of the Estimation of a Sinusoid in Noise (con-tinued).

Cramér-Rao bound. The accuracy of any unbiased estimator is limited by theCramér-Rao bound (CRB). In other words, for a given statistical model thereis no unbiased estimator having variance smaller than the CRB. In practice,the CRB is obtained from the elements on the main diagonal of the matrixinverse of I(θ).In our example, the computation of I−1(θ) is straightforward since the matrixI(θ) is diagonal. From the elements on the main diagonal of I

−1(θ), it isfound that the variance of amplitude and phase estimates are lower boundedas

E

(α− Eα)2

≥ 2σ2

K

E

(ϕ− Eϕ)2

≥ 2σ2

αK.

This analytic result provides insights useful for the design of the experimentand a benchmark that allows to evaluate the performance of an estimationalgorithm.

4.3 Theoretical Performance Analysis

In this section, we discuss the advantages and potential of the joint processingof rotational and translational measurements from a theoretical standpoint. Tothis aim, we use several ideas from estimation theory. A reader unfamiliar withthis branch of statistics may refer to the box “An Example of the Estimationof a Sinusoid in Noise” included in this article.

First, we derive an expression of the Fisher information matrix (FIM) foreach wave models considered. Then we look at the issue of the identifiabil-ity of statistical models concerning Love wave and Rayleigh wave for threecomponents (translational) single sensor, six components (translational androtational) single sensors, and arrays of sensors. Following, we find the small-est achievable mean-squared estimation error (MSEE) of an unbiased estimatorusing the Cramér-Rao bound (CRB). The contribution to the parameter esti-mation of the different measurements and parameters is also understood. Atlast, we briefly discuss the estimator performance at lower SNR, in the thresh-old region.

Introduction

The MSEE of an estimation algorithm can be computed numerically usingMonte Carlo methods. For example, it is sufficient to repeat a large number oftimes the estimation of the wavefield parameters of a known wave with differentnoise realizations to compute the MSEE. In this way, it is possible to quantita-tively compare the estimation accuracy of two different estimation algorithmsor the estimation accuracy of the same estimation algorithm under differentconditions. The CRB, provides a lower bound on estimator variance and isindependent of estimation technique and algorithm implementation. Therefore

73

−20 −15 −10 −5 0 5 1010

−9

10−8

10−7

10−6

10−5

10−4

10−3

No information

Threshold zone region

Asymptotic region

MSE

E

SNR [dB]

Figure 4.1: An example of the MSEE of a ML estimator. The MSEE is depictedwith a blue dashed line. In the no information region the MSEE is very largeand constrained by the implementation of the algorithm. In the thresholdregion the occurrence of outliers keep the MSEE significantly larger than theCRB. At last, in the asymptotic region, the MSEE is well described by theCRB, which is shown with the black dashed line.

the MSEE of any algorithm can be compared with the CRB. Fig. 4.1 illustratesthese concepts with an example.

It is known from literature that non-linear estimators exhibit an abruptincrease in the MSEE below a certain SNR or sample size. This behavior iscalled threshold effect and is due to a transition from local to global estimationerrors (Van Trees, 2001). Three operation regions for the estimator are definedat different SNR ranges, see also Fig. 4.1. At very low SNR, the noise dominatesover the signal of interest, this is called no information region. At larger SNR,is found the threshold region where MSEE is still considerably large as globalestimation errors occur. Global estimation are also known as outliers. At evenlarger SNR is found the asymptotic region. Local estimation error occurs inthis region and the MSEE of a ML estimator is well described by the CRB.

Preliminary definitions: Consider the following definitions related to the geo-metrical layout of the array. We introduce the coordinate system (a, b), whichis related to (x, y) as

(

a

b

)

=

(

cosψ sinψ

− sinψ cosψ

)(

x

y

)

, (4.13)

where the angle of rotation is the azimuth ψ. Therefore a is the axis alongthe direction of propagation of the wave and b the axis perpendicular toit. In this rotated coordinate system we consider the new sensor positions(an, bn)n=1,...,Ns

. The moment of inertia (MOI) of the array in the coordi-

74

nate system (a, b) are defined as

Qaa =

Ns∑

n=1

(an − a)2 (4.14)

Qbb =

Ns∑

n=1

(bn − b)2 (4.15)

Qab =

Ns∑

n=1

(an − a)(bn − b) , (4.16)

where a = 1Ns

∑Ns

n=1 an and b = 1Ns

∑Ns

n=1 bn define the phase center of thearray.

We observe that the MOIs are invariant to a translation of the array andthat for the single sensor setting (Ns = 1) all the MOIs are equal to zero.

Fisher Information

When combining the measurements of translational and rotational motions,one question that arises naturally is how and to which extent the differentmeasurements contribute to the parameter estimation of the statistical model.

The Fisher information (FI) conveys the amount of information about astatistical parameter carried by the PDF of the observations (Fisher, 1922).For a statistical model with multiple parameters the Fisher information matrix(FIM) is given by

I(θ) = E

−∂2 ln pY (y |θ)

∂θ2

, (4.17)

where E· denotes the expectation operation. The vector θ collects the un-known wavefield parameters of either Love wave or Rayleigh wave. The matrixI is a square symmetric matrix with as many columns as the elements in thevector θ.

Measurement of translational and rotational motions are independent cor-rupted by additive white Gaussian noise, as in (4.11). Throughout this sectionwe consider the translational and the rotational components to be subject todifferent noise levels, with power σ2

t and σ2r respectively.

For independent observations, the FIM is additive. Let It(θ) and Ir(θ) bethe FIM pertaining the translational and the rotational components. The FIMaccounting for all the observations is obtained as

I(θ) = It(θ) + Ir(θ) . (4.18)

In Appendix 4.A the expressions of the FIMs, together with an outline ofthe derivation, are provided. The FIMs for the model of a single Love waveare given in (4.40) and in (4.41) for translational and rotational measurements,respectively. The FIMs for a single Rayleigh wave are given in (4.42) andin (4.43) for translational and rotational measurements, respectively.

75

We observe that the diagonal elements of the FIM correspond to the FI ofa certain parameter when all the other parameters are known. In other words,the uncertainty associated with the other unknown parameters is neglected ifa single element on the diagonal is considered.

Identifiability

Consider a statistical model described in terms of its PDF pY (y |θ) parametrizedwith a vector θ ∈ Θ. A statistical model is said to be identifiable when themapping θ → pY (y |θ) is bijective (Rothenberg, 1971)

pY (y |θ1) = pY (y |θ2)⇔ θ1 = θ2 ∀θ1, θ2 ∈ Θ . (4.19)

This definition means that two distinct parameter vectors which specify thesame statistical model do not exist. Whenever condition (4.19) does not hold,the model is said to be unidentifiable. The analysis in this section is limitedto the local identifiability, i.e., to a neighborhood of the maximum likelihoodpoint.

In addition, a statistical model is identifiable if and only if the correspondingFIM is non-singular (Rothenberg, 1971).

Love wave, single sensor

Consider the problem of estimating wavefield parameters θ(L) = (α, ϕ, κ, ψ)T

for a Love wave from the measurements of a single three-components (trans-lational) sensor. From (4.7), we understand that this model is not iden-tifiable as several parameters specify the same PDF. Consider, for exam-

ple, the parameter vectors θ(L)1 = (α, ϕ, κ, ψ)T, θ

(L)2 = (α, ϕ, γκ, ψ)T with

γ ∈ R+, and θ(L)3 = (α, ϕ, κ, ψ + π)T, they specify the same distribution, i.e.

pY (y |θ(L)1 ) = pY (y |θ(L)

2 ) = pY (y |θ(L)3 ). Indeed, with a single translational

sensor it is not possible to determine the wavevector, and thus the velocity ofpropagation, of the Love wave. Moreover, there is an ambiguity of 180 aboutthe direction of propagation.

The related problem of estimating wavefield parameters θ(L) for a Love

wave from a single six (translational and rotational) sensor is however well-posed. From (4.7) and (4.8), we understand that this model is identifiable as

two distinct parameter vectors θ(L)1 6= θ

(L)2 specifying the same distribution do

not exist. This fact can be verified by checking the non-singularity of the FIMin (4.38) for Ns = 1. The same conclusion has been reached using differentarguments in Ferreira & Igel (2009) and Fichtner & Igel (2009).

Rayleigh wave, single sensor

We now consider the estimation of the wavefield parameters θ(R) = (α, ϕ, κ, ψ, ξ)T

for a Rayleigh wave from the measurements of a single three-components (trans-lational) sensor. From (4.9), we understand that this model is not identifiableas several parameters specify the same distribution. Indeed the parameter

vectors θ(R)1 = (α, ϕ, κ, ψ, ξ)T, θ

(R)2 = (α, ϕ, γκ, ψ, ξ)T with γ ∈ R+, and

θ(R)3 = (α, ϕ, κ, ψ+π,−ξ)T specify the same PDF. Again, from a single sensor

76

is not possible to retrieve any information concerning wave velocity of propa-gation. Moreover, there is an ambiguity involving direction of propagation andthe prograde/retrograde sense of rotation.

For a six components sensor the estimation of θ(R) is well-posed. This canbe understood from (4.9) and (4.10). Again, this can be verified by checkingthe non-singularity of the FIM of (4.39) for Ns = 1.

Array of sensors

It is well known that by means of an array of three components (translational)sensors it is possible to estimate wavefield parameters of either a Love wave ora Rayleigh wave. The only exception is the case of collinear sensors. Indeed alinear array cannot resolve the wavenumber for a wave propagating perpendic-ular to the array. When employing an array of six components (translationaland rotational) the limitation of the linear array is no more present. Both thesefacts can be verified by testing the singularity of the FIMs.

Interestingly, with an array of three components (rotational) sensors it ispossible to identify the parameters of a Love wave but not the parameters ofa Rayleigh wave. Indeed an array of sole rotational sensors it is not capableof estimating correctly Rayleigh wave amplitude, phase, and ellipticity. The

parameter vectors θ(R)1 = (α, ϕ, κ, ψ, ξ), θ

(R)2 = (γα, ϕ, κ, ψ, arccos(cos ξ/γ))

with γ ∈ R+, and θ(R)3 = (α, ϕ+π, κ, ψ,−ξ) specify the same statistical model.

Cramér-Rao Bound

The Cramér-Rao bound (CRB) is a lower bound on the variance of unbiasedestimators (Cramér, 1946; Rao, 1945). Knowledge of a lower bound on theestimator variance has at least two practical implications. First, it allows usto evaluate the performance of an estimation algorithm, by enabling a quanti-tative comparison between the mean-squared estimation error (MSEE) of thealgorithm under test and the smallest achievable variance. Second, the ana-lytic expression of the CRB enables us to design the experiment set up in orderto reduce the lower bound and therefore increase the amount of informationgathered by the experiment.

The information inequality states that the MSEE of an unbiased estimatoris lower bounded as

E

(

θ − Eθ)(

θ − Eθ)T

(I(θ))−1 . (4.20)

where A B means that the matrix A−B is positive semidefinite (PSD).In particular, we are interested in the diagonal elements of I−1 as they providea lower bound on the MSEEs of the corresponding parameters.

In high SNR regime, the CRB well describes the performance of ML es-timator. Thus in order to increase estimation accuracy, one is interested toreduce the CRB. This can be achieved by tuning the value of some determin-istic parameters of the model as, for example, increasing the number of sensorsor optimizing the array geometry.

77

To derive the CRB for the wavefield parameters of interest is necessaryto invert the FIM I. The CRB is obtained from the elements on the maindiagonal of I−1.

Since we are interested in the elements on the main diagonal of I−1 corre-sponding to wavenumber and ellipticity angle, we avoid the complete inversionof I as follows. We partition the FIM as

I(θ) =

(

c dT

d G

)

, (4.21)

where c is a scalar, d is a vector, and D is a matrix of suitable sizes. Theelement in the first position of I−1 is then found using the Woodbury matrixidentity to be

[

(I(θ))−1]

1,1=(

c− dTG

−1d)−1

, (4.22)

where [·]i,j denotes the element of the matrix in position (i, j) (Horn & Johnson,1990).

In (4.22), the quantity c − dTG

−1d has the dimension of FI and has beenreferred to by some authors as equivalent Fisher information (EFI) (Shen &Win, 2010). In contrast with FI, the EFI accounts for the uncertainty intro-duced by the other unknown parameters of the statistical model. The term c isexactly the FI of the parameter of interest. The term dT

G−1d is non-negative

since G is PSD being a diagonal sub-block of a PSD matrix. This last quantityaccounts for the uncertainty due to the other parameters.

It is now clear that reducing the CRB is equivalent to increase the EFI. Inother words, increasing the EFI is desirable as better estimation accuracy canbe achieved.

In order to use (4.22) effectively, it may be necessary to permute the row andcolumns of I such that the element of interest is in the top-left-most position.This can be accomplished using a permutation matrix P and consider the re-arranged I

′ obtained as I′ = P

TIP .

In the following, we restrict ourselves to the analysis of the CRB of wavenum-ber and ellipticity angle as these are the parameters of greater practical interest.

Love wave wavenumber

The CRB on Love wavenumber for translational measurements is obtainedusing (4.22) and (4.40). The MSEE of Love wave wavenumber is lower boundedas

E

(κ− Eκ)2

≥(

α2K

2σ2t

(

Qaa −Q2ab

Qbb +Ns/κ2

))−1

. (4.23)

The CRB is directly proportional to noise power σ2t , inversely proportional to

the amplitude of the wave α and to the number of samples K. We observe thatK cannot be arbitrarily increased as the validity of the model described in (4.7)may be no longer valid for long observations because of the time variabilityof real seismic sources. We emphasize that the CRB depends on the sensorpositions pnn=1,...,Ns

only trough the MOIs. The term Qaa is representative

78

of the information contribution due to the spatial sampling of the wavefield.A large Qaa can be obtained with a large aperture array along the directionof wave propagation a, cf. (4.14). However observe that a large aperture mayinvalidate the plane wave assumption. The last term is due to the uncertaintyof the other wavefield parameters and increases the CRB. It can be eliminatedby choosing an array geometry such that Qab = 0.

The CRB on Love wavenumber for rotational measurements is obtainedusing (4.22) and (4.41). The MSEE of Love wave wavenumber is lower boundedas

E

(κ− Eκ)2

≥(

α2κ2K

8σ2r

(

Qaa −Q2ab

Qbb

))−1

. (4.24)

The CRB is similar to the expression in (4.23). One difference is the pres-ence of a factor 4/κ2. This is due to the different overall amplitude of thesignal measured on ωz. Concerning seismic surface waves, the κ is generallya small quantity (smaller than one) thus the CRB is increased. In addition,the smaller the wavenumber, the less information is obtained from this type ofmeasurement.

The CRB on Love wavenumber for joint translational and rotational mea-surements is obtained using (4.22) with the FIM for the joint measurements (4.38).The MSEE of Love wave wavenumber is lower bounded as in (4.25). The firstand the third addends of the sum are, similarly to (4.23) and (4.24), represen-tative of the information contributed by the spatial sampling of the wavefieldand the uncertainty due to the other parameters weighted by the quality ofthe signal on the translational and the rotational components. The secondaddend is representative of the information gain due to the joint processing ofthe translational and rotational measurements. This term is proportional toNs and does not go to zero in the single sensor case, so that a single rotationalsensors carries information about the wavenumber.

E

(κ− Eκ)2

≥

(

(

Ct + κ2Cr/4)

Qaa +CtCrNs/4

Ct + Crκ2/4−

Q2ab

(

Ct + κ2Cr/4)2

CtNs/κ2 + (Ct + κ2Cr/4)

)

−1

(4.25)

E

(κ− Eκ)2

≥

(

ΦQaa +CtCrNs cos2 ξ

Ct + κ2Cr−

Q2abΦ

2

Ct sin2 ξNs/κ2 + Cr cos2 ξNs +ΦQbb

)

−1

(4.26)

E

(ξ − Eξ)2

≥ (CtNs

+κ2Φ

(

QaaΨNs − κ2(Q2ab −QaaQbb)Φ

)

NsCrCt cos2 ξ(ΨNs + κ2ΦQbb) +QaaΨΦ2Ns − κ2(Q2ab −QaaQbb)Φ3

)

−1

(4.27)

withCt = α2K/2σ2

t

Cr = α2K/2σ2r

andΦ = Ct + Crκ

2 cos2 ξ

Ψ = Ct sin2 ξ + Crκ

2 cos2 ξ .

79

Rayleigh wave wavenumber

The CRB on Rayleigh wavenumber for translational measurements is obtainedusing (4.22) and (4.42). The MSEE of Rayleigh wave wavenumber is lowerbounded as

E

(κ− Eκ)2

≥(

α2K

2σ2t

(

Qaa −Q2ab

Qbb +Ns sin2 ξ/κ2

))−1

. (4.28)

This result is similar to (4.23) and the same considerations apply.The CRB on Rayleigh wavenumber for rotational measurements is obtained

using (4.22) and (4.43). The MSEE of Rayleigh wave wavenumber is lowerbounded as

E

(κ− Eκ)2

≥(

α2κ2 cos2(ξ)K

2σ2r

(

Qaa −Q2ab

Qbb +Ns/κ2

))−1

(4.29)

This result is similar to (4.24) and the same considerations apply. Observethat when the Rayleigh wave is horizontally polarized (ξ = ±π/2) the EFI iszero since no rotations are induced by the Rayleigh wave.

The CRB on Rayleigh wavenumber for joint translational and rotationalmeasurements is obtained using (4.22) with the FIM for the joint measure-ments (4.39). The MSEE of Rayleigh wave wavenumber is lower bounded asin (4.26). This result is similar to (4.25) and the same considerations apply.We observe that when the Rayleigh wave is horizontally polarized (ξ = ±π/2)then (4.26) reduces to (4.28) since no rotations are induced by the Rayleighwave and thus no information is added by the rotational measurements.

Rayleigh wave ellipticity angle

The CRB on Rayleigh ellipticity angle for translational measurements is ob-tained using (4.22) and (4.42). The MSEE of Rayleigh wave ellipticity angle islower bounded as

E

(ξ − Eξ)2

≥(

α2K

2σ2t

Ns

)−1

. (4.30)

This quantity is related to the number of sensors and it is not affected by thegeometry of the array.

As discussed earlier in this section, an array of sole rotational sensors is notable to estimate Rayleigh wave ellipticity angle as the model is unidentifiable.

The CRB on Rayleigh ellipticity angle for joint translational and rotationalmeasurements is obtained using (4.22) with the FIM for the joint measure-ments (4.39). The MSEE of Rayleigh wave ellipticity angle is lower boundedas in (4.27). This latter expression is however not immediate to interpret.

Threshold Zone Performance

Benefits of processing jointly multiple components in terms of reduction ofglobal errors are well known in literature (Hawkes & Nehorai, 1998; Cox &

80

Lai, 2007). Performance in the threshold zone of direction of arrival estimatorsis studied in detail in Athley (2008). In this work, this issue is not addresseddirectly, however we emphasize that the use of additional measurements reducesthe magnitude of local maxima other than the true maximum of the LF andthus of global errors. Improvement in accuracy are to be expected in thelow SNR regime, i.e. threshold zone. Performance in the threshold zone arenot easily quantifiable analytically and we limit ourselves in presenting somenumerical examples in Section 4.5 and Section 4.5.

4.4 Processing Technique

In this work we employ an extension of the method presented in Maranò et al.(2012). The method allows us to perform ML estimation of wavefield param-eters for Love waves and Rayleigh waves relying on observation from seismicsensors. The method models jointly measurements from all sensor componentsmaking optimal use of the available information. The wavefield parametersare also estimated jointly. The noise variance on each channel is estimatedadaptively. Information from the different channel is merged according to thedifferent noise levels on the different sensor components.

In our approach, we model the system by means of a probabilistic graphi-cal model. A complex system where a large number of random variables andstatistical parameters interact with complex relationships can be effectivelyrepresented by a graphical model. Within the graphical model, observed ran-dom variables (measurements), unobserved random variables, and parametersof the statistical model are represented in a unique framework together withthe functional relationships occurring among them. The probabilistic graphcan be used to perform inference tasks in an efficient manner. As an exam-ple, likelihood of the observations and thus ML estimation can be performedexploiting the structure of the graph. By using the graph it is possible to un-derstand the relationship between the different parts of the stochastic systemand then, for example, derive sufficient statistics which enable to efficientlycompute statistical quantities of interest. In our approach we rely on factorgraphs, one flavor among many graphical modeling techniques (Kschischanget al., 2001; Loeliger, 2004; Loeliger et al., 2007).

Using (4.12) it is possible to compute the likelihood of the observations y

for a specific wavefield parameter vector θ directly. A maximization over theparameter space allows us to obtain a ML estimate θ

θ = argmaxθ

pY (y |θ) . (4.31)

In this context, different sensor technologies are used and the amplitudesof the measured signals are expected to vary greatly. It would be surely notoptimal to assume equal noise variance on every channel. Thus, after estimatingthe wavefield parameters the noise variances are also estimated as

(σ21 , . . . , σ

2L) = argmax

(σ21 ,...,σ

2L)

pY (y |θ, σ21 , . . . , σ

2L) , (4.32)

where θ is the estimated wavefield parameter vector obtained from (4.31).

81

The maximizations in (4.31) and in (4.32) are repeated alternatively andthe estimation of the wavefield parameters accounts for the different noise levelon the different sensors. Since the likelihood is a finite value the alternatingmaximizations are guaranteed to converge.

In our implementation, the maximizations in (4.31) and in (4.32) are notcomputed directly from (4.12). Details concerning the functioning and the im-plementation of our algorithm are found in Maranò et al. (2012) and referencestherein. In particular, in Reller et al. (2011) and in Maranò et al. (2011),we explain in detail the design of the factor graph which allows us to derivea sufficient statistic. This allows to perform ML parameter estimation in acomputationally attractive manner.

4.5 Numerical Results

Introduction

We provide some details about processing and the presentation of some results.Frequencies are processed independently. Unless differently noted, we apply nopreprocessing to the recordings other than mean removal. The whole signal issplit in time windows where the signal is assumed to be stationary. The lengthof such time windows is non-adaptive and not dependent on frequency.

We define the SNR as

SNR =α2

2σ2t

(4.33)

where σ2t is the noise variance on the translational components.

It is clear from equations (4.8) and (4.10) that rotational motions havesignificantly smaller amplitude than translational motions. Depending on thevalue of the wavenumber κ, rotational motions can be even one or two orderof magnitude smaller than the translational counterparts.

In the numerical examples that follow, we choose the true value of thevariance on the rotational components to be σ2

r = κ2σ2t . This choice is moti-

vated by the fact that the different noise level allows to obtain measurementsof comparable SNR on translational and rotational components. Both in thesynthetic examples and in the real dataset the noise variances are unknown tothe algorithm and are estimated with the proposed algorithm as in (4.32).

In Sec. 4.5 and 4.5 we present numerical examples to illustrate the potentialand the benefit introduced by the joint processing of translational and rota-tional components over the processing of the sole translational components. InSec. 4.5 we quantify increased estimation accuracy, in terms of MSEE, achievedby employing the rotational measurements and compare with the CRB. At lastin Sec. 4.5 and 4.5 we show two applications from translational and rotationalrecordings of a building demolition and an explosion.

Example Likelihood Functions for Single Sensor

By means of numerical examples, we show how the joint processing of transla-tional components and rotational components enables us to identify statisticalmodels for surface waves and to estimate correctly the wave parameters.

82

−0.1 0 0.1 −0.1

0

0.1

Wavenumber along x [1/m]

Wavenum

ber

alo

ngy

[1/m

]

(a) A line across the origin individuate themaxima point of the LLF reflecting the inabilityto determine velocity of propagation and direc-tion of propagation.

−0.1 0 0.1 −0.1

0

0.1

Wavenumber along x [1/m]

Wavenum

ber

alo

ngy

[1/m

]

(b) The LLF exhibits a single maxima.

Figure 4.2: The log-likelihood functions (LLFs) of observations from a singlesensor of a single Love wave as a function of wavenumber along x, κ cosψ,and wavenumber along y, κ sinψ. Comparison of analysis of sole translationalcomponents (left) and joint translational and rotational components (right).Large LL values are shown with colors towards red and low LL values withcolors towards blue. White crosses and lines mark the maxima point.

The figures shown in this section and in the following should be seen asexplanatory examples. In first place, a different noise realization will lead toa different LF . More importantly, a different choice of σ2

t and σ2r could lead

to a substantially different shape of the LF. To reduce this effects and toensure a fair comparison, we use a high SNR (SNR = 10 dB) and σ2

r = κ2σ2t .

We consider 1 s of observation, sampling at 100 Hz monochromatic waves offrequency ω = 2π. Maxima points of the LF are marked with white crosses.

In Fig. 4.2 the LLFs of observations of a noisy Love wave are shown. A singleLove wave with θ(L) = (1, 0, 0.05, π/4)T is considered. Fig. 4.2(a) depicts theLLF obtained from a single three components (translational) sensor. A wholeset of points, namely the line defined by the set (α, ϕ, κ, ψ) : α = 1, ϕ =0, κ ≥ 0, ψ ∈ π/4, 5π/4, maximize the likelihood of the observations. Thisreflects the inability to determine wavenumber and azimuth from a single threecomponents sensor.

Fig. 4.2(b) depicts the LLF obtained from a single six components (trans-lational and rotational) sensor. The global maximum point is seen in corre-spondence of the true wavefield parameters. Indeed a single six componentssensor allows the determination of velocity of propagation and direction ofpropagation without ambiguity.

A similar setup is repeated for a single Rayleigh wave with parametersθ(R) = (1, 0, 0.05, π/4,−π/4)T . In Fig. 4.3 the LLFs of observations of anoisy Rayleigh wave are shown. We are interested in showing the shape of theLLF as a function of three parameters, namely wavenumber, ellipticity, andazimuth. Thus we depict three slices of the LLFs, each slice is a function oftwo parameters for a fixed value of the third parameter equal to the true value.

Figures 4.3(a), 4.3(c), and 4.3(e) depicts slices of the LLF obtained from a

83

single three components (translational) sensor. A whole set of points, namelythe set (α, ϕ, κ, ψ, ξ) : α = 1, ϕ = 0, κ ≥ 0,(ψ = π/4, ξ = −π/4) ∨ (ψ =5π/4, ξ = π/4), maximize the likelihood of the observations. This reflectsthe inability to determine unambiguously wavenumber, azimuth, and sense ofrotation of the Rayleigh wave from a single three components sensor.

In contrast, from a single six components (translational and rotational)sensor is possible to estimate wavefield parameters correctly. Figures 4.3(b),4.3(d), and 4.3(f) depicts the same slices obtained from a single six compo-nents (translational and rotational) sensor. The global maximum point is seenin correspondence of the true wavefield parameters. Indeed a single six com-ponents sensor allows the determination of velocity of propagation, directionof propagation, and Rayleigh wave ellipticity without ambiguity.

From the previous pictures we can empirically confirm the theoretical find-ings about model identifiability of Sec. 4.3 and that, at least under the goodconditions of high SNR, the joint processing of translational motions and rota-tional motions allows to estimate all the wavefield parameters correctly. Thebroadness of the main peak of the LF suggests that the κ remains difficult toestimate accurately with a single six components sensor. This latter aspect isquantified by the CRB analysis Sec. 4.5.

Example Likelihood Functions for Array of Sensors

We now compare the shape of LLFs obtained from array of three componentssensors and six components sensors.

We consider an array of five sensors arranged on a circle of radius 20 mas shown in Fig. 4.4. The same choice of wavefield parameters and SNR ofSec. 4.5 is used in this section.

In Fig. 4.5 the LLFs of observations of a single Love wave are shown. Itis shown that the local maxima (sidelobes) of the LLF are smaller in the sixcomponents case than in the three components case.

In Fig. 4.6 different slices of the LLFs of observations of a single Rayleighwave are shown. In this example, the reduction of the local maxima is some-what limited. The largest improvement is seen in comparing Fig. 4.6(c) withFig. 4.6(d).

By comparing the LLFs in this Section with the corresponding LLFs ofSec. 4.5, we see that using the five sensors array the wavenumber and theazimuth are more easily determined, as witnessed by the peakiness of the LLFsin the two different setups. The reason is that estimation from an array ofsensors relies on the very important information extracted from the spatialsampling of the signal. This aspect is quantified in Sec. 4.5.

Cramér-Rao Bound Analysis

We are interested in comparing the MSEE obtained processing three com-ponents and six components with the theoretical bounds given by the CRBderived in Sec. 4.3.

Fig. 4.7 portrays the MSEE obtained by means of Monte-Carlo simulationswith different processing settings as a function of SNR. The five sensors arraydepicted in Fig. 4.4 is considered, with three components sensors and with six

84

−0.1 0 0.1 −0.1

0

0.1

Wavenumber along x [1/m]

Wavenum

ber

alo

ngy

[1/m

]

(a) Slice of the LLF for ξ = −π/4 (retrogrademotion).

−0.1 0 0.1 −0.1

0

0.1

Wavenumber along x [1/m]

Wavenum

ber

alo

ngy

[1/m

]

(b) Slice of the LLF for ξ = −π/4. The LLFexhibit a single maxima.

Ellip

ticity

angleξ

[rad]

π2

π4

0

-π2

-π4

Azimuth ψ [rad]0 π/2 π 3π/2 2π

(c) Slice of the LLF as a function of ξ and ψfor κ = 0.05. The likelihood is maximized foropposite direction of propagation with differ-ent sense of rotation, prograde (π/2) or ret-rograde (−π/2).

Ellip

ticity

angleξ

[rad]

π2

π4

0

-π2

-π4

Azimuth ψ [rad]0 π/2 π 3π/2 2π

(d) Slice of the LLF as a function of ξ and ψfor κ = 0.05. Direction of propagation andsense of rotation are pinpointed correctly.

0.02 0.04 0.06 0.08 0.1

Wavenumber [1/m]

Ellip

ticity

angleξ

[rad]

π2

π4

0

-π2

-π4

0

(e) Slice of the LLF as a function of κ and ξfor ψ = π/4. The function is constant for dif-ferent wavenumbers, because any wavenum-ber value fits the data equally well.

0.02 0.04 0.06 0.08 0.1

Ellip

ticity

angleξ

[rad]

π2

π4

0

-π2

-π4

0

Wavenumber [1/m]

(f) Slice of the LLF as a function of κ and ξfor ψ = π/4. The wavenumber can be cor-rectly estimated using a six components sen-sor.

Figure 4.3: LLFs of observations from a single sensor of a single Rayleigh waveas a function of wavenumber κ, azimuth ψ, and ellipticity angle ξ. Comparisonof analysis of sole translational components (left) and joint translational androtational components (right). Large LL values are shown with colors towardsred and low LL values with colors towards blue. White crosses and lines markthe maxima point.

85

−15 −10 −5 0 5 10 15 20

−20

−15

−10

−5

0

5

10

15

20

x [m]

y[m

]

Figure 4.4: The layout of the five sensors array used in the numerical examples.

−0.1 0 0.1 −0.1

0

0.1

Wavenumber along x [1/m]

Wavenum

ber

alo

ngy

[1/m

]

(a) LLF obtained from translational compo-nents only.

−0.1 0 0.1 −0.1

0

0.1

Wavenumber along x [1/m]

Wavenum

ber

alo

ngy

[1/m

]

(b) LLF obtained from translational and rota-tional components jointly.

Figure 4.5: LLFs of observations from a five sensors array of a single Lovewave as a function of wavenumber κ and azimuth ψ. Comparison of analysisof sole translational components (left) and joint translational and rotationalcomponents (right). Large LL values are shown with colors towards red andlow LL values with colors towards blue. White crosses mark the maxima point.

components sensors. We also consider the performances of a single six com-ponents sensor. The wavefield parameters are the same used in the numericalexamples of the previous section and are unknown to the algorithm. Both σ2

t

and σ2r are unknown to the algorithm and estimated. The true values σ2

t andσ2

r are chosen as explained in the introduction of this section in order to havecomparable SNR on both sensor types.

In Fig. 4.7(a) the estimation of Love wave wavenumber is analyzed. Atvery low SNR, where the noise dominates, the estimate is substantially ran-dom. The MSEE saturates for decreasing SNR since the wavenumber estimateis constrained by the algorithm implementation to belong to a finite interval.As the SNR increases, the ML method using six components always exhibitsthe smaller MSEE. In the threshold region, approximately in the interval(−16,−3)dB, the performance gain due to the reduction of the outliers of the

86

−0.1 0 0.1 −0.1

0

0.1

Wavenumber along x [1/m]

Wavenum

ber

alo

ngy

[1/m

]

(a) Slice of the LLF as a function for ξ = −π/4.

−0.1 0 0.1 −0.1

0

0.1

Wavenumber along x [1/m]

Wavenum

ber

alo

ngy

[1/m

]

(b) Slice of the LLF for ξ = −π/4.

Ellip

ticity

angleξ

[rad]

π2

π4

0

-π2

-π4

Azimuth ψ [rad]0 π/2 π 3π/2 2π

(c) Slice of the LLF as a function of ξ and ψfor κ = 0.05.

Ellip

ticity

angleξ

[rad]

π2

π4

0

-π2

-π4

Azimuth ψ [rad]0 π/2 π 3π/2 2π

(d) Slice of the LLF as a function of ξ and ψfor κ = 0.05.

0.02 0.04 0.06 0.08 0.1

Wavenumber [1/m]

Ellip

ticity

angleξ

[rad]

π2

π4

0

-π2

-π4

0

(e) Slice of the LLF as a function of κ and ξfor ψ = π/4.

0.02 0.04 0.06 0.08 0.1

Ellip

ticity

angleξ

[rad]

π2

π4

0

-π2

-π4

0

Wavenumber [1/m]

(f) Slice of the LLF as a function of κ and ξfor ψ = π/4.

Figure 4.6: LLFs of observations from a five sensor array of a single Rayleighwave as a function of wavenumber κ, azimuth ψ, and ellipticity angle ξ. Com-parison of analysis of sole translational components (left) and joint translationaland rotational components (right). Large LL values are shown with colors to-wards red and low LL values with colors towards blue. White crosses mark themaxima point.

87

six components array over the three components array is substantial. This as-pect was discussed in Sec. 4.3. For sufficiently large SNR, the ML method usingsix components achieves the CRB. Even for high SNR the three componentsarray does not achieve the CRB as it disregards the rotational measurements.

It is worth noting that the performance gap observed when comparing thethree components array with the six components array in the asymptotic re-gion is strongly dependent on the choice of σ2

t and σ2r . For very large σ2

r , therotational measurements become uninformative and the performance gap willnarrows to zero. On the other hand, for very large σ2

t the translational mea-surement become uninformative and therefore the three components array willexhibit poor performance while the six components will still provide meaningfulestimates.The performance gap between single sensor and any of the two fivesensors arrays is considerably large. In order to achieve with a single six com-ponents sensor the same MSEE achieved with an array of sensors it is wouldbe necessary a SNR of several decibels higher. This gap is explained by thefact that the single sensor only relies on amplitude information and disregardsthe phase information relative to wave propagation. This is also quantifiedanalytically by the expression of the CRBs in Section 4.3.

In Fig. 4.7(b) the estimation of Rayleigh wave wavenumber is analyzed. Inthis scenario the considerations are similar to the previous case.

In Fig. 4.7(c) the estimation of Rayleigh wave ellipticity angle is analyzed.In this scenario we observe how the single six component sensor exhibits smallerMSEE than the three components array over a certain SNR range.Concerningthe Rayleigh ellipticity angle estimation, the performance gap is smaller thanthe previous case. Indeed the CRB on Rayleigh wave ellipticity for a transla-tional array does not rely on phase information, cf. (4.30).

Fig. 4.7 also shows that our implementation of the ML estimator achieves,for sufficiently large SNR, the CRB in all the considered cases.

Analysis of the Agfa Dataset

The Agfa dataset consist of recordings of an explosion from building demoli-tion in southern Germany (Wassermann et al., 2009). The seismic motion isrecorded by an array of seven translational sensors and one rotational sensor.The array layout is depicted in Fig. 4.8. The rotational sensor is collocated witha translational sensor at the central location labeled ’BW01’. The translationalvelocimeters are of different make and model as explained in Wassermann et al.(2009), the rotational sensor is a eentec R1 (Bernauer et al., 2012).

The portion of interest of the recording is only 10 seconds long and it issplit in 0.75 seconds window which are processed independently with a 50%overlap.

We apply the considered ML method and process the signals recorded bythe seven translational sensors modeling the presence of a single Rayleigh wave.The retrieved dispersion curve is depicted in Fig. 4.9(a). Estimated parameters(wavenumber and ellipticity angle) are shown with a scale of grays, with darkercolors corresponding to a value being more often estimated. The red dashedlines represent a manual pick of upper and lower boundaries of the dispersioncurve as identified by visual inspection. The red dots depict the estimateddispersion curve starting from such manual selection obtained as the medianof the values in the selection. The blue lines represent constant velocity lines.

88

−25 −20 −15 −10 −5 0 5 10 15 2010

−9

10−8

10−7

10−6

10−5

10−4

10−3

MSE

E[m

−2]

SNR [dB]

3C ML (Ns = 5)

6C ML (Ns = 5)

6C ML (Ns = 1)

Cramér-Rao L.B.

(a) Love wavenumber MSEE.

−25 −20 −15 −10 −5 0 5 10 15 2010

−9

10−8

10−7

10−6

10−5

10−4

10−3

MSE

E[m

−2]

SNR [dB]

3C ML (Ns = 5)

6C ML (Ns = 5)

6C ML (Ns = 1)

Cramér-Rao L.B.

(b) Rayleigh wavenumber MSEE.

−25 −20 −15 −10 −5 0 5 10 15 20

10−4

10−3

10−2

10−1

100

MSE

E[rad2]

SNR [dB]

3C ML (Ns = 5)

6C ML (Ns = 5)

6C ML (Ns = 1)

Cramér-Rao L.B.

(c) Rayleigh ellipticity angle MSEE.

Figure 4.7: Comparison of the MSEE from different processing setups with theCRB at different SNR. The processing of three components (3C) translationalsensors is compared with six components (6C) translational and rotationalsensors. Different number of sensors Ns is also compared.

89

−40 −20 0 20 40

−40

−20

0

20

40

BW01

BW02

BW03

BW05

BW06

BW07

BW08

x [m]

y[m

]

Figure 4.8: Layout of the array of the Agfa dataset. In the central location,at (x, y) = (0, 0), a translational seismometer is co-located with a rotationalsensors. The array is centered in 48.108589 N 11.582967 E.

Empirical array resolution limits according to Asten & Henstridge (1984) aredepicted with thin dashed black lines.

The stripes visible in the figure are due to the fact that the wavenumberestimates within the same time window and at neighboring frequencies arestrongly correlated. In other words, the maximum of the LF changes onlyslightly when the frequency ω is changed slightly.

The ellipticity angle is shown in Fig. 4.9(b). Rayleigh particle motion isretrograde, i.e. ξ ∈ (−π/2, 0), for frequencies above 4 Hz. Below this frequencythe wave appears to be close to horizontally polarized (ξ = ±π/2). It shouldbe also considered that at low frequencies the wavenumber estimates are belowthe conventional resolution limit.

For the single sensor setting, we process the recording of the co-locatedtranslational sensor and rotational sensor at position BW01. The wavenumberestimates are shown in Fig. 4.9(c). From this picture is possible to recognize ageneral increase in the values of the estimated wavenumbers. Compared to theresults obtained from the array (Fig. 4.9(a)), it is possible to see a shift of theestimated wavenumbers toward lower values (faster velocities). One possibleexplanation is that the sensor are, due to physical constraints, not exactly co-located while the algorithm assume they are both located at the same position.Moreover, the estimation of the wavenumber from single station is affected bythe possible presence of higher modes of propagation, as suggested in Kurrleet al. (2010). In the single sensor setting, the dispersion curve does not appearto be reliable below 4 Hz and manual selection is not performed.

Estimates of the ellipticity angle are shown in Fig. 4.9(d) and are in goodagreement with the results obtained from the array processing of Fig. 4.9(b).As expected from the CRB analysis the scatter of the estimates is to someextent larger in the single sensor setting than in the array setting.

90

400m

/s50

0m

/s70

0m

/s12

00m

/s

1 2 3 4 5 6 7 80

0.005

0.01

0.015

0.02

0.025

Frequency [Hz]

Wavenum

ber

[1/m

]

(a) Rayleigh wavenumber estimated usingthe array of translational sensors.

1 2 3 4 5 6 7 8

Ellip

ticity

angleξ

[rad]

π2

π4

0

-π2

-π4

Frequency [Hz]

(b) Rayleigh ellipticity angle estimated usingthe array of translational sensors.

400m

/s50

0m

/s70

0m

/s12

00m

/s

1 2 3 4 5 6 7 80

0.005

0.01

0.015

0.02

0.025

Wavenum

ber

[1/m

]

Frequency [Hz]

(c) Rayleigh wavenumber estimated using asingle six components sensor.

1 2 3 4 5 6 7 8

Ellip

ticity

angleξ

[rad]

π2

π4

0

-π2

-π4

Frequency [Hz]

(d) Rayleigh ellipticity angle estimated usinga single six components sensor.

Figure 4.9: Analysis of Rayleigh waves for the Agfa dataset. On the topprocessing from seven translational sensors, on the bottom from a single sixcomponents (translational and rotational) sensor.

The same recording is processed modeling the presence of a single Lovewave. In Fig. 4.10(a) wavenumber estimates obtained from the processing of theseven translational sensors are shown. In Fig. 4.10(b) wavenumber estimatesobtained from the processing of the sensors co-located at BW01 are shown.Similarly to the case for Rayleigh wave, a shift towards faster velocities isobserved.

Love wave wavenumber estimates are more scattered than Rayleigh wavewavenumber estimates and the dispersion curve is more difficult to observe.An explanation could be that the Love waves are not as strong because of thenature of the source. Indeed an explosion excites mostly compressional wavesand only to a lesser extent Shear waves. The branching of the dispersion curvethat can be observed below 4 Hz in Fig. 4.10(a) could also be explained by thelittle energy of the Love wave.

Analysis of the TAIGER Dataset

The TAIGER dataset includes recordings of two explosions in north-easternTaiwan (Lin et al., 2009). Recordings from an array of eleven accelerometers

91

400m

/s50

0m

/s70

0m

/s12

00m

/s

1 2 3 4 5 6 7 80

0.005

0.01

0.015

0.02

0.025W

avenum

ber

[1/m

]

Frequency [Hz]

(a) Love wavenumber estimated using the ar-ray of translational sensors.

400m

/s50

0m

/s70

0m

/s12

00m

/s

1 2 3 4 5 6 7 80

0.005

0.01

0.015

0.02

0.025

Wavenum

ber

[1/m

]

Frequency [Hz]

(b) Love wavenumber estimated using a sin-gle six components sensor.

Figure 4.10: Analysis of Love waves for the Agfa dataset. On the left processingfrom seven translational sensors, on the right from a single six components(translational and rotational) sensor.

and five rotational sensors are used. The array layout is depicted in Fig. 4.11(a).The rotational sensors are co-located with the accelerometers in the five innerlocations. In this work, we use recording from the explosion N3P, as named inthe referenced paper.

The total duration is around 7 seconds, the recording is subdivided in 50%overlapping windows of one seconds. The only preprocessing step we purse isto convert the velocity recordings from the rotational sensors to acceleration(rad/s2).

Fig. 4.11(b) shows the wavenumber estimates obtained processing the sig-nals from the translational sensors. It is difficult to determine the wavenumberchange with frequency. Associated ellipticity angle estimates are very scatteredand not shown.

In Fig. 4.11(c) all the available sensors, both translational and rotationalare processed jointly. Here the dispersion curve is well identified and the im-provement on the previous picture is significant. At last, Fig. 4.11(d) showsthe estimated ellipticity angle obtained from the processing of all the sensors.Despite the scatter it is possible to identify a retrograde particle motion atfrequencies above 10 Hz. Below this frequency the wave is substantially hori-zontally polarized.

The estimation of the wavenumber and ellipticity angle is significantly im-proved by the joint processing of translational and rotational sensors over theprocessing of the sole translational sensors. We observe that the array is rathersmall, 20 m in diameter, and this might be a limiting factor in the estimationof the wavenumber of seismic waves with wavelengths between 10 m and 50 m.We speculate that in this setting the amplitude information provided by therotational sensors is particularly important.

Comparing the estimated wavenumber at different time windows (not shown)it is possible to see a tendency of a shift of wavenumbers towards smaller val-ues, i.e. slower velocity with increasing time. This shift could be explainedby the nonlinear behaviors induced by the large strains. This observationhowever remains speculative, due to missing independent observation such as

92

−10 −5 0 5 10

−10

−5

0

5

10 N02A N02B

N03

N04 N05 N06 N07 N08

N09

N10A N10B

x [m]

y[m

]

(a) Array layout for the TAIGER dataset.Five rotational sensors are co-located withtranslational sensors in the inner five loca-tions of the array. The array is centered at24.5792222 N 121.4818722 E.

200m

/s30

0m

/s50

0m

/s90

0m

/s

5 10 15 20 250

0.02

0.04

0.06

0.08

0.1

0.12

Wavenum

ber

[1/m

]

Frequency [Hz]

(b) Rayleigh wavenumber estimated usingthe array of translational sensors.

200m

/s30

0m

/s50

0m

/s90

0m

/s

5 10 15 20 250

0.02

0.04

0.06

0.08

0.1

0.12

Wavenum

ber

[1/m

]

Frequency [Hz]

(c) Rayleigh wavenumber estimated from thejoint processing of translational and rota-tional sensors.

5 10 15 20 25

Ellip

ticity

angleξ

[rad]

π2

π4

0

-π2

-π4

Frequency [Hz]

(d) Rayleigh ellipticity angle estimated fromjoint processing of translational and rota-tional sensors.

Figure 4.11: Analysis of Rayleigh waves for the TAIGER dataset.

pore-pressure build-up in the soils.

4.6 Conclusions

In this paper, we study different aspects of the processing of translational mo-tions and rotational motions for surface waves. Using tools from statistics, weinvestigate the contribution of the different measurements types to the accu-racy of wavefield parameters estimation. Advantages and limitations of singleand array of sensors are outlined quantitatively with respect to identifiabilityof the statistical models and lower bounds on the estimation accuracy. Thesefindings are also useful for experiment design and to compare estimation al-gorithms with an implementation-independent benchmark. We show severalnumerical examples clarifying the theoretical aspects for both single sensorand array settings.

93

A method for ML estimation of wavefield parameters is considered. Themethod extends a previous work of the authors and accounts for all the mea-surements and all the wavefield parameters within a single statistical model. Inthis context we show the estimation of wave parameters of both Love waves andRayleigh waves from arrays of sensors and single sensors using jointly transla-tional and rotational recordings. In addition, the method accounts for differentnoise level on each sensor.

Firstly, using Monte Carlo simulations we show that our method achieves,for sufficiently large SNR, the theoretical lower bounds on estimation accuracy.We also show that the performance loss in wavenumber estimation is significantwhen using a single six components sensor instead of a five sensors array. Thisis due to the lack of information extracted from spatial sampling of the signaland is also explained by our theoretical findings.

Secondly we demonstrate, on real recordings, the applicability of the pro-posed method for the estimation of Love wave and Rayleigh wave parametersfor both single sensor and array settings. In the Agfa dataset, we retrieve Loveand Rayleigh dispersions curves from single six component sensor and comparewith the dispersion curve retrieved from an array of translational sensors. Usinga single six component sensor, we observe a shift of the estimated wavenum-bers towards faster velocities when compared to the array retrieved dispersion.For the same dataset we also retrieve Rayleigh wave ellipticity angle from thesingle sensor and find agreement with the same quantity estimated from thearray of sensors. Concerning the TAIGER dataset, we compare Rayleigh wavedispersion curve obtained from a three component (translational) array andan array of mixed three- and six-components sensors. We find that the jointanalysis of translational and rotational sensors greatly improve the retrieveddispersion curve.

It is expected that the interest of the seismological community in this areato grow further in the coming years. As sensor technology will develop furtherimproving the quality and the availability of rotational measurements a widerange of applications will be possible.

4.7 Acknowledgments

We wish to thank the authors of Wassermann et al. (2009) and Lin et al. (2009)for making the recordings of the Agfa and the TAIGER datasets available.The authors also wish to thank Dr. Edwards for the careful reading of themanuscript. The authors also would like to thank the anonymous reviewers fortheir comments and helping to improve the manuscript.

This work is supported in part by the Swiss Commission for Technologyand Innovation under project 9260.1 PFIW-IW and with funds of the SwissSeismological Service.

4.A Derivation of Fisher Information Matrices

The FIM can be derived using (4.17) and (4.12) together with one of the wavemodel presented in Sec. 4.2. Six distinct FIMs are presented in this section,corresponding to the wave type Love or Rayleigh and the measurement typetranslational, rotational, or both.

94

The noise variances σℓℓ=1,...,L are in general also unknown parameters ofthe statistical model. However, for the purposes of this discussion, we assumethe noise variances to be known and derive the FIM only for the wavefieldparameters θ. This choice is supported by the fact that

Iθi,σ2ℓ= E

−∂2 ln pY

(

y∣

∣θ, σ21 , . . . , σ

2L

)

∂θi∂σ2ℓ

= 0 ,

implying that the wavefield parameters and the noise variances are decoupled.Thus, all the derivation to follow can accommodate for the case of unknownnoise variances σℓℓ=1,...,L with trivial modifications.

With the measurement model presented in (4.12) and assuming the varianceon channel ℓ to be known and equal to σ2

ℓ , then (4.17) reduces to a simplerexpression (Kay, 1993). The element in position i,j of I is obtained as

[I(θ)]i,j =

L∑

ℓ=1

1

σ2ℓ

K∑

k=1

∂u(ℓ)k

∂θi

∂u(ℓ)k

∂θj. (4.34)

We further assume that the noise variances are equal to σ2t for translational

measurement and are equal to σ2r for rotational measurements.

In the derivation of the FIMs the two following approximations are used(see (Kay, 1993, example 3.14) or Stoica et al. (1989))

K∑

k=1

cos2(ωk + β) ≈ K

2(4.35)

K∑

k=1

sin2(ωk + β) ≈ K

2(4.36)

K∑

k=1

sin(ωk + β) cos(ωk + β) ≈ 0, (4.37)

which are valid for ω being not near 0 or 1/2 and are exact when ω = 2πmK , m ∈

Z.According to the model of (4.23) and (4.24) a Love wave is parametrized

with the vector θ(L) = (α, ϕ, κ, ψ)T, thus the corresponding FIM is I(θ(L)) ∈R4×4. From (4.7) and using (4.34) is derived the FIM for the model of a singleLove wave and translational measurements, which is given in (4.40).

From (4.8) and using (4.34) is derived the FIM the model of a single Lovewave and rotational measurements, which is given in (4.41).

The FIM for the model of a single Love wave using both translational androtational measurements is obtained adding the two FIM of (4.40) and (4.41)as in (4.18)

I(θ(L)) = It(θ(L)) + Ir(θ

(L)) . (4.38)

According to (4.28) and (4.29) a Rayleigh wave is parametrized with the

vector θ(R) = (α, ϕ, κ, ψ, ξ)T, thus the corresponding FIM is I(θ(R)) ∈ R5×5.

95

From (4.9) and using (4.34) is derived the FIM for the model of a single Rayleighwave and translational measurements, which is given in (4.42).

From (4.10) and using (4.34) is derived the FIM for the model of a singleRayleigh wave and rotational measurements, which is given in (4.43).

The FIM for the model of a single Rayleigh wave using both transla-tional and rotational measurements is obtained adding the two FIM of (4.42)and (4.43) as in (4.18)

I(θ(R)) = It(θ(R)) + Ir(θ

(R)) . (4.39)

It(θ(L)) =

α2K

2σ2t

Ns

α2 0 0 0

0 Ns

∑Ns

n=1∂Φn∂κ

∑Ns

n=1∂Φn∂ψ

0∑Ns

n=1∂Φn∂κ

∑Ns

n=1

(

∂Φn∂κ

)2 ∑Ns

n=1∂Φn∂ψ

∂Φn∂κ

0∑Ns

n=1∂Φn∂ψ

∑Ns

n=1∂Φn∂ψ

∂Φn∂κ

Ns +∑Ns

n=1

(

∂Φn∂ψ

)2

(4.40)

Ir(θ(L)) =

α2κ2K

8σ2r

Ns

α2 0 Ns

ακ0

0 Ns

∑Ns

n=1∂Φn∂κ

∑Ns

n=1∂Φn∂ψ

Ns

ακ

∑Ns

n=1∂Φn∂κ

Ns

κ2+

∑Ns

n=1

(

∂Φn∂κ

)2 ∑Ns

n=1∂Φn∂ψ

∂Φn∂κ

0∑Ns

n=1∂Φn∂ψ

∑Ns

n=1∂Φn∂ψ

∂Φn∂κ

∑Ns

n=1

(

∂Φn∂ψ

)2

(4.41)

It(θ(R)) =

α2K

2σ2t

·

·

Ns

α2 0 0 0 0

0 Ns

∑Ns

n=1∂Φn∂κ

∑Ns

n=1∂Φn∂ψ

0

0∑Ns

n=1∂Φn∂κ

∑Ns

n=1

(

∂Φn∂κ

)2 ∑Ns

n=1

(

∂Φn∂κ

∂Φn∂ψ

)

0

0∑Ns

n=1∂Φn∂ψ

∑Ns

n=1

(

∂Φn∂κ

∂Φn∂ψ

)

Ns sin2 ξ +

∑Ns

n=1

(

∂Φn∂ψ

)20

0 0 0 0 Ns

(4.42)

Ir(θ(R)) =

α2κ2 cos2 ξK

2σ2r

·

·

Ns

α2 0 Ns

ακ0 − tan ξNs

α

0 Ns

∑Ns

n=1∂Φn∂κ

∑Ns

n=1∂Φn∂ψ

0

Ns

ακ

∑Ns

n=1∂Φn∂κ

Ns

κ2+

∑Ns

n=1

(

∂Φn∂κ

)2 ∑Ns

n=1∂Φn∂ψ

∂Φn∂κ

− tan ξNs

κ

0∑Ns

n=1∂Φn∂ψ

∑Ns

n=1∂Φn∂ψ

∂Φn∂κ

Ns +∑Ns

n=1

(

∂Φn∂ψ

)20

− tan ξNs

α0 − tan ξNs

κ0 tan2 ξNs

(4.43)

with Φn = −κ · pn + ϕ

Chapter 5

Sensor Placement for the Analysis of Seismic

Surface Waves:

Sources of Error, Design Criterion, and Array Design

Algorithms

Stefano Maranò1, Donat Fäh1, and Yue M. Lu2

1 ETH Zurich, Swiss Seismological Service, 8092 Zürich, Switzerland.2 Harvard University, School of Engineering and Applied Sciences, Cambridge, MA

02138, USA.

Submitted to Geophys. J. Int.

Abstract

Seismic surface waves can be measured by deploying an array of seis-mometers on the surface of the earth. The goal of such measurementsurveys is, usually, to estimate the velocity of propagation and the direc-tion of arrival of the seismic waves.

In this paper, we address the issue of sensor placement for the anal-ysis of seismic surface waves from ambient vibration wavefields. First,we explain in detail how the array geometry affects the MSEE of pa-rameters of interest, such as the velocity and direction of propagation,both at low and high SNRs. Second, we propose a cost function suit-able for the design of the array geometry with particular focus on theestimation of the wavenumber of both Love and Rayleigh waves. Third,we present and compare several computational approaches to minimizethe proposed cost function. Numerical experiments verify the effective-ness of our cost function and resulting array geometry designs, leadingto greatly improved estimation performance in comparison to arbitraryarray geometries, both at low and high SNR levels.

5.1 Introduction

Sensor arrays are used in numerous applications, including radar, underwatersource location, astronomical imaging, and geophysical surveying. Since the

96

97

geometry of the sensor array has a major impact on the performance of thearray processing system, the design of optimal array geometries is an importanttask in many applications (Tokimatsu, 1997; Van Trees, 2002).

The motivation of this work arises from the analysis of seismic surface waves.In particular, our interest lies in the analysis of ambient vibrations from arrayrecordings. Ambient vibrations span a broad range of frequencies and mayhave natural or anthropic origin (Bonnefoy-Claudet et al., 2006b). Propertiesof the wavefield, such as the velocity of propagation and polarization, are usedto infer a structural model for the site. This has application in microzonationand in geotechnical investigations (Tokimatsu, 1997; Okada, 2006).

The wavefield of ambient vibrations is primarily composed of Love wavesand Rayleigh waves. Array recordings of ambient vibrations are used to es-timate the dispersion curve, i.e. the relationship between the velocity andfrequency, of such waves.

Fig. 5.1(a) shows the location and the geometry of an array deployed bythe Swiss Seismological Service near Brigerbad, in southwestern Switzerland.In this survey, the ground displacement produced by ambient vibrations isrecorded for around two hours. A maximum likelihood (ML) method is usedto estimate the wavenumbers of Love and Rayleigh waves. Additional detailsconcerning the survey and the processing are given in Maranò et al. (2012).Fig. 5.1(b) depicts a large number of ML estimates of the wavenumber ofRayleigh waves at different frequencies. Darker regions indicate the presence ofseveral wavenumber estimates having the same value. The dark curve extendingacross the whole figure, from bottom-left to top-right, identifies the dispersioncurve of the fundamental mode. The first higher mode is also visible between8 Hz and 12Hz, just below the fundamental mode.

An ML estimator suffers from two distinct types of error, namely, gross er-rors (or outliers) and fine errors (Vertatschitsch & Haykin, 1991; Athley, 2008).Both types of errors are influenced by array geometry. At low signal-to-noiseratio (SNR), the presence of local maxima in the likelihood function (LF) leadsto large estimation errors. At high SNR, errors are smaller and the variance ofthe estimator is well described by the Cramér-Rao bound (CRB) (Rao, 1945;Cramér, 1946).

In Fig. 5.1(b), it is possible to see that, toward higher frequencies and largerwavenumber, there is a significant amount of wavenumber estimates that donot belong to the dispersion curve. These are the gross errors or outliers. Onthe other hand, the thickness of the dispersion curve is related to fine errors,i.e., the variance of the estimator at high SNRs.

Using sensor arrays to study seismic wavefields has a long history and severaldifferent array geometries have been used. In Horike (1985) L-shaped and cross-shaped arrays with a regular sensor spacing have been employed. Irregularlyspaced crosses were used in Asten & Henstridge (1984); Milana et al. (1996);Ohori et al. (2002) and Rost & Thomas (2002). In other works, sensors werearranged as several triangles centred around a common point (Satoh et al.,2001b,a). In Gaffet et al. (1998) and Cornou et al. (2003b), concentric circleswere used.

The limitations of different array geometries have been investigated by dif-ferent authors. In particular, the interest has been to identify a range ofwavenumbers, or a related quantity such as velocity or slowness, where theresult of the array processing is more reliable. The largest and the smallest

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(a) Geometry of the sensor array. The in-let pinpoints the location of the array withinSwitzerland. The geographic coordinates areSwiss coordinates (CH1903).

2 4 6 8 10 12 140

0.05

0.1

Frequency [Hz]

Wavenum

ber

[1/m

]

(b) Rayleigh wave dispersion curve. Fun-damental mode and first higher mode arevisible.

Figure 5.1: Array deployment and Rayleigh wave dispersion curve from theBrigerbad survey.

resolvable wavenumbers have been related either to the array aperture and thesmallest interstation distance or to the height of the sidelobes of the array re-sponse function (Woods & Lintz, 1973; Asten & Henstridge, 1984; Tokimatsu,1997; Kind et al., 2005; Wathelet et al., 2008; Poggi & Fäh, 2010).

The design of array geometries for the analysis of ambient vibrations hasalso been investigated by the community. Qualitative guidelines, based onempirical evidences, for array design are provided in Rost & Thomas (2002)and Kind et al. (2005).

In this work, we present quantitative criteria and computational proceduresfor designing array geometries for measuring ambient vibrations. Our goal isto improve the performance of the ML estimator of wavefield parameters byoptimizing the sensor positions. Given the nature of ambient vibrations andthat we mostly rely on very noisy measurements, we are mainly interested in thelow SNR regime, focusing primarily on reducing the occurrence of gross errors.In addition, we deal with small scale arrays deployed at the earth surface, andthus optimize the geometry of a planar array, given a budget on the numberof sensors and indication about the frequency support of the signals.

The contributions of this paper are three-fold:

• We rigorously derive the relationship between array geometry and grosserrors and fine errors in parameter estimations. We show how the shapeof the average LF is related to sensor positions through the Fourier trans-form of the sampling pattern.

• We propose a quantitative design criterion for improving estimation per-formance by means of sensor placement. By reformulating and relaxingthe proposed optimization problem, we propose a practical array designalgorithm based on a mixed integer program (MIP) with linear objectivefunction and linear constraints. The proposed sensor placement algo-rithm generates arrays composed of simple regular geometries and arethus suitable for field deployment.

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• We compare the proposed sensor placement techniques with several otheroptimization techniques both in terms of array design and of estimationperformance. We show through numerical experiments how the estima-tion performance can be increased by using the proposed array designcriterion and algorithm.

The rest of this paper is organized as follows. In Section 5.2, wave equations,a measurement model, and an estimation method are presented. The distinc-tion between gross and fine errors, together with a rigorous derivation of therelationship between sensor position and LF are given in Section 5.3. In Sec-tion 5.4, we outline the quantities relevant to the sensor placement problem andpropose a design criterion. In Section 5.5, we consider array design methodsand in Section 5.6 we compare the results of different techniques. The findingsof the paper are summarized in Section 5.7.

5.2 System Model

Seismic surface waves propagate along the surface of the earth (Aki & Richards,1980) and can be measured using an array of seismometers. In seismic survey-ing, a typical goal is to estimate the velocity of propagation and the directionof arrival of such waves.

In this section, we briefly describe the model for seismic surface waves,the noise model for sensor measurements, and the ML approach to parameterestimation.

Notation. The vector θ indicates a generic value of the wavefield pa-rameters. The vector θ indicates the true, and possibly unknown, wavefieldparameters. The vector θ indicates an estimate of θ. When necessary, a su-perscript will specify whether the parameter vector describes a Love wave or aRayleigh wave, i.e., θ(L) and θ(R), respectively. The same convention is usedfor each element of the wavefield parameter vector.

Seismic Surface Waves

The perturbation induced on a ground particle by a seismic wave is describedby a vector quantity. Indeed, at each spatial and temporal location it is possi-ble to use, for example, a three component displacement vector to describe themovement of a ground particle in space. In addition, other quantities are avail-able to describe the seismic wavefield, including, for example, ground rotationsand strains (Aki & Richards, 1980).

Different instrumentation can be used to measure the seismic wavefield anddictates which equations are the more appropriate to model the ground motioninduced by the wavefield. Certain instruments can measure the sole verticaldisplacement of the seismic wavefield, in which case the scalar wave model issuitable. A common type of instrument, the triaxial seismometer, measuresthe ground displacement vector u along the three axes x, y, z. We use a right-handed Cartesian coordinate system having the z-axis pointing upwards.

Consider the displacement at the earth surface (z = 0) induced by a Lovewave with a planar wavefront. At the surface of the earth, Love waves exhibit atranslational particle motion confined to the horizontal plane, and the particle

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oscillates perpendicular to the direction of propagation. The particle displace-ment u = (ux, uy, uz) generated by a single monochromatic Love wave withfrequency ω at position p ∈ R2 and time t is

ux(p, t) = −α sinψ cos(ωt− κ · p+ ϕ)

uy(p, t) = α cosψ cos(ωt− κ · p+ ϕ) (5.1)

uz(p, t) = 0 ,

where the wavevector κ = κ(cosψ, sinψ)T = (κx, κy)T is a vector pointing

in the direction of propagation. The magnitude of the wavevector is calledwavenumber κ = ‖κ‖. The velocity of propagation is ω/κ. The azimuthψ, measured counterclockwise from the x-axis, is related to the direction-of-arrival (DOA). The amplitude and phase of the source are denoted by α ∈ R+

and ϕ, respectively. We say that a Love wave is parametrized by a wavefieldparameter vector θ(L) = (α, ϕ, κ, ψ).

Rayleigh waves exhibit a translational particle motion having an ellipticalpattern and confined to the vertical plane perpendicular to the surface of theearth and containing the direction of propagation of the wave. The particledisplacement generated by a single Rayleigh wave is

ux(p, t) = α sin ξ cosψ cos(ωt− κ · p+ ϕ)

uy(p, t) = α sin ξ sinψ cos(ωt− κ · p+ ϕ) (5.2)

uz(p, t) = α cos ξ cos(ωt− κ · p+ π/2 + ϕ) .

The angle ξ ∈ [−π/2, π/2) is called ellipticity angle of the Rayleigh wave anddetermines the eccentricity and the sense of rotation of the particle motion. Thequantity |tan ξ| is known as the ellipticity of the Rayleigh wave. See Maranòet al. (2012) for a detailed description of this parametrization. A Rayleigh wave

is parametrized by a wavefield parameter vector θ(R) = (α, ϕ, κ, ψ, ξ)T.

Scalar Plane Wave

We also consider a simpler scalar wave model. This model will be used in partsof this paper to introduce certain ideas before extending them to Love waveand Rayleigh wave models. This wave model is analogous to an acoustic wavemeasured by a scalar pressure sensor (i.e., a microphone).

Let u(p, t) denote the scalar value of the wavefield at position p and timet. For a monochromatic source at frequency ω the wavefield is

u(p, t) = α0 cos(ωt− κ · p+ ϕ0) . (5.3)

Suitable parametrization of α0 and ϕ0 makes the scalar wave model equivalentto a given component of a vector wave model as given in (5.1) or (5.2).

Measurement Model

To measure seismic waves, we deploy an array ofNs sensors on the surface of theearth positioned at locations pnn=1,...,Ns

. We restrict our interest to smallaperture arrays and work with a flat earth model, thus consider planar arrays.The signal at each sensor component is sampled at K instants tkk=1,...,K . In

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general, each sensor measures a vector quantity. Let L be the total numberof channels recorded by the array. In the case of scalar sensors, then simplyL = Ns.

Each measurement Y(ℓ)k is corrupted by additive white Gaussian noise and

is modeled as

Y(ℓ)k = u

(ℓ)k (θ) + Z

(ℓ)k , (5.4)

for each channel ℓ = 1, . . . , L where Z(ℓ)k ∼ N (0, σ2

ℓ ). The noise variance , σ2ℓ is,

in general, different on each channel. The quantities u(ℓ)k (θ) are deterministic

functions of wavefield parameters θ as described in (5.1)-(5.3).It follows that the joint probability density function (PDF) of the measure-

ments is

pY (y |θ) =L∏

ℓ=1

K∏

k=1

1√

2πσ2ℓ

exp

(

− (y(ℓ)k − u

(ℓ)k (θ))2

2σ2ℓ

)

, (5.5)

where we grouped the measurements as Y = Y (ℓ)k ℓ=1,...L

k=1,...,K.

Parameter Estimation

Wavefield parameters can be found by using maximum likelihood (ML) esti-mation. ML estimation is a useful method for estimating the parameters ofa statistical model. It is a widely-used estimation technique due to its broadapplicability and to the optimal performances in many settings (Fisher, 1922).We refer the interested reader to Kay (1993) for additional details on MLestimation. An implementation this method for the estimation of wavefieldparameters of surface waves has been proposed by the authors (Maranò et al.,2012).

The LF of the observations is obtained from the PDF of the measure-ments (5.5). We denote the LF by pY (y |θ), where y are the observations andθ is the vector of the wavefield parameters argument of the LF. We stress thatthe LF is a function of the model parameters θ while the measurements y arefixed.

An ML estimate θ of the true wavefield parameters θ is found by maximizingthe LF, i.e.,

θ = argmaxθ

pY (y |θ) . (5.6)

The LF can be thought of as a utility function which is legitimised by the statis-tical model of the observations. The point of maximum of the LF correspondsto the ML estimate of the parameters.

5.3 Sources of Error

In this section we describe in detail how the sensor positions affect the perfor-mance of the ML parameter estimation. We make a distinction between two

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−20 −15 −10 −5 0 5 1010

−9

10−8

10−7

10−6

10−5

10−4

10−3

No information

Threshold zone region

Asymptotic region

MSE

E[m

−2]

SNR [dB]

Figure 5.2: An example of the MSEE of a ML estimator. The MSEE is depictedwith a blue dashed line. In the no information region the MSEE is very largeand constrained by the implementation of the algorithm. In the thresholdregion the occurrence of outliers keep the MSEE significantly larger than theCRB. At last, in the asymptotic region, the MSEE is well described by theCRB, which is shown with the black dashed line.

types of errors: gross errors and fine errors (Athley, 2008). Array geometryaffects both types of error.

Fig. 5.2 shows the typical behavior of the mean-squared estimation error(MSEE) of an ML estimator. The figure is obtained by repeating the estimationof the wavenumber of an unknown wave with several different noise realizationand for different SNRs. The SNR is defined as the ratio of signal power overnoise power, i.e., SNR = α2

0/2σ2. Three operation regions of the estimator are

recognized at different SNR ranges. The approximate extent of the regions isshown in Fig. 5.2. At very low SNR, the noise dominates the signal of interest,and this is called the no information region. In this region, the estimates arecompletely random and carry no information about the value of the parameterestimated. At larger SNR, there is a region called the threshold region. Inthis region, the estimated value may be often close to the true value, howeverthe MSEE is still considerably large as gross estimation errors occur. Grossestimation errors are also known as global errors or outliers. Further increasingthe SNR, we approach the asymptotic region. Fine estimation error occurs inthis region and the MSEE of an ML estimator is well described by the Cramér-Rao bound (CRB) (Rao, 1945; Cramér, 1946). Fine estimation errors are alsoknown as local errors.

The abrupt increase in the MSEE below a certain SNR is known in literatureas the threshold effect and is due to a transition from fine errors to grosserrors (Van Trees, 2001).

Gross Errors

Gross errors are due to the presence of local maxima (sidelobes) other thanthe true maximum in the LF (Athley, 2008). In this section, we establish therelationship between the LF and the array geometry.

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Consider an array of Ns sensors at positions pnn=1,...,Ns. The spatial

sampling pattern is given by a sum of Dirac delta located at the sensor positions

h(x, y) =

Ns∑

n=1

δ(p− pn) . (5.7)

Its Fourier transform is given by

H(κx, κy) =

∫

R2

h(x, y)e−i(κx,κy)·pdp (5.8)

=

Ns∑

n=1

e−i(κx, κy)·pn , (5.9)

where κx and κy are the wavenumber along the x and y coordinate axes,respectively.

We will show that H(κx, κy) is important to explain the occurrence of grosserrors and to establish a strategy to mitigate them.

It is possible to verify that the function |H(κx, κy)| always exhibits a globalmaximum at (κx, κy) = (0, 0). Other maxima also exist but, in general, areof smaller amplitudes. Moreover, the function is symmetric around the origin,i.e. |H(κx, κy)| = |H(−κx,−κy)|.

Similarly, the temporal sampling pattern g and its Fourier transform G aregiven by

g(t) =

K∑

k=1

δ(t− tk) (5.10)

G(ω) =

∫

R

g(t)e−iωtdt =K∑

k=1

e−iωtk , (5.11)

where tkk=1,...,K are the sampling times.After introducing these quantities, we return to our main interest, that is

the analysis of the shape of the LF. To this aim, we compute the expectationof the LF from the model of (5.5).

We consider the observations y of a single scalar wave (cf. (5.3)) with true

wavefield parameter vector θ. The ML estimates of α0 and ϕ0 can be foundexplicitly as a function of the observations y and of the wavevector κ. Thereforewe write the LF of the observations as a function of κ as

pY (y |κ) = maxα0,ϕ0

pY (y |θ) , (5.12)

where the maximization is achieved by inserting into pY (y |θ) the ML estimatesα0 and ϕ0. In Appendix 5.A, analytic expressions for α0 and ϕ0 are given.

Since we are interested only in the shape of the LF we compute its loga-rithm, and drop multiplicative and additive constants. We use the expectationoperator E· to obtain an indication about the average shape of the LF. Aftersome manipulations, explained in detail in Appendix 5.A, we obtain

E ln(pY (y |κ)) ∝ |G(ω − ω)H(κ− κ)|2 , (5.13)

104

where the symbol ∝ denotes equality up to an affine transform, i.e. f(x) ∝g(x) if f(x) = C1g(x) + C2 and C1, C2 do not depend on x. From (5.13),we understand that the average shape of the log-likelihood function (LLF) isrelated to a translation of the Fourier transform of the sampling pattern.

The quantity G(·) is of marginal importance concerning the occurrence ofgross errors. Indeed it is usually possible to sample the signal with sufficientlysmall sampling time and with enough samples. We observe that for ω exactlyknown G(ω− ω) = G(0) is a real constant and thus does not change the shapeof the LF. Hereafter we will omit the factor G(·).

The quantity |H(κ)|, or quantities closely related to it, is known in literaturewith several names such as array response (Woods & Lintz, 1973; Asten & Hen-stridge, 1984; Van Trees, 2002; Rost & Thomas, 2002; Wathelet et al., 2008),array factor (Bevelacqua & Balanis, 2007), and array transfer function (Gaffetet al., 1998).

It is now clear how the Fourier transform of the sampling pattern H(κ)affects the shape of the LF. At low SNR, outliers tend to accumulate aroundthe local maxima of |H(κ)|. In order to reduce the occurrence of the grosserrors it is necessary to reduce the height of the local maxima (Vertatschitsch& Haykin, 1991; Athley, 2008).

As mentioned in the previous section, seismic waves are measured and mod-eled as vector quantities. Therefore it is necessary to extend the findings con-cerning gross errors obtained for the scalar wave case to the vector wave case.

When considering vector measurements, the PDF of the observations needsto be augmented with the contribution of all the sensor components. Withthe assumption of independent observations, this is achieved by increasing L

in (5.5) and by choosing the appropriate wave model u(ℓ)k .

Concerning gross errors, the shape of the LF of observations of Love andRayleigh waves is influenced differently by the different components of eachsensor but remains a function of the Fourier transform of the sampling patternH(κ). The derivation of this relationships are detailed in Appendix 5.A andin this section we only present the final results.

Love Wave

For Love waves as in (5.1), the LLF is related to H as

E ln(pY (y |κ, ψ)) ∝ fL(ψ, ψ) |H(κ− κ)|2 , (5.14)

where

fL(ψ, ψ) = cos2(ψ − ψ) . (5.15)

From the previous expressions it is possible to understand that, similar to thescalar wave setting, the LLF for Love waves is directly related to the Fouriertransform of the sampling pattern. The factor fL(ψ, ψ) influences the shape ofthe LLF as a function of azimuth.

Rayleigh Wave

For Rayleigh waves as in (5.2), the LLF is related to H as

E ln(pY (y |κ, ψ, ξ)) ∝ fR(ψ, ψ, ξ, ξ) |H(κ− κ)|2 , (5.16)

105

where

fR(ψ, ψ, ξ, ξ) =(

sin ξ sin ξ cos(ψ − ψ) + cos ξ cos ξ)2

. (5.17)

We observe that for a fixed ellipticity angle ξ the relationship is similar to theLove wave setting. For fixed azimuth ψ and wavenumber κ, the occurrence oflocal maxima is described by trigonometric functions of ξ. We observe thatthe local maxima due to (5.17) are independent of the sensor positions andtherefore are outside the scope of this work.

Fine Errors

At high SNR, the performance of the ML estimator is well described by theCRB. The CRB is a lower bound on the variance of all unbiased estimators.For example, Fig. 5.2 shows how the MSEE matches the CRB in the asymptoticregion.

To compute the CRB, we first need to introduce the notion of Fisher in-formation (FI). The FI conveys the amount of information about a statisticalparameter carried by the PDF of the observations (Fisher, 1922).

For a statistical model with multiple parameters the Fisher informationmatrix (FIM) is given by

I(θ) = E

−∂2 ln pY (y |θ)

∂θ2

. (5.18)

The matrix I is a square symmetric matrix with as many columns as theelements in the vector θ. The diagonal terms of the matrix correspond to theFI of each parameter in the parameter vector θ. These elements should beinterpreted with care, as they disregard the uncertainty due to the other modelparameters being unknown. The off-diagonal terms are sometimes referred toas cross-information terms.

The information inequality (Cramér, 1946; Rao, 1945) states that the mean-squared estimation error (MSEE) of an unbiased estimator is lower boundedas

E

(

θ − Eθ)(

θ − Eθ)T

(I(θ))−1 , (5.19)

where A B means that the matrix A−B is positive semidefinite (PSD). The

left-hand side of (5.19) represents the covariance matrix of the vector θ andthe right-hand side is the matrix inverse of the FIM. Following the informationinequality, we are interested in the diagonal elements of I−1 as they provide alower bound on the MSEEs of the corresponding parameters.

The CRB on wavenumber for the scalar wave model is obtained using (5.3)and (5.5). The FIM is derived and then inverted analytically as in (5.18)and (5.19). From the corresponding entry of I−1, the MSEE of wavenumberis lower bounded as

E

(κ− Eκ)2

≥(

α20K

2σ2

(

Qaa(ψ)−Q2ab(ψ)

Qbb(ψ)

))−1

. (5.20)

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The CRB is directly proportional to noise power σ2, inversely proportional tothe amplitude of the wave α0 and to the number of samples K. The CRBdepends on the sensor positions through Qaa, Qbb and Qab. We also observethat the wavenumber CRB is independent of the temporal frequency ω, thuswe expect fine errors to have comparable variance at any frequency.

The quantities Qaa, Qbb, and Qab are called moment of inertia (MOI) ofthe array. These quantities are independent of array translations, but are ingeneral dependent on the azimuth. We introduce the coordinate system (a, b),which is related to the coordinate system (x, y) as

(

a

b

)

=

(

cosψ sinψ

− sinψ cosψ

)(

x

y

)

, (5.21)

where the angle of rotation is the azimuth ψ. Therefore a is the axis along thedirection of propagation of the wave and b the axis perpendicular to it. Thesensor positions in the rotated coordinate system are (an, bn)n=1,...,Ns

. TheMOI of the array in the coordinate system (a, b) are defined as

Qaa(ψ) =

Ns∑

n=1

(an − a)2 (5.22)

Qbb(ψ) =

Ns∑

n=1

(bn − b)2 (5.23)

Qab(ψ) =

Ns∑

n=1

(an − a)(bn − b) , (5.24)

where a = 1Ns

∑Ns

n=1 an and b = 1Ns

∑Ns

n=1 bn define the phase centre of thearray.

With reference to the CRB in (5.20), large Qaa and Qbb are desirable inorder to reduce the CRB and thus the MSEE in the asymptotic region. Also asmall Q2

ab is advantageous.A large Qaa can in general be obtained with a large aperture array. How-

ever, observe that a large aperture may invalidate the plane wave assumptionwhich is of critical importance in practical applications. Moreover, it is possi-ble to choose an array geometry such that Qab = 0 and thus eliminating theterm −Q2

ab/Qbb from (5.20). Some further remarks on the MOIs are given inAppendix 5.B.

Love Wave and Rayleigh Wave

In the vector case, the derivation of the CRB in Section 5.3 needs to be ex-tended. However, for translational sensors the dependence of the CRB on arraygeometry remain similar to the one presented in (5.20) and therefore we do notreview this aspect. A detailed analysis of the CRB of parameters of Love waveand Rayleigh wave measured with translational and/or rotational sensors isfound in Maranò & Fäh (2013).

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5.4 Problem Statement and Design Criterion

The aim of sensor placement is to improve the performance of parameter es-timation, by an appropriate choice of the array geometry. In particular, weconsider the setting where two design requirements are given:

• A budget of Ns sensors to be placed on a plane surface;

• An indication about the spatial bandwidth of the signals of interest.

In several applications, only a limited number of sensors is available. Thisis often the case for the analysis of seismic wavefields where each individualinstrument can be quite expensive and/or difficult to install at the measurementsite.

Spatial bandwidth is defined by the spatial frequency content of the wave-field. Particularly important is the largest wavenumber present in the wavefield,denoted by κmax. Knowledge of spatial bandwidth is a reasonable assumptionin many applications. In the seismic case the wavenumber is the goal of theestimation process and thus an exact knowledge of the signal bandwidth isnot available. However, prior knowledge on the geology of the site or previoussurveys may suggest a meaningful bound of the spatial bandwidth of the signal.

We have seen that two different criteria applies at low and high SNR inorder to reduce gross errors and fine errors, respectively. At low SNR, arraydesign focuses on the design of the LF in order to reduce the height of localmaxima and thus the occurrence of gross errors. We have shown as the Fouriertransform of the spatial sampling pattern H(κ) plays a central role. At highSNR, array design targets the reduction of fine errors. This can be achievedby designing an array geometry that reduces the CRB in (5.20).

Reduction of Gross Errors

It is shown in the literature that the probability of occurrence of gross errorsis related to the amplitude of the local maxima relative to the true maximum,see, for example, Athley (2008). Therefore, in order to decrease gross errors itis necessary to reduce the amplitude of local maxima of the LF over a certainregion of the κx κy-plane. We choose this region as an annulus defined byκmin and 2κmax, i.e., the region bounded by two concentric circles with radiiκmin and 2κmax, respectively. We now explain the rationale behind this designchoice.

The circular symmetry of the considered region is motivated by the finalapplication, the analysis of ambient vibration wavefields. In particular, by thefact that seismic waves may traverse the array of sensors from any DOA.

We choose a value of κmin related to the smallest spacing, in the κx κy-plane,between two signals that we wish to resolve. The smallest spacing between tworesolvable signals is known in literature as the Rayleigh resolution limit (VanTrees, 2001). The exact Rayleigh resolution limit can be computed from agiven sampling pattern and it is in general slightly different from κmin. Inpractice, the quantity κmin is also related to the smallest wavenumber that canbe reliably estimated in field measurements.

We recall that the LF is related to the Fourier transform of the samplingpattern H(κ) through a translation as in (5.13). We also observe that the ML

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estimate of the wavevector is found as the largest value of the LF in a disk ofradius κmax centred in the origin. A geometrical argument suggests that, inorder to reduce gross errors, it is necessary to reduce the height of the localmaxima of |H(κ)| in a disk of radius 2κmax.

We illustrate this argument with an example. In Fig. 5.3, we show graph-ically how the LF is related to a translation of the Fourier transform of thesampling pattern. Fig. 5.3(a) shows the magnitude of the Fourier transform ofan optimised sampling pattern. The two magenta circles have radius κmin and2κmax, respectively. In this example κmin = 1/4 and κmax = 1. We observe thepresence of a global maximum at (0, 0) inside the inner circle, that the localmaxima in the annulus region are small in amplitude, and that outside theouter ring there are several local maxima with large amplitude.

Fig. 5.3(b) depicts the expected LLF of a wave with wavenumber κ =(0.5, 0) as an highlighted disk of radius κmax = 1. The global maxima ofthe expected LLF is located at (0.5, 0). Observe that the shaded region ofFig. 5.3(b) is exactly the Fourier transform of the sampling pattern shown inFig. 5.3(a). Fig. 5.3(c) depicts the expected LLF of a wave with wavenumberκ = (1, 0). This wave has the largest admissible wavenumber since ‖κ‖ = κmax

and therefore the maxima of the LLF lies at the edge of the parameter space.Fig. 5.3(d) depicts the expected LLF of a wave with wavenumber κ = (1.5, 0).This wave has a wavenumber larger than what was assumed for the array designsince ‖κ‖ > κmax. The global maximum of the expected LLF corresponds toa local maxima of |H(κ)| outside the minimization region. As results, the MLestimate of the wavevector is not corresponding to the true wavevector.

We formulate the following minimization program, aimed at minimizing thelargest value of |H(κ)| over the annulus defined by κmin ≤ ‖κ‖2 ≤ 2κmax as

minp1,p2,...,pNs

maxκ

∣

∣H(κ,p1,p2, . . . ,pNs)∣

∣ (5.25a)

s. t.κmin ≤ ‖κ‖2 ≤ 2κmax , (5.25b)

where we emphasized the dependence of H on sensor positions. In this op-timization problem, the minimization variables are the sensor positions. Thetarget of the minimization, maxκ

∣

∣H(κ,p1,p2, . . . ,pNs)∣

∣, is a function of thesensor positions pnn=1,...,Ns

. To the best of our knowledge, the use of suchcost function is new to the seismological community.

This minimization problem becomes harder as the extent of the annulus isincreased. In particular we observe that large values of the ratio κmax/κmin

lead to harder optimization problems.

Reduction of Fine Errors

Fine errors can be reduced by decreasing the CRB. The CRB derived in (5.20)depends on sensor positions pnn=1,...,Ns

and on the azimuth ψ. Indeed, ingeneral, different MSEE values are expected for different DOAs of the incomingwave.

We choose to maximize the inverse of the CRB for the DOA leading to the

109

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(a) Fourier transform of an optimised sam-pling pattern.

−1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) Expected LLF, κ = (0.5, 0).

−1 0 1 2 3

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(c) Expected LLF, κ = (1, 0).

0 1 2 3

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(d) Expected LLF, κ = (1.5, 0).

Figure 5.3: Illustration of the relationship between the magnitude of the Fouriertransform of the sampling pattern and expected LLFs of different waves.

worst performance

maxp1,p2,...,pNs

minψ

(

Qaa(ψ,p1, . . . ,pNs)− Q2

ab(ψ,p1, . . .)

Qbb(ψ,p1, . . .)

)

. (5.26)

Such maximization problem is unbounded. Indeed, it is possible to verifythat a uniform circular array (UCA) with sufficiently large radius can makethe objective function arbitrarily large. Moreover, an array with a very largeaperture could void the assumption of planar wavefronts.

Discussion

Elaborating on the two distinct sources of errors discussed in Section 5.3, weshowed how two different sensor placement criteria exist. Depending on theapplication, interest may lie in reducing one or the other type of error. It is

110

also possible to combine both optimization problems (5.25) and (5.26) in orderto account for both types of errors.

For ambient vibration wavefields, typically the waves have very small am-plitudes compared to the level of the background noise. Therefore we chooseto primarily focus on improving performance at low SNR and thus on thereduction of gross errors.

Although we choose not to address fine errors explicitly, we consider arrayswith Qab = 0. This additional constraint is beneficial for reducing the CRBof (5.20) and it also ensures that the resulting array has no preferential DOA,i.e., Qaa(ψ) is equal for every azimuth.

5.5 Array Design Methods

To the best of our knowledge, there is no algorithm with polynomial complexitythat can always find the global optimal solution to the program in (5.25). Theobjective function (5.25a) has many local minima and any solution found by,for example, a gradient descent method will strongly depend on the initialstarting point. In this section, we propose an algorithm using a mixed integerprogram (MIP), which is shown empirically to be effective. We also compare itwith two other possible approaches addressing the same optimization problem.

Mixed Integer Program

The program in (5.25) is difficult to solve because the minimization variablespnn=1,...,Ns

are arguments of the complex exponentials. To overcome thislimitation, instead of representing the sensor positions with continuous vari-ables we consider a finite number of possible sensor positions. We representthe discrete sensor positions with the binary vector x ∈ 0, 1N . An analogousminimization problem in this new variable is

minx‖Fx‖∞ (5.27a)

s. t.Asx = bs (5.27b)

x ∈ 0, 1N , (5.27c)

where ‖v‖∞ = max(|v1| , |v2| , . . . , |vN |).The vector x is a binary vector representing the presence or the absence of

a sensor at given spatial locations on the plane. Fig. 5.4(a) shows a possiblechoice of N feasible spatial locations for positioning the Ns available sensors.The linear operator F : RN → CM computes the two dimensional Fouriertransform of the array positions, restricted to M discrete spatial frequencies ofinterest. The operator F can be thought as a discretized version of H(κ). TheM frequencies lie in the annulus defined by κmin ≤ ‖κ‖2 ≤ 2κmax . Fig. 5.4(b)shows a possible choice of the M frequencies computed by F. Observe thatbecause of the symmetry |H(κ)| = |H(−κ)| the choice of the M frequenciescan be limited only to half of the κx κy-plane.

We choose to arrange both the N possible sensor positions and and theM spatial frequencies on circles around the origin, as in Fig. 5.4. This arbi-trary choice is supported by the radial symmetry of the problem as expressed

111

−0.5 0 0.5

−0.5

0

0.5

x [m]

y[m

]

(a) Possible N locations available in thex y-plane for the placement of Ns sensors.

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) Position of the M spatial frequenciesin the κx κy-plane. The magenta circlesdepict κmin and 2κmax.

Figure 5.4: A possible choice of the N spatial locations and of the M spatialfrequencies used in the construction of the operator F.

in (5.25). Concerning the sensor positions, this choice also makes it easier toenforce numerically any constraint on the MOIs.

The equalities in (5.27b) enforce a number of linear constraints. The numberof sensors is enforced to be equal to Ns. The vector x has exactly Ns elementsequal to one, corresponding to the sensor positions. The remaining elementsare zero. As an extension, using linear equalities is also possible to enforce thepresence or the absence of a sensor in a specific position.

In (5.27b) we also enforce the linear constraint Qab = 0. This constraint al-lows to eliminate the term −Q2

ab/Qbb from the expression of the CRB in (5.20),thus lowering the CRB. It also ensures that the array performance, in terms offine errors, would have no preferential direction, i.e., Qaa and Qbb are constantfor every azimuth.

Comparing the original problem in (5.25) and the discretized formulationin (5.27), we observe that the discretization of the sensor positions causesthe optimal value of (5.25) to be smaller than or equal to the optimal valueof (5.27). This is especially important when the vector x has small dimensionN and the possible sensor positions are coarse.

Observe that the objective function (5.27a) is a convex function of theminimization variable x. However, due to the binary constraint (5.27c) theminimization is an integer programming problem. Integer programs are gen-erally considered to be NP-hard, i.e. there is no polynomial time algorithm tosolve them.

We relax the optimization problem in (5.27) to make it more tractablefor implementation. We replace the convex objective function with a linearobjective function as

‖Fx‖∞ →∥

∥

∥

∥

∥

(

|ReFx||ImFx|

)∥

∥

∥

∥

∥

∞

.

112

This modification enables us to formulate the problem as a MIP with linearobjective function and linear constraints

minyy (5.28a)

s. t.Asx = bs (5.28b)

ReF

ImF

−ReF

− ImF

x 1y (5.28c)

y ∈ R (5.28d)

x ∈ 0, 1N , (5.28e)

where 1 is a vector of ones of size 4M×1 and denotes element-wise ≤. In thisprogram there are N binary variables and one continuous real variable. Thiscan be addressed using general purpose MIP algorithms (Gurobi Optimization,Inc., 2013). It is in theory possible to find the optimal solution to (5.28) usingthe branch and bound algorithm (Land & Doig, 1960). However, for a largenumber of possible sensor positions N , finding the optimal solution becomesnot always practical.

Genetic Algorithm

We also attempt a direct minimization of (5.25) using the genetic algorithm(GA) method (Goldberg, 1989). Such algorithm attempts to find good solu-tions using some random search pattern and there is no guarantee to find theoptimal solution. In our implementation we do not enforce any constraint onthe MOIs.

The constraint on the Qab = 0 is nonlinear in the variables pnn=1,...,Ns.

We do not enforce such constraint in the considered GA technique since weobserved that it makes the minimization considerably harder.

Uniform Circular Array

In addition, we compare with the best uniform circular array (UCA). In anUCA sensors are uniformly spaced on a ring of radius r. A line search over thepossible r allows us to obtain the radius of the best UCA for given κmin, κmax,and Ns

minr

maxκ

∣

∣H(κ,p1,p2, . . . ,pNs)∣

∣ (5.29a)

s. t.pn = r(cos(2πn/Ns), sin(2πn/Ns)) (5.29b)

κmin ≤ ‖κ‖2 ≤ 2κmax . (5.29c)

The objective function is the same of the original problem (5.25). Sensors arerestricted to be uniformly spaced on a circle of radius r.

We state again that all UCA with Ns ≥ 3 have Qab = 0. Moreover Qaa andQbb are constants for all azimuths. Details are provided in Appendix 5.B.

113

5.6 Numerical Results

In this section, we compare the arrays designed using the considered tech-niques and we quantify the impact different geometries have on the estimationproblem. In Section 5.6, the outcomes of the different array design methodsare compared in terms of the design criteria presented in Section 5.4. In Sec-tion 5.6, the estimation performance achieved using different array layouts isanalyzed by means of Monte Carlo simulations.

Array Design

The three array design techniques considered are: i) The proposed approach,i.e., the MIP of (5.28); ii) The direct minimization of (5.25) using a GA; andiii) The best UCA obtained from the program in (5.29).

Gross Errors

In terms of sensor design and minimization of the original problem (5.25), thegoodness of a solution is quantified with the amplitude of the largest localmaxima of |H(κ)| compared to the central maximum |H(0, 0)|. Therefore, weconsider the quantity

Hmax = maxκ

(

|H(κ)|2N2

s

)

(5.30)

for κmin ≤ ‖κ‖2 ≤ 2κmax . Observe that the quantity Hmax is smaller than 1,since the largest value of |H(κ)| is Ns.

Fig. 5.5 shows the value of Hmax achieved with different design techniquesfor different number of sensorsNs and κmax/κmin . In general, the value ofHmax

decreases for increasing number of sensors Ns. In particular, such decrease issteeper for few sensors. This indicates that the marginal benefit of an additionalsensor is greater for few sensors.

Different values of κmax/κmin are also considered. Fig. 5.5(a), 5.5(b), and 5.5(c)depict the minimization results for κmax = 1 and κmin equal to 1/2, 1/4, and1/6, respectively. A large value of κmax/κmin corresponds to a larger extensionof the annulus involved in the minimization. For larger values of κmax/κmin theminimization problem is harder and the Hmax values found are indeed larger.

In Fig. 5.5, we observe that the MIP technique typically achieves val-ues of Hmax smaller than the other design techniques, for all the consideredκmax/κmin.

We observed that the quality of the MIP solutions and the time necessaryto find good solutions depend on the choice of the N possible sensor locations,cf. Fig. 5.4(a). In particular, we observed that the number of possible sensorlocations on each concentric circle has to be related to Ns. One motivationis that due to the Qab = 0 constraint and the arrangement of the possiblesensor locations, the MIP solutions exhibit sensors placed on concentric circles.Therefore certain choices of the number of possible locations on each circle mayturn the MIP problem infeasible, and other choices may be more convenient.

114

For certain choices of Ns and arrangement of the possible spatial locationsthe globally optimal solution was found. For larger Ns and N the algorithmwas terminated after a given time limit.

The GA technique is able to consistently decrease Hmax for increasing num-ber of sensors. In contrast to the MIP technique, there is no unfavorable sensornumber when the GA technique is used. The GA solutions are in general worsethan the MIP solutions in terms of Hmax.

In terms of minimization, an intrinsic advantage of the GA technique isthat the minimization variables pnn=1,...,Ns

are continuous, this may help thealgorithm to optimize the solution, at least locally. The GA solution presentedis the best out of hundreds of runs performed with different initialization ofthe GA algorithm.

The UCAs technique performs similarly or worse than the other techniquesdepending on the specific design parameters. We observe that for κmax/κmin =2, there is little or no decrease in the value of Hmax for increasing number ofsensors above 10. For κmax/κmin equals to 4 and 6, the UCA is often largelyoutperformed by the other two design techniques.

Fine Errors

At high SNR, the performance of the ML estimator is well characterized bythe CRB. From (5.20) it is clear how the CRB performance depends on theazimuth. We define the quantity Qmin, related to the CRB at the azimuth ψexhibiting the worst performance as

Qmin = minψ

(

Qaa(ψ)−Q2ab(ψ)

Qbb(ψ)

)

. (5.31)

Note that Qmin is not explicitly taken into account in any of the strategiesconsidered for array design. However, the MIP and the UCA geometries areguaranteed to have Qab = 0 for any azimuth.

Fig. 5.6 depicts the value ofQmin for the same arrays considered in Fig. 5.5(b).As expected, the value of Qmin is generally an increasing function of the numberof sensors.

Array Geometry

Example array geometries are depicted in Fig. 5.7 and Fig. 5.8. The arraygeometries obtained for Ns = 14 and κmin = 1/4 with different techniques are

compared together with a representation of the corresponding |H(κ)|2.The array obtained using the proposed MIP method, in Fig. 5.7(a), exhibits

symmetry around the origin due to the constraint Qab = 0. The GA array ofFig. 5.7(b) exhibit a completely irregular pattern since the sensor positions areunconstrained. The UCA array is shown in Fig. 5.7(c). A spiral shaped array isshown in Fig. 5.8(a). From the large width of the central lobe, a large MSEE athigh SNR is to be expected for the spiral array. A cross shaped array is shownin Fig. 5.8(b). Due to the regular geometry, the cross array exhibits manymaxima having the same magnitude as the central lobe. In these two latterarrays, sensor spacings are chosen arbitrarily and the arrays are not designedto follow the proposed criteria.

115

4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Hm

ax

Number of sensors Ns

MIP

GA

UCA

(a) κmin = 1/2 and κmax = 1.

4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Hm

ax

Number of sensors Ns

MIP

GA

UCA

(b) κmin = 1/4 and κmax = 1.

4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Hm

ax

Number of sensors Ns

MIP

GA

UCA

(c) κmin = 1/6 and κmax = 1.

Figure 5.5: Values of Hmax achieved with different array design strategies fordifferent number of sensors. Different κmax/κmin are considered. A small Hmax

is desirable in order to reduce the occurrence of gross errors.

116

4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

Qm

in

Number of sensors Ns

MIP

GA

UCA

Figure 5.6: Values of Qmin obtained from different array design strategies fordifferent number of sensors and κmax/κmin = 4. A large Qmin is desirable inorder to reduce the MSEE at high SNR.

Values of Hmax and Qmin for the arrays shown in Fig. 5.7(a)-5.7(c) can befound in Fig. 5.5(b) and Fig. 5.6, respectively.

The symmetry of the MIP array and the irregular pattern of the GA arrayare also observed for other sensor numbers and values of κmax/κmin. Thisfact suggests that the actual deployment in the field of a MIP array may besignificantly easier that the deployment of a GA array. Additional array layoutsobtained with the MIP technique are shown in Appendix 5.C.

Estimator Performance

In order to quantify the impact of the different arrays geometry on the actualestimation problem we resort to the analysis of the MSEE using Monte Carlosimulations.

Arrays obtained with the sensor placement techniques considered in theprevious section are compared. Moreover, we compare with an array havingsensors arranged on a spiral and with another array with sensors arranged ona cross. These two latter arrays are designed without following the designcriterion proposed in this work.

The MSEE is computed as follows. A wavevector is drawn randomly fromthe uniform distribution having as support a disk with radius κmax. For eachconsidered array geometry and SNR, random noise is added to the wavefieldas in (5.5). The ML estimation method of Maranò et al. (2012) is used toestimate the parameters of the wave and the MSEE is obtained by repeatingthe procedure 3000 times. The MSEEs are also compared with the CRBs.

In Fig. 5.9, we compare the MSEE of the ML estimate of the wavenumberfor different array geometries. The MIP, GA, and UCA arrays are designed

117

−1.4 −0.7 0 0.7 1.4

−1.4

−0.7

0

0.7

1.4

x [m]

y[m

]

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

](a) MIP array.

−2.4 −1.6 −0.8 0 0.8 1.6 2.4

−0.8

0

0.8

1.6

2.4

3.2

x [m]

y[m

]

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) GA array.

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

x [m]

y[m

]

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(c) UCA array.

Figure 5.7: Array geometries h(x, y) and the corresponding |H(κx, κy)|2. Thearrays are obtained by choosing Ns = 14, κmin = 0.25, and κmax = 1. Themagenta circles delimit the annulus κmin < ‖κ‖2 < 2κmax.

118

−0.9 −0.6 −0.3 0 0.3 0.6 0.9−0.9

−0.6

−0.3

0

0.3

0.6

0.9

x [m]

y[m

]

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(a) Spiral array.

−1.8 −0.9 0 0.9 1.8

−1.8

−0.9

0

0.9

1.8

x [m]

y[m

]

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) Cross array.

Figure 5.8: Array geometries h(x, y) and the corresponding |H(κx, κy)|2. Themagenta circles delimit the annulus κmin < ‖κ‖2 < 2κmax.

with κmax/κmin = 4. In general, the MSEE decreases as the SNR increasesand it achieves the CRB for sufficiently large SNR.

Fig. 5.9(a) shows the MSEE for an array of Ns = 7 sensors. The MIP, theGA, and the UCA exhibit similar performances in the threshold region. At highSNR, in the asymptotic region, certain arrays perform better than others. Weobserve how the best performing arrays are those with larger Qmin, cf. Fig. 5.6.The spiral array shows a large MSEE over all the SNR range considered. Thecross array does not achieve its CRB because the local maxima of the LLFare indistinguishable from the right maximum. This ambiguity is due to theregular array geometry, cf. Fig. 5.8(b).

Fig. 5.9(b) shows the MSEE for an array of Ns = 14 sensors. The geometryof these arrays is shown in Fig. 5.7. The MIP, the GA, and the UCA arraysexhibit similar performance both at low and high SNR. The spiral array andthe cross array again exhibit the larger MSEE.

In Fig. 5.10 the MSEE of the ML estimate of the wavenumber for different

119

−35 −30 −25 −20 −15 −10 −5

10−4

10−3

10−2

10−1

MIP

GA

UCA

Spiral Array

Cross Array

MSE

E[m

−2]

SNR [dB]

(a) Ns = 7.

−35 −30 −25 −20 −15 −10 −5

10−4

10−3

10−2

10−1

MIP

GA

UCA

Spiral Array

Cross Array

MSE

E[m

−2]

SNR [dB]

(b) Ns = 14.

Figure 5.9: Comparison of the MSEE of the wavenumber obtained with dif-ferent array geometries. The CRBs are also depicted with black thin dottedlines.

array geometries and number of sensors is depicted. Only the MIP geome-tries, with κmax/κmin = 4, are considered for different values of the SNR. Asexpected, the MSEE decreases when the number of sensors is increased. ForSNR = −25 dB, the MSEE is constantly well above the CRB because the SNRis low and the estimator operates in the threshold region, thus gross errorsoccur. For SNR = −20 dB, the MSEE achieves the CRB for a sufficiently largenumber of sensors. At last, a SNR of −15 dB is sufficient for the estimator tooperate in the asymptotic region for any number of sensors.

In Fig. 5.10, it is also observed that the marginal benefit of an additionalsensor also depends on the SNR. When the estimator operates in the asymp-totic region (e.g. SNR = −15 dB), most of the MSEE reduction is achievedwith the first 8 sensors. However for SNR = −20 dB, the MSEE reductionis still significant until Ns = 11 because the threshold effect of the estimator.

120

4 6 8 10 12 14 16 18 20

10−4

10−3

10−2

10−1

SNR -25 [dB]

SNR -20 [dB]

SNR -15 [dB]M

SE

E[m

−2]

Number of sensors Ns

Figure 5.10: Comparison of the MSEE of the wavenumber obtained with dif-ferent array geometries having a different number of sensors. The CRBs arealso depicted with black thin dotted lines.

These considerations may also vary for different κmax/κmin as suggested byFig. 5.5.

5.7 Conclusions

In this paper, we described in detail the occurrence of gross errors and fineerrors in the estimation of parameters of Love waves and Rayleigh waves. Wederived a relationship between the Fourier transform of the sampling patternand the average likelihood function of the observations. Sensor placementcriteria for the reduction of gross errors and fine errors are then proposed.

An array design algorithm employing the proposed criterion with linear ob-jective function and linear constraints is formulated as a mixed integer program.The proposed algorithm is compared with the genetic algorithm technique andwith the uniform circular array technique using the same design criterion. Theproposed algorithm achieves superior design for most choices of the sensornumber and spatial frequency requirements. In addition, the proposed sensorplacement algorithm generates arrays composed of simple regular geometriesand are thus suitable for field deployment.

The mean-squared estimation errors of the maximum likelihood estimatorusing different array geometries are compared by means of Monte Carlo simu-lations. We find that the proposed design criterion is suitable for the optimisedplacement of seismic sensors for the analysis of Love waves and Rayleigh waves.We show that the sensor placement techniques considered can greatly reducethe mean-squared estimation error when compared with non optimised arrays.

We emphasize that prior knowledge of the spatial frequency support ofthe wavefield is useful to design an array able to achieve better performance.From our findings, we suggest that a minimum number of around 10 sensors isdesirable.

Future extensions of this work will address the presence of physical obstruc-tions limiting the available space to deploy sensors.

121

5.8 Acknowledgments

This work is supported in part by the Swiss Commission for Technology andInnovation under project 9260.1 PFIW-IW and with funds of the Swiss Seis-mological Service. The first author would like to thank Dr. Michel Baes andDr. Michael Bürgisser for the helpful discussions.

5.A Relationship Between Likelihood Function and Sam-pling Pattern

In this section, we show the relationship between the LF pY (y |θ) and theFourier transform of the sampling pattern H(κx, κy).

Scalar case

We consider the scalar wave equation of (5.3) and the measurement modelof (5.4). We parametrize the wave equation as a function of α1, α2 insteadof α0, ϕ0 using

α1 = α0 cosϕ0

α2 = −α0 sinϕ0 .

This parametrization is equivalent because the transformation between the twoparameters pairs is bijective.

The measurement Y(n)k , at the time instant tk and location pn is therefore

Y(n)k = u(pn, tk) + Z

(n)k

= α0 cos(ωt− κ · p+ ϕ0) + Z(n)k

= α1 cos(ωt− κ · p) + α2 sin(ωt− κ · p) + Z(n)k .

For a given κ, the ML estimates α1 and α2 can be found analytically andare

α1 =2

NsK

Ns∑

n=1

K∑

k=1

Y(n)k cos(ωtk − κ · pn)

α2 = − 2

NsK

Ns∑

n=1

K∑

k=1

Y(n)k sin(ωtk − κ · pn) ,

and their expected values are

Eα1 ≈2

NsK

Ns∑

n=1

K∑

k=1

(α1 cos(ωtk − κ · pn) cos(ωtk − κ · pn)

+α2 sin(ωtk − κ · pn) cos(ωtk − κ · pn))

Eα2 ≈ −2

NsK

Ns∑

n=1

K∑

k=1

(α1 cos(ωtk − κ · pn) sin(ωtk − κ · pn)

+α2 sin(ωtk − κ · pn) sin(ωtk − κ · pn)) ,

122

where the superscript˘denotes the true value of the parameter and the lack ofsuperscript denotes search parameters, e.g., the argument of the LF. When κ =κ, then Eα1 ≈ α1 and Eα2 ≈ α2. The symbol≈ denotes an approximation.We used the following trigonometric approximations

K∑

k=1

cos2(ωk + γ) ≈ K

2

K∑

k=1

sin2(ωk + γ) ≈ K

2

K∑

k=1

sin(ωk + γ) cos(ωk + γ) ≈ 0 ,

which are valid for ω being not near 0 or 1/2 and are exact when ω = 2πmK m ∈ Z

(see example 3.14 in Kay (1993) or Stoica et al. (1989)).The second moments of α1 and α2 are

Eα21 ≈ Eα12 +

2

NsKσ2

and

Eα22 ≈ Eα22 +

2

NsKσ2 ,

respectively.From the previous derivations, it is clear to see the dependence on κ of

α1(κ) and α2(κ). Therefore, given observations y and plugging α1 and α2

into the PDF (5.5), it is possible to rewrite the LF of the observation as a solefunction of the κ

pY (y |κ) = maxα0,ϕ0

pY (y |θ)

= pY (y |(α1(κ), α2(κ),κ)) .

In order to investigate the shape of the LF, we take the natural logarithmof the LF and drop all the additive and multiplicative constants not dependingon κ

ln(pY (y |κ)) ∝ −Ns∑

n=1

K∑

k=1

(Y(n)k − u(n)k (θ))2

= −Ns∑

n=1

K∑

k=1

(

u(n)k (θ) + Z

(n)k − u(n)k (θ)

)2

,

where again we distinguish between the true and unknown wavefield u(n)k =

u(n)k (θ) and the wavefield as function of the search parameters u

(n)k = u

(n)k (θ).

The symbol ∝ denotes equality up to an affine transform.In order to get insight about the average shape of the function, we take the

expectation of such quantity

123

E ln(pY (y |κ)) ∝ E

−Ns∑

n=1

K∑

k=1

(

u(n)k + Z

(n)k − u(n)k

)2

∝ Eα12 + Eα22

≈ |Eα1+ iEα2|2

=

∣

∣

∣

∣

∣

Ns∑

n=1

K∑

k=1

u(n)k e−i(ωtk−κ·pn)

∣

∣

∣

∣

∣

2

=∣

∣α0e−iϕ0G(ω − ω)H(κ− κ)

∣

∣

2.

Where the quantities E

∑Ns

n=1

∑Kk=1(u

(n)k )2

and E

∑Ns

n=1

∑Kk=1(Z

(n)k )2

were

dropped since they are constants with respect to κ. Moreover we used

Ns∑

n=1

K∑

k=1

u(n)k u

(n)k =

Ns∑

n=1

K∑

k=1

(α1 cos(ωtk − κ · pn) + α2 sin(ωtk − κ · pn))

· (α1 cos(ωtk − κ · pn) + α2 sin(ωtk − κ · pn))

≈ NsL

2(α1 E α1+ α2 E α2)

and

Ns∑

n=1

K∑

k=1

(u(n)k )2 =

Ns∑

n=1

K∑

k=1

(α1 cos(ωtk − κ · pn) + α2 sin(ωtk − κ · pn))2

≈ NsL

2

(

α21 + α2

2

)

.

Vector case

In the vector case, each sensor component may experience a different ampli-tude scalings or phase delay on each component, as shown in (5.1) and (5.2).The amplitude scalings β(ℓ)ℓ=1,...,L and the phase delays γ(ℓ)ℓ=1,...,L are,in general, functions of the wavefield parameters θ except for α and ϕ. The

measurement Y(ℓ)k , from the ℓ-th channel, at the time instant tk and at position

pℓ is

Y(ℓ)k = u(pℓ, tk) + Z

(ℓ)k

= α0β(ℓ) cos(ωtk − κ · pℓ + ϕ0 + γ(ℓ)) + Z

(ℓ)k

= α1β(ℓ) cos(ωtk − κ · pℓ + γ(ℓ)) + α2β

(ℓ) sin(ωtk − κ · pℓ + γ(ℓ)) + Z(ℓ)k .

With a suitable parametrization of β(ℓ), γ(ℓ)ℓ=1,...,L, this model is able to cap-ture all the amplitude scaling and phase delays as in the seismic wave equationsof (5.1) and (5.2).

Similar to the scalar case, it is possible to obtain analytic expressions ofboth ML estimates α1 and α2 as

124

α1 =2

K

∑Lℓ=1

∑Kk=1 Y

(ℓ)k β(ℓ) cos(ωtk − κ · pℓ + γ(ℓ))∑L

ℓ=1(β(ℓ))2

.

Here and in what follows, we omit the formulas concerning α2 because of theirsimilarity with the derivations of α1. Details on the estimation of α1 and α2

in this setting can be found in Reller et al. (2011).The first and the second moments of α1 are

Eα1 ≈2

K

1∑Lℓ=1(β

(ℓ))2

L∑

ℓ=1

K∑

k=1

β(ℓ)(

α1β(ℓ) cos(ωtk − κ · pℓ + γ(ℓ))

+α2β(ℓ) sin(ωtk − κ · pℓ + γ(ℓ))

)

cos(ωtk − κ · pℓ + γ(ℓ))

Eα21 ≈ Eα12 +

2

NsKσ2 .

As for the scalar case, we are interested in the average shape. Thereforewe take the logarithm of the LF, take the expectation, and drop multiplicativeand additive constants to get

E ln(pY (y |θ)) ∝ Eα12 + Eα22

≈∣

∣

∣

∣

∣

∑Lℓ=1

∑Kk=1 β

(ℓ)u(n)k e−i(ωtk−κ·pℓ+γ

(ℓ))

∑Lℓ=1(β

(ℓ))2

∣

∣

∣

∣

∣

2

.

We observe that, in the wave models of (5.1) and (5.2) the amplitude scal-ings and the phase delays are identical on the same component of differentsensors. We denote with βx, γx, βy, γy, and βz, γz the scalings and delays on

the x,y, and z components, respectively. Moreover the quantity∑L

ℓ=1(β(ℓ))2

is constant with respect to κ.The expression can be further simplified by grouping identical scalings

E ln(pY (y |θ)) ∝∣

∣

∣(βxβxe

i(γx−γx) + βyβyei(γy−γy) + βzβye

i(γz−γz))

·αe−iϕG(ω − ω)H(κ− κ)∣

∣

2.

5.B Some Remarks on the MOIs of an Array

In this section, we show that any array with Qab = 0 has uniform properties, interm of fine errors, at any azimuth. In addition, we also show that all uniformcircular array (UCA) belong to this class of arrays.

Without loss of generality we consider arrays centred in the origin, i.e.,∑Ns

n=1 pn = (0, 0).

For any array, if Qab = 0 then Qaa and Qbb are constant. Consider the arraywith sensor positions

pn = (xn, yn) = rn (cosψn, sinψn) .

125

Using (5.22), (5.23), (5.24), and (5.21) the MOIs are found to be

Qaa(ψ) =

Ns∑

n=1

a2n =

Ns∑

n=1

r2n cos2(ψn − ψ)

Qbb(ψ) =

Ns∑

n=1

b2n =

Ns∑

n=1

r2n sin2(ψn − ψ)

Qab(ψ) =

Ns∑

n=1

anbn =

Ns∑

n=1

r2n cos(ψn − ψ) sin(ψn − ψ) .

The derivatives of Qaa and Qbb with respect to ψ are

∂Qaa(ψ)

∂ψ=

Ns∑

n=1

2r2n cos(ψn − ψ) sin(ψn − ψ)

∂Qbb(ψ)

∂ψ= −

Ns∑

n=1

2r2n sin(ψn − ψ) cos(ψn − ψ) .

From the previous expressions we conclude that if Qab(ψ) = 0 for every ψ, thenQaa and Qbb are constant for every ψ.

All UCAs have Qab = 0. A UCA is composed of Ns sensors equally spacedon a circle of radius r. Sensor positions are given by

pn = (xn, yn) = r

(

cos2πn

Ns, sin

2πn

Ns

)

.

Using (5.24) and (5.21), Qab for an UCA can be computed as

Qab(ψ) =

Ns∑

n=1

anbn

= r2Ns∑

n=1

cos

(

2πn

Ns− ψ

)

sin

(

2πn

Ns− ψ

)

=r2

2

Ns∑

n=1

sin

(

2

(

2πn

Ns− ψ

))

= 0 ,

for any ψ and Ns ≥ 3. The last equality can be proved using de Moivre’sformula (Abramowitz & Stegun, 1964).

All UCAs have Qaa and Qbb constants at every azimuth. This fact followsdirectly from the two previous considerations.

5.C MIP Array Layouts

In this section, we provide the layouts found using the MIP method for dif-ferent number of sensors. We explain how it is possible to use the presented

126

geometries by considering the largest wavenumber in the wavefield in practicalapplications.

Using the scaling property of the Fourier transform it is possible to stretch orcompress the array layout according to the largest wavenumber in the wavefieldand therefore adapt the Fourier transform of the sampling pattern to the actualfrequency content of the wavefield. The array layouts presented in this sectionare obtained using κmax = 1 and different values of κmin. Let κ⋆max be thelargest wavenumber in the wavefield. Let h⋆(x, y) and H⋆(κx, κy) denote thedesired sampling pattern and its Fourier Transform, respectively. They arerelated to the h(x, y) and H(κx, κy) provided in Fig. 5.11 and Fig. 5.12 as

h⋆(x, y) = h (xκ⋆max, yκ⋆max)

H⋆(κx, κy) =1

κ⋆max

H⋆

(

κxκ⋆max

,κyκ⋆max

)

.

The effective κ⋆min is also changed as κ⋆min = κminκ⋆max. Choice of which

κmin to choose should be related in particular to the smallest wavenumber ofinterest in the analysis. In fact, we observe how a small κmin leads to arrayswith larger aperture.

In practice, stretching the spatial sampling pattern by 1/κ⋆max allows us toobtain a H⋆(κx, κy) with local maxima minimized in the annulus κminκ

⋆max ≤

‖κ‖2 ≤ 2κ⋆max.The electronic supplement to this article provides tables with coordinates

of each array. The supplement is found in Appendix D.

127

−0.4 0 0.4

−0.4

0

0.4

−2 0 2

−2

0

2

(a) Ns = 7, κmax/κmin = 2.

−1.2 0 1.2

0

1.2

−2 0 2

−2

0

2

(b) Ns = 7, κmax/κmin = 4.

−1.2 0 1.2

0

1.2

−2 0 2

−2

0

2

(c) Ns = 7, κmax/κmin = 6.

−0.5 0 0.5

−0.5

0

0.5

−2 0 2

−2

0

2

(d) Ns = 8, κmax/κmin = 2.

−1.1 0 1.1

−1.1

0

1.1

−2 0 2

−2

0

2

(e) Ns = 8, κmax/κmin = 4.

−1.4 0 1.4

−1.4

0

1.4

−2 0 2

−2

0

2

(f) Ns = 8, κmax/κmin = 6.

−0.6 0 0.6

−0.6

0

0.6

−2 0 2

−2

0

2

(g) Ns = 9, κmax/κmin = 2.

−1.1 0 1.1

−1.1

0

1.1

−2 0 2

−2

0

2

(h) Ns = 9, κmax/κmin = 4.

−1.5 0 1.5

−1.5

0

−2 0 2

−2

0

2

(i) Ns = 9, κmax/κmin = 6.

−0.6 0 0.6

−0.6

0

0.6

−2 0 2

−2

0

2

(j) Ns = 10, κmax/κmin = 2.

0 1.3

−1.3

0

1.3

−2 0 2

−2

0

2

(k) Ns = 10, κmax/κmin = 4.

−1.9 0 1.9

0

1.9

−2 0 2

−2

0

2

(l) Ns = 10, κmax/κmin = 6.

−0.6 0 0.6

−0.6

0

0.6

−2 0 2

−2

0

2

(m) Ns = 11, κmax/κmin = 2.

−1.1 0 1.1

−1.1

0

1.1

−2 0 2

−2

0

2

(n) Ns = 11, κmax/κmin = 4.

−1.8 0

−1.8

0

1.8

−2 0 2

−2

0

2

(o) Ns = 11, κmax/κmin = 6.

−0.8 0 0.8

−0.8

0

−2 0 2

−2

0

2

(p) Ns = 12, κmax/κmin = 2.

−1.3 0

−1.3

0

1.3

−2 0 2

−2

0

2

(q) Ns = 12, κmax/κmin = 4.

−2.2 0

−2.2

0

2.2

−2 0 2

−2

0

2

(r) Ns = 12, κmax/κmin = 6.

Figure 5.11: Array layouts found with the MIP method for Ns = 7, . . . , 12.

128

0 0.7

−0.7

0

0.7

−2 0 2

−2

0

2

(a) Ns = 13, κmax/κmin = 2.

−1.3 0 1.3

−1.3

0

1.3

−2 0 2

−2

0

2

(b) Ns = 13, κmax/κmin = 4.

−1.9 0 1.9

−1.9

0

−2 0 2

−2

0

2

(c) Ns = 13, κmax/κmin = 6.

−0.8 0 0.8

−0.8

0

0.8

−2 0 2

−2

0

2

(d) Ns = 14, κmax/κmin = 2.

−1.5 0 1.5

−1.5

0

1.5

−2 0 2

−2

0

2

(e) Ns = 14, κmax/κmin = 4.

−1.9 0 1.9

−1.9

0

1.9

−2 0 2

−2

0

2

(f) Ns = 14, κmax/κmin = 6.

−0.7 0 0.7

−0.7

0

0.7

−2 0 2

−2

0

2

(g) Ns = 15, κmax/κmin = 2.

−1.5 0 1.5

−1.5

0

−2 0 2

−2

0

2

(h) Ns = 15, κmax/κmin = 4.

−2.1 0 2.1

−2.1

0

2.1

−2 0 2

−2

0

2

(i) Ns = 15, κmax/κmin = 6.

−0.8 0 0.8

−0.8

0

0.8

−2 0 2

−2

0

2

(j) Ns = 16, κmax/κmin = 2.

−1.6 0 1.6

−1.6

0

1.6

−2 0 2

−2

0

2

(k) Ns = 16, κmax/κmin = 4.

−1.8 0 1.8

−1.8

0

1.8

−2 0 2

−2

0

2

(l) Ns = 16, κmax/κmin = 6.

−0.8 0 0.8

−0.8

0

0.8

−2 0 2

−2

0

2

(m) Ns = 17, κmax/κmin = 2.

−1.4 0 1.4

−1.4

0

1.4

−2 0 2

−2

0

2

(n) Ns = 17, κmax/κmin = 4.

−2.4 0 2.4

−2.4

0

2.4

−2 0 2

−2

0

2

(o) Ns = 17, κmax/κmin = 6.

−0.8 0 0.8

−0.8

0

0.8

−2 0 2

−2

0

2

(p) Ns = 18, κmax/κmin = 2.

−1.5 0 1.5

−1.5

0

1.5

−2 0 2

−2

0

2

(q) Ns = 18, κmax/κmin = 4.

0 2.1

−2.1

0

2.1

−2 0 2

−2

0

2

(r) Ns = 18, κmax/κmin = 6.

Figure 5.12: Array layouts found with the MIP method for Ns = 13, . . . , 18.

Chapter 6

Conclusions

In this section we summarize the main contributions of this thesis.

6.1 A Method for the Analysis of Surface Waves

We developed a method to perform maximum likelihood (ML) estimation ofthe parameters of a monochromatic wavefield from measurements corrupted byadditive white Gaussian noise. The approach accounts for all the measurementsand all the wavefield parameters jointly. It also allows to model different typesof waves and to combine measurements from different types of sensors. Thetechnique can cope with arbitrary sensor positions and different sampling ratesin each sensor. This method was presented in Chapter 2 and Chapter 3.

The method makes an optimal use of the available measurements accordingto the ML criterion. All the wavefield parameters and all the measurementare modelled jointly within this estimation framework. We showed that ourimplementation is asymptotically efficient, i.e., for sufficiently large signal-to-noise ratio (SNR) or sample size the estimates are unbiased and the mean-squared estimation error (MSEE) achieves the theoretical limit on estimationaccuracy given by the Cramér-Rao bound (CRB).

The simultaneous presence of multiple waves, of possibly different type, isalso addressed. The wave type and number are chosen by the algorithm accord-ing to the Bayesian information criterion (BIC). By accounting for multiplewaves, the estimates of the parameters of each wave are refined as parame-ters are iteratively re-estimated. This allows to identify waves with smalleramplitude that would not be visible by modelling a single wave.

We showed how to account for different noise variance on each channel, byproperly merging the information from sensors with different noise level. Theproposed technique estimates the noise variance in each channel. This bringsabout various advantages. It enables us to use sensors of different type withdifferent noise levels. A misplaced or badly working sensor, may exhibit ahigher noise level and will be automatically given less weight in the estimationprocess.

The factor graph approach has been used to define the statistical modeland to design an algorithm for the computation of the likelihood of the ob-servations. Thanks to this framework it has been possible to derive sufficient

129

130

statistics enabling us to reduce the computational complexity of the estimationalgorithm.

We showed how it is possible to extend the method to different wave typesand measurement types. We showed applications to the analysis of Love wavesand Rayleigh waves from recordings of either translational or rotational mo-tions.

Application to the Analysis of Ambient Vibrations

In Chapter 3, we evaluated the performance of the proposed method on high-fidelity synthetic dataset from the SESAME project (Bard, P.-Y., 2008) andon real data from seismic surveys in Switzerland. We showed how this methodimproves the identification of Rayleigh wave and Love wave dispersion curveswhen compared with existing methods. We have also shown that modellingmultiple waves enables us to detect weaker waves that are not visible withtraditional methods.

Rayleigh wave ellipticity curve is also retrieved including information aboutthe prograde or retrograde particle motion. This is useful in mode separationand in the identification of singularities of the ellipticity (i.e., peaks and minimaof the H/V representation of the ellipticity).

We present another example to demonstrate the capabilities of the pro-posed method by showing the unpublished results from the Yverdon survey.The survey is performed within the framework of the Swiss Strong Motion Net-work (SSMNet) to characterize the sediments under the Yverdon-Philosophes(SYVP) station (Michel et al., 2012).

An array of twelve stations was deployed around the SYVP station. Thearray layout is depicted in Fig. 6.1. The array analysed has an aperture of240 m and records for 2.5 hours. The field measurement are processed withthe ML method presented in this thesis. The simultaneous presence of up tothree Love or Rayleigh waves is modelled.

In Fig. 6.2, the estimated wavefield parameters of Rayleigh waves are pre-sented. The dispersion curve of the fundamental mode and the first highermode of Rayleigh wave are depicted in Fig. 6.2(a) and in Fig. 6.2(b), respec-tively. The black dashed curves depict a manual selection of the dispersioncurve. The solid red curves are found automatically as the median value of theestimates in the selection.

The Rayleigh ellipticity angle for both modes is retrieved over a broad rangeof frequencies. The ellipticity of the fundamental mode is shown in Fig. 6.2(c).A singularity is found just above 1 Hz, where the Rayleigh particle motion ishorizontally polarized (V=0). Just above 2 Hz another singularity is foundwhere the Rayleigh wave particle motion changes from prograde to retrogradeand the motion is vertically polarized (H=0).

The ellipticity of the first higher mode is shown in Fig. 6.2(d). A singularityis found just below 3 Hz, where the Rayleigh particle motion is horizontallypolarized (V=0) and the particle motion changes from retrograde to prograde.

In Fig. 6.3, the estimated wavenumbers of Love waves are presented. Thefundamental mode and the first higher mode of Love wave are depicted inFig. 6.3(a) and in Fig. 6.3(b), respectively.

131

Figure 6.1: Geometry of the sensor array used in the Yverdon survey. Theinlet pinpoints the location of the array within Switzerland. The geographiccoordinates are Swiss coordinates (CH1903).

1 2 3 4 5 60

0.005

0.01

0.015

0.02

0.025

0.03

Wav

enum

ber

[1/m

]

Frequency [Hz]

(a) Rayleigh wavenumber, fundamental mode.

1 2 3 4 5 60

0.005

0.01

0.015

0.02

0.025

0.03

Wav

enum

ber

[1/m

]

Frequency [Hz]

(b) Rayleigh wavenumber, first higher mode.

1 2 3 4 5 6

Ellip

tici

tyangleξ

[rad]

π2

π4

0

-π2

-π4

Frequency [Hz]

(c) Rayleigh ellipticity angle, fundamentalmode.

1 2 3 4 5 6

Ellip

tici

tyangleξ

[rad]

π2

π4

0

-π2

-π4

Frequency [Hz]

(d) Rayleigh ellipticity angle, first highermode.

Figure 6.2: Estimated Rayleigh wavenumber and ellipticity angle from theYverdon survey.

132

1 2 3 4 5 60

0.005

0.01

0.015

0.02

0.025

0.03W

aven

um

ber

[1/m

]

Frequency [Hz]

(a) Love wave fundamental mode.

1 2 3 4 5 60

0.005

0.01

0.015

0.02

0.025

0.03

Wav

enum

ber

[1/m

]

Frequency [Hz]

(b) Love wave first higher mode.

Figure 6.3: Estimated Love wavenumber from the Yverdon survey.

Application to the Analysis of Translational and Rotational

Motions

In Chapter 4, we showed how the proposed ML method deals with the jointprocessing translational motions and rotational motions measurements. In thisapplication, sensors of different type are used together and is therefore neces-sary to consider the different noise level.

For the single sensor setting, we showed an example of the retrieval of bothLove wave and Rayleigh wave dispersion curves. Rayleigh wave ellipticity curveis also retrieved. These quantities were compared with the same quantitiesobtained from an array of translational sensors.

For the array of sensors setting, we showed an application to the process-ing of recordings from an array of mixed three- and six- components sensors.We found that the joint analysis of translational and rotational measurementsgreatly improves the retrieved dispersion curve.

Analysis of Fisher information (FI) enables us to understand and quantifythe sources of uncertainty affecting the accuracy of wavefield parameter estima-tion. In particular, we derive analytical expressions of CRB for the parametersof interest.

A Monte Carlo simulation suggest that, in the considered setup, the MSEEobtained using a single six-components sensor is significantly larger that theMSEE obtained using an array of five sensors.

6.2 Sensor Placement for the Analysis of Seismic SurfaceWaves

Chapter 5 dealt with the design of array geometries for the analysis of Lovewaves and Rayleigh waves.

We explained in detail how the array geometry affects the MSEE of parame-ters of Love waves and Rayleigh waves. We distinguished between gross errors,occurring at low SNR and fine errors occurring at high SNR. We showed howthe Fourier transform of the sensor positions is related with the average shape

133

of the log-likelihood function (LLF) of the observations of a Love wave or of aRayleigh wave.

Relying on the considerations concerning the sources of error, we proposea cost function suitable for the design of the array geometry with particularfocus on the estimation of the wavenumber of both Love and Rayleigh waves.

The proposed cost function is however difficult to minimize. To circumventthis, we consider a relaxation of the original optimization problem and wepropose an algorithm to minimize the relaxed problem.

Numerical experiments verify the effectiveness of our cost function and re-sulting array geometry designs, leading to greatly improved estimation per-formance in comparison to arbitrary array geometries, both at low and highSNR levels. Moreover, the optimized array geometries obtained with the pro-posed sensor placement algorithm exhibit a symmetry suitable for the fielddeployment.

6.3 Outlook

Further developments and extensions of the proposed methods are envisioned.The proposed method can be extended to the analysis of standing waves

occurring in resonances. An application of particular interest is, for example,the analysis of two-dimensional resonances in Alpine valleys filled with alluvialsediments, see e.g., Roten et al. (2006).

At the present state, a long recording is split in intervals of determinedduration and each interval is processed separately. This approach does notconsider that certain waves may have longer duration than other, possiblydepending on the nature of their source. We argue that a strategy able to findadaptively the duration of a wave may be both interesting from a scientificpoint of view and improve the estimation results.

Moreover, wavefield parameter from each interval of a long recording areestimated independently. Wavefield parameters from each frequency analysedwithin the same interval are also estimated independently. This approach ne-glects two facts. First, at a certain frequency the wavenumber of a given wavetype and propagation mode is the same during the whole recording. Second, fora given time interval the wavenumber of a certain wave type and propagationmode are strongly correlated at neighbouring frequencies, i.e., the dispersioncurves are continuous and typically do not exhibit sudden variations. We be-lieve that these two facts may be exploited to significantly improve the accuracyof the estimated parameters.

Concerning the proposed algorithm for sensor placement, we envision twodifferent extensions. First, the presence of physical obstructions in the fieldshould be considered. In fact, in practical applications there are often limita-tions to where it is possible to position the seismic sensors. Second, possibilityof sensor failures should be considered. The robustness of the proposed sensorplacement algorithm should be evaluate and, if necessary, a suitable strategyto address this problem should be developed.

The algorithms developed in this thesis also need to be implemented asstandalone software packages and distributed to interested parties within theSwiss Seismological Service. In this way, the methods proposed in this the-

134

sis will contribute to the site-characterization tasks performed at the SwissSeismological Service.

Appendix A

Seismic Waves Estimation and Wavefield

Decomposition with Factor Graphs

Stefano Maranò1, Christoph Reller2, Donat Fäh1, and Hans-AndreaLoeliger2

1 ETH Zurich, Swiss Seismological Service, 8092 Zürich, Switzerland.2 ETH Zurich, Dept. of Information Technology & Electrical Engineering, 8092

Zürich, Switzerland.

Published in Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing,Prague, Czech Republic, May 2011, pp. 2748–2751.

Abstract

Physical wavefields are often described by means of a vector field.Advances in sensor technology enable us to collect an increasing numberof measurement at the same location (e.g. direction, polarization, trans-lation, and rotation). One question arising naturally is how to properlyprocess such large and diverse information, possibly from sensors of dif-ferent kinds. In this paper we propose a technique for the analysis ofvector wavefields and show an application to the seismic wavefield. Thecontributions of this paper are the following: i) We provide a frameworkto perform maximum likelihood parameter estimation of any wave type,modeling jointly all the measurements and parameters; ii) In the sameframework, we address wave superposition by gradually decomposing thewavefield; iii) We also propose an iterative algorithm for noise varianceestimation.

A.1 Introduction and System Model

In this paper, we propose a technique for the analysis of vector wave fields.Our primary goal is the estimation of wavefield parameters based on discrete-time observations from a sensor array. In particular, we are concerned withsensors measuring vector quantities such as direction, polarization, translation,and rotation. In practical applications several waves may be simultaneously

135

136

present. Our further goal is to decompose the wavefield by separating thecontributions of different waves.

In the analysis of physical wavefields we are typically interested in studyingvector fields of the form u(p, t) : R4 → RC . The quantity u ∈ RC dependson the position p ∈ R3, time t, and wavefield parameters θ. The value ofC depends on the sensor we use to measure the wavefield. The wavefield ismeasured by means of an array of N sensors. We call L = NC the overallnumber of channels. In presence of multiple sources, M waves coexist at thesame time in the wavefield. Assuming a linear medium, the superposition ofthese waves is measured in each channel. We now restrict our analysis to amonochromatic wavefield with known angular frequency ω. In this setting, wemodel the signal on each channel as the sum of scaled and delayed copies ofunderlying reference sinusoids

u(ℓ)(p, t) =M∑

m=1

ρ(m)ℓ ρ

(m)0 cos

(

ωt+ φ(m)ℓ + φ

(m)0

)

. (A.1)

With this signal model, the m-th wave is parametrized by the amplitudes and

phases ηm , ρ(m)ℓ , φ

(m)ℓ ℓ=0,...,L. The m-th reference sinusoid ρ

(m)0 cos(ωt +

φ(m)0 ) is defined by ρ

(m)0 and φ

(m)0 . However, our interest lies in the estimation

of wavefield parameters θm governing the amplitudes and the phases measuredat the different sensors. The parameters η and θ are related by means of amapping Γ : θ → η. In general, ρℓ, φℓℓ=1,...,L can be influenced by differentquantities such as velocity and direction of propagation (i.e., by the wave vec-tor), wave polarization, wave attenuation, sensor directional gain, instrumentresponse, and others.

Each signal is measured as

Y(ℓ)k = u(ℓ)(p, tk) + Z

(ℓ)k (A.2)

at times tk and is corrupted by Gaussian noise Z(ℓ)k

iid∼ N(

0, σ2ℓ

)

.

A.2 Seismic Wavefield

The seismic wavefield (i.e., elastic waves propagating through the earth) offersan interesting example since it presents the simultaneous presence of function-ally different and completely decoupled wave types Aki & Richards (1980).

To measure seismic waves, we deploy an array of triaxial (C = 3) seismome-ters on the surface of the earth. Each sensor measures the ground velocity alongthe direction of the axes of the coordinate system x, y, and z. For the sake ofsimplicity, we provide wave equations of the displacement field u, despite theactual measurement is the velocity field ∂u

∂t . The displacement can be describedby the vector field

u(p, t) , (ux(p, t), uy(p, t), uz(p, t)) : R4 → R

3 .

We restrict our interest to small aperture arrays and work with a flat earth model. Weuse a three-dimensional, right-handed Cartesian coordinate system with the z axis pointingupward. The azimuth ψ is measured counterclockwise from the x axis.

137

In this paper, we study waves propagating near the surface of the earthand having a direction of propagation lying on the horizontal plane z = 0.The wavefield is composed of the superposition of several Rayleigh and Lovewaves. The wave equations we describe hereafter are valid for z = 0 and forplane wave fronts. Rayleigh waves exhibit an elliptical particle motion confinedin the vertical plane perpendicular to the earth’s surface and defined by thedirection of propagation of the wave. The particle displacement generated bya single Rayleigh wave at position and time (p, t) is

ux(p, t) = α sin ξ cosψ cos(ωt− κTp+ ϕ)

uy(p, t) = α sin ξ sinψ cos(ωt− κTp+ ϕ) (A.3)

uz(p, t) = α cos ξ cos(ωt− κTp+π

2+ ϕ) .

The direction of propagation of a wave is given by the wave vector κ ,

κ (cosψ, sinψ, 0)T, whose magnitude κ is the wavenumber. The quantity tan ξis called ellipticity of the Rayleigh wave and determines the eccentricity andthe sense of rotation of the particle motion.

Love waves exhibit a particle motion confined on the horizontal plane, theparticle oscillates transversely with respect to the direction of propagation.The particle displacement generated by a single Love wave is

ux(p, t) = −α sinψ cos(ωt− κTp+ ϕ)

uy(p, t) = α cosψ cos(ωt− κTp+ ϕ) (A.4)

uz(p, t) = 0 .

With these wave equations in mind, we can now give an explicit expressionfor the mapping. Let pn be the known position of the n-th sensor. The mapping

Γ(R) : θ(R) → η, with θ(R)

, (α, ϕ, κ, ψ, ξ), specialized to the Rayleigh wave,is

(ρ0, φ0) = (α, ϕ)(

ρ(n,1), φ(n,1))

= (sin ξ cosψ,−κTpn)(

ρ(n,2), φ(n,2))

= (sin ξ sinψ,−κTpn)(

ρ(n,3), φ(n,3))

=(

cos ξ,−κTpn +π

2

)

for n = 1, . . . , N . Analogously, for a Love wave we define the mapping Γ(L) :θ(L) → η, with θ(L)

, (α, ϕ, κ, ψ) as

(ρ0, φ0) = (α, ϕ)(

ρ(n,1), φ(n,1))

= (− sinψ,−κTpn)(

ρ(n,2), φ(n,2))

= (cosψ,−κTpn)(

ρ(n,3), φ(n,3))

= (0, 0) .

We use (n, c) to refer to the c-th component of the n-th sensor instead of (ℓ) for the ℓ-thchannel. The mapping between (n, c) and (ℓ) is bijective.

138

A.3 Proposed Technique

Factor Graph

The probability density function of the observations y is

p(y |η) =L∏

ℓ=1

K∏

k=1

1√

2πσ2ℓ

e−(

y(ℓ)k

−u(ℓ)k

)2/2σ2

ℓ , (A.5)

where we rely on K discrete-time observations for each channel and define

y , y(ℓ)k k=1,...,Kℓ=1,...L and u

(ℓ)k , u(ℓ)(pℓ, tk).

Instead of computing (A.5), we model it by means of a factor graph Loeliger

et al. (2007). For every signal Y(ℓ)k , we consider a second-order state space

model with state X(ℓ)k ∈ R

2

X(ℓ)k−1 = AkX

(ℓ)k

Y(ℓ)k = CX

(ℓ)k + Z

(ℓ)k ,

where Ak , rotm(ω(tk−1 − tk)) is a clockwise rotation matrix rotm(β) ,(

cos β − sin βsin β cos β

)

, and the measurement matrix C , (0,−ω) accounts for the

derivative ∂u∂t . The corresponding factor graph is depicted in Fig. A.1(a).

Using a glue factor, we constrain the final states of every channel with thefollowing L equations

X(ℓ)K =

M∑

m=1

H(m)ℓ um =

M∑

m=1

S(m)ℓ , (A.6)

where H(m)ℓ , ρ

(m)ℓ rotm(φ

(m)ℓ ) are the constraint matrices, S

(m)ℓ , H

(m)ℓ um is

the contribution on the ℓ-th channel of the m-th wave, and the state vector of

the m-th reference sinusoid is um , ρ(m)0 (cos(ωtK + φ

(m)0 ), sin(ωtK + φ

(m)0 ))T .

The corresponding factor graph is shown in Fig. A.1(b), where S6mℓ represents

the contribution on the ℓ-th channel of all but the m-th wave. The overallgraph is shown in Fig. A.1(c).

Using the sum-product algorithm on the factor graph we can compute thelikelihood function p(y |η). We use Gaussian messages parametrized by meanvector, covariance matrix, and scale factor. We provide a detailed descriptionin Reller et al. (2011).

Parameter Estimation

We now focus on the estimation of wavefield parameters θ, in the case of asingle wave (M = 1) and known noise variance.

We introduce the set G of all the parameter mappings of interest. Themaximum likelihood (ML) estimate of wavefield parameter is given by

(Γ, θ) ∈ argmaxΓ∈G,θ∈domΓ

p(

y∣

∣Γ(θ))

. (A.7)

This maximization allows us to find the most likely wave type with the mostlikely parameters.

139

= Ak

X(ℓ)k−1X

(ℓ)k

C

+ NZ

(ℓ)k

Y(ℓ)k = y

(ℓ)k

(a) State space model.

= H(m)1

+S

(m)1

S( 6m)1

X(1)K

um

H(m)L

+S

(m)L

S( 6m)L

X(L)K

(b) Glue factor g.

g

X(1)K X

(1)k

Y(1)k = y

(1)k

X(1)k−1 X

(1)1

Y(1)1 = y

(1)1

X(L)K X

(L)k

Y(L)k = y

(L)k

X(L)k−1 X

(L)1

Y(L)1 = y

(L)1

(c) Overall factor graph.

Figure A.1: Building blocks and overall view of the factor graph of (A.5).

Wavefield Decomposition

When M > 1, Eq. A.6 captures the presence of multiple waves. However,because of the larger number wavefield parameters, a joint maximization ofthe likelihood function might be impractical.

We propose to gradually increase the number of waves modeled by the graphand perform smaller maximizations on the wavefield parameters of each wave

separately. In practice, once we insert an estimate s6mℓ of S

6m

ℓ , we perform themaximization over θm as in (A.7). Each maximization increases the likelihoodand convergence is guaranteed.

Noise Variance Estimation

The ML noise variance estimate is given by

σ2ℓ =

1

K

K∑

k=1

(

y(ℓ)k −C rotm (ω(tk − tK))−→m

X(ℓ)K

)2

,

with −→mX

(ℓ)K

=∑M

m=1 H(m)ℓ−→mUm

. Since the messages −→mUmdepend on σ2

ℓ , this

leads to an iterative algorithm where noise variance and messages are estimatediteratively.

140

2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

α(R)1

α(R)2

α(L)3

α(L)4

Am

plitu

de

[m]

Iteration

Figure A.2: Estimated amplitude at different iterations. The graph accountsfor an additional wave at iteration 1, 6, 10, and 15.

A.4 Numerical Examples

In the first example, we generate a synthetic wavefield composed of two Rayleighand two Love waves. The number of waves M = 4 is known. Waves aremonochromatic at known frequency of 1 Hz. We use an array of 14 triaxial sen-sors, 500 samples, and 5 seconds of observation. Measurements are corruptedby additive white Gaussian noise, with different variance in each channel. Welook for both Rayleigh and Love waves, i.e., G , Γ(R),Γ(L). True wave-

field parameters are θ(R)1 = (0.9, 0, 0.03, π4 ,

π4 )T, θ

(R)2 = (0.8, π4 , 0.03,

π2 ,

π4 )T,

θ(L)3 = (0.7, π3 , 0.04,−π4 )T, and θ

(L)4 = (0.2, π, 0.04, π)T. Noise variance and

wavefield parameters are unknown to the algorithm.Fig. A.2 shows how the estimates of the amplitudes α converge toward their

true values (dotted lines) after a sufficient number of iterations. The factorgraph is enlarged to account for additional waves at iterations 6, 10, and 15as the likelihood (not shown) converges to a stable value. Similarly, Fig. A.3shows the the estimates of noise variance σ2

ℓ . Sudden decrease in estimatedvariance in the graph correspond to the inclusion of an additional wave in thegraph.

Fig. A.4 depicts log p(

y∣

∣Γ(L)(θ(L)))

, as a function of wavenumber κ andazimuth ψ in polar coordinates (κ, ψ). In Fig. A.4(a) it is possible to see at

ψ3 = −π4 one stronger peak associated with the wave parametrized by θ(L)3

and no other strong peaks are visible. At iteration 14, only one Love wave

remains in the wavefield (the wave parametrized by θ(L)4 ) and the associated

peak, located at ψ4 = π, is now clearly visible, as shown in Fig. A.4(b). At thelast iteration, no more waves remain in the the residual wavefield, Fig. A.4(c).

We now use a more sophisticated synthetic wavefield developed in theSESAME project Bard, P.-Y. (2008); Bonnefoy-Claudet et al. (2006a). Thissynthetic dataset captures the complexity of the seismic wavefield, accountingfor the simultaneous presence of several seismic sources, emitting both shortburst of energy and longer harmonic excitations. It is a wavefield of ambientvibrations, where the wavefield is dominated by surface waves (i.e., Rayleigh

141

2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ21

σ22

σ23

σ24

σ25

σ26

Nois

eVariance

[m2]

Iteration

Figure A.3: Estimated noise variance at different iterations. Only six channelsare shown.

−0.06 0 0.06 −0.06

0

0.06

κ cosψ [m−1]

κsinψ

[m−1]

(a) Iteration 1

−0.06 0 0.06 −0.06

0

0.06

κ cosψ [m−1]

κsinψ

[m−1]

(b) Iteration 14

−0.06 0 0.06 −0.06

0

0.06

κ cosψ [m−1]

κsinψ

[m−1]

(c) Iteration 18

Figure A.4: Log-likelihood function of a Love wave.

and Love waves) but also other waves are present (e.g., body waves and stand-ing waves). We use an earth model of a layer with low seismic velocities overa half-space with higher velocities. We use 38 sensors and solely 10 secondsof recording. Different frequencies are processed independently. Of practicalinterest is the phase velocity dispersion of surface waves Aki & Richards (1980).

We define G , Γ(R) and initially model a single Rayleigh wave, i.e.,M = 1. In Fig. A.5, the estimates of the wavenumbers κ (black dots) suggestthe Rayleigh wave dispersion curves. For comparison, the theoretical dispersioncurves are depicted by lines. In Fig. A.6, the number of waves modeled isincreased to M = 3. It is shown that increasing the number of waves modeledby the factor graph, allows to better retrieve the fundamental and the highermodes.

A.5 Conclusions

We have developed a technique to perform ML estimation of wavefield param-eter of any wave type. The technique accounts for different noise variance oneach channel, by properly merging the information from sensors with differentnoise level. In the same framework, we address the superposition of multiple

142

2 3 4 5 6 7 8 9 10 11 120

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Fundamental mode

First higher mode

Second higher mode

Est. wavenumber κ

Frequency [Hz]

Wav

enum

ber

[m−1]

Figure A.5: Rayleigh wave dispersion curve, M = 1.

2 3 4 5 6 7 8 9 10 11 120

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Fundamental mode

First higher mode

Second higher mode

Est. wavenumber κ

Frequency [Hz]

Wav

enum

ber

[m−1]

Figure A.6: Rayleigh wave dispersion curve, M = 3.

waves and show that wavefield decomposition enables to detect weaker waves.We also propose an iterative algorithm for noise variance estimation. The tech-nique accounts for arbitrary sensor positions and arbitrary sampling instants.

We show numerical examples on monochromatic signals and on a well-established dataset of the seismic wavefield.

Appendix B

Multi-Sensor Estimation and Detection of

Phase-Locked Sinusoids

Christoph Reller1, Hans-Andrea Loeliger1, and Stefano Maranò2

1 ETH Zurich, Dept. of Information Technology & Electrical Engineering, 8092

Zürich, Switzerland.2 ETH Zurich, Swiss Seismological Service, 8092 Zürich, Switzerland.

Published in Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing,Prague, Czech Republic, May 2011, pp. 3872–3875.

Abstract

This paper proposes a method to compute the likelihood functionfor the amplitudes and phase shifts of noisily observed phase-locked andamplitude-constrained sinusoids. The sinusoids are assumed to be cou-pled based on a set of parameters as, e.g., measurements of a monochro-matic wave field. A factor graph is used to formulate the probabilitydensity function of the observations given the parameters. The factorgraph consists of one second-order state-space model per signal and oneadditional factor connecting all the final states. Because the parametersappear only in this latter factor, we are able to formulate a sufficientstatistic for parameter estimation and signal detection in terms of mes-sages in the factor graph. In special cases, the general form of the suf-ficient statistic reduces to the discrete Fourier transform. As extensionswe provide iterative algorithms for approximate maximum likelihood es-timation of the noise variances and the parameters of superposed waves.

B.1 Introduction

Phase-coupled and amplitude-constrained sinusoids play an important role inmany fields ranging from sensor arrays in seismology (Maranò et al., 2011),acoustics, and electromagnetics to multi-terminal communication. Parameterestimation for, and detection of, uncoupled sinusoids is described e.g. in (Kay,

143

144

gθ

X(1)K

, S1

X(1)k+1

y(1)k

X(1)k X

(1)1

y(1)0

X(L)K

, SL

X(L)k+1

y(L)k

X(L)k X

(L)1

y(L)0

Figure B.1: Overall factor graph

=X

(ℓ)k

A

C

+N (0, σ2ℓ )

Z(ℓ)k

Y(ℓ)k = y

(ℓ)k

X(ℓ)k+1

Figure B.2: State-space model for sinu-soid

=

U = u(θ)

H1(θ)S1

HL(θ)SL

Figure B.3: Glue factor gθ

1993) and (Kay, 1998). However, as soon as we consider some coupling, a linearmodel does not apply anymore. This paper provides a unified and generalapproach to estimation and detection of coupled sinusoids based on a factorgraph representation (Loeliger et al., 2007).

Consider L discrete-time sinusoidal signals

ξ(ℓ)k = αℓ cos(Ωk + ψℓ) , (B.1)

where ℓ = 1, . . . , L enumerates the signals and k = 0, . . . , K − 1 is the timeindex. All L signals have the same, known frequency Ω but differ in amplitude

αℓ and phase ψℓ. We observe the noisy signal Y(ℓ)k = ξ

(ℓ)k + Z

(ℓ)k , where Z

(ℓ)k

are zero-mean white Gaussian noises with noise variances σ2ℓ for ℓ = 1, . . . , L.

Note that we allow σ2ℓ to differ in each signal.

Unconstrained estimation of αℓ and ψℓ is a well known problem (Kay, 1993).This paper, however, deals with coupled signals. Specifically we assume thatthe amplitudes αℓ and phase shifts ψℓ are constrained by some parameter vectorθ via some mapping

Γ : θ 7→(

(α1, ψ1), . . . , (αL, ψL))

. (B.2)

As a toy example assume that we have two signals (B.1) with unconstrainedamplitudes α1 and α2 but with the same phase ψ , ψ1 = ψ2. We are interestedin α1, α2 and ψ. In this example one possible choice is θ = (ρ0, φ0, β) andΓ(θ) =

(

(ρ0, φ0), (βρ0, φ0))

.

145

For a more relevant example consider the estimation of seismic wavefieldsmeasured by a sensor array. In this example, θ may contain wavefield param-eters such as wave type, velocity of propagation, angle of arrival, etc. Themapping Γ would include sensor characteristics and positions. (This setting istreated in more detail in (Maranò et al., 2011).)

We use the term wave for ξ , ξ(ℓ)k ℓ=1,...,L,k=0,...,K−1 induced by the

parameters θ given the mapping (B.2). For the noisy signal we define Y (ℓ),

(

Y(ℓ)0 , . . . , Y

(ℓ)K−1

)

and Y ,(

Y (1), . . . ,Y (L))

. Given observations Y = y, wewant to formulate the likelihood function f(y |θ) in order to make ML estimates

θ = argmaxθ

f(y |θ) (B.3)

and to compute the generalized log-likelihood ratio (LLR)

lnf(

y∣

∣θ)

f(y |H0), (B.4)

where the hypothesis H0 is the presence of only noise. (The LLR is usuallyused for signal detection (Kay, 1998).)

Clearly, the likelihood can be written as

f(y |θ) =L∏

ℓ=1

K−1∏

k=0

1√

2πσ2ℓ

e−(

y(ℓ)k

−ξ(ℓ)k

(θ))2/

(2σ2ℓ ) . (B.5)

In Section B.2 we show that f(y |θ) is a function of the quantities Hℓ(θ) ∈R2×2,

←−WSℓ(σ

2ℓ ) ∈ R2×2, and ←−mSℓ(y

(ℓ)) ∈ R for ℓ = 1, . . . , L. The complexity

for computing←−WSℓ and ←−mSℓ is linear in K and for fixed Hℓ,

←−WSℓ ,

←−mSℓ the

complexity for computing this function is linear in L. Note that←−WSℓ and

←−mSℓ do not depend on θ and form a sufficient statistic. In Section B.3 weshow how this sufficient statistic simplifies in restricted settings and we showa connection with the discrete Fourier transform (DFT). In Section B.4 aniterative algorithm for approximating the ML estimate of the noise variances ineach signal is given. Finally, in Section B.5 we formulate an iterative algorithmto compute the likelihood function and LLR of several superposed waves.

B.2 Computing Likelihoods with Factor Graphs

State-Space Model

For each signal Y (ℓ) we formulate a second-order linear state-space model withstate vector Xk as

X(ℓ)k = AX

(ℓ)k+1 , (B.6)

Y(ℓ)k = CX

(ℓ)k + Z

(ℓ)k , (B.7)

where A is a rotation matrix

A , rotm(−Ω) , (B.8)

rotmα ,

(

cosα − sinα

sinα cosα

)

, (B.9)

146

and C , (1, 0). The corresponding (Forney) factor graph is depicted inFig. B.2. Note that the time progression in Figs. B.2 and B.1 is from right

to left. We use the abbreviation Sℓ , X(ℓ)K for the final system state.

Glue Factor

Without loss of generality we can assume that θ contains the overall amplitudeρ0 6= 0 and overall phase φ0 of an unknown reference sinusoid ρ0 cos(Ωk+φ0).By letting

u(θ) , ρ0

(

cos(ΩK + φ0)

sin(ΩK + φ0)

)

(B.10)

be the state of this reference sinusoid at time K we can express the couplingbetween the signals as

Sℓ = Hℓ(θ)u(θ) , (B.11)

where Hℓ(θ) , ρℓ rotm(φℓ), ρℓ , αℓ/ρ0, and φℓ , ψℓ − φ0.Both Hℓ(θ) and u(θ) depend on the parameters θ via the mapping Γ

in (B.2). In our toy example we have H1 = I2 and H2 = β I2, where I2

denotes the 2× 2 identity matrix.We define a glue factor (Fig. B.3)

gθ(s1, . . . , sL) =

L∏

ℓ=1

δ(

sℓ −Hℓ(θ)u(θ))

, (B.12)

modeling the relations (B.11) as constraints, where δ(·) denotes the Dirac delta.(See (Loeliger et al., 2009) for the concept of a glue factor.)

Likelihood Function and LLR

The overall factor graph in Fig. B.1 consists of L state-space models (Fig. B.2)connected by the glue factor (Fig. B.3). This factor graph is tree-shaped andrepresents the probability density function

f(y,x|θ) = f(

y,x∣

∣u(θ),H(θ))

, (B.13)

with X , X(ℓ)k ℓ=1,...,L,k=1,...,K and H(θ) ,

(

H1(θ)T, . . . , HL(θ)

T)T

.Instead of using the brute force calculation (B.5), we use sum-product mes-

sage passing (Loeliger et al., 2007) in the factor graph to compute the likelihoodfunction. We use arrows to distinguish between forward messages (−→· ) in thesame direction as the edge and backward messages (←−· ) in the opposite direc-tion.

By marginalization and due to the definition of the sum-product rule wecan write

f(y |θ) =∫

f(y,x|θ) dx = ←−µU (u(θ)) . (B.14)

Since the state-space models (Fig. B.2) do not depend on θ, we can computethe messages←−µSℓ without specifying θ. The likelihood (B.14) is then calculated

147

from ←−µSℓ and θ. It immediately follows that ←−µSℓ for ℓ = 1, . . . , L is a sufficientstatistic.

All messages in the factor graph at hand are (potentially scaled and degen-erate) multivariate Gaussian probability density functions. In this paper weprefer to write a message (e.g. for an edge X in forward direction) in the form

−→µX(x) , −→γX e−xT−→WXx/2+xT−→

WX−→mX , (B.15)

where−→VX =

−→W

−1X is the covariance matrix (if it exists), −→mX is the mean vector

and the scale factor is defined as

−→γX ,−→µX(0) . (B.16)

The maximization in (B.3) can be done by first maximizing over u(θ).Since for Gaussian messages max-product message passing coincides with sum-product message passing the ML estimate of U(θ) is u = ←−mU . We set u(θ) = u

in (B.14) and get the partially maximized likelihood function from messageupdate rules for Gaussian messages (Loeliger et al., 2007) as

ln f(

y∣

∣u(θ) = u,H(θ))

=1

2←−mTU

←−WU←−mU + ln←−γU , (B.17)

where

←−WU =

L∑

ℓ=1

Hℓ(θ)T←−WSℓ Hℓ(θ) , (B.18)

←−WU←−mU =

L∑

ℓ=1

Hℓ(θ)T←−WSℓ

←−mSℓ , (B.19)

ln←−γU =

L∑

ℓ=1

ln←−γSℓ . (B.20)

The remaining maximization over Hℓ(θ) is in general non-convex and dependslargely on the mapping Γ. We do not treated it here.

For the noise hypothesis H0 we constrain X to be zero by setting u = 0

in (B.12). From (B.14) and (B.16) we get

ln f(y |H0) = ln←−γU , (B.21)

so that the partially maximized LLR is

LLR(θ) , lnf(y |u(θ) = u,H(θ))

f(y |H0)=

1

2←−mTU

←−WU←−mU . (B.22)

Note that (B.20) does not depend on θ and therefore can be neglected when

finding ML estimates θ. The sufficient statistic thus consists of

←−mSℓ ,←−WSℓ

ℓ=1,...,L.

It is easy to generalize the state-space models to non-uniform sampling. For

this we substitute A in (B.6) by time-varying matrices A(ℓ)k , rotm

(

(t(ℓ)k−1 −

t(ℓ)k )ω

)

, where ω is the continuous time frequency and t(ℓ)k are time stamps

of y(ℓ)k . In the same fashion we can easily accommodate time varying noise

variances. The likelihood function and the LLR can still be computed bymessage passing as in (B.17) and (B.22).

148

B.3 Connection with the Discrete Fourier Transform

In the case of uniform sampling as assumed in (B.1), the following analyticsolution for the messages ←−µSℓ can be proven.

←−WSℓ =

K

2σ2ℓ

I2 +sin(ΩK)

2σ2ℓ sinΩ

rotm(Ω)R , (B.23)

←−WSℓ←−mSℓ =

1

σ2ℓ

R

K−1∑

k=0

y(ℓ)k

(

cos(kΩ)

sin(kΩ)

)

, (B.24)

where R , rotm(ΩK)(

1 00 −1

)

. When viewing the sum in (B.24) as a function ofΩ we recognize the real and imaginary parts of the finite-length discrete-time

Fourier transform of(

y(ℓ)0 , . . . , y

(ℓ)K−1

)

.

If we further restrict the frequency Ω to be Ωn , 2πn/K for n = 0, . . . , K−1, then R =

(

1 00 −1

)

and the expressions above simplify to

←−WSℓ =

K

2σ2ℓ

I2 , (B.25)

←−mSℓ =2

K

K−1∑

k=0

y(ℓ)k

(

cos(2πkn/K)

− sin(2πkn/K)

)

. (B.26)

The latter consists of the real and the imaginary parts of the DFT of(

y(ℓ)0 , . . . , y

(ℓ)K−1

)

,scaled by 2/K. If, in addition, the noise variances are the same in all signals,i.e., if σ2

ℓ = σ2 for ℓ = 1, . . . , L, then (B.22) simplifies to

LLRn(θ) =yTn H(θ)H(θ)T yn

Kσ2∑L

ℓ=1 ρ2ℓ

, (B.27)

where yn , K2

(

←−mTS1, . . . ,←−mT

SL

)Tcontains the n’th component of the DFTs

of the signals. Equation (B.27) is a standard beam-forming result. (See e.g.(Krim & Viberg, 1996).)

B.4 Noise Variance Estimation

In this section we consider the case where the noise variances η ,(

σ21 , . . . , σ

2L

)

are not given a-priori but are estimated in an ML sense for both the “signalpresent” (H1) and the “noise present” (H0) hypothesis.

The joint maximization for H1 over (θ,η) is non-convex. We propose touse cyclic maximization (Stoica & Selen, 2004) by alternating

θ = argmaxθ

f(

y∣

∣θ, η)

, (B.28)

η = argmaxη

f(

y∣

∣θ,η)

. (B.29)

Since the likelihood in every iteration cannot decrease, cyclic maximizationalgorithms are guaranteed to converge.

The maximization (B.28) is the same as (B.3) and hence the procedure inSection B.2 applies. To start the algorithm we propose an initial estimate η

149

based on assuming that the signals are decoupled, i.e., the glue factor (B.12)is replaced by

gθ(s1, . . . , sL) =L∏

ℓ=1

δ(sℓ −←−msℓ) . (B.30)

With (B.30), these initial ML noise variance estimates are

σ2ℓ =

1

K

K−1∑

k=0

(

y(ℓ)k −CA

K−k←−mSℓ

)2

. (B.31)

Once we have an estimate θ we can calculate −→mSℓ by apply θ in the gluefactor (B.12) and get the coupled ML noise variance estimates as

σ2ℓ =

1

K

K−1∑

k=0

(

y(ℓ)k − CAK−k−→mSℓ

)2

. (B.32)

If the signals are long we might want to avoid the direct computation of σ2ℓ .

We can approximate (B.31) and (B.32) by

σ2ℓ ≈ ζ2ℓ −←−mT

Sℓ←−mSℓ/2 , (B.33)

σ2ℓ ≈ ζ2ℓ + −→mT

Sℓ−→mSℓ/2−−→mT

Sℓ←−mSℓ (B.34)

respectively, where ζ2ℓ , 1K

∑K−1k=0

(

y(ℓ)k

)2are the signal powers. Using these

approximations, the only input to the algorithm is ←−mSℓ ,←−WSℓ , ζ

2ℓ ℓ=1,...,L. It

can be shown that ←−mSℓ does not depend on σ2ℓ and

←−WSℓ depends linearly on

σ2ℓ .

Under the noise hypothesis H0 the ML estimate of the noise variances areη(0)

, (ζ21 , . . . , ζ2L). The generalized LLR can easily be calculated from (B.5)

as (Kay, 1998)

lnf(

y∣

∣θ, η,H1

)

f(

y

∣

∣

∣η(0),H0

) =K

2

L∑

ℓ=1

lnζ2ℓσ2ℓ

. (B.35)

B.5 Extension to Wave Superposition

Assume that we observe a linear superposition of M waves ξ(m) with same fre-

quency Ω, parameters θm, and mappings Γm : θm 7→(

(

α(m)1 , ψ

(m)1

)

, . . . ,(

α(m)L , ψ

(m)L

)

)

for m = 1, . . . ,M . Our signal model now is

Y(ℓ)k =

M∑

m=1

(

α(m)ℓ cos

(

Ωk + ψ(m)ℓ

)

)

+ Z(ℓ)k . (B.36)

We collect the parameters in a vector θ , (θ1, . . . ,θM ). It is straight for-ward to model all waves simultaneously by using extended matrices Hℓ(θ) ,(

Hℓ(θ1), . . . ,Hℓ(θM ))

and state vectors u(θ) ,(

u(θ1)T, . . . ,u(θM )T

)Tin

(B.11) and (B.12). However, the space over which to maximize in (B.17) in-creases approximately M fold.

150

As an alternative we propose an iterative algorithm based on cyclic max-imization (Stoica & Selen, 2004). Assume that we have an estimate θ. Wepick some m ∈ 1, . . . ,M and update the estimate of θm while fixing θj =θjj∈1,...,M\m. This leads to the following glue factor

gθm(s1, . . . , sL) =

L∏

ℓ=1

δ(

sℓ − s6mℓ −Hℓ(θm)u(θm)

)

, (B.37)

where s6mℓ ,

∑

j∈1,...,M\m Hℓ(θj)u(θj) is the estimated state due to all

the waves except for the m-th. The corresponding θm can again be found bymaximizing (B.17) where ←−µU is calculated using the glue factor (B.37).

To apply this algorithm we propose the following greedy-type procedure.Initially, set M = 1 and use the glue factor (B.12) to find θ1. Then repeatedlydo the following. Increase M , use the glue factor (B.37) with m = M tofind θM , and iterate finding θm for m ∈ 1, . . . ,M until convergence. Thisalgorithm is applied successfully in (Maranò et al., 2011).

B.6 Conclusion

We have used a factor graph to derive a sufficient statistic (in simple cases thisis the DFT) for the ML estimation of wave parameters. The sufficient statisticcan be used to devise iterative algorithms for the estimation of superposedwaves and of the noise variances.

Appendix C

Estimation of Wavefield Parameters of a Single

P Wave at the Free Surface

C.1 Introduction

We are interested in the estimation of body waves from array recordings ofambient vibrations. We consider the case of a single incident P wave.

At the free surface, where the array is positioned, a body wave will ex-hibit an apparent velocity larger than the true velocity of propagation. Theapparent velocity depends on the velocity of the wave in the uppermost layerand the incidence angle. Assuming an homogeneous first layer with thicknessmuch larger than the wavelength, the ground motion observed at the surfaceis explained by considering the superposition of the incident P wave togetherwith the reflected P wave and S wave.

In the following sections, we describe the wave equations used to model theground displacement, the approach for wavefield parameters estimation, andshow some numerical results.

C.2 System Model

Our interest lies in the estimation of P wave parameters from the analysis ofthe ground displacement induced by a single incident P wave. The grounddisplacement is measured by a planar array of Ns sensors positioned at thesurface of the earth. In such setting, it is necessary to consider the P-SVcoupling at the free surface.

We model the wavefield u measured at the surface as the sum of the threewavefields u(1), u(2), and u(3) due to the incident P wave, the reflected P wave,and the reflected S wave, respectively.

Let vP and vS be the velocity of P and S waves, respectively. The wavenum-bers for P waves and S waves are then κP = ω/vP and κS = ω/vS.

The displacement due to the incident P wave, with incidence angle η andwavevector κ1 = κP(cosψ sin η, sinψ sin η, cos η) is

151

152

u(1)x (p, t) = α cosψ sin η cos(ωt− κ1 · p+ ϕ) (C.1)

u(1)y (p, t) = α sinψ sin η cos(ωt− κ1 · p+ ϕ) (C.2)

u(1)z (p, t) = α cos η cos(ωt− κ1 · p+ ϕ) . (C.3)

The angle of the reflected P wave is the same as the incidence angle. Am-plitude of the reflected wave is scaled by the reflection coefficient RPP

RPP =4 sin2 η cos η

√

γ − sin2 η − (γ − 2 sin2 η)2

4 sin2 η cos η√

γ − sin2 η + (γ − 2 sin2 η)2, (C.4)

where γ = v2P/v2S. We observe that γ > 2. The displacement at the free surface

due to the reflected P wave is given by

u(2)x (p, t) = RPPα cosψ sin η cos(ωt− κ2 · p+ ϕ) (C.5)

u(2)y (p, t) = RPPα sinψ sin η cos(ωt− κ2 · p+ ϕ) (C.6)

u(2)z (p, t) = −RPPα cos η cos(ωt− κ2 · p+ ϕ) , (C.7)

where κ2 = κP(cosψ sin η, sinψ sin η,− cos η).Using Snell’s law the angle η′ of the reflected SV wave is found

sin η

vP=

sin η′

vS. (C.8)

The reflection coefficient RPS for the SV wave is

RPS =κS

κP· −4 sin η cos η(γ − 2 sin2 η)

4 sin2 η cos η√

γ − 2 sin2 η + (γ − 2 sin2 η)2. (C.9)

The ground displacement due to the reflected SV wave is

u(3)x (p, t) = −RPSα cosψ cos η′ cos(ωt− κ3 · p+ ϕ) (C.10)

u(3)y (p, t) = −RPSα sinψ cos η′ cos(ωt− κ3 · p+ ϕ) (C.11)

u(3)z (p, t) = −RPSα sin η′ cos(ωt− κ3 · p+ ϕ) , (C.12)

where κ3 = κP(cosψ sin η, sinψ sin η,−√

κ2S/κ2P − sin2 η).

With the assumption of a linear medium, the total wavefield observed atthe surface is given by the sum of the three displacements u(1), u(2), and u(3).Considering receivers at the free surface, having p = (x, y, 0), then we canreplace κ1, κ2, and κ3 with κ = κP(cosψ sin η, sinψ sin η, 0) and obtain

ux(p, t) = α cosψ((1 +RPP) sin η −RPS cos η′) cos(ωt− κ · p+ ϕ)

uy(p, t) = α sinψ((1 +RPP) sin η −RPS cos η′) cos(ωt− κ · p+ ϕ) (C.13)

uz(p, t) = α((1 −RPP) cos η −RPS sin η′) cos(ωt− κ · p+ ϕ) ,

which describes the total ground displacement induced by the incident P wave,the reflected P wave, and the reflected SV wave. In particular, we observe thatthe oscillation angle of a particle is not equal to the incidence angle.

153

C.3 Parameter Estimation

Using the wave equations presented in the previous section, it is possible to per-form ML estimation of wavefield parameters. The wavefield parameter vectorfor the considered scenario is θ(P) = (α, ϕ, κP, ψ, η, κS), cf. (C.13).

We extend the ML method proposed in Maranò et al. (2011) to model anincident P wave.

We remark that the P wavenumber κP and the S wavenumber κS that areestimated are the actual wavenumbers in the first layer and are not related tothe apparent velocity of the incident P wave. In addition, the inclination angleη corresponds to the incidence angle of the P wave and not to the oscillationangle observed at the surface.

C.4 Numerical Results

Monochromatic Wavefield

We simulate the ground displacement induced by a single monochromatic in-coming P wave. The ground displacement accounts for the P-SV couplingdescribed in the following section. In particular, we consider an incident Pwave with azimuth ψ = π/5, and different incidence angle (or inclination an-gle). The body wave velocity in the uppermost layer are vP = 500 m/s andvS = 200 m/s. Therefore, considering a monochromatic wave at 5 Hz, the truevalues for P wave wavenumber and S wave wavenumber are κP = 0.01 1/m andκS = 0.025 1/m, respectively. Parameters are estimated using the proposedML method.

First, we consider a P wave with a shallow incidence angle of 80. InFig. C.1, two slices of the LLF are shown. In Fig.C.1(a) the LLF as a functionof κP and κS is depicted. In Fig.C.1(b) the LLF as a function of ψ and η isdepicted. The true maximum is marked with a white cross. In both figures itis possible to see how the influence of the parameters on the LLF is different.In particular, it is possible to see that κS and η only marginally affect theLLF around the maximum value. The maximum value, pinpointed by a whitecross, is correctly found and the ML estimate of the parameters matches withthe true values.

Second, we also consider a steeper incidence angle corresponding to thecritical angle between the first layer and the half-space. We let the P wavevelocity in the half-space be 2000 m/s. The critical angle is then

ηc = arcsin

(

500

2000

)

≈ 14.5 . (C.14)

In Fig. C.2, slices of the LLF are depicted. Similarly to the previous exam-ple, the maximum of the LLF identifies the true wavefield parameters.

We showed that, under the good conditions of large SNR the retrieval ofwave parameters from a single incident P wave is possible.

From the shape of the LLF we draw the following considerations on esti-mation accuracy. These statements can be verified using FI analysis.

Firstly, the statistical model that follows from (C.13) is identifiable. In-tuitively, this can be understood from the fact that the LLF exhibits a single

154

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Pwavenum

berκP

[1/m

]

S wavenumber κS [1/m]

(a) Slice of the LLF as a function of κP andκS.

Azimuth ψ [rad]0 π/2 π 3π/2 2π

Incl

inationη

[rad]

0

+π4

+π2

(b) Slice of the LLF as a function of Azimuthand Inclination.

Figure C.1: Slices of the LLF for the estimation of wavefield parameters of anincoming P wave. Shallow incidence angle (η = 80).

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Pwavenum

berκP

[1/m

]

S wavenumber κS [1/m]

(a) Slice of the LLF as a function of κP andκS.

Azimuth ψ [rad]0 π/2 π 3π/2 2π

Incl

inationη

[rad]

0

+π4

+π2

(b) Slice of the LLF as a function of Azimuthand Inclination.

Figure C.2: Slices of the LLF for the estimation of wavefield parameters of anincoming P wave. Critical incidence angle (η = 14.5).

maxima. It can be verified by testing the non-singularity of the Fisher infor-mation matrix (FIM).

Secondly, the model carries very little information about vS. Intuitively, thiscan be understood by the little change in the value of likelihood for differentκP, cf. Fig. C.1(a) and Fig. C.2(a). This could also be verified by inspectionof the FIM. The reason for this is that vS only affects the angle of oscillationat the surface, which is difficult to estimated accurately. As a consequence, weexpect to retrieve κS with much less accuracy than κP.

SESAME M2.1 Dataset

The proposed method is also tested on the synthetic dataset M2.1 from theSESAME project (Bard, P.-Y., 2008). The structural model is a layer over ahalf-space and it is depicted in Fig. C.3(a). The P wave and S wave velocities

155

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200−30

−25

−20

−15

−10

−5

0

Depth

[m]

Velocity [m/s]

P wave

S wave

(a) Velocity profile.

−40 −20 0 20

0

20

40

60

x [m]

y[m

]

(b) Array layout.

2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

900

1000

1100

Velo

city

[m/s]

Frequency [Hz]

(c) Rayleigh phase velocity for differentmodes.

2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

900

1000

1100

Velo

city

[m/s]

Frequency [Hz]

(d) Rayleigh group velocity for differentmodes.

Figure C.3: Details of the M2.1 dataset.

in the layer are vP = 500 m/s and vS = 200 m/s, respectively. The P waveand S wave velocities in the half-space are vP = 2000 m/s and vS = 1000 m/s,respectively.

We use 14 three-components sensors, the array layout is shown in Fig. C.3(b).We model the presence of Rayleigh waves and P waves and perform ML pa-rameter estimation. The recording is split in windows of duration 0.5 seconds.

Phase velocity of the different modes of surface waves can be computedfrom the structural model using the the propagator matrix method (Gilbert &Backus, 1966). Phase velocity of several modes of Rayleigh wave is shown inFig. C.3(c). The Rayleigh wave group velocity is shown in Fig. C.3(d). At anyangular frequency ω, the group velocity vg is related to the phase velocity v as

vg =dω

dκ= v + κ

dv

dκ. (C.15)

In Fig. C.4, the wavenumber and the ellipticity angle estimated for Rayleighwaves are shown. The estimated parameters are depicted in gray and black.The lines represent the theoretical values of the parameters for the differentmodes. The fundamental mode and part of the first higher mode of the Rayleighwave are correctly estimated as shown in Fig. C.4(a). In Fig. C.4(b), it ispossible to follow the ellipticity angle of the Rayleigh wave especially for the

156

2 4 6 8 10 12 14 16 18 200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Wavenum

ber

[1/m

]

Frequency [Hz]

(a) Rayleigh wave wavenumber.

2 4 6 8 10 12 14 16 18 20

Ellip

ticity

angleξ

[rad]

π2

π4

0

-π2

-π4

Frequency [Hz]

(b) Rayleigh wave ellipticity angle.

Figure C.4: Summary of the estimated Rayleigh wave parameters for the M2.1dataset.

fundamental mode. In both figures, there are no estimates above 14 Hz due tolimitations of the dataset.

The estimated wavenumbers of P wave and S wave are shown in Fig. C.5(a)and in Fig. C.5(b), respectively. The theoretical wavenumber for P wave andS wave is also depicted as a straight line (points of constant velocity).

Equivalently, the estimated P wave and S wave velocities are plotted in inFig. C.5(c) and Fig. C.5(d). A line depicts the theoretical velocity for the bodywaves.

Both in the wavenumber and in the velocity representation, the estimatedvalues do not resemble the expected values.

The estimates of the incidence angle and of the azimuth angle for theP waves are shown in Fig. C.5(e) and Fig. C.5(f), respectively. Very shal-low incidence angles and vertical incidence angles have been removed from allthe results shown.

The resonance frequency of a P wave in the layer is f0 = vP/4h = 5 Hz,where h = 25m is the thickness of the layer. Therefore it is expected thatno P wave should appear below such frequency. We explain the estimatedwavenumbers below this frequency as due to noise or surface waves.

At 10 Hz the wavelength of a P wave in the layer is λ = 500/10 = 50m.This is double the thickness of the layer. Therefore the ray theory is not validand the assumptions of the proposed method are not valid.

With these considerations in mind, we conclude that the estimation of wave-field parameters of a P wave was not possible on the M2.1 model. In particular,we consider that the estimated wavenumbers are not reliable because the con-sidered wavelengths are too large when compared with the thickness of thelayer.

It would be interesting to analyze higher frequencies, where wavelengthsare shorter and ray theory is applicable. However, the M2.1 dataset is notmodeling frequency higher than those considered.

157

2 4 6 8 10 12 14 16 18 20

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Wavenum

ber

[1/m

]

Frequency [Hz]

(a) P wave wavenumber κP. TheoreticalP wave wavenumber is depicted with the redline.

2 4 6 8 10 12 14 16 18 20

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Wavenum

ber

[1/m

]

Frequency [Hz]

(b) S wave wavenumber κS. TheoreticalS wave wavenumber is depicted with the blueline.

2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

900

1000

1100

Velo

city

[m/s]

Frequency [Hz]

(c) P wave velocity vP. Theoretical P wavevelocity is depicted with the red line.

2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

900

1000

1100

Velo

city

[m/s]

Frequency [Hz]

(d) S wave velocity vS. Theoretical S wavevelocity is depicted with the red line.

2 4 6 8 10 12 14 16 18 20

Inclinati

onη

[rad]

0

+π4

+π2

Frequency [Hz]

(e) Incidence angle η.

2 4 6 8 10 12 14 16 18 20

Azim

uthψ

[rad]

0

π2

π

3π2

2π

Frequency [Hz]

(f) Azimuth angle ψ.

Figure C.5: Summary of the P wave parameters for the M2.1 dataset.

158

C.5 Discussion

We developed a model for the parameter estimation of a single incident P wave.We applied the ML method for the retrieval of wavefield parameters of P waves.Noticeably, from the noisy recordings of the wavefield induced by a singleP wave it is possible to estimate the wavenumber of both P wave and S wave.In fact, the angle of oscillation observed at the surface is influenced by bothquantities.

A simple numerical example with a monochromatic wave following the P-SV system shows that the retrieval of all the wavefield parameters of interest ispossible. From the shape of the likelihood function (LF), we expect that certainparameters are associated with a large uncertainty. This can be formalizedwith analysis of Fisher information (FI) and derivation of Cramér-Rao bound(CRB).

From the analysis of the M2.1 SESAME synthetic the estimation of param-eters of a P wave was not successful due to the limited high-frequency contentof the wavefield, as discussed in Section C.4.

Appendix D

Electronic Supplement to: “Sensor Placement for

the Analysis of Seismic Surface Waves: Sources

of Error, Design Criterion, and Array Design

Algorithms”

Stefano Maranò1, Donat Fäh1, and Yue M. Lu2

1 ETH Zurich, Swiss Seismological Service, 8092 Zürich, Switzerland.2 Harvard University, School of Engineering and Applied Sciences, Cambridge, MA

02138, USA.

Submitted to Geophys. J. Int.

D.1 Overview

In this electronic supplement we provide the details about the optimized sen-sor arrays for the analysis of seismic surface waves waves that were obtainedin Maranò et al. (2013).

The arrays described in this supplement are designed for unitary largestwavenumber, i.e., κmax = 1. Guidelines to adapt the physical extent of thearray to the actual spatial frequency content of the wavefield are provided inan appendix of the article. The array are obtained using the mixed integerprogram (MIP) algorithm presented in the article for different number of sen-sors Ns and different values of κmax/κmin. In particular Ns = 6, . . . , 20 andκmax/κmin = 2, 4, 6 are considered.

We provide sensor coordinates in both Cartesian (xn, yn) and polar coordi-nates (rn, ψn). The azimuth ψn is measured counter-clockwise from the x-axis.

159

160

−0.4 −0.2 0 0.2 0.4 0.6 0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

1

2

3

4

5

6

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 -0.12564 0.21762 0.25128 1202 -0.12564 -0.21762 0.25128 2403 0.25128 0 0.25128 04 -0.35897 0.62176 0.71795 1205 -0.35897 -0.62176 0.71795 2406 0.71795 0 0.71795 0

(c) Table of coordinates.

Figure D.1: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 6 for κmin = 1/2 and κmax = 1.

161

−0.4 −0.2 0 0.2 0.4 0.6−0.6

−0.4

−0.2

0

0.2

0.4

0.6

1

2

3

4

5

6

7

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0.32249 0.4044 0.51724 51.42 -0.1151 0.50427 0.51724 102.93 -0.46602 0.22442 0.51724 154.34 -0.46602 -0.22442 0.51724 205.75 -0.1151 -0.50427 0.51724 257.16 0.32249 -0.4044 0.51724 308.67 0.51724 0 0.51724 0

(c) Table of coordinates.

Figure D.2: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 7 for κmin = 1/2 and κmax = 1.

162

−0.6 −0.4 −0.2 0 0.2 0.4 0.6

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

1

2

3

4

5

6

7

8

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0.36472 0.36472 0.51579 452 0 0.51579 0.51579 903 -0.36472 0.36472 0.51579 1354 -0.51579 0 0.51579 1805 -0.36472 -0.36472 0.51579 2256 0 -0.51579 0.51579 2707 0.36472 -0.36472 0.51579 3158 0.51579 0 0.51579 0

(c) Table of coordinates.

Figure D.3: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 8 for κmin = 1/2 and κmax = 1.

163

−0.6 −0.3 0 0.3 0.6

−0.6

−0.3

0

0.3

0.6

0.9

1

2

3

4

5

6

7

8

9

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0.11111 0.19245 0.22222 602 -0.22222 0 0.22222 1803 0.11111 -0.19245 0.22222 3004 0.33333 0.57735 0.66667 605 -0.66667 0 0.66667 1806 0.33333 -0.57735 0.66667 3007 -0.045224 0.77646 0.77778 93.38 -0.64982 -0.4274 0.77778 213.39 0.69505 -0.34907 0.77778 333.3

(c) Table of coordinates.

Figure D.4: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 9 for κmin = 1/2 and κmax = 1.

164

−0.6 −0.3 0 0.3 0.6

−0.6

−0.3

0

0.3

0.6

1

2

3

4

5

6

7

8

9

10

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0.1553 0.47797 0.50256 722 -0.40658 0.2954 0.50256 1443 -0.40658 -0.2954 0.50256 2164 0.1553 -0.47797 0.50256 2885 0.50256 0 0.50256 06 0.55179 0.4009 0.68205 367 -0.21077 0.64867 0.68205 1088 -0.68205 0 0.68205 1809 -0.21077 -0.64867 0.68205 25210 0.55179 -0.4009 0.68205 324

(c) Table of coordinates.

Figure D.5: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 10 for κmin = 1/2 and κmax = 1.

165

−0.6 −0.3 0 0.3 0.6

−0.6

−0.3

0

0.3

0.6

0.9

1

2

3

4

5

6 7

8

9 10

11

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0 0 0 02 0.36464 0.21053 0.42105 303 -0.21053 0.36464 0.42105 1204 -0.36464 -0.21053 0.42105 2105 0.21053 -0.36464 0.42105 3006 0.19071 0.71173 0.73684 757 -0.19071 0.71173 0.73684 1058 -0.71173 -0.19071 0.73684 1959 -0.52103 -0.52103 0.73684 22510 0.52103 -0.52103 0.73684 31511 0.71173 -0.19071 0.73684 345

(c) Table of coordinates.

Figure D.6: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 11 for κmin = 1/2 and κmax = 1.

166

−0.8 −0.4 0 0.4 0.8−1.2

−0.8

−0.4

0

0.4

0.8

1

2

3

4

5

6

7

8

9

10

11

12

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 -0.11111 0.19245 0.22222 1202 -0.11111 -0.19245 0.22222 2403 0.22222 0 0.22222 04 0.42558 0.3571 0.55556 405 -0.52205 0.19001 0.55556 1606 0.096471 -0.54712 0.55556 2807 -0.33333 0.57735 0.66667 1208 -0.33333 -0.57735 0.66667 2409 0.66667 0 0.66667 010 0.76604 0.64279 1 4011 -0.93969 0.34202 1 16012 0.17365 -0.98481 1 280

(c) Table of coordinates.

Figure D.7: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 12 for κmin = 1/2 and κmax = 1.

167

−0.6 −0.3 0 0.3 0.6 0.9

−0.9

−0.6

−0.3

0

0.3

0.6

0.9

1

2

3

4

5

6

7

8

9

10

11

12

13

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0 0 0 02 -0.086824 0.4924 0.5 1003 -0.38302 -0.32139 0.5 2204 0.46985 -0.17101 0.5 3405 0.275 0.47631 0.55 606 -0.55 0 0.55 1807 0.275 -0.47631 0.55 3008 0.67615 0.18117 0.7 159 -0.49497 0.49497 0.7 13510 -0.18117 -0.67615 0.7 25511 -0.45 0.77942 0.9 12012 -0.45 -0.77942 0.9 24013 0.9 0 0.9 0

(c) Table of coordinates.

Figure D.8: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 13 for κmin = 1/2 and κmax = 1.

168

−0.8 −0.4 0 0.4 0.8

−0.8

−0.4

0

0.4

0.8

1

2

3

4

5

6

7

8

9

10

11

12

13

14

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0.32815 0.41149 0.52632 51.42 -0.11712 0.51312 0.52632 102.93 -0.47419 0.22836 0.52632 154.34 -0.47419 -0.22836 0.52632 205.75 -0.11712 -0.51312 0.52632 257.16 0.32815 -0.41149 0.52632 308.67 0.52632 0 0.52632 08 0.75871 0.36538 0.84211 25.79 0.18739 0.82099 0.84211 77.110 -0.52504 0.65838 0.84211 128.611 -0.84211 0 0.84211 18012 -0.52504 -0.65838 0.84211 231.413 0.18739 -0.82099 0.84211 282.914 0.75871 -0.36538 0.84211 334.3

(c) Table of coordinates.

Figure D.9: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 14 for κmin = 1/2 and κmax = 1.

169

−0.6 −0.3 0 0.3 0.6 0.9

−0.9

−0.6

−0.3

0

0.3

0.6

0.9

1 2

3

4 5

6

7

8

9

10

11

12

13

14

15

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0.225 0.38971 0.45 602 -0.225 0.38971 0.45 1203 -0.45 0 0.45 1804 -0.225 -0.38971 0.45 2405 0.225 -0.38971 0.45 3006 0.45 0 0.45 07 0.64952 0.375 0.75 308 0 0.75 0.75 909 -0.64952 0.375 0.75 15010 -0.64952 -0.375 0.75 21011 0 -0.75 0.75 27012 0.64952 -0.375 0.75 33013 -0.45 0.77942 0.9 12014 -0.45 -0.77942 0.9 24015 0.9 0 0.9 0

(c) Table of coordinates.

Figure D.10: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 15 for κmin = 1/2 and κmax = 1.

170

−0.8 −0.4 0 0.4 0.8

−0.8

−0.4

0

0.4

0.8

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0 0 0 02 0.33867 0.28418 0.44211 403 -0.076771 0.43539 0.44211 1004 -0.41544 0.15121 0.44211 1605 -0.33867 -0.28418 0.44211 2206 0.076771 -0.43539 0.44211 2807 0.41544 -0.15121 0.44211 3408 0.27722 0.76165 0.81053 709 -0.79821 -0.14075 0.81053 19010 0.521 -0.6209 0.81053 31011 -0.15354 0.87078 0.88421 10012 -0.67734 -0.56836 0.88421 22013 0.83089 -0.30242 0.88421 34014 0.73379 0.61572 0.95789 4015 -0.90013 0.32762 0.95789 16016 0.16634 -0.94334 0.95789 280

(c) Table of coordinates.

Figure D.11: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 16 for κmin = 1/2 and κmax = 1.

171

−0.8 −0.4 0 0.4 0.8

−0.8

−0.4

0

0.4

0.8

1

2

3

4

5 6

7 8

9

10 11

12

13

14

15 16

17

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0 0.44 0.44 902 -0.44 0 0.44 1803 0 -0.44 0.44 2704 0.44 0 0.44 05 0.33941 0.33941 0.48 456 -0.33941 0.33941 0.48 1357 -0.33941 -0.33941 0.48 2258 0.33941 -0.33941 0.48 3159 0 0.8 0.8 9010 -0.69282 -0.4 0.8 21011 0.69282 -0.4 0.8 33012 -0.42 0.72746 0.84 12013 -0.42 -0.72746 0.84 24014 0.84 0 0.84 015 0.7621 0.44 0.88 3016 -0.7621 0.44 0.88 15017 0 -0.88 0.88 270

(c) Table of coordinates.

Figure D.12: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 17 for κmin = 1/2 and κmax = 1.

172

−0.8 −0.4 0 0.4 0.8 1.2

−0.8

−0.4

0

0.4

0.8

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0.125 0.21651 0.25 602 -0.25 0 0.25 1803 0.125 -0.21651 0.25 3004 -0.3 0.51962 0.6 1205 -0.3 -0.51962 0.6 2406 0.6 0 0.6 07 0.16823 0.62785 0.65 758 -0.62785 -0.16823 0.65 1959 0.45962 -0.45962 0.65 31510 0.49497 0.49497 0.7 4511 -0.67615 0.18117 0.7 16512 0.18117 -0.67615 0.7 28513 0.92924 0.19752 0.95 1214 -0.63567 0.70599 0.95 13215 -0.29357 -0.9035 0.95 25216 -0.30902 0.95106 1 10817 -0.66913 -0.74314 1 22818 0.97815 -0.20791 1 348

(c) Table of coordinates.

Figure D.13: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 18 for κmin = 1/2 and κmax = 1.

173

−0.8 −0.4 0 0.4 0.8 1.2

−0.8

−0.4

0

0.4

0.8

1.2

1

2

3

4

5

6 7

8

9

10 11

12

13

14

15

16

17

18

19

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0.25525 0.14737 0.29474 302 -0.14737 0.25525 0.29474 1203 -0.25525 -0.14737 0.29474 2104 0.14737 -0.25525 0.29474 3005 0.50801 0.42627 0.66316 406 0.11516 0.65308 0.66316 807 -0.33158 0.57431 0.66316 1208 -0.62316 0.22681 0.66316 1609 -0.62316 -0.22681 0.66316 20010 -0.33158 -0.57431 0.66316 24011 0.11516 -0.65308 0.66316 28012 0.50801 -0.42627 0.66316 32013 0.66316 0 0.66316 014 0.47895 0.82956 0.95789 6015 -0.95789 0 0.95789 18016 0.47895 -0.82956 0.95789 30017 -0.26699 0.99643 1.0316 10518 -0.72944 -0.72944 1.0316 22519 0.99643 -0.26699 1.0316 345

(c) Table of coordinates.

Figure D.14: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 19 for κmin = 1/2 and κmax = 1.

174

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

1

2

3

4 5

6 7

8

9

10

11

12

13

14 15

16

17

18

19

20

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0.33657 0.14985 0.36842 242 -0.03851 0.3664 0.36842 963 -0.36037 0.076599 0.36842 1684 -0.18421 -0.31906 0.36842 2405 0.24652 -0.27379 0.36842 3126 0.44374 0.49282 0.66316 487 -0.33158 0.57431 0.66316 1208 -0.64867 -0.13788 0.66316 1929 -0.069319 -0.65953 0.66316 26410 0.60582 -0.26973 0.66316 33611 0.71369 0.18325 0.73684 14.412 0.046267 0.73539 0.73684 86.413 -0.6851 0.27125 0.73684 158.414 -0.46968 -0.56775 0.73684 230.415 0.39482 -0.62214 0.73684 302.416 0.34155 1.0512 1.1053 7217 -0.89418 0.64966 1.1053 14418 -0.89418 -0.64966 1.1053 21619 0.34155 -1.0512 1.1053 28820 1.1053 0 1.1053 0

(c) Table of coordinates.

Figure D.15: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 20 for κmin = 1/2 and κmax = 1.

175

−0.8 −0.4 0 0.4 0.8 1.2−1.2

−0.8

−0.4

0

0.4

0.8

1.2

1

2

3

4

5

6

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 -0.15789 0.27348 0.31579 1202 -0.15789 -0.27348 0.31579 2403 0.31579 0 0.31579 04 -0.57895 1.0028 1.1579 1205 -0.57895 -1.0028 1.1579 2406 1.1579 0 1.1579 0

(c) Table of coordinates.

Figure D.16: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 6 for κmin = 1/4 and κmax = 1.

176

−1.2 −0.6 0 0.6 1.2

−0.6

0

0.6

1.2

1.8

1

2

3 4

5

6 7

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0 0 0 02 0 0.5 0.5 903 -0.43301 -0.25 0.5 2104 0.43301 -0.25 0.5 3305 0 1.5 1.5 906 -1.299 -0.75 1.5 2107 1.299 -0.75 1.5 330

(c) Table of coordinates.

Figure D.17: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 7 for κmin = 1/4 and κmax = 1.

177

−1.5 −1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

1

2

3

4

5

6

7

8

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0.16904 0.18774 0.25263 482 -0.24711 0.052525 0.25263 1683 0.078067 -0.24027 0.25263 2884 1.0219 0.74247 1.2632 365 -0.39034 1.2013 1.2632 1086 -1.2632 0 1.2632 1807 -0.39034 -1.2013 1.2632 2528 1.0219 -0.74247 1.2632 324

(c) Table of coordinates.

Figure D.18: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 8 for κmin = 1/4 and κmax = 1.

178

−1.5 −1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

1

2

3

4

5

6

7

8

9

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0.13161 0.74638 0.75789 802 -0.71219 -0.25922 0.75789 2003 0.58058 -0.48717 0.75789 3204 0.088103 1.1757 1.1789 85.75 -1.0622 -0.51153 1.1789 205.76 0.97409 -0.66412 1.1789 325.77 0.96764 0.81194 1.2632 408 -1.187 0.43203 1.2632 1609 0.21935 -1.244 1.2632 280

(c) Table of coordinates.

Figure D.19: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 9 for κmin = 1/4 and κmax = 1.

179

−1.2 −0.6 0 0.6 1.2 1.8

−1.2

−0.6

0

0.6

1.2

1

2

3

4

5

6

7

8

9

10

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0 0 0 02 -0.1931 0.33446 0.38621 1203 -0.1931 -0.33446 0.38621 2404 0.38621 0 0.38621 05 -0.12688 1.2071 1.2138 966 -0.98198 -0.71345 1.2138 2167 1.1089 -0.49369 1.2138 3368 1.5844 0.22268 1.6 89 -0.98506 1.2608 1.6 12810 -0.59937 -1.4835 1.6 248

(c) Table of coordinates.

Figure D.20: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 10 for κmin = 1/4 and κmax = 1.

180

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

1

2

3

4

5

6

7

8

9

10

11

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0 0 0 02 0.31227 0.96107 1.0105 723 -0.81753 0.59397 1.0105 1444 -0.81753 -0.59397 1.0105 2165 0.31227 -0.96107 1.0105 2886 1.0105 0 1.0105 07 0.80296 0.89177 1.2 488 -0.6 1.0392 1.2 1209 -1.1738 -0.24949 1.2 19210 -0.12543 -1.1934 1.2 26411 1.0963 -0.48808 1.2 336

(c) Table of coordinates.

Figure D.21: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 11 for κmin = 1/4 and κmax = 1.

181

−1.8 −1.2 −0.6 0 0.6 1.2

−1.2

−0.6

0

0.6

1.2

1

2

3

4

5

6

7

8

9

10

11

12

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

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enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0.15435 0.87538 0.88889 802 -0.83528 -0.30402 0.88889 2003 0.68093 -0.57137 0.88889 3204 -0.19294 1.0942 1.1111 1005 -0.85116 -0.71421 1.1111 2206 1.0441 -0.38002 1.1111 3407 1.0214 0.85705 1.3333 408 -1.2529 0.45603 1.3333 1609 0.23153 -1.3131 1.3333 28010 0.92891 1.2477 1.5556 53.311 -1.545 0.18059 1.5556 173.312 0.61612 -1.4283 1.5556 293.3

(c) Table of coordinates.

Figure D.22: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 12 for κmin = 1/4 and κmax = 1.

182

−1.2 −0.6 0 0.6 1.2

−1.8

−1.2

−0.6

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0.6

1.2

1

2

3

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6

7

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9

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11

12

13

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y[m

]

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2

Wavenumber κx [1/m]

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[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0 0 0 02 -0.44444 0.7698 0.88889 1203 -0.44444 -0.7698 0.88889 2404 0.88889 0 0.88889 05 0.85116 0.71421 1.1111 406 -1.0441 0.38002 1.1111 1607 0.19294 -1.0942 1.1111 2808 -0.66667 1.1547 1.3333 1209 -0.66667 -1.1547 1.3333 24010 1.3333 0 1.3333 011 1.1916 0.99989 1.5556 4012 -1.4617 0.53203 1.5556 16013 0.27012 -1.5319 1.5556 280

(c) Table of coordinates.

Figure D.23: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 13 for κmin = 1/4 and κmax = 1.

183

−1.4 −0.7 0 0.7 1.4

−1.4

−0.7

0

0.7

1.4

1 2

3

4

5 6

7

8

9

10

11

12

13

14

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0.38699 0.48527 0.62069 51.42 -0.13812 0.60513 0.62069 102.93 -0.55922 0.26931 0.62069 154.34 -0.55922 -0.26931 0.62069 205.75 -0.13812 -0.60513 0.62069 257.16 0.38699 -0.48527 0.62069 308.67 0.62069 0 0.62069 08 1.032 1.2941 1.6552 51.49 -0.36831 1.6137 1.6552 102.910 -1.4913 0.71815 1.6552 154.311 -1.4913 -0.71815 1.6552 205.712 -0.36831 -1.6137 1.6552 257.113 1.032 -1.2941 1.6552 308.614 1.6552 0 1.6552 0

(c) Table of coordinates.

Figure D.24: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 14 for κmin = 1/4 and κmax = 1.

184

−1.4 −0.7 0 0.7 1.4

−2.1

−1.4

−0.7

0

0.7

1.4

1

2

3

4 5

6

7

8

9

10

11

12

13

14

15

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0.78705 0.35042 0.86154 242 -0.697 0.5064 0.86154 1443 -0.090055 -0.85682 0.86154 2644 -0.11257 1.071 1.0769 965 -0.53846 0.93264 1.0769 1206 -0.87125 -0.633 1.0769 2167 -0.53846 -0.93264 1.0769 2408 0.98382 -0.43802 1.0769 3369 1.0769 0 1.0769 010 0.37716 1.1608 1.2205 7211 -1.1938 -0.25376 1.2205 19212 0.81668 -0.90702 1.2205 31213 1.7053 0.75924 1.8667 2414 -1.5102 1.0972 1.8667 14415 -0.19512 -1.8564 1.8667 264

(c) Table of coordinates.

Figure D.25: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 15 for κmin = 1/4 and κmax = 1.

185

−1.6 −0.8 0 0.8 1.6

−1.6

−0.8

0

0.8

1.6

1

2

3

4

5

6

7

8

9

10

11

12 13

14

15 16

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0 0 0 02 0.16923 0.29312 0.33846 603 -0.33846 0 0.33846 1804 0.16923 -0.29312 0.33846 3005 -0.50769 0.87935 1.0154 1206 -0.50769 -0.87935 1.0154 2407 1.0154 0 1.0154 08 -0.67692 1.1725 1.3538 1209 -0.67692 -1.1725 1.3538 24010 1.3538 0 1.3538 011 1.7981 0.4818 1.8615 1512 1.3163 1.3163 1.8615 4513 -1.3163 1.3163 1.8615 13514 -1.7981 0.4818 1.8615 16515 -0.4818 -1.7981 1.8615 25516 0.4818 -1.7981 1.8615 285

(c) Table of coordinates.

Figure D.26: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 16 for κmin = 1/4 and κmax = 1.

186

−1.4 −0.7 0 0.7 1.4

−1.4

−0.7

0

0.7

1.4

1

2 3

4 5

6

7

8

9

10

11

12

13

14

15 16

17

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0 0 0 02 0.24244 0.24244 0.34286 453 -0.24244 0.24244 0.34286 1354 -0.24244 -0.24244 0.34286 2255 0.24244 -0.24244 0.34286 3156 -0.6 1.0392 1.2 1207 -0.6 -1.0392 1.2 2408 1.2 0 1.2 09 1.4903 0.39932 1.5429 1510 1.3362 0.77143 1.5429 3011 -0.39932 1.4903 1.5429 10512 -1.091 1.091 1.5429 13513 -1.3362 0.77143 1.5429 15014 -1.091 -1.091 1.5429 22515 -0.39932 -1.4903 1.5429 25516 0 -1.5429 1.5429 27017 1.4903 -0.39932 1.5429 345

(c) Table of coordinates.

Figure D.27: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 17 for κmin = 1/4 and κmax = 1.

187

−1.4 −0.7 0 0.7 1.4

−1.4

−0.7

0

0.7

1.4

1

2

3

4

5

6

7 8

9

10 11

12

13

14

15

16

17

18

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0.25128 0.43523 0.50256 602 -0.50256 0 0.50256 1803 0.25128 -0.43523 0.50256 3004 -0.13714 0.77775 0.78974 1005 -0.60498 -0.50764 0.78974 2206 0.74212 -0.27011 0.78974 3407 0.21194 1.202 1.2205 808 -0.21194 1.202 1.2205 1009 -1.1469 -0.41744 1.2205 20010 -0.93497 -0.78453 1.2205 22011 0.93497 -0.78453 1.2205 32012 1.1469 -0.41744 1.2205 34013 1.5517 0.56477 1.6513 2014 -0.28674 1.6262 1.6513 10015 -1.265 1.0614 1.6513 14016 -1.265 -1.0614 1.6513 22017 -0.28674 -1.6262 1.6513 26018 1.5517 -0.56477 1.6513 340

(c) Table of coordinates.

Figure D.28: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 18 for κmin = 1/4 and κmax = 1.

188

−1.4 −0.7 0 0.7 1.4

−1.4

−0.7

0

0.7

1.4

1

2

3

4

5

6

7

8

9 10

11

12

13

14

15

16

17

18

19

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0 0 0 02 0.44211 0.76575 0.88421 603 -0.88421 0 0.88421 1804 0.44211 -0.76575 0.88421 3005 -0.51579 0.89337 1.0316 1206 -0.51579 -0.89337 1.0316 2407 1.0316 0 1.0316 08 0.66316 1.1486 1.3263 609 0.23031 1.3062 1.3263 8010 -0.23031 1.3062 1.3263 10011 -1.3263 0 1.3263 18012 -1.2463 -0.45363 1.3263 20013 -1.016 -0.85254 1.3263 22014 0.66316 -1.1486 1.3263 30015 1.016 -0.85254 1.3263 32016 1.2463 -0.45363 1.3263 34017 1.1463 1.1463 1.6211 4518 -1.5658 0.41956 1.6211 16519 0.41956 -1.5658 1.6211 285

(c) Table of coordinates.

Figure D.29: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 19 for κmin = 1/4 and κmax = 1.

189

−1.4 −0.7 0 0.7 1.4

−1.4

−0.7

0

0.7

1.4

1

2

3

4 5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0.67314 0.2997 0.73684 242 -0.077021 0.73281 0.73684 963 -0.72074 0.1532 0.73684 1684 -0.36842 -0.63812 0.73684 2405 0.49304 -0.54758 0.73684 3126 0.36431 1.1212 1.1789 727 -0.95379 0.69297 1.1789 1448 -0.95379 -0.69297 1.1789 2169 0.36431 -1.1212 1.1789 28810 1.1789 0 1.1789 011 1.4274 0.36649 1.4737 14.412 0.78964 1.2443 1.4737 57.613 0.092533 1.4708 1.4737 86.414 -0.93936 1.1355 1.4737 129.615 -1.3702 0.5425 1.4737 158.416 -1.3702 -0.5425 1.4737 201.617 -0.93936 -1.1355 1.4737 230.418 0.092533 -1.4708 1.4737 273.619 0.78964 -1.2443 1.4737 302.420 1.4274 -0.36649 1.4737 345.6

(c) Table of coordinates.

Figure D.30: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 20 for κmin = 1/4 and κmax = 1.

190

−1 −0.5 0 0.5 1 1.5

−1

−0.5

0

0.5

1

1

2

3

4

5

6

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0 0 0 02 0.38058 1.1713 1.2316 723 -0.99637 0.7239 1.2316 1444 -0.99637 -0.7239 1.2316 2165 0.38058 -1.1713 1.2316 2886 1.2316 0 1.2316 0

(c) Table of coordinates.

Figure D.31: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 6 for κmin = 1/6 and κmax = 1.

191

−1.2 −0.6 0 0.6 1.2

−1.2

−0.6

0

0.6

1.2

1.8

1

2

3 4

5

6 7

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0 0 0 02 0 1.0345 1.0345 903 -0.89589 -0.51724 1.0345 2104 0.89589 -0.51724 1.0345 3305 0 1.5517 1.5517 906 -1.3438 -0.77586 1.5517 2107 1.3438 -0.77586 1.5517 330

(c) Table of coordinates.

Figure D.32: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 7 for κmin = 1/6 and κmax = 1.

192

−1.4 −0.7 0 0.7 1.4

−1.4

−0.7

0

0.7

1.4

1

2

3

4

5

6

7

8

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0 0 0 02 0.96477 1.2098 1.5474 51.43 -0.34432 1.5086 1.5474 102.94 -1.3941 0.67138 1.5474 154.35 -1.3941 -0.67138 1.5474 205.76 -0.34432 -1.5086 1.5474 257.17 0.96477 -1.2098 1.5474 308.68 1.5474 0 1.5474 0

(c) Table of coordinates.

Figure D.33: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 8 for κmin = 1/6 and κmax = 1.

193

−1.4 −0.7 0 0.7 1.4

−2.1

−1.4

−0.7

0

0.7

1.4

1

2

3

4

5

6

7

8

9

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 -0.53846 0.93264 1.0769 1202 -0.53846 -0.93264 1.0769 2403 1.0769 0 1.0769 04 1.0725 0.8999 1.4 405 -1.3156 0.47883 1.4 1606 0.24311 -1.3787 1.4 2807 1.4025 1.1768 1.8308 408 -1.7204 0.62616 1.8308 1609 0.31791 -1.803 1.8308 280

(c) Table of coordinates.

Figure D.34: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 9 for κmin = 1/6 and κmax = 1.

194

−1.8 −0.9 0 0.9 1.8

−1.8

−0.9

0

0.9

1.8

2.7

1

2

3

4

5

6

7

8

9 10

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0 0 0 02 0.15344 0.47225 0.49655 723 -0.4857 -0.10324 0.49655 1924 0.33226 -0.36901 0.49655 3125 -0.14706 1.3992 1.4069 966 -1.1382 -0.82695 1.4069 2167 1.2853 -0.57224 1.4069 3368 0.083759 2.3985 2.4 889 -2.1191 -1.1267 2.4 20810 2.0353 -1.2718 2.4 328

(c) Table of coordinates.

Figure D.35: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 10 for κmin = 1/6 and κmax = 1.

195

−1.8 −0.9 0 0.9 1.8

−1.8

−0.9

0

0.9

1.8

1

2

3

4

5

6

7

8

9

10

11

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0.24396 0.75083 0.78947 722 -0.6387 0.46404 0.78947 1443 -0.6387 -0.46404 0.78947 2164 0.24396 -0.75083 0.78947 2885 0.78947 0 0.78947 06 0.53671 1.6518 1.7368 727 -1.6989 -0.36111 1.7368 1928 1.1622 -1.2907 1.7368 3129 0.68309 2.1023 2.2105 7210 -2.1622 -0.45959 2.2105 19211 1.4791 -1.6427 2.2105 312

(c) Table of coordinates.

Figure D.36: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 11 for κmin = 1/6 and κmax = 1.

196

−2.2 −1.1 0 1.1 2.2

−2.2

−1.1

0

1.1

2.2

1 2

3

4

5

6

7 8

9

10

11

12

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0.46632 0.26923 0.53846 302 -0.46632 0.26923 0.53846 1503 0 -0.53846 0.53846 2704 0.48462 0.83938 0.96923 605 -0.96923 0 0.96923 1806 0.48462 -0.83938 0.96923 3007 1.772 1.0231 2.0462 308 -1.772 1.0231 2.0462 1509 0 -2.0462 2.0462 27010 1.3462 2.3316 2.6923 6011 -2.6923 0 2.6923 18012 1.3462 -2.3316 2.6923 300

(c) Table of coordinates.

Figure D.37: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 12 for κmin = 1/6 and κmax = 1.

197

−1.8 −0.9 0 0.9 1.8

−2.7

−1.8

−0.9

0

0.9

1.8

1

2

3

4

5

6

7

8

9

10

11

12

13

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0 0 0 02 1.0214 0.85705 1.3333 403 0.23153 1.3131 1.3333 804 -1.2529 0.45603 1.3333 1605 -1.2529 -0.45603 1.3333 2006 0.23153 -1.3131 1.3333 2807 1.0214 -0.85705 1.3333 3208 1.8794 0.68404 2 209 -1.5321 1.2856 2 14010 -0.3473 -1.9696 2 26011 2.0851 1.0472 2.3333 26.712 -1.9495 1.2822 2.3333 146.713 -0.13567 -2.3294 2.3333 266.7

(c) Table of coordinates.

Figure D.38: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 13 for κmin = 1/6 and κmax = 1.

198

−1.8 −0.9 0 0.9 1.8

−1.8

−0.9

0

0.9

1.8

1

2 3

4

5

6

7

8

9

10

11

12

13

14

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 1.7549 0.40054 1.8 12.92 0.78099 1.6217 1.8 64.33 -0.78099 1.6217 1.8 115.74 -1.7549 0.40054 1.8 167.15 -1.4073 -1.1223 1.8 218.66 0 -1.8 1.8 2707 1.4073 -1.1223 1.8 321.48 1.247 1.5637 2 51.49 -0.44504 1.9499 2 102.910 -1.8019 0.86777 2 154.311 -1.8019 -0.86777 2 205.712 -0.44504 -1.9499 2 257.113 1.247 -1.5637 2 308.614 2 0 2 0

(c) Table of coordinates.

Figure D.39: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 14 for κmin = 1/6 and κmax = 1.

199

−2 −1 0 1 2

−2

−1

0

1

2

1

2 3

4

5

6

7

8

9

10

11

12

13

14

15

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 -0.091225 0.86795 0.87273 962 -0.70605 -0.51298 0.87273 2163 0.79728 -0.35497 0.87273 3364 -0.54545 0.94475 1.0909 1205 -0.54545 -0.94475 1.0909 2406 1.0909 0 1.0909 07 1.7651 1.2824 2.1818 368 -1.0909 1.8895 2.1818 1209 -1.9932 0.88743 2.1818 15610 -1.0909 -1.8895 2.1818 24011 0.22806 -2.1699 2.1818 27612 2.1818 0 2.1818 013 2.1925 0.97617 2.4 2414 -1.9416 1.4107 2.4 14415 -0.25087 -2.3869 2.4 264

(c) Table of coordinates.

Figure D.40: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 15 for κmin = 1/6 and κmax = 1.

200

−1.8 −0.9 0 0.9 1.8

−1.8

−0.9

0

0.9

1.8

1

2

3

4

5 6 7

8

9

10 11

12

13

14

15

16

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0 0 0 02 1.1667 0.97901 1.5231 403 -1.4312 0.52092 1.5231 1604 0.26448 -1.4999 1.5231 2805 0.4599 1.7164 1.7769 756 0 1.7769 1.7769 907 -0.4599 1.7164 1.7769 1058 -1.7164 -0.4599 1.7769 1959 -1.5389 -0.88846 1.7769 21010 -1.2565 -1.2565 1.7769 22511 1.2565 -1.2565 1.7769 31512 1.5389 -0.88846 1.7769 33013 1.7164 -0.4599 1.7769 34514 1.0154 1.7587 2.0308 6015 -2.0308 0 2.0308 18016 1.0154 -1.7587 2.0308 300

(c) Table of coordinates.

Figure D.41: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 16 for κmin = 1/6 and κmax = 1.

201

−2.4 −1.2 0 1.2 2.4

−2.4

−1.2

0

1.2

2.4

1

2 3

4

5

6

7 8

9

10 11

12

13

14

15

16

17

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 -0.125 0.21651 0.25 1202 -0.125 -0.21651 0.25 2403 0.25 0 0.25 04 1.1419 0.50842 1.25 245 -0.13066 1.2432 1.25 966 -1.2227 0.25989 1.25 1687 -0.625 -1.0825 1.25 2408 0.83641 -0.92893 1.25 3129 2.2839 1.0168 2.5 2410 0.77254 2.3776 2.5 7211 -1.25 2.1651 2.5 12012 -2.0225 1.4695 2.5 14413 -2.4454 -0.51978 2.5 19214 -1.25 -2.1651 2.5 24015 -0.26132 -2.4863 2.5 26416 1.6728 -1.8579 2.5 31217 2.5 0 2.5 0

(c) Table of coordinates.

Figure D.42: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 17 for κmin = 1/6 and κmax = 1.

202

−2 −1 0 1 2 3

−2

−1

0

1

2

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0.225 0.38971 0.45 602 -0.45 0 0.45 1803 0.225 -0.38971 0.45 3004 -0.825 1.4289 1.65 1205 -0.825 -1.4289 1.65 2406 1.65 0 1.65 07 1.5588 0.9 1.8 308 0 1.8 1.8 909 -1.5588 0.9 1.8 15010 -1.5588 -0.9 1.8 21011 0 -1.8 1.8 27012 1.5588 -0.9 1.8 33013 -1.05 1.8187 2.1 12014 -1.05 -1.8187 2.1 24015 2.1 0 2.1 016 -1.275 2.2084 2.55 12017 -1.275 -2.2084 2.55 24018 2.55 0 2.55 0

(c) Table of coordinates.

Figure D.43: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 18 for κmin = 1/6 and κmax = 1.

203

−2.4 −1.2 0 1.2 2.4

−2.4

−1.2

0

1.2

2.4

1

2 3

4

5

6 7

8

9 10

11

12

13

14

15

16

17

18

19

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 0 0 0 02 0.45963 0.38567 0.6 403 -0.56382 0.20521 0.6 1604 0.10419 -0.59088 0.6 2805 1.4095 0.51303 1.5 206 1.1491 0.96418 1.5 407 -1.1491 0.96418 1.5 1408 -1.4095 0.51303 1.5 1609 -0.26047 -1.4772 1.5 26010 0.26047 -1.4772 1.5 28011 1.05 1.8187 2.1 6012 -2.1 0 2.1 18013 1.05 -1.8187 2.1 30014 1.4964 1.8764 2.4 51.415 -2.3732 0.3577 2.4 171.416 0.87682 -2.2341 2.4 291.417 -0.60081 2.6323 2.7 102.918 -1.9792 -1.8365 2.7 222.919 2.58 -0.79584 2.7 342.9

(c) Table of coordinates.

Figure D.44: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 19 for κmin = 1/6 and κmax = 1.

204

−2 −1 0 1 2

−2

−1

0

1

2

1

2

3

4 5

6

7

8

9

10

11 12

13

14

15

16

17

18

19 20

x [m]

y[m

]

(a) h(x, y)

−2 −1 0 1 2

−2

−1

0

1

2

Wavenumber κx [1/m]

Wav

enum

berκy

[1/m

]

(b) |H(κx, κy)|2

n xn [m] yn [m] rn [m] ψn [ ]1 1.1746 0.52295 1.2857 242 -0.13439 1.2787 1.2857 963 -1.2576 0.26732 1.2857 1684 -0.64286 -1.1135 1.2857 2405 0.86031 -0.95547 1.2857 3126 1.2135 0.88168 1.5 367 -0.46353 1.4266 1.5 1088 -1.5 0 1.5 1809 -0.46353 -1.4266 1.5 25210 1.2135 -0.88168 1.5 32411 1.0076 1.3869 1.7143 5412 -1.0076 1.3869 1.7143 12613 -1.6304 -0.52974 1.7143 19814 0 -1.7143 1.7143 27015 1.6304 -0.52974 1.7143 34216 2.2418 0.7284 2.3571 1817 0 2.3571 2.3571 9018 -2.2418 0.7284 2.3571 16219 -1.3855 -1.907 2.3571 23420 1.3855 -1.907 2.3571 306

(c) Table of coordinates.

Figure D.45: Array layout, magnitude of the Fourier transform of the sensorpositions, and sensor coordinates for Ns = 20 for κmin = 1/6 and κmax = 1.

Appendix E

Derivation of the Cramér-Rao Bounds for Love

and Rayleigh Waves Parameters

In this chapter, details of the derivation the CRBs for the wavefield parametersof geophysical interest are given. To this aim, the FIMs are first derived.

E.1 Overview

In the measurement model used throughout this thesis the measurement at thetime instants tk on the ℓ-th channel is modeled as a random variable

Y(ℓ)k = u

(ℓ)k (θ) + Z

(ℓ)k , (E.1)

where u(ℓ)k (θ) is a deterministic function of the wavefield parameters θ and

Z(ℓ)k represents additive Gaussian noise, Z

(ℓ)k ∼ N

(

0, σ2ℓ

)

. Let K the numberof sampling times tk and L the number of channels.

Assuming independent noise, the probability density function (PDF) of the

measurements Y = Y (ℓ)k k=1,...,K

ℓ=1,...,Lis given by the product of Gaussian densities

pY (y |θ) =L∏

ℓ=1

K∏

k=1

1√

2πσ2ℓ

exp

(

− (y(ℓ)k − u

(ℓ)k (θ))2

2σ2ℓ

)

, (E.2)

and its logarithm is

ln pY (y |θ) = −LK2

ln(2π)− K

2

L∑

ℓ=1

lnσ2ℓ −

L∑

ℓ=1

1

2σ2ℓ

K∑

k=1

(Y(ℓ)k − u(ℓ)k )2 . (E.3)

Fisher Information with Known Noise Variance

The FIM is defined Fisher (1922); Rao (1945); Cramér (1946) as

I(θ) = E

−∂2 ln pY (y |θ)

∂θ2

. (E.4)

205

206

We assume the variances be known and not dependent on θ, then the elementin position i j of the FIM is

[I(θ)]i,j = E

−∂2 ln pY (y |θ)∂θi∂θj

(E.5)

= E

− ∂2

∂θi∂θj

L∑

ℓ=1

K∑

k=1

lnN(

Y(ℓ)k |u

(ℓ)k , σ2

ℓ

)

(E.6)

= E

+∂2

∂θi∂θj

L∑

ℓ=1

K∑

k=1

1

2σ2ℓ

(Y(ℓ)k − u(ℓ)k )2

(E.7)

= E

− ∂

∂θj

L∑

ℓ=1

K∑

k=1

1

σ2ℓ

(Y(ℓ)k − u(ℓ)k )

∂u(ℓ)k

∂θi

(E.8)

= E

−L∑

ℓ=1

K∑

k=1

1

σ2ℓ

(

−∂u(ℓ)k

∂θi

∂u(ℓ)k

∂θj+ (Y

(ℓ)k − u(ℓ)k )

∂2u(ℓ)k

∂θi∂θj

)

(E.9)

=

L∑

ℓ=1

1

σ2ℓ

K∑

k=1

∂u(ℓ)k

∂θi

∂u(ℓ)k

∂θj. (E.10)

The FIM is a positive semidefinite (PSD) matrix of size as the number ofelements of the parameter θ,

I(θ) =

Iθ1,θ1 Iθ2,θ1 · · ·Iθ1,θ2 Iθ2,θ2 · · ·

......

. . .

. (E.11)

Fisher Information with Unknown Noise Variance

The setting of unknown noise variance is now considered. It will be shownthat this new case is accounted for with simple modifications from the knownvariance setting.

We distinguish two cases. In the first case, all channels share the samenoise variance σ2. In the second case, each channel has different noise variance,σ21 , σ

22 , . . . , σ

2L.

Equal Noise Variance on Each Channel

In the first case, we consider the extended parameter vector η = (θT, σ2)T.The partial derivatives are

∂ ln pY (y |η)∂θi

= − 1

σ2

L∑

ℓ=1

K∑

k=1

(Y(ℓ)k − u(ℓ)k )

∂u(ℓ)k

∂θi(E.12)

∂ ln pY (y |η)∂σ2

= −KL2σ2

+1

2σ4

L∑

ℓ=1

K∑

k=1

(Y(ℓ)k − u(ℓ)k )2 . (E.13)

207

It follows that the FIM elements are

Iθi,θj =1

σ2

L∑

ℓ=1

K∑

k=1

∂u(ℓ)k

∂θi

∂u(ℓ)k

∂θj(E.14)

Iθi,σ2 = E

1

2σ4

L∑

ℓ=1

K∑

k=1

Z(ℓ)k

∂u(ℓ)k

∂θi

− 1

2σ6

L∑

ℓ1=1

K∑

k1=1

L∑

ℓ2=1

K∑

k2=1

(Z(ℓ1)k1

)2Z(ℓ2)k2

∂u(ℓ2)k2

∂θi

(E.15)

= 0 (E.16)

Iσ2,σ2 = E

K2L2

4σ4− KL

4σ6

L∑

ℓ=1

K∑

k=1

(Z(ℓ)k )2

+1

4σ8

L∑

ℓ1=1

K∑

k1=1

L∑

ℓ2=1

K∑

k2=1

(Z(ℓ1)k1

)2(Z(ℓ2)k2

)2

(E.17)

=K2L2

4σ4− K2L2

2σ4+

3LK

4σ4+LK(LK − 1)

4σ4(E.18)

=KL

2σ4, (E.19)

where we used the fact that the third moment is E(Z(ℓ)k )3 = 0 and the fourth

moment is E(Z(ℓ)k )4 = 3σ4.

The FIM is

I(η) =

(

I(θ) 0

0 Iσ2,σ2

)

. (E.20)

The FIM is block diagonal. This implies that the wavefield parameters θ andthe variance are decoupled, i.e., the uncertainty about θi does not affect theestimation accuracy of σ2 and vice versa.

Different Noise Variance on Each Channel

In the second case, the noise variances are allowed to be different on eachchannel. We consider the extended parameter vector η = (θT,σT)T withσ = (σ2

1 , σ22 , . . . , σ

2L)

T. The partial derivatives are

∂ ln pY (y |η)∂θi

= −L∑

ℓ=1

1

σ2ℓ

K∑

k=1

(Y(ℓ)k − u(ℓ)k )

∂u(ℓ)k

∂θi(E.21)

∂ ln pY (y |η)∂σ2

ℓ

= − K

2σ2ℓ

+1

2σ4ℓ

K∑

k=1

(Y(ℓ)k − u(ℓ)k )2 . (E.22)

208

It follows that each element of the FIM is

Iθi,θj =L∑

ℓ=1

1

σ2ℓ

K∑

k=1

∂u(ℓ)k

∂θi

∂u(ℓ)k

∂θj(E.23)

Iθi,σ2ℓ= E

1

2σ4

L∑

ℓ=1

K∑

k=1

Z(ℓ)k

∂u(ℓ)k

∂θi

− 1

2σ6

L∑

ℓ1=1

K∑

k1=1

L∑

ℓ2=1

K∑

k2=1

(Z(ℓ1)k1

)2Z(ℓ2)k2

∂u(ℓ2)k2

∂θi

(E.24)

= 0 (E.25)

Iσ2ℓ,σ2ℓ= E

K2

4σ4ℓ

− K

4σ6ℓ

K∑

k=1

(Z(ℓ)k )2 +

1

4σ8ℓ

K∑

k1=1

K∑

k2=1

(Z(ℓ)k1

)2(Z(ℓ)k2

)2

(E.26)

=K2

4σ4ℓ

− K2

4σ4ℓ

+3K

4σ4ℓ

+K(K − 1)

4σ4ℓ

(E.27)

=K

2σ4(E.28)

Iσ2ℓ,σ2p= E

K2

4σ2ℓσ

2p

+

∑Kk=1(Z

(ℓ)k )2

∑Kk=1(Z

(p)k )2

4σ4ℓ , σ

4p

− K∑Kk=1(Z

(p)k )2

4σ2ℓσ

4p

−K∑Kk=1(Z

(ℓ)k )2

4σ2pσ

4ℓ

(E.29)

=K2

4σ2ℓσ

2p

+K2

4σ2ℓ , σ

2p

− K2

4σ2ℓσ

2p

− K2

4σ2ℓσ

2p

(E.30)

= 0 . (E.31)

The FIM is

I(η) =

I(θ) 0 0 0

0 Iσ21 ,σ

21

0 0

0 0. . . 0

0 0 0 Iσ2L,σ

2L

. (E.32)

Similarly to the previous case, the FIM exhibits a block diagonal structure.

Discussion

As observed, in both the settings with unknown variance the FIM is blockdiagonal. Therefore we proceed with our further derivations assuming knownvariance. It is immediate to extend the results by augmenting appropriatelythe FIMs.

E.2 Derivation of Fisher Information Matrices

In this section we derive the FIMs for different setups. We consider the sta-tistical models involving a single Love wave or a single Rayleigh wave using

209

translational measurements, rotational measurements, or both types of mea-surements.

Approximations

Throughout the document we use the following approximations

K∑

k=1

cos2(ωk + ϕ) ≈ K

2(E.33)

K∑

k=1

sin2(ωk + ϕ) ≈ K

2(E.34)

K∑

k=1

sin(ωk + ϕ) cos(ωk + ϕ) ≈ 0 , (E.35)

which are valid for ω being not near 0 or 1/2 and are exact when ω = 2πmK m ∈ Z

(see (Kay, 1993, example 3.14) or Stoica et al. (1989)).

Useful Quantities

We define Φn = −xnκ cosψ − ynκ sinψ + ϕ, then we have

∂Φn∂κ

= −xn cosψ − yn sinψ (E.36)

∂Φn∂ψ

= +xnκ sinψ − ynκ cosψ (E.37)

∂Φn∂ϕ

= 1 . (E.38)

Parameter Space Transformation

Let θ and η be two different parameterizations of a statistical model. The twoFIMs are related as

I(η) = JTI(θ(η))J , (E.39)

where J is the Jacobian matrix with elements [J]i,j = ∂θi/∂ηj .In all the equations and the derivations of this section, it is assumed that

the wavenumber κ is measured in rad/m. If one consider the wavenumberκ′ = κ/2π measured in 1/m should use the parameters space transformationas in (E.39). In practice, let η = (α, ϕ, κ/2π, ψ)T

J =

1 0 0 0

0 1 0 0

0 0 2π 0

0 0 0 1

. (E.40)

210

Moreover it is also useful, to the aim of computing the CRB of a subset of theparameters, to permute the order of the parameters. For example, to move thewavenumber in the first position one can use the transformation of the FIM

J =

0 0 1 0

0 1 0 0

1 0 0 0

0 0 0 1

. (E.41)

Fisher Information Matrix for a Single Love Wave, Transla-

tional Measurements

The displacements induced by a single Love wave are

ux(p, t) = −α sinψ cos(ωt− κTp+ ϕ) (E.42)

uy(p, t) = α cosψ cos(ωt− κTp+ ϕ) (E.43)

uz(p, t) = 0 . (E.44)

The derivatives of the displacements with respect to the wavefield parametersare

∂ux∂α

= − sinψ cosΦn (E.45)

∂uy∂α

= cosψ cosΦn (E.46)

∂uz∂α

= 0 , (E.47)

∂ux∂ψ

= −α(

cosψ cosΦn − sinψ sinΦn∂Φn∂ψ

)

(E.48)

∂uy∂ψ

= α

(

− sinψ cosΦn − cosψ sinΦn∂Φn∂ψ

)

(E.49)

∂uz∂ψ

= 0 , (E.50)

∂ux∂ϕ

= α sinψ sinΦn∂Φn∂ϕ

(E.51)

∂uy∂ϕ

= −α cosψ sinΦn∂Φn∂ϕ

(E.52)

∂uz∂ϕ

= 0 , (E.53)

211

and

∂ux∂κ

= α sinψ sinΦn∂Φn∂κ

(E.54)

∂uy∂κ

= −α cosψ sinΦn∂Φn∂κ

(E.55)

∂uz∂κ

= 0 . (E.56)

The elements of the FIM are obtained using (E.10). The diagonal elements are

Iα,α =NsK

2σ2(E.57)

Iϕ,ϕ =α2NsK

2σ2(E.58)

Iκ,κ =α2K

2σ2

Ns∑

n=1

(

∂Φn∂κ

)2

(E.59)

Iψ,ψ =α2K

2σ2

(

Ns +

Ns∑

n=1

(

∂Φn∂ψ

)2)

. (E.60)

The off-diagonal elements are

Iα,ϕ = 0 (E.61)

Iα,κ = 0 (E.62)

Iα,ψ = 0 (E.63)

Iϕ,κ =α2K

2σ2

Ns∑

n=1

∂Φn∂κ

(E.64)

Iϕ,ψ =α2K

2σ2

Ns∑

n=1

∂Φn∂ψ

(E.65)

Iκ,ψ =α2K

2σ2

Ns∑

n=1

∂Φn∂ψ

∂Φn∂κ

. (E.66)

The FIM is

I(θ) =

Iα,α 0 0 0

0 Iϕ,ϕ Iϕ,κ Iϕ,ψ0 Iϕ,κ Iκ,κ Iκ,ψ0 Iϕ,ψ Iκ,ψ Iψ,ψ

. (E.67)

From (E.67), we observe that the amplitude is decoupled from the other wave-field parameters.

Fisher Information Matrix for a Single Love Wave, Rotational

Measurements

The rotations induced by a single Love wave are

212

ωx(p, t) =0 (E.68)

ωy(p, t) =0 (E.69)

ωz(p, t) =1

2ακ sin(ωt− κTp+ ϕ) . (E.70)

The derivatives of the rotations with respect to the wavefield parameters are

∂ωz∂α

=1

2κ sin(ωt− κTp+ ϕ) (E.71)

∂ωz∂ϕ

=1

2ακ cos(ωt− κTp+ ϕ) (E.72)

∂ωz∂κ

=1

2α sin(ωt− κTp+ ϕ) +

1

2ακ cos(ωt− κTp+ ϕ)

∂Φn∂κ

(E.73)

∂ωz∂ψ

=1

2ακ cos(ωt− κTp+ ϕ)

∂Φn∂ψ

. (E.74)

The elements of the FIM are obtained using (E.10). The elements on thediagonal are

Iα,α =κ2KNs

8σ2(E.75)

Iϕ,ϕ =α2κ2KNs

8σ2(E.76)

Iκ,κ =α2K

8σ2

(

Ns + κ2Ns∑

n=1

(

∂Φn∂κ

)2)

(E.77)

Iψ,ψ =α2κ2K

8σ2

Ns∑

n=1

(

∂Φn∂ψ

)

,2 (E.78)

and the off-diagonal elements are

Iα,ϕ = 0 (E.79)

Iα,κ =ακKNs

8σ2(E.80)

Iα,ψ = 0 (E.81)

Iϕ,κ =α2κ2K

8σ2

Ns∑

n=1

∂Φn∂κ

(E.82)

Iϕ,ψ =α2κ2K

8σ2

Ns∑

n=1

∂Φn∂ψ

(E.83)

Iκ,ψ =α2κ2K

8σ2

Ns∑

n=1

∂Φn∂ψ

∂Φn∂κ

. (E.84)

The FIM is

213

I(θ) =

Iα,α 0 Iα,κ 0

0 Iϕ,ϕ Iϕ,κ Iϕ,ψIα,κ Iϕ,κ Iκ,κ Iψ,κ0 Iϕ,ψ Iψ,κ Iψ,ψ

. (E.85)

Comparing (E.85) and (E.67), we observe that the amplitude is now coupledwith the wavenumber.

Fisher Information Matrix for a Single Rayleigh Wave, Trans-

lational Measurements

The displacements induced by a single Rayleigh wave are

ux(p, t) =α cosψ sin ξ cos(ωt− κTp+ ϕ) (E.86)

uy(p, t) =α sinψ sin ξ cos(ωt− κTp+ ϕ) (E.87)

uz(p, t) =α cos ξ cos(ωt− κTp+π

2+ ϕ) . (E.88)

The derivatives of the displacements with respect to the wavefield parametersare

∂ux∂α

= cosψ sin ξ cos(ωtk +Φn) (E.89)

∂uy∂α

= sinψ sin ξ cos(ωtk +Φn) (E.90)

∂uz∂α

= cos ξ cos(ωtk +Φn +π

2) , (E.91)

∂ux∂ψ

= α sin ξ

(

− sinψ cosΦn − cosψ sinΦn∂Φn∂ψ

)

(E.92)

∂uy∂ψ

= α sin ξ

(

cosψ cosΦn − sinψ sinΦn∂Φn∂ψ

)

(E.93)

∂uz∂ψ

= −α cos ξ sin(

Φn +π

2

) ∂Φn∂ψ

, (E.94)

∂ux∂ξ

= α cosψ cos ξ cosΦn (E.95)

∂uy∂ξ

= α sinψ cos ξ cosΦn (E.96)

∂uz∂ξ

= −α sin ξ cos(Φn +π

2) , (E.97)

214

∂ux∂ϕ

= −α cosψ sin ξ sinΦn (E.98)

∂uy∂ϕ

= −α sinψ sin ξ sinΦn (E.99)

∂uz∂ϕ

= −α cos ξ sin(Φn +π

2) , (E.100)

and

∂ux∂κ

= −α cosψ sin ξ sinΦn∂Φn∂κ

(E.101)

∂uy∂κ

= −α sinψ sin ξ sinΦn∂Φn∂κ

(E.102)

∂uz∂κ

= −α cos ξ sin(Φn +π

2)∂Φn∂κ

. (E.103)

The elements of the FIM are obtained using (E.10). The elements on thediagonal are

Iα,α =NsK

2σ2(E.104)

Iξ,ξ =α2KNs

2σ2(E.105)

Iψ,ψ =Kα2

2σ2

(

Ns sin2 ξ +

Ns∑

n=1

(

∂Φn∂ψ

)2)

(E.106)

Iϕ,ϕ =NKα2

2σ2(E.107)

Iκ,κ =Kα2

2σ2

Ns∑

n=1

(

∂Φn∂κ

)2

. (E.108)

Off-diagonal elements are

Iα,ϕ = 0 (E.109)

Iξ,ϕ = 0 (E.110)

Iξ,κ = 0 (E.111)

Iα,ψ = 0 (E.112)

Iα,ξ = 0 (E.113)

Iα,κ = 0 (E.114)

Iψ,ξ = 0 (E.115)

Iψ,ϕ =Kα2

2σ2

Ns∑

n=1

∂Φn∂ψ

(E.116)

Iψ,κ =Kα2

2σ2

Ns∑

n=1

(

∂Φn∂κ

∂Φn∂ψ

)

(E.117)

Iκ,ϕ =Kα2

2σ2

Ns∑

n=1

∂Φn∂κ

. (E.118)

215

The FIM is

=

Iα,α 0 0 0 0

0 Iϕ,ϕ Iϕ,κ Iϕ,ψ 0

0 Iϕ,κ Iκ,κ Iψ,κ 0

0 Iϕ,ψ Iψ,κ Iψ,ψ 0

0 0 0 0 Iξ,ξ

. (E.119)

Similarly to (E.67), also in E.119 the amplitude is decoupled from the otherwavefield parameters. In addition, also the ellipticity is decoupled from theother wavefield parameters.

Fisher Information Matrix for a Single Rayleigh Wave, Rota-

tional Measurements

The rotations induced by a single Rayleigh wave are

ωx(p, t) =ακ sinψ cos ξ cos(ωt− κTp+ ϕ) (E.120)

ωx(p, t) =− ακ cosψ cos ξ cos(ωt− κTp+ ϕ) (E.121)

ωx(p, t) =0 (E.122)

∂ωx∂α

= κ sinψ cos ξ cos(ωtk +Φn) (E.123)

∂ωy∂α

= −κ cosψ cos ξ cos(ωtk +Φn) (E.124)

∂ωz∂α

= 0 , (E.125)

∂ωx∂ϕ

= −ακ sinψ cos ξ sin(ωtk +Φn) (E.126)

∂ωy∂ϕ

= ακ cosψ cos ξ sin(ωtk +Φn) (E.127)

∂ωz∂ϕ

= 0 , (E.128)

∂ωx∂ξ

= −ακ sinψ sin ξ cos(ωtk +Φn) (E.129)

∂ωy∂ξ

= ακ cosψ sin ξ cos(ωtk +Φn) (E.130)

∂ωz∂ξ

= 0 , (E.131)

216

∂ωx∂κ

= α sinψ cos ξ

(

cos(ωtk +Φn)− κ sin(ωtk +Φn)∂Φn∂κ

)

(E.132)

∂ωy∂κ

= α cosψ cos ξ

(

− cos(ωtk +Φn) + κ sin(ωtk +Φn)∂Φn∂κ

)

(E.133)

∂ωz∂κ

= 0 , (E.134)

and

∂ωx∂ψ

= ακ cos ξ

(

cosψ cos(ωtk +Φn)− sinψ sin(ωtk +Φn)∂Φn∂ψ

)

(E.135)

∂ωy∂ψ

= ακ cos ξ

(

sinψ cos(ωtk +Φn) + cosψ sin(ωtk +Φn)∂Φn∂ψ

)

(E.136)

∂ωz∂ψ

= 0 . (E.137)

The elements of the FIM are obtained using (E.10). The elements on thediagonal are

Iα,α =κ2 cos2 ξNsK

2σ2(E.138)

Iϕ,ϕ =α2κ2 cos2 ξNsK

2σ2(E.139)

Iξ,ξ =α2κ2 sin2 ξNsK

2σ2(E.140)

Iκ,κ =α2 cos2 ξK

2σ2

(

Ns + κ2Ns∑

n=1

(

∂Φn∂κ

)2)

(E.141)

Iψ,ψ =α2κ2 cos2 ξK

2σ2

(

Ns +

Ns∑

n=1

(

∂Φn∂ψ

)2)

. (E.142)

The off-diagonal elements are

217

Iα,ϕ = 0 (E.143)

Iα,ξ =−ακ2 sin ξ cos ξNsK

2σ2(E.144)

Iα,κ =ακ cos2 ξNsK

2σ2(E.145)

Iα,ψ = 0 (E.146)

Iϕ,ξ = 0 (E.147)

Iϕ,κ =α2κ2 cos2 ξK

2σ2

Ns∑

n=1

∂Φn∂κ

(E.148)

Iϕ,ψ =α2κ2 cos2 ξK

2σ2

Ns∑

n=1

∂Φn∂ψ

(E.149)

Iξ,κ =−α2κ sin ξ cos ξNsK

2σ2(E.150)

Iξ,ψ = 0 (E.151)

Iκ,ψ =α2κ2 cos2 ξK

2σ2

Ns∑

n=1

∂Φn∂ψ

∂Φn∂κ

. (E.152)

The FIM is

I(θ) =

Iα,α 0 Iα,κ 0 Iα,ξ0 Iϕ,ϕ Iϕ,κ Iϕ,ψ 0

Iα,κ Iϕ,κ Iκ,κ Iκ,ψ Iξ,κ0 Iϕ,ψ Iκ,ψ Iψ,ψ 0

Iα,ξ 0 Iξ,κ 0 Iξ,ξ

. (E.153)

Similarly to (E.85), also in E.153 there are no decoupled parameters.

Fisher Information Matrices for Joint Translational and Rota-

tional Measurements

Translational and rotational measurements are independent. Thus the FIMsare additive.

I(θ) = It(θ) + Ir(θ) , (E.154)

where It and Ir are the FIMs for the translational and rotational measure-ments, respectively.

E.3 Derivation of Cramér-Rao Bounds

The Cramér-Rao bound (CRB) is a lower bound on the variance of unbiasedestimators Cramér (1946); Rao (1945). Knowledge of a lower bound on the

218

estimator variance has at least two practical implications. First, it allows usto evaluate the performance of an estimation algorithm, by enabling a quanti-tative comparison between the mean-squared estimation error (MSEE) of thealgorithm under test and the smallest achievable variance. Second, the ana-lytic expression of the CRB enables us to design the experiment set up in orderto reduce the lower bound and therefore increase the amount of informationgathered by the experiment.

The information inequality states that the MSEE of an unbiased estimatoris lower bounded as

E

(

θ − Eθ)(

θ − Eθ)T

(I(θ))−1 . (E.155)

where A B means that the matrix A−B is PSD. In particular, we areinterested in the diagonal elements of I−1 as they provide a lower bound onthe MSEEs of the corresponding parameters.

Analytical inversion of the FIM derived in the previous sections is a tedioustask. A possible approach is to invert the matrix numerically. This approachhowever, gives no insights of the dependency of the CRB on the parameters.To circumvent this limitation, in the following section we obtain analyticalexpression of the CRB for the parameters of interest through the equivalentFisher information (EFI) .

Equivalent Fisher Information

Since we are interested in the elements on the main diagonal of I−1 corre-

sponding to wavenumber and ellipticity angle, we avoid the complete inversionof I as follows. We partition the FIM as

I(θ) =

(

c dT

d G

)

, (E.156)

where c is a scalar, d is a vector, and D is a matrix of suitable sizes. Theelement in the first position of I−1 is then found using the Woodbury matrixidentity to be

[

(I(θ))−1]

1,1=(

c− dTG

−1d)−1

, (E.157)

where [·]i,j denotes the element of the matrix in position (i, j) Horn & Johnson(1990).

In (E.157), the quantity c−dTG

−1d has the dimension of FI and has beenreferred to by some authors as EFI Shen & Win (2010). In contrast with FI, theEFI accounts for the uncertainty introduced by the other unknown parametersof the statistical model. The term c is exactly the FI of the parameter ofinterest. The term dT

G−1d is non-negative since G is PSD being a diagonal

sub-block of a PSD matrix. This last quantity accounts for the uncertaintydue to the other parameters.

It is now clear that reducing the CRB is equivalent to increase the EFI. Inother words, increasing the EFI is desirable as better estimation accuracy canbe achieved.

219

In order to use (E.157) effectively, it may be necessary to permute therow and columns of I such that the element of interest is in the top-left-mostposition. This can be accomplished using a permutation matrix P and considerthe re-arranged I

′ obtained as I′ = P

TIP .

In the following, we restrict ourselves to the analysis of the CRB of wavenum-ber and ellipticity angle as these are the parameters of greater practical interest.

Moment of Inertia

We consider the following definitions from mechanics. We study properties ofthe array in the coordinate system (a, b) instead of (x, y). The two coordinatesystems are related as

(

a

b

)

=

(

cosψ sinψ

− sinψ cosψ

)(

x

y

)

, (E.158)

where the angle or rotation is the azimuth ψ. Therefore a is the axis along thedirection of propagation of the wave and b the axis perpendicular to it.

In this rotated coordinate system we consider the new sensor positions(an, bn)n=1,...,Ns

introduce the center of gravity, or phase center, or the array

a =1

Ns

Ns∑

n=1

an (E.159)

b =1

Ns

Ns∑

n=1

bn . (E.160)

The following quantities are called moment of inertia of the array

Qaa =

Ns∑

n=1

(an − a)2 (E.161)

Qbb =

Ns∑

n=1

(bn − b)2 (E.162)

Qab =

Ns∑

n=1

(an − a)(bn − b) , (E.163)

where the sensor are associated with unitary mass. An important remark isthat the moment of inertia are invariant to a translation of the array.

We observe the the moment of inertias (MOIs) are related to certain quan-tities that appear as elements of the FIMs. Recall the definition Φn = ωtk −

220

xnκ cosψ − ynκ sinψ + ϕ, we find observe that

Ns∑

n=1

∂Φn∂ψ

= −κb (E.164)

Ns∑

n=1

∂Φn∂κ

= −Nsa (E.165)

Ns∑

n=1

∂Φn∂ψ

∂Φn∂κ

=

Ns∑

n=1

anbn (E.166)

Ns∑

n=1

(

∂Φn∂ψ

)2

= κ2(

Qbb +Nsb2)

(E.167)

Ns∑

n=1

(

∂Φn∂κ

)2

= Qaa +Nsa2 . (E.168)

Cramér-Rao Bounds Expressions

We permute the rows/columns of the matrix as explained in (E.39) and thenuse (E.157) to obtain the following expressions. Further discussion on theseexpressions can be found in Maranò & Fäh (2013).

We define

Ct = α2K/2σ2t (E.169)

Cr = α2K/2σ2r , (E.170)

and

Φ = Ct + Crκ2 cos2 ξ (E.171)

Ψ = Ct sin2 ξ + Crκ

2 cos2 ξ . (E.172)

Love Wave

The MSEE of Love wave wavenumber, for translational measurements, is lowerbounded as

E

(κ− Eκ)2

≥ 2σ2t

α2K

(

Qaa −Q2ab

Qbb +Ns/κ2

)−1

. (E.173)

The MSEE of Love wave wavenumber, for rotational measurements, is lowerbounded as

E

(κ− Eκ)2

≥ 8σ2r

α2κ2K

(

Qaa −Q2ab

Qbb

)−1

. (E.174)

The MSEE of Love wave wavenumber, for joint translational and rotational, islower bounded as

E

(κ− Eκ)2

≥(

(

Ct + κ2Cr/4)

Qaa +CtCrNs/4

Ct + Crκ2/4

− Q2ab

(

Ct + κ2Cr/4)2

CtNs/κ2 + (Ct + κ2Cr/4)

)−1

. (E.175)

221

Rayleigh Wave

The MSEE of Rayleigh wave wavenumber, for translational measurements, islower bounded as

E

(κ− Eκ)2

≥ 2σ2t

α2K

(

Qaa −Q2ab

Qbb +Ns sin2 ξ/κ2

)−1

. (E.176)

The MSEE of Rayleigh wave ellipticity angle, for translational measurements,is lower bounded as

E

(ξ − Eξ)2

≥ 2σ2t

α2KNs. (E.177)

The MSEE of Rayleigh wave wavenumber, for rotational measurements, is lowerbounded as

E

(κ− Eκ)2

≥ 2σ2r

α2κ2 cos2(ξ)K

(

Qaa −Q2ab

Qbb +Ns/κ2

)−1

. (E.178)

The MSEE of Rayleigh wave wavenumber, for joint translational and rotationalmeasurements, is lower bounded as

E

(κ− Eκ)2

≥(

ΦQaa +CtCrNs cos

2 ξ

Ct + Crκ2

− Q2abΦ

2

Ct sin2 ξNs/κ2 + Cr cos2 ξNs +ΦQbb

)−1

. (E.179)

The MSEE of Rayleigh wave ellipticity angle, for joint translational and rota-tional measurements, is lower bounded as

E

(ξ − Eξ)2

≥

(

CtNs

+κ2Φ

(

QaaΨNs − κ2(Q2ab −QaaQbb)Φ

)

NsCrCt cos2 ξ(ΨNs + κ2ΦQbb) +QaaΨΦ2Ns − κ2(Q2ab −QaaQbb)Φ3

)

−1

.

(E.180)

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Acknowledgements

I would like to express my sincere gratitude to my advisor Prof. Donat Fähfor his supervision during the development of this thesis. His guidance and hisenduring support truly helped me throughout the development of this work.

I am truly thankful to Prof. Hans-Andrea Loeliger for co-advising this the-sis. I learned a lot from his carefully thought advice. His comments were alwayshitting the essence of problems with a surgical precision.

I thank Prof. Domenico Giardini for making my PhD possible and for ac-cepting to referee this thesis.

I wish to thank Prof. Yue M. Lu for the hospitality at Harvard University.I appreciated his great can-do attitude when tackling challenging problems. Ihave fond memories of the weeks spent in Cambridge.

I wish to thank Prof. Heiner Igel for serving as the external examiner ofthis thesis. I also wish to thank Prof. Johan Robertsson for accepting to chairthe thesis committee.

I thank my colleagues both at the Swiss Seismological Service and the Signaland Information Processing Laboratory for making the work days productiveand enjoyable. Above everyone I would like to thank Dr. Christoph Reller withwhom I shared a lot of my time thinking about this project. His input andeffort was very much appreciated.

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About the Author

Short Biography

Stefano Maranò received the B.Sc. and M.Sc. degrees in TelecommunicationsEngineering from the University of Trento, Italy, in 2005 and 2008, respectively.

From September 2007 to June 2008, he was with Laboratory for Informationand Decision Systems at the Massachusetts Institute of Technology. Since 2009,he has been working towards a Ph.D. with the Swiss Seismological Service, ETHZurich.

In 2009, he was co-recipient of the Best Paper Award at the IEEE GlobalCommunications Conference.

His research interests lie in the broad area of signal processing. In particularthe application of mathematical and statistical theories to challenging real-world problems.

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